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+# GLOBAL OPTIMALITY CONDITIONS FOR DEEP NEURAL NETWORKS
+
+Chulhee Yun, Suvrit Sra & Ali Jadbabaie Massachusetts Institute of Technology Cambridge, MA 02139, USA {chulheey,suvrit,jadbabai}@mit.edu
+
+# ABSTRACT
+
+We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and sufficient conditions for a critical point of the risk function to be a global minimum. Surprisingly, our conditions provide an efficiently checkable test for global optimality, while such tests are typically intractable in nonconvex optimization. We further extend these results to deep nonlinear neural networks and prove similar sufficient conditions for global optimality, albeit in a more limited function space setting.
+
+# 1 INTRODUCTION
+
+Since the advent of AlexNet (Krizhevsky et al., 2012), deep neural networks have surged in popularity, and have redefined the state-of-the-art across many application areas of machine learning and artificial intelligence, such as computer vision, speech recognition, and natural language processing. However, a concrete theoretical understanding of why deep neural networks work well in practice remains elusive. From the perspective of optimization, a significant barrier is imposed by the nonconvexity of training neural networks. Moreover, it was proved by Blum & Rivest (1988) that training even a 3-node neural network to global optimality is NP-Hard in the worst case, so there is little hope that neural networks have properties that make global optimization tractable.
+
+Despite the difficulties of optimizing weights in neural networks, the empirical successes suggest that the local minima of their loss surfaces could be close to global minima; and several papers have recently appeared in the literature attempting to provide a theoretical justification for the success of these models. For example, by relating neural networks to spherical spin-glass models from statistical physics, Choromanska et al. (2015) provided some empirical evidence that the increase of size of neural networks makes local minima close to global minima.
+
+Another line of results (Yu & Chen, 1995; Soudry & Carmon, 2016; Xie et al., 2016; Nguyen & Hein, 2017) provides conditions under which a critical point of the empirical risk is a global minimum. Such results roughly involve proving that if full rank conditions of certain matrices (as well as some additional technical conditions) are satisfied, derivative of the risk being zero implies loss being zero. However, these results are obtained under restrictive assumptions; for example, Nguyen & Hein (2017) require the width of one of the hidden layers to be as large as the number of training examples. Soudry & Carmon (2016) and Xie et al. (2016) require the product of widths of two adjacent layers to be at least as large as the number of training examples, meaning that the number of parameters in the model must grow rapidly as we have more training data available. Another recent paper (Haeffele & Vidal, 2017) provides a sufficient condition for global optimality when the neural network is composed of subnetworks with identical architectures connected in parallel and a regularizer is designed to control the number of parallel architectures.
+
+Towards obtaining a more precise characterization of the loss-surfaces, a valuable conceptual simplification of deep nonlinear networks is deep linear neural networks, in which all activation functions are linear and the output of the entire network is a chained product of weight matrices with the input vector. Although at first sight a deep linear model may appear overly simplistic, even its optimization is nonconvex, and only recently theoretical results on this problem have started emerging. Interestingly, already in 1989, Baldi & Hornik (1989) showed that some shallow linear neural networks have no local minima. More recently, Kawaguchi (2016) extended this result to deep linear networks and proved that any local minimum is also global while any other critical point is a saddle point. Subsequently, Lu & Kawaguchi (2017) provided a simpler proof that any local minimum is also global, with fewer assumptions than (Kawaguchi, 2016). Motivated by the success of deep residual networks (He et al., 2016a;b), Hardt & Ma (2017) investigated loss surfaces of deep linear residual networks and showed every critical point is a global minimum in a near-identity region; subsequently, Bartlett et al. (2017) extended this result to a nonlinear function space setting.
+
+# 1.1 OUR CONTRIBUTIONS
+
+Inspired by this recent line of work, we study deep linear and nonlinear networks, in settings either similar to or more general than existing work. We summarize our main contributions below.
+
+• We provide both necessary and sufficient conditions for a critical point of the empirical risk to be a global minimum. Specifically, Theorem 2.1 shows that if the hidden layers are wide enough, then a critical point of the risk function is a global minimum if and only if the product of all parameter matrices is full-rank. In Theorem 2.2, we consider the case where some hidden layers have smaller width than both the input and output layers, and again provide necessary and sufficient conditions for global optimality. In comparison, Kawaguchi (2016) only proves that every critical point of the risk is either a global minimum or a saddle; it is an “existence” result without any computational implication. In contrast, we present efficiently checkable conditions for distinguishing the two different types of critical points; we can even use these conditions while running optimization to test whether the critical points we encounter are saddle points or not, if desired. It is also worth noting that such tests are intractable for general nonconvex optimization (Murty & Kabadi, 1987). Under the same assumption as (Hardt & Ma, 2017) on the data distribution, namely, a linear model with Gaussian noise, we can modify Theorem 2.1 to handle the population risk. As a corollary, we not only recover Theorem 2.2 in (Hardt & Ma, 2017), but also extend it to a strictly larger set, while removing their assumption that the true underlying linear model has a positive determinant. • Motivated by (Bartlett et al., 2017), we extend our results on deep linear networks to obtain sufficient conditions for global optimality in deep nonlinear networks, although only via a function space view; these are presented in Theorems 4.1 and 4.2.
+
+# 2 GLOBAL OPTIMALITY CONDITIONS FOR DEEP LINEAR NEURAL NETWORKS
+
+In this section, we describe the problem formulation and notations for deep linear neural networks, state main results (Theorems 2.1 and 2.2), and explain their implication.
+
+# 2.1 PROBLEM FORMULATION AND NOTATION
+
+Suppose we have $m$ input-output pairs, where the inputs are of dimension $d _ { x }$ and outputs of dimension $d _ { y }$ . Let $\boldsymbol { X } \in \mathbb { R } ^ { d _ { \boldsymbol { x } } \times m }$ be the data matrix and $\bar { Y } \in \mathbb { R } ^ { d _ { y } \times m }$ be the output matrix. Suppose we have $H$ hidden layers in the network, each having width $d _ { 1 } , \dots , d _ { H }$ . For notational simplicity we let $d _ { 0 } = d _ { x }$ and $d _ { H + 1 } = d _ { y }$ . The weights between adjacent layers can be represented as matrices $W _ { k } \in \mathbb { R } ^ { d _ { k } \times d _ { k - 1 } }$ , for $k = 1 , \ldots , H + 1$ , and the output of the network can be written as the product of weight matrices $W _ { H + 1 } , \dots , W _ { 1 }$ and data matrix $X$ : $W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X$ . We consider minimizing the summation of squared error loss over all data points (i.e. empirical risk),
+
+$$
+\begin{array} { r } { \mathrm { m i n i m i z e } \quad L ( W ) : = \frac { 1 } { 2 } \left\| W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X - Y \right\| _ { \mathrm { F } } ^ { 2 } , } \end{array}
+$$
+
+where $W$ is a shorthand notation for the tuple $( W _ { 1 } , \dots , W _ { H + 1 } )$ .
+
+Assumptions. We assume that $d _ { x } \leq m$ and $d _ { y } \leq m$ , and that $X X ^ { T }$ and $Y X ^ { T }$ have full ranks. These assumptions are common when we consider supervised learning problems with deep neural networks (e.g. Kawaguchi (2016)). We also assume that the singular values of $Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X$ are all distinct, which is made for notational simplicity and can be relaxed without too much difficulty.
+
+Notation. Given a matrix $A$ , let $\sigma _ { \mathrm { m a x } } ( A )$ and $\sigma _ { \mathrm { m i n } } ( A )$ denote the largest and smallest singular values of A, respectively. Let row $( A )$ , $\operatorname { c o l } ( A )$ , $\operatorname { n u l l } ( A )$ , $\operatorname { r a n k } ( A )$ , and $\Vert A \Vert _ { \mathrm { F } }$ be respectively the row space, column space, null space, rank, and Frobenius norm of matrix $A$ . Given a subspace $V$ of $\mathbb { R } ^ { n }$ , we denote $V ^ { \perp }$ as its orthogonal complement. Given a set $\nu$ , let $\mathcal { V } ^ { c }$ denote the complement of $\nu$ .
+
+Let us denote $k : = \mathrm { m i n } _ { i \in \{ 0 , \dots , H + 1 \} } d _ { i }$ , and define $p \in { \mathrm { ~ a r g m i n } } _ { i \in \{ 0 , . . . , H + 1 \} } d _ { i }$ . That is, $p$ is any layer with the smallest width, and $k = d _ { p }$ is the width of that layer. Here, $p$ might not be unique, but our results hold for any layer $p$ with smallest width. Notice also that the product $W _ { H + 1 } \cdots W _ { 1 }$ can have rank at most $k$ . Let $\mathbf { \dot { \boldsymbol { Y } } } \bar { X ^ { T } } ( \bar { X } \bar { X ^ { T } } ) ^ { - 1 } \boldsymbol { X } = \boldsymbol { U } \Sigma \boldsymbol { V } ^ { T }$ be the singular value decomposition of $Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \in \mathbb { R } ^ { d _ { y } \times d _ { x } }$ . Let $\hat { U } \in \mathbb { R } ^ { d _ { y } \times k }$ be a matrix consisting of the first $k$ columns of $U$ .
+
+# 2.2 NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY
+
+We now present two main theorems for deep linear neural networks. The theorems describe two sets, one for the case $k = \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ and the other for $k < \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ , inside which every critical point of $L ( W )$ is a global minimum. Moreover, the sets have another remarkable property that every critical point outside of these sets is a saddle point. Previous works (Kawaguchi, 2016; Lu & Kawaguchi, 2017) showed that any critical point is either a global minimum or a saddle point, without providing any condition to distinguish between the two; here, we take a step further and partition the domain of $L ( W )$ into two sets clearly delineating one set which only contains global minima and the other set with only saddle points.
+
+Theorem 2.1. If $k = \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ , define the following set
+
+$$
+\mathcal { V } _ { 1 } : = \left\{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \operatorname { r a n k } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } ) = k \right\} .
+$$
+
+Then, every critical point of $L ( W )$ in $\mathcal { V } _ { 1 }$ is a global minimum. Moreover, every critical point of $L ( W )$ in $\mathcal { V } _ { 1 } ^ { c }$ is a saddle point.
+
+Theorem 2.2. If $k < \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ , define the following set
+
+$$
+\mathcal { V } _ { 2 } : = \Big \{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \mathrm { r a n k } \big ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } \big ) = k , \mathrm { c o l } \big ( W _ { H + 1 } \cdot \cdot \cdot W _ { p + 1 } \big ) = \mathrm { c o l } ( \hat { U } ) \Big \} .
+$$
+
+Then, every critical point of $L ( W )$ in $\nu _ { 2 }$ is a global minimum. Moreover, every critical point of $L ( W )$ in $\mathcal { V } _ { 2 } ^ { c }$ is a saddle point.
+
+Theorems 2.1 and 2.2 provide necessary and sufficient conditions for a critical point of $L ( W )$ to be globally optimal. From an algorithmic perspective, they provide easily checkable conditions, which we can use to determine if the critical point the algorithm encountered is a global optimum or not. Given that $L ( W )$ is nonconvex, it is interesting to have such efficient tests for global optimality, which is not possible in general (Murty & Kabadi, 1987).
+
+In Hardt & Ma (2017), the authors consider minimizing population risk of linear residual networks:
+
+$$
+\begin{array} { r l } { \mathrm { m i n i m i z e } } & { { } \frac { 1 } { 2 } \mathbb { E } _ { \boldsymbol { x } , \boldsymbol { y } } \left[ \| ( I + W _ { H + 1 } ) \cdot \cdot \cdot ( I + W _ { 1 } ) \boldsymbol { x } - \boldsymbol { y } \| _ { \mathrm { F } } ^ { 2 } \right] , } \end{array}
+$$
+
+where $d _ { x } = d _ { 1 } = \cdot \cdot \cdot = d _ { H } = d _ { y } = d$ . They assume that $x$ is drawn from a zero-mean distribution with a fixed covariance matrix, and $y = R x + \xi$ where $\xi$ is iid standard Gaussian noise and $R$ is the true underlying matrix with $\operatorname* { d e t } ( R ) > 0$ . With these assumptions they prove that whenever $\sigma _ { \operatorname* { m a x } } ( W _ { i } ) < 1$ for all $i$ , any critical point is a global minimum (Hardt & Ma, 2017, Theorem 2.2).
+
+Under the same assumptions on data distribution, we can slightly modify Theorem 2.1 to derive a population risk counterpart, and in fact notice that the result proved in Hardt & Ma (2017) is a corollary of this modification because having $\sigma _ { \operatorname* { m a x } } ( W _ { i } ) < 1$ for all $i$ is a sufficient condition for $( I + W _ { H + 1 } ) \cdot \cdot \cdot ( I + W _ { 1 } )$ having full rank. Moreover, notice that we can remove the assumption $\operatorname* { d e t } ( R ) > 0$ which was required by Hardt & Ma (2017). We state this special case as a corollary:
+
+Corollary 2.3 (Theorem 2.2 of Hardt & Ma (2017)). Under assumptions on data distribution as described above, any critical point of $\begin{array} { r } { \frac { 1 } { 2 } \mathbb { E } _ { x , y } \left[ \left\| ( I + W _ { H + 1 } ) \cdot \cdot \cdot ( I + W _ { 1 } ) x - y \right\| _ { \mathrm { F } } ^ { 2 } \right] } \end{array}$ is a global minimum if $\sigma _ { \operatorname* { m a x } } ( W _ { i } ) < 1$ for all $i$ .
+
+We also note in passing that the classical problem of matrix factorization $\begin{array} { r } { \operatorname* { m i n } _ { U , V } \left\| U V ^ { T } - Y \right\| _ { \mathrm { F } } ^ { 2 } } \end{array}$ is a special case of deep linear neural networks, so our theorems can also be directly applied.
+
+Remarks. The previous result (Kawaguchi, 2016) assumed $d _ { y } \leq d _ { x }$ and showed that: 1) every local minimum is a global minimum, and 2) any other critical point is a saddle point. A subsequent paper by Lu & Kawaguchi (2017) proved 1) without the assumption $d _ { y } \ \leq \ d _ { x }$ , but as far as we know there is no result showing 2) in the case of $d _ { y } \ > \ d _ { x }$ . We provide the proof for this case in Lemma B.1. In fact, we propose an alternative proof technique for handling degenerate critical points, which is much simpler than the technique presented by Kawaguchi (2016).
+
+# 3 ANALYSIS OF DEEP LINEAR NETWORKS
+
+In this section, we provide proofs for Theorems 2.1 and 2.2.
+
+# 3.1 SOLUTIONS OF THE RELAXED PROBLEM
+
+We first analyze the globally optimal solution of a “relaxation” of $L ( W )$ , which turns out to be very useful while proving Theorems 2.1 and 2.2. Consider the relaxed risk function
+
+$$
+L _ { 0 } ( R ) = { \frac { 1 } { 2 } } \left\| R X - Y \right\| _ { \mathrm { F } } ^ { 2 } ,
+$$
+
+where $R \in \mathbb { R } ^ { d _ { y } \times d _ { x } }$ and $\operatorname { r a n k } ( R ) \leq k$ . For any $W$ , the product $W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 }$ has rank at most $k$ and setting $R$ to be this product gives the same loss values: $L _ { 0 } ( W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } ) = L ( W )$ . Therefore, $L _ { 0 }$ is a relaxation of $L$ and
+
+$$
+\operatorname* { i n f } _ { R : \operatorname { r a n k } ( R ) \leq k } L _ { 0 } ( R ) \leq \operatorname* { i n f } _ { W } L ( W ) .
+$$
+
+This means that if there exists $W$ such that $L ( W ) = \operatorname* { i n f } _ { R : \operatorname { r a n k } ( R ) \leq k } L _ { 0 } ( R )$ , then $W$ is a global minimum of the function $L$ . This observation is very important in proofs; we will show that inside certain sets, any critical point $W$ of $L ( W )$ must satisfy $R ^ { * } = W _ { H + 1 } \cdot \cdot \cdot W _ { 1 }$ , where $R ^ { * }$ is a global optimum of $L _ { 0 } ( R )$ . This proves that $L ( W ) = L _ { 0 } ( R ^ { * } ) = \operatorname* { i n f } _ { R : \operatorname { r a n k } ( R ) \leq k } L _ { 0 } ( R )$ , thus showing that $W$ is a global minimum of $L$ .
+
+By restating this observation as an optimization problem, the solution of problem in (1) is bounded below by the minimum value of the following:
+
+$$
+\begin{array} { l l } { \mathrm { m i n i m i z e } } & { \frac { 1 } { 2 } \left\| R X - Y \right\| _ { \mathrm { F } } ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { \mathrm { r a n k } ( R ) \leq k . } \end{array}
+$$
+
+In case where $k = \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ , (2) is actually an unconstrained optimization problem. Note that $L _ { 0 }$ is a convex function of $R$ , so any critical point is a global minimum. By differentiating and setting the derivative to zero, we can easily get the unique globally optimal solution
+
+$$
+R ^ { * } = Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .
+$$
+
+In case of $k < \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ , the problem becomes non-convex because of the rank constraint, but its exact solution can still be computed easily. We present the solution of this case as a proposition and defer the proof to Appendix C due to its technicalities.
+
+Proposition 3.1. Suppose $k < \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ . Then the optimal solution to (2) is
+
+$$
+R ^ { * } = \hat { U } \hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } ,
+$$
+
+which is the orthogonal projection of $Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }$ onto the column space of $\hat { U }$
+
+# 3.2 PARTIAL DERIVATIVES OF $L ( W )$
+
+By simple matrix calculus, we can calculate the derivatives of $L ( W )$ with respect to $W _ { i }$ , for $i =$ $1 , \ldots , H + 1$ . We present the result as the following lemma, and defer the details to Appendix C.
+
+Lemma 3.2. The partial derivative of $L ( W )$ with respect to $W _ { i }$ is given as
+
+$$
+\frac { \partial L } { \partial W _ { i } } = W _ { i + 1 } ^ { T } \cdot \cdot \cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \cdot \cdot \cdot W _ { i - 1 } ^ { T } ,
+$$
+
+for $i = 1 , \ldots , H + 1$
+
+This result will be used throughout the proof of Theorems 2.1 and 2.2. For clarity in notation, note $i = 1$ $W _ { 1 } ^ { T } \cdots W _ { 0 } ^ { \breve { T } }$ is just an matrix in matrix in . $\mathbb { R } ^ { d _ { x } \times d _ { x } }$ . Similarly, when $i = H + 1$ $W _ { H + 2 } ^ { T } \cdot \cdot \cdot W _ { H + 1 } ^ { T }$ $\mathbb { R } ^ { d _ { y } \times d _ { y } ^ { * } }$
+
+We also state an elementary lemma which proves useful in our proofs, whose proof we defer to Appendix C.
+
+Lemma 3.3. 1. For any $A \in \mathbb { R } ^ { m \times n }$ and $\boldsymbol { B } \in \mathbb { R } ^ { n \times l }$ where $m \geq n ,$ ,
+
+$$
+\left\| A B \right\| _ { \mathrm { F } } ^ { 2 } \geq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( A ) \left\| B \right\| _ { \mathrm { F } } ^ { 2 } .
+$$
+
+2. For any $A \in \mathbb { R } ^ { m \times n }$ and $\boldsymbol { B } \in \mathbb { R } ^ { n \times l }$ where $n \leq l$
+
+$$
+\left\| A B \right\| _ { \mathrm { F } } ^ { 2 } \geq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( B ) \left\| A \right\| _ { \mathrm { F } } ^ { 2 } .
+$$
+
+# 3.3 PROOF OF THEOREM 2.1
+
+We prove Theorem 2.1, which addresses the case $k = \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ . First, recall that the set defined in Theorem 2.1 is
+
+$$
+\mathcal { V } _ { 1 } : = \left\{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \operatorname { r a n k } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } ) = k \right\} .
+$$
+
+As seen in (3), the unique minimum point of $L _ { 0 }$ has rank $k$ . So, no point $W \in \mathcal { V } _ { 1 } ^ { c }$ can be a global minimum of $L$ . Therefore, by Kawaguchi (2016, Theorem 2.3.(iii)) and Lemma B.1, any critical point in $\mathcal { V } _ { 1 } ^ { c }$ must be a saddle point.
+
+For the rest of our proof, we need to consider two cases: $d _ { y } \leq d _ { x }$ and $d _ { x } \leq d _ { y }$ . If $d _ { x } = d _ { y }$ , both cases work. The outline of the proof is as follows: we define a new set $\mathcal { W } _ { \epsilon }$ , show that any critical point in the set $\mathcal { W } _ { \epsilon }$ is a global minimum, and then show that every $W \in \mathcal { V } _ { 1 }$ is also in $\mathcal { W } _ { \epsilon }$ for some $\epsilon > 0$ . This proves that any critical point of $L ( W )$ in $\nu _ { 1 }$ is also a critical point in $\mathcal { W } _ { \epsilon }$ for some $\epsilon > 0$ , hence a global minimum.
+
+The following proposition proves the first step:
+
+Proposition 3.4. Assume that $k = \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ . For any $\epsilon > 0$ , define the following set:
+
+$$
+\mathcal { W } _ { \epsilon } : = \left\{ \begin{array} { l l } { \{ ( W _ { 1 } , \dots , W _ { H + 1 } ) : \sigma _ { \operatorname* { m i n } } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 2 } ) \geq \epsilon \} , } & { \mathrm { i f ~ } d _ { y } \leq d _ { x } , } \\ { \{ ( W _ { 1 } , \dots , W _ { H + 1 } ) : \sigma _ { \operatorname* { m i n } } ( W _ { H } \cdot \cdot \cdot W _ { 1 } ) \geq \epsilon \} , } & { \mathrm { i f ~ } d _ { x } \leq d _ { y } . } \end{array} \right.
+$$
+
+Then any critical point of $L ( W )$ in $\mathcal { W } _ { \epsilon }$ is a global minimum point.
+
+Proof. (If $d _ { y } \leq d _ { x } ,$ ) Consider (5) in the case of $i = 1$ . We can observe that $W _ { 2 } ^ { T } \cdot \cdot \cdot W _ { H + 1 } ^ { T } \in \mathbb { R } ^ { d _ { 1 } \times d _ { y } }$ and that $d _ { 1 } \geq d _ { y }$ . Then by Lemma 3.3.1,
+
+$$
+\begin{array} { r l } {  { \| \frac { \partial L } { \partial W _ { 1 } } \| _ { \mathrm { F } } ^ { 2 } \geq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 2 } ) \| ( W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } \| _ { \mathrm { F } } ^ { 2 } } } \\ & { \geq \epsilon ^ { 2 } \| ( W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } \| _ { \mathrm { F } } ^ { 2 } . } \end{array}
+$$
+
+By the above inequality, any critical point in $\mathcal { W }$ satisfies
+
+$$
+\forall i , \frac { \partial L } { \partial W _ { i } } = 0 \Rightarrow ( W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } = 0 ,
+$$
+
+which means that $W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } = Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }$ . The product is the unique globally optimal solution (3) of the relaxed problem in (2), so $W$ is a global minimum point of $L$ .
+
+(If $d _ { x } \leq d _ { y } ,$ ) Consider (5) for $i = H + 1$ . We can observe that $W _ { 1 } ^ { T } \cdot \cdot \cdot W _ { H } ^ { T } \in \mathbb { R } ^ { d _ { x } \times d _ { H } }$ and that $d _ { x } \leq d _ { H }$ . Then by Lemma 3.3.2,
+
+$$
+\left\| \frac { \partial \cal { L } } { \partial W _ { H + 1 } } \right\| _ { \mathrm { F } } ^ { 2 } \geq \epsilon ^ { 2 } \left\| ( W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } \right\| _ { \mathrm { F } } ^ { 2 } ,
+$$
+
+and the rest of the proof flows in a similar way as the previous case.
+
+The next proposition proves the theorem:
+
+Proposition 3.5. For any point $W \in \nu _ { 1 }$ , there exists an $\epsilon > 0$ such that $W \in \mathcal { W } _ { \epsilon }$
+
+Proof. Define a new set $\mathcal { W }$ , a “limit” version (as $\epsilon  0$ ) of $\mathcal { W } _ { \epsilon }$ , as
+
+$$
+\begin{array} { r } { \mathcal { W } : = \{ \{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \mathrm { r a n k } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 2 } ) = d _ { y } \} , \mathrm { i f } \ d _ { y } \leq d _ { x } , } \\ { \{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \mathrm { r a n k } ( W _ { H } \cdot \cdot \cdot W _ { 1 } ) = d _ { x } \} , \mathrm { i f } \ d _ { x } \leq d _ { y } . } \end{array}
+$$
+
+We show that $\nu _ { 1 } \subset \mathcal { W }$ by showing that $\mathcal { W } ^ { c } \subset \mathcal { V } _ { 1 } ^ { c }$ . Consider
+
+$$
+\mathcal { W } ^ { c } = \left\{ \begin{array} { l l } { \{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \operatorname { r a n k } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 2 } ) < d _ { y } \} , } & { \mathrm { i f ~ } d _ { y } \leq d _ { x } , } \\ { \{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \operatorname { r a n k } ( W _ { H } \cdot \cdot \cdot W _ { 1 } ) < d _ { x } \} , } & { \mathrm { i f ~ } d _ { x } \leq d _ { y } . } \end{array} \right.
+$$
+
+Then any $W \in \mathcal { W } ^ { c }$ must have $\mathrm { r a n k } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } ) < \mathrm { m i n } \{ d _ { x } , d _ { y } \} = k$ , so $W \in \mathcal { V } _ { 1 } ^ { c }$ . Thus, any $W \in \nu _ { 1 }$ is also in $\mathcal { W }$ , so either $\operatorname { r a n k } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 2 } ) = d _ { y }$ or $\operatorname { r a n k } ( W _ { H } \cdot \cdot \cdot W _ { 1 } ) = d _ { x }$ , depending on the cases. Then, we can set
+
+$$
+\epsilon = \left\{ \begin{array} { l l } { \sigma _ { \operatorname* { m i n } } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 2 } ) , } & { \mathrm { i f ~ } d _ { y } \leq d _ { x } , } \\ { \sigma _ { \operatorname* { m i n } } ( W _ { H } \cdot \cdot \cdot W _ { 1 } ) , } & { \mathrm { i f ~ } d _ { x } \leq d _ { y } . } \end{array} \right.
+$$
+
+We always have $\epsilon > 0$ because the matrices are full rank, and we can see that $W \in \mathcal { W } _ { \epsilon }$
+
+# 3.4 PROOF OF THEOREM 2.2
+
+In this section we prove Theorem 2.2, which tackles the case $k < \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ . Note that this assumption also implies that $1 \leq p \leq H$ .
+
+As for the proof of Theorem 2.1, define
+
+$$
+\mathcal { V } _ { 1 } : = \left\{ ( W _ { 1 } , \ldots , W _ { H + 1 } ) : \operatorname { r a n k } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } ) = k \right\} .
+$$
+
+The globally optimal point of the relaxed problem (2) has rank $k$ , as seen in (4). Thus, any point outside of $\nu _ { 1 }$ cannot be a global minimum. Then, by Kawaguchi (2016, Theorem 2.3.(iii)) and Lemma B.1, it follows that any critical point in $\mathcal { V } _ { 1 } ^ { c }$ must be a saddle point. The remaining proof considers points in $\mathcal { V } _ { 1 }$ .
+
+For this section, let us introduce some additional notations to ease presentation. Define
+
+$$
+\begin{array} { r l } & { E : = ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } \in \mathbb { R } ^ { d _ { y } \times d _ { x } } , } \\ & { A _ { i } : = W _ { i + 1 } ^ { T } \cdot \cdot \cdot W _ { H + 1 } ^ { T } \in \mathbb { R } ^ { d _ { i } \times d _ { y } } , \ B _ { i } : = W _ { 1 } ^ { T } \cdot \cdot \cdot W _ { i - 1 } ^ { T } \in \mathbb { R } ^ { d _ { x } \times d _ { i - 1 } } , \ i = 1 , \dots , H + 1 , } \end{array}
+$$
+
+so that $\begin{array} { r } { \frac { \partial L } { \partial W _ { i } } = A _ { i } E B _ { i } } \end{array}$ . Notice that $A _ { H + 1 }$ and $B _ { 1 }$ are identity matrices.
+
+Now consider any tuple $W \in \mathcal { V } _ { 1 }$ . Since the full product $W _ { H + 1 } \cdots W _ { 1 }$ has rank $k$ , any partial products $A _ { i }$ and $B _ { i }$ must have $\operatorname { r a n k } ( A _ { i } ) \geq k$ and $\operatorname { r a n k } ( B _ { i } ) \geq k$ , for all $i$ . Then, consider $A _ { p } \in$ $\mathbb { R } ^ { k \times d _ { y } }$ and $B _ { p + 1 } \in \mathbb { R } ^ { d _ { x } \times k }$ . Since $\operatorname { r a n k } ( A _ { p } ) \leq k$ and $\operatorname { r a n k } ( B _ { p + 1 } ) \leq k$ , we can see that ra $\operatorname { l k } ( A _ { p } ) =$ $\mathrm { r a n k } ( B _ { p + 1 } ) \dot { = } k$ . Also, notice that $A _ { i } = W _ { i + 1 } A _ { i + 1 }$ and $B _ { i + 1 } = B _ { i } W _ { i }$ , so that
+
+$$
+\operatorname { r a n k } ( A _ { 1 } ) \leq \operatorname { r a n k } ( A _ { 2 } ) \leq \cdots \leq \operatorname { r a n k } ( A _ { p } ) { \mathrm { ~ a n d ~ } } \operatorname { r a n k } ( B _ { H + 1 } ) \leq \operatorname { r a n k } ( B _ { H } ) \leq \cdots \leq \operatorname { r a n k } ( B _ { p + 1 } ) .
+$$
+
+However, we have $k \leq \mathrm { r a n k } ( A _ { 1 } )$ and $k \leq \mathrm { r a n k } ( B _ { H + 1 } )$ , so the ranks are all identically $k$ . Also,
+
+$$
+\operatorname { r o w } ( A _ { 1 } ) \subset \operatorname { r o w } ( A _ { 2 } ) \subset \cdots \subset \operatorname { r o w } ( A _ { p } ) { \mathrm { ~ a n d ~ } } \operatorname { c o l } ( B _ { H + 1 } ) \subset \operatorname { c o l } ( B _ { H } ) \subset \cdots \subset \operatorname { c o l } ( B _ { p + 1 } ) ,
+$$
+
+ut it was just shown that the these spaces have the same dimensions, which equals $k$ , meaning
+
+$$
+\operatorname { r o w } ( A _ { 1 } ) = \operatorname { r o w } ( A _ { 2 } ) = \cdots = \operatorname { r o w } ( A _ { p } ) { \mathrm { ~ a n d ~ } } \operatorname { c o l } ( B _ { H + 1 } ) = \operatorname { c o l } ( B _ { H } ) = \cdots = \operatorname { c o l } ( B _ { p + 1 } ) .
+$$
+
+Using this observation, we can now state a proposition showing necessary and sufficient conditions for a tuple $W \in \mathcal { V } _ { 1 }$ to be a critical point of $L ( W )$ .
+
+Proposition 3.6. A tuple $W \in \nu _ { 1 }$ is a critical point of $L$ if and only if $A _ { p } E = 0$ and $E B _ { p + 1 } = 0$ .
+
+Proof. (If part) $A _ { p } E = 0$ implies that $\operatorname { c o l } ( E ) \subset \operatorname { r o w } ( A _ { p } ) ^ { \perp } = \cdot \cdot \cdot = \operatorname { r o w } ( A _ { 1 } ) ^ { \perp }$ , so $\begin{array} { r } { \frac { \partial L } { \partial W _ { i } } \ = } \end{array}$ $A _ { i } E B _ { i } = 0 \cdot B _ { i } = 0 .$ , for $i = 1 , \ldots , p$ . Similarly, $E B _ { p + 1 } = 0$ implies $\operatorname { r o w } ( E ) \subset \operatorname { c o l } ( B _ { p + 1 } ) ^ { \perp } =$ $\cdot \cdot \cdot = \mathrm { c o l } ( B _ { H + 1 } ) ^ { \perp }$ , so $\begin{array} { r } { \frac { \partial L } { \partial W _ { i } } = A _ { i } E B _ { i } = A _ { i } \cdot 0 = 0 } \end{array}$ for $i = p + 1 , \ldots , H + 1$ .
+
+(Only if part) We have $\begin{array} { r } { \frac { \partial L } { \partial W _ { i } } = A _ { i } E B _ { i } = 0 } \end{array}$ for all $i$ . This means that
+
+$$
+\begin{array} { r l } & { \quad \mathrm { c o l } ( E B _ { i } ) \subset \mathrm { r o w } ( A _ { i } ) ^ { \perp } = \mathrm { r o w } ( A _ { p } ) ^ { \perp } \mathrm { f o r } i = 1 , \ldots , p } \\ & { \quad \mathrm { r o w } ( A _ { i } E ) \subset \mathrm { c o l } ( B _ { i } ) ^ { \perp } = \mathrm { c o l } ( B _ { p + 1 } ) ^ { \perp } \mathrm { f o r } i = p + 1 , \ldots , H + 1 . } \end{array}
+$$
+
+Now recall that $B _ { 1 }$ and $A _ { H + 1 }$ are identity matrices, so $\mathrm { c o l } ( E ) \subset \mathrm { r o w } ( A _ { p } ) ^ { \perp }$ and $\operatorname { r o w } ( E ) ~ \subset$ $\mathrm { c o l } ( B _ { p + 1 } ) ^ { \perp }$ , which proves $A _ { p } E = 0$ and $E B _ { p + 1 } = 0$ .
+
+Now we present a proposition that specifies the necessary and sufficient condition in which a critical point of $L ( W )$ in $\nu _ { 1 }$ is a global minimum. Recall that when we take the SVD ${ \cal Y } X ^ { T } ( X X ^ { T } ) ^ { - 1 } X =$ $U { \boldsymbol { \Sigma } } V ^ { T }$ , $\hat { U } \in \mathbb { R } ^ { d _ { y } \times k }$ is defined to be a matrix consisting of the first $k$ columns of $U$ .
+
+Proposition 3.7. A critical point $W \in \mathsf { V } _ { 1 }$ of $L ( W )$ is a global minimum point if and only $i f$ $\operatorname { c o l } ( W _ { H + 1 } \cdot \cdot \cdot W _ { p + 1 } ) = \operatorname { r o w } ( A _ { p } ) = \operatorname { c o l } ( { \hat { U } } )$ .
+
+Proof. Since $W$ is a critical point, by Proposition 3.6 we have $A _ { p } E = 0$ . Also note from the definitions of $A _ { i }$ ’s and $B _ { i }$ ’s that $W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } = A _ { p } ^ { T } B _ { p + 1 } ^ { T }$ , so
+
+$$
+A _ { p } E = A _ { p } ( A _ { p } ^ { T } B _ { p + 1 } ^ { T } X - Y ) X ^ { T } = A _ { p } A _ { p } ^ { T } B _ { p + 1 } ^ { T } X X ^ { T } - A _ { p } Y X ^ { T } = 0 .
+$$
+
+Because $\operatorname { r a n k } ( A _ { p } ) = k$ , and $A _ { p } A _ { p } ^ { T } \in \mathbb { R } ^ { k \times k }$ is invertible, so $B _ { p + 1 }$ is determined uniquely as
+
+$$
+B _ { p + 1 } ^ { T } = ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } ,
+$$
+
+thus
+
+$$
+W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } = A _ { p } ^ { T } B _ { p + 1 } ^ { T } = A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .
+$$
+
+Comparing this with (4), $W$ is a global minimum solution if and only if
+
+$$
+\hat { U } \hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } = W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } = A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .
+$$
+
+This equation holds if and only if $A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } \ = \ \hat { U } \hat { U } ^ { T }$ , meaning that they are projecting $Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }$ onto the same subspace. The projection matrix $A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p }$ is onto $\operatorname { r o w } ( A _ { p } )$ , while $\hat { U } \hat { U } ^ { T }$ is onto $\mathrm { c o l } ( { \hat { U } } )$ . From this, we conclude that $W$ is a global minimum point if and only if $\operatorname { r o w } ( A _ { p } ) = \operatorname { c o l } ( \hat { U } )$ . □
+
+From Proposition 3.7, we can define the set $\nu _ { 2 }$ that appeared in Theorem 2.2, and conclude that every critical point of $L ( W )$ in $\nu _ { 2 }$ is a global minimum, and any other critical points are saddle points.
+
+# 4 EXTENSION TO DEEP NONLINEAR NEURAL NETWORKS
+
+In this section, we present some sufficient conditions for global optimality for deep nonlinear neural networks via a function space view. Given a smooth nonlinear function $h ^ { * }$ that maps input to output, Bartlett et al. (2017) described a method to decompose it into a number of smooth nonlinear functions $h ^ { * } = h _ { H + 1 } \circ \cdot \cdot \cdot \circ h _ { 1 }$ where $h _ { i }$ ’s are close to identity. Using Frechet derivatives of the ´ population risk with respect to each function $h _ { i }$ , they showed that when all $h _ { i }$ ’s are close to identity, any critical point of the population risk is a global minimum. One can see that these results are direct generalization of Theorems 2.1 and 2.2 of Hardt & Ma (2017) to nonlinear networks and utilize the classical “small gain” arguments often used in nonlinear analysis and control (Khalil, 1996; Zames, 1966). Motivated by this result, we extended Theorem 2.1 to deep nonlinear neural networks and obtained sufficient conditions for global optimality in function space.
+
+# 4.1 PROBLEM FORMULATION AND NOTATION
+
+Suppose the data $X \in \mathbb { R } ^ { d _ { x } }$ and its corresponding label $Y \in \mathbb { R } ^ { d _ { y } }$ are drawn from some distribution. Notice that in this section, $X$ and $Y$ are random vectors instead of matrices. We want to predict $Y$ given $X$ with a deep nonlinear neural network that has $H$ hidden layers. We express each layer of the network as functions $h _ { i } : \mathbb { R } ^ { d _ { i - 1 } }  \mathbb { R } ^ { d _ { i } }$ , so the entire network can be expressed as a composition of functions: $h _ { H + 1 } \circ h _ { H } \circ \cdot \cdot \cdot \circ h _ { 1 }$ . Our goal is to obtain functions $h _ { 1 } , \ldots , h _ { H + 1 }$ that minimize the population risk functional:
+
+$$
+L ( h ) = L ( h _ { 1 } , \dots , h _ { H + 1 } ) : = { \frac { 1 } { 2 } } \mathbb { E } \left[ \left\| h _ { H + 1 } \circ \cdot \cdot \cdot \circ h _ { 1 } ( X ) - Y \right\| _ { 2 } ^ { 2 } \right] ,
+$$
+
+where $h$ is a shorthand notation for $\left( h _ { 1 } , \ldots , h _ { H + 1 } \right)$ . It is well-known that the minimizer of squared error risk is the conditional expectation of $Y$ given $X$ , which we will denote $h ^ { * } ( x ) = \mathbb { E } [ Y \mid X = x ]$ . With this, we can separate the risk functional into two terms
+
+$$
+L ( h ) = \frac { 1 } { 2 } \mathbb { E } \left[ \| h _ { H + 1 } \circ \cdot \cdot \cdot \circ h _ { 1 } ( X ) - h ^ { * } ( X ) \| _ { 2 } ^ { 2 } \right] + C ,
+$$
+
+where the constant $C$ denotes the variance that is independent of $h _ { 1 } , \ldots , h _ { H + 1 }$ . Note that if $h _ { H + 1 } \circ$ $\cdots \circ h _ { 1 } = h ^ { * }$ almost surely, the first term in $L ( h )$ vanishes and the optimal value $L ^ { * }$ of $L ( h )$ is $C$ .
+
+Assumptions. Define the function spaces as the following:
+
+$$
+\begin{array} { r l } & { \mathcal { F } : = \{ h : \mathbb { R } ^ { d _ { x } }  \mathbb { R } ^ { d _ { y } } \ | \ h \mathrm { ~ i s ~ d i f f e r e n t i a b l e } , h ( 0 ) = 0 , \ \mathrm { a n d ~ } \underset { x } { \operatorname* { s u p } } \ \frac { \| h ( x ) \| _ { 2 } } { \| x \| _ { 2 } } < \infty \} , } \\ & { \mathcal { F } _ { i } : = \{ h : \mathbb { R } ^ { d _ { i } - 1 }  \mathbb { R } ^ { d _ { i } } \ | \ h \mathrm { ~ i s ~ d i f f e r e n t i a b l e } , h ( 0 ) = 0 , \ \mathrm { a n d ~ } \underset { x } { \operatorname* { s u p } } \ \frac { \| h ( x ) \| _ { 2 } } { \| x \| _ { 2 } } < \infty \} , } \end{array}
+$$
+
+where ${ \mathcal { F } } _ { i }$ are defined for all $i = 1 , \ldots , H + 1$ . Assume that $h ^ { \ast } \in \mathcal { F }$ , and that we are optimizing $L ( h )$ with $h _ { 1 } \in \mathcal { F } _ { 1 } , \ldots , h _ { H + 1 } \in \mathcal { F } _ { H + 1 }$ . In other words, the functions in $\mathcal { F } , \mathcal { F } _ { 1 } , \ldots , \mathcal { F } _ { H + 1 }$ are differentiable and show sublinear growth starting from 0. Notice that $h _ { H + 1 } \circ \cdot \cdot \cdot \circ h _ { 1 } \in \mathcal { F }$ , because a composition of differentiable functions is also differentiable, and a composition of sublinear functions is also sublinear. We also assume that $d _ { i } \geq \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ for all $i = 1 , \ldots , H + 1$ , which is identical to the assumption $k = \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ in Theorem 2.1.
+
+Notation. To simplify multiple composition of functions, we denote $h _ { i : j } = h _ { i } \circ h _ { i - 1 } \circ \cdot \cdot \cdot \circ h _ { j + 1 } \circ$ $h _ { j }$ . As in the matrix case, $h _ { 0 : 1 }$ and $h _ { H + 1 : H + 2 }$ mean identity maps in $\mathbb { R } ^ { d _ { x } }$ and $\mathbb { R } ^ { d _ { y } }$ , respectively. Given a function $f$ , let $J [ f ] ( x )$ be the Jacobian matrix of function $f$ evaluated at $x$ . Let $D _ { h _ { i } } [ L ( h ) ]$ be the Frechet derivative of ´ $L ( h )$ with respect to $h _ { i }$ evaluated at $h$ . The Frechet derivative ´ $D _ { h _ { i } } [ L ( h ) ]$ is a linear functional that maps a function (direction) $\eta \in \mathcal { F } _ { i }$ to a real number (directional derivative).
+
+# 4.2 SUFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY
+
+Here, we present two theorems which give sufficient conditions for a critical point $( D _ { h _ { i } } [ L ( h ) ] = 0$ for all $i$ ) in the function space to be a global optimum. The proofs are deferred to Appendix A.
+
+Theorem 4.1. Consider the case $d _ { x } \geq d _ { y }$ . If there exists $\epsilon > 0$ such that
+
+1. $J [ h _ { H + 1 : 2 } ] ( z ) \in \mathbb { R } ^ { d _ { y } \times d _ { 1 } }$ has $\sigma _ { \operatorname* { m i n } } ( J [ h _ { H + 1 : 2 } ] ( z ) ) \geq \epsilon f o r \ : a l l \ : z \in \mathbb { R } ^ { d _ { 1 } } ,$   
+2. $h _ { H + 1 : 2 } ( z )$ is twice-differentiable,
+
+then any critical point of $L ( h )$ , in terms of $D _ { h _ { 1 } } [ L ( h ) ] , \ldots , D _ { h _ { H + 1 } } [ L ( h ) ] ,$ , is a global minimum.
+
+Theorem 4.2. Consider the case $d _ { x } \leq d _ { y }$ . Assume that there exists some $j \in \{ 1 , \dots , H + 1 \}$ such that $d _ { x } = d _ { j - 1 }$ and $d _ { y } \leq d _ { j }$ . If there exist $\epsilon _ { 1 } , \epsilon _ { 2 } > 0$ such that
+
+1. $h _ { j - 1 : 1 } : \mathbb { R } ^ { d _ { x } }  \mathbb { R } ^ { d _ { j - 1 } } = \mathbb { R } ^ { d _ { x } }$ is invertible,
+
+2. $h _ { j - 1 : 1 }$ satisfies $\left\| h _ { j - 1 : 1 } ( u ) \right\| _ { 2 } \geq \epsilon _ { 1 } \left\| u \right\| _ { 2 }$ for all $u \in \mathbb { R } ^ { d _ { x } }$ ,
+
+3. $J [ h _ { H + 1 : j + 1 } ] ( z ) \in \mathbb { R } ^ { d _ { y } \times d _ { j } }$ has $\sigma _ { \mathrm { m i n } } ( J [ h _ { H + 1 : j + 1 } ] ( z ) ) \geq \epsilon _ { 2 }$ for all $z \in \mathbb { R } ^ { d _ { j } }$
+
+4. $h _ { H + 1 : j + 1 } ( z )$ is twice-differentiable,
+
+then any critical point of $L ( h )$ , in terms of $D _ { h _ { 1 } } [ L ( h ) ] , \ldots , D _ { h _ { H + 1 } } [ L ( h ) ] ,$ , is a global minimum.
+
+Note that these theorems give sufficient conditions, whereas Theorems 2.1 and 2.2 provide necessary and sufficient conditions. So, if the sets we are describing in Theorems 4.1 and $4 . 2 \ \mathrm { d o }$ not contain any critical point, the claims would be vacuous. We ensure that there are critical points in the sets, by presenting the following proposition, whose proof is also deferred to Appendix A.
+
+Proposition 4.3. For each of Theorems 4.1 and 4.2, there exists at least one global minimum solution of $L ( h )$ satisfying the conditions of the theorem.
+
+Discussion and Future work. Theorems 4.1 and 4.2 state that in certain sets of $\left( h _ { 1 } , \ldots , h _ { H + 1 } \right)$ , any critical point in function space a global minimum. However, this does not imply that any critical point for a fixed sigmoid or arctan network is a global minimum. As noted in (Bartlett et al., 2017), there is a downhill direction in function space at any suboptimal point, but this direction might be orthogonal to the function space represented by a fixed network, and may hence result in local minima in the parameter space of the fixed architecture.
+
+Understanding the connection between the function space and parameter space of commonly used architectures is an open direction for future research, and we believe that these results can be good initial steps from the theoretical point of view. For example, we can see that one of the sufficient conditions for global optimality is the Jacobian matrix being full rank. Given that a nonlinear function can locally be linearly approximated using Jacobians, this connection is already interesting. An extension of the function space viewpoint to cover different architectures or design new architectures (that have “better” properties when viewed via the function space view) should also be possible and worth studying.
+
+# ACKNOWLEDGMENTS
+
+This research project was supported in parts by DARPA DSO’s Fundamental Limits of Learning program.
+
+# REFERENCES
+
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+
+Peter Bartlett, Steve Evans, and Phil Long. Deep residual networks: Representation and optimization properties, 2017. Talk by Peter Bartlett at the Computational Challenges in Machine Learning Workshop at Simons Institute for the Theory of Computing, Berkeley, CA, USA.
+
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+
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+
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+
+Hassan K Khalil. Noninear Systems. Prentice-Hall, New Jersey, 1996.
+
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+
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+
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+
+George Zames. On the input-output stability of time-varying nonlinear feedback systems part one: Conditions derived using concepts of loop gain, conicity, and positivity. IEEE transactions on automatic control, 11(2):228–238, 1966.
+
+# A ANALYSIS OF DEEP NONLINEAR NETWORKS
+
+# A.1 NOTATION
+
+In this section, we introduce additional notation that is used in the proofs. To emphasize that the Frechet derivative ´ $D _ { h _ { i } } [ L ( h ) ]$ is a linear functional that outputs a real number, we will write $D _ { h _ { i } } [ L ( h ) ] ( \eta )$ in an inner-product form $\langle D _ { h _ { i } } [ L ( h ) ] , \eta \rangle$ . This notation also helps avoiding confusion coming from multiple parentheses and square brackets.
+
+There are many different kinds of norms that appear in the proofs. Given a finite-dimensional real vector $v$ , $\lVert \boldsymbol { v } \rVert _ { 2 }$ denotes its $\ell _ { 2 }$ norm. For a matrix $A$ , its operator norm is defined as $\| A \| _ { \mathrm { o p } } =$ $\operatorname* { s u p } _ { x } { \frac { \| A x \| _ { 2 } } { \| x \| _ { 2 } } }$ . Let $h \in { \mathcal { F } }$ . Then define a “generalized” induced norm for nonlinear functions with sublinear growth: khknl = supx $\begin{array} { r } { \| h \| _ { \mathrm { n l } } = \operatorname* { s u p } _ { x } \frac { \| h ( x ) \| _ { 2 } } { \| x \| _ { 2 } } } \end{array}$ kh(x)k2kxk , where the subscript nl is used to emphasize that this norm is for nonlinear functions. The norm $\lVert \cdot \rVert _ { \mathrm { n l } }$ is defined in the same way for ${ \mathcal { F } } _ { i }$ ’s. Now, given a linear functional $G$ that maps a function $f \in \mathcal { F } _ { i }$ to a real number $\langle G , f \rangle$ , define the operator norm kGkop = supf∈Fi kfknl .
+
+# A.2 FRECHET ´ DERIVATIVES
+
+By definition of Frechet derivatives, we have ´
+
+$$
+\langle D _ { h _ { i } } [ L ( h ) ] , \eta \rangle = \operatorname* { l i m } _ { \epsilon \to 0 } \frac { L ( h _ { 1 } , \dots , h _ { i } + \epsilon \eta , \dots , h _ { H + 1 } ) - L ( h ) } { \epsilon } ,
+$$
+
+where $\eta \in \mathcal { F } _ { i }$ is the direction of perturbation and $\langle D _ { h _ { i } } [ L ( h ) ] , \eta \rangle$ is the directional derivative along that direction $\eta$ . From the definition of $L ( h )$ ,
+
+$$
+\begin{array} { l } { { \displaystyle { \cal L } ( h _ { 1 } , \ldots , h _ { i } + \epsilon \eta , \ldots , h _ { H + 1 } ) } } \\ { { \displaystyle = \frac { 1 } { 2 } \mathbb { E } [ \| h _ { H + 1 : i + 1 } \circ ( h _ { i } + \epsilon \eta ) \circ h _ { i - 1 : 1 } ( X ) - h ^ { * } ( X ) \| _ { 2 } ^ { 2 } ] + C } } \\ { { \displaystyle = \frac { 1 } { 2 } \mathbb { E } [ \| h _ { H + 1 : i + 1 } ( h _ { i : 1 } ( X ) + \epsilon \eta ( h _ { i - 1 : 1 } ( X ) ) ) - h ^ { * } ( X ) \| _ { 2 } ^ { 2 } ] + C } } \\ { { \displaystyle = \frac { 1 } { 2 } \mathbb { E } [ \| h _ { H + 1 : 1 } ( X ) + \epsilon J [ h _ { H + 1 : i + 1 } ] ( h _ { i : 1 } ( X ) ) \eta ( h _ { i - 1 : 1 } ( X ) ) + O ( \epsilon ^ { 2 } ) - h ^ { * } ( X ) \| _ { 2 } ^ { 2 } ] + C } } \\ { { \displaystyle = { \cal L } ( h ) + \epsilon \mathbb { E } [ ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) ^ { T } J [ h _ { H + 1 : i + 1 } ] ( h _ { i : 1 } ( X ) ) \eta ( h _ { i - 1 : 1 } ( X ) ) ] + O ( \epsilon ^ { 2 } ) } } \end{array}
+$$
+
+Therefore,
+
+$$
+\langle D _ { h _ { i } } [ L ( h ) ] , \eta \rangle = \mathbb { E } \left[ ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) ^ { T } J [ h _ { H + 1 : i + 1 } ] ( h _ { i : 1 } ( X ) ) \eta ( h _ { i - 1 : 1 } ( X ) ) \right] .
+$$
+
+This equation (6) will be used in the proof of Theorems 4.1 and 4.2.
+
+# A.3 PROOF OF THEOREM 4.1
+
+From (6), consider $D _ { h _ { 1 } } [ L ( h ) ]$ . For any $\eta \in \mathcal { F } _ { 1 }$
+
+$$
+\langle D _ { h _ { 1 } } [ L ( h ) ] , \eta \rangle = \mathbb { E } \left[ ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) ^ { T } J [ h _ { H + 1 : 2 } ] ( h _ { 1 } ( X ) ) \eta ( X ) \right] .
+$$
+
+Let $A ( X ) = J [ h _ { H + 1 : 2 } ] ( h _ { 1 } ( X ) )$ . Since $A ( X )$ has full row rank by assumption, $A ( X ) A ( X ) ^ { T }$ is invertible. Then define a particular direction
+
+$$
+\tilde { \eta } ( X ) = A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) ,
+$$
+
+so that
+
+$$
+\begin{array} { r } { \langle D _ { h _ { 1 } } [ L ( h ) ] , \tilde { \eta } \rangle = \mathbb { E } \left[ \| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \| _ { 2 } ^ { 2 } \right] . } \end{array}
+$$
+
+It remains to check if $\tilde { \eta } \in \mathcal { F } _ { 1 }$ . It is easily checked that $\tilde { \eta } ( 0 ) = 0$ because $h _ { H + 1 : 1 } ( 0 ) - h ^ { * } ( 0 ) = \underline { { 0 } }$ . Since $J [ h _ { H + 1 : 2 } ]$ is differentiable by assumption and $h _ { 1 } \in \mathcal { F } _ { 1 }$ , $A ( X )$ is differentiable and ${ \dot { A ( X ) } } ^ { T }$ , $( A ( X ) \bar { A } ( X ) ^ { T } ) ^ { \bar { - } 1 }$ are differentiable functions. Also, $h _ { H + 1 : 1 } - h ^ { * } \in \mathcal { F }$ , so we can conclude that $\tilde { \eta }$ is differentiable.
+
+Moreover, if we decompose $A ( X )$ with SVD, $A ( X ) = U \Sigma V ^ { T }$ , $\Sigma$ is of the form $\Sigma = [ \Sigma _ { 1 } \ : 0 ]$ and
+
+$$
+\begin{array} { r l } & { A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } = V \Sigma ^ { T } U ^ { T } ( U \Sigma V ^ { T } V \Sigma ^ { T } U ^ { T } ) ^ { - 1 } = V \Sigma ^ { T } U ^ { T } ( U \Sigma _ { 1 } ^ { 2 } U ^ { T } ) ^ { - 1 } } \\ & { \qquad = V \Sigma ^ { T } U ^ { T } U \Sigma _ { 1 } ^ { - 2 } U ^ { T } = V \left[ \Sigma _ { 0 } ^ { - 1 } \right] U ^ { T } , } \end{array}
+$$
+
+from which we can see that
+
+$$
+\begin{array} { r } { \big \| { A ( X ) } ^ { T } ( A ( X ) { A ( X ) } ^ { T } ) ^ { - 1 } \big \| _ { \mathrm { o p } } = \sigma _ { \operatorname* { m a x } } ( { A ( X ) } ^ { T } ( A ( X ) { A ( X ) } ^ { T } ) ^ { - 1 } ) \leq 1 / \epsilon , } \end{array}
+$$
+
+by our assumption. Note that, for any $X \in \mathbb { R } ^ { d _ { x } }$ ,
+
+$$
+\begin{array} { r l } & { \| \tilde { \eta } ( X ) \| _ { 2 } = \left\| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) \right\| _ { 2 } } \\ & { \qquad \leq \left\| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } \right\| _ { \mathrm { o p } } \left\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \right\| _ { 2 } } \\ & { \qquad \leq \left\| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } \right\| _ { \mathrm { o p } } \left\| h _ { H + 1 : 1 } - h ^ { * } \right\| _ { \mathrm { n l } } \left\| X \right\| _ { 2 } . } \end{array}
+$$
+
+Since this holds for any $X$ , we have
+
+$$
+\begin{array} { r } { \| \tilde { \eta } \| _ { \mathrm { n l } } \leq \| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } \| _ { \mathrm { o p } } \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } \leq \frac { \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } } { \epsilon } , } \end{array}
+$$
+
+which ensures that $\tilde { \eta } \in \mathcal { F } _ { 1 }$ . Finally,
+
+$$
+\| D _ { h _ { 1 } } [ L ( h ) ] \| _ { \mathrm { o p } } \ge \frac { \langle D _ { h _ { 1 } } [ L ( h ) ] , \tilde { \eta } \rangle } { \| \tilde { \eta } \| _ { \mathrm { n l } } } \ge \frac { \epsilon \mathbb { E } \left[ \| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \| _ { 2 } ^ { 2 } \right] } { \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } } = \frac { \epsilon ( L ( h ) - L ^ { * } ) } { \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } } ,
+$$
+
+which yields
+
+$$
+\begin{array} { r } { \| D _ { h _ { 1 } } [ L ( h ) ] \| _ { \mathrm { o p } } \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } \ge \epsilon ( L ( h ) - L ^ { * } ) . } \end{array}
+$$
+
+From this we can see that if we have a critical point of $L ( h )$ , then $\| D _ { h _ { 1 } } [ L ( h ) ] \| _ { \mathrm { o p } } = 0$ implies $L ( h ) = L ^ { * }$ , which means that the critical point is a global minimum of $L ( h )$ .
+
+# A.4 PROOF OF THEOREM 4.2
+
+Recall that by assumption we have $j \in \{ 1 , \dots , H + 1 \}$ such that $d _ { x } = d _ { j - 1 }$ and $d _ { y } \leq d _ { j }$ . Consider $D _ { h _ { j } } [ L ( h ) ]$ , then for any $\eta \in \mathcal { F } _ { j }$ ,
+
+$$
+\left. D _ { h _ { j } } [ L ( h ) ] , \eta \right. = \mathbb { E } \left[ \left( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \right) ^ { T } J [ h _ { H + 1 : j + 1 } ] ( h _ { j : 1 } ( X ) ) \eta ( h _ { j - 1 : 1 } ( X ) ) \right] .
+$$
+
+As done in the previous theorem, for any $w \in \mathbb { R } ^ { d _ { j - 1 } }$ , let $A ( w ) = J [ h _ { H + 1 : j + 1 } ] ( h _ { j } ( w ) )$ . Since $A ( w )$ has full row rank by assumption, $A ( w ) A ( w ) ^ { T }$ is invertible. Then define
+
+$$
+\tilde { \eta } ( w ) = A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } - h ^ { * } ) \circ h _ { j - 1 : 1 } ^ { - 1 } ( w ) ,
+$$
+
+so that
+
+$$
+\begin{array} { r } { \left. D _ { h _ { j } } [ L ( h ) ] , \tilde { \eta } \right. = \mathbb { E } \left[ \left\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \right\| _ { 2 } ^ { 2 } \right] . } \end{array}
+$$
+
+We need to check if $\tilde { \eta } \in \mathcal { F } _ { j }$ . It is easily checked that $\tilde { \eta } ( 0 ) = 0$ . Since $J [ h _ { H + 1 : j + 1 } ]$ is differentiable by assumption and $h _ { j } \in { \mathcal { F } } _ { j }$ , $A ( w )$ is differentiable, and so are $A ( w ) ^ { T }$ and $( \dot { A ( w ) } \dot { A ( w ) } ^ { T } ) ^ { - 1 }$ . The inverse function of a differentiable and invertible function is also differentiable, so $\left( h _ { H + 1 : 1 } - h ^ { * } \right) \circ$ $h _ { j - 1 : 1 } ^ { - 1 }$ is differentiable. Hence, we can conclude that $\tilde { \eta }$ is differentiable.
+
+As seen in the previous section,
+
+$$
+\begin{array} { r } { \left\| \boldsymbol A ( \boldsymbol w ) ^ { T } ( \boldsymbol A ( \boldsymbol w ) \boldsymbol A ( \boldsymbol w ) ^ { T } ) ^ { - 1 } \right\| _ { \mathrm { o p } } = \sigma _ { \mathrm { m a x } } ( \boldsymbol A ( \boldsymbol w ) ^ { T } ( \boldsymbol A ( \boldsymbol w ) \boldsymbol A ( \boldsymbol w ) ^ { T } ) ^ { - 1 } ) \le 1 / \epsilon _ { 2 } . } \end{array}
+$$
+
+By the assumption that $h _ { j - 1 : 1 }$ is invertible and $\left\| h _ { j - 1 : 1 } ( u ) \right\| _ { 2 } \geq \epsilon _ { 1 } \left\| u \right\| _ { 2 }$ ,
+
+$$
+\left\| v \right\| _ { 2 } \geq \epsilon _ { 1 } \left\| h _ { j - 1 : 1 } ^ { - 1 } ( v ) \right\| _ { 2 } ,
+$$
+
+for all $v \in \mathbb { R } ^ { d _ { j - 1 } }$ . From this, we can see that $\| h _ { j - 1 : 1 } ^ { - 1 } \| _ { \mathrm { n l } } \leq 1 / \epsilon _ { 1 }$ . For any $w \in \mathbb { R } ^ { d _ { j - 1 } }$
+
+$$
+\begin{array} { r l } & { \left. \tilde { \eta } ( w ) \right. _ { 2 } = \left. A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } - h ^ { * } ) \circ h _ { j - 1 : 1 } ^ { - 1 } ( w ) \right. _ { 2 } } \\ & { \qquad \leq \left. A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } \right. _ { \mathrm { o p } } \left. \left( h _ { H + 1 : 1 } - h ^ { * } \right) \circ h _ { j - 1 : 1 } ^ { - 1 } ( w ) \right. _ { 2 } } \\ & { \qquad \leq \left. A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } \right. _ { \mathrm { o p } } \left. h _ { H + 1 : 1 } - h ^ { * } \right. _ { \mathrm { n l } } \left. h _ { j - 1 : 1 } ^ { - 1 } ( w ) \right. _ { 2 } } \\ & { \qquad \leq \left. A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } \right. _ { \mathrm { o p } } \left. h _ { H + 1 : 1 } - h ^ { * } \right. _ { \mathrm { n l } } \left. h _ { j - 1 : 1 } ^ { - 1 } \right. _ { \mathrm { n l } } \left. w \right. _ { 2 } . } \end{array}
+$$
+
+From this, we have
+
+$$
+\begin{array} { r } { \| \tilde { \eta } \| _ { \mathrm { n l } } \leq \| A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } \| _ { \mathrm { o p } } \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } \left\| h _ { j - 1 : 1 } ^ { - 1 } \right\| _ { \mathrm { n l } } \leq \frac { \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } } { \epsilon _ { 1 } \epsilon _ { 2 } } . } \end{array}
+$$
+
+Finally,
+
+$$
+\left\| \boldsymbol { D } _ { h _ { j } } [ L ( h ) ] \right\| _ { \mathrm { o p } } \ge \frac { \left. \boldsymbol { D } _ { h _ { j } } [ L ( h ) ] , \tilde { \eta } \right. } { \| \tilde { \eta } \| _ { \mathrm { n l } } } \ge \frac { \epsilon _ { 1 } \epsilon _ { 2 } \mathbb { E } \left[ \| h _ { H + 1 : 1 } ( \boldsymbol { X } ) - h ^ { * } ( \boldsymbol { X } ) \| _ { 2 } ^ { 2 } \right] } { \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } } = \frac { \epsilon _ { 1 } \epsilon _ { 2 } \left( L ( h ) - L ^ { * } \right) } { \left\| h _ { H + 1 : 1 } - h ^ { * } \right\| _ { \mathrm { n l } } } ,
+$$
+
+which yields
+
+$$
+\begin{array} { r } { \left\| D _ { h _ { j } } [ L ( h ) ] \right\| _ { \mathrm { o p } } \| h _ { H + 1 : 1 } - h ^ { * } \| _ { \mathrm { n l } } \geq \epsilon _ { 1 } \epsilon _ { 2 } ( L ( h ) - L ^ { * } ) . } \end{array}
+$$
+
+# A.5 PROOF OF PROPOSITION 4.3
+
+(Theorem 4.1) By assumption, we have $d _ { 1 } \geq d _ { y }$ . Set $h _ { 1 } ( x ) = ( h ^ { * } ( x ) , 0 , \ldots , 0 )$ where for every $x \in \mathbb { R } ^ { d _ { x } }$ , the first $d _ { y }$ components of $h _ { 1 } ( x )$ are identical to $h ^ { \ast } ( x )$ , and all other components are zero. For the rest of $h _ { i }$ ’s, define $h _ { i } : \mathbb { R } ^ { d _ { i - 1 } }  \mathbb { R } ^ { d _ { i } }$ to be
+
+$$
+h _ { i } ( w ) = \left\{ \begin{array} { l l } { ( w _ { 1 } , \ldots , w _ { d _ { i } } ) , } & { \mathrm { i f ~ } d _ { i } \leq d _ { i - 1 } , } \\ { ( w _ { 1 } , \ldots , w _ { d _ { i - 1 } } , 0 , \ldots , 0 ) , } & { \mathrm { i f ~ } d _ { i } > d _ { i - 1 } , } \end{array} \right.
+$$
+
+for all $w \in \mathbb { R } ^ { d _ { i - 1 } }$ . Since $d _ { i } \geq d _ { y }$ for all $i$ , we can check that $h _ { H + 1 } \circ \cdot \cdot \cdot \circ h _ { 1 } = h ^ { * }$ , and $h _ { i } \in$ ${ \mathcal { F } } _ { i }$ for all $i$ . Moreover, for all $z ~ \in ~ \mathbb { R } ^ { d _ { 1 } }$ , $J [ h _ { H + 1 : 2 } ] ( z )$ is all 0 except 1’s in diagonal entries, so $\sigma _ { \operatorname* { m i n } } ( J [ h _ { H + 1 : 2 } ] ( z ) ) \geq 1$ and $h _ { H + 1 : 2 } ( z )$ is twice-differentiable.
+
+(Theorem 4.2) It is given that we have $j \in \{ 1 , \dots , H + 1 \}$ such that $d _ { x } = d _ { j - 1 }$ and $d _ { y } \leq d _ { j }$ . Set $h _ { j } ( x ) = ( h ^ { * } ( x ) , 0 , \ldots , 0 )$ , where the first $d _ { y }$ components are $h ^ { \ast } ( x )$ and the rest are zero. All the rest of $h _ { i }$ are set as in (7). Then, it can be easily checked that $h _ { i } \in { \mathcal { F } } _ { i }$ for all $i$ and all the conditions of the theorem are satisfied.
+
+# B DEFERRED LEMMA
+
+Lemma B.1. Suppose we are given a data matrix $X \in \mathbb { R } ^ { d _ { x } \times m }$ and an output matrix $Y \in \mathbb { R } ^ { d _ { y } \times m }$ , where $d _ { x } \ < \ d _ { y }$ . Assume $X \breve { X } ^ { T }$ and $Y X ^ { T }$ have full ranks. Consider minimizing the empirical squared error risk:
+
+$$
+L ( W _ { 1 } , \dots , W _ { H + 1 } ) : = \frac { 1 } { 2 } \left\| W _ { H + 1 } W _ { H } \cdot \cdot \cdot W _ { 1 } X - Y \right\| _ { \mathrm { F } } ^ { 2 } ,
+$$
+
+where $W _ { k } \in \mathbb { R } ^ { d _ { k } \times d _ { k - 1 } }$ , $k = 1 , \ldots , H + 1$ are weight matrices of the linear neural network, and $d _ { 0 } = d _ { x }$ and $d _ { H + 1 } = d _ { y }$ for simplicity in notation. Also let $W$ denote the tuple $( W _ { 1 } , \dots , W _ { H + 1 } )$ . Then, any critical point of $L ( W )$ that is not a local minimum is a saddle point.
+
+Proof. For this lemma, we separate the proof into two cases: $W _ { H } \cdot \cdot \cdot W _ { 1 } \neq 0$ and $W _ { H } \cdot \cdot \cdot W _ { 1 } = 0$ . The crux of the proof is to show that any critical point cannot be a local maximum. Then, any critical point is either a local minimum or a saddle point, so the conclusion of this lemma follows.
+
+In case of $W _ { H } \cdot \cdot \cdot W _ { 1 } \neq 0$ , we use some of the results in Kawaguchi (2016) and examine the Hessian of $L ( W )$ with respect to $\mathrm { v e c } ( W _ { H + 1 } ^ { T } )$ , where $\operatorname { v e c } ( A )$ denotes vectorization of matrix $A$ .
+
+Let $D _ { \mathrm { v e c } ( W _ { H + 1 } ^ { T } ) } L ( W )$ be the partial derivative of $L ( W )$ with respect to $\mathrm { v e c } ( W _ { H + 1 } ^ { T } )$ in numerator layout. It was shown by Kawaguchi (2016, Lemma 4.3) that the Hessian matrix
+
+$$
+\begin{array} { r l r } & { } & { \mathcal { H } ( W ) = D _ { \mathrm { v e c } ( W _ { H + 1 } ^ { T } ) } \left( D _ { \mathrm { v e c } ( W _ { H + 1 } ^ { T } ) } L ( W ) \right) ^ { T } = \left( I \otimes ( W _ { H } \cdot \cdot \cdot W _ { 1 } X ) ( W _ { H } \cdot \cdot \cdot W _ { 1 } X ) ^ { T } \right) } \\ & { } & { \quad = \left( I \otimes W _ { H } \cdot \cdot \cdot W _ { 1 } X X ^ { T } W _ { 1 } ^ { T } \cdot \cdot \cdot W _ { H } ^ { T } \right) , } \end{array}
+$$
+
+where $\otimes$ denotes the Kronecker product of two matrices. Notice that $\mathcal { H } ( W )$ is positive semidefinite. Since $X X ^ { T }$ is full rank, whenever $W _ { H } \cdot \cdot \cdot W _ { 1 } \neq 0$ there exists a strictly positive eigenvalue in $\mathcal { H } ( W )$ , which means that there exists an increasing direction. So $W$ cannot be a local maximum.
+
+The case where $W _ { H } \cdot \cdot \cdot W _ { 1 } = 0$ requires a bit more careful treatment. Note that this case corresponds to where we have degenerate critical points, which are in many cases much harder to handle.
+
+For any arbitrary $\epsilon > 0$ , we describe a procedure that perturbs the matrices $W _ { 1 } , \dots , W _ { H + 1 }$ by perturbations sampled from Frobenius norm balls of radius $\epsilon$ centered at 0, which we will denote as $\bar { \boldsymbol { B } } _ { i } ( \epsilon )$ , $i = 1 , \ldots , H + 1$ . Let $\mathcal { U } ( B _ { i } ( \epsilon ) )$ be the uniform distribution over the ball $B _ { i } ( \epsilon )$ . The algorithm goes as the following:
+
+1. For $i \in \{ 1 , \ldots , H + 1 \}$
+
+1.1. Sample $\Delta _ { i } \sim \mathcal { U } ( B _ { i } ( \epsilon ) )$ , and define $V _ { i } = W _ { i } + \Delta _ { i }$
+
+1.2. If $W _ { H + 1 } \cdot \cdot \cdot W _ { i + 1 } V _ { i } \cdot \cdot \cdot V _ { 1 } \neq 0$ , stop and return $i ^ { * } = i$ .
+
+First, recall that the set of rank-deficient matrices have Lebesgue measure zero, so for any sample $\Delta _ { i } \sim \mathcal { U } ( B _ { i } ( \epsilon ) )$ , $V _ { i } = W _ { i } + \Delta _ { i }$ has full rank with probability 1. If we proceed the for loop until $i = H + 1$ , we have a full-rank $V _ { H + 1 } \cdots V _ { 1 }$ with probability 1, which means that the algorithm must return $i ^ { * } \in \{ 1 , \ldots , H + 1 \}$ with probability 1. Notice that before and after the $i ^ { * }$ -th iteration, we have
+
+$$
+\begin{array} { r l } & { W _ { H + 1 } \cdots W _ { i ^ { * } } V _ { i ^ { * } - 1 } \cdots V _ { 1 } = 0 , } \\ & { W _ { H + 1 } \cdots W _ { i ^ { * } + 1 } V _ { i ^ { * } } \cdots V _ { 1 } = W _ { H + 1 } \cdot \cdot \cdot W _ { i ^ { * } + 1 } ( W _ { i ^ { * } } + \Delta _ { i ^ { * } } ) V _ { i ^ { * } - 1 } \cdot \cdot \cdot V _ { 1 } \not = 0 . } \end{array}
+$$
+
+This means that if we define $\hat { \Delta } = { \cal W } _ { H + 1 } \cdot \cdot \cdot { \cal W } _ { i ^ { * } + 1 } \Delta _ { i ^ { * } } V _ { i ^ { * } - 1 } \cdot \cdot \cdot V _ { 1 }$ , then $\hat { \Delta } \neq 0$ . Also, notice that
+
+$$
+W _ { H + 1 } \cdot \cdot \cdot W _ { i ^ { * } + 1 } ( W _ { i ^ { * } } - \Delta _ { i ^ { * } } ) V _ { i ^ { * } - 1 } \cdot \cdot \cdot V _ { 1 } = - \hat { \Delta } .
+$$
+
+Now, define two points
+
+$$
+\begin{array} { r l } & { U ^ { ( 1 ) } = ( V _ { 1 } , \ldots , V _ { i ^ { * } - 1 } , W _ { i ^ { * } } + \Delta _ { i ^ { * } } , W _ { i ^ { * } + 1 } , \ldots , W _ { H + 1 } ) , } \\ & { U ^ { ( 2 ) } = ( V _ { 1 } , \ldots , V _ { i ^ { * } - 1 } , W _ { i ^ { * } } - \Delta _ { i ^ { * } } , W _ { i ^ { * } + 1 } , \ldots , W _ { H + 1 } ) , } \end{array}
+$$
+
+and notice that they are all in the neighborhood of $W$ , that is, the Cartesian product of $\epsilon$ -radius balls centered at $W _ { 1 } , \dots , W _ { H + 1 }$ . Moreover, we have
+
+$$
+\begin{array} { l } { \displaystyle { L ( W ) = \frac { 1 } { 2 } \left\| 0 \cdot X - Y \right\| _ { \mathrm { F } } ^ { 2 } = \frac { 1 } { 2 } \left\| Y \right\| _ { \mathrm { F } } ^ { 2 } , } } \\ { \displaystyle { L ( U ^ { ( 1 ) } ) = \frac { 1 } { 2 } \left\| \hat { \Delta } X - Y \right\| _ { \mathrm { F } } ^ { 2 } = \frac { 1 } { 2 } \left\| Y \right\| _ { \mathrm { F } } ^ { 2 } + \frac { 1 } { 2 } \left\| \hat { \Delta } X \right\| _ { \mathrm { F } } ^ { 2 } - \left. \hat { \Delta } X , Y \right. , } } \\ { \displaystyle { L ( U ^ { ( 2 ) } ) = \frac { 1 } { 2 } \left\| - \hat { \Delta } X - Y \right\| _ { \mathrm { F } } ^ { 2 } = \frac { 1 } { 2 } \left\| Y \right\| _ { \mathrm { F } } ^ { 2 } + \frac { 1 } { 2 } \left\| \hat { \Delta } X \right\| _ { \mathrm { F } } ^ { 2 } + \left. \hat { \Delta } X , Y \right. , } } \end{array}
+$$
+
+from which we can see that at least one of $L ( W ) < L ( U ^ { ( 1 ) } )$ or $L ( W ) < L ( U ^ { ( 2 ) } )$ must hold. This shows that for any $\epsilon > 0$ , there is a point $U$ in $\epsilon$ -neighborhood of $W$ with a strictly greater function value $L ( U )$ . This proves that $W$ cannot be a local maximum. □
+
+# C DEFERRED PROOFS
+
+# C.1 PROOF OF PROPOSITION 3.1
+
+In case of $k < \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$ , we can decompose the loss function in the following way:
+
+$$
+\begin{array} { r l } & { \left\| R X - Y \right\| _ { \mathrm { F } } ^ { 2 } = \left\| R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X + Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y \right\| _ { \mathrm { F } } ^ { 2 } } \\ & { \qquad = \left\| R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \right\| _ { \mathrm { F } } ^ { 2 } + \left\| Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y \right\| _ { \mathrm { F } } ^ { 2 } } \\ & { \qquad + 2 \operatorname { t r } ( ( Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y ) ( R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X ) ^ { T } ) . } \end{array}
+$$
+
+Let us take a close look into the last term in the RHS. Note that $Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X$ is the orthogonal projection of $Y$ onto row $( X )$ , so each row of $Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y$ must be in $\operatorname { n u l l } ( X )$ . Also,
+
+$$
+( R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X ) ^ { T } = X ^ { T } ( R ^ { T } - ( X X ^ { T } ) ^ { - 1 } X Y ^ { T } ) .
+$$
+
+It is $X ^ { T }$ right-multiplied with some matrix, so its columns must lie in $\operatorname { c o l } ( X ^ { T } ) = \operatorname { r o w } ( X )$ . By the fact that $\mathrm { n u l l } ( X ) ^ { \perp } \overset { \cdot } { = } \operatorname { r o w } ( X )$ ,
+
+$$
+( Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y ) ( R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X ) ^ { T } = 0 .
+$$
+
+thus
+
+$$
+L _ { 0 } ( R ) = { \frac { 1 } { 2 } } \left\| R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \right\| _ { \mathrm { F } } ^ { 2 } + { \frac { 1 } { 2 } } \left\| Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y \right\| _ { \mathrm { F } } ^ { 2 }
+$$
+
+holds.
+
+Now, (2) becomes a problem of minimizing $\begin{array} { r } { \left\| R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \right\| _ { \mathrm { F } } ^ { 2 } } \end{array}$ subject to the rank constraint $\mathrm { r a n k } ( R ) \leq k$ . The optimal solution for this is obtained when $R X$ is the $k$ -rank approximation of $\overset { \cdot } { Y X ^ { T } } ( X X ^ { T } ) ^ { - 1 } X$ . Then, $k$ -rank approximation of $Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X$ can be expressed as $\hat { U } \hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X$ , where $\hat { U }$ is unique due to our assumption that all singular values are distinct. Therefore,
+
+$$
+R ^ { * } = \hat { U } \hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }
+$$
+
+is the unique global minimum solution of (2) when $k < \operatorname* { m i n } \{ d _ { x } , d _ { y } \}$
+
+# C.2 PROOF OF LEMMA 3.2
+
+$$
+\begin{array} { l } { { \displaystyle \begin{array} { l } { { \displaystyle L ( W _ { 1 } , \ldots , W _ { i - 1 } , W _ { i } + \Delta _ { i } , W _ { i + 1 } , \ldots , W _ { H + 1 } ) } } \\ { { \displaystyle = \frac { 1 } { 2 } \| W _ { H + 1 } \cdot \cdot \cdot W _ { i + 1 } ( W _ { i } + \Delta _ { i } ) W _ { i - 1 } \cdot \cdot \cdot W _ { 1 } X - Y \| _ { \mathrm { F } } ^ { 2 } } } \\ { { \displaystyle = \frac { 1 } { 2 } \| W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } X - Y + W _ { H + 1 } \cdot \cdot \cdot W _ { i + 1 } \Delta _ { i } W _ { i - 1 } \cdot \cdot \cdot W _ { 1 } X \| _ { \mathrm { F } } ^ { 2 } } } \\ { { \displaystyle = L ( W ) + \mathrm { t r } ( ( W _ { H + 1 } \cdot \cdot \cdot W _ { i + 1 } \Delta _ { i } W _ { i - 1 } \cdot \cdot \cdot W _ { 1 } X ) ^ { T } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } X - Y ) ) + O ( \| \Delta _ { i } \| _ { \mathrm { F } } ^ { 2 } ) } } \end{array} } } \\ { { \displaystyle = L ( W ) + \mathrm { t r } ( W _ { i + 1 } ^ { T } \cdot \cdot \cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \cdot \cdot \cdot W _ { i - 1 } ^ { T } \Delta _ { i } ^ { T } ) + O ( \| \Delta _ { i } \| _ { \mathrm { F } } ^ { 2 } ) . } } \end{array}
+$$
+
+From this, we can conclude that
+
+$$
+\frac { \partial L } { \partial W _ { i } } = W _ { i + 1 } ^ { T } \cdot \cdot \cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } \cdot \cdot \cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \cdot \cdot \cdot W _ { i - 1 } ^ { T } .
+$$
+
+# C.3 PROOF OF LEMMA 3.3
+
+1. Since $A ^ { T } A \succeq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( A ) I , B ^ { T } A ^ { T } A B \succeq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( A ) B ^ { T } B$ . Then
+
+$$
+\left\| A B \right\| _ { \mathrm { F } } ^ { 2 } = \mathrm { t r } ( B ^ { T } A ^ { T } A B ) \geq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( A ) \mathrm { t r } ( B ^ { T } B ) = \sigma _ { \operatorname* { m i n } } ^ { 2 } ( A ) \left\| B \right\| _ { \mathrm { F } } ^ { 2 } .
+$$
+
+2. Since $B B ^ { T } \succeq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( B ) I , A B B ^ { T } A ^ { T } \succeq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( B ) A A ^ { T }$ . Then
+
+$$
+\left\| A B \right\| _ { \mathrm { F } } ^ { 2 } = \operatorname { t r } ( B ^ { T } A ^ { T } A B ) = \operatorname { t r } ( A B B ^ { T } A ^ { T } ) \geq \sigma _ { \operatorname* { m i n } } ^ { 2 } ( B ) \operatorname { t r } ( A A ^ { T } ) = \sigma _ { \operatorname* { m i n } } ^ { 2 } ( B ) \left\| A \right\| _ { \mathrm { F } } ^ { 2 } .
+$$

parse/train/BJlnmgrFvS/BJlnmgrFvS.md

@@ -0,0 +1,373 @@
+# BAIL: BEST-ACTION IMITATION LEARNING FOR BATCH DEEP REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The field of Deep Reinforcement Learning (DRL) has recently seen a surge in research in batch reinforcement learning, which aims for sample-efficient learning from a given data set without additional interactions with the environment. In the batch DRL setting, commonly employed off-policy DRL algorithms can perform poorly and sometimes even fail to learn altogether. In this paper we propose a new algorithm, Best-Action Imitation Learning (BAIL), which unlike many offpolicy DRL algorithms does not involve maximizing Q functions over the action space. Striving for simplicity as well as performance, BAIL first selects from the batch the actions it believes to be high-performing actions for their corresponding states; it then uses those state-action pairs to train a policy network using imitation learning. Although BAIL is simple, we demonstrate that BAIL achieves state of the art performance on the Mujoco benchmark.
+
+# 1 INTRODUCTION
+
+The field of Deep Reinforcement Learning (DRL) has recently seen a surge in research in batch reinforcement learning, which is the problem of sample-efficient learning from a given data set without additional interactions with the environment. Batch reinforcement learning is appealing because it dis-entangles policy optimization (exploitation) from data collection (exploration). This enables reusing the data collected by a policy to possibly improve the policy without further interactions with the environment. Furthermore, a batch learning reinforcement learning algorithm can potentially be deployed as part of a growing-batch algorithm, where the batch algorithm seeks a high-performing exploitation policy using the data in an experience replay buffer, combines this policy with exploration to add fresh data to the buffer, and then repeats the whole process (Lange et al., 2012).
+
+Fujimoto et al. (2018a) recently made the critical observation that commonly employed off-policy algorithms based on Deep Q-Learning (DQL) often perform poorly and sometimes even fail to learn altogether. Indeed, off-policy DRL algorithms typically involve maximizing an approximate Qfunction over the action space (Lillicrap et al., 2015; Fujimoto et al., 2018b; Haarnoja et al., 2018a), leading to an extrapolation error, particularly for state-action pairs that are not in the batch distribution. Batch-Constrained deep Q-learning (BCQ), which obtains good performance for many of the Mujoco environments (Todorov et al., 2012), avoids the extrapolation error problem by constraining the set of actions over which the approximate Q-function is optimized (Fujimoto et al., 2018a).
+
+We propose a new algorithm, Best-Action Imitation Learning (BAIL), which strives for both simplicity and performance. BAIL does not suffer from the extrapolation error problem since it does not maximize over the action space in any step of the algorithm. BAIL is simple, thereby satisfying the principle of Occam’s razor.
+
+The BAIL algorithm has two steps. In the first step, it selects from the batch a subset of state-action pairs for which the actions are believed to be good actions for their corresponding states. In the second step, it simply trains a policy network with imitation learning using the selected actions from the first step. To find the best actions, we train a neural network to obtain the “upper envelope” of the Monte Carlo returns in the batch data, and then we select from the batch the state-action pairs that are near the upper envelope. We believe the concept of the upper-envelope of a data set is also novel and interesting in its own right.
+
+Because the BCQ code is publicly available, we are able to make a careful comparison of the performance of BAIL and BCQ. We do this for batches generated by training DDPG (Lillicrap et al., 2015) for the Half-Cheetah, Walker, and Hopper environments, and for batches generated by training Soft Actor Critic (SAC) for the Ant environment (Haarnoja et al., 2018a;b). Although BAIL is simple, we demonstrate that BAIL achieves state of the art performance on the Mujoco benchmark, often outperforming Batch Constrained deep Q-Learning (BCQ) by a wide-margin. We also provide anonymized code for reproducibility1.
+
+# 2 RELATED WORK
+
+Batch reinforcement learning in both the tabular and functional approximator settings has long been studied (Lange et al., 2012; Strehl et al., 2010) and continues to be a highly active area of research (Swaminathan & Joachims, 2015; Jiang & Li, 2015; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Irpan et al., 2019; Jaques et al., 2019). Imitation learning is also a well-studied problem (Schaal, 1999; Argall et al., 2009; Hussein et al., 2017) and also continues to be a highly active area of research (Kim et al., 2013; Piot et al., 2014; Chemali & Lazaric, 2015; Hester et al., 2018; Ho et al., 2016; Sun et al., 2017; 2018; Cheng et al., 2018; Gao et al., 2018).
+
+This paper relates most closely to (Fujimoto et al., 2018a), which made the critical observation that when conventional DQL-based algorithms are employed for batch reinforcement learning, performance can be very poor, with the algorithm possibly not learning at all. Off-policy DRL algorithms involve maximizing an approximate action-value function $Q ( s , a )$ over all actions in the action space. (Or over the actions in the manifold of the parameterized policy.) The approximate action-value function can be very inaccurate, particularly for state-action pairs that are not in the state-action distribution of the batch (Fujimoto et al., 2018a). Due to this extrapolation error, poorperforming actions can be chosen when optimizing $Q ( s , a )$ over all actions. With traditional offpolicy DRL algorithms (such as DDPG (Lillicrap et al., 2015), TD3 (Fujimoto et al., 2018b) and SAC (Haarnoja et al., 2018a)), if the action-value function over-estimates a state-action pair, the policy will subsequently collect new data in the over-estimated region, and the estimate will get corrected. In the batch setting, however, where there is no further interaction with the environment, the extrapolation error is not corrected, and the poor choice of action persists in the policy (Fujimoto et al., 2018a).
+
+Batch-Constrained deep Q-learning (BCQ) avoids the extrapolation error problem by constraining the set of actions over which the approximate Q-function is optimized (Fujimoto et al., 2018a). More specifically, BCQ first trains a state-dependent Variational Auto Encoder (VAE) using the state action pairs in the batch data. When optimizing the approximate Q-function over actions, instead of optimizing over all actions, it optimizes over a subset of actions generated by the VAE. The BCQ algorithm is further complicated by introducing a perturbation model, which employs an additional neural network that outputs an adjustment to an action. BCQ additionally employs a modified version of clipped-Double Q-Learning to obtain satisfactory performance. We show experimentally that our much simpler BAIL algorithm typically performs better than BCQ by a wide margin.
+
+Kumar et al. (2019) recently proposed BEAR for batch DRL. BEAR is also complex, employing Maximum Mean Discrepancy (Gretton et al., 2012), kernel selection, a parametric model that fits a tanh-Gaussian distribution, and a test policy that is different from the learned actor policy. In this paper we do not experimentally compare BAIL with BEAR since the code for BEAR is not publicly available at the time of writing.
+
+Agarwal et al. (2019) recently proposed another algorithm for batch DRL called Random Ensemble Mixture (REM), an ensembling scheme which enforces optimal Bellman consistency on random convex combinations of the Q-heads of a multi-headed Q-network. For the Atari 2600 games, batch REM can out-perform the policies used to collect the data. REM and BAIL are orthogonal, and it may be possible to combine them in the future to achieve even higher performance. No experimental results are provided for REM applied to the Mujoco benchmark (Agarwal et al., 2019).
+
+# 3 BATCH DEEP REINFORCEMENT LEARNING
+
+We represent the environment with a Markov Decision Process (MDP) defined by a tuple $( S , { \mathcal { A } } , g , r , \rho , \gamma )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $\rho$ is the initial state distribution, and $\gamma$ is the discount factor. The functions $g ( s , a )$ and $r ( s , a )$ represent the dynamics and reward function, respectively. In this paper we assume that the dynamics of the environment are deterministic, that is, there are real-valued functions $g ( s , a )$ and $r ( s , a )$ such that when in state $s$ and action $a$ is chosen, then the next state is $s ^ { \prime } = g ( s , a )$ and the reward received is $r ( s , a )$ . We note that all the simulated robotic locomotion environments in the Mujoco benchmark are deterministic, and many robotic tasks are expected to be deterministic environments. Furthermore, many of the Atari game environments are deterministic (Bellemare et al., 2013). Thus, from an applications perspective, the class of deterministic environments is a large and important class. Although we assume that the environment is deterministic, as is typically the case with reinforcement learning, we do not assume the functions $g ( s , a )$ and $r ( s , a )$ are known.
+
+In batch reinforcement learning, we are provided a batch of $m$ data points $\begin{array} { r } { B = \{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i } ^ { \prime } ) , i = } \end{array}$ $1 , . . . , m \}$ . We assume $\boldsymbol { B }$ is fixed and given, and there is no further interaction with the environment. Often the batch $\boldsymbol { B }$ is training data, generated in some episodic fashion. However, in the batch reinforcement learning problem, we do not have knowledge of the algorithm, models, or seeds that were used to generate the episodes in the batch $\boldsymbol { B }$ .
+
+Typically the batch data is generated during training with a non-stationary DRL policy. After training, the original DRL algorithm produces a final-DRL policy, with exploration turned off. In our numerical experiments, we will compare the performance of policies obtained by batch algorithms with the performance of the final-DRL policy. Ideally, we would like the performance of the batchderived policy to be as good or better than the final-DRL policy.
+
+The case where batch data is generated from a non-stationary training policy is of particular interest because it is typically a rich data set from which it may be possible to derive high-performing policies. Furthermore, a batch learning algorithm can potentially be deployed as part of a growingbatch algorithm, where the batch algorithm seeks a high-performing exploitation policy using the current data in an experience replay buffer, combines this policy with exploration to add fresh data to the buffer, and then repeats the whole process (Lange et al., 2012).
+
+# 4 BEST-ACTION IMITATION LEARNING (BAIL)
+
+In this paper we present BAIL, an algorithm that not only performs well on simulated robotic locomotion tasks, but is also conceptually and algorithmically simple. BAIL has two steps. In the first step, it selects from the batch data $\boldsymbol { B }$ the state-action pairs for which the actions are believed to be good actions for their corresponding states. In the second step, we simply train a policy network with imitation learning using the selected actions from the first step.
+
+Many approaches could be employed to select the best actions. In this paper we propose training a single neural network to create an upper envelope of the Monte Carlo returns, and then selecting the state-action pairs in the batch $\boldsymbol { B }$ that have returns near the upper envelope.
+
+# 4.1 UPPER ENVELOPE
+
+We first define a $\lambda$ -smooth upper envelope, and then provide an algorithm for finding it. To the best of our knowledge, the notion of the upper envelope of a data set is novel.
+
+Recall that we have a batch of data $\boldsymbol { B } = \{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i } ^ { \prime } ) , i = 1 , . . . , m \}$ . Although we do not assume we know what algorithm was used to generate the batch, we make the natural assumption that the data in the batch was generated in an episodic fashion, and that the data in the batch is ordered accordingly. For each data point $i \in \{ 1 , \ldots , m \}$ , we calculate an approximate Monte Carlo return $G _ { i }$ as the sum of the discounted returns from state $s _ { i }$ to the end of the episode:
+
+$$
+G _ { i } = \sum _ { t = i } ^ { T _ { i } } \gamma ^ { t - i } r _ { t }
+$$
+
+where $T _ { i }$ denotes the time at which the episode ends for the data point $s _ { i }$ . The Mujoco environments are naturally infinite-horizon non-episodic continuing-task environments (Sutton & Barto, 2018). During training, however, researchers typically create artificial episodes of length 1000 time steps; after 1000 time steps, a random initial state is chosen and a new episode begins. Because the Mujoco environments are continuing tasks, it is desirable to approximate the return over the infinite horizon, particularly for $i$ values that are close to the (artificial) end of an episode. To do this, we note that the data-generation policy from one episode to the next typically changes slowly. We therefore apply a simple augmentation heuristic of concatenating the subsequent episode to the current episode, and running the sum in (1) to infinity. (In practice, we end the sum when the discounting reduces the contribution of the rewards to a negligible amount.) Our ablation study in the Appendix shows that this simple heuristic can significantly improve performance. Note this approach also obviates the need for knowing when new episodes begin in the data set $\boldsymbol { B }$ .
+
+Having defined the return for each data point in the batch, we now seek an upper-envelope $V ( s )$ for the data $\mathcal { G } : = \{ ( s _ { i } , G _ { i } ) , i = 1 , . . . , m \}$ . Let $V _ { \phi } ( s )$ be a neural network with parameters $\phi$ that takes as input a state $s$ and outputs a real number. We say that $V _ { \phi ^ { * } } ( s )$ is a $\lambda$ -smooth upper envelope for $\mathcal { G }$ if it has the following properties:
+
+1. $V _ { \phi ^ { * } } ( s _ { i } ) \geq G _ { i }$ for all $i = 1 , \ldots , m$
+
+2. Among all the parameterized functions $V _ { \phi } ( s )$ satisfying condition 1, it minimizes:
+
+$$
+L ( \phi ) = \sum _ { i = 1 } ^ { m } [ V _ { \phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } + \lambda | | \phi | | ^ { 2 }
+$$
+
+where $\lambda$ is a non-negative constant. An upper-envelope is thus a smooth function that lies above all the data points, but is nevertheless close to the data points.
+
+Theorem 4.1. Suppose that $V _ { \phi ^ { * } } ( s )$ is a $\lambda$ -smooth upper envelope for $\mathcal { G }$ . Then,
+
+$$
+V _ { \phi ^ { * } } ( s ) = \operatorname* { m a x } \{ G _ { i } : i = 1 , 2 , \ldots , m \} a s \lambda  \infty .
+$$
+
+(2) If there is sufficient capacity in the network and $\lambda = 0$ , then the $V _ { \phi ^ { * } }$ interpolates the data in $\mathcal { G }$ . For example, if $\lambda = 0$ and $V _ { \phi } ( s )$ is a neural network with ReLU activation functions with at least $2 m + d$ weights and two layers, where $d$ is the dimension of the state space $s$ , then $V _ { \phi ^ { * } } ( s _ { i } ) = G _ { i }$ for all $i = 1 , 2 , \dots , m$ .
+
+From the above theorem, we see that when $\lambda$ is very small, the upper envelope aims to interpolate the data, and when $\lambda$ is large, the upper envelope approaches a constant going through the highest data point. Just as in classical regression, there is a sweet-spot for $\lambda$ , the one that provides the best generalization.
+
+We note that there are other natural approaches for defining an upper-envelope, some based on alternative loss functions, others based on data clustering without making use of function approximators. Also, it may be possible to combine episodes to generate improved upper envelopes. These are all questions for future research.
+
+To obtain an approximate upper envelope of the data $\mathcal { G }$ , we employ classic regression with a modified loss function, namely,
+
+$$
+L ( \phi ) = \sum _ { i = 1 } ^ { m } ( V _ { \phi } ( s _ { i } ) - G _ { i } ) ^ { 2 } \{ \mathbb { 1 } _ { ( V _ { \phi } ( s _ { i } ) > G _ { i } ) } + K \mathbb { 1 } _ { ( V _ { \phi } ( s _ { i } ) < G _ { i } ) } \} + \lambda \| \phi \| ^ { 2 }
+$$
+
+where $K > > 1$ and $\mathbb { 1 } _ { ( \cdot ) }$ is the indicator function.
+
+For a finite $K$ value, the above loss function will only produce an approximate upper envelope, since it is possible $V ( s _ { i } )$ may be slightly less than $G _ { i }$ for a few data points. In practice, we find $K = 1 0 , 0 0 0$ works well for all environments tested. When $K  \infty$ , the approximation becomes exact, as stated in the following:
+
+Theorem 4.2. Let $\phi ^ { * }$ be an optimal solution that minimizes $L ( \phi )$ . Then, when $K  \infty$ , $V _ { \phi ^ { * } } ( s )$ will be an exact $\lambda$ -smooth upper envelope.
+
+Also, instead of L2 regularization, in practice we employ the simpler early-stopping regularization, thereby obviating a search for the parameter $\lambda$ . We also clip the upper envelope at values near $\operatorname* { m a x } _ { i } G _ { i }$ , as described in the appendix, which can potentially provide further gains in performance.
+
+# 4.2 SELECTING THE BEST ACTIONS
+
+The BAIL algorithm employs the upper envelope to select the best $( s , a )$ pairs from the batch data $\boldsymbol { B }$ . It then uses ordinary imitation learning (behavioral cloning) to train a policy network using the selected actions. Let $V ( s )$ denote the upper envelope obtained from minimizing the loss function (3) for a fixed value of $K$ .
+
+We consider two approaches for selecting the best actions. In the first approach, which we call BAIL-border, we choose all $( s _ { i } , a _ { i } )$ pairs from the batch data set $\boldsymbol { B }$ such that
+
+$$
+G _ { i } > x V ( s _ { i } )
+$$
+
+We set $x$ such that $p \%$ of the data points are selected, where $p$ is a hyper-parameter. In this paper we use $p = 2 5$ for all environments and batches. Thus BAIL-border chooses state-action pairs whose returns are near the upper envelope. The pseudo-code for the Bail-border algorithm is given in the appendix.
+
+In the second approach, which we refer to as BAIL-TD, we select a pair $( s _ { i } , a _ { i } )$ if
+
+$$
+r _ { i } + \gamma V ( s _ { i } ^ { \prime } ) > x V ( s _ { i } )
+$$
+
+where $x$ is a hyper-parameter close to 1. Thus BAIL-TD chooses state-action pairs for which backedup estimated return $r _ { i } + \gamma V ( s _ { i } ^ { \prime } )$ is close to the upper envelope value $V ( s _ { i } )$ .
+
+In summary, BAIL employs two neural networks. The first neural network is used to approximate a value function based on the data in the batch $\boldsymbol { B }$ . The second neural network is the policy network, which is trained with imitation learning. This simple approach does not suffer from extrapolation error since it does not perform any optimization over the action space. An algorithmic description of BAIL is given in Algorithm 1.
+
+# Algorithm 1 BAIL
+
+1: Initialize upper envelope parameter2: Compute return for each data point $\phi$ $\theta$ , obtain batch data $\boldsymbol { B }$ .   
+$i$ $\begin{array} { r } { \dot { G _ { i } } = \dot { \sum _ { t = i } ^ { T _ { i } } } \gamma ^ { t - i } r _ { t } } \end{array}$   
+3: Obtain upper envelope by minimizing the loss:   
+4: for $j = 1 , \dots , J$ do   
+5: sample a minibatch of data B from the batch $\boldsymbol { B }$   
+6: minimize over $\phi$ : $\begin{array} { r } { L ( \phi ) = \sum _ { i = 1 } ^ { | B | } ( V _ { \phi } ( s _ { i } ) - G _ { i } ) ^ { 2 } \{ \mathbb { 1 } _ { ( V _ { \phi } ( s _ { i } ) > G _ { i } ) } + K \mathbb { 1 } _ { ( V _ { \phi } ( s _ { i } ) < G _ { i } ) } \} + \lambda \| \phi \| ^ { 2 } } \end{array}$   
+7: Select data $i$ for $G _ { i } > x V _ { \phi } ( s _ { i } )$ , select $x$ so that $p \%$ data in $\boldsymbol { B }$ are selected, let $\mathcal { U }$ be set of selected   
+data   
+8: for $l = 1 , \ldots , L$ do   
+9: sample a minibatch $U$ of data from $\mathcal { U }$   
+10: minimize over $\theta$ : $\begin{array} { r } { L ( \theta ) = \sum _ { i = 1 } ^ { | U | } ( \pi _ { \theta } ( s _ { i } ) - a _ { i } ) ^ { 2 } } \end{array}$
+
+# 5 EXPERIMENTAL RESULTS
+
+We carried out experiments with four of the most challenging environments in the Mujoco benchmark (Todorov et al., 2012) of OpenAI Gym. For the environments Hopper-v2, Walker-v2 and HalfCheetah-v2, we used the “Final Buffer” batch exactly as described in Fujimoto et al. (2018a) to allow for a fair comparison with BCQ. Specifically, we trained DDPG for one million time steps with $\sigma = 0 . 5$ to generate a batch. For the environment Ant-v2, we trained adaptive SAC (Haarnoja et al., 2018b) again for one million time steps to generate a batch.
+
+In our experiments, we found that different batches generated with different seeds but with the same algorithm in the same environment can lead to surprisingly different results for batch DRL algorithms. To address this, for each of the environments we generated four batches, giving a total of 16 data sets.
+
+Figure 1 provides visualizations of 4 of the 16 upper envelopes, one for each of the 4 environments. In each visualization, the data points in the corresponding batch are ordered according to their upperenvelope values $V ( s _ { i } )$ . With this new ordering, the figure plots $( s _ { i } , G _ { i } )$ for each of the one million data points. The monotonically increasing blue line is the the upper envelope obtained by minimizing $V ( s )$ . Note that in all of the figures, a small fraction of the data points are above upper envelopes due to the finite value of $K = 1 0$ , 000. But also note that the upper envelope mostly hugs the data. The constant black line is the clipping value. The final upper envelope is the minimum of the blue and black lines. All 16 upper envelopes are shown in the appendix.
+
+![](images/52db04581c9ae2b7247c5495cd26f266d38553f38741170149bd321c8095952a.jpg)  
+Figure 1: Five illustrative upper envelopes trained from data with adaptive clipping
+
+Figure 2 compares the performance of BAIL, BCQ, Behavioral Cloning (BC), and the final DDPG/SAC policy for the four environments. When training with BCQ, we used the code provided by the authors (Fujimoto et al., 2018a). Because at the time of writing the code for BEAR was not available, we do not compare our results with BEAR (Kumar et al., 2019). Also, all the results presented in this section are for BAIL-border, which we simply refer to as BAIL. In the appendix we provide results for BAIL-TD. The x-axis is the number of parameter updates and the y-axis is the test return averaged over 10 episodes. BAIL, BCQ, and BC are each trained with five seeds. The figure shows the mean and standard deviation confidence intervals for these values. The figure also shows test result of the final-DDPG/SAC policy. This value is obtained by averaging test results from the last 100,000 timesteps (of one million time steps). During this period, test performance of SAC and particularly DDPG can greatly fluctuate with relatively small improvement on average. We calculate the mean and standard deviation of the test results over this period, plot the mean as a straight line, and use the transparent green background to show the confidence intervals. This enables us to fairly compare the performance of the final test policy obtained with the behavioral algorithm with the test policies from the batch algorithms.
+
+We make the following observations. For Hopper, Walker and Ant, BAIL always beats BCQ usually by a wide margin. For HalfCheetah, BAIL does better than BCQ for half of the batches. In almost all of the curves, BAIL has a much lower confidence interval than BCQ. Perhaps more importantly, BAIL’s performance is stable over training, whereas BCQ can vary dramatically. (This is a serious issue for batch reinforcement learning, since it cannot interact with the environment to find the best stopping point.) Importantly, BAIL also performs as well or better than the Final-DDPG/SAC policy in all but of the 16 batches. This gives promise that BAIL, or a future variation of BAIL, could also be employed within a growing-batch algorithm.
+
+Table 1: Performance comparison at one million samples (mean and std over batches and random seeds). Last column shows percentage improvement of BAIL over BCQ.   
+
+<table><tr><td>Environment</td><td>Final-DDPG/S AC</td><td>BCQ</td><td>BAIL</td><td>Improvement</td></tr><tr><td>Hopper-v2</td><td>2547.7 ± 750.4</td><td>1468.9 ± 552.6</td><td>2437.5 ± 489.7</td><td>65.9%</td></tr><tr><td>Walker2d-v2</td><td>1742.1 ± 656.3</td><td>2020.2 ± 699.3</td><td>2496.7 ± 409.9</td><td>23.6%</td></tr><tr><td>HalfCheetah-v2</td><td>2612.4±342.2</td><td>2449.7 ± 267.7</td><td>2660.0± 77.7</td><td>8.6%</td></tr><tr><td>Ant-v2</td><td>4506.0 ± 483.6</td><td>4315.6 ± 416.4</td><td>4630.7 ± 310.9</td><td>7.3%</td></tr></table>
+
+We also summarize the results in Table 1. For this table, we average the performance of each algorithm over four batches, using the performance values at one million updates. Table 1 shows that BAIL’s performance is better than that of BCQ for all four environments, with a $66 \%$ and $23 \%$ average improvement for Hopper and Walker, respectively. BAIL also beats the Final-DDPG/SAC policies in three of the environments, and has significantly lower variance.
+
+![](images/2d26fdca3bbcbda2acdc35295c042f6351ae22c378de4a9badaafcb71ba2fbef.jpg)  
+Figure 2: Performance comparison of BAIL, BCQ, BC and Final-DDPG/SAC
+
+In the appendix we also provide experimental results for DDPG batches generated with $\sigma = 0 . 1$ , which is similar to the “Concurrent” dataset in Fujimoto et al. (2018a). For this low noise level 0.1, BAIL continues to beat BCQ by a wide margin for Hopper and Walker, and continues to beat Final-DDPG for half of the batches. However, in the low noise case for HalfCheetah, BCQ beats BAIL for 3 of the 4 batches.
+
+# 5.1 ABLATION STUDY
+
+BAIL uses an upper envelope to select the “best” data points for training a policy network with imitation learning. We have shown that BAIL typcially beats ordinary behavioral cloning and BCQ by a wide margin, and often performs better than the Final-DDPG and Final-SAC policies. But it is natural to ask how BAIL performs when using more naive approaches for selecting the best actions. We consider two naive approaches. The first approach, “Highest Returns,” is to select from the batch the $2 5 \%$ of data points that have the highest $G _ { i }$ values. The second approach, “Recent Data,” is to select the last $2 5 \%$ data points from the batch. Figure (3) shows the results for all four environments. We see that for each environment, the upper envelope approach is the winner for most of the batches: for Hopper, the upper envelope wins for all four batches by a wide margin; for Walker the upper-envelope approach wins by a wide margin for two batches, and ties Highest Returns for two batches; for HalfCheetah, the upper-envelope approach wins for three batches and ties Highest Returns for one batch; and for Ant, the upper-envelope approach wins for three batches and ties the other two approaches for the other batch. We conclude, for the environments and batch generation mechanisms considered in this paper, the upper envelope approach performs significantly better and is more robust than both naive approaches.
+
+![](images/f74fce8b0118f892e70bd56b5fe44ed31cca7c084c470c95ec64f97bdc660ed9.jpg)  
+Figure 3: Comparison of BAIL scheme with Highest Returns and with Recent Samples schemes. All schemes use $2 5 \%$ of the data in the batch.
+
+In the Appendix we provide additional ablation studies. Our experimental results show that modifying the returns to approximate infinite horizon returns is often useful for BAIL’s performance, and that clipping the upper envelope also provides gains although much more modest.
+
+In summary, our experimental results show that BAIL achieves state-of-the-art performance, and often beats BCQ by a wide margin. Moreover, BAIL’s performance is stable over training, whereas BCQ typically varies dramatically over training. Finally, BAIL achieves this superior performance with an algorithm that is much simpler than BCQ.
+
+# 6 CONCLUSION
+
+Although BAIL as described in this paper is simple and gives state-of-the-art performance, there are several directions that could be explored in the future for extending BAIL. One avenue is generating multiple upper envelopes from the same batch, and then ensembling or using a heuristic to pick the upper envelope which we believe would give the best performance. A second avenue is to optimize the policy by modifying the best actions. A third avenue is to assign weights to the state-action pairs when training with imitation learning. And a fourth avenue is to explore designing a growing batch algorithm which uses BAIL as a subroutine for finding a high-performing exploitation policy for each batch iteration.
+
+# REFERENCES
+
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+Nan Jiang and Lihong Li. Doubly robust off-policy value evaluation for reinforcement learning. arXiv preprint arXiv:1511.03722, 2015.
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+Beomjoon Kim, Amir-massoud Farahmand, Joelle Pineau, and Doina Precup. Learning from limited demonstrations. In Advances in Neural Information Processing Systems, pp. 2859–2867, 2013.
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+Aviral Kumar, Justin Fu, George Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. arXiv preprint arXiv:1906.00949, 2019.
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+Bilal Piot, Matthieu Geist, and Olivier Pietquin. Boosted bellman residual minimization handling expert demonstrations. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 549–564. Springer, 2014.
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+Stefan Schaal. Is imitation learning the route to humanoid robots? Trends in cognitive sciences, 3 (6):233–242, 1999.
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+Alex Strehl, John Langford, Lihong Li, and Sham M Kakade. Learning from logged implicit exploration data. In Advances in Neural Information Processing Systems, pp. 2217–2225, 2010.
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+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ICLR, 2017.
+
+# A PROOFS
+
+# A. Proof of Theorem 4.1
+
+Proof. First, let us consider the case when $\lambda \to + \infty$ . We can re-write the definition of the upper envelope as a constrained optimization problem:
+
+$$
+\operatorname* { m i n } _ { \phi } \sum _ { i = 1 } ^ { m } [ V _ { \phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } + \lambda \| \phi \| ^ { 2 }
+$$
+
+$$
+s . t . \qquad G _ { i } - V _ { \phi } ( s _ { i } ) \leq 0 , \qquad i = 1 , 2 \dots , m
+$$
+
+where $V _ { \phi ^ { * } }$ is the optimal solution to the above optimization problem. We write the Lagrangian function:
+
+$$
+L ( \phi , \mu ) = \sum _ { i = 1 } ^ { m } [ V _ { \phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } + \lambda \| \phi \| ^ { 2 } + \sum _ { i = 1 } ^ { m } \mu _ { i } [ G _ { i } - V _ { \phi } ( s _ { i } ) ]
+$$
+
+As the optimal solution, $V _ { \phi ^ { * } }$ must satisfy the KKT conditions specified below:
+
+$$
+\left\{ \begin{array} { c } { \frac { \mathrm { d } L ( \phi , \mu ) } { \mathrm { d } \phi } | _ { \phi = \phi ^ { * } } = 0 } \\ { \mu _ { i } [ G _ { i } - V _ { \phi ^ { * } } ( s _ { i } ) ] = 0 , i = 1 , 2 , . . . , m } \\ { \mu _ { i } \geq 0 , i = 1 , 2 , . . . , m } \end{array} \right.
+$$
+
+Suppose $\phi = ( \phi _ { 1 } , \ldots , \phi _ { n } )$ , by the first KKT condition, we have
+
+$$
+\frac { \partial } { \partial \phi _ { j } } \sum _ { i = 1 } ^ { m } [ V _ { \phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } | _ { \phi _ { j } = \phi _ { j } ^ { * } } + 2 \lambda \phi _ { j } ^ { * } + \frac { \partial } { \partial \phi _ { j } } \sum _ { i = 1 } ^ { m } \mu _ { i } [ G _ { i } - V _ { \phi } ( s _ { i } ) ] | _ { \phi _ { j } = \phi _ { j } ^ { * } } = 0 , \qquad j = 1 , 2 , \ldots , n .
+$$
+
+So we have:
+
+$$
+\nu _ { j } ^ { * } = - \frac { 1 } { 2 \lambda } \frac { \partial } { \partial \phi _ { j } } \sum _ { i = 1 } ^ { m } [ V _ { \phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } | _ { \phi _ { j } = \phi _ { j } ^ { * } } - \frac { 1 } { 2 \lambda } \frac { \partial } { \partial \phi _ { j } } \sum _ { i = 1 } ^ { m } \mu _ { i } [ G _ { i } - V _ { \phi } ( s _ { i } ) ] | _ { \phi _ { j } = \phi _ { j } ^ { * } } , \qquad j = 1 , 2 , \ldots , n .
+$$
+
+When $\lambda  \infty$ , we have $\phi ^ { * } = 0$ . In this case, it follows that $V _ { \phi ^ { * } } = C$ for some constant $C$ . As $V _ { \phi ^ { * } } ( s _ { i } ) \geq G _ { i }$ , in order to minimize (3) we must have $C = \operatorname* { m a x } \{ G _ { i } , \ i = 1 , 2 , \ldots , m \}$ .
+
+For the case of $\lambda = 0$ , notice that we only have finitely many input $s _ { i }$ to be the input of the neural network. Therefore, this is a typical problem regarding the finite-sample expressivity of the neural networks, and the proof directly follows from the work done by Zhang et al. (Zhang et al., 2017).
+
+# B. Proof of Theorem 4.2
+
+Proof. Let $\phi ^ { * }$ be the optimal value that minimizes $L ( \phi )$ . Let’s proceed by contradiction and assume that there exists some $k$ such that
+
+$$
+V _ { \phi ^ { * } } ( s _ { k } ) < G _ { k }
+$$
+
+Let $\phi ^ { \prime }$ be an arbitrary given value such that $V _ { \phi ^ { \prime } } ( s _ { i } ) \geq G _ { i }$ for all $i \in \{ 1 , 2 , \dots , m \}$ . Then we have
+
+$$
+\begin{array} { c } { { \displaystyle \sum _ { i = 1 } ^ { m } ( V _ { \phi ^ { * } } ( s _ { i } ) - G _ { i } ) ^ { 2 } \mathbb { 1 } _ { ( V _ { \phi ^ { * } } ( s _ { i } ) > G _ { i } ) } + K ( V _ { \phi ^ { * } } ( s _ { k } ) - G _ { k } ) ^ { 2 } + \lambda \| \phi ^ { * } \| ^ { 2 } \leq L ( \phi ^ { * } ) } } \\ { { \displaystyle L ( \phi ^ { * } ) \leq L ( \phi ^ { \prime } ) = \sum _ { i = 1 } ^ { m } ( V _ { \phi ^ { \prime } } ( s _ { i } ) - G _ { i } ) ^ { 2 } + \| \phi ^ { \prime } \| ^ { 2 } } } \end{array}
+$$
+
+This implies that
+
+$\displaystyle \sum _ { = 1 } ^ { m } ( V _ { \phi ^ { * } } ( s _ { i } ) - G _ { i } ) ^ { 2 } \mathbb { 1 } _ { ( V _ { \phi ^ { * } } ( s _ { i } ) > G _ { i } ) } + K ( V _ { \phi ^ { * } } ( s _ { k } ) - G _ { k } ) ^ { 2 } + \lambda \| \phi ^ { * } \| ^ { 2 } \leq \displaystyle \sum _ { i = 1 } ^ { m } ( V _ { \phi ^ { \prime } } ( s _ { i } ) - G _ { i } ) ^ { 2 } + \| \phi ^ { \prime } \| ^ { 2 }$ i   
+which is impossible when $K  \infty$ . Therefore, we must have $V _ { \phi ^ { * } } ( s _ { i } ) \geq G _ { i }$ for all $i \in \{ 1 , 2 , \dots , m \}$ as $K  \infty$ . In this way, when $K  \infty$ , $\phi ^ { * }$ actually minimizes
+
+$$
+L ( \phi ) = \sum _ { i = 1 } ^ { m } ( V _ { \phi } ( s _ { i } ) - G _ { i } ) ^ { 2 } + \| \phi \| ^ { 2 }
+$$
+
+which completes the proof.
+
+# B IMPLEMENTATION DETAILS AND HYPERPARAMETERS
+
+B.1 IMPLEMENTATION OF BAIL ALGORITHM
+
+Table 2: Upper Envelope Hyperparameters   
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td></td><td>Adam (Kingma &amp; Ba,2014)</td></tr><tr><td>optimizer learning rate</td><td>3.10-3</td></tr><tr><td>discount (γ)</td><td>0.99</td></tr><tr><td>regularization constant 入</td><td>2.10-2</td></tr><tr><td>K</td><td>10,000</td></tr><tr><td>number of hidden units</td><td>128 ×128</td></tr></table>
+
+Table 3: BAIL Hyperparameters   
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>data in batch</td><td>106</td></tr><tr><td>optimizer</td><td>Adam (Kingma &amp; Ba,2014)</td></tr><tr><td>learning rate</td><td>10-3</td></tr><tr><td>regularization constant 入</td><td>0</td></tr><tr><td>mini-batch size</td><td>100</td></tr><tr><td>BAIL-border p%</td><td>25%</td></tr><tr><td>BAIL-TD x</td><td>0.96</td></tr><tr><td>number of hidden units</td><td>400 × 300</td></tr></table>
+
+# B.2 IMPLEMENTATION OF COMPETING ALGORITHMS
+
+For the behavioral DDPG algorithm, we used the implementation of Fujimoto et al. (2018b). For the behavioral SAC algorithm, we implemented it in Pytorch, mainly following the pseudocode provided by (Achiam), and used hyperparameters in Haarnoja et al. (2018b). For the BCQ algorithm, we used the authors’ implementation (Fujimoto et al., 2018a). For behavioral cloning and its variants in the ablation study section, the network structure, learning rate, mini-batch size, and so on are identical to those in Table 3 for BAIL.
+
+For the upper envelope network, our network has two hidden layers as does the Q network in BCQ and SAC. However, the number of hidden units in our network is less than those used in BCQ $( 4 0 0 \times 3 0 0 )$ and SAC $( 2 5 6 \times 2 5 6 )$ . In future work we will see if we can obtain further improvements with BAIL using larger networks for the upper envelope.
+
+# B.3 CLIPPING HEURISTICS
+
+As is shown in Figure 1, in practice the trained upper envelope does not always fit well the data points on the right side of the plots, where the upper envelope can become very large. In that region, the data points with the highest returns will not be selected as “best actions” and therefore not used for imitation learning step. We observe that if we plot the upper envelope values for all the states in the buffer in ascending order as is shown in Figure 1, the upper envelope value $V ( s _ { i } )$ starts to deviate from the Monte Carlo return $G _ { i }$ at a point where $V ( s _ { i } ) \approx \operatorname* { m a x } \{ G _ { i } \}$ . We therefore use the following heuristic. We say that the upper envelope value begins to deviate from the Monte Carlo return at state $s _ { i }$ if $V ( s _ { j } ) > G _ { j }$ for $i \leq j < i + 1 0 0 0 0$ . We set the clipping value $C = V ( s ^ { \prime } )$ where $s ^ { \prime }$ is the starting point of this deviation. Then the actual UE values used to select data is $\operatorname* { m i n } \{ V ( s ) , C \}$ . In practice, the clipping heuristic gives a small boost in performance as is shown in Figure 5.
+
+# C ADDITIONAL EXPERIMENTAL RESULTS
+
+# C.1 EXPERIMENTS ON BAIL-TD
+
+We present some of the experimental results for BAIL-TD in this subsection. Figure 4 shows that the performance of BAIL-TD is similar to the performance of naive Behavioral Cloning in Hopperv2 and Walker2d-v2 environments. The result implies that the BAIL-TD approach is limited in the ability to distinguish between good data points and the bad ones. This is possibly due to the inaccuracy of the trained upper envelope. Recall that in BAIL-TD algorithm, we select state action pairs based on the difference between the values of $r + V ( s _ { i } ^ { \prime } )$ and $V ( s _ { i } )$ . Since we only use one-step Monte Carlo return, and the value of $r$ is very small compared with the value of $v ( s _ { i } )$ , the selection is very sensitive to the accuracy of $V ( s _ { i } )$ .
+
+![](images/d26f46af57a5316ce66f9bc954b2be321ff21d0804d5148b3d9eceb1ffebf218.jpg)  
+Figure 4: Performance comparison of BAIL-border, BAIL-TD, BCQ, BC and Behavioral Policy (DDPG)
+
+# C.2 ABLATION STUDY FOR BAIL
+
+We do additional ablation studies for BAIL, where we focus on two heuristic features in the BAIL algorithm: clipping of upper envelope, and return augmentation to approximate non-episodic continuous tasks. We removed each of these features one at a time from the BAIL algorithm and compare with the original BAIL algorithm. Each performance curve is again averaged over 5 independent runs.
+
+![](images/cb39a0d4716f35152a6abda2ff1f3bc29daaaaca60eb219cc9a25d4597cb4302.jpg)  
+Figure 5: Ablation of BAIL without Upper Envelope clipping
+
+We found that clipping often, but not always, gives a small boost in performance. As for return augmentation, we found it does not harm the performance of BAIL, and sometimes gives a considerable improvement, particularly for Hopper.
+
+![](images/dede49c1dbcbdf6760a93a48b80b7c96bca7d1ebfe21303dfd2e9a23bdfa2782.jpg)  
+Figure 6: Ablation of BAIL without return augmentation
+
+# C.3 BAIL FOR LOW NOISE-LEVEL DATA
+
+In the main body of the paper we present BAIL in the ”Final Buffer” case as described in Fujimoto et al. (2018a), where the exploration noise $\sigma = 0 . 5$ added to behavior policy is relatively large. In this section we examine the performance of BAIL in a low-noise scenario. To this end, we set $\sigma = 0 . 1$ and do a similar experiments as were done $\sigma = 0 . 5$ for the Hopper, Walker and HalfCheetah environments. (Recall that for Ant we used adaptive SAC, which does not have an explicit noise parameter.)
+
+The results are shown in Figure 7. We see that even in this low-noise scenario, BAIL out-performs BCQ by a wide margin for Hopper and Walker, and BAIL continues to out-perform the Final-DDPG policy in most batches. For HalfCheetah, where the Final-DDPG policy gives greatly different results depending on the batch, BAIL’s performance is stable and typically near that of the of FinalDDPG policy. After sufficient training, however, BCQ can often do better than the Final-DDPG policy in this environment.
+
+![](images/090dd7cb80be99621216571aa31d07269a80f42830c6bf6c6a8cb4ec6ec66c15.jpg)  
+Figure 7: Performance comparison of BAIL, BCQ, BC and Final-DDPG with noise level $\sigma = 0 . 1$
+
+# C.4 VISUALIZATION OF UPPER ENVELOPES USED
+
+In this section we present the upper envelopes used in the training of BAIL in Figure 2.
+
+![](images/856ef1d7387b230a0b8ea9c407649fbc0f4e7f792685b42d4895796482918c0c.jpg)  
+Figure 8: Upper Envelopes of BAIL (DDPG data)
+
+![](images/e8a35e6bdfc4e4ee27333559f7b327108a4ae2943a3ec0962daaec77a013a6c7.jpg)  
+Figure 9: Upper Envelopes of BAIL (SAC data)

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+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The field of Deep Reinforcement Learning (DRL) has recently seen a surge in research in batch reinforcement learning, which aims for sample-efficient learning from a given data set without additional interactions with the environment. In the batch DRL setting, commonly employed off-policy DRL algorithms can perform poorly and sometimes even fail to learn altogether. In this paper we propose a new algorithm, Best-Action Imitation Learning (BAIL), which unlike many offpolicy DRL algorithms does not involve maximizing Q functions over the action space. Striving for simplicity as well as performance, BAIL first selects from the batch the actions it believes to be high-performing actions for their corresponding states; it then uses those state-action pairs to train a policy network using imitation learning. Although BAIL is simple, we demonstrate that BAIL achieves state of the art performance on the Mujoco benchmark. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The field of Deep Reinforcement Learning (DRL) has recently seen a surge in research in batch reinforcement learning, which is the problem of sample-efficient learning from a given data set without additional interactions with the environment. Batch reinforcement learning is appealing because it dis-entangles policy optimization (exploitation) from data collection (exploration). This enables reusing the data collected by a policy to possibly improve the policy without further interactions with the environment. Furthermore, a batch learning reinforcement learning algorithm can potentially be deployed as part of a growing-batch algorithm, where the batch algorithm seeks a high-performing exploitation policy using the data in an experience replay buffer, combines this policy with exploration to add fresh data to the buffer, and then repeats the whole process (Lange et al., 2012). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Fujimoto et al. (2018a) recently made the critical observation that commonly employed off-policy algorithms based on Deep Q-Learning (DQL) often perform poorly and sometimes even fail to learn altogether. Indeed, off-policy DRL algorithms typically involve maximizing an approximate Qfunction over the action space (Lillicrap et al., 2015; Fujimoto et al., 2018b; Haarnoja et al., 2018a), leading to an extrapolation error, particularly for state-action pairs that are not in the batch distribution. Batch-Constrained deep Q-learning (BCQ), which obtains good performance for many of the Mujoco environments (Todorov et al., 2012), avoids the extrapolation error problem by constraining the set of actions over which the approximate Q-function is optimized (Fujimoto et al., 2018a). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a new algorithm, Best-Action Imitation Learning (BAIL), which strives for both simplicity and performance. BAIL does not suffer from the extrapolation error problem since it does not maximize over the action space in any step of the algorithm. BAIL is simple, thereby satisfying the principle of Occam’s razor. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The BAIL algorithm has two steps. In the first step, it selects from the batch a subset of state-action pairs for which the actions are believed to be good actions for their corresponding states. In the second step, it simply trains a policy network with imitation learning using the selected actions from the first step. To find the best actions, we train a neural network to obtain the “upper envelope” of the Monte Carlo returns in the batch data, and then we select from the batch the state-action pairs that are near the upper envelope. We believe the concept of the upper-envelope of a data set is also novel and interesting in its own right. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Because the BCQ code is publicly available, we are able to make a careful comparison of the performance of BAIL and BCQ. We do this for batches generated by training DDPG (Lillicrap et al., 2015) for the Half-Cheetah, Walker, and Hopper environments, and for batches generated by training Soft Actor Critic (SAC) for the Ant environment (Haarnoja et al., 2018a;b). Although BAIL is simple, we demonstrate that BAIL achieves state of the art performance on the Mujoco benchmark, often outperforming Batch Constrained deep Q-Learning (BCQ) by a wide-margin. We also provide anonymized code for reproducibility1. ",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Batch reinforcement learning in both the tabular and functional approximator settings has long been studied (Lange et al., 2012; Strehl et al., 2010) and continues to be a highly active area of research (Swaminathan & Joachims, 2015; Jiang & Li, 2015; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Irpan et al., 2019; Jaques et al., 2019). Imitation learning is also a well-studied problem (Schaal, 1999; Argall et al., 2009; Hussein et al., 2017) and also continues to be a highly active area of research (Kim et al., 2013; Piot et al., 2014; Chemali & Lazaric, 2015; Hester et al., 2018; Ho et al., 2016; Sun et al., 2017; 2018; Cheng et al., 2018; Gao et al., 2018). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper relates most closely to (Fujimoto et al., 2018a), which made the critical observation that when conventional DQL-based algorithms are employed for batch reinforcement learning, performance can be very poor, with the algorithm possibly not learning at all. Off-policy DRL algorithms involve maximizing an approximate action-value function $Q ( s , a )$ over all actions in the action space. (Or over the actions in the manifold of the parameterized policy.) The approximate action-value function can be very inaccurate, particularly for state-action pairs that are not in the state-action distribution of the batch (Fujimoto et al., 2018a). Due to this extrapolation error, poorperforming actions can be chosen when optimizing $Q ( s , a )$ over all actions. With traditional offpolicy DRL algorithms (such as DDPG (Lillicrap et al., 2015), TD3 (Fujimoto et al., 2018b) and SAC (Haarnoja et al., 2018a)), if the action-value function over-estimates a state-action pair, the policy will subsequently collect new data in the over-estimated region, and the estimate will get corrected. In the batch setting, however, where there is no further interaction with the environment, the extrapolation error is not corrected, and the poor choice of action persists in the policy (Fujimoto et al., 2018a). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Batch-Constrained deep Q-learning (BCQ) avoids the extrapolation error problem by constraining the set of actions over which the approximate Q-function is optimized (Fujimoto et al., 2018a). More specifically, BCQ first trains a state-dependent Variational Auto Encoder (VAE) using the state action pairs in the batch data. When optimizing the approximate Q-function over actions, instead of optimizing over all actions, it optimizes over a subset of actions generated by the VAE. The BCQ algorithm is further complicated by introducing a perturbation model, which employs an additional neural network that outputs an adjustment to an action. BCQ additionally employs a modified version of clipped-Double Q-Learning to obtain satisfactory performance. We show experimentally that our much simpler BAIL algorithm typically performs better than BCQ by a wide margin. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Kumar et al. (2019) recently proposed BEAR for batch DRL. BEAR is also complex, employing Maximum Mean Discrepancy (Gretton et al., 2012), kernel selection, a parametric model that fits a tanh-Gaussian distribution, and a test policy that is different from the learned actor policy. In this paper we do not experimentally compare BAIL with BEAR since the code for BEAR is not publicly available at the time of writing. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Agarwal et al. (2019) recently proposed another algorithm for batch DRL called Random Ensemble Mixture (REM), an ensembling scheme which enforces optimal Bellman consistency on random convex combinations of the Q-heads of a multi-headed Q-network. For the Atari 2600 games, batch REM can out-perform the policies used to collect the data. REM and BAIL are orthogonal, and it may be possible to combine them in the future to achieve even higher performance. No experimental results are provided for REM applied to the Mujoco benchmark (Agarwal et al., 2019). ",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "3 BATCH DEEP REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "We represent the environment with a Markov Decision Process (MDP) defined by a tuple $( S , { \\mathcal { A } } , g , r , \\rho , \\gamma )$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $\\rho$ is the initial state distribution, and $\\gamma$ is the discount factor. The functions $g ( s , a )$ and $r ( s , a )$ represent the dynamics and reward function, respectively. In this paper we assume that the dynamics of the environment are deterministic, that is, there are real-valued functions $g ( s , a )$ and $r ( s , a )$ such that when in state $s$ and action $a$ is chosen, then the next state is $s ^ { \\prime } = g ( s , a )$ and the reward received is $r ( s , a )$ . We note that all the simulated robotic locomotion environments in the Mujoco benchmark are deterministic, and many robotic tasks are expected to be deterministic environments. Furthermore, many of the Atari game environments are deterministic (Bellemare et al., 2013). Thus, from an applications perspective, the class of deterministic environments is a large and important class. Although we assume that the environment is deterministic, as is typically the case with reinforcement learning, we do not assume the functions $g ( s , a )$ and $r ( s , a )$ are known. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In batch reinforcement learning, we are provided a batch of $m$ data points $\\begin{array} { r } { B = \\{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i } ^ { \\prime } ) , i = } \\end{array}$ $1 , . . . , m \\}$ . We assume $\\boldsymbol { B }$ is fixed and given, and there is no further interaction with the environment. Often the batch $\\boldsymbol { B }$ is training data, generated in some episodic fashion. However, in the batch reinforcement learning problem, we do not have knowledge of the algorithm, models, or seeds that were used to generate the episodes in the batch $\\boldsymbol { B }$ . ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Typically the batch data is generated during training with a non-stationary DRL policy. After training, the original DRL algorithm produces a final-DRL policy, with exploration turned off. In our numerical experiments, we will compare the performance of policies obtained by batch algorithms with the performance of the final-DRL policy. Ideally, we would like the performance of the batchderived policy to be as good or better than the final-DRL policy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The case where batch data is generated from a non-stationary training policy is of particular interest because it is typically a rich data set from which it may be possible to derive high-performing policies. Furthermore, a batch learning algorithm can potentially be deployed as part of a growingbatch algorithm, where the batch algorithm seeks a high-performing exploitation policy using the current data in an experience replay buffer, combines this policy with exploration to add fresh data to the buffer, and then repeats the whole process (Lange et al., 2012). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "4 BEST-ACTION IMITATION LEARNING (BAIL) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper we present BAIL, an algorithm that not only performs well on simulated robotic locomotion tasks, but is also conceptually and algorithmically simple. BAIL has two steps. In the first step, it selects from the batch data $\\boldsymbol { B }$ the state-action pairs for which the actions are believed to be good actions for their corresponding states. In the second step, we simply train a policy network with imitation learning using the selected actions from the first step. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Many approaches could be employed to select the best actions. In this paper we propose training a single neural network to create an upper envelope of the Monte Carlo returns, and then selecting the state-action pairs in the batch $\\boldsymbol { B }$ that have returns near the upper envelope. ",
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+    {
+        "type": "text",
+        "text": "4.1 UPPER ENVELOPE ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We first define a $\\lambda$ -smooth upper envelope, and then provide an algorithm for finding it. To the best of our knowledge, the notion of the upper envelope of a data set is novel. ",
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+    {
+        "type": "text",
+        "text": "Recall that we have a batch of data $\\boldsymbol { B } = \\{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i } ^ { \\prime } ) , i = 1 , . . . , m \\}$ . Although we do not assume we know what algorithm was used to generate the batch, we make the natural assumption that the data in the batch was generated in an episodic fashion, and that the data in the batch is ordered accordingly. For each data point $i \\in \\{ 1 , \\ldots , m \\}$ , we calculate an approximate Monte Carlo return $G _ { i }$ as the sum of the discounted returns from state $s _ { i }$ to the end of the episode: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/a8b453ebaff95c37eca3ca93975ea5cd81240cde460beea27928b7ecdb416495.jpg",
+        "text": "$$\nG _ { i } = \\sum _ { t = i } ^ { T _ { i } } \\gamma ^ { t - i } r _ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $T _ { i }$ denotes the time at which the episode ends for the data point $s _ { i }$ . The Mujoco environments are naturally infinite-horizon non-episodic continuing-task environments (Sutton & Barto, 2018). During training, however, researchers typically create artificial episodes of length 1000 time steps; after 1000 time steps, a random initial state is chosen and a new episode begins. Because the Mujoco environments are continuing tasks, it is desirable to approximate the return over the infinite horizon, particularly for $i$ values that are close to the (artificial) end of an episode. To do this, we note that the data-generation policy from one episode to the next typically changes slowly. We therefore apply a simple augmentation heuristic of concatenating the subsequent episode to the current episode, and running the sum in (1) to infinity. (In practice, we end the sum when the discounting reduces the contribution of the rewards to a negligible amount.) Our ablation study in the Appendix shows that this simple heuristic can significantly improve performance. Note this approach also obviates the need for knowing when new episodes begin in the data set $\\boldsymbol { B }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having defined the return for each data point in the batch, we now seek an upper-envelope $V ( s )$ for the data $\\mathcal { G } : = \\{ ( s _ { i } , G _ { i } ) , i = 1 , . . . , m \\}$ . Let $V _ { \\phi } ( s )$ be a neural network with parameters $\\phi$ that takes as input a state $s$ and outputs a real number. We say that $V _ { \\phi ^ { * } } ( s )$ is a $\\lambda$ -smooth upper envelope for $\\mathcal { G }$ if it has the following properties: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. $V _ { \\phi ^ { * } } ( s _ { i } ) \\geq G _ { i }$ for all $i = 1 , \\ldots , m$ ",
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+    {
+        "type": "text",
+        "text": "2. Among all the parameterized functions $V _ { \\phi } ( s )$ satisfying condition 1, it minimizes: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec77d60737b14b3e3c5a2595f8a50696781180d0d30afd947157a3d7b5a0a71d.jpg",
+        "text": "$$\nL ( \\phi ) = \\sum _ { i = 1 } ^ { m } [ V _ { \\phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } + \\lambda | | \\phi | | ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda$ is a non-negative constant. An upper-envelope is thus a smooth function that lies above all the data points, but is nevertheless close to the data points. ",
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+    {
+        "type": "text",
+        "text": "Theorem 4.1. Suppose that $V _ { \\phi ^ { * } } ( s )$ is a $\\lambda$ -smooth upper envelope for $\\mathcal { G }$ . Then, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/70e52692d98c0bf67f81db97b0787cb3ae7c58520ebe0b5adf262fd023d357a5.jpg",
+        "text": "$$\nV _ { \\phi ^ { * } } ( s ) = \\operatorname* { m a x } \\{ G _ { i } : i = 1 , 2 , \\ldots , m \\} a s \\lambda  \\infty .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "(2) If there is sufficient capacity in the network and $\\lambda = 0$ , then the $V _ { \\phi ^ { * } }$ interpolates the data in $\\mathcal { G }$ . For example, if $\\lambda = 0$ and $V _ { \\phi } ( s )$ is a neural network with ReLU activation functions with at least $2 m + d$ weights and two layers, where $d$ is the dimension of the state space $s$ , then $V _ { \\phi ^ { * } } ( s _ { i } ) = G _ { i }$ for all $i = 1 , 2 , \\dots , m$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From the above theorem, we see that when $\\lambda$ is very small, the upper envelope aims to interpolate the data, and when $\\lambda$ is large, the upper envelope approaches a constant going through the highest data point. Just as in classical regression, there is a sweet-spot for $\\lambda$ , the one that provides the best generalization. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We note that there are other natural approaches for defining an upper-envelope, some based on alternative loss functions, others based on data clustering without making use of function approximators. Also, it may be possible to combine episodes to generate improved upper envelopes. These are all questions for future research. ",
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+    {
+        "type": "text",
+        "text": "To obtain an approximate upper envelope of the data $\\mathcal { G }$ , we employ classic regression with a modified loss function, namely, ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bc072bda585802d38df07613fe82a11eabf8aa4af5f8fc20f5f478ca5c01b02.jpg",
+        "text": "$$\nL ( \\phi ) = \\sum _ { i = 1 } ^ { m } ( V _ { \\phi } ( s _ { i } ) - G _ { i } ) ^ { 2 } \\{ \\mathbb { 1 } _ { ( V _ { \\phi } ( s _ { i } ) > G _ { i } ) } + K \\mathbb { 1 } _ { ( V _ { \\phi } ( s _ { i } ) < G _ { i } ) } \\} + \\lambda \\| \\phi \\| ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $K > > 1$ and $\\mathbb { 1 } _ { ( \\cdot ) }$ is the indicator function. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For a finite $K$ value, the above loss function will only produce an approximate upper envelope, since it is possible $V ( s _ { i } )$ may be slightly less than $G _ { i }$ for a few data points. In practice, we find $K = 1 0 , 0 0 0$ works well for all environments tested. When $K  \\infty$ , the approximation becomes exact, as stated in the following: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.2. Let $\\phi ^ { * }$ be an optimal solution that minimizes $L ( \\phi )$ . Then, when $K  \\infty$ , $V _ { \\phi ^ { * } } ( s )$ will be an exact $\\lambda$ -smooth upper envelope. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Also, instead of L2 regularization, in practice we employ the simpler early-stopping regularization, thereby obviating a search for the parameter $\\lambda$ . We also clip the upper envelope at values near $\\operatorname* { m a x } _ { i } G _ { i }$ , as described in the appendix, which can potentially provide further gains in performance. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4.2 SELECTING THE BEST ACTIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The BAIL algorithm employs the upper envelope to select the best $( s , a )$ pairs from the batch data $\\boldsymbol { B }$ . It then uses ordinary imitation learning (behavioral cloning) to train a policy network using the selected actions. Let $V ( s )$ denote the upper envelope obtained from minimizing the loss function (3) for a fixed value of $K$ . ",
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+    {
+        "type": "text",
+        "text": "We consider two approaches for selecting the best actions. In the first approach, which we call BAIL-border, we choose all $( s _ { i } , a _ { i } )$ pairs from the batch data set $\\boldsymbol { B }$ such that ",
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+        "text": "$$\nG _ { i } > x V ( s _ { i } )\n$$",
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+        "text": "We set $x$ such that $p \\%$ of the data points are selected, where $p$ is a hyper-parameter. In this paper we use $p = 2 5$ for all environments and batches. Thus BAIL-border chooses state-action pairs whose returns are near the upper envelope. The pseudo-code for the Bail-border algorithm is given in the appendix. ",
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+        "type": "text",
+        "text": "In the second approach, which we refer to as BAIL-TD, we select a pair $( s _ { i } , a _ { i } )$ if ",
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+        "type": "equation",
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+        "text": "$$\nr _ { i } + \\gamma V ( s _ { i } ^ { \\prime } ) > x V ( s _ { i } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $x$ is a hyper-parameter close to 1. Thus BAIL-TD chooses state-action pairs for which backedup estimated return $r _ { i } + \\gamma V ( s _ { i } ^ { \\prime } )$ is close to the upper envelope value $V ( s _ { i } )$ . ",
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+        "text": "In summary, BAIL employs two neural networks. The first neural network is used to approximate a value function based on the data in the batch $\\boldsymbol { B }$ . The second neural network is the policy network, which is trained with imitation learning. This simple approach does not suffer from extrapolation error since it does not perform any optimization over the action space. An algorithmic description of BAIL is given in Algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 BAIL ",
+        "text_level": 1,
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+        "text": "1: Initialize upper envelope parameter2: Compute return for each data point $\\phi$ $\\theta$ , obtain batch data $\\boldsymbol { B }$ .   \n$i$ $\\begin{array} { r } { \\dot { G _ { i } } = \\dot { \\sum _ { t = i } ^ { T _ { i } } } \\gamma ^ { t - i } r _ { t } } \\end{array}$   \n3: Obtain upper envelope by minimizing the loss:   \n4: for $j = 1 , \\dots , J$ do   \n5: sample a minibatch of data B from the batch $\\boldsymbol { B }$   \n6: minimize over $\\phi$ : $\\begin{array} { r } { L ( \\phi ) = \\sum _ { i = 1 } ^ { | B | } ( V _ { \\phi } ( s _ { i } ) - G _ { i } ) ^ { 2 } \\{ \\mathbb { 1 } _ { ( V _ { \\phi } ( s _ { i } ) > G _ { i } ) } + K \\mathbb { 1 } _ { ( V _ { \\phi } ( s _ { i } ) < G _ { i } ) } \\} + \\lambda \\| \\phi \\| ^ { 2 } } \\end{array}$   \n7: Select data $i$ for $G _ { i } > x V _ { \\phi } ( s _ { i } )$ , select $x$ so that $p \\%$ data in $\\boldsymbol { B }$ are selected, let $\\mathcal { U }$ be set of selected   \ndata   \n8: for $l = 1 , \\ldots , L$ do   \n9: sample a minibatch $U$ of data from $\\mathcal { U }$   \n10: minimize over $\\theta$ : $\\begin{array} { r } { L ( \\theta ) = \\sum _ { i = 1 } ^ { | U | } ( \\pi _ { \\theta } ( s _ { i } ) - a _ { i } ) ^ { 2 } } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+        "text": "We carried out experiments with four of the most challenging environments in the Mujoco benchmark (Todorov et al., 2012) of OpenAI Gym. For the environments Hopper-v2, Walker-v2 and HalfCheetah-v2, we used the “Final Buffer” batch exactly as described in Fujimoto et al. (2018a) to allow for a fair comparison with BCQ. Specifically, we trained DDPG for one million time steps with $\\sigma = 0 . 5$ to generate a batch. For the environment Ant-v2, we trained adaptive SAC (Haarnoja et al., 2018b) again for one million time steps to generate a batch. ",
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+        "text": "In our experiments, we found that different batches generated with different seeds but with the same algorithm in the same environment can lead to surprisingly different results for batch DRL algorithms. To address this, for each of the environments we generated four batches, giving a total of 16 data sets. ",
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+        "text": "Figure 1 provides visualizations of 4 of the 16 upper envelopes, one for each of the 4 environments. In each visualization, the data points in the corresponding batch are ordered according to their upperenvelope values $V ( s _ { i } )$ . With this new ordering, the figure plots $( s _ { i } , G _ { i } )$ for each of the one million data points. The monotonically increasing blue line is the the upper envelope obtained by minimizing $V ( s )$ . Note that in all of the figures, a small fraction of the data points are above upper envelopes due to the finite value of $K = 1 0$ , 000. But also note that the upper envelope mostly hugs the data. The constant black line is the clipping value. The final upper envelope is the minimum of the blue and black lines. All 16 upper envelopes are shown in the appendix. ",
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+        "text": "",
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+    {
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+        "img_path": "images/52db04581c9ae2b7247c5495cd26f266d38553f38741170149bd321c8095952a.jpg",
+        "image_caption": [
+            "Figure 1: Five illustrative upper envelopes trained from data with adaptive clipping "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 2 compares the performance of BAIL, BCQ, Behavioral Cloning (BC), and the final DDPG/SAC policy for the four environments. When training with BCQ, we used the code provided by the authors (Fujimoto et al., 2018a). Because at the time of writing the code for BEAR was not available, we do not compare our results with BEAR (Kumar et al., 2019). Also, all the results presented in this section are for BAIL-border, which we simply refer to as BAIL. In the appendix we provide results for BAIL-TD. The x-axis is the number of parameter updates and the y-axis is the test return averaged over 10 episodes. BAIL, BCQ, and BC are each trained with five seeds. The figure shows the mean and standard deviation confidence intervals for these values. The figure also shows test result of the final-DDPG/SAC policy. This value is obtained by averaging test results from the last 100,000 timesteps (of one million time steps). During this period, test performance of SAC and particularly DDPG can greatly fluctuate with relatively small improvement on average. We calculate the mean and standard deviation of the test results over this period, plot the mean as a straight line, and use the transparent green background to show the confidence intervals. This enables us to fairly compare the performance of the final test policy obtained with the behavioral algorithm with the test policies from the batch algorithms. ",
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+    {
+        "type": "text",
+        "text": "We make the following observations. For Hopper, Walker and Ant, BAIL always beats BCQ usually by a wide margin. For HalfCheetah, BAIL does better than BCQ for half of the batches. In almost all of the curves, BAIL has a much lower confidence interval than BCQ. Perhaps more importantly, BAIL’s performance is stable over training, whereas BCQ can vary dramatically. (This is a serious issue for batch reinforcement learning, since it cannot interact with the environment to find the best stopping point.) Importantly, BAIL also performs as well or better than the Final-DDPG/SAC policy in all but of the 16 batches. This gives promise that BAIL, or a future variation of BAIL, could also be employed within a growing-batch algorithm. ",
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+    {
+        "type": "table",
+        "img_path": "images/d7ac7eb7343205280d8a341bdc19bdcfa32495f8af350e8b7f17fe3234a8e61a.jpg",
+        "table_caption": [
+            "Table 1: Performance comparison at one million samples (mean and std over batches and random seeds). Last column shows percentage improvement of BAIL over BCQ. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Environment</td><td>Final-DDPG/S AC</td><td>BCQ</td><td>BAIL</td><td>Improvement</td></tr><tr><td>Hopper-v2</td><td>2547.7 ± 750.4</td><td>1468.9 ± 552.6</td><td>2437.5 ± 489.7</td><td>65.9%</td></tr><tr><td>Walker2d-v2</td><td>1742.1 ± 656.3</td><td>2020.2 ± 699.3</td><td>2496.7 ± 409.9</td><td>23.6%</td></tr><tr><td>HalfCheetah-v2</td><td>2612.4±342.2</td><td>2449.7 ± 267.7</td><td>2660.0± 77.7</td><td>8.6%</td></tr><tr><td>Ant-v2</td><td>4506.0 ± 483.6</td><td>4315.6 ± 416.4</td><td>4630.7 ± 310.9</td><td>7.3%</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also summarize the results in Table 1. For this table, we average the performance of each algorithm over four batches, using the performance values at one million updates. Table 1 shows that BAIL’s performance is better than that of BCQ for all four environments, with a $66 \\%$ and $23 \\%$ average improvement for Hopper and Walker, respectively. BAIL also beats the Final-DDPG/SAC policies in three of the environments, and has significantly lower variance. ",
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+    {
+        "type": "image",
+        "img_path": "images/2d26fdca3bbcbda2acdc35295c042f6351ae22c378de4a9badaafcb71ba2fbef.jpg",
+        "image_caption": [
+            "Figure 2: Performance comparison of BAIL, BCQ, BC and Final-DDPG/SAC "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the appendix we also provide experimental results for DDPG batches generated with $\\sigma = 0 . 1$ , which is similar to the “Concurrent” dataset in Fujimoto et al. (2018a). For this low noise level 0.1, BAIL continues to beat BCQ by a wide margin for Hopper and Walker, and continues to beat Final-DDPG for half of the batches. However, in the low noise case for HalfCheetah, BCQ beats BAIL for 3 of the 4 batches. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5.1 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "BAIL uses an upper envelope to select the “best” data points for training a policy network with imitation learning. We have shown that BAIL typcially beats ordinary behavioral cloning and BCQ by a wide margin, and often performs better than the Final-DDPG and Final-SAC policies. But it is natural to ask how BAIL performs when using more naive approaches for selecting the best actions. We consider two naive approaches. The first approach, “Highest Returns,” is to select from the batch the $2 5 \\%$ of data points that have the highest $G _ { i }$ values. The second approach, “Recent Data,” is to select the last $2 5 \\%$ data points from the batch. Figure (3) shows the results for all four environments. We see that for each environment, the upper envelope approach is the winner for most of the batches: for Hopper, the upper envelope wins for all four batches by a wide margin; for Walker the upper-envelope approach wins by a wide margin for two batches, and ties Highest Returns for two batches; for HalfCheetah, the upper-envelope approach wins for three batches and ties Highest Returns for one batch; and for Ant, the upper-envelope approach wins for three batches and ties the other two approaches for the other batch. We conclude, for the environments and batch generation mechanisms considered in this paper, the upper envelope approach performs significantly better and is more robust than both naive approaches. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/f74fce8b0118f892e70bd56b5fe44ed31cca7c084c470c95ec64f97bdc660ed9.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of BAIL scheme with Highest Returns and with Recent Samples schemes. All schemes use $2 5 \\%$ of the data in the batch. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the Appendix we provide additional ablation studies. Our experimental results show that modifying the returns to approximate infinite horizon returns is often useful for BAIL’s performance, and that clipping the upper envelope also provides gains although much more modest. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In summary, our experimental results show that BAIL achieves state-of-the-art performance, and often beats BCQ by a wide margin. Moreover, BAIL’s performance is stable over training, whereas BCQ typically varies dramatically over training. Finally, BAIL achieves this superior performance with an algorithm that is much simpler than BCQ. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although BAIL as described in this paper is simple and gives state-of-the-art performance, there are several directions that could be explored in the future for extending BAIL. One avenue is generating multiple upper envelopes from the same batch, and then ensembling or using a heuristic to pick the upper envelope which we believe would give the best performance. A second avenue is to optimize the policy by modifying the best actions. A third avenue is to assign weights to the state-action pairs when training with imitation learning. And a fourth avenue is to explore designing a growing batch algorithm which uses BAIL as a subroutine for finding a high-performing exploitation policy for each batch iteration. ",
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+    {
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+        "text": "Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ICLR, 2017. ",
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+            173,
+            686,
+            825,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            277,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A. Proof of Theorem 4.1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            132,
+            343,
+            147
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. First, let us consider the case when $\\lambda \\to + \\infty$ . We can re-write the definition of the upper envelope as a constrained optimization problem: ",
+        "bbox": [
+            173,
+            161,
+            823,
+            190
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a017735e3a800dce964d7aec86ec69f6c0ee8736d0b5e1161ff9cf4febfa81e.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\phi } \\sum _ { i = 1 } ^ { m } [ V _ { \\phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } + \\lambda \\| \\phi \\| ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            193,
+            611,
+            233
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14e6f30aa98e819a771b0bdb4456545c7599f206ced3375aad31599b865e7b57.jpg",
+        "text": "$$\ns . t . \\qquad G _ { i } - V _ { \\phi } ( s _ { i } ) \\leq 0 , \\qquad i = 1 , 2 \\dots , m\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            236,
+            653,
+            252
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $V _ { \\phi ^ { * } }$ is the optimal solution to the above optimization problem. We write the Lagrangian function: ",
+        "bbox": [
+            173,
+            252,
+            823,
+            279
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a922013a6a90de2295220e260737fbb2ddc61bc2a61a157b41e93b52bf55cfd2.jpg",
+        "text": "$$\nL ( \\phi , \\mu ) = \\sum _ { i = 1 } ^ { m } [ V _ { \\phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } + \\lambda \\| \\phi \\| ^ { 2 } + \\sum _ { i = 1 } ^ { m } \\mu _ { i } [ G _ { i } - V _ { \\phi } ( s _ { i } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            275,
+            705,
+            316
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "As the optimal solution, $V _ { \\phi ^ { * } }$ must satisfy the KKT conditions specified below: ",
+        "bbox": [
+            171,
+            316,
+            689,
+            332
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37b99255c65ba5417ec623f8735cfc4e54adecbc49f2cf75adfc5967763de852.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { c } { \\frac { \\mathrm { d } L ( \\phi , \\mu ) } { \\mathrm { d } \\phi } | _ { \\phi = \\phi ^ { * } } = 0 } \\\\ { \\mu _ { i } [ G _ { i } - V _ { \\phi ^ { * } } ( s _ { i } ) ] = 0 , i = 1 , 2 , . . . , m } \\\\ { \\mu _ { i } \\geq 0 , i = 1 , 2 , . . . , m } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            333,
+            642,
+            385
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Suppose $\\phi = ( \\phi _ { 1 } , \\ldots , \\phi _ { n } )$ , by the first KKT condition, we have ",
+        "bbox": [
+            174,
+            386,
+            596,
+            401
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/debaeb08bba6b0d911ef0af38270326125b1587bcd47520fe05ae50eab9062e2.jpg",
+        "text": "$$\n\\frac { \\partial } { \\partial \\phi _ { j } } \\sum _ { i = 1 } ^ { m } [ V _ { \\phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } | _ { \\phi _ { j } = \\phi _ { j } ^ { * } } + 2 \\lambda \\phi _ { j } ^ { * } + \\frac { \\partial } { \\partial \\phi _ { j } } \\sum _ { i = 1 } ^ { m } \\mu _ { i } [ G _ { i } - V _ { \\phi } ( s _ { i } ) ] | _ { \\phi _ { j } = \\phi _ { j } ^ { * } } = 0 , \\qquad j = 1 , 2 , \\ldots , n .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            404,
+            825,
+            444
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "So we have: ",
+        "bbox": [
+            173,
+            446,
+            254,
+            459
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23b49078079ba759c8ab448a5c707e781d4adaac6566d18c7311a8cab7f40131.jpg",
+        "text": "$$\n\\nu _ { j } ^ { * } = - \\frac { 1 } { 2 \\lambda } \\frac { \\partial } { \\partial \\phi _ { j } } \\sum _ { i = 1 } ^ { m } [ V _ { \\phi } ( s _ { i } ) - G _ { i } ] ^ { 2 } | _ { \\phi _ { j } = \\phi _ { j } ^ { * } } - \\frac { 1 } { 2 \\lambda } \\frac { \\partial } { \\partial \\phi _ { j } } \\sum _ { i = 1 } ^ { m } \\mu _ { i } [ G _ { i } - V _ { \\phi } ( s _ { i } ) ] | _ { \\phi _ { j } = \\phi _ { j } ^ { * } } , \\qquad j = 1 , 2 , \\ldots , n .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            459,
+            826,
+            501
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "When $\\lambda  \\infty$ , we have $\\phi ^ { * } = 0$ . In this case, it follows that $V _ { \\phi ^ { * } } = C$ for some constant $C$ . As $V _ { \\phi ^ { * } } ( s _ { i } ) \\geq G _ { i }$ , in order to minimize (3) we must have $C = \\operatorname* { m a x } \\{ G _ { i } , \\ i = 1 , 2 , \\ldots , m \\}$ . ",
+        "bbox": [
+            171,
+            502,
+            823,
+            531
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the case of $\\lambda = 0$ , notice that we only have finitely many input $s _ { i }$ to be the input of the neural network. Therefore, this is a typical problem regarding the finite-sample expressivity of the neural networks, and the proof directly follows from the work done by Zhang et al. (Zhang et al., 2017). ",
+        "bbox": [
+            173,
+            536,
+            825,
+            580
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B. Proof of Theorem 4.2 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            594,
+            343,
+            609
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $\\phi ^ { * }$ be the optimal value that minimizes $L ( \\phi )$ . Let’s proceed by contradiction and assume that there exists some $k$ such that ",
+        "bbox": [
+            173,
+            625,
+            825,
+            652
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0bd7a748ca838a7694d86eb3d07c5a35d424908f16b84322ffc792bb4a8bd75b.jpg",
+        "text": "$$\nV _ { \\phi ^ { * } } ( s _ { k } ) < G _ { k }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            449,
+            651,
+            549,
+            667
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Let $\\phi ^ { \\prime }$ be an arbitrary given value such that $V _ { \\phi ^ { \\prime } } ( s _ { i } ) \\geq G _ { i }$ for all $i \\in \\{ 1 , 2 , \\dots , m \\}$ . Then we have ",
+        "bbox": [
+            171,
+            667,
+            813,
+            684
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3b8fe24444ffc1ed9ba632e4f376ac523bc12a94fb75657c53b6cf9886d6edb.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\displaystyle \\sum _ { i = 1 } ^ { m } ( V _ { \\phi ^ { * } } ( s _ { i } ) - G _ { i } ) ^ { 2 } \\mathbb { 1 } _ { ( V _ { \\phi ^ { * } } ( s _ { i } ) > G _ { i } ) } + K ( V _ { \\phi ^ { * } } ( s _ { k } ) - G _ { k } ) ^ { 2 } + \\lambda \\| \\phi ^ { * } \\| ^ { 2 } \\leq L ( \\phi ^ { * } ) } } \\\\ { { \\displaystyle L ( \\phi ^ { * } ) \\leq L ( \\phi ^ { \\prime } ) = \\sum _ { i = 1 } ^ { m } ( V _ { \\phi ^ { \\prime } } ( s _ { i } ) - G _ { i } ) ^ { 2 } + \\| \\phi ^ { \\prime } \\| ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            686,
+            754,
+            770
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This implies that ",
+        "bbox": [
+            174,
+            768,
+            287,
+            784
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "$\\displaystyle \\sum _ { = 1 } ^ { m } ( V _ { \\phi ^ { * } } ( s _ { i } ) - G _ { i } ) ^ { 2 } \\mathbb { 1 } _ { ( V _ { \\phi ^ { * } } ( s _ { i } ) > G _ { i } ) } + K ( V _ { \\phi ^ { * } } ( s _ { k } ) - G _ { k } ) ^ { 2 } + \\lambda \\| \\phi ^ { * } \\| ^ { 2 } \\leq \\displaystyle \\sum _ { i = 1 } ^ { m } ( V _ { \\phi ^ { \\prime } } ( s _ { i } ) - G _ { i } ) ^ { 2 } + \\| \\phi ^ { \\prime } \\| ^ { 2 }$ i   \nwhich is impossible when $K  \\infty$ . Therefore, we must have $V _ { \\phi ^ { * } } ( s _ { i } ) \\geq G _ { i }$ for all $i \\in \\{ 1 , 2 , \\dots , m \\}$ as $K  \\infty$ . In this way, when $K  \\infty$ , $\\phi ^ { * }$ actually minimizes ",
+        "bbox": [
+            173,
+            784,
+            826,
+            857
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c2fec58bb8dbc0cb703c07eb181de1c587da73f0c1aa2221d9b00be91487f284.jpg",
+        "text": "$$\nL ( \\phi ) = \\sum _ { i = 1 } ^ { m } ( V _ { \\phi } ( s _ { i } ) - G _ { i } ) ^ { 2 } + \\| \\phi \\| ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            858,
+            617,
+            900
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "which completes the proof. ",
+        "bbox": [
+            174,
+            901,
+            354,
+            916
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B IMPLEMENTATION DETAILS AND HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            656,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1 IMPLEMENTATION OF BAIL ALGORITHM ",
+        "bbox": [
+            174,
+            133,
+            501,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f7c8000fcfddb657d4215291c25683e18ac931519e603a4417f2bad81e5a755a.jpg",
+        "table_caption": [
+            "Table 2: Upper Envelope Hyperparameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameter</td><td>Value</td></tr><tr><td></td><td>Adam (Kingma &amp; Ba,2014)</td></tr><tr><td>optimizer learning rate</td><td>3.10-3</td></tr><tr><td>discount (γ)</td><td>0.99</td></tr><tr><td>regularization constant 入</td><td>2.10-2</td></tr><tr><td>K</td><td>10,000</td></tr><tr><td>number of hidden units</td><td>128 ×128</td></tr></table>",
+        "bbox": [
+            303,
+            194,
+            692,
+            310
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/4a01c0bbff5735b6cd394720b38bbac4de8fdd05bf3189e1cab181f975c5f6d5.jpg",
+        "table_caption": [
+            "Table 3: BAIL Hyperparameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>data in batch</td><td>106</td></tr><tr><td>optimizer</td><td>Adam (Kingma &amp; Ba,2014)</td></tr><tr><td>learning rate</td><td>10-3</td></tr><tr><td>regularization constant 入</td><td>0</td></tr><tr><td>mini-batch size</td><td>100</td></tr><tr><td>BAIL-border p%</td><td>25%</td></tr><tr><td>BAIL-TD x</td><td>0.96</td></tr><tr><td>number of hidden units</td><td>400 × 300</td></tr></table>",
+        "bbox": [
+            305,
+            367,
+            692,
+            512
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.2 IMPLEMENTATION OF COMPETING ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            546,
+            550,
+            559
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the behavioral DDPG algorithm, we used the implementation of Fujimoto et al. (2018b). For the behavioral SAC algorithm, we implemented it in Pytorch, mainly following the pseudocode provided by (Achiam), and used hyperparameters in Haarnoja et al. (2018b). For the BCQ algorithm, we used the authors’ implementation (Fujimoto et al., 2018a). For behavioral cloning and its variants in the ablation study section, the network structure, learning rate, mini-batch size, and so on are identical to those in Table 3 for BAIL. ",
+        "bbox": [
+            174,
+            570,
+            825,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the upper envelope network, our network has two hidden layers as does the Q network in BCQ and SAC. However, the number of hidden units in our network is less than those used in BCQ $( 4 0 0 \\times 3 0 0 )$ and SAC $( 2 5 6 \\times 2 5 6 )$ . In future work we will see if we can obtain further improvements with BAIL using larger networks for the upper envelope. ",
+        "bbox": [
+            174,
+            661,
+            825,
+            718
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.3 CLIPPING HEURISTICS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            734,
+            374,
+            748
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As is shown in Figure 1, in practice the trained upper envelope does not always fit well the data points on the right side of the plots, where the upper envelope can become very large. In that region, the data points with the highest returns will not be selected as “best actions” and therefore not used for imitation learning step. We observe that if we plot the upper envelope values for all the states in the buffer in ascending order as is shown in Figure 1, the upper envelope value $V ( s _ { i } )$ starts to deviate from the Monte Carlo return $G _ { i }$ at a point where $V ( s _ { i } ) \\approx \\operatorname* { m a x } \\{ G _ { i } \\}$ . We therefore use the following heuristic. We say that the upper envelope value begins to deviate from the Monte Carlo return at state $s _ { i }$ if $V ( s _ { j } ) > G _ { j }$ for $i \\leq j < i + 1 0 0 0 0$ . We set the clipping value $C = V ( s ^ { \\prime } )$ where $s ^ { \\prime }$ is the starting point of this deviation. Then the actual UE values used to select data is $\\operatorname* { m i n } \\{ V ( s ) , C \\}$ . In practice, the clipping heuristic gives a small boost in performance as is shown in Figure 5. ",
+        "bbox": [
+            174,
+            761,
+            825,
+            914
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            537,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.1 EXPERIMENTS ON BAIL-TD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            416,
+            147
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We present some of the experimental results for BAIL-TD in this subsection. Figure 4 shows that the performance of BAIL-TD is similar to the performance of naive Behavioral Cloning in Hopperv2 and Walker2d-v2 environments. The result implies that the BAIL-TD approach is limited in the ability to distinguish between good data points and the bad ones. This is possibly due to the inaccuracy of the trained upper envelope. Recall that in BAIL-TD algorithm, we select state action pairs based on the difference between the values of $r + V ( s _ { i } ^ { \\prime } )$ and $V ( s _ { i } )$ . Since we only use one-step Monte Carlo return, and the value of $r$ is very small compared with the value of $v ( s _ { i } )$ , the selection is very sensitive to the accuracy of $V ( s _ { i } )$ . ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/d26f46af57a5316ce66f9bc954b2be321ff21d0804d5148b3d9eceb1ffebf218.jpg",
+        "image_caption": [
+            "Figure 4: Performance comparison of BAIL-border, BAIL-TD, BCQ, BC and Behavioral Policy (DDPG) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            286,
+            813,
+            396
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.2 ABLATION STUDY FOR BAIL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            419,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We do additional ablation studies for BAIL, where we focus on two heuristic features in the BAIL algorithm: clipping of upper envelope, and return augmentation to approximate non-episodic continuous tasks. We removed each of these features one at a time from the BAIL algorithm and compare with the original BAIL algorithm. Each performance curve is again averaged over 5 independent runs. ",
+        "bbox": [
+            173,
+            128,
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+            199
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb39a0d4716f35152a6abda2ff1f3bc29daaaaca60eb219cc9a25d4597cb4302.jpg",
+        "image_caption": [
+            "Figure 5: Ablation of BAIL without Upper Envelope clipping "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            214,
+            808,
+            657
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We found that clipping often, but not always, gives a small boost in performance. As for return augmentation, we found it does not harm the performance of BAIL, and sometimes gives a considerable improvement, particularly for Hopper. ",
+        "bbox": [
+            174,
+            700,
+            826,
+            742
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/dede49c1dbcbdf6760a93a48b80b7c96bca7d1ebfe21303dfd2e9a23bdfa2782.jpg",
+        "image_caption": [
+            "Figure 6: Ablation of BAIL without return augmentation "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            276,
+            808,
+            719
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.3 BAIL FOR LOW NOISE-LEVEL DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            470,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In the main body of the paper we present BAIL in the ”Final Buffer” case as described in Fujimoto et al. (2018a), where the exploration noise $\\sigma = 0 . 5$ added to behavior policy is relatively large. In this section we examine the performance of BAIL in a low-noise scenario. To this end, we set $\\sigma = 0 . 1$ and do a similar experiments as were done $\\sigma = 0 . 5$ for the Hopper, Walker and HalfCheetah environments. (Recall that for Ant we used adaptive SAC, which does not have an explicit noise parameter.) ",
+        "bbox": [
+            173,
+            128,
+            825,
+            213
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The results are shown in Figure 7. We see that even in this low-noise scenario, BAIL out-performs BCQ by a wide margin for Hopper and Walker, and BAIL continues to out-perform the Final-DDPG policy in most batches. For HalfCheetah, where the Final-DDPG policy gives greatly different results depending on the batch, BAIL’s performance is stable and typically near that of the of FinalDDPG policy. After sufficient training, however, BCQ can often do better than the Final-DDPG policy in this environment. ",
+        "bbox": [
+            173,
+            219,
+            825,
+            304
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/090dd7cb80be99621216571aa31d07269a80f42830c6bf6c6a8cb4ec6ec66c15.jpg",
+        "image_caption": [
+            "Figure 7: Performance comparison of BAIL, BCQ, BC and Final-DDPG with noise level $\\sigma = 0 . 1$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            321,
+            808,
+            651
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C.4 VISUALIZATION OF UPPER ENVELOPES USED ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            532,
+            117
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "In this section we present the upper envelopes used in the training of BAIL in Figure 2. ",
+        "bbox": [
+            176,
+            128,
+            743,
+            145
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/856ef1d7387b230a0b8ea9c407649fbc0f4e7f792685b42d4895796482918c0c.jpg",
+        "image_caption": [
+            "Figure 8: Upper Envelopes of BAIL (DDPG data) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            157,
+            810,
+            492
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/e8a35e6bdfc4e4ee27333559f7b327108a4ae2943a3ec0962daaec77a013a6c7.jpg",
+        "image_caption": [
+            "Figure 9: Upper Envelopes of BAIL (SAC data) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            546,
+            808,
+            659
+        ],
+        "page_idx": 16
+    }
+]

parse/train/ByxQB1BKwH/ByxQB1BKwH.md

@@ -0,0 +1,313 @@
+# ABSTRACT DIAGRAMMATIC REASONING WITH MULTIPLEX GRAPH NETWORKS
+
+Duo Wang ∗& Mateja Jamnik & Pietro Lio
+
+Department of Computer Science and Technology   
+University of Cambridge   
+Cambridge, United Kingdom   
+{Duo.Wang,Mateja.Jamnik,Pietro.Lio}@cl.cam.ac.uk
+
+# ABSTRACT
+
+Abstract reasoning, particularly in the visual domain, is a complex human ability, but it remains a challenging problem for artificial neural learning systems. In this work we propose MXGNet, a multilayer graph neural network for multi-panel diagrammatic reasoning tasks. MXGNet combines three powerful concepts, namely, object-level representation, graph neural networks and multiplex graphs, for solving visual reasoning tasks. MXGNet first extracts object-level representations for each element in all panels of the diagrams, and then forms a multi-layer multiplex graph capturing multiple relations between objects across different diagram panels. MXGNet summarises the multiple graphs extracted from the diagrams of the task, and uses this summarisation to pick the most probable answer from the given candidates. We have tested MXGNet on two types of diagrammatic reasoning tasks, namely Diagram Syllogisms and Raven Progressive Matrices (RPM). For an Euler Diagram Syllogism task MXGNet achieves state-of-the-art accuracy of $9 9 . 8 \%$ . For PGM and RAVEN, two comprehensive datasets for RPM reasoning, MXGNet outperforms the state-of-the-art models by a considerable margin.
+
+# 1 INTRODUCTION
+
+Abstract reasoning has long been thought of as a key part of human intelligence, and a necessary component towards Artificial General Intelligence. When presented in complex scenes, humans can quickly identify elements across different scenes and infer relations between them. For example, when you are using a pile of different types of LEGO bricks to assemble a spaceship, you are actively inferring relations between each LEGO brick, such as in what ways they can fit together. This type of abstract reasoning, particularly in the visual domain, is a crucial key to human ability to build complex things.
+
+Many tests have been proposed to measure human ability for abstract reasoning. The most popular test in the visual domain is the Raven Progressive Matrices (RPM) test (Raven (2000)). In the RPM test, the participants are asked to view a sequence of contextual diagrams, usually given as a $3 \times 3$ matrices of diagrams with the bottom-right diagram left blank. Participants should infer abstract relationships in rows or columns of the diagram, and pick from a set of candidate answers the correct one to fill in the blank. Figures 1 (a) shows an example of RPM tasks containing XOR relations across diagrams in rows. More examples can be found in Appendix C. Another widely used test for measuring reasoning in psychology is Diagram Syllogism task (Sato et al. (2015)), where participants need to infer conclusions based on 2 given premises. Figure 1c shows an example of Euler Diagram Syllogism task.
+
+Barrett et al. (2018) recently published a large and comprehensive RPM-style dataset named Procedurally Generated Matrices ‘PGM’, and proposed Wild Relation Network (WReN), a state-of-the-art neural net for
+
+![](images/d9f2288353f35bdf6658e8f9a6f26755f7b3970e082d7fba1eaf4cc042ef3b87.jpg)  
+Figure 1: (a) shows an example of RPM tasks containing XOR relations across diagrams in rows and the overview of MXGNet architecture. Here $F _ { \rho }$ is object representation module, $E _ { \gamma }$ is edge embeddings module, $G _ { \phi }$ is graph summarization module and $R _ { \theta }$ is reasoning network. (b) shows an example of a multilayer graph formed from objects in the first row of diagrams in the example. (c) An example of syllogism represented in Euler diagrams.
+
+RPM-style tasks. While WReN outperforms other state-of-the-art vision models such as Residual Network He et al. (2016), the performance is still far from deep neural nets’ performance on other vision or natural language processing tasks. Recently, there has been a focus on object-level representations (Yi et al. (2018); Hu et al. (2017); Hudson & Manning (2018); Mao et al. (2019); Teney et al. (2017); Zellers et al. (2018)) for visual reasoning tasks, which enable the use of inductive-biased architectures such as symbolic programs and scene graphs to directly capture relations between objects. For RPM-style tasks, symbolic programs are less suitable as these programs are generated from given questions in the Visual-Question Answering setting. In RPM-style tasks there are no explicit questions. Encoding RPM tasks into graphs is a more natural choice. However, previous works on scene graphs (Teney et al. (2017); Zellers et al. (2018)) model a single image as graphs, which is not suitable for RPM tasks as there are many different layers of relations across different subsets of diagrams in a single task.
+
+In this paper we introduce MXGNet, a multi-layer multiplex graph neural net architecture for abstract diagram reasoning. Here ’Multi-layer’ means the graphs are built across different diagram panels, where each diagram is a layer. ‘Multiplex’ means that edges of the graphs encode multiple relations between different element attributes, such as colour, shape and position. Multiplex networks are discussed in detail by Kao & Porter (2018). We first tested the application of multiplex graph on a Diagram Syllogism dataset (Wang et al. (2018a)), and confirmed that multiplex graph improves performance on the original model. For RPM task, MXGNet encodes subsets of diagram panels into multi-layer multiplex graphs, and combines summarisation of several graphs to predict the correct candidate answer. With a hierarchical summarisation scheme, each graph is summarised into feature embeddings representing relationships in the subset. These relation embeddings are then combined to predict the correct answer.
+
+For PGM dataset (Barrett et al. (2018)), MXGNet outperforms WReN, the previous state-of-the-art model, by a considerable margin. For ‘neutral’ split of the dataset, MXGNet achieves $8 9 . 6 \%$ test accuracy, $1 2 . 7 \%$ higher than WReN’s $7 6 . 9 \%$ . For other splits MXGNet consistently performs better with smaller margins. For the RAVEN dataset (Zhang et al. (2019)), MXGNet, without any auxiliary training with additional labels, achieves $8 3 . 9 1 \%$ test accuracy, outperforming $5 9 . 5 6 \%$ accuracy by the best model with auxiliary training for the RAVEN dataset. We also show that MXGNet is robust to variations in forms of object-level representations. Both variants of MXGNet achieve higher test accuracies than existing best models for the two datasets.
+
+# 2 RELATED WORK
+
+Raven Progressive Matrices: Hoshen & Werman (2017) proposed a neural network model on Raven-style reasoning tasks that are a subset of complete RPM problems. Their model is based on Convolutional Network, and is demonstrated to be ineffective in complete RPM tasks (Barrett et al. (2018)). Mandziuk & Zychowski also experimented with an auto-encoder based neural net on simple single-shape RPM tasks. Barrett et al. (2018) built PGM, a complete RPM dataset, and proposed WReN, a neural network architecture based on Relation Network (Santoro et al. (2017)).Steenbrugge et al. (2018) replace CNN part of WReN with a pre-trained Variational Auto Encoder and slightly improved performance. Zhang et al. (2019) built RAVEN, a RPM-style dataset with structured labels of elements in the diagrams in the form of parsing trees, and proposed Dynamic Residual Trees, a simple tree neural network for learning with these additional structures. Anonymous (2020) applies Multi-head attention (Vaswani et al. (2017)), originally developed for Language model, on RPM tasks.
+
+Visual Reasoning: RPM test falls in the broader category of visual reasoning. One widely explored task of visual reasoning is Visual Question Answering(VQA). Johnson et al. (2017) built CLEVR dataset, a VQA dataset that focuses on visual reasoning instead of information retrieval in traditional VQA datasets. Current leading approaches (Yi et al. (2018); Mao et al. (2019)) on CLEVR dataset generate synthetic programs using questions in the VQA setting, and use these programs to process object-level representations extracted with objection detection models (Ren et al. (2015)). This approach is not applicable to RPM-style problems as there is no explicit question present for program synthesis.
+
+Graph Neural Networks: Recently there has been a surge of interest in applying Graph Neural Networks (GNN) for datasets that are inherently structured as graphs, such as social networks. Many variants of GNNs (Li et al. (2015); Hamilton et al. (2017); Kipf & Welling (2016); Velickovi ˇ c et al. (2017)) have been ´ proposed, which are all based on the same principle of learning feature representations of nodes by recursively aggregating information from neighbour nodes and edges. Recent methods (Teney et al. (2017); Zellers et al. (2018)) extract graph structures from visual scenes for visual question answering. These methods build scene graphs in which nodes represent parts of the scene, and edges capture relations between these parts. Such methods are only applied to scenes of a single image. For multi-image tasks such as video classification, Wang et al. (2018b) proposed non-local neural networks, which extract dense graphs where pixels in feature maps are connected to all other feature map pixels in the space-time dimensions.
+
+# 3 REASONING TASKS
+
+# 3.1 DIAGRAM SYLLOGISM
+
+Syllogism is a reasoning task where conclusion is drawn from two given assumed propositions (premises). One well-known example is ’Socrates is a man, all man will die, therefore Socrates will die’. Syllogism can be conveniently represented using many types of diagrams (Al-Fedaghi (2017)) such as Euler diagrams and Venn diagrams. Figure 1 (c) shows an example of Euler diagram syllogism. Wang et al. (2018a) developed Euler-Net, a neural net architecture that tackles Euler diagram syllogism tasks. However Euler-Net is just a simple Siamese Conv-Net, which does not guarantee scalability to more entities in diagrams. We show that the addition of multiplex graph both improves performance and scalability to more entities.
+
+# 3.2 RAVEN PROGRESSIVE MATRICES
+
+In this section we briefly describe Raven Progressive Matrices (RPM) in the context of the PGM dataset (Barrett et al. (2018)) and the RAVEN dataset (Zhang et al. (2019)). RPM tasks usually have 8 context diagrams and 8 answer candidates. The context diagrams are laid out in a $3 \times 3$ matrix $\mathbf { C }$ where $c _ { 1 , 1 } , . . c _ { 3 , 2 }$ are context diagrams and $^ { c _ { 3 , 3 } }$ is a blank diagram to be filled with 1 of the 8 answer candidates $\mathbf { A } = \{ a _ { 1 } , \ldots , a _ { 8 } \}$ One or more relations are present in rows or/and columns of the matrix. For example, in Figure 1 (a), there is $X O R$ relation of positions of objects in rows of diagrams. With the correct answer filled in, the third row and column must satisfy all relations present in the first 2 rows and columns (in the RAVEN dataset, relations are only present in rows). In addition to labels of correct candidate choice, both datasets also provide labels of meta-targets for auxiliary training. The meta-target of a task is a multi-hot vector encoding tuples of $( r , o , a )$ where $r$ is the type of a relation present, $o$ is the object type and $a$ is the attribute. For example, the meta-target for Figure 1 (a) encodes $( X O R , S h a p e , P o s i t i o n )$ . The RAVEN dataset also provides additional structured labels of relations in the diagram. However, we found that structured labels do not improve results, and therefore did not use them in our implementation.
+
+# 4 METHOD
+
+MXGNet is comprised of three main components: an object-level representation module, a graph processing module and a reasoning module. Figure 1a shows an overview of the MXGNet architecture. The object-level representation module $F _ { \rho }$ , as the name suggests, extracts representations of objects in the diagrams as nodes in a graph. For each diagram $d _ { i } \subset \mathbf { C } \cup \mathbf { A }$ , a set of nodes $v _ { i , j } ; i = 1 \ldots L , j = 1 \ldots N$ is extracted where $L$ is the number of layers and $N$ is the number of nodes per layer. We experimented with both fixed and dynamically learnt $N$ values. We also experimented with an additional ‘background’ encoder that encodes background lines (See Appendix C for an example containing background lines) into a single vector, which can be considered as a single node. The multiplex graph module $G _ { \phi }$ , for a subset of diagrams, learns the multiplex edges capturing multiple parallel relations between nodes in a multi-layer graph where each layer corresponds to one diagram in the subset, as illustrated in Figure 1 (c). In MXGNet, we consider a subset of cardinality 3 for $3 \times 3$ diagram matrices. While prior knowledge of RPM rules allows us to naturally treat rows and columns in RPM as subsets, this prior does not generalise to other types of visual reasoning problems. Considering all possible diagram combinations as subsets is computationally expensive. To tackle this, we developed a relatively quick pre-training method to greatly reduce the search space of subsets, as described below.
+
+Search Space Reduction: We can consider each diagram as node $v _ { i } ^ { d }$ in a graph, where relations between adjacent diagrams are embedded as edges $e _ { i j } ^ { d }$ . Note here we are considering the graph of ’diagrams’, which is different from the graph of ’objects’ in the graph processing modules. Each subset of 3 diagrams in this case can be considered as subset of 2 edges. We here make weak assumptions that edges exist between adjacent diagrams (including vertical, horizontal and diagonal direction) and edges in the same subset must be adjacent (defined as two edges linking the same node), which are often used in other visual reasoning problems. We denote the subset of edges as $\{ e _ { i j } ^ { d } , e _ { j k } ^ { d } \}$ . We use 3 neural nets to embed nodes, edges and subsets. We use CNNs to embed diagram nodes into feature vectors, and MLPs to embed edges based on node embeddings and subsets based on edge embeddings. While it is possible to include graph architectures for better accuracy, we found that simple combinations of CNNs and MLPs train faster while still achieving the search space reduction results. This architecture first embeds nodes, then embeds edges based on node embedding, and finally embed subsets based on edge embedding. The subset embeddings are summed and passed through a reasoning network to predict answer probability, similar to WReN (Barrett et al. (2018)). For the exact configuration of the architecture used please refer to Appendix A. For each subset $\{ e _ { i j } ^ { d } , e _ { j k } ^ { d } \}$ we define a gating variable $G _ { i j k }$ , controlling how much does each subset contributes to the final result. In practice we use tanh function, which allows a subset to contribute both positively and negatively to the final summed embeddings. In training we put L1 regularization constraint on the gating variables to suppress $G _ { i j k }$ of non-contributing subsets close to zero. This architecture can quickly discover rows and columns as contributing subsets while leaving gating variables of other subsets not activated. We describe the experiment results in section 5.1. While this method is developed for discovering reasoning rules for RPM task, it can be readily applied to any other multi-frame reasoning task for search space reduction. In the rest of the paper, we hard-gate subsets by rounding the gating variables, thereby reducing subset space to only treat rows and columns as valid subsets.
+
+We treat the first 2 rows and columns as contextual subsets $c _ { i , j }$ where $i$ and $j$ are row and column indices. For the last row and column, where the answers should be filled in, we fill in each of the 8 answer candidates, and make 8 row subsets $a _ { i } , i \subset [ 1 , 8 ]$ and 8 column subsets $a _ { i } , i \subset [ 1 , 8 ]$ .
+
+The graph module then summarises the graph of objects in a subset into embeddings representing relations present in the subset. The reasoning module $R _ { \theta }$ takes embeddings from context rows/columns and last rows/columns with different candidate answers filled in, and produce normalised probability of each answer being true. It also predicts meta-target for auxiliary training using context rows/columns. Next, we describe each module in detail.
+
+# 4.1 OBJECT-LEVEL REPRESENTATION
+
+In the PGM dataset there are two types of objects, namely ‘shapes’ and background ‘lines’. While it is a natural choice to use object-level representation on shapes as they are varying in many attributes such as position and size, it is less efficient on background lines as they only vary in colour intensity. In this section we first describe object-level representation applied to ‘shapes’ objects, and then discuss object-level representation on ’lines’ and an alternative background encoder which performs better.
+
+In MXGNet we experiment with two types of object-level representations for ‘shapes’, namely CNN grid features and representation obtained with spatial attention. For CNN grid features, we use each spatial location in the final CNN feature map as the object feature vector. Thus for each feature maps of width $W$ and height $H$ , $N = W \times H$ object representations are extracted. This type of representation is used widely, such as in Relation Network (Santoro et al. (2017)) and VQ-VAE (van den Oord et al. (2017)). For representation obtained with attention, we use spatial attention to attend to locations of objects, and extract representations for each object attended. This is similar to objection detection models such as faster R-CNN (Ren et al. (2015)), which use a Region Proposal Network to propose bounding boxes of objects in the input image. For each attended location a presence variable $z _ { p r e s }$ is predicted by attention module indicating whether an object exists in the location. Thus the total number of objects $N$ can vary depending on the sum of $z _ { p r e s }$ variables. As object-level representation is not the main innovation of this paper, we leave exact details for Appendix A.1.
+
+For background ‘lines’ objects, which are not varying in position and size, spatial attention is not needed. We experimented with a recurrent encoder with Long-Short Term Memory (Hochreiter & Schmidhuber (1997)) on the output feature map of CNN, outputting $M$ number of feature vectors. However, in the experiment we found that this performs less well than just feature map embeddings produced by feed-forward conv-net encoder.
+
+![](images/b2267ee8575c5399c27e3c0ee85bad73a8f7e945bd9839fe28469263cf9348f9.jpg)  
+Figure 2: Illustration of multiplex edge embeddings and cross-gating function. Each edge contains a set of different sub-connections (colored differently). Multiplex edges connecting to each node in the last layer are aggregated according to its originating layer. Aggregated embeddings are then passed to a gating function $G$ , which outputs gating variables from each aggregated embeddings.
+
+# 4.2 MULTIPLEX GRAPH NETWORK
+
+Multiplex Edge Embedding:The object-level representation module outputs a set of representations $v _ { i , j } ; i \subset$ $[ 1 , L ] , j \subset [ 1 , N ]$ for ‘shapes’ objects, where $L$ is the number of layers (cardinality of subset of diagrams) and $N$ is the number of nodes per layer. MXGNet uses an multiplex edge-embedding network $E _ { \gamma }$ to generate edge embeddings encoding multiple parallel relation embeddings:
+
+$$
+e _ { ( i , j ) , ( l , k ) } ^ { t } = E _ { \gamma } ^ { t } ( P ^ { k } ( v _ { i , j } , v _ { l , k } ) ) ; i \neq l , t = 1 \dots T
+$$
+
+Here $P ^ { t }$ is a projection layer projecting concatenated node embeddings to $T$ different embeddings. $E ^ { t }$ is a small neural net processing $\mathbf { \chi } _ { t ^ { t h } } ^ { * }$ projections to produce the $t ^ { t h }$ sub-layer of edge embeddings. Here, we restricted the edges to be inter-layer only, as we found using intra-layer edges does not improve performance but increases computational costs. Figure 2 illustrates these multiplex edge embeddings between nodes of different layers. We hypothesise that different layers of the edge embeddings encode similarities/differences in different feature spaces. Such embeddings of similarities/differences are useful in comparing nodes for subsequent reasoning tasks. For example,for P rogessive relation of object sizes, part of embeddings encoding size differences can be utilized to check if nodes in later layers are larger in size. This is similar to Mixture of Experts layers (Eigen et al. (2013); Shazeer et al. (2017)) introduced in Neural Machine Translation tasks. However, in this work we developed a new cross-multiplexing gating function at the node message aggregation stage, which is described below.
+
+Graph Summarisation: After edge embeddings are generated, the graph module then summarises the graph into a feature embedding representing relations present in the subset of diagrams. We aggregate information in the graph to nodes of the last layer corresponding to the third diagram in a row or column, because in RPM tasks the relations are in the form $D i a g r a m 3 = F u n c t i o n ( D i a g r a m 1 , D i a g r a m 2 )$ ). All edges connecting nodes in a particular layer $v _ { i , j } ; i \ne L$ , to a node $v _ { L , k }$ in the last layer $L$ are aggregated by a function $F _ { a g }$ composed of four different types of set operations, namely max, min, sum and mean:
+
+$$
+f v _ { i , k } = F _ { a g } ( e _ { ( i , 1 ) , ( L , k ) } \cdot \cdot \cdot e _ { ( i , 1 ) , ( L , k ) } ) ; F _ { a g } = c o n c a t ( m a x ( ) , m i n ( ) , s u m ( ) , m e a n ( ) )
+$$
+
+We use multiple aggregation functions together because different sub-tasks in reasoning may require different types of summarization. For example, counting number of objects is better suited for sum while checking if there is a object with the same size is better suited for max. The aggregated node information from each layer is then combined with a cross-multiplexing gating function. It is named ’cross-multiplexing’ because each embeddings in the set are ’multiplexing’ other embeddings in the set with gating variables that regulate which stream of information pass through. This gating function accepts a set of summarised node embeddings $\{ f v _ { 1 , k } \ldots f v _ { N , k } \}$ as input, and output gating variables for each layer of node embeddings in the set:
+
+$$
+\mathbf { g } _ { 1 , k } \ldots \mathbf { g } _ { N , k } = G ( f v _ { 1 , k } \ldots f v _ { N , k } ) ; \mathbf { g } _ { i , k } = \{ g _ { i , k } ^ { 1 } \ldots g _ { i , k } ^ { T } \}
+$$
+
+In practice $G$ is implemented as an MLP with multi-head outputs for different embeddings, and Sigmoid activation which constrains gating variable $g$ within the range of 0 to 1. The node embeddings of different layers are then multiplied with the gating variables, concatenated and passed through a small MLP to produce the final node embeddings: $f v _ { k } = M \bar { L } P ( c o n c a t ( \{ f v _ { i , k } \times g _ { ( } i , k ) | \bar { i } = 1 \ldots N \} )$ ). Node embeddings and background embeddings are then concatenated and processed by a residual neural block to produce final relation feature embeddings $r$ of the diagram subset.
+
+# 4.3 REASONING NETWORK
+
+The reasoning network takes relation feature embeddings $r$ from all graphs, and infers the correct answer based on these relation embeddings. We denote the relation embeddings for context rows as $r _ { i } ^ { c r } ; i = 1 , 2$ and context columns as $r _ { i } ^ { c c } ; i = 1 , 2$ . The last row and column filled with each answer candidate $a _ { i }$ are denoted $r _ { i } ^ { a r } ; i = 1 , . . . , 8$ and $r _ { i } ^ { a c } ; i = 1 , \ldots , 8$ . For the RAVEN dataset, only row relation embeddings $r ^ { c r }$ and $r ^ { a r }$ are used, as discussed in Section 3.2. The reasoning network $R _ { \theta }$ is a multi-layer residual neural net with a softmax output activation that processes concatenated relation embeddings and outputs class probabilities for each answer candidate. The exact configuration of the reasoning network can be found in Appendix A.3.
+
+For meta-target prediction, all relation information is contained in the context rows and columns of the RPM task. Therefore, we apply a meta-predicting network $R _ { m e t a }$ with Sigmoid output activation to all context rows and columns to obtain probabilities of each meta-target categories:
+
+$$
+p _ { m e t a } = R _ { m e t a } ( r _ { 1 } ^ { c r } + r _ { 2 } ^ { c r } + r _ { 1 } ^ { c c } + r _ { 2 } ^ { c c } )
+$$
+
+# 4.4 TRAINING
+
+The full pipeline of MXGNet is end-to-end trainable with any gradient descent optimiser. In practice, we used RAdam optimiser (Liu et al. (2019)) for its fast convergence and robustness to learning rate differences. The loss function for the PGM dataset is the same as used in WReN (Barrett et al. (2018)): $\mathcal { L } = \mathcal { L } _ { a n s } + \beta \mathcal { L } _ { m e t a - t a r g e t }$ where $\beta$ balances the training between answer prediction and meta-target prediction. For the RAVEN dataset, while the loss function can include auxiliary meta-target and structured labels as $\mathcal { L } = \mathcal { L } _ { a n s } + \alpha \mathcal { L } _ { s t r u c t } + \beta \mathcal { L } _ { m e t a - t a r g e t }$ , we found that both auxiliary targets do not improve performance, and thus set $\alpha$ and $\beta$ to 0.
+
+# 5 EXPERIMENTS
+
+# 5.1 SEARCH SPACE REDUCTION
+
+The Search Space Reduction model is applied on both PGM and RAVEN dataset to reduce the subset space. After 10 epochs, only gating variables of rows and columns subset for PGM and of rows for RAVEN have value larger than 0.5. The Gating variables for three rows are 0.884, 0.812 and 0.832. The gating variables for three columns are 0.901, 0.845 and 0.854. All other gating variables are below the threshold value of 0.5. Interestingly all activated (absolute value $> 0 . 5$ ) gating variables are positive. This is possibly because it is easier for the neural net to learn an aggregation function than a comparator function. Exact experiment statistics can be found in Appendix D.
+
+# 5.2 DIAGRAM SYLLOGISM PERFORMANCE
+
+We first test how well can the multiplex graph network capture relations for the simple Diagram Syllogism task. We simply add the multiplex graph to the original Conv-Net used in (Wang et al. (2018a)). MXGNet achieved $9 9 . 8 \%$ accuracy on both 2-contour and 3-contour tasks, higher than the original paper’s $9 9 . 5 \%$ and $9 9 . 4 \%$ accuracies. The same performance on 2-contour and 3-contour tasks also show that MXGNet scales better for more entities in the diagram. For more details please refer to Appendix E.
+
+# 5.3 RPM TASK PERFORMANCES
+
+In this section we compare all variants of MXGNet against the state-of-the-art models for the PGM and the RAVEN datasets. For the PGM dataset, we tested against results of WReN (Barrett et al. (2018)) in the auxiliary training setting with $\beta$ value of 10. In addition, we also compared MXGNet with VAE-WReN (Steenbrugge et al. (2018))’s result without auxiliary training. For the RAVEN dataset, we compared with WReN and ResNet model’s performance as reported in the original paper (Zhang et al. (2019)). We evaluated MXGNet with different object-level representations (Section 4.1) on the test data in the ‘neutral’ split of the PGM dataset.
+
+Table 1 (a) shows test accuracies of model variants compared with WReN and VAE-WReN for the case without auxiliary training $( \beta = 0$ ) and with auxiliary training ( $\beta = 1 0$ ) for the PGM dataset. Both model variants of MXGNet outperform other models by a considerable margin, showing that the multi-layer graph is indeed a more suitable way to capture relations in the reasoning task. Model variants using grid features from the CNN feature maps slightly outperform model using spatial-attention-based object representations for both with and without auxiliary training settings. This is possibly because the increased number of parameters for the spatial attention variant leads to over-fitting, as the training losses of both model variants are very close. In our following experiments for PGM we will use model variants using CNN features to report performances.
+
+Table 1 (b) shows test accuracies of model variants compared with $\mathrm { W R e N }$ the best performing ResNet models for RAVEN dataset. WReN surprisingly only achieves $1 4 . 6 9 \%$ as tested by Zhang et al. (2019). We include results of the ResNet model with or without Dynamic Residual Trees (DRT) which utilise additional structure labels of relations. We found that for the RAVEN dataset, auxiliary training of MXGNet with meta-target or structure labels does not improve performance. Therefore, we report test accuracies of models trained only with the target-prediction objective. Both variants of MXGNet significantly outperform the ResNet models. Models with spatial attention object-level representations under-perform simpler CNN features slightly, most probably due to overfitting, as the observed training losses of spatial attention models are in fact lower than CNN feature models.
+
+# 5.4 GENERALISATION EVALUATION FOR PGM
+
+In the PGM dataset, other than the neutral data regime in which test dataset’s sampling space is the same as the training dataset, there are also other data regimes which restrict the sampling space of training or test data to evaluate the generalisation capability of a neural network. In the main paper, due to space limitations, we selected 2 representative regimes, the ‘interpolation’ regime and the ‘extrapolation’ regime to report results. For results of other data splits of PGM, please refer to Appendix G. For ‘interpolation’ regime, in the training dataset, when attribute $a = c o l o r$ and $a = s i z e$ , the values of $a$ are restricted to even-indexed values in the spectrum of $a$ values. This tests how well can a model ‘interpolate’ for missing values. For ‘Extrapolation’ regime, in the training dataset, the value of $a$ is restricted to be the lower half of the value spectrum. This tests how well can a model ‘extrapolate’ outside of the value range in the training dataset. Table 2 shows validation and test accuracies for all three data regimes with and without auxiliary training. In addition, differences between validation and test accuracies are also presented to show how well can models generalise. MXGNet models consistently perform better than WReN for all regimes tested. Interesting for ’Interpolation’ regime, while validation accuracy of MXGNet is lower than WReN, the test accuracy is higher. In addition, for regime
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>WReNBarrett  et  al.(2018)</td><td rowspan=1 colspan=1>VAE-WReNSteenbrugge et al.(2018)</td><td rowspan=1 colspan=1>ARNeAnonymous (2020)</td><td rowspan=1 colspan=1>MXGNetCNN Sp-Attn</td></tr><tr><td rowspan=1 colspan=1>acc.(%)β= 10</td><td rowspan=1 colspan=1>76.9</td><td rowspan=1 colspan=1>N/A</td><td rowspan=1 colspan=1>88.2</td><td rowspan=1 colspan=1>89.6    88.8</td></tr><tr><td rowspan=1 colspan=1>acc.(%)β= 0</td><td rowspan=1 colspan=1>62.6</td><td rowspan=1 colspan=1>64.2</td><td rowspan=1 colspan=1>N/A</td><td rowspan=1 colspan=1>66.7    66.1</td></tr></table>
+
+(a) PGM   
+(b) RAVEN   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>WReNZhang et al. (2019)</td><td rowspan=1 colspan=1>ResNetZhang et al. (2019)</td><td rowspan=1 colspan=1>ResNet+DRTZhang et al. (2019)</td><td rowspan=1 colspan=1>ARNeAnonymous (2020)</td><td rowspan=1 colspan=1>MXGNetCNN  Sp-Attn</td></tr><tr><td rowspan=1 colspan=1>acc. (%)</td><td rowspan=1 colspan=1>14.69</td><td rowspan=1 colspan=1>53.43</td><td rowspan=1 colspan=1>59.56</td><td rowspan=1 colspan=1>19.67</td><td rowspan=1 colspan=1>83.91   82.61</td></tr></table>
+
+Table 1: (a) shows results comparing MXGNet model variants against $\mathrm { W R e N }$ for the PGM dataset. (b) shows results comparing MXGNet model variants against ResNet models for the RAVEN dataset. The object-level representation has two variations which are (o1) CNN features and (o2) Spatial Attention features (Section 4.1).   
+‘Interpolation’ and ‘Extrapolation’, MXGNet also shows a smaller difference between validation and test accuracy. These results show that MXGNet has better capability of generalising outside of the training space.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Regime</td><td colspan="3">β=0</td><td colspan="3">β=10</td></tr><tr><td>Val.(%)</td><td> test%</td><td>Diff.</td><td>Val.(%)</td><td> test%</td><td>Diff.</td></tr><tr><td rowspan="3">WReN</td><td>Neutral</td><td>63.0</td><td>62.6</td><td>-0.4</td><td>77.2</td><td>76.9</td><td>-0.3</td></tr><tr><td>Interpolation</td><td>79.0</td><td>64.4</td><td>-14.6</td><td>92.3</td><td>67.4</td><td>-24.9</td></tr><tr><td>Extrapolation</td><td>69.3</td><td>17.2</td><td>-52.1</td><td>93.6</td><td>15.5</td><td>-79.1</td></tr><tr><td rowspan="3">MXGNet</td><td>Neutral</td><td>67.1</td><td>66.7</td><td>-0.4</td><td>89.9</td><td>89.6</td><td>-0.3</td></tr><tr><td>Interpolation</td><td>74.2</td><td>65.4</td><td>-8.8</td><td>91.5</td><td>84.6</td><td>-6.9</td></tr><tr><td>Extrapolation</td><td>69.1</td><td>18.9</td><td>-50.2</td><td>94.3</td><td>18.4</td><td>-75.9</td></tr></table>
+
+Table 2: Generalisation performance comparing MXGNet model variants against WReN. ‘Diff.’ is the difference between the test and the validation performances.
+
+# 6 DISCUSSION AND CONCLUSION
+
+We presented MXGNet, a new graph-based approach to diagrammatic reasoning problems in the style of Raven Progressive Matrices (RPM). MXGNet combines three powerful ideas, namely, object-level representation, graph neural networks and multiplex graphs, to capture relations present in the reasoning task. Through experiments we showed that MXGNet performs better than previous models on two RPM datasets. We also showed that MXGNet has better generalisation performance.
+
+One important direction for future work is to make MXGNet interpretable, and thereby extract logic rules from MXGNet. Currently, the learnt representations in MXGNet are still entangled, providing little in the way of understanding its mechanism of reasoning. Rule extraction can provide people with better understanding of the reasoning problem, and may allow neural networks to work seamlessly with more programmable traditional logic engines.
+
+While the multi-layer multiplex graph neural network is designed for RPM style reasoning task, it can be readily extended to other diagrammatic reasoning tasks where relations are present between multiple elements across different diagrams. One example of a real-world application scenario is robots assembling parts of an object into a whole, such as building a LEGO model from a room of LEGO blocks. MXGNet provides a suitable way of capturing relations between parts, such as ways of piecing and locking two parts together.
+
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+
+# A ARCHITECTURE
+
+In this section we present exact configurations of all model variants of MXGNet. Due to the complexity of architectures, we will describe each modules in sequence. The object-level representation has two variations which are (o1) CNN features and (o2) Spatial Attention features. Also the models for PGM and RAVEN dataset differ in details. Unless otherwise stated, in all layers we apply Batch Normalization Ioffe & Szegedy (2015) and use Rectified Linear Unit as activation function.
+
+# A.1 OBJECT-LEVEL REPRESENTATION ARCHITECTURE
+
+CNN features: The first approach applies a CNN on the input image and use each spatial location in the final CNN feature map as the object feature vector. This type of representation is used widely, such as in Relation Network Santoro et al. (2017) and VQ-VAE van den Oord et al. (2017). Formally, the output of a CNN is a feature map tensor of dimension $H \times W \times D$ where $H$ , $W$ and $D$ are respectively height, width and depth of the feature map. At each $H$ and $W$ location, an object vector is extracted. This type of object representation is simple and fast, but does not guarantee that the receptive field at each feature map location fully bounds objects in the image.
+
+We use a residual module He et al. (2016) with two residual blocks to extract CNN features, as shown in figure 4.This is because Residual connections show better performance in experiments. The structure of a single Residual Convolution Block is shown in figure 3.Unless otherwise stated, convolutional layer in residual blocks has kernel size of $3 \times 3$ . The output feature map processed by another residual block is treated as background encoding because we found that convolutional background encoding gives better results than feature vectors.
+
+![](images/a2c5ce0d4847ea7b46bb33da416635d22b4c956b822b83e6fce0f13406826ad0.jpg)  
+Figure 3: Architecture of a single Residual Convolution Block.
+
+Spatial Attention Object-level representation: The second approach is to use spatial attention to attend to locations of objects, and extract representations for each object attended. This is similar to object detection models such as faster R-CNN Ren et al. (2015), which use a Region Proposal Network to propose bounding boxes of objects in the input image. In practice, we use Spatial Transformer Jaderberg et al. (2015) as our spatial attention module. Figure 5 shows the architecture used for extracting object-level representation using spatial attention. A CNN composed of 1 conv layr and 2 residual blocks is first applied to the input image, and the last layer feature map is extracted. This part is the same as CNN grid feature module. A spatial attention network composed of 2 conv layer then processes information at each spatial location on the feature map, and outputs $k$ numbers of $z = ( \dot { z } ^ { p r e s } , z ^ { w \hat { h } e r e } )$ , corresponding to $k$ possible objects at each location. Here, $z ^ { p r e s }$ is a binary value indicating if an object exists in this location, and $z ^ { w h e r \bar { e } }$ is an affine transformation matrix specifying a sampling region on the feature maps. $z ^ { p r e s }$ , the binary variable, is sampled from Gumbel-Sigmoid distribution Maddison et al. (2016); Jang et al. (2016), which approximates the Bernoulli distribution. We set Gumbel temperature to 0.7 throughout the experiments. For the PGM dataset we restricted $k$ to be 1 and $z ^ { w h e r e }$ to be a translation and scaling matrix as ‘shapes’ objects do not overlap and do not have affine transformation attributes other than scaling and translation. For all $z _ { i } ; i \subset [ 1 , H \times \bar { W } ]$ , if $z _ { i } ^ { p r e s }$ is 1, an object encoder network samples a patch from location specified by $z _ { i } ^ { w h e r e }$ using a grid sampler with a fixed window size of pixels. More details of the grid sampler can be found in Jaderberg et al. (2015). The sampled patches are then processed by a conv-layer to generate object embeddings.
+
+![](images/f352e955bb44246b024725ad05a01c37c8807f81f397e4cb2dc0a623b5e7d640.jpg)  
+Figure 4: CNN feature object-level representation module. ’Conv’ is convolution layers, ’Max-Pooling’ is max-pooling layer and ’ResConv Block’ is Residual Convolutional Block.
+
+![](images/930d44d182c4d735f6b116edc17c39f12cbb68f371a9c1c822e2b2f2a5fd6e48.jpg)  
+Figure 5: Spatial attention based feature object-level representation module. ’Conv’ is convolution layers, ’Max-Pooling’ is max-pooling layer and ’ResConv Block’ is Residual Convolutional Block. $z$ is the spatial attention variable $( z ^ { p r e \bar { s } } , z ^ { w h \bar { e } r e } )$ . Sampler is a grid sampler which samples grid of points from given feature maps.
+
+# A.2 GRAPH NETWORKS
+
+Multiplex Edge Embeddings:Figure 2 in the main paper shows an overview of the multiplex graph architecture. While motivation and overview of architecture is explained in section 4.2 of the main paper, in this section we provide exact configurations for each part of the model. Each sub-layer of the multiplex edge is embedded by a small MLP. For PGM dataset, we use 6 parallel layers for each multiplex edge embeddings , with each layer having 32 hidden units and 8 output units. For RAVEN dataset we use 4 layers with 16 hidden units and 8 output units because RAVEN dataset contains fewer relations types than PGM dataset. Gating function is implemented as one Sigmoid fully connected layer with hidden size equal to the length of concatenated aggregated embeddings. Gating variables are element-wise multiplied with concatenated embeddings for gating effects. Gated embeddings are then processed with a final fully connected layer with hidden size 64.
+
+Graph Summarization: This module summarizes all node summary embeddings and background embeddings to produce a diagram subset embedding representing relations present in the set of diagrams. We experimented with various approaches and found that keeping embeddings as feature maps and processing them with residual blocks yields the best results. Background feature map embeddings are generated with one additional residual block of 48 on top of lower layer feature-extracting resnet. For object representations obtained from CNN-grid features, we can simply reshape node embeddings into a feature map, and process it with additional conv-nets to generate a feature map embeddings of the same dimension to background feature map embeddings. For object representations with spatial attention, we can use another Spatial Transformer to write node summary embeddings to its corresponding locations on a canvas feature map. Finally we concatenate node summary embeddings and background embeddings and process it with 2 residual blocks of size 64 to produce the relation embeddings.
+
+# A.3 REASONING NETWORK
+
+Figure 6 shows the reasoning network configuration for RPM tasks. We experimented with the approach introduced in Barrett et al. (2018), which compute scores for each answer candidates and finally normalize the scores. We found this approach leads to severe overfitting on the RAVEN dataset, and therefore used a simpler approach to just concatenate all relation embeddings and process them with a neural net. In practice we used two residual blocks of size 128 and 256, and a final fully connected layer with 8 units corresponding to 8 answer candidates. The output is normalized with softmax layer. For Meta-target prediction, all context relation embeddings (context rows and columns for PGM while only rows for RAVEN dataset) are summed and fed into a fully connected prediction layer with Sigmoid activation. For PGM there are 12 different meta-targets while for RAVEN there are 9.
+
+![](images/2aa33a733d1badb8845c69ca5558e0454d9221b714de16f54361ff559e995da9.jpg)  
+Figure 6: Architecture overview of reasoning module. ’RelEmbed’ is relation embeddings, ’Concat’ is concatenation layer. ’ResBlock’ is Residual Convolutional Block. ’FC’ is fully connected layer.
+
+# B TRAINING DETAILS
+
+The architecture is implemented in Pytorch framework. During training, we used RAdam optimizer Liu et al. (2019) with learning rate 0.0001, $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 9 9 .$ We used batch size of 64, and distributed the training across 2 Nvidia Geforce Titan X GPUs. We early-stop training when validation accuracy stops increasing.
+
+# C MORE DETAILS OF RPM DATASETS
+
+In PGM dataset there are two types of elements present in the diagram, namely shapes and lines. These elements have different attributes such as colour and size. In the PGM dataset, five types of relations can be present in the task: $\{ P r o g r e s s i o n , A N D , O R , X O R , C o n s i s t e n t U n i o n \}$ . The RAVEN dataset, compared to PGM, does not have logic relations $A N D , O R , X O R$ , but has additional relations Arithmetic, Constant. In addition RAVEN dataset only allow relations to be present in rows.
+
+Figure 7a and 7b show two examples from the PGM dataset(Image courtesy Barrett et al. (2018)). The first example contains a ’Progression’ relation of the number of objects across diagrams in columns. The second examples contains a ’XOR’ relation of position of objects across diagrams in rows.
+
+In addition to shape objects, diagrams in the PGM dataset can also contain background line objects that appear at fixed locations. Figure 8a and 8b show two examples of PGM tasks containing line objects.
+
+# D MORE DETAILS ON SEARCH SPACE REDUCTION
+
+In this section we provide detailed architecture used for Search Space reduction, and present additional experimental results.
+
+The node embeddings are generated by applying a Conv-Net of 4 convolutional layer (32 filters in each layer) of kernel size 3, and a fully connected layer mapping flattened final-layer feature maps to a feature vector of size 256. Edge embeddings are generated by a 3-layer MLP of $5 1 2 - 5 1 2 - 2 5 6$ hidden units. Subset embeddings are generated by a fully connected layer of 512 units. The subset embeddings are gated with the gating variables and summed into a feature vector, which is then feed into the reasoning net, a 3-layer MLP with $2 5 6 - 2 5 6 - 1 3$ . The output layer contains 13 units. The first unit gives probability of currently combined answer choice being true. The rest 12 units give meta-target prediction probabilities. This is the same as Barrett et al. (2018). The training loss function is:
+
+$$
+\mathcal { L } = \mathcal { L } _ { a n s } + \beta \mathcal { L } _ { m e t a - t a r g e t } + \lambda \left\| \sum _ { ( i , j , k ) \subset S } G _ { i , j , k } \right\| _ { L 1 }
+$$
+
+In our experiment we have tested various values of $\lambda$ , and found 0.01 to be the best. This model is trained with RAdam optimizer with learning rate of 0.0001 and batch size of 64. After 10 epochs of training, only gating variables of subsets that are rows and columns are above the 0.5 threshold. The Gating variables for three rows are 0.884, 0.812 and 0.832. The gating variables for three columns are 0.901, 0.845 and 0.854. All other gating variables are below 0.5. Among these, the one with highest absolute value is 0.411. Table 3 shows the top-16 ranked subsets, with each subset indexed by 2 connecting edges in the subset. Figure 9 illustrates this way of indexing the subset. For example, the first column with red inter-connecting arrows is indexed as 0-3-6. This indicates that there two edges, one connecting diagram 0 and 3, and the other connecting diagram 3-6. Similarly the subset connected by blue arrows is indexed as 1-2-5. Note that 1-2-5 and 2-1-5 is different because the 1-2-5 contains edge 1-2 and 2-5 while 2-1-5 contains edges 1-2 and 1-5.
+
+![](images/b939ff955ed972f136fc62ca2ca38f20c2eb71ba1c81cb8075d086f336d818f2.jpg)  
+Figure 7: Two examples in PGM dataset. (a) task contains a ’Progression’ relation of the number of objects across diagrams in columns while (b) contains a ’XOR’ relation of position of objects across diagrams in rows.
+
+# E MORE DETAILS ON EULER DIAGRAM SYLLOGISM
+
+The original model in Wang et al. (2018a) uses a Siamese Conv-Net model to process two input premise diagrams and output all consistent conclusions. Convolutional layers with shared weights are first applied to two input diagrams. The top layer feature maps are then flattened and fed into a reasoning network to make predictions. We simply use CNN grid features of the top layer feature maps as object-level representations, and use the multi-layer multiplex graph to capture object relations between the two input premise diagrams. We use a multiplex edge embeddings of 4 layers, with each layer of dimension 32. The cross-multiplexing here becomes self-multiplexing as there are only 2 diagrams (Only 1 embedding of node summary for edges from first diagram to second diagram). Final node embeddings are processed by a convolutional layer to produce the final embedding, which is also fed into the reasoning network along with the conv-net embeddings.
+
+![](images/c2db232a1632fe25d692a0beb8798f602a4e8ca891866674fda34455238278cf.jpg)  
+Figure 8: Two examples in PGM dataset containing background line objects.
+
+# F ABLATION STUDY
+
+We performed ablation study experiments to test how much does the multiplex edges affects performance. We have tested two model variants, one without any graph modules, and the other model graphs using vanilla edge embeddings produced by MLPs, on PGM dataset. We found that without graph modules, the model only achieved $8 3 . 2 \%$ test accuracy. While this is lower than MXGNet’s $8 9 . 6 \%$ , it is still higher than WReN’s $7 6 . 9 \%$ . This is possibly because the search space reduction, by trimming away non-contributing subsets, allow the model to learn more efficiently. The graph model with vanilla edge embeddings achieves $8 8 . 3 \%$ accuracy, only slightly lower than MXGNet with multiplex edge embeddings. This shows that while general graph neural network is a suitable model for capturing relations between objects, the multiplex edge embedding does so more efficiently by allowing parallel relation multiplexing.
+
+# G ADDITIONAL GENERALIZATION PERFORMANCE ON PGM DATASET
+
+Table 4 shows performance of MXGNet on other splits of PGM dataset. MXGNet consistently outperforms WReN for test accuracy, except for H.O. Triple Pairs and H.O. shape-color in the case $\beta = 0$ Additionally
+
+![](images/b20325ffcd29352a51b8e65f99e0bc4aeddcd95a3bd33496918ee25d5b6a0ab3.jpg)  
+Figure 9: Illustration of diagram ordering in the matrix and numbered representation of subsets.
+
+<table><tr><td rowspan=2 colspan=1>Rank1</td><td rowspan=1 colspan=1>Diagram subsets</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>GatingVariablel</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>0-3-6</td><td rowspan=1 colspan=2>0.901</td><td rowspan=2 colspan=1></td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0-1-2</td><td rowspan=1 colspan=2>0.884</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>2-5-8</td><td rowspan=1 colspan=2>0.854</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>1-4-7</td><td rowspan=1 colspan=2>0.845</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>6-7-8</td><td rowspan=1 colspan=2>0.832</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>3-4-5</td><td rowspan=2 colspan=3>0.8120.411</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>1-2-5</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>2-1-5</td><td rowspan=1 colspan=3>0.384</td></tr><tr><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>3-6-7</td><td rowspan=1 colspan=3>0.381</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>3-7-4</td><td rowspan=1 colspan=3>0.364</td></tr><tr><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>6-3-7</td><td rowspan=1 colspan=3>0.360</td></tr><tr><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>1-5-4</td><td rowspan=1 colspan=3>0.357</td></tr><tr><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>0-4-6</td><td rowspan=1 colspan=3>0.285</td></tr><tr><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>3-4-7</td><td rowspan=1 colspan=3>0.282</td></tr><tr><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>1-3-4</td><td rowspan=2 colspan=3>0.2730.271</td></tr><tr><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>1-4-5</td></tr></table>
+
+Table 3: All subsets ranked by the absolute value of their corresponding gating variables.
+
+here we provide the analysis according to Sec 4.2 and Sec 4.6 in Barrett et al. (2018). unfortunately sec 4.3 of this paper, namely the analysis of distractors, cannot be performed as the publicly available dataset does not include any ground truth labels about distractors, nor any labels of present objects that can be used to synthesize distractor labels. For Meta-target prediction, MXG-Net achieves $8 4 . 1 \%$ accuracy. When Metatarget is correctly predicted, the model’s target prediction accuracy increases to $9 2 . 4 \%$ . When Meta-target is incorrectly predicted, the model only has $7 5 . 6 \%$ accuracy. For three logical relations the model performs best for $O R$ relation $( 9 5 . 3 \% )$ , and worst for $X O R$ relation $( 9 2 . 6 \% )$ . Accuracy for line-type tasks $( 8 6 . 5 \% )$ is only slightly better than for shape tasks $( 8 0 . 1 \% )$ , showing that object representation with graph modeling does improve on relations between shapes. The type of relation with worst performance is ConsistentU nion, with only $7 5 . 1 \%$ accuracy. This is expected as ConsistentUnion is in fact a memory task instead of relational reasoning task.
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Regime</td><td colspan="3">β=0</td><td colspan="3">β=10</td></tr><tr><td>Val.(%)</td><td>test%</td><td>Diff.</td><td>Val.(%)</td><td>test%</td><td>Diff.</td></tr><tr><td rowspan="5">WReN</td><td>H.O.Attribute Pairs</td><td>46.7</td><td>27.2</td><td>-19.5</td><td>73.4</td><td>51.7</td><td>-21.7</td></tr><tr><td>H.O. Triple Pairs</td><td>63.9</td><td>41.9</td><td>-22.0</td><td>74.5</td><td>56.3</td><td>-18.2</td></tr><tr><td>H.O. Triples</td><td>63.4</td><td>19.0</td><td>-44.4</td><td>80.0</td><td>20.1</td><td>-59.9</td></tr><tr><td>H.O.line-type</td><td>59.5</td><td>14.4</td><td>-45.1</td><td>78.1</td><td>16.4</td><td>-61.7</td></tr><tr><td>H.O. shape-color</td><td>69.3</td><td>17.2</td><td>-52.1</td><td>93.6</td><td>15.5</td><td>-78.1</td></tr><tr><td rowspan="5">MXGNet</td><td>H.O.Attribute Pairs</td><td>68.3</td><td>33.6</td><td>-34.7</td><td>81.9</td><td>69.3</td><td>-12.6</td></tr><tr><td>H.O. Triple Pairs</td><td>67.1</td><td>43.3</td><td>-23.8</td><td>78.1</td><td>64.2</td><td>-13.9</td></tr><tr><td>H.O. Triples</td><td>63.7</td><td>19.9</td><td>-43.8</td><td>80.5</td><td>20.2</td><td>-60.3</td></tr><tr><td>H.0.line-type</td><td>60.1</td><td>16.7</td><td>-43.4</td><td>85.2</td><td>16.8</td><td>-61.5</td></tr><tr><td>H.O. shape-color</td><td>68.5</td><td>16.6</td><td>-51.9</td><td>89.2</td><td>15.6</td><td>-73.6</td></tr></table>
+
+Table 4: Generalisation performance comparing MXGNet model variants against $\mathrm { W R e N }$ . ‘Diff.’ is the difference between the test and the validation performances.

parse/train/ByxQB1BKwH/ByxQB1BKwH_content_list.json

@@ -0,0 +1,1728 @@
+[
+    {
+        "type": "text",
+        "text": "ABSTRACT DIAGRAMMATIC REASONING WITH MULTIPLEX GRAPH NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            116,
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+            162
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Duo Wang ∗& Mateja Jamnik & Pietro Lio ",
+        "bbox": [
+            158,
+            186,
+            454,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science and Technology   \nUniversity of Cambridge   \nCambridge, United Kingdom   \n{Duo.Wang,Mateja.Jamnik,Pietro.Lio}@cl.cam.ac.uk ",
+        "bbox": [
+            156,
+            202,
+            624,
+            257
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            292,
+            544,
+            309
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract reasoning, particularly in the visual domain, is a complex human ability, but it remains a challenging problem for artificial neural learning systems. In this work we propose MXGNet, a multilayer graph neural network for multi-panel diagrammatic reasoning tasks. MXGNet combines three powerful concepts, namely, object-level representation, graph neural networks and multiplex graphs, for solving visual reasoning tasks. MXGNet first extracts object-level representations for each element in all panels of the diagrams, and then forms a multi-layer multiplex graph capturing multiple relations between objects across different diagram panels. MXGNet summarises the multiple graphs extracted from the diagrams of the task, and uses this summarisation to pick the most probable answer from the given candidates. We have tested MXGNet on two types of diagrammatic reasoning tasks, namely Diagram Syllogisms and Raven Progressive Matrices (RPM). For an Euler Diagram Syllogism task MXGNet achieves state-of-the-art accuracy of $9 9 . 8 \\%$ . For PGM and RAVEN, two comprehensive datasets for RPM reasoning, MXGNet outperforms the state-of-the-art models by a considerable margin. ",
+        "bbox": [
+            207,
+            319,
+            792,
+            512
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            529,
+            310,
+            544
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract reasoning has long been thought of as a key part of human intelligence, and a necessary component towards Artificial General Intelligence. When presented in complex scenes, humans can quickly identify elements across different scenes and infer relations between them. For example, when you are using a pile of different types of LEGO bricks to assemble a spaceship, you are actively inferring relations between each LEGO brick, such as in what ways they can fit together. This type of abstract reasoning, particularly in the visual domain, is a crucial key to human ability to build complex things. ",
+        "bbox": [
+            148,
+            550,
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+            633
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many tests have been proposed to measure human ability for abstract reasoning. The most popular test in the visual domain is the Raven Progressive Matrices (RPM) test (Raven (2000)). In the RPM test, the participants are asked to view a sequence of contextual diagrams, usually given as a $3 \\times 3$ matrices of diagrams with the bottom-right diagram left blank. Participants should infer abstract relationships in rows or columns of the diagram, and pick from a set of candidate answers the correct one to fill in the blank. Figures 1 (a) shows an example of RPM tasks containing XOR relations across diagrams in rows. More examples can be found in Appendix C. Another widely used test for measuring reasoning in psychology is Diagram Syllogism task (Sato et al. (2015)), where participants need to infer conclusions based on 2 given premises. Figure 1c shows an example of Euler Diagram Syllogism task. ",
+        "bbox": [
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+            641,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Barrett et al. (2018) recently published a large and comprehensive RPM-style dataset named Procedurally Generated Matrices ‘PGM’, and proposed Wild Relation Network (WReN), a state-of-the-art neural net for ",
+        "bbox": [
+            147,
+            773,
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+            801
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9f2288353f35bdf6658e8f9a6f26755f7b3970e082d7fba1eaf4cc042ef3b87.jpg",
+        "image_caption": [
+            "Figure 1: (a) shows an example of RPM tasks containing XOR relations across diagrams in rows and the overview of MXGNet architecture. Here $F _ { \\rho }$ is object representation module, $E _ { \\gamma }$ is edge embeddings module, $G _ { \\phi }$ is graph summarization module and $R _ { \\theta }$ is reasoning network. (b) shows an example of a multilayer graph formed from objects in the first row of diagrams in the example. (c) An example of syllogism represented in Euler diagrams. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            150,
+            119,
+            854,
+            556
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "RPM-style tasks. While WReN outperforms other state-of-the-art vision models such as Residual Network He et al. (2016), the performance is still far from deep neural nets’ performance on other vision or natural language processing tasks. Recently, there has been a focus on object-level representations (Yi et al. (2018); Hu et al. (2017); Hudson & Manning (2018); Mao et al. (2019); Teney et al. (2017); Zellers et al. (2018)) for visual reasoning tasks, which enable the use of inductive-biased architectures such as symbolic programs and scene graphs to directly capture relations between objects. For RPM-style tasks, symbolic programs are less suitable as these programs are generated from given questions in the Visual-Question Answering setting. In RPM-style tasks there are no explicit questions. Encoding RPM tasks into graphs is a more natural choice. However, previous works on scene graphs (Teney et al. (2017); Zellers et al. (2018)) model a single image as graphs, which is not suitable for RPM tasks as there are many different layers of relations across different subsets of diagrams in a single task. ",
+        "bbox": [
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+            669,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper we introduce MXGNet, a multi-layer multiplex graph neural net architecture for abstract diagram reasoning. Here ’Multi-layer’ means the graphs are built across different diagram panels, where each diagram is a layer. ‘Multiplex’ means that edges of the graphs encode multiple relations between different element attributes, such as colour, shape and position. Multiplex networks are discussed in detail by Kao & Porter (2018). We first tested the application of multiplex graph on a Diagram Syllogism dataset (Wang et al. (2018a)), and confirmed that multiplex graph improves performance on the original model. For RPM task, MXGNet encodes subsets of diagram panels into multi-layer multiplex graphs, and combines summarisation of several graphs to predict the correct candidate answer. With a hierarchical summarisation scheme, each graph is summarised into feature embeddings representing relationships in the subset. These relation embeddings are then combined to predict the correct answer. ",
+        "bbox": [
+            148,
+            119,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For PGM dataset (Barrett et al. (2018)), MXGNet outperforms WReN, the previous state-of-the-art model, by a considerable margin. For ‘neutral’ split of the dataset, MXGNet achieves $8 9 . 6 \\%$ test accuracy, $1 2 . 7 \\%$ higher than WReN’s $7 6 . 9 \\%$ . For other splits MXGNet consistently performs better with smaller margins. For the RAVEN dataset (Zhang et al. (2019)), MXGNet, without any auxiliary training with additional labels, achieves $8 3 . 9 1 \\%$ test accuracy, outperforming $5 9 . 5 6 \\%$ accuracy by the best model with auxiliary training for the RAVEN dataset. We also show that MXGNet is robust to variations in forms of object-level representations. Both variants of MXGNet achieve higher test accuracies than existing best models for the two datasets. ",
+        "bbox": [
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+            266,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            393,
+            318,
+            410
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Raven Progressive Matrices: Hoshen & Werman (2017) proposed a neural network model on Raven-style reasoning tasks that are a subset of complete RPM problems. Their model is based on Convolutional Network, and is demonstrated to be ineffective in complete RPM tasks (Barrett et al. (2018)). Mandziuk & Zychowski also experimented with an auto-encoder based neural net on simple single-shape RPM tasks. Barrett et al. (2018) built PGM, a complete RPM dataset, and proposed WReN, a neural network architecture based on Relation Network (Santoro et al. (2017)).Steenbrugge et al. (2018) replace CNN part of WReN with a pre-trained Variational Auto Encoder and slightly improved performance. Zhang et al. (2019) built RAVEN, a RPM-style dataset with structured labels of elements in the diagrams in the form of parsing trees, and proposed Dynamic Residual Trees, a simple tree neural network for learning with these additional structures. Anonymous (2020) applies Multi-head attention (Vaswani et al. (2017)), originally developed for Language model, on RPM tasks. ",
+        "bbox": [
+            147,
+            419,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Visual Reasoning: RPM test falls in the broader category of visual reasoning. One widely explored task of visual reasoning is Visual Question Answering(VQA). Johnson et al. (2017) built CLEVR dataset, a VQA dataset that focuses on visual reasoning instead of information retrieval in traditional VQA datasets. Current leading approaches (Yi et al. (2018); Mao et al. (2019)) on CLEVR dataset generate synthetic programs using questions in the VQA setting, and use these programs to process object-level representations extracted with objection detection models (Ren et al. (2015)). This approach is not applicable to RPM-style problems as there is no explicit question present for program synthesis. ",
+        "bbox": [
+            148,
+            578,
+            851,
+            676
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Graph Neural Networks: Recently there has been a surge of interest in applying Graph Neural Networks (GNN) for datasets that are inherently structured as graphs, such as social networks. Many variants of GNNs (Li et al. (2015); Hamilton et al. (2017); Kipf & Welling (2016); Velickovi ˇ c et al. (2017)) have been ´ proposed, which are all based on the same principle of learning feature representations of nodes by recursively aggregating information from neighbour nodes and edges. Recent methods (Teney et al. (2017); Zellers et al. (2018)) extract graph structures from visual scenes for visual question answering. These methods build scene graphs in which nodes represent parts of the scene, and edges capture relations between these parts. Such methods are only applied to scenes of a single image. For multi-image tasks such as video classification, Wang et al. (2018b) proposed non-local neural networks, which extract dense graphs where pixels in feature maps are connected to all other feature map pixels in the space-time dimensions. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 REASONING TASKS ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            118,
+            341,
+            135
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 DIAGRAM SYLLOGISM ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            143,
+            346,
+            159
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Syllogism is a reasoning task where conclusion is drawn from two given assumed propositions (premises). One well-known example is ’Socrates is a man, all man will die, therefore Socrates will die’. Syllogism can be conveniently represented using many types of diagrams (Al-Fedaghi (2017)) such as Euler diagrams and Venn diagrams. Figure 1 (c) shows an example of Euler diagram syllogism. Wang et al. (2018a) developed Euler-Net, a neural net architecture that tackles Euler diagram syllogism tasks. However Euler-Net is just a simple Siamese Conv-Net, which does not guarantee scalability to more entities in diagrams. We show that the addition of multiplex graph both improves performance and scalability to more entities. ",
+        "bbox": [
+            148,
+            166,
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+            263
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 RAVEN PROGRESSIVE MATRICES ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            277,
+            416,
+            292
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section we briefly describe Raven Progressive Matrices (RPM) in the context of the PGM dataset (Barrett et al. (2018)) and the RAVEN dataset (Zhang et al. (2019)). RPM tasks usually have 8 context diagrams and 8 answer candidates. The context diagrams are laid out in a $3 \\times 3$ matrix $\\mathbf { C }$ where $c _ { 1 , 1 } , . . c _ { 3 , 2 }$ are context diagrams and $^ { c _ { 3 , 3 } }$ is a blank diagram to be filled with 1 of the 8 answer candidates $\\mathbf { A } = \\{ a _ { 1 } , \\ldots , a _ { 8 } \\}$ One or more relations are present in rows or/and columns of the matrix. For example, in Figure 1 (a), there is $X O R$ relation of positions of objects in rows of diagrams. With the correct answer filled in, the third row and column must satisfy all relations present in the first 2 rows and columns (in the RAVEN dataset, relations are only present in rows). In addition to labels of correct candidate choice, both datasets also provide labels of meta-targets for auxiliary training. The meta-target of a task is a multi-hot vector encoding tuples of $( r , o , a )$ where $r$ is the type of a relation present, $o$ is the object type and $a$ is the attribute. For example, the meta-target for Figure 1 (a) encodes $( X O R , S h a p e , P o s i t i o n )$ . The RAVEN dataset also provides additional structured labels of relations in the diagram. However, we found that structured labels do not improve results, and therefore did not use them in our implementation. ",
+        "bbox": [
+            148,
+            299,
+            852,
+            479
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            497,
+            254,
+            513
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "MXGNet is comprised of three main components: an object-level representation module, a graph processing module and a reasoning module. Figure 1a shows an overview of the MXGNet architecture. The object-level representation module $F _ { \\rho }$ , as the name suggests, extracts representations of objects in the diagrams as nodes in a graph. For each diagram $d _ { i } \\subset \\mathbf { C } \\cup \\mathbf { A }$ , a set of nodes $v _ { i , j } ; i = 1 \\ldots L , j = 1 \\ldots N$ is extracted where $L$ is the number of layers and $N$ is the number of nodes per layer. We experimented with both fixed and dynamically learnt $N$ values. We also experimented with an additional ‘background’ encoder that encodes background lines (See Appendix C for an example containing background lines) into a single vector, which can be considered as a single node. The multiplex graph module $G _ { \\phi }$ , for a subset of diagrams, learns the multiplex edges capturing multiple parallel relations between nodes in a multi-layer graph where each layer corresponds to one diagram in the subset, as illustrated in Figure 1 (c). In MXGNet, we consider a subset of cardinality 3 for $3 \\times 3$ diagram matrices. While prior knowledge of RPM rules allows us to naturally treat rows and columns in RPM as subsets, this prior does not generalise to other types of visual reasoning problems. Considering all possible diagram combinations as subsets is computationally expensive. To tackle this, we developed a relatively quick pre-training method to greatly reduce the search space of subsets, as described below. ",
+        "bbox": [
+            147,
+            522,
+            851,
+            729
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Search Space Reduction: We can consider each diagram as node $v _ { i } ^ { d }$ in a graph, where relations between adjacent diagrams are embedded as edges $e _ { i j } ^ { d }$ . Note here we are considering the graph of ’diagrams’, which is different from the graph of ’objects’ in the graph processing modules. Each subset of 3 diagrams in this case can be considered as subset of 2 edges. We here make weak assumptions that edges exist between adjacent diagrams (including vertical, horizontal and diagonal direction) and edges in the same subset must be adjacent (defined as two edges linking the same node), which are often used in other visual reasoning problems. We denote the subset of edges as $\\{ e _ { i j } ^ { d } , e _ { j k } ^ { d } \\}$ . We use 3 neural nets to embed nodes, edges and subsets. We use CNNs to embed diagram nodes into feature vectors, and MLPs to embed edges based on node embeddings and subsets based on edge embeddings. While it is possible to include graph architectures for better accuracy, we found that simple combinations of CNNs and MLPs train faster while still achieving the search space reduction results. This architecture first embeds nodes, then embeds edges based on node embedding, and finally embed subsets based on edge embedding. The subset embeddings are summed and passed through a reasoning network to predict answer probability, similar to WReN (Barrett et al. (2018)). For the exact configuration of the architecture used please refer to Appendix A. For each subset $\\{ e _ { i j } ^ { d } , e _ { j k } ^ { d } \\}$ we define a gating variable $G _ { i j k }$ , controlling how much does each subset contributes to the final result. In practice we use tanh function, which allows a subset to contribute both positively and negatively to the final summed embeddings. In training we put L1 regularization constraint on the gating variables to suppress $G _ { i j k }$ of non-contributing subsets close to zero. This architecture can quickly discover rows and columns as contributing subsets while leaving gating variables of other subsets not activated. We describe the experiment results in section 5.1. While this method is developed for discovering reasoning rules for RPM task, it can be readily applied to any other multi-frame reasoning task for search space reduction. In the rest of the paper, we hard-gate subsets by rounding the gating variables, thereby reducing subset space to only treat rows and columns as valid subsets. ",
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+        "type": "text",
+        "text": "We treat the first 2 rows and columns as contextual subsets $c _ { i , j }$ where $i$ and $j$ are row and column indices. For the last row and column, where the answers should be filled in, we fill in each of the 8 answer candidates, and make 8 row subsets $a _ { i } , i \\subset [ 1 , 8 ]$ and 8 column subsets $a _ { i } , i \\subset [ 1 , 8 ]$ . ",
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+        "type": "text",
+        "text": "The graph module then summarises the graph of objects in a subset into embeddings representing relations present in the subset. The reasoning module $R _ { \\theta }$ takes embeddings from context rows/columns and last rows/columns with different candidate answers filled in, and produce normalised probability of each answer being true. It also predicts meta-target for auxiliary training using context rows/columns. Next, we describe each module in detail. ",
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+    {
+        "type": "text",
+        "text": "4.1 OBJECT-LEVEL REPRESENTATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the PGM dataset there are two types of objects, namely ‘shapes’ and background ‘lines’. While it is a natural choice to use object-level representation on shapes as they are varying in many attributes such as position and size, it is less efficient on background lines as they only vary in colour intensity. In this section we first describe object-level representation applied to ‘shapes’ objects, and then discuss object-level representation on ’lines’ and an alternative background encoder which performs better. ",
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+        "type": "text",
+        "text": "In MXGNet we experiment with two types of object-level representations for ‘shapes’, namely CNN grid features and representation obtained with spatial attention. For CNN grid features, we use each spatial location in the final CNN feature map as the object feature vector. Thus for each feature maps of width $W$ and height $H$ , $N = W \\times H$ object representations are extracted. This type of representation is used widely, such as in Relation Network (Santoro et al. (2017)) and VQ-VAE (van den Oord et al. (2017)). For representation obtained with attention, we use spatial attention to attend to locations of objects, and extract representations for each object attended. This is similar to objection detection models such as faster R-CNN (Ren et al. (2015)), which use a Region Proposal Network to propose bounding boxes of objects in the input image. For each attended location a presence variable $z _ { p r e s }$ is predicted by attention module indicating whether an object exists in the location. Thus the total number of objects $N$ can vary depending on the sum of $z _ { p r e s }$ variables. As object-level representation is not the main innovation of this paper, we leave exact details for Appendix A.1. ",
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+        "type": "text",
+        "text": "For background ‘lines’ objects, which are not varying in position and size, spatial attention is not needed. We experimented with a recurrent encoder with Long-Short Term Memory (Hochreiter & Schmidhuber (1997)) on the output feature map of CNN, outputting $M$ number of feature vectors. However, in the experiment we found that this performs less well than just feature map embeddings produced by feed-forward conv-net encoder. ",
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+        "type": "image",
+        "img_path": "images/b2267ee8575c5399c27e3c0ee85bad73a8f7e945bd9839fe28469263cf9348f9.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of multiplex edge embeddings and cross-gating function. Each edge contains a set of different sub-connections (colored differently). Multiplex edges connecting to each node in the last layer are aggregated according to its originating layer. Aggregated embeddings are then passed to a gating function $G$ , which outputs gating variables from each aggregated embeddings. "
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+    {
+        "type": "text",
+        "text": "4.2 MULTIPLEX GRAPH NETWORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Multiplex Edge Embedding:The object-level representation module outputs a set of representations $v _ { i , j } ; i \\subset$ $[ 1 , L ] , j \\subset [ 1 , N ]$ for ‘shapes’ objects, where $L$ is the number of layers (cardinality of subset of diagrams) and $N$ is the number of nodes per layer. MXGNet uses an multiplex edge-embedding network $E _ { \\gamma }$ to generate edge embeddings encoding multiple parallel relation embeddings: ",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a1687e353169bd629ee59a52396611eddc8a45545c0b59c486193f6a58a05615.jpg",
+        "text": "$$\ne _ { ( i , j ) , ( l , k ) } ^ { t } = E _ { \\gamma } ^ { t } ( P ^ { k } ( v _ { i , j } , v _ { l , k } ) ) ; i \\neq l , t = 1 \\dots T\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "Here $P ^ { t }$ is a projection layer projecting concatenated node embeddings to $T$ different embeddings. $E ^ { t }$ is a small neural net processing $\\mathbf { \\chi } _ { t ^ { t h } } ^ { * }$ projections to produce the $t ^ { t h }$ sub-layer of edge embeddings. Here, we restricted the edges to be inter-layer only, as we found using intra-layer edges does not improve performance but increases computational costs. Figure 2 illustrates these multiplex edge embeddings between nodes of different layers. We hypothesise that different layers of the edge embeddings encode similarities/differences in different feature spaces. Such embeddings of similarities/differences are useful in comparing nodes for subsequent reasoning tasks. For example,for P rogessive relation of object sizes, part of embeddings encoding size differences can be utilized to check if nodes in later layers are larger in size. This is similar to Mixture of Experts layers (Eigen et al. (2013); Shazeer et al. (2017)) introduced in Neural Machine Translation tasks. However, in this work we developed a new cross-multiplexing gating function at the node message aggregation stage, which is described below. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Graph Summarisation: After edge embeddings are generated, the graph module then summarises the graph into a feature embedding representing relations present in the subset of diagrams. We aggregate information in the graph to nodes of the last layer corresponding to the third diagram in a row or column, because in RPM tasks the relations are in the form $D i a g r a m 3 = F u n c t i o n ( D i a g r a m 1 , D i a g r a m 2 )$ ). All edges connecting nodes in a particular layer $v _ { i , j } ; i \\ne L$ , to a node $v _ { L , k }$ in the last layer $L$ are aggregated by a function $F _ { a g }$ composed of four different types of set operations, namely max, min, sum and mean: ",
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+        "type": "equation",
+        "img_path": "images/f3809686c1d76352d09d46260ac51cfd031a5bc30a008324ab679be2f0bf3187.jpg",
+        "text": "$$\nf v _ { i , k } = F _ { a g } ( e _ { ( i , 1 ) , ( L , k ) } \\cdot \\cdot \\cdot e _ { ( i , 1 ) , ( L , k ) } ) ; F _ { a g } = c o n c a t ( m a x ( ) , m i n ( ) , s u m ( ) , m e a n ( ) )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We use multiple aggregation functions together because different sub-tasks in reasoning may require different types of summarization. For example, counting number of objects is better suited for sum while checking if there is a object with the same size is better suited for max. The aggregated node information from each layer is then combined with a cross-multiplexing gating function. It is named ’cross-multiplexing’ because each embeddings in the set are ’multiplexing’ other embeddings in the set with gating variables that regulate which stream of information pass through. This gating function accepts a set of summarised node embeddings $\\{ f v _ { 1 , k } \\ldots f v _ { N , k } \\}$ as input, and output gating variables for each layer of node embeddings in the set: ",
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+        "page_idx": 6
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+    {
+        "type": "equation",
+        "img_path": "images/c2293941df842a206db194bc13066edd953337f1cfbcb582e83b17025f7deec1.jpg",
+        "text": "$$\n\\mathbf { g } _ { 1 , k } \\ldots \\mathbf { g } _ { N , k } = G ( f v _ { 1 , k } \\ldots f v _ { N , k } ) ; \\mathbf { g } _ { i , k } = \\{ g _ { i , k } ^ { 1 } \\ldots g _ { i , k } ^ { T } \\}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In practice $G$ is implemented as an MLP with multi-head outputs for different embeddings, and Sigmoid activation which constrains gating variable $g$ within the range of 0 to 1. The node embeddings of different layers are then multiplied with the gating variables, concatenated and passed through a small MLP to produce the final node embeddings: $f v _ { k } = M \\bar { L } P ( c o n c a t ( \\{ f v _ { i , k } \\times g _ { ( } i , k ) | \\bar { i } = 1 \\ldots N \\} )$ ). Node embeddings and background embeddings are then concatenated and processed by a residual neural block to produce final relation feature embeddings $r$ of the diagram subset. ",
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+    {
+        "type": "text",
+        "text": "4.3 REASONING NETWORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The reasoning network takes relation feature embeddings $r$ from all graphs, and infers the correct answer based on these relation embeddings. We denote the relation embeddings for context rows as $r _ { i } ^ { c r } ; i = 1 , 2$ and context columns as $r _ { i } ^ { c c } ; i = 1 , 2$ . The last row and column filled with each answer candidate $a _ { i }$ are denoted $r _ { i } ^ { a r } ; i = 1 , . . . , 8$ and $r _ { i } ^ { a c } ; i = 1 , \\ldots , 8$ . For the RAVEN dataset, only row relation embeddings $r ^ { c r }$ and $r ^ { a r }$ are used, as discussed in Section 3.2. The reasoning network $R _ { \\theta }$ is a multi-layer residual neural net with a softmax output activation that processes concatenated relation embeddings and outputs class probabilities for each answer candidate. The exact configuration of the reasoning network can be found in Appendix A.3. ",
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+    {
+        "type": "text",
+        "text": "For meta-target prediction, all relation information is contained in the context rows and columns of the RPM task. Therefore, we apply a meta-predicting network $R _ { m e t a }$ with Sigmoid output activation to all context rows and columns to obtain probabilities of each meta-target categories: ",
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+    {
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+        "img_path": "images/18ddd0fe80439d5fca4bf02dda18ff15928442fb493189c33e8f7a39eec43c98.jpg",
+        "text": "$$\np _ { m e t a } = R _ { m e t a } ( r _ { 1 } ^ { c r } + r _ { 2 } ^ { c r } + r _ { 1 } ^ { c c } + r _ { 2 } ^ { c c } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4.4 TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The full pipeline of MXGNet is end-to-end trainable with any gradient descent optimiser. In practice, we used RAdam optimiser (Liu et al. (2019)) for its fast convergence and robustness to learning rate differences. The loss function for the PGM dataset is the same as used in WReN (Barrett et al. (2018)): $\\mathcal { L } = \\mathcal { L } _ { a n s } + \\beta \\mathcal { L } _ { m e t a - t a r g e t }$ where $\\beta$ balances the training between answer prediction and meta-target prediction. For the RAVEN dataset, while the loss function can include auxiliary meta-target and structured labels as $\\mathcal { L } = \\mathcal { L } _ { a n s } + \\alpha \\mathcal { L } _ { s t r u c t } + \\beta \\mathcal { L } _ { m e t a - t a r g e t }$ , we found that both auxiliary targets do not improve performance, and thus set $\\alpha$ and $\\beta$ to 0. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 SEARCH SPACE REDUCTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The Search Space Reduction model is applied on both PGM and RAVEN dataset to reduce the subset space. After 10 epochs, only gating variables of rows and columns subset for PGM and of rows for RAVEN have value larger than 0.5. The Gating variables for three rows are 0.884, 0.812 and 0.832. The gating variables for three columns are 0.901, 0.845 and 0.854. All other gating variables are below the threshold value of 0.5. Interestingly all activated (absolute value $> 0 . 5$ ) gating variables are positive. This is possibly because it is easier for the neural net to learn an aggregation function than a comparator function. Exact experiment statistics can be found in Appendix D. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 DIAGRAM SYLLOGISM PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first test how well can the multiplex graph network capture relations for the simple Diagram Syllogism task. We simply add the multiplex graph to the original Conv-Net used in (Wang et al. (2018a)). MXGNet achieved $9 9 . 8 \\%$ accuracy on both 2-contour and 3-contour tasks, higher than the original paper’s $9 9 . 5 \\%$ and $9 9 . 4 \\%$ accuracies. The same performance on 2-contour and 3-contour tasks also show that MXGNet scales better for more entities in the diagram. For more details please refer to Appendix E. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 RPM TASK PERFORMANCES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we compare all variants of MXGNet against the state-of-the-art models for the PGM and the RAVEN datasets. For the PGM dataset, we tested against results of WReN (Barrett et al. (2018)) in the auxiliary training setting with $\\beta$ value of 10. In addition, we also compared MXGNet with VAE-WReN (Steenbrugge et al. (2018))’s result without auxiliary training. For the RAVEN dataset, we compared with WReN and ResNet model’s performance as reported in the original paper (Zhang et al. (2019)). We evaluated MXGNet with different object-level representations (Section 4.1) on the test data in the ‘neutral’ split of the PGM dataset. ",
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+    {
+        "type": "text",
+        "text": "Table 1 (a) shows test accuracies of model variants compared with WReN and VAE-WReN for the case without auxiliary training $( \\beta = 0$ ) and with auxiliary training ( $\\beta = 1 0$ ) for the PGM dataset. Both model variants of MXGNet outperform other models by a considerable margin, showing that the multi-layer graph is indeed a more suitable way to capture relations in the reasoning task. Model variants using grid features from the CNN feature maps slightly outperform model using spatial-attention-based object representations for both with and without auxiliary training settings. This is possibly because the increased number of parameters for the spatial attention variant leads to over-fitting, as the training losses of both model variants are very close. In our following experiments for PGM we will use model variants using CNN features to report performances. ",
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+    {
+        "type": "text",
+        "text": "Table 1 (b) shows test accuracies of model variants compared with $\\mathrm { W R e N }$ the best performing ResNet models for RAVEN dataset. WReN surprisingly only achieves $1 4 . 6 9 \\%$ as tested by Zhang et al. (2019). We include results of the ResNet model with or without Dynamic Residual Trees (DRT) which utilise additional structure labels of relations. We found that for the RAVEN dataset, auxiliary training of MXGNet with meta-target or structure labels does not improve performance. Therefore, we report test accuracies of models trained only with the target-prediction objective. Both variants of MXGNet significantly outperform the ResNet models. Models with spatial attention object-level representations under-perform simpler CNN features slightly, most probably due to overfitting, as the observed training losses of spatial attention models are in fact lower than CNN feature models. ",
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+    {
+        "type": "text",
+        "text": "5.4 GENERALISATION EVALUATION FOR PGM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the PGM dataset, other than the neutral data regime in which test dataset’s sampling space is the same as the training dataset, there are also other data regimes which restrict the sampling space of training or test data to evaluate the generalisation capability of a neural network. In the main paper, due to space limitations, we selected 2 representative regimes, the ‘interpolation’ regime and the ‘extrapolation’ regime to report results. For results of other data splits of PGM, please refer to Appendix G. For ‘interpolation’ regime, in the training dataset, when attribute $a = c o l o r$ and $a = s i z e$ , the values of $a$ are restricted to even-indexed values in the spectrum of $a$ values. This tests how well can a model ‘interpolate’ for missing values. For ‘Extrapolation’ regime, in the training dataset, the value of $a$ is restricted to be the lower half of the value spectrum. This tests how well can a model ‘extrapolate’ outside of the value range in the training dataset. Table 2 shows validation and test accuracies for all three data regimes with and without auxiliary training. In addition, differences between validation and test accuracies are also presented to show how well can models generalise. MXGNet models consistently perform better than WReN for all regimes tested. Interesting for ’Interpolation’ regime, while validation accuracy of MXGNet is lower than WReN, the test accuracy is higher. In addition, for regime ",
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+    {
+        "type": "table",
+        "img_path": "images/8baf640a4b95f87e0c50a905faf5b732504d547501ac423916e7f9378d7db770.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>WReNBarrett  et  al.(2018)</td><td rowspan=1 colspan=1>VAE-WReNSteenbrugge et al.(2018)</td><td rowspan=1 colspan=1>ARNeAnonymous (2020)</td><td rowspan=1 colspan=1>MXGNetCNN Sp-Attn</td></tr><tr><td rowspan=1 colspan=1>acc.(%)β= 10</td><td rowspan=1 colspan=1>76.9</td><td rowspan=1 colspan=1>N/A</td><td rowspan=1 colspan=1>88.2</td><td rowspan=1 colspan=1>89.6    88.8</td></tr><tr><td rowspan=1 colspan=1>acc.(%)β= 0</td><td rowspan=1 colspan=1>62.6</td><td rowspan=1 colspan=1>64.2</td><td rowspan=1 colspan=1>N/A</td><td rowspan=1 colspan=1>66.7    66.1</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/26f436c345fae7d1f25f41a46aae600ee21b08c216bbb654c36023da0e2d556a.jpg",
+        "table_caption": [
+            "(a) PGM ",
+            "(b) RAVEN "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>WReNZhang et al. (2019)</td><td rowspan=1 colspan=1>ResNetZhang et al. (2019)</td><td rowspan=1 colspan=1>ResNet+DRTZhang et al. (2019)</td><td rowspan=1 colspan=1>ARNeAnonymous (2020)</td><td rowspan=1 colspan=1>MXGNetCNN  Sp-Attn</td></tr><tr><td rowspan=1 colspan=1>acc. (%)</td><td rowspan=1 colspan=1>14.69</td><td rowspan=1 colspan=1>53.43</td><td rowspan=1 colspan=1>59.56</td><td rowspan=1 colspan=1>19.67</td><td rowspan=1 colspan=1>83.91   82.61</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/83aec426321134d268016c2f0dd180cb9174dddbf0baf343fba0313625842489.jpg",
+        "table_caption": [
+            "Table 1: (a) shows results comparing MXGNet model variants against $\\mathrm { W R e N }$ for the PGM dataset. (b) shows results comparing MXGNet model variants against ResNet models for the RAVEN dataset. The object-level representation has two variations which are (o1) CNN features and (o2) Spatial Attention features (Section 4.1). ",
+            "‘Interpolation’ and ‘Extrapolation’, MXGNet also shows a smaller difference between validation and test accuracy. These results show that MXGNet has better capability of generalising outside of the training space. "
+        ],
+        "table_footnote": [
+            "Table 2: Generalisation performance comparing MXGNet model variants against WReN. ‘Diff.’ is the difference between the test and the validation performances. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Regime</td><td colspan=\"3\">β=0</td><td colspan=\"3\">β=10</td></tr><tr><td>Val.(%)</td><td> test%</td><td>Diff.</td><td>Val.(%)</td><td> test%</td><td>Diff.</td></tr><tr><td rowspan=\"3\">WReN</td><td>Neutral</td><td>63.0</td><td>62.6</td><td>-0.4</td><td>77.2</td><td>76.9</td><td>-0.3</td></tr><tr><td>Interpolation</td><td>79.0</td><td>64.4</td><td>-14.6</td><td>92.3</td><td>67.4</td><td>-24.9</td></tr><tr><td>Extrapolation</td><td>69.3</td><td>17.2</td><td>-52.1</td><td>93.6</td><td>15.5</td><td>-79.1</td></tr><tr><td rowspan=\"3\">MXGNet</td><td>Neutral</td><td>67.1</td><td>66.7</td><td>-0.4</td><td>89.9</td><td>89.6</td><td>-0.3</td></tr><tr><td>Interpolation</td><td>74.2</td><td>65.4</td><td>-8.8</td><td>91.5</td><td>84.6</td><td>-6.9</td></tr><tr><td>Extrapolation</td><td>69.1</td><td>18.9</td><td>-50.2</td><td>94.3</td><td>18.4</td><td>-75.9</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We presented MXGNet, a new graph-based approach to diagrammatic reasoning problems in the style of Raven Progressive Matrices (RPM). MXGNet combines three powerful ideas, namely, object-level representation, graph neural networks and multiplex graphs, to capture relations present in the reasoning task. Through experiments we showed that MXGNet performs better than previous models on two RPM datasets. We also showed that MXGNet has better generalisation performance. ",
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+    {
+        "type": "text",
+        "text": "One important direction for future work is to make MXGNet interpretable, and thereby extract logic rules from MXGNet. Currently, the learnt representations in MXGNet are still entangled, providing little in the way of understanding its mechanism of reasoning. Rule extraction can provide people with better understanding of the reasoning problem, and may allow neural networks to work seamlessly with more programmable traditional logic engines. ",
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+    {
+        "type": "text",
+        "text": "While the multi-layer multiplex graph neural network is designed for RPM style reasoning task, it can be readily extended to other diagrammatic reasoning tasks where relations are present between multiple elements across different diagrams. One example of a real-world application scenario is robots assembling parts of an object into a whole, such as building a LEGO model from a room of LEGO blocks. MXGNet provides a suitable way of capturing relations between parts, such as ways of piecing and locking two parts together. ",
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+    {
+        "type": "text",
+        "text": "A ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we present exact configurations of all model variants of MXGNet. Due to the complexity of architectures, we will describe each modules in sequence. The object-level representation has two variations which are (o1) CNN features and (o2) Spatial Attention features. Also the models for PGM and RAVEN dataset differ in details. Unless otherwise stated, in all layers we apply Batch Normalization Ioffe & Szegedy (2015) and use Rectified Linear Unit as activation function. ",
+        "bbox": [
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+        "page_idx": 12
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+    {
+        "type": "text",
+        "text": "A.1 OBJECT-LEVEL REPRESENTATION ARCHITECTURE ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "CNN features: The first approach applies a CNN on the input image and use each spatial location in the final CNN feature map as the object feature vector. This type of representation is used widely, such as in Relation Network Santoro et al. (2017) and VQ-VAE van den Oord et al. (2017). Formally, the output of a CNN is a feature map tensor of dimension $H \\times W \\times D$ where $H$ , $W$ and $D$ are respectively height, width and depth of the feature map. At each $H$ and $W$ location, an object vector is extracted. This type of object representation is simple and fast, but does not guarantee that the receptive field at each feature map location fully bounds objects in the image. ",
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+    {
+        "type": "text",
+        "text": "We use a residual module He et al. (2016) with two residual blocks to extract CNN features, as shown in figure 4.This is because Residual connections show better performance in experiments. The structure of a single Residual Convolution Block is shown in figure 3.Unless otherwise stated, convolutional layer in residual blocks has kernel size of $3 \\times 3$ . The output feature map processed by another residual block is treated as background encoding because we found that convolutional background encoding gives better results than feature vectors. ",
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+    {
+        "type": "image",
+        "img_path": "images/a2c5ce0d4847ea7b46bb33da416635d22b4c956b822b83e6fce0f13406826ad0.jpg",
+        "image_caption": [
+            "Figure 3: Architecture of a single Residual Convolution Block. "
+        ],
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+        "type": "text",
+        "text": "Spatial Attention Object-level representation: The second approach is to use spatial attention to attend to locations of objects, and extract representations for each object attended. This is similar to object detection models such as faster R-CNN Ren et al. (2015), which use a Region Proposal Network to propose bounding boxes of objects in the input image. In practice, we use Spatial Transformer Jaderberg et al. (2015) as our spatial attention module. Figure 5 shows the architecture used for extracting object-level representation using spatial attention. A CNN composed of 1 conv layr and 2 residual blocks is first applied to the input image, and the last layer feature map is extracted. This part is the same as CNN grid feature module. A spatial attention network composed of 2 conv layer then processes information at each spatial location on the feature map, and outputs $k$ numbers of $z = ( \\dot { z } ^ { p r e s } , z ^ { w \\hat { h } e r e } )$ , corresponding to $k$ possible objects at each location. Here, $z ^ { p r e s }$ is a binary value indicating if an object exists in this location, and $z ^ { w h e r \\bar { e } }$ is an affine transformation matrix specifying a sampling region on the feature maps. $z ^ { p r e s }$ , the binary variable, is sampled from Gumbel-Sigmoid distribution Maddison et al. (2016); Jang et al. (2016), which approximates the Bernoulli distribution. We set Gumbel temperature to 0.7 throughout the experiments. For the PGM dataset we restricted $k$ to be 1 and $z ^ { w h e r e }$ to be a translation and scaling matrix as ‘shapes’ objects do not overlap and do not have affine transformation attributes other than scaling and translation. For all $z _ { i } ; i \\subset [ 1 , H \\times \\bar { W } ]$ , if $z _ { i } ^ { p r e s }$ is 1, an object encoder network samples a patch from location specified by $z _ { i } ^ { w h e r e }$ using a grid sampler with a fixed window size of pixels. More details of the grid sampler can be found in Jaderberg et al. (2015). The sampled patches are then processed by a conv-layer to generate object embeddings. ",
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+        "image_caption": [
+            "Figure 4: CNN feature object-level representation module. ’Conv’ is convolution layers, ’Max-Pooling’ is max-pooling layer and ’ResConv Block’ is Residual Convolutional Block. "
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+        "image_caption": [
+            "Figure 5: Spatial attention based feature object-level representation module. ’Conv’ is convolution layers, ’Max-Pooling’ is max-pooling layer and ’ResConv Block’ is Residual Convolutional Block. $z$ is the spatial attention variable $( z ^ { p r e \\bar { s } } , z ^ { w h \\bar { e } r e } )$ . Sampler is a grid sampler which samples grid of points from given feature maps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            145,
+            479,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 GRAPH NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            757,
+            328,
+            771
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Multiplex Edge Embeddings:Figure 2 in the main paper shows an overview of the multiplex graph architecture. While motivation and overview of architecture is explained in section 4.2 of the main paper, in this section we provide exact configurations for each part of the model. Each sub-layer of the multiplex edge is embedded by a small MLP. For PGM dataset, we use 6 parallel layers for each multiplex edge embeddings , with each layer having 32 hidden units and 8 output units. For RAVEN dataset we use 4 layers with 16 hidden units and 8 output units because RAVEN dataset contains fewer relations types than PGM dataset. Gating function is implemented as one Sigmoid fully connected layer with hidden size equal to the length of concatenated aggregated embeddings. Gating variables are element-wise multiplied with concatenated embeddings for gating effects. Gated embeddings are then processed with a final fully connected layer with hidden size 64. ",
+        "bbox": [
+            148,
+            780,
+            852,
+            823
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            119,
+            851,
+            218
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Graph Summarization: This module summarizes all node summary embeddings and background embeddings to produce a diagram subset embedding representing relations present in the set of diagrams. We experimented with various approaches and found that keeping embeddings as feature maps and processing them with residual blocks yields the best results. Background feature map embeddings are generated with one additional residual block of 48 on top of lower layer feature-extracting resnet. For object representations obtained from CNN-grid features, we can simply reshape node embeddings into a feature map, and process it with additional conv-nets to generate a feature map embeddings of the same dimension to background feature map embeddings. For object representations with spatial attention, we can use another Spatial Transformer to write node summary embeddings to its corresponding locations on a canvas feature map. Finally we concatenate node summary embeddings and background embeddings and process it with 2 residual blocks of size 64 to produce the relation embeddings. ",
+        "bbox": [
+            147,
+            231,
+            851,
+            385
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.3 REASONING NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            419,
+            354,
+            433
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows the reasoning network configuration for RPM tasks. We experimented with the approach introduced in Barrett et al. (2018), which compute scores for each answer candidates and finally normalize the scores. We found this approach leads to severe overfitting on the RAVEN dataset, and therefore used a simpler approach to just concatenate all relation embeddings and process them with a neural net. In practice we used two residual blocks of size 128 and 256, and a final fully connected layer with 8 units corresponding to 8 answer candidates. The output is normalized with softmax layer. For Meta-target prediction, all context relation embeddings (context rows and columns for PGM while only rows for RAVEN dataset) are summed and fed into a fully connected prediction layer with Sigmoid activation. For PGM there are 12 different meta-targets while for RAVEN there are 9. ",
+        "bbox": [
+            147,
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+            579
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/2aa33a733d1badb8845c69ca5558e0454d9221b714de16f54361ff559e995da9.jpg",
+        "image_caption": [
+            "Figure 6: Architecture overview of reasoning module. ’RelEmbed’ is relation embeddings, ’Concat’ is concatenation layer. ’ResBlock’ is Residual Convolutional Block. ’FC’ is fully connected layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            215,
+            599,
+            782,
+            772
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            118,
+            348,
+            135
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The architecture is implemented in Pytorch framework. During training, we used RAdam optimizer Liu et al. (2019) with learning rate 0.0001, $\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 9 .$ We used batch size of 64, and distributed the training across 2 Nvidia Geforce Titan X GPUs. We early-stop training when validation accuracy stops increasing. ",
+        "bbox": [
+            148,
+            146,
+            852,
+            189
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C MORE DETAILS OF RPM DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            210,
+            482,
+            228
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In PGM dataset there are two types of elements present in the diagram, namely shapes and lines. These elements have different attributes such as colour and size. In the PGM dataset, five types of relations can be present in the task: $\\{ P r o g r e s s i o n , A N D , O R , X O R , C o n s i s t e n t U n i o n \\}$ . The RAVEN dataset, compared to PGM, does not have logic relations $A N D , O R , X O R$ , but has additional relations Arithmetic, Constant. In addition RAVEN dataset only allow relations to be present in rows. ",
+        "bbox": [
+            148,
+            239,
+            852,
+            309
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 7a and 7b show two examples from the PGM dataset(Image courtesy Barrett et al. (2018)). The first example contains a ’Progression’ relation of the number of objects across diagrams in columns. The second examples contains a ’XOR’ relation of position of objects across diagrams in rows. ",
+        "bbox": [
+            148,
+            318,
+            851,
+            361
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In addition to shape objects, diagrams in the PGM dataset can also contain background line objects that appear at fixed locations. Figure 8a and 8b show two examples of PGM tasks containing line objects. ",
+        "bbox": [
+            147,
+            369,
+            849,
+            397
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D MORE DETAILS ON SEARCH SPACE REDUCTION ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            420,
+            581,
+            436
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this section we provide detailed architecture used for Search Space reduction, and present additional experimental results. ",
+        "bbox": [
+            148,
+            446,
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+            477
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The node embeddings are generated by applying a Conv-Net of 4 convolutional layer (32 filters in each layer) of kernel size 3, and a fully connected layer mapping flattened final-layer feature maps to a feature vector of size 256. Edge embeddings are generated by a 3-layer MLP of $5 1 2 - 5 1 2 - 2 5 6$ hidden units. Subset embeddings are generated by a fully connected layer of 512 units. The subset embeddings are gated with the gating variables and summed into a feature vector, which is then feed into the reasoning net, a 3-layer MLP with $2 5 6 - 2 5 6 - 1 3$ . The output layer contains 13 units. The first unit gives probability of currently combined answer choice being true. The rest 12 units give meta-target prediction probabilities. This is the same as Barrett et al. (2018). The training loss function is: ",
+        "bbox": [
+            147,
+            484,
+            851,
+            597
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b166c91f90b76a4740166f98c721ee78ffdc65d726b039785d74de0d322adac2.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { a n s } + \\beta \\mathcal { L } _ { m e t a - t a r g e t } + \\lambda \\left\\| \\sum _ { ( i , j , k ) \\subset S } G _ { i , j , k } \\right\\| _ { L 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            617,
+            673,
+            669
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In our experiment we have tested various values of $\\lambda$ , and found 0.01 to be the best. This model is trained with RAdam optimizer with learning rate of 0.0001 and batch size of 64. After 10 epochs of training, only gating variables of subsets that are rows and columns are above the 0.5 threshold. The Gating variables for three rows are 0.884, 0.812 and 0.832. The gating variables for three columns are 0.901, 0.845 and 0.854. All other gating variables are below 0.5. Among these, the one with highest absolute value is 0.411. Table 3 shows the top-16 ranked subsets, with each subset indexed by 2 connecting edges in the subset. Figure 9 illustrates this way of indexing the subset. For example, the first column with red inter-connecting arrows is indexed as 0-3-6. This indicates that there two edges, one connecting diagram 0 and 3, and the other connecting diagram 3-6. Similarly the subset connected by blue arrows is indexed as 1-2-5. Note that 1-2-5 and 2-1-5 is different because the 1-2-5 contains edge 1-2 and 2-5 while 2-1-5 contains edges 1-2 and 1-5. ",
+        "bbox": [
+            147,
+            683,
+            851,
+            823
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b939ff955ed972f136fc62ca2ca38f20c2eb71ba1c81cb8075d086f336d818f2.jpg",
+        "image_caption": [
+            "Figure 7: Two examples in PGM dataset. (a) task contains a ’Progression’ relation of the number of objects across diagrams in columns while (b) contains a ’XOR’ relation of position of objects across diagrams in rows. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            151,
+            119,
+            846,
+            569
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "E MORE DETAILS ON EULER DIAGRAM SYLLOGISM ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            656,
+            598,
+            672
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The original model in Wang et al. (2018a) uses a Siamese Conv-Net model to process two input premise diagrams and output all consistent conclusions. Convolutional layers with shared weights are first applied to two input diagrams. The top layer feature maps are then flattened and fed into a reasoning network to make predictions. We simply use CNN grid features of the top layer feature maps as object-level representations, and use the multi-layer multiplex graph to capture object relations between the two input premise diagrams. We use a multiplex edge embeddings of 4 layers, with each layer of dimension 32. The cross-multiplexing here becomes self-multiplexing as there are only 2 diagrams (Only 1 embedding of node summary for edges from first diagram to second diagram). Final node embeddings are processed by a convolutional layer to produce the final embedding, which is also fed into the reasoning network along with the conv-net embeddings. ",
+        "bbox": [
+            147,
+            695,
+            852,
+            823
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/c2db232a1632fe25d692a0beb8798f602a4e8ca891866674fda34455238278cf.jpg",
+        "image_caption": [
+            "Figure 8: Two examples in PGM dataset containing background line objects. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            166,
+            117,
+            831,
+            549
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "F ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            601,
+            331,
+            617
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We performed ablation study experiments to test how much does the multiplex edges affects performance. We have tested two model variants, one without any graph modules, and the other model graphs using vanilla edge embeddings produced by MLPs, on PGM dataset. We found that without graph modules, the model only achieved $8 3 . 2 \\%$ test accuracy. While this is lower than MXGNet’s $8 9 . 6 \\%$ , it is still higher than WReN’s $7 6 . 9 \\%$ . This is possibly because the search space reduction, by trimming away non-contributing subsets, allow the model to learn more efficiently. The graph model with vanilla edge embeddings achieves $8 8 . 3 \\%$ accuracy, only slightly lower than MXGNet with multiplex edge embeddings. This shows that while general graph neural network is a suitable model for capturing relations between objects, the multiplex edge embedding does so more efficiently by allowing parallel relation multiplexing. ",
+        "bbox": [
+            147,
+            626,
+            852,
+            752
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "G ADDITIONAL GENERALIZATION PERFORMANCE ON PGM DATASET ",
+        "text_level": 1,
+        "bbox": [
+            151,
+            768,
+            743,
+            785
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Table 4 shows performance of MXGNet on other splits of PGM dataset. MXGNet consistently outperforms WReN for test accuracy, except for H.O. Triple Pairs and H.O. shape-color in the case $\\beta = 0$ Additionally ",
+        "bbox": [
+            147,
+            794,
+            848,
+            823
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/b20325ffcd29352a51b8e65f99e0bc4aeddcd95a3bd33496918ee25d5b6a0ab3.jpg",
+        "image_caption": [
+            "Figure 9: Illustration of diagram ordering in the matrix and numbered representation of subsets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            321,
+            117,
+            676,
+            392
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/dc8f576bee682d3e2e1df310c7137ce83d8b1cf8263de0d6ecee1ff2364afe0e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Rank1</td><td rowspan=1 colspan=1>Diagram subsets</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>GatingVariablel</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>0-3-6</td><td rowspan=1 colspan=2>0.901</td><td rowspan=2 colspan=1></td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0-1-2</td><td rowspan=1 colspan=2>0.884</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>2-5-8</td><td rowspan=1 colspan=2>0.854</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>1-4-7</td><td rowspan=1 colspan=2>0.845</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>6-7-8</td><td rowspan=1 colspan=2>0.832</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>3-4-5</td><td rowspan=2 colspan=3>0.8120.411</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>1-2-5</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>2-1-5</td><td rowspan=1 colspan=3>0.384</td></tr><tr><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>3-6-7</td><td rowspan=1 colspan=3>0.381</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>3-7-4</td><td rowspan=1 colspan=3>0.364</td></tr><tr><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>6-3-7</td><td rowspan=1 colspan=3>0.360</td></tr><tr><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>1-5-4</td><td rowspan=1 colspan=3>0.357</td></tr><tr><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>0-4-6</td><td rowspan=1 colspan=3>0.285</td></tr><tr><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>3-4-7</td><td rowspan=1 colspan=3>0.282</td></tr><tr><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>1-3-4</td><td rowspan=2 colspan=3>0.2730.271</td></tr><tr><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>1-4-5</td></tr></table>",
+        "bbox": [
+            333,
+            433,
+            663,
+            671
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Table 3: All subsets ranked by the absolute value of their corresponding gating variables. ",
+        "bbox": [
+            202,
+            680,
+            789,
+            695
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "here we provide the analysis according to Sec 4.2 and Sec 4.6 in Barrett et al. (2018). unfortunately sec 4.3 of this paper, namely the analysis of distractors, cannot be performed as the publicly available dataset does not include any ground truth labels about distractors, nor any labels of present objects that can be used to synthesize distractor labels. For Meta-target prediction, MXG-Net achieves $8 4 . 1 \\%$ accuracy. When Metatarget is correctly predicted, the model’s target prediction accuracy increases to $9 2 . 4 \\%$ . When Meta-target is incorrectly predicted, the model only has $7 5 . 6 \\%$ accuracy. For three logical relations the model performs best for $O R$ relation $( 9 5 . 3 \\% )$ , and worst for $X O R$ relation $( 9 2 . 6 \\% )$ . Accuracy for line-type tasks $( 8 6 . 5 \\% )$ is only slightly better than for shape tasks $( 8 0 . 1 \\% )$ , showing that object representation with graph modeling does improve on relations between shapes. The type of relation with worst performance is ConsistentU nion, with only $7 5 . 1 \\%$ accuracy. This is expected as ConsistentUnion is in fact a memory task instead of relational reasoning task. ",
+        "bbox": [
+            147,
+            723,
+            851,
+            823
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            119,
+            852,
+            176
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/779663e6ae3a445d72ca96f0ad278668a98a439d9d7d9877265f93a11d1f823b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Regime</td><td colspan=\"3\">β=0</td><td colspan=\"3\">β=10</td></tr><tr><td>Val.(%)</td><td>test%</td><td>Diff.</td><td>Val.(%)</td><td>test%</td><td>Diff.</td></tr><tr><td rowspan=\"5\">WReN</td><td>H.O.Attribute Pairs</td><td>46.7</td><td>27.2</td><td>-19.5</td><td>73.4</td><td>51.7</td><td>-21.7</td></tr><tr><td>H.O. Triple Pairs</td><td>63.9</td><td>41.9</td><td>-22.0</td><td>74.5</td><td>56.3</td><td>-18.2</td></tr><tr><td>H.O. Triples</td><td>63.4</td><td>19.0</td><td>-44.4</td><td>80.0</td><td>20.1</td><td>-59.9</td></tr><tr><td>H.O.line-type</td><td>59.5</td><td>14.4</td><td>-45.1</td><td>78.1</td><td>16.4</td><td>-61.7</td></tr><tr><td>H.O. shape-color</td><td>69.3</td><td>17.2</td><td>-52.1</td><td>93.6</td><td>15.5</td><td>-78.1</td></tr><tr><td rowspan=\"5\">MXGNet</td><td>H.O.Attribute Pairs</td><td>68.3</td><td>33.6</td><td>-34.7</td><td>81.9</td><td>69.3</td><td>-12.6</td></tr><tr><td>H.O. Triple Pairs</td><td>67.1</td><td>43.3</td><td>-23.8</td><td>78.1</td><td>64.2</td><td>-13.9</td></tr><tr><td>H.O. Triples</td><td>63.7</td><td>19.9</td><td>-43.8</td><td>80.5</td><td>20.2</td><td>-60.3</td></tr><tr><td>H.0.line-type</td><td>60.1</td><td>16.7</td><td>-43.4</td><td>85.2</td><td>16.8</td><td>-61.5</td></tr><tr><td>H.O. shape-color</td><td>68.5</td><td>16.6</td><td>-51.9</td><td>89.2</td><td>15.6</td><td>-73.6</td></tr></table>",
+        "bbox": [
+            181,
+            189,
+            813,
+            363
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Generalisation performance comparing MXGNet model variants against $\\mathrm { W R e N }$ . ‘Diff.’ is the difference between the test and the validation performances. ",
+        "bbox": [
+            147,
+            380,
+            849,
+            410
+        ],
+        "page_idx": 19
+    }
+]

parse/train/Wga_hrCa3P3/Wga_hrCa3P3.md

@@ -0,0 +1,615 @@
+# CONTRASTIVE LEARNING WITH ADVERSARIAL PERTURBATIONS FOR CONDITIONAL TEXT GENERATION
+
+Seanie Lee1∗, Dong Bok Lee1∗, Sung Ju Hwang1,2 KAIST1, AITRICS2, South Korea {lsnfamily02, markhi, sjhwang82}@kaist.ac.kr
+
+# ABSTRACT
+
+Recently, sequence-to-sequence (seq2seq) models with the Transformer architecture have achieved remarkable performance on various conditional text generation tasks, such as machine translation. However, most of them are trained with teacher forcing with the ground truth label given at each time step, without being exposed to incorrectly generated tokens during training, which hurts its generalization to unseen inputs, that is known as the “exposure bias” problem. In this work, we propose to mitigate the conditional text generation problem by contrasting positive pairs with negative pairs, such that the model is exposed to various valid or incorrect perturbations of the inputs, for improved generalization. However, training the model with na¨ıve contrastive learning framework using random non-target sequences as negative examples is suboptimal, since they are easily distinguishable from the correct output, especially so with models pretrained with large text corpora. Also, generating positive examples requires domain-specific augmentation heuristics which may not generalize over diverse domains. To tackle this problem, we propose a principled method to generate positive and negative samples for contrastive learning of seq2seq models. Specifically, we generate negative examples by adding small perturbations to the input sequence to minimize its conditional likelihood, and positive examples by adding large perturbations while enforcing it to have a high conditional likelihood. Such “hard” positive and negative pairs generated using our method guides the model to better distinguish correct outputs from incorrect ones. We empirically show that our proposed method significantly improves the generalization of the seq2seq on three text generation tasks — machine translation, text summarization, and question generation.
+
+# 1 INTRODUCTION
+
+The sequence-to-sequence (seq2seq) models (Sutskever et al., 2014), which learn to map an arbitrary-length input sequence to another arbitrary-length output sequence, have successfully tackled a wide range of language generation tasks. Early seq2seq models have used recurrent neural networks to encode and decode sequences, leveraging attention mechanism (Bahdanau et al., 2015) that allows the decoder to attend to a specific token in the input sequence to capture long-term dependencies between the source and target sequences. Recently, the Transformer (Vaswani et al., 2017), which is an all-attention model that effectively captures long-term relationships between tokens in the input sequence as well as across input and output sequences, has become the de facto standard for most of the text generation tasks due to its impressive performance. Moreover, Transformerbased language models trained on large text corpora (Dong et al., 2019; Raffel et al., 2020; Lewis et al., 2020) have shown to significantly improve the model performance on text generation tasks.
+
+However, a crucial limitation of seq2seq models is that they are mostly trained only with teacher forcing, where ground truth is provided at each time step and thus never exposed to incorrectly generated tokens during training (Fig. 1-(a)), which hurts its generalization. This problem is known as the “exposure bias” problem (Ranzato et al., 2016) and often results in the generation of lowquality texts on unseen inputs. Several prior works tackle the problem, such as using reinforcement learning (RL) to maximize non-differentiable reward (Bahdanau et al., 2017; Paulus et al., 2018).
+
+![](images/3152b03d6604aaa99473aaf4ecace4d924367395828fc4b243fd8556ad897871.jpg)  
+Figure 1: Concept. (a) Training seq2seq with teacher forcing. (b) Na¨ıve contrastive learning with randomly sampled negative examples. (c) Our method, CLAPS, which generates hard negative and positive examples.
+
+Another approach is to use RL or gumbel softmax (Jang et al., 2017) to match the distribution of generated sentences to that of the ground truth, in which case the reward is the discriminator output from a Generative Adversarial Network (GAN) (Zhang et al., 2018; 2017; Yu et al., 2017). Although the aforementioned approaches improve the performance of the seq2seq models on text generation tasks, they either require a vast amount of effort in tuning hyperparameters or stabilize training.
+
+In this work, we propose to mitigate the exposure bias problem with a simple yet effective approach, in which we contrast a positive pair of input and output sequence to negative pairs, to expose the model to various valid or incorrect sentences. Na¨ıvely, we can construct negative pairs by simply using random nontarget sequences from the batch (Chen et al., 2020). However, such a na¨ıve construction yields meaningless negative examples that are already well-discriminated in the embedding space (Fig. 1-(b)), which we highlight as the reason why existing methods (Chen et al., 2020) require large batch size. This is clearly shown in Fig. 2, where a large portion of positive-negative pairs can be easily discriminated without any training, which gets worse as the batch size decreases as it will reduce the chance to have meaningfully difficult examples in the batch. Moreover, discriminating positive and na¨ıve negative pairs becomes even more easier for models pretrained on large text corpora.
+
+![](images/876f3fbd65a3f50ad0917ec686970cfba6abaa84b103c8f8f2f38c365038e298.jpg)  
+Figure 2: Accuracy of classifying a positive pair from negative pairs varying batch size without any training.
+
+To resolve this issue, we propose principled approaches to automatically generate negative and positive pairs for constrastive learning, which we refer to as Contrastive Learning with Adversarial Perturbation for Seq2seq learning (CLAPS). Specifically, we generate a negative example by adding a small perturbation to the hidden representation of the target sequence, such that its conditional likelihood is minimized (Denoted as the red circle in Fig. 1-(c)). Conversely, we construct an additional positive example (Denoted as green circle in Fig. 1-(c)) by adding a large amount of perturbation to the hidden representation of target sequence such that the perturbed sample is far away from the source sequence in the embedding space, while enforcing it to have high conditional likelihood by minimizing Kullback-Leibler (KL) divergence between the original conditional distribution and perturbed conditional distribution. This will yield a negative example that is very close to the original representation of target sequence in the embedding space but is largely dissimilar in the semantics, while the generated positive example is far away from the original input sequence but has the same semantic as the target sequence. This will generate difficult examples that the model fails to correctly discriminate (Fig. 1-(c), Fig.2), helping it learn with more meaningful pairs.
+
+To verify the efficacy of our method, we empirically show that it significantly improves the performance of seq2seq model on three conditional text generation tasks, namely machine translation, text summarization and question generation. Our contribution in this work is threefold:
+
+• To mitigate the exposure bias problem, we propose a contrastive learning framework for conditional sequence generation, which contrasts a positive pair of source and target sentence to negative pairs in the latent embedding space, to expose the model to various valid or incorrect outputs. • To tackle the ineffectiveness of conventional approach for constructing negative and positive examples for contrastive learning, we propose a principled method to automatically generate negative and positive pairs, that are more difficult and allows to learn more meaningful representations.
+
+• We show that our proposed method, CLAPS, significantly improves the performance of seq2seq model on three different tasks: machine translation, text summarization, and question generation.
+
+# 2 RELATED WORK
+
+Exposure Bias There are several prior works to tackle the exposure bias (Ranzato et al., 2016). Bengio et al. (2015) introduce scheduled sampling where the model is initially guided with the true previous tokens but uses the tokens generated by the seq2seq model as the conditional input for the next token, as training goes on. Paulus et al. (2018); Bahdanau et al. (2017) leverage RL to maximize non-differentiable rewards, so it enables to penalize the model for incorrectly generated sentences. Another works (Zhang et al., 2017; 2018; Yu et al., 2017) train GANs to match the distribution of generated sequences to that of ground truth. Since sampling tokens from the generator is not differentiable, they resort RL or gumbel-softmax to train the networks in end-to-end fashion. However, they require either a large amount of effort to tune hyperparameters or stabilize training. However, Choshen et al. (2020) show that RL for machine translation does not optimize the expected reward and the performance gain is attributed to the unrelated effects such as increasing the peakiness of the output distribution. Moreover, (Caccia et al., 2019) show that by tuning the temperature parameter, the language models trained with MLE can be tuned to outperform GAN-based text generation models.
+
+Adversarial Perturbation Many existing works, such as (Madry et al., 2018), address the robustness of neural networks to adversarial examples, which are generated by applying a small perturbations to the input samples. While adversarial robustness has been mostly explored in image domains, Miyato et al. (2017) adopted adversarial training to text domains. However instead of targeting robustness to perturbed samples, they utilize the adversarial examples as augmented data, and enforce consistency across the predictions across original unlabeled example and its perturbation, for semisupervised learning. Recently Zhu et al. (2019) and Jiang et al. (2020) leverage adversarial training to induce the smoothness of text classifiers, to prevent overfitting to training samples. While they are relevant to ours, these methods do not have the notion of positive and negative examples as they do not consider contrastive learning, and only target text classification. Moreover, they are computationally prohibitive since they use PGD for adversarial training, which requires iterative optimization for each individual sample. Recently, Aghajanyan et al. (2020) propose a simpler yet effective method based on Gaussian noise perturbation to regularize neural networks without expensive PGD steps, which is shown to outperform methods from Zhu et al. (2019) and Jiang et al. (2020). Although our work is similar to these prior works in that we add perturbations to the text embeddings, note that we used the adversarially-generated samples as negative examples of our contrastive learning framework rather than trying to learn the model to be robust to them.
+
+Contrastive Learning Contrastive learning has been widely used. It is to learn a representation by contrasting positive pairs and negative pairs. Chopra et al. (2005); Weinberger & Saul (2009); Schroff et al. (2015) leverage a triplet loss to separate positive examples from negative examples in metric learning. Chen et al. (2020) show that contrastive learning can boost the performance of selfsupervised and semi-supervised learning in computer vison tasks. In natural language processing (NLP), contrastive learning has been widely used. In Word2Vec (Mikolov et al., 2013), neighbouring words are predicted from context with noise-contrastive estimation (Gutmann & Hyvarinen, 2012). ¨ Beyond word representation, Logeswaran & Lee (2018) sample two contiguous sentences for positive pairs and the sentences from other document as negative pairs. They constrast positive and negative pairs to learn sentence representation. Moreover, contrastive learning has been investigated in various NLP tasks — language modeling (Huang et al., 2018), unsupervised word alignment (Liu & Sun, 2015), caption generation (Mao et al., 2016; Vedantam et al., 2017), and machine translation (Yang et al., 2019).
+
+# 3 METHOD
+
+# 3.1 BACKGROUND: CONDITIONAL TEXT GENERATION
+
+The goal of conditional text generation with a seq2seq model is to generate an output text sequence $\mathbf { y } ^ { ( i ) } = ( y _ { 1 } ^ { ( i ) } , \dots , y _ { T } ^ { ( i ) } )$ with length $T$ conditioned on the input text sequence $\mathbf { x } ^ { ( i ) } \stackrel { - } { = } ( x _ { 1 } ^ { ( i ) } , \cdot \cdot \cdot , x _ { L } ^ { ( i ) } )$
+
+with length $L$ . A typical approach to the conditional text generation is to leverage the encoderdecoder architecture to parameterize the conditional distribution. We maximize the conditional log likelihood $\log p _ { \theta } ( \mathbf { y } | \mathbf { x } )$ for a given $N$ observations $\{ ( \mathbf { x } ^ { ( i ) } , \mathbf { y } ^ { ( i ) } ) \} _ { i = 1 } ^ { N }$ as follows:
+
+$$
+\begin{array} { c } { { \displaystyle \mathcal { L } _ { M L E } ( \theta ) = \sum _ { i = 1 } ^ { N } \log p _ { \theta } ( \mathbf { y } ^ { ( i ) } | \mathbf { x } ^ { ( i ) } ) } } \\ { { { } } } \\ { { \displaystyle p _ { \theta } ( y _ { 1 } ^ { ( i ) } , \dots , y _ { T } ^ { ( i ) } | \mathbf { x } ^ { ( i ) } ) = \prod _ { t = 1 } ^ { T } p _ { \theta } ( y _ { t } ^ { ( i ) } | \mathbf { y } _ { < t } ^ { ( i ) } , \mathbf { x } ^ { ( i ) } ) } } \\ { { { } } } \\ { { \displaystyle p _ { \theta } ( y _ { t } ^ { ( i ) } | \mathbf { y } _ { < t } ^ { ( i ) } , \mathbf { x } ^ { ( i ) } ) = \mathrm { s o f t m a x } ( \mathbf { W h } _ { t } ^ { ( i ) } + \mathbf { b } ) } } \\ { { { } } } \\ { { { \mathbf { h } } _ { t } ^ { ( i ) } = g ( y _ { t - 1 } ^ { ( i ) } , \mathbf { M } ^ { ( i ) } ; \theta ) , ~ \mathbf { M } ^ { ( i ) } = f ( \mathbf { x } ^ { ( i ) } ; \theta ) } } \end{array}
+$$
+
+where $f , g$ denote the encoder and the decoder respectively and $\mathbf { M } ^ { ( i ) } = [ \mathbf { m } _ { 1 } ^ { ( i ) } \cdot \cdot \cdot \mathbf { m } _ { L } ^ { ( i ) } ] \in \mathbb { R } ^ { d \times L }$ is the concatenation of the hidden representations of the source tokens $\mathbf { x } ^ { ( i ) }$ .
+
+# 3.2 CONTRASTIVE LEARNING WITH ADVERSARIAL PERTURBATIONS FOR SEQ2SEQ
+
+Since most of the seq2seq models are trained with teacher forcing where the ground truth tokens are provided to maximize Eq. 1, they are never exposed to incorrectly generated tokens during training, which is known as the “expousre bias” problem. In order to tackle the problem, we propose a contrastive learning framework to expose the model to various valid or incorrect output sequences for a given input sentence. Following the contrastive learning framework (Chen et al., 2020), we can train the model to learn the representations of the ground truth sentence by contrasting the positive pairs with the negative pairs, where we select the negative pairs as a random non-target output sequence from the same batch. As shown in Fig. 3-(a), we project the source and target text sequences onto the latent embedding space. Then we maximize the similarity between the pair of source and target sequence, while minimizing the similarity between the negative pairs as follows:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { c o n t } ( \theta ) = \displaystyle \sum _ { i = 1 } ^ { N } \log \frac { \exp \left( \sin ( \mathbf { z } _ { \mathbf { x } } ^ { ( i ) } , \mathbf { z } _ { \mathbf { y } } ^ { ( i ) } ) / \tau \right) } { \sum _ { \mathbf { z } _ { \mathbf { y } } ^ { ( j ) } \in S } \exp \left( \sin ( \mathbf { z } _ { \mathbf { x } } ^ { ( i ) } , \mathbf { z } _ { \mathbf { y } } ^ { ( j ) } ) / \tau \right) } } \\ & { \qquad \mathbf { z } _ { \mathbf { x } } ^ { ( i ) } = \xi ( \mathbf { M } ^ { ( i ) } ; \theta ) , \mathbf { z } _ { \mathbf { y } } ^ { ( i ) } = \xi ( \mathbf { H } ^ { ( i ) } ; \theta ) } \\ & { \mathbf { \cdot } _ { 1 } \cdot \cdot \mathbf { \cdot v } _ { T } ] ; \theta ) : = \mathbf { A } \mathrm { v g P o o l } ( [ \mathbf { u } _ { 1 } \cdot \cdot \cdot \mathbf { u } _ { T } ] ) , \mathrm { w h e r e } \mathbf { u } _ { t } = \mathrm { R e L U } ( \mathbf { W } ^ { ( 1 ) } \mathbf { v } _ { t } + \mathbf { b } ^ { ( 1 ) } ) } \end{array}
+$$
+
+where $\xi$ denotes the composition of affine transformation with the ReLU (Nair & Hinton, 2010) and average pooling to compute the fixed sized representation of a sentence $\textbf { z } \in \ \mathbb { R } ^ { d }$ , $\begin{array} { r l } { \mathbf { H } ^ { ( i ) } } & { { } = } \end{array}$ $[ { \bf h } _ { 1 } ^ { ( i ) } \cdot \cdot \cdot { \bf h } _ { T } ^ { \bar { ( i ) } } ] \in \mathbb { R } ^ { \bar { d } \times T }$ is a concatenation of the decoder hidden states of the target sentence $\mathbf { y } ^ { ( i ) }$ across all the time steps. Furthermore, $S = \{ \mathbf { z } _ { \mathbf { y } } ^ { ( j ) } : j \neq i \}$ is a set of hidden representations of target sentences (the objects other than circles in Fig. 3-(a)) that are randomly sampled and not paired with the source sentence $\mathbf { x } ^ { ( i ) }$ , and $\sin ( \cdot , \cdot )$ is a cosine similarity function.
+
+However, training the model with na¨ıve contrastive learning framework using random non-target sequences as negative examples is highly suboptimal, as described in the introduction and shown in Fig. 1. Many of such na¨ıve negative examples are often located far away from the positive examples in the embedding space from the beginning, when using the pretrained language model. Therefore, simply using the examples from the same batch as done in Chen et al. (2020) will result in trivial negative examples and require very large batch size to enable sampling meaningful negative pairs within the same batch. Moreover, generating positive examples for text sequences is not a trivial problem either since for text domains, we do not have a well-defined set of augmentation methods that preserves the input semantics, unlike with the image domains. To tackle such difficulties, we propose a principled method to automatically construct the adversarial negative and positive examples, such that the samples are difficult for the model to classify correctly. These adversarial positive/negative pairs can guide the model to learn a more accurate representation of the target text sequence, by identifying which features make the output positive or negative (See Fig. 1-(c)).
+
+![](images/bf63475b3841ed84076bdf161013a042e960c1737dd40ffe340a99b5ed4b73a6.jpg)  
+Figure 3: Generation of imposters and distant-targets with perturbation. (a) We add small perturbation $\delta _ { t }$ to $\mathbf { h } _ { t }$ for $\tilde { \mathbf { z } } _ { \mathbf { y } }$ so that its conditional likelihood is minimized to generate an invalid sentence. (b) We add large perturbation $\zeta _ { t }$ to $\mathbf { h } _ { t }$ for $\hat { \mathbf { z } } _ { \mathbf { y } }$ by maximizing the distance from $\mathbf { z _ { x } }$ , the representation of source sentence but enforcing its likelihood high to preserve the original semantics.
+
+# 3.3 GENERATION OF IMPOSTERS
+
+As shown in Fig. 3-(b), to generate a negative example, we add a small perturbation $\begin{array} { r l } { \delta ^ { ( i ) } = } \end{array}$ $[ \delta _ { 1 } ^ { ( i ) } \cdot \cdot \cdot \delta _ { T } ^ { ( i ) } ] \in \mathbb { R } ^ { d \times T }$ to the $\mathbf { H } ^ { ( i ) }$ , which is the hidden representation of target sequence $\mathbf { y } ^ { ( i ) }$ , such that its conditional likelihood is minimized as follows:
+
+$$
+p _ { \theta } ( \mathbf { y } ^ { ( i ) } | \mathbf { x } ^ { ( i ) } ; \mathbf { H } ^ { ( i ) } + \delta ) = \prod _ { t = 1 } ^ { T } p _ { \theta } ( y _ { t } ^ { ( i ) } | \mathbf { y } _ { < t } ^ { ( i ) } , \mathbf { x } ^ { ( i ) } ; \mathbf { h } _ { t } ^ { ( i ) } + \delta _ { t } )
+$$
+
+$p _ { \theta } ( y _ { t } ^ { ( i ) } | \mathbf { y } _ { < t } ^ { ( i ) } , \mathbf { x } ^ { ( i ) } ; \mathbf { h } _ { t } ^ { ( i ) } + \delta _ { t } ) = \mathrm { s o f t m a x } \{ \mathbf { W } ( \mathbf { h } _ { t } ^ { ( i ) } + \delta _ { t } ) + \mathbf { b } \}$ where $\delta _ { t } \in \mathbb { R } ^ { d }$
+
+The exact minimization of the conditional log likelihood with respect to $\delta$ is intractable for deep neural networks. Following Goodfellow et al. (2015), we approximate it by linearizing $\log p _ { \theta } ( \mathbf { y } ^ { ( i ) } | \mathbf { x } ^ { ( i ) } )$ around $\mathbf { H } ^ { ( i ) }$ as follows:
+
+$$
+\tilde { \mathbf { H } } ^ { ( i ) } = \mathbf { H } ^ { ( i ) } - \epsilon \frac { g } { | | g | | _ { 2 } } , \mathrm { w h e r e } g = \nabla _ { \mathbf { H } ^ { ( i ) } } \log p _ { \theta } ( \mathbf { y } ^ { ( i ) } | \mathbf { x } ^ { ( i ) } )
+$$
+
+We add small perturbation to the hidden representation of each token of target sentence $\mathbf { y } ^ { ( i ) }$ such that its conditional likelihood is minimized. Thus, the perturbed $\tilde { \mathbf { H } } ^ { ( i ) }$ , which we call an imposter (inspired by Weinberger & Saul (2009)), is semantically very dissimilar to $\mathbf { y } ^ { ( i ) }$ , but very close to the hidden representation $\mathbf { H } ^ { ( i ) }$ in the embedding space (Fig. 3-(a)). This will make it non-trivial for the sequence-to-sequence model to distinguish it from the representation of true target sequence $\mathbf { y } ^ { ( i ) }$ . Please note while adversarial perturbations are generated similarly as in Miyato et al. (2017), we use them in a completely different way. While they train the model to be invariant to adversarial samples within the $\epsilon$ -ball, we push them far away from the source sentence while pulling the ground truth target sentence to the input sentence. In other words, we use the perturbed representation as an additional negative sample for contrastive learning as follows:
+
+$$
+\mathcal { L } _ { c o n t - n e g } ( \theta ) = \sum _ { i = 1 } ^ { N } \log \frac { \exp ( \sin ( \mathbf { z } _ { \mathbf { x } } ^ { ( i ) } , \mathbf { z } _ { \mathbf { y } } ^ { ( i ) } ) / \tau ) } { \sum _ { \mathbf { z } _ { \mathbf { y } } ^ { ( k ) } \in S \cup \{ \bar { \mathbf { z } } _ { \mathbf { y } } ^ { ( i ) } \} } \exp ( \sin ( \mathbf { z } _ { \mathbf { x } } ^ { ( i ) } , \mathbf { z } _ { \mathbf { y } } ^ { ( k ) } ) / \tau ) } , \mathrm { w h e r e ~ } \tilde { \mathbf { z } } _ { \mathbf { y } } ^ { ( i ) } = \xi ( \tilde { \mathbf { H } } ^ { ( i ) } ; \theta _ { \mathbf { z } _ { \mathbf { x } } } ^ { ( i ) } ) ,
+$$
+
+Alternatively, we can generate an imposter by perturbing the hidden representation of target sentence $\mathbf { y }$ so that its conditional likelihood is minimized but very close to the source sentence $\mathbf { x }$ in the embedding space. However, we empirically find that such a variation yields less performance gain.
+
+# 3.4 GENERATION OF DISTANT-TARGETS
+
+Moreover, as shown in Fig. 3-(c), we construct an additional positive pair of source sequence $\mathbf { x } ^ { ( i ) }$ by adding large perturbation $\boldsymbol { \zeta } ^ { ( i ) } = [ \zeta _ { 1 } ^ { ( i ) } \cdot \cdot \cdot \zeta _ { T } ^ { ( i ) } ] \in \mathbb { R } ^ { d \times T }$ to $\mathbf { H } ^ { ( i ) }$ the hidden state of target sequence $\mathbf { y } ^ { ( i ) }$ , such that cosine similarity from $\mathbf { z } _ { \mathbf { x } } ^ { ( i ) }$ is minimized, but the conditional likelihood is enforced to remain high. However, the exact computation of $\zeta ^ { ( i ) }$ with such constraints is intractable. We approximate it with the following two separate stages. First, we add perturbation to $\mathbf { H } ^ { ( i ) }$ such that it minimizes the contrastive learning objective $\mathcal { L } _ { c o n t } ( \theta )$ as shown in Eq. 6. Then we add another perturbation to minimize the KL divergence between perturbed conditional distribution $p _ { \theta } ( \hat { y } _ { t } ^ { ( i ) } | \hat { \mathbf { y } } _ { < t } ^ { ( i ) } , \mathbf { x } ^ { ( i ) } )$ and the original conditional distribution $p _ { \theta } ( y _ { t } ^ { ( i ) } | \mathbf { y } _ { < t } ^ { ( i ) } , \mathbf { x } ^ { ( i ) } )$ as shown in Eq. 7, where $\overline { { \mathbf { H } } } = [ \overline { { \mathbf { h } } } _ { 1 } \cdot \cdot \cdot \overline { { \mathbf { h } } } _ { T } ] \in \mathbb { R } ^ { d \times T }$ , $\hat { \mathbf { H } } = [ \hat { \mathbf { h } } _ { 1 } \cdot \cdot \cdot \hat { \mathbf { h } } _ { T } ] \in \mathbb { R } ^ { d \times T }$ , and $\eta \in \mathbb { R }$ . Note that $\theta ^ { * }$ denotes the copied of the model parameter $\theta$ and is considered to be constant to prevent it from being updated through back-propagation.
+
+$$
+\begin{array} { l } { { \displaystyle { \overline { { { \bf { H } } } } } ^ { ( i ) } = { \bf { H } } ^ { ( i ) } - \eta \frac { { \bf { g } } } { | | { \bf { g } } | | _ { 2 } } } \mathrm { { w h e r e } ~ } { \bf { g } } = \nabla _ { { \bf { H } } ^ { ( i ) } } \mathcal { L } _ { c o n t } ( \theta ) } \\ { { p _ { \theta } } ( \hat { y } _ { t } ^ { ( i ) } | \hat { \bf { y } } _ { < t } ^ { ( i ) } , { \bf { x } } ^ { ( i ) } ) = \mathrm { s o f t m a x } ( \mathbf { W } { \bf { \overline { { { h } } } } } _ { t } ^ { ( i ) } + { \bf { b } } ) } \\ { \mathcal { L } _ { K L } ( \theta ) = \displaystyle { \sum _ { i = 1 } ^ { N } \sum _ { t = 1 } ^ { T } D _ { K L } \big ( p _ { \theta ^ { * } } ( y _ { t } ^ { ( i ) } | \mathbf { y } _ { < t } ^ { ( i ) } , { \bf { x } } ^ { ( i ) } ) | | p _ { \theta } ( \hat { y } _ { t } ^ { ( i ) } | \hat { \bf { y } } _ { < t } ^ { ( i ) } , { \bf { x } } ^ { ( i ) } ) } \\ { \hat { \bf { H } } ^ { ( i ) } = { \bf { \overline { { { H } } } } } ^ { ( i ) } - \eta \frac { { \bf { f } } } { | | { \bf { f } } | | _ { 2 } } } , \mathrm { { w h e r e } ~ } { \bf { f } } = \nabla _ { { \bf { \overline { { { H } } } } } _ { 1 } ^ { ( i ) } } \mathcal { L } _ { K L } ( \theta ) } \end{array}
+$$
+
+We consider the perturbed hidden state $\hat { \mathbf { H } } ^ { ( i ) }$ as an additional positive example for source sequence $\mathbf { x } ^ { ( i ) }$ , which we refer to as a distant-target. We can use a distant-target to augment contrastive learning and minimize ${ \mathcal { L } } _ { K L } ( \theta )$ as follows:
+
+$$
+\mathcal { L } _ { c o n t - p o s } ( \theta ) = \sum _ { i = 1 } ^ { N } \log \frac { \exp ( \sin ( \mathbf { z } _ { \mathbf { x } } ^ { ( i ) } , \hat { \mathbf { z } } _ { \mathbf { y } } ^ { ( i ) } ) / \tau ) } { \sum _ { \mathbf { z } _ { \mathbf { y } } ^ { ( k ) } \in S \cup \{ \hat { \mathbf { z } } _ { \mathbf { y } } ^ { ( i ) } \} } \exp ( \sin ( \mathbf { z } _ { \mathbf { x } } ^ { ( i ) } , \mathbf { z } _ { \mathbf { y } } ^ { ( k ) } ) / \tau ) } , \mathrm { w h e r e ~ } \hat { \mathbf { z } } _ { \mathbf { y } } ^ { ( i ) } = \xi ( \hat { \mathbf { H } } ^ { ( i ) } ; \theta )
+$$
+
+CLAPS objective Incorporating the loss on the imposter and the distant target introduced above, we estimate the parameters of the seq2seq model $\theta$ by maximizing the following objective, where $\alpha , \beta$ are hyperparameters which control the importance of contrastive learning and KL divergence:
+
+$$
+\displaystyle \operatorname* { m a x } _ { \theta } \mathcal { L } _ { M L E } ( \theta ) - \alpha \mathcal { L } _ { K L } ( \theta ) + \beta \{ \mathcal { L } _ { c o n t - n e g } ( \theta ) + \mathcal { L } _ { c o n t - p o s } ( \theta ) \}
+$$
+
+For all the experiments, we set $\alpha$ and $\beta$ as 1, which we search through cross-validation. Note that after training is done, we remove the pooling layer $\xi$ and generate text with the decoder $g$ , given an input encoded with the encoder $f$ .
+
+# 4 EXPERIMENT
+
+We validate our method on benchmark datasets on three conditional text generation tasks.
+
+# 4.1 TASKS
+
+Machine Translation (MT) For machine translation, we use WMT16 Romanian-English parallel corpus (WMT’16 RO-EN) to train the model. We tokenize the pairs of source and target sequences with the same tokenizer as Raffel et al. (2020). We finetune the pretrained T5-small model for 20 epochs with the batch size of 128 and Adafactor (Shazeer & Stern, 2018). For contrastive learning, we set the norm of perturbation, $\eta$ and $\epsilon$ as 3.0.
+
+Text Summarization (Sum.) For text summarization, we use XSum dataset (Narayan et al., 2018) of which summaries are highly abstractive, thus extractive summarization models under-perform abstractive models. We follow the most of the experimental settings for machine translation as described above, except that we set the norm of perturbation, $\eta$ and $\epsilon$ as 1.0 and 1.0, respectively.
+
+Question Generation (QG) For question generation, we aim to generate a question from a given answer and paragraph, i.e., we model conditional distribution $p _ { \theta } ( \mathbf { y } | \mathbf { x } , \mathbf { a } )$ where $\mathbf x , \mathbf y$ , a denote a paragraph, question and answer, respectively. We concatenate the answer and paragraph with special tokens to generate the question conditioned on both of the answer and paragraph. As the previous experimental settings, we finetune T5-small model on SQuAD dataset (Rajpurkar et al., 2016) for 20 epochs with batch size 128 and set the norm of perturbation, $\eta$ as 3.0 and $\epsilon$ as 1.0. Since the test set of $\mathrm { S Q u A D }$ is only accessible via leader board, we randomly split the validation set into a validation set and a test set.
+
+<table><tr><td>Method</td><td>Aug.</td><td>BLEU-1</td><td>BLEU-2</td><td>BLEU-3</td><td>BLEU-4</td><td>BLEU</td><td>F1/EM</td></tr><tr><td colspan="8">Question Generation - SQuAD</td></tr><tr><td>Harvesting-QG</td><td></td><td></td><td>=</td><td>20.90</td><td>15.16</td><td>=</td><td>66.05/54.62</td></tr><tr><td>T5-MLE</td><td></td><td>41.26</td><td>30.30</td><td>23.38</td><td>18.54</td><td>21.00</td><td>67.64/55.91</td></tr><tr><td>α-T5-MLE(α=0.7)</td><td></td><td>40.82</td><td>29.79</td><td>22.84</td><td>17.99</td><td>20.50</td><td>68.04/56.30</td></tr><tr><td>α-T5-MLE(α= 2.0)</td><td></td><td>37.35</td><td>27.20</td><td>20.79</td><td>16.36</td><td>18.41</td><td>65.74/54.76</td></tr><tr><td>T5-SSMBA</td><td>Pos.</td><td>41.67</td><td>30.59</td><td>23.53</td><td>18.57</td><td>21.07</td><td>68.47/56.37</td></tr><tr><td>T5-WordDropout Contrastive</td><td>Neg.</td><td>41.37</td><td>30.50</td><td>23.58</td><td>18.71</td><td>21.19</td><td>68.16/56.41</td></tr><tr><td>R3F</td><td>-</td><td>41.00</td><td>30.15</td><td>23.26</td><td>18.44</td><td>20.97</td><td>65.84/54.10</td></tr><tr><td>T5-MLE-contrastive</td><td>-</td><td>41.23</td><td>30.28</td><td>23.33</td><td>18.45</td><td>20.91</td><td>67.32/55.25</td></tr><tr><td>T5-CLAPS w/o negative</td><td>Pos.</td><td>41.87</td><td>30.93</td><td>23.90</td><td>18.92</td><td>21.38</td><td></td></tr><tr><td>T5-CLAPS w/o positive</td><td>Neg.</td><td>41.65</td><td>30.69</td><td>23.71</td><td>18.81</td><td>21.25</td><td>68.26/56.41</td></tr><tr><td>T5-CLAPS</td><td>Pos.+Neg.</td><td>42.33</td><td>31.29</td><td>24.22</td><td>19.19</td><td>21.55</td><td>69.01/57.06</td></tr><tr><td>ERNIE-GEN (Xiao et al.,2020)</td><td></td><td></td><td></td><td>=</td><td>26.95</td><td>=</td><td></td></tr><tr><td>Info-HCVAE (Lee et al.,2020)</td><td></td><td>-</td><td>=</td><td>=</td><td>-</td><td>=</td><td>81.51/71.18</td></tr><tr><td colspan="8">MachineTranslation-WMT&#x27;16RO-EN</td></tr><tr><td>Transformer</td><td></td><td>50.36</td><td>37.18</td><td>28.42</td><td>22.21</td><td>26.17</td><td></td></tr><tr><td>Scratch-T5-MLE</td><td></td><td>51.62</td><td>37.22</td><td>27.26</td><td>21.13</td><td>25.34</td><td></td></tr><tr><td>Scratch-CLAPS</td><td>Pos.+Neg.</td><td>53.42</td><td>39.57</td><td>30.24</td><td>23.59</td><td>27.61</td><td></td></tr><tr><td>T5-MLE</td><td></td><td>57.76</td><td>44.45</td><td>35.12</td><td>28.21</td><td>32.43</td><td></td></tr><tr><td>α-T5-MLE(α = 0.7)</td><td></td><td>57.63</td><td>44.23</td><td>33.84</td><td>27.90</td><td>32.14</td><td></td></tr><tr><td>α-T5-MLE(α = 2.0)</td><td>-</td><td>56.03</td><td>42.59</td><td>33.29</td><td>26.45</td><td>30.72</td><td></td></tr><tr><td>T5-SSMBA</td><td>Pos.</td><td>58.23</td><td>44.87</td><td>35.50</td><td>28.48</td><td>32.81</td><td></td></tr><tr><td>T5-WordDropout Contrastive</td><td>Neg.</td><td>57.77</td><td>44.45</td><td>35.12</td><td>28.21</td><td>32.44</td><td></td></tr><tr><td>R3F</td><td>-</td><td>58.07</td><td>44.86</td><td>35.57</td><td>28.66</td><td>32.99</td><td></td></tr><tr><td>T5-MLE-contrastive</td><td>-</td><td>57.64</td><td>44.12</td><td>34.74</td><td>27.79</td><td>32.03</td><td></td></tr><tr><td>T5-CLAPS w/o negative</td><td>Pos.</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>T5-CLAPS w/o positive</td><td>Neg.</td><td>58.81 57.90</td><td>45.52 44.60</td><td>36.20 35.27</td><td>29.23 28.34</td><td>33.50</td><td></td></tr><tr><td>T5-CLAPS</td><td>Pos.+Neg.</td><td>58.98</td><td>45.72</td><td>36.39</td><td>29.41</td><td>32.55 33.96</td><td></td></tr><tr><td>Conneau &amp; Lample (2019)</td><td>1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>38.5</td><td></td></tr></table>
+
+Table 1: BLEU scores on WMT’16 RO-EN and SQuAD for machine translation and question generation. EM/F1 scores with BERT-base QA model for question generation.
+
+# 4.2 EXPERIMENTAL SETUPS
+
+Implementation Details For the encoder $f$ , and decoder $g$ , we use T5-small model, which is based on transformer with the hidden dimension, $d = 5 1 2$ . We set the temperature, $\tau$ as 0.1 for all the experiments. At test time, we use beam search of width 4 to generate the target sequences. Common Baselines We compare our method against relevant baselines.
+
+1. T5-MLE: A pretrained T5 model fine-tuned to maximize $\mathcal { L } _ { M L E } ( \theta )$ .   
+2. Scratch-T5-MLE: A random initialized Transformer model that has the identical architecture to T5, trained by maximizing $\mathcal { L } _ { M L E } ( \theta )$ .   
+3. $\alpha$ -T5-MLE: T5 model trained with MLE, with varying temperature $\alpha$ in the softmax function when decoding the target sentences, as done in Caccia et al. (2019)   
+4. T5-SSMBA: This is the T5 model trained to maximize $\mathcal { L } _ { M L E } ( \theta )$ , with additional examples generated by the technique proposed in $\mathrm { N g }$ et al. (2020). which are generated by corrupting the target sequences and reconstructs them using a masked language model, BERT.   
+5. T5-WordDropout Contrastive: This is a T5 model trained with the contrastive learning framework proposed in Yang et al. (2019), which heuristically generates negative examples by removing the most frequent word from the target sequence. We pretrain T5-small to maximize $\mathcal { L } _ { M L E } ( \theta )$ and further train the model to assign higher probability to the ground truth target sentence than a negative example with max-margin loss.   
+6. R3F: This is a T5 model that minimizes the negative log likelihood and symmetric KL-divergence between original conditional log likelihood $p _ { \boldsymbol { \theta } } ( \mathbf { y } | \mathbf { x } )$ and $p _ { \boldsymbol { \theta } } ( \mathbf { y } | \tilde { \mathbf { x } } )$ to enforce the function to be smooth, where $\tilde { \mathbf { x } } =$ WordEmbedding $\mathbf { \Phi } _ { \cdot } ( \mathbf { x } ) + \mathbf { z } , \mathbf { z } = ( z _ { 1 } , \dots , z _ { L } ) , z _ { i } \overset { i . i . d } { \sim } \mathcal { N } ( \mathbf { 0 } , \mathrm { d i a g } ( \sigma _ { 1 } , \dots , \sigma _ { d } ) )$ .   
+7. T5-MLE-contrastive: This is a naive constrastive learning framework with positive/negative pairs, which maximizes the contrastive learning objective from Eq. 2.   
+8. T5-CLAPS w/o positive (negative): Our proposed model which jointly maximizes the log likelihood and the contrastive learning objective with imposters but does not use any distant-targets or imposters.
+
+Table 2: Rouge and Meteor on Xsum test set for text summarization.   
+
+<table><tr><td>Method</td><td>Aug.</td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td><td>METEOR</td></tr><tr><td colspan="6">Text Summarization - XSum</td></tr><tr><td>PTGEN-COVG</td><td>一</td><td>28.10</td><td>8.02</td><td>21.72</td><td>12.46</td></tr><tr><td>CONVS2S</td><td></td><td>31.89</td><td>11.54</td><td>25.75</td><td>13.20</td></tr><tr><td>Scratch-T5-MLE</td><td></td><td>31.44</td><td>11.07</td><td>25.18</td><td>13.01</td></tr><tr><td>Stcratch-CLAPS</td><td>Pos.+Neg.</td><td>33.52</td><td>12.59</td><td>26.91</td><td>14.18</td></tr><tr><td>T5-MLE</td><td></td><td>36.10</td><td>14.72</td><td>29.16</td><td>15.78</td></tr><tr><td>α-T5-MLE(α= 0.7)</td><td>=</td><td>36.68</td><td>15.10</td><td>29.72</td><td>15.78</td></tr><tr><td>α-T5-MLE (α= 2.0)</td><td>=</td><td>34.18</td><td>13.53</td><td>27.35</td><td>14.51</td></tr><tr><td>T5-SSMBA</td><td>Pos.</td><td>36.58</td><td>14.81</td><td>29.68</td><td>15.38</td></tr><tr><td>T5-WordDropout Contrastive</td><td>Neg.</td><td>36.88</td><td>15.11</td><td>29.79</td><td>15.77</td></tr><tr><td>R3F</td><td>-</td><td>36.96</td><td>15.12</td><td>29.76</td><td>15.68</td></tr><tr><td>T5-MLE-contrastive</td><td>-</td><td>36.34</td><td>14.81</td><td>29.41</td><td>15.85</td></tr><tr><td>T5-CLAPS w/o negative</td><td>Pos.</td><td>37.49</td><td>15.31</td><td>30.42</td><td>16.36</td></tr><tr><td>T5-CLAPS w/o positive</td><td>Neg.</td><td>37.72</td><td>15.49</td><td>30.74</td><td>16.06</td></tr><tr><td>T5-CLAPS</td><td>Pos.+Neg.</td><td>37.89</td><td>15.78</td><td>30.59</td><td>16.38</td></tr><tr><td>PEGASUS (Zhang et al.,2020)</td><td>=</td><td>47.21</td><td>24.56</td><td>39.25</td><td>-</td></tr></table>
+
+9. T5-CLAPS: Our full model which jointly maximizes the log likelihood, contrastive learning objective, and KL-divergence as described in the Eq. 9.
+
+10. Scratch-CLAPS: Our full model as T5-CLAPS but with randomly initialized T5 architecture.
+
+Task specific baselines For machine translation, we use the Transformer (Vaswani et al., 2017) which consists of 6 layers of self-attention layer with 8 multi-head attention and 512 dimension, as an additional baseline. For QG, we additionally compare our models against Harvesting-QG (Du & Cardie, 2018), which is a LSTM model with copy mechanism. For text summarization, we use PTGEN-COVG (See et al., 2017) as a baseline, which uses copy mechanism and coverage to handle out of vocabulary word and prevent word repetition, and CONVS2S (Narayan et al., 2018) which uses convolutional networks as the encoder and decoder.
+
+Evaluation Metric Following the conventional evaluation metrics, we adopt n-gram BLEU and BLEU (Papineni et al., 2002) for MT and QG. For text summarization, we use Rouge (Lin & Hovy, 2002) and Meteor (Banerjee & Lavie, 2005). As an additional performance measure for question generation, we evaluate a BERT QA model on the SQuAD test set, where the QA model is trained with the questions generated by each QG methods from the contexts and answers of HarvestingQA dataset (Du & Cardie, 2018), and report the F1 and Exact Match (EM).
+
+# 4.3 EXPERIMENTAL RESULTS
+
+Quantitative Results We compare our model with the baseline models on WMT’16 RO-En, XSum, SQuAD dataset for machine translation, text summarization and question generation, respectively. Table 1 shows that our proposed method CLAPS significantly outperforms the other baseline, with the performance gain of more than $1 \%$ on all tasks according to the BLEU scores. Moreover our proposed method improves the performance of the randomly initialized T5 model (Scratch-CLAPS). For question generation, our proposed method also improves F1/EM as well as BLEU scores. It shows that our proposed model is able to generate semantically valid questions that are beneficial for training the QA model. Note that naively constructing the negative examples for contrastive learning on the both tasks, by randomly shuffling the association of $\displaystyle ( \mathbf { x } , \mathbf { y } )$ from a given mini-batch, degrades the performance. Increasing the batch size to a large value, using larger memory, may increase its performance as observed in SimCLR (Chen et al., 2020). However, such an approach will be highly sample-inefficient. On the contrary, our model outperforms all the other baseline models on Xsum dataset for text summarization, as shown in Table 2. For summarization, we observe that contrastive learning with imposters alone can improve the performance by a large margin.
+
+Visualization To examine our model with proposed contrastive learning framework learns meaningful representation of sentences, we encode a pair of sequences $\displaystyle ( \mathbf { x } , \mathbf { y } )$ into M, H with encoder $f$ and decoder $g$ . Then, we add perturbations to $\mathbf { H }$ to construct an imposter $\tilde { \bf H }$ and an additional positive
+
+![](images/733b12af4b7726f4664721ec16f095a20ba6155c9de025921b94f7a542040235.jpg)  
+Figure 4: Visualization. (a) Embedding space without contrastive learning. (b) Embedding space with our proposed contrastive learning, CLAPS.
+
+<table><tr><td>(MT) Lupta lui Hilary a fost mai atractiva. =&gt;(GT): Hillary&#x27;s struggle was more attractive =&gt;(Dist.): Hilary&#x27;s fight was more attractive =&gt;(Imp.): Thearies&#x27;batle fight has attractive appealing</td></tr><tr><td>(QG)...Von Miller... recording five solo tackles,.. =&gt;(GT): How many solo tackles did Von Miler make at Super Bowl 50? =&gt;(Dist.): How many solo tackles did Von Miller record at Super Bowl 50? =&gt;(Imp.):What much tackle did was Miller record at Super Bowl 50?</td></tr><tr><td>(Sum.) Pieces from the board game...have been found in... China.... =&gt;(GT): An ancient board game has been found in a Chinese Tomb. =&gt;(Dist.):An ancient board game has been discovered in a Chinese Tomb. =&gt;(Imp.):America&#x27;s gained vast Africa most well geographical countries, 22</td></tr></table>
+
+Table 3: Greedy decoding from hidden representation of imposters and distant-targets. The answer span is highlighted for QG.
+
+example $\hat { \bf H }$ as shown in Eq. 3 and 6, 7. We apply average pooling to $\mathbf { M } , \mathbf { H } , \tilde { \mathbf { H } }$ , and $\hat { \bf H }$ and project them onto two dimensional space with t-SNE (Maaten & Hinton, 2008). As shown in Fig. 4-(b), the model pushes away the imposter from the embedding of target sequence and pulls the embedding of the distant-targets to the embedding of the source sequence. For the model without contrastive learning, however, the embeddings of both target sequences and distant targets are far away from those of source sequences and the imposters are very close to them as shown in Fig. 4-(a).
+
+Qualitative Examples For qualitative analysis, we examine the texts that are represented by the distant-target and imposter from our method, CLAPS. To decode them into output sequences, we apply affine transformation and softmax to $\tilde { \bf H }$ and $\hat { \bf H }$ and select the most likely token at each time step. As shown in Table 3, the distant-target example (Dist.), preserves the semantic of the original target sequence (GT) with a single word replaced by a synonym (colored in green). However, the imposters (Imp.) have completely different semantics, and often are gramatically incorrect (colored in red). This shows that the model are exposed to those various valid or incorrect sentences with our proposed contrastive learning framework with adversarial perturbations.
+
+Human Evaluation We further conduct a human evaluation of the 20 summaries and 20 questions generated by our CLAPS and T5-MLE trained for text summarization and QG task. Specifically, 20 human judges perform blind quality assessment of two sentences generated by the two models, that are presented in a random order. For text summarization, $70 \%$ of the human annotators chose the sentences generated by our model as better than the baseline, and for QG, $85 \%$ favored the sentences generated by our model over that of the baseline.
+
+# 5 CONCLUSION
+
+To mitigate the exposure bias problem in sequence-to-sequence learning, we proposed a contrastive learning framework which maximizes the similarity between ground truth input and output sequence, and minimize the similarity between the input and an incorrect output sequence. Moreover, since conventional approach to sample random non-target examples from the batch as negative examples for contrastive learning results in trivial pairs that are well-discriminated from the beginning, we propose a new principled approach to automatically construct “hard” negative and positive examples, where the former is semantically dissimilar but close to the input embedding, and the latter is far from the input embedding but semantically similar. This adversarial learning enables the model to learn both the correct and incorrect variations of the input, and generalize better to unseen inputs. We empirically showed that our method improved the performance of seq2seq model on machine translation, question generation, and text summarization tasks. While we specifically targeted the exposure bias problem with seq2seq models for conditional text generation, our method may be applicable to seq2seq learning for tasks from other domains, such as automatic speech recognition, text-to-speech generation, or video captioning.
+
+Acknowledgements This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2020- 0-00153), Samsung Advanced Institute of Technology (SAIT), Samsung Electronics (IO201214- 08145-01), Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST)), and the Engineering Research Center Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MSIT (NRF2018R1A5A1059921).
+
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+Ramakrishna Vedantam, Samy Bengio, Kevin Murphy, Devi Parikh, and Gal Chechik. Contextaware captions from context-agnostic supervision. IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017, 2017.
+
+Kilian Q Weinberger and Lawrence K Saul. Distance metric learning for large margin nearest neighbor classification. Journal of Machine Learning Research, 2009.
+
+Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick ´ von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. Huggingface’s transformers: Stateof-the-art natural language processing. ArXiv, abs/1910.03771, 2019.
+
+Dongling Xiao, Han Zhang, Yukun Li, Yu Sun, Hao Tian, Hua Wu, and Haifeng Wang. Ernie-gen: An enhanced multi-flow pre-training and fine-tuning framework for natural language generation. IJCAI, 2020.
+
+Zonghan Yang, Yong Cheng, Yang Liu, and Maosong Sun. Reducing word omission errors in neural machine translation: A contrastive learning approach. Annual Meeting of the Association for Computational Linguistics, ACL, 2019.
+
+Lantao Yu, Weinan Zhang, Jun Wang, and Yong Yu. Seqgan: sequence generative adversarial nets with policy gradient. Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, 2017.
+
+Jingqing Zhang, Yao Zhao, Mohammad Saleh, and Peter J Liu. Pegasus: Pre-training with extracted gap-sentences for abstractive summarization. ICML, 2020.
+
+Yizhe Zhang, Zhe Gan, Kai Fan, Zhi Chen, Ricardo Henao, Dinghan Shen, and Lawrence Carin. Adversarial feature matching for text generation. International Conference on Machine Learning, ICML, 2017.
+
+Yizhe Zhang, Michel Galley, Jianfeng Gao, Zhe Gan, Xiujun Li, Chris Brockett, and Bill Dolan. Generating informative and diverse conversational responses via adversarial information maximization. Advances in Neural Information Processing Systems, 2018.
+
+Chen Zhu, Yu Cheng, Zhe Gan, Siqi Sun, Tom Goldstein, and Jingjing Liu. Freelb: Enhanced adversarial training for natural language understanding. In International Conference on Learning Representations, 2019.
+
+Table 4: The statistics and the data source of WMT’16 RO-EN, Xsum, and SQuAD.   
+
+<table><tr><td>Datasets</td><td>Train (#)</td><td>Valid (#)</td><td>Test (#)</td><td>Source</td></tr><tr><td>WMT&#x27;16 RO-EN</td><td>610,320</td><td>1,999</td><td>1,999</td><td>Romanian-English Parallel corpus.</td></tr><tr><td>Xsum</td><td>204,045</td><td>11,332</td><td>11,334</td><td>One-sentence summary of BBC news articles.</td></tr><tr><td>SQuAD</td><td>86.588</td><td>5,192</td><td>5,378</td><td>Crowd-sourced questions from Wikipedia paragraph</td></tr></table>
+
+# A EXPERIMENTAL DETAILS
+
+Dataset For machine translation, text summarization, and question generation, we use WMT’16 RO-EN, Xsum, SQuAD dataset, for each task. The number of train/validation/test set and its source is shown in Table 4. Note that the number of validation and test set for SQuAD is different from the original dataset. Since the original test set is only accessible via the leader board of $\mathrm { S Q u A D ^ { 1 } }$ , we split the original validation set into our new validation and test set, following the conventions of question generation communities.
+
+Preprocessing For machine translation, we download the raw text2, not the tokenized text, and use the same T5-tokenizer as Raffel et al. (2020) to tokenize both Romanian and English sentences. We limit the input and output length to 128 tokens. For text summarization, we also use the T5-tokenizer as before, and limit the input length to 512 tokens and output length to 128 tokens. For question generation, we set the maximum length of question as 64 tokens and input which is concatenation of answer and context as 384 tokens.
+
+Implementation We finetune the pretrained T5-small model provided from the transformers library (Wolf et al., 2019)3 with Adafactor optimizer. We set the batch size 128 and follow the default setting of Adafactor optimizer to finetune the T5-small models. However, the number of negative examples from the batch is 16 or 32 (total batch size divided by the number of GPUs), because we split the batch into smaller batches and distribute them to each GPU machines. We use 8 GPUs for text summarization, and 4 GPUs for machine translation and question generation. The dimension of hidden state of T5 model, $d$ is 512, so we set the hidden size of $\mathbf { z }$ as the same.
+
+Evaluation We use beam search with beam width 4 to generate the target sentences from the source sentences of the test set. Some of the examples are shown in Table 5,6,A. After the generation, we convert the tokens into the raw texts and compare them to the raw text of ground truth target sentences with the automatic evaluation metrics. For n-gram BLEU and Meteor, we use the implementation by Sharma et al. $( 2 0 1 7 ) ^ { 4 }$ . For BLEU score, we adopt the implementation by Post $( 2 0 1 8 ) ^ { 5 }$ .
+
+Article: The US military says a strike targeting Taliban in the northern city of Kunduz may have caused ”collateral damage”. Offering his ”deepest condolences”, Mr Obama said he expected a ”full accounting of the facts” and would then make a definitive judgement. . . .
+
+GT: President Barack Obama says the US has launched a ”full investigation” into air strikes that killed 19 people at an MSF-run Afghan hospital on Saturday.
+
+CLAPS: US President Barack Obama has called for an inquiry into air strikes in Afghanistan that killed dozens of medical workers.
+
+Article: Forecasts were for quarterly growth of between $0 . 5 \%$ and $0 . 7 \%$ . Official statistics also showed that household consumption expenditure boosted the quarterly growth numbers. But economist Shane Oliver told the BBC the numbers were “well below potential”. On an annual basis the economy expanded $2 . 3 \%$ , beating expectations for $2 . 1 \%$ . Economic growth in the March quarter of 2014 was $2 . 9 \%$ . “The March quarter GDP [gross domestic product] growth was far better than feared just a few days ago,” said Mr Oliver, who is chief economist with AMP Capital in Sydney. “However, Australia is still not out of the woods, as annual growth at $2 . 3 \%$ is well below potential, and a full $0 . 8 \%$ percentage points of the $0 . 9 \%$ growth came from higher inventories and trade.” He said domestic demand remained “very weak with consumer spending and home construction only just offsetting the ongoing slump in mining investment”. ...
+
+GT: Australia’s economy grew at a better-than-expected $0 . 9 \%$ in the first quarter of 2015, compared to the previous quarter, boosted by mining together with financial and insurance services.
+
+CLAPS: Australia’s economy grew faster than expected in the first three months of the year, according to official figures.
+
+Article: After the problems last week, many doubt the system will cope.   
+Transport for London (TfL) remains confident, although it admits there will be breakdowns.   
+The trick will be in getting the system back up and running quickly. So here’s some friendly advice for tourists and Olympic visitors to try and make the transport experience as easy as possible.   
+If anyone thinks of any more please post below.
+
+GT: The busiest summer ever looms for London’s transport system.
+
+CLAPS: London’s transport system has been a pretty busy week.
+
+Article: The outgoing vice-president spoke during a state dinner and took the opportunity to praise America’s northern neighbour. ”The world is going to spend a lot of time looking to you, Mr Prime Minister”, he told the Canadian leader. Mr Biden has been highly critical of US President-elect Donald Trump. ”Vive le Canada because we need you very, very badly,” he told the dinner guests. He went on to describe the self-doubt that liberal leaders across the world are currently experiencing after several political defeats. But he praised ”genuine leaders” including German Chancellor Angela Merkel, saying such statesmen and women are in short supply. Mr Trudeau reportedly became emotional during Mr Biden’s remarks when the American spoke of his late father, former Prime Minister Pierre Trudeau. ”You’re a successful father when your children turn out better than you,” Mr Biden said. ...
+
+GT: US Vice-President Joe Biden told an audience in Ottawa that the world needs ”genuine leaders” such as Canadian Prime Minister Justin Trudeau.
+
+CLAPS: Vice-President Joe Biden has praised Canadian Prime Minister Vive le Canada for his visit to the country.
+
+Article: The Swedish giant asked customers who bought   
+any model of the Mysingso chair to return it for a full refund.   
+The global recall comes after Ikea received reports from   
+Finland, Germany, the US, Denmark and Australia   
+that users had received injuries to their fingers that needed medical treatment.   
+Ikea’s statement said the chair had a ”risk of falling or finger entrapment”.   
+It said: ”After washing the fabric seat it is possible to re-assemble the chair incorrectly leading to risks of falls or finger entrapments.   
+”Ikea has received five incident reports in which a Mysingso beach chair collapsed during use due to incorrect re-assembly.   
+All five reports included injuries to fingers and required medical attention.   
+It added that a full investigation had led to an improved design   
+”to further mitigate the risks of incorrect re-assembly and injuries”   
+and the updated chair would be available from next month.   
+Ikea has more than 300 stores in 27 countries.
+
+GT: Ikea is recalling a beach chair sold in the UK after reports that it can collapse and cause injury.
+
+CLAPS: Ikea is recalling a popular beach chair that collapsed during use because of incorrect re-assemblies.   
+
+<table><tr><td>Article: Spending on the NHS should also be paid for by a dedicated tax marked on every payslip,the former health minister suggested. Under Mr Lamb&#x27;s plan,taxes would not be increased as the new levy would be offset by deductions to income tax or national insurance. He has warned the NHS faces collapse without an urgent cash injection. The plans are not yet party policy and will not be put to this year&#x27;s conference in Bournemouth.But Mr Lamb,the party&#x27;s health spokesman, told party members he was &quot;very interested in the idea of a dedicated NH S and care contribution - separating it out from the rest of taxation, clearly identified on your payslip.&quot;And Iam really interested in the idea of the right for local areas to raise additional funds for the NHS and care if theychoose.’The Lib Dems say he would like to implement the ideas across theUK,although,as health and social care are devolved, it is unclear how this would be enforced. Mr Lamb - who lost out to Tim Farron in a leadership election in July - proposes a cross-party commission to explore the ideas. Heintends to consult health bodies and professionals, patients,trade unions and academics.Ministers have pledged f2bn in this financial year for the NHS,and an extra f8bn by 2020. But MrLamb told the BBC that this was insuffcient and, having &quot;seen the books”as a minister in the last government, he feared the NHS could face a funding shortfall of f3Obn by 2020. &quot;The bottom line is with rising demand because of an ageing population we need more investment,” he said.MrLamb also warned that the social care system was &quot;on its knees&quot; and could collapse without a cash injection of f5bn. &quot;I&#x27;ve been in the department. Ihave seen the books and Iam deeply concerned. If we carry on regardless, the system will crash.&quot; Taxpayers are already shown how much they have contributed to the health service in annual personal tax statements.An attempt to establish a cross-party commission on social care before the 2O1O election -led in part by MrLamb -collapsed in acrimony.</td></tr></table>
+
+GT: English councils should be allowed to put up taxes to fund the NHS, Norman Lamb has told the Lib Dem conference.
+
+CLAPS:A new levy on the NHS and social care should be introduced by the Liberal Democrats, Norman Lamb has said.
+
+Article: Yorkshire, Lancashire and Derbyshire have been worst affected,   
+after $2 { \cdot } 5 \mathrm { c m }$ fell overnight, with $1 0 \mathrm { c m }$ reported on higher ground.   
+Passengers waiting to depart Manchester Airport have reported   
+being stuck on the runway for hours due to a lack of de-icers.   
+Leeds Bradford Airport suspended all morning flights but has since reopened.   
+Manchester Airport reported ”minor delays to departing aircraft”   
+- but passengers told the BBC they had been stuck on board outbound flights.   
+Shirley Hale said her Jet2 flight to Tenerife   
+had been waiting to depart for over four hours.   
+”We have been told that there are not enough de-icers at the airport,”   
+she said. The airport apologised and said de-icing was the responsibility of   
+airlines and their ground teams. More than 100 schools were closed across East Lancashire and Oldham, with 80 shut in West Yorkshire.   
+BBC Weather said Buxton in Derbyshire saw up to $1 7 \mathrm { c m }$ of snow,   
+the deepest measured on Friday. The avalanche risk in the Peak District was   
+currently extremely high, Buxton Mountain Rescue Team said.   
+Parts of Staffordshire have been affected, with several centimetres of   
+snow reported in Flash, England’s highest village.   
+Commuters have been urged to allow extra journey time,   
+and the Met Office has issued snow and ice warnings.   
+More on the snow and other stories in West Yorkshire Weather   
+updates for Lancashire and Greater Manchester BBC Weather   
+presenter Kay Crewdson said conditions were due to slowly   
+improve into Saturday. Molly Greenwood reported $1 0 \mathrm { c m }$ of snow   
+in the Huddersfield area. ”Don’t think I’m going anywhere,” she said.   
+Zulfi Hussain said the snow was causing ”traffic chaos” in Woodhall Road,   
+Calverley, near Leeds. Elliott Hudson, another West Yorkshire resident,   
+said: ”Looks like I have woken up in Narnia.”   
+West Yorkshire’s Liversedge FC, who have had to cancel every home   
+game for the last four months due to bad weather,   
+tweeted a picture of snow with the caption:   
+”It’s not looking good for Liversedge FC’s home game with Worksop Town tomorrow.” The A628 Woodhead, A57 Snake Pass and A537 Cat and   
+Fiddle roads are all affected, with delays reported on   
+the M65 motorway. Highways England said the A57 eastbound   
+in Great Manchester is closed between M67/A560 and B6174   
+due to severe weather conditions. It said teams were working   
+to clear the road. Tony Hallwood, from Leeds Bradford Airport,   
+said it reopened at about 09:00 GMT after crews used ploughs   
+to clear snow from the runway. He said: ”We are asking passengers   
+to make their way to the airport as early as they can given the difficult   
+conditions.” Bus operators are also reporting delays to all services   
+across West Yorkshire. Oldham Council has said 48 schools had closed   
+this morning as a result of the snow and severe weather.   
+Drivers are also being asked to take extra care after snow   
+fell overnight in some parts of Northern Ireland.   
+A Met Office yellow warning for ice and snow in   
+northern England and Wales ended at 15:00.
+
+GT: Heavy snowfall has caused travel disruption in parts of northern England.
+
+CLAPS: Flights have been disrupted after a large avalanche hit parts of England.
+
+Article: But once the votes are counted, what can residents expect to pay in co Below are the figures for a Band D property for every council area   
+in Wales for the current financial year of 2017/18,   
+how much that has gone up by for the current year,   
+and what the average property in the area actually pays.   
+They are grouped here by police force region -   
+council tax includes the police precept which is added to   
+the overall bill paid by homes. Local government is not fully   
+funded by council tax. Much of the funding for councils comes   
+in the form of grants from the Welsh Government,   
+which in turn gets its funding from the UK government in London.   
+In 2017/18 a total of $\pm 4 . 1$ bn is being divided among Wales’ 22 councils. The lions share of council cash goes on schools   
+- with social services following behind, as shown in the graph above.   
+Residents pay council tax based on which band their property is in,   
+based on its worth. Band D has historically been used as   
+the standard for comparing council tax levels between and across local   
+authorities. It is used to charge tax to a property that, in Wales,   
+was worth between $\varepsilon 9 1 \mathrm { , 0 0 1 }$ to $\mathcal { L } 1 2 3 \mathrm { , 0 0 0 }$ on April 2003 values.   
+Council tax gets lower the cheaper a property is,   
+and higher the more expensive a property is.   
+Council tax figures source: Welsh Government
+
+GT: Voters will go to the polls on Thursday to determine who will represent them on local councils.
+
+CLAPS: The people of Wales are voting in a referendum on whether or not to pay council tax.
+
+Article: The side’s appearance in France will be its first at a major tournament since the 1958 World Cup. Players and coaches left their base at the Vale Resort, Vale of Glamorgan, on Saturday and headed to Cardiff Airport. After a send-off from pupils from Ysgol Treganna, Cardiff, the team took off for a friendly in Sweden on Sunday. They will then head to France ahead of the team’s first game of the tournament against Slovakia on 11 June.
+
+GT: Wales’ football team has departed the country as their Euro 2016 preparations reach a climax.
+
+CLAPS: Wales’ Euro 2016 squad have arrived in France for the first time since 1958.
+
+Article: The 40-year-old, from the South Bank area of Teesside, was discovered on the A66 in the early hours ”in a distressed state” with wounds to his groin after the attack. The road, from Greystones Roundabout to Church Lane in Middlesbrough, was shut earlier while searches of the area were carried out. It has now reopened. A 22-year-old man was arrested on suspicion of assault and later bailed. Cleveland Police said the injured man had been placed in an induced coma in hospital. The force said in a statement: ”Police can confirm that the man found this morning on the A66 had wounds to his groin area. ”Officers are continuing to investigate and are appealing for anyone with information to contact them.”
+
+GT: A man has been found by the side of a road with his penis cut off.
+
+CLAPS: A man is in an induced coma after being found with serious injuries on a Teesside road.
+
+<table><tr><td>Article:In July,a majorbugwas discovered in the software that could let hijackers access data on up to a billion phones. Manufacturers have been slow to roll out a fix because many variations of Android are widely used. One Android expert said it was &quot;about time&quot; phone makers issued security fixes more quickly. Android has been working to patch a vulnerability,known as Stagefright, which could let hackers access a phone&#x27;s data simply by sending somebody a video message.&quot;My guess is that this is the single largest software update the world has ever seen,”said Adrian Ludwig, Android&#x27;s lead engineer for security,at hacking conference Black Hat. LG,Samsung and Google have all said a number of their handsets will get the fix,with further updates every month. Android is an open source operating system,with the software freelyavailable for phone manufacturers to modify and use on their handsets. The Google-led project does provide security fixes for the software, but phone manufacturers are responsible for sending the updates to their devices. Some phones running old versions of Android are no longer updated by the manufacturer. Many companies also deploy customised versions of Android which take time to rebuild with the security changes. Apple and BlackBerry can patch security problems more quickly because they develop both the software and the hardware for their devices. BlackBerry&#x27;s software is reviewed by mobile networks before being sent to handsets,while Apple can push updates to its phones whenever it wants. &quot;The very nature of Android is that manufacturers add their own software on top,so there have been delays in software roll-outs,”said Jack Parsons,editor of Android Magazine. &quot;In the US it&#x27;s even worse because mobile carriers often add their own software too,adding another layer of bureaucracy holding up security fixes. &quot;There&#x27;s no real villain here,that&#x27;s just how the system works. But there will always be security concerns with software,</td><td>so it&#x27;s right that some of the manufacturers are stepping up to deal with this now.&quot;</td></tr></table>
+
+GT: Samsung, LG and Google have pledged to provide monthly security updates for smartphones running the Android operating system.
+
+CLAPS: The world’s largest software update is to be issued by Google-led Android.
+
+Article: The move follows a claim by Crossmaglen Rangers player Aaron Cunningham that he was the victim of verbal abuse during the 2 December Ulster football final. The Ulster Council carried out an investigation and BBC Sport understands one Kilcoo player is to be banned for six months and another for four months. Kilcoo said they had not been notified, and the players could appeal. The two suspensions have yet to be officially confirmed by the Ulster Council. It is believed the case was the first time an allegation of racial abuse had been lodged with the provincial governing body. When an investigation was announced, Ulster GAA president Aogan O Fearghail, said anyone found ´ guilty of racism would be dealt with severely. Kilcoo released a statement saying the club condemned abuse and would co-operate with the Ulster Council’s investigation. The Gaelic Athletic Association, which governs the sport in Ireland, is to discuss how to deal with racism at its annual congress in March.
+
+GT: Two Kilcoo players are to be suspended by Ulster GAA chiefs following allegations of racial abuse.
+
+CLAPS: Two Kilcoo players have been suspended by the Ulster GAA for alleged racial abuse.
+
+Table 6: Generated Questions by CLAPS from SQuAD. Answer spans are highlighted.
+
+Context: ... The Broncos finished the regular season with a 12-4 record, and denied the New England Patriots a chance to defend their title from Super Bowl XLIX by defeating them 20-18 in the AFC Championship Game. They joined the Patriots, Dallas Cowboys, and Pittsburgh Steelers as one of four teams that have made eight appearances in the Super Bowl.
+
+GT: How many appearances have the Denver Broncos made in the Super Bowl?
+
+CLAPS: How many Super Bowl appearances have the Broncos made?
+
+Context: In late November 2015, reports surfaced stating that “multiple acts” would perform during the halftime show. On December 3, the league confirmed that the show would be headlined by the British rock group Coldplay. On January 7, 2016, Pepsi confirmed to the Associated Press that Beyonce, who headlined the Super Bowl XLVII halftime show and ´ collaborated with Coldplay on the single “Hymn for the Weekend”, would be making an appearance. Bruno Mars, who headlined the Super Bowl XLVIII halftime show, and Mark Ronson also performed.
+
+GT: What nationality is the band Coldplay?
+
+CLAPS: What nationality was Coldplay?
+
+Context: There are 13 natural reserves in Warsaw - among others, Bielany Forest, Kabaty Woods, Czerniakow Lake. About 15 kilometres (9 miles) from Warsaw, the Vistula river’s environment ´ changes strikingly and features a perfectly preserved ecosystem, with a habitat of animals that includes the otter, beaver and hundreds of bird species. There are also several lakes in Warsaw - mainly the oxbow lakes, like Czerniakow Lake, the lakes in the Łazienki or Wilan ´ ow Parks, ´ Kamionek Lake. There are lot of small lakes in the parks, but only a few are permanent - the majority are emptied before winter to clean them of plants and sediments.
+
+GT: What animals does the Vistula river’s ecosystem include?
+
+CLAPS: What animals are included in the Vistula river’s habitat?
+
+Context: ”The FSO Car Factory was established in 1951. A number of vehicles have been assembled there over the decades, including the Warszawa, Syrena, Fiat 125p (under license from Fiat, later renamed FSO $1 2 5 \mathrm { p }$ when the license expired) and the Polonez. The last two models listed were also sent abroad and assembled in a number of other countries, including Egypt and Colombia. In 1995 the factory was purchased by the South Korean car manufacturer Daewoo, which assembled the Tico, Espero,Nubia, Tacuma, Leganza, Lanos and Matiz there for the European market. In 2005 the factory was sold to AvtoZAZ, a Ukrainian car manufacturer which assembled there the Chevrolet Aveo. The license for the production of the Aveo expired in February 2011 and has since not been renewed. Currently the company is defunct.”
+
+GT: Who bought the factory in 2005?
+
+CLAPS: To whom was the factory sold in 2005?
+
+<table><tr><td>Context: The Scotland Act 1998,which was passed by the Parliament of the United Kingdom and given royal assent by Queen Elizabeth II on 19 November 1998, governs the functions and role of the Scottish Parliament and delimits its legislative competence. The Scotland Act 2O12 extends the devolved competencies. For the purposes of parliamentary sovereignty, the Parliament of the United Kingdom at Westminster continues to constitute the supreme legislature of Scotland. However, under the terms of the Scotland Act, Westminster agreed to devolve some of its responsibilities over Scottish domestic policy to the Scottish Parliament. Such devolved mattersinclude education, health,agriculture and justice.</td></tr></table>
+
+GT: What does the Scotland Act of 2012 extend?
+
+CLAPS: What does the Scotland Act 2012 extend?
+
+Context: Stage 1 is the first, or introductory stage of the bill,   
+where the minister or member in charge of the bill will formally introduce it to Parliament together with its accompanying documents-Explanatory Notes, a Policy Memorandum   
+setting out the policy underlying the bill,   
+and a Financial Memorandum setting out the costs and savings associated with it.   
+Statements from the Presiding Officer and   
+the member in charge of the bill are also lodged indicating whether the bill is   
+within the legislative competence of the Parliament.   
+Stage 1 usually takes place, initially, in the relevant committee or committees   
+and is then submitted to the whole Parliament   
+for a full debate in the chamber on the general principles of the bill.   
+If the whole Parliament agrees in a vote to the general principles of the bill, it then proceeds to Stage 2.
+
+GT: Where are bills typically gestated in Stage 1?
+
+CLAPS: Where does Stage 1 usually take place?
+
+Context: Moderate and reformist Islamists who accept and   
+work within the democratic process include parties like the Tunisian Ennahda Movement. Jamaat-e-Islami of Pakistan is basically a socio-political   
+and democratic Vanguard party but has also gained political influence   
+through military coup d’etat in past. ´   
+The Islamist groups like Hezbollah in Lebanon and Hamas in Palestine participate in democratic and political process as well as armed attacks,   
+seeking to abolish the state of Israel.   
+Radical Islamist organizations like al-Qaeda and the Egyptian Islamic Jihad,   
+and groups such as the Taliban, entirely reject democracy,   
+often declaring as kuffar those Muslims who support it (see takfirism),   
+as well as calling for violent/offensive jihad   
+or urging and conducting attacks on a religious basis.
+
+GT: Where does Hamas originate?
+
+CLAPS: Where are Hamas located?
+
+Context: Sayyid Abul Ala Maududi was an important early twentieth-century figure   
+in the Islamic revival in India, and then after independence from Britain, in Pakistan.   
+Trained as a lawyer he chose the profession of journalism,   
+and wrote about contemporary issues and most importantly about Islam and Islamic law.   
+Maududi founded the Jamaat-e-Islami party in 1941 and remained its leader until 1972.   
+However, Maududi had much more impact through his writing   
+than through his political organising. His extremely influential books (translated into many languages) placed Islam in a modern context, and influenced not only conservative ulema   
+but liberal modernizer Islamists such as al-Faruqi,   
+whose ¨Islamization of Knowledgecarried forward some of Maududi’s key principles. ¨
+
+GT: Where did Maududi exert the most impact?
+
+CLAPS: How did Maududi have more impact on Islam than his political organising?
+
+<table><tr><td>Context: ByLike many other mainline Protestant denominations in the United States, the United Methodist Church has experienced significant membership losses in recent decades. At the time of its formation,the UMC had about 11 million members in nearly 42.000 congregations. In 1975,membership dropped below 1O million for the first time. In 2005,there were about 8 million members in over 34,OoO congregations. Membership is concentrated primarily in the Midwest and in the South. Texas has the largest number of members,with about 1 million.</td></tr></table>
+
+Context: Celoron’s expedition force consisted of about 200 Troupes de la marine and 30 Indians. The expedition covered about 3,000 miles $( 4 , 8 0 0 \mathrm { k m } )$ between June and November 1749.   
+It went up the St. Lawrence, continued along the northern shore of Lake Ontario,   
+crossed the portage at Niagara, and followed the southern shore of Lake Erie.   
+At the Chautauqua Portage (near present-day Barcelona, New York),   
+the expedition moved inland to the Allegheny River, which it followed to the site of present-day Pittsburgh. There Celoron buried lead plates engraved with the French claim to the Ohio Country. ´   
+Whenever he encountered British merchants or fur-traders,   
+Celoron informed them of the French claims on the territory and told them to leave.
+
+GT: How did Celeron handle business on trip?
+
+CLAPS: What did Celoron do when he encountered the British?
+
+Context: Like many cities in Central and Eastern Europe, infrastructure in Warsaw suffered considerably during its time as an Eastern Bloc economy - though it is worth mentioning that the initial Three-Year Plan to rebuild Poland (especially Warsaw) was a major success, but what followed was very much the opposite. However, over the past decade Warsaw has seen many improvements due to solid economic growth, an increase in foreign investment as well as funding from the European Union. In particular, the city’s metro, roads, sidewalks, health care facilities and sanitation facilities have improved markedly. answer:improved markedly
+
+GT: Warsaw’s sidewalks and sanitation facilities are some examples of things which have what?
+
+CLAPS: What has happened to Warsaw’s infrastructure in the past decade?
+
+Context: Several commemorative events take place every year.   
+Gatherings of thousands of people on the banks of the Vistula   
+on Midsummer’s Night for a festival called Wianki (Polish for Wreaths) have become a tradition and a yearly event in the programme of cultural events in Warsaw. The festival traces its roots to a peaceful pagan ritual   
+where maidens would float their wreaths of herbs on the water to predict when they would be married, and to whom.   
+By the 19th century this tradition had become a festive event,   
+and it continues today. The city council organize concerts and other events. Each Midsummer’s Eve, apart from the official floating of wreaths,   
+jumping over fires, looking for the fern flower,   
+there are musical performances, dignitaries’ speeches, fairs   
+and fireworks by the river bank.
+
+GT: How man people gather along the banks of the Vistula for the Wianki festival?
+
+CLAPS: How many people gather on the banks of the Vistula on Midsummer’s Night for a festival called Wianki?
+
+Context: The origin of the legendary figure is not fully known. The best-known legend, by Artur Oppman, is that long ago two of Triton’s daughters set out on a journey through the depths of the oceans and seas. One of them decided to stay on the coast of Denmark and can be seen sitting at the entrance to the port of Copenhagen. The second mermaid reached the mouth of the Vistula River and plunged into its waters. She stopped to rest on a sandy beach by the village of Warszowa, where fishermen came to admire her beauty and listen to her beautiful voice. A greedy merchant also heard her songs; he followed the fishermen and captured the mermaid.
+
+GT: What did a greedy merchant do to the mermaid?
+
+CLAPS: What did Oppman do to the mermaid?
+
+Context: Warsaw remained the capital of the Polish-Lithuanian   
+Commonwealth until 1796, when it was annexed by the Kingdom of Prussia to become the capital of the province of South Prussia.   
+Liberated by Napoleon’s army in 1806, Warsaw was made   
+the capital of the newly created Duchy of Warsaw.   
+Following the Congress of Vienna of 1815, Warsaw became the centre of the Congress Poland, a constitutional monarchy under a personal union   
+with Imperial Russia. The Royal University of Warsaw was established in 1816.
+
+GT: How long was Warsaw the capital of the Polish-Lithuanian Commonwealth?
+
+CLAPS: How long did Warsaw remain the capital of the Polish-Lithuanian Commonwealth?
+
+Context: John Paul II’s visits to his native country in 1979 and 1983 brought support to the budding solidarity movement and encouraged the growing anti-communist fervor there. In 1979, less than a year after becoming pope, John Paul celebrated Mass in Victory Square in Warsaw and ended his sermon with a call to ”renew the face” of Poland: Let Thy Spirit descend! Let Thy Spirit descend and renew the face of the land! This land! These words were very meaningful for the Polish citizens who understood them as the incentive for the democratic changes. GT: What is St. John’s Cathedral an example of, stylistically?
+
+CLAPS: St. John’s Cathedral is a typical example of what style?
+
+Context: Gothic architecture is represented in the majestic churches   
+but also at the burgher houses and fortifications.   
+The most significant buildings are St. John’s Cathedral (14th century),   
+the temple is a typical example of the so-called Masovian gothic style,   
+St. Mary’s Church (1411), a town house of Burbach family (14th century),   
+Gunpowder Tower (after 1379) and the Royal Castle Curia Maior (140720131410). ˘ The most notable examples of Renaissance architecture in the city   
+are the house of Baryczko merchant family (1562), building called ”The Negro” (early 17th century) and Salwator tenement (1632). The most interesting examples of mannerist architecture are the Royal Castle (159620131619) and ˘   
+the Jesuit Church (160920131626) at Old Town. ˘   
+Among the first structures of the early baroque the most important   
+are St. Hyacinth’s Church (1603-1639) and Sigismund’s Column (1644).
+
+GT: What is St. John’s Cathedral an example of, stylistically?
+
+CLAPS: St. John’s Cathedral is a typical example of what style?
+
+Table 7: Translation of Romanian by CLAPS from WMT’16 RO-EN.   
+
+<table><tr><td>RO: De partea cealalta, 47% dintre alegatoriirepublicani afirma ca ar fi,nemultumiti” sau ,suparati” daca favoritul Jeb Bush castigä cursa pentru nominalizare.</td></tr><tr><td>GT: On the flip side,47 percent of Republican Voters say they would be &quot;dissatisfied&quot; or &quot;upset&quot;if establishment favorite Jeb Bush becomes the nominee.</td></tr><tr><td>CLAPS: On the other hand,47% of Republican voters say they would be &quot;unsatisfied&quot;or &quot;uneasy&quot; if the winer</td></tr><tr><td>Jeb Bush wins the nomination race. RO:Datoria va deveni o problema importanta.</td></tr><tr><td>GT: Debt will become a big issue.</td></tr><tr><td>CLAPS: Debt will become an important issue.</td></tr><tr><td>RO: Se pare ca Moreno a avut noroc si a scapat de cartonasul rosu si de concesia unei lovituri de pedeapsa.</td></tr><tr><td>GT: Moreno appeared fortunate to escape a red card and the concession of a spot-kick.</td></tr><tr><td> CLAPS: It seems that Moreno has been lucky and escaped the red card and the concession of a penalty kick.</td></tr><tr><td>RO: Astfel de decizii nu se iau niciodata usor.</td></tr><tr><td>GT: Such decisions are never taken lightly.</td></tr><tr><td>CLAPS: Such decisions are never made easily.</td></tr><tr><td>RO: Toate astea au cu certitudine un cost.</td></tr><tr><td>GT: All that stuff sure does take a toll.</td></tr><tr><td>CLAPS: All of this certainly has a cost.</td></tr><tr><td>RO:Astfel de decizii nu se iau niciodata usor.</td></tr><tr><td>GT: Such decisions are never taken lightly.</td></tr><tr><td>CLAPS: Such decisions are never made easily. RO:Deci, nu am mai gasit niciun sens de a continua pe acest drum.</td></tr><tr><td>GT: So,I had no reason to continue on this path.</td></tr><tr><td>CLAPS: So there is no point in continuing on this road.</td></tr><tr><td>RO: Profitul a crescut cu 5%, pana la 12,3 miliarde dolari.</td></tr><tr><td>GT: Revenue rose 5 percent, to $12.3 billion.</td></tr><tr><td>CLAPS: Profit has increased by between 5% and $12.3 billion.</td></tr><tr><td>RO:,Este imposibil ca toata lumea sa vina in Europa”- Dalai Lama - RT news</td></tr><tr><td>GT: &quot;It is impossible for everyone to come to Europe&quot; - Dalai Lama - RT News</td></tr><tr><td>CLAPS: &quot;It is impossible for everyone to come to Europe&quot; - the Dalai Lama - RT news</td></tr><tr><td>RO: Spune povestea detinutului asa cum isi doreste acesta,obtine acces.</td></tr><tr><td>GT: Tellthe inmate&#x27;s story the way he likes it, get access.</td></tr><tr><td>CLAPS: It tells the detainee&#x27;s story as he wants, it gets acess.</td></tr><tr><td>RO: Peter Moody a afirmat cä stewarzii Racing Victoria sau incercat sa infiltreze un spion la grajdurile sale anul trecut si a amenintat cä se retrage imediat din curse.</td></tr><tr><td>GT: Peter Moody has alleged Racing Victoria stewards attmpted to plant a spy in his stables last year and threatened to quit racing immediately.</td></tr><tr><td>CLAPS: Peter Moody said stewarzia Racing Victoria or tried to infiltrate a spy at her track last year and</td></tr></table>
+
+RO: Corbyn, Tsipras și Syriza în Grecia, Podemos în Spania, chiar Bernie Sanders în Statele Unite își alimentează retorica populistă din frustrările acumulate în societățile oocidentale.
+
+GT: Corbyn, Tsipras and Syriza in Greece, Podemos in Spain, even Bernie Sanders in the United States feed their populist rhetoric with the frustrations accumulated in the Western world.
+
+CLAPS: Corbyn, Tsipras and Syriza in Greece, Podemos in Spain, even Bernie Sanders in the United States are fuelling their populist rhetoric from the frustrations gained in the occident societies.
+
+RO: Pentru România, chiar și cota voluntară propusă de București, în cuantum de circa 1500 de suflete, ne depășește cu mult bunele intenții de solidaritate cu Uniunea Europeană exprimate în ultimele luni.
+
+GT: For Romania, even the voluntary quota proposed by Bucharest, amounting to about 1,500 souls, surpasses by
+
+<table><tr><td>far our good intentions of solidarity with the European Union expressed in recent months.</td></tr><tr><td>CLAPS: For Romania, even Bucharest&#x27;s proposed voluntary quota,amounting to around 1 50O souls,goes far beyond our good intentions of solidarity with the European Union expressed in recent months.</td></tr><tr><td>RO: 7.0oo de euro pe an pentru a inchiria Clubul Pogor.</td></tr><tr><td>GT: 7,00o Euro per year to rent Pogor Club.</td></tr><tr><td> CLAPS: 7,0o0 euros per year to rent the Pogor Club.</td></tr><tr><td>RO: Crina a fost internata impreuna cubunicaei la spitalul municipal din Pascani,iar tot atunci tatal fetei, Costica Balcan, a plecat la munca in orasul Alexandria, judetul Teleorman.</td></tr><tr><td>GT: Crina and her grandmother were admited to the hospital of the municipalityof Pascani,and on the same day her father, Costica Balcan went to work in the city of Alexandria, Teleorman County.</td></tr><tr><td>CLAPS:The crisis was admited with her grandmother to the municipal hospital in Pascani,and the girl&#x27;s father, Costica Balcan, also went to work in the town of Alexandria, Teleorman County.</td></tr><tr><td>RO: In tara, parintii Crinei, Alina si Costica,au continuat separat cautarile, fiecare pe unde a putut.</td></tr><tr><td>GT: I was just a little girl when I found out.</td></tr><tr><td>CLAPS:Iwas small when I found out.</td></tr><tr><td>RO: Cu o saptamana inainte de intalnire nici nu prea mai putea sä doarma.</td></tr><tr><td>GT: A week before the meeting he hadn&#x27;t really been able to sleep.</td></tr><tr><td>CLAPS: A week before the meeting, we could not even sleep much longer.</td></tr><tr><td>RO: &quot;Casa iti vezi copilul dupa atatia ani,esteceva&quot;,ne spune onepoata,venita si eacu Costica de la Ruginoasa. GT: &quot;It&#x27;s something extraordinary to see your baby after so many years&quot;,says a niece who came with Costica from</td></tr><tr><td>Ruginoasa. CLAPS: &quot;To see your child after so many years,it is something,&quot; says a litle girl, who also comes with Costica</td></tr><tr><td>from Ruginoasa.</td></tr><tr><td>RO: In Piata Uniri, subcerul inceputului de toamna,familia din Ruginoasa isiuneste din nou destinul cu fata lor din Palermo. GT: In Piata Uniri,beneath the autumn sky,the family from Ruginoasa joins theirdestiny with their daughter in</td></tr><tr><td>Palermo. CLAPS: In the Square of the Union,under the skies ofthe early autumn,the family of Ruginoasa is once again</td></tr><tr><td>uniting their destiny with their mother in Palermo.</td></tr><tr><td>RO: &quot;Sper sa am o amintire frumoasa cu ei, pentru ca acum sunt atat de multe lucruri de spus&quot;, ne marturiseste Crina. GT: &quot;I hope to gain a beautiful memory after meting them, because now there are so many things to say,&quot;</td></tr><tr><td>confesses Crina. CLAPS: &quot;Ihope to have a nice memory with them,because there is so much to say now,&quot; Crina tels us.</td></tr><tr><td>RO: In weekendul care a trecut,Alina si Costica si-au vazut pentru prima data fata pierduta pe holurile spitalului</td></tr><tr><td>din Iasi,in urma cu 20 ani. GT: Last weekend,Alina and Costica saw the girl who was lost on thehals of the hospitalin Iasi for the first time</td></tr><tr><td>in 20 years. CLAPS: Forthe firsttime in the weekend,Alina and Costica saw their girllostin hospital rooms in Jasmine 20</td></tr><tr><td>yearsago. RO: &quot;A durat pana la urma cam oluna de zile pana sa il gasim pe tatal biologic al fetei&quot;, ne explica comisarul sef</td></tr><tr><td>Romica Ichim. GT: Then they gave me some pointers,only to find that the person I was directed to wasn&#x27;t he person I was looking for.</td></tr></table>

parse/train/ryxB0Rtxx/ryxB0Rtxx.md

@@ -0,0 +1,483 @@
+# IDENTITY MATTERS IN DEEP LEARNING
+
+# Moritz Hardt
+
+# Tengyu Ma
+
+Google Brain   
+1600 Amphitheatre Parkway, Mountain View, CA, 94043 m@mrtz.org
+
+Department of Computer Sciene Princeton University 35 Olden Street, Princeton, 08540 tengyu@cs.princeton.edu
+
+# ABSTRACT
+
+An emerging design principle in deep learning is that each layer of a deep artificial neural network should be able to easily express the identity transformation. This idea not only motivated various normalization techniques, such as batch normalization, but was also key to the immense success of residual networks.
+
+In this work, we put the principle of identity parameterization on a more solid theoretical footing alongside further empirical progress. We first give a strikingly simple proof that arbitrarily deep linear residual networks have no spurious local optima. The same result for feed-forward networks in their standard parameterization is substantially more delicate. Second, we show that residual networks with ReLu activations have universal finite-sample expressivity in the sense that the network can represent any function of its sample provided that the model has more parameters than the sample size.
+
+Directly inspired by our theory, we experiment with a radically simple residual architecture consisting of only residual convolutional layers and ReLu activations, but no batch normalization, dropout, or max pool. Our model improves significantly on previous all-convolutional networks on the CIFAR10, CIFAR100, and ImageNet classification benchmarks.
+
+# 1 INTRODUCTION
+
+Traditional convolutional neural networks for image classification, such as AlexNet (Krizhevsky et al. (2012)), are parameterized in such a way that when all trainable weights are 0, a convolutional layer represents the 0-mapping. Moreover, the weights are initialized symmetrically around 0. This standard parameterization makes it non-trivial for a convolutional layer trained with stochastic gradient methods to preserve features that were already good. Put differently, such convolutional layers cannot easily converge to the identity transformation at training time.
+
+This shortcoming was observed and partially addressed by Ioffe & Szegedy (2015) through batch normalization, i.e., layer-wise whitening of the input with a learned mean and covariance. But the idea remained somewhat implicit until residual networks (He et al. (2015); He et al. (2016)) explicitly introduced a reparameterization of the convolutional layers such that when all trainable weights are 0, the layer represents the identity function. Formally, for an input $x$ , each residual layer has the form $x + h ( x )$ , rather than $h ( x )$ . This simple reparameterization allows for much deeper architectures largely avoiding the problem of vanishing (or exploding) gradients. Residual networks, and subsequent architectures that use the same parameterization, have since then consistently achieved state-of-the-art results on various computer vision benchmarks such as CIFAR10 and ImageNet.
+
+# 1.1 OUR CONTRIBUTIONS
+
+In this work, we consider identity parameterizations from a theoretical perspective, while translating some of our theoretical insight back into experiments. Loosely speaking, our first result underlines how identity parameterizations make optimization easier, while our second result shows the same is true for representation.
+
+Linear residual networks. Since general non-linear neural networks, are beyond the reach of current theoretical methods in optimization, we consider the case of deep linear networks as a simplified model. A linear network represents an arbitrary linear map as a sequence of matrices $A _ { \ell } \cdots A _ { 2 } A _ { 1 }$ . The objective function is $\mathbb { E } \| y - A _ { \ell } \cdot \cdot \cdot A _ { 1 } x \| ^ { 2 }$ , where $y = R x$ for some unknown linear transformation $R$ and $x$ is drawn from a distribution. Such linear networks have been studied actively in recent years as a stepping stone toward the general non-linear case (see Section 1.2). Even though $A _ { \ell } \cdots A _ { 1 }$ is just a linear map, the optimization problem over the factored variables $( A _ { \ell } , \ldots , A _ { 1 } )$ is non-convex.
+
+In analogy with residual networks, we will instead parameterize the objective function as
+
+$$
+\operatorname* { m i n } _ { A _ { 1 } , \dots , A _ { \ell } } \mathbb { E } \| y - ( I + A _ { \ell } ) \cdot \cdot \cdot ( I + A _ { 1 } ) x \| ^ { 2 } .
+$$
+
+To give some intuition, when the depth $\ell$ is large enough, we can hope that the target function $R$ has a factored representation in which each matrix $A _ { i }$ has small norm. Any symmetric positive semidefinite matrix $O$ can, for example, be written as a product $O = O _ { \ell } \cdot \cdot \cdot O _ { 1 }$ , where each $O _ { i } =$ $O ^ { 1 / \ell }$ is very close to the identity for large $\ell$ so that $A _ { i } = O _ { i } - I$ has small spectral norm. We first prove that an analogous claim is true for all linear transformations $R$ . Specifically, we prove that for every linear transformation $R$ , there exists a global optimizer $( A _ { 1 } , \ldots , A _ { \ell } )$ of (1.1) such that for large enough depth $\ell$ ,
+
+$$
+\operatorname* { m a x } _ { 1 \leq i \leq \ell } \| A _ { i } \| \leq O ( 1 / \ell ) .
+$$
+
+Here, $\| A \|$ denotes the spectral norm of $A$ . The constant factor depends on the conditioning of $R$ . We give the formal statement in Theorem 2.1. The theorem has the interesting consequence that as the depth increases, smaller norm solutions exist and hence regularization may offset the increase in parameters.
+
+Having established the existence of small norm solutions, our main result on linear residual networks shows that the objective function (1.1) is, in fact, easy to optimize when all matrices have sufficiently small norm. More formally, letting $A = ( A _ { 1 } , \ldots , A _ { \ell } ) $ and $f ( A )$ denote the objective function in (1.1), we can show that the gradients of vanish only when $f ( A ) = 0$ provided that maxi $\| A _ { i } \| \leq$ $O ( 1 / \ell )$ . See Theorem 2.2. This result implies that linear residual networks have no critical points other than the global optimum. In contrast, for standard linear neural networks we only know, by work of Kawaguchi (2016) that these networks don’t have local optima except the global optimum, but it doesn’t rule out other critical points. In fact, setting $A _ { i } = 0$ will always lead to a bad critical point in the standard parameterization.
+
+Universal finite sample expressivity. Going back to non-linear residual networks with ReLU activations, we can ask: How expressive are deep neural networks that are solely based on residual layers with ReLU activations? To answer this question, we give a very simple construction showing that such residual networks have perfect finite sample expressivity. In other words, a residual network with ReLU activations can easily express any functions of a sample of size $n$ , provided that it has sufficiently more than $n$ parameters. Note that this requirement is easily met in practice. On CIFAR 10 $\mathit { n } = 5 0 0 0 0 $ ), for example, successful residual networks often have more than $1 0 ^ { 6 }$ parameters. More formally, for a data set of size $n$ with $r$ classes, our construction requires $O ( n \log \bar { n } + r ^ { 2 } )$ parameters. Theorem 3.2 gives the formal statement.
+
+Each residual layer in our construction is of the form $x + V \mathrm { R e L U } ( U x )$ , where $U$ and $V$ are linear transformations. These layers are significantly simpler than standard residual layers, which typically have two ReLU activations as well as two instances of batch normalization.
+
+The power of all-convolutional residual networks. Directly inspired by the simplicity of our expressivity result, we experiment with a very similar architecture on the CIFAR10, CIFAR100, and ImageNet data sets. Our architecture is merely a chain of convolutional residual layers each with a single ReLU activation, but without batch normalization, dropout, or max pooling as are common in standard architectures. The last layer is a fixed random projection that is not trained. In line with our theory, the convolutional weights are initialized near 0, using Gaussian noise mainly as a symmetry breaker. The only regularizer is standard weight decay $\ell _ { 2 }$ -regularization) and there is no need for dropout. Despite its simplicity, our architecture reaches $6 . 3 8 \%$ top-1 classification error on the CIFAR10 benchmark (with standard data augmentation). This is competitive with the best residual network reported in He et al. (2015), which achieved $6 . 4 3 \%$ . Moreover, it improves upon the performance of the previous best all-convolutional network, $7 . 2 5 \%$ , achieved by Springenberg et al. (2014). Unlike ours, this previous all-convolutional architecture additionally required dropout and a non-standard preprocessing (ZCA) of the entire data set. Our architecture also improves significantly upon Springenberg et al. (2014) on both Cifar100 and ImageNet.
+
+# 1.2 RELATED WORK
+
+Since the advent of residual networks (He et al. (2015); He et al. (2016)), most state-of-the-art networks for image classification have adopted a residual parameterization of the convolutional layers. Further impressive improvements were reported by Huang et al. (2016) with a variant of residual networks, called dense nets. Rather than adding the original input to the output of a convolutional layer, these networks preserve the original features directly by concatenation. In doing so, dense nets are also able to easily encode an identity embedding in a higher-dimensional space. It would be interesting to see if our theoretical results also apply to this variant of residual networks.
+
+There has been recent progress on understanding the optimization landscape of neural networks, though a comprehensive answer remains elusive. Experiments in Goodfellow et al. (2014) and Dauphin et al. (2014) suggest that the training objectives have a limited number of bad local minima with large function values. Work by Choromanska et al. (2015) draws an analogy between the optimization landscape of neural nets and that of the spin glass model in physics (Auffinger et al. (2013)). Soudry & Carmon (2016) showed that 2-layer neural networks have no bad differentiable local minima, but they didn’t prove that a good differentiable local minimum does exist. Baldi & Hornik (1989) and Kawaguchi (2016) show that linear neural networks have no bad local minima. In contrast, we show that the optimization landscape of deep linear residual networks has no bad critical point, which is a stronger and more desirable property. Our proof is also notably simpler illustrating the power of re-parametrization for optimization. Our results also indicate that deeper networks may have more desirable optimization landscapes compared with shallower ones.
+
+# 2 OPTIMIZATION LANDSCAPE OF LINEAR RESIDUAL NETWORKS
+
+Consider the problem of learning a linear transformation $R \colon  { \mathbb { R } } ^ { d } \to  { \mathbb { R } } ^ { d }$ from noisy measurements $y = R x + \xi$ , where $\xi \in \mathcal { N } ( 0 , \operatorname { I d } _ { d } )$ is a $d$ -dimensional spherical Gaussian vector. Denoting by $\mathcal { D }$ the distribution of the input data $x$ , let $\Sigma = \mathbb { E } _ { x \sim { \mathcal { D } } } [ x x ^ { \top } ]$ be its covariance matrix.
+
+There are, of course, many ways to solve this classical problem, but our goal is to gain insights into the optimization landscape of neural nets, and in particular, residual networks. We therefore parameterize our learned model by a sequence of weight matrices $A _ { 1 } , \ldots , A _ { \ell } \in \mathbb { R } ^ { d \times d }$ ,
+
+$$
+h _ { 0 } = x , \qquad h _ { j } = h _ { j - 1 } + A _ { j } h _ { j - 1 } , \qquad \hat { y } = h _ { \ell } .
+$$
+
+Here $h _ { 1 } , \ldots , h _ { \ell - 1 }$ are the $\ell - 1$ hidden layers and $\hat { y } = h _ { \ell }$ are the predictions of the learned model on input $x$ . More succinctly, we have
+
+$$
+\hat { y } = ( \operatorname { I d } _ { d } + A _ { \ell } ) \dots ( \operatorname { I d } + A _ { 1 } ) x .
+$$
+
+It is easy to see that this model can express any linear transformation $R$ . We will use $A$ as a shorthand for all of the weight matrices, that is, the $\ell \times d \times d$ -dimensional tensor the contains $A _ { 1 } , \ldots , A _ { \ell }$ as slices. Our objective function is the maximum likelihood estimator,
+
+$$
+f ( A , ( x , y ) ) = \| { \hat { y } } - y \| ^ { 2 } = \| ( \operatorname { I d } + A _ { \ell } ) \dots ( \operatorname { I d } + A _ { 1 } ) x - R x - \xi \| ^ { 2 } .
+$$
+
+We will analyze the landscape of the population risk, defined as,
+
+$$
+f ( A ) : = \mathbb { E } \left[ f ( A , ( x , y ) ) \right] .
+$$
+
+Recall that $\left. A _ { i } \right.$ is the spectral norm of $A _ { i }$ . We define the norm $\left\| \cdot \right\|$ for the tensor $A$ as the maximum of the spectral norms of its slices,
+
+$$
+\| A \| : = \operatorname* { m a x } _ { 1 \leq i \leq \ell } \left\| A _ { i } \right\| .
+$$
+
+The first theorem of this section states that the objective function $f$ has an optimal solution with small $\left\| \cdot \right\|$ -norm, which is inversely proportional to the number of layers $\ell$ . Thus, when
+
+the architecture is deep, we can shoot for fairly small norm solutions. We define $\gamma \quad : =$ $\operatorname* { m a x } \{ | \log \sigma _ { \operatorname* { m a x } } ( R ) | , | \log \sigma _ { \operatorname* { m i n } } ( R ) | \}$ . Here $\sigma _ { \operatorname* { m i n } } ( \cdot ) , \sigma _ { \operatorname* { m a x } } ( \cdot )$ denote the least and largest singular values of $R$ respectively.
+
+Theorem 2.1. Suppose $\ell \geq 3 \gamma$ . Then, there exists a global optimum solution $A ^ { \star }$ of the population risk $f ( \cdot )$ with norm
+
+$$
+\left\| | A ^ { \star } | \right\| \leq 2 ( \sqrt { \pi } + \sqrt { 3 \gamma } ) ^ { 2 } / \ell .
+$$
+
+Here $\gamma$ should be thought of as a constant since if $R$ is too large (or too small), we can scale the data properly so that $\sigma _ { \operatorname* { m i n } } ( R ) \leq 1 \leq \sigma _ { \operatorname* { m a x } } ( R )$ . Concretely, if $\sigma _ { \mathrm { m a x } } ( R ) / \sigma _ { \mathrm { m i n } } ( R ) = \kappa ,$ , then we can scaling for the outputs properly so that √ $\sigma _ { \mathrm { m i n } } ( R ) = 1 / \bar { \sqrt { \kappa } }$ and $\sigma _ { \operatorname* { m a x } } ( R ) = \sqrt { \kappa }$ . In this case, we have $\gamma = \log \sqrt { \kappa }$ , which will remain a small constant for fairly large condition number $\kappa$ . We also point out that we made no attempt to optimize the constant factors here in the analysis. The proof of Theorem 2.1 is rather involved and is deferred to Section A.
+
+Given the observation of Theorem 2.1, we restrict our attention to analyzing the landscape of $f ( \cdot )$ in the set of $A$ with $\left\| \cdot \right\|$ -norm less than $\tau$ ,
+
+$$
+\begin{array} { r } { B _ { \tau } = \left\{ A \in \mathbb { R } ^ { \ell \times d \times d } : \left\| \left| A \right\| \right\| \leq \tau \right\} . } \end{array}
+$$
+
+Here using Theorem 2.1, the radius $\tau$ should be thought of as on the order of $1 / \ell$ . Our main theorem in this section claims that there is no bad critical point in the domain $B _ { \tau }$ for any $\tau < 1$ . Recall that a critical point has vanishing gradient.
+
+Theorem 2.2. For any $\tau < 1$ , we have that any critical point $A$ of the objective function $f ( \cdot )$ inside the domain $B _ { \tau }$ must also be a global minimum.
+
+Theorem 2.2 suggests that it is sufficient for the optimizer to converge to critical points of the population risk, since all the critical points are also global minima.
+
+Moreover, in addition to Theorem 2.2, we also have that any $A$ inside the domain $B _ { \tau }$ satisfies that
+
+$$
+\begin{array} { r } { \| \nabla f ( A ) \| _ { F } ^ { 2 } \geq 4 \ell ( 1 - \tau ) ^ { \ell - 1 } \sigma _ { \operatorname* { m i n } } ( \Sigma ) ^ { 2 } ( f ( A ) - C _ { \mathrm { o p t } } ) . } \end{array}
+$$
+
+Here $C _ { \mathrm { o p t } }$ is the global minimal value of $f ( \cdot )$ and $\| \nabla f ( A ) \| _ { F }$ denotes the euclidean norm1 of the ${ \ell \times d \times \bar { d } }$ -dimensional tensor $\nabla f ( A )$ . Note that $\sigma _ { \mathrm { m i n } } ( \Sigma )$ denote the minimum singular value of $\Sigma$ .
+
+Equation (2.3) says that the gradient has fairly large norm compared to the error, which guarantees convergence of the gradient descent to a global minimum (Karimi et al. (2016)) if the iterates stay inside the domain $B _ { \tau }$ , which is not guaranteed by Theorem 2.2 by itself.
+
+Towards proving Theorem 2.2, we start off with a simple claim that simplifies the population risk.   
+We also use $\left\| \cdot \right\| _ { F }$ to denote the Frobenius norm of a matrix.
+
+Claim 2.3. In the setting of this section, we have,
+
+$$
+f ( A ) = \left\| ( ( \operatorname { I d } + A _ { \ell } ) \ldots ( \operatorname { I d } + A _ { 1 } ) - R ) \Sigma ^ { 1 / 2 } \right\| _ { F } ^ { 2 } + C .
+$$
+
+Here $C$ is a constant that doesn’t depend on $A$ , and $\Sigma ^ { 1 / 2 }$ denote the square root of $\Sigma$ , that is, the unique symmetric matrix $B$ that satisfies $B ^ { 2 } = \Sigma$ .
+
+Proof of Claim 2.3. Let $\operatorname { t r } ( A )$ denotes the trace of the matrix $A$ . Let $E = ( \mathbf { I d } + A _ { \ell } ) \dots ( \mathbf { I d } + A _ { 1 } ) - R$ . Recalling the definition of $f ( A )$ and using equation (2.2), we have
+
+$$
+\begin{array} { r l r } { f ( A ) = \mathbb { E } \left[ \| E x - \xi \| ^ { 2 } \right] } & { { \mathrm { ( b y ~ e q u a t i o n ~ } ( 2 . 2 ) ) } } \\ { = \mathbb { E } \left[ \| E x \| ^ { 2 } + \| \xi \| ^ { 2 } - 2 \langle E x , \xi \rangle \right] } & { } \\ { = \mathbb { E } \left[ \mathrm { t r } ( E x x ^ { \top } E ^ { \top } ) \right] + \mathbb { E } \left[ \| \xi \| ^ { 2 } \right] } & { { \mathrm { ( s i n c e ~ } } \mathbb { E } \left[ \langle E x , \xi \rangle \right] = \mathbb { E } \left[ \langle E x , \mathbb { E } \left[ \xi | x \right] \rangle \right] = 0 ) } \\ { } & { = \mathrm { t r } \left( E \mathbb { E } \left[ x x ^ { \top } \right] E ^ { \top } \right) + C } & { { \mathrm { ( w h e r e ~ } } C = \mathbb { E } [ x x ^ { \top } ] ) } \\ { } & { = \mathrm { t r } ( E \Sigma E ^ { \top } ) + C = \| E \Sigma ^ { 1 / 2 } \| _ { F } ^ { 2 } + C . } & { { \mathrm { ( s i n c e ~ } } \mathbb { E } \left[ x x ^ { \top } \right] = \Sigma ) } \end{array}
+$$
+
+Next we compute the gradients of the objective function $f ( \cdot )$ from straightforward matrix calculus.   
+We defer the full proof to Section A.
+
+Lemma 2.4. The gradients of $f ( \cdot )$ can be written as,
+
+$$
+\frac { \partial f } { \partial A _ { i } } = 2 \big ( \mathbf { I d } + A _ { \ell } ^ { \top } \big ) \ldots \big ( \mathbf { I d } + A _ { i + 1 } ^ { \top } \big ) E \Sigma \big ( \mathbf { I d } + A _ { i - 1 } ^ { \top } \big ) \ldots \big ( \mathbf { I d } + A _ { 1 } ^ { \top } \big ) ,
+$$
+
+where $E = \left( \operatorname { I d } + A _ { \ell } \right) \ldots \left( \operatorname { I d } + A _ { 1 } \right) - R$ .
+
+Now we are ready to prove Theorem 2.2. The key observation is that each matric $A _ { j }$ has small norm and cannot cancel the identity matrix. Therefore, the gradients in equation (2.5) is a product of non-zero matrices, except for the error matrix $E$ . Therefore, if the gradient vanishes, then the only possibility is that the matrix $E$ vanishes, which in turns implies $A$ is an optimal solution.
+
+Proof of Theorem 2.2. Using Lemma 2.4, we have,
+
+$$
+\left\| \frac { \partial f } { \partial A _ { i } } \right\| _ { F } = 2 \left\| ( \mathbf { I d } + A _ { \ell } ^ { \top } ) \ldots ( \mathbf { I d } + A _ { i + 1 } ^ { \top } ) E \Sigma ( \mathbf { I d } + A _ { i - 1 } ^ { \top } ) \ldots ( \mathbf { I d } + A _ { 1 } ^ { \top } ) \right\| _ { F } \quad \mathrm { ( b y ~ L e m m a ~ 2 . 4 ) }
+$$
+
+It follows that
+
+$$
+\begin{array} { r l r } {  { \| \nabla f ( A ) \| _ { F } ^ { 2 } = \sum _ { i = 1 } ^ { \ell } \| \frac { \partial f } { \partial A _ { i } } \| _ { F } ^ { 2 } \geq 4 \ell ( 1 - \tau ) ^ { \ell - 1 } \sigma _ { \operatorname* { m i n } } ( \Sigma ) ^ { 2 } \| E \| ^ { 2 } } } \\ & { } & { \geq 4 \ell ( 1 - \tau ) ^ { \ell - 1 } \sigma _ { \operatorname* { m i n } } ( \Sigma ) ^ { 2 } ( f ( A ) - C ) \qquad \mathrm { ( b y ~ t h e ~ d e f i n i t i o n ~ o f ~ } E \mathrm { ~ a n d ~ C l a i m ~ } 2 . 3 ) } \\ & { } & { \geq 4 \ell ( 1 - \tau ) ^ { \ell - 1 } \sigma _ { \operatorname* { m i n } } ( \Sigma ) ^ { 2 } ( f ( A ) - C _ { \operatorname* { m i n } } ) . \qquad . \qquad . \qquad . \qquad . \qquad . \qquad . } \end{array}
+$$
+
+Therefore we complete the proof of equation (2.3). Finally, if $A$ is a critical point, namely, $\nabla f ( A ) =$ 0, then by equation (2.3) we have that $f ( A ) = C _ { \mathrm { o p t } }$ . That is, $A$ is a global minimum.
+
+# 3 REPRESENTATIONAL POWER OF THE RESIDUAL NETWORKS
+
+In this section we characterize the finite-sample expressivity of residual networks. We consider a residual layers with a single ReLU activation and no batch normalization. The basic residual building block is a function $\mathcal { T } _ { U , V , s } ( \cdot ) : \mathbb { R } ^ { k } \to \mathbb { R } ^ { k }$ that is parameterized by two weight matrices $U \in \mathbb { R } ^ { \times k } , V \in \mathbb { R } ^ { k \times k }$ and a bias vector $s \in \mathbb { R } ^ { k }$ ,
+
+$$
+\begin{array} { r } { \mathcal { T } _ { U , V , s } ( h ) = V \mathrm { R e L u } ( U h + s ) . } \end{array}
+$$
+
+A residual network is composed of a sequence of such residual blocks. In comparison with the full pre-activation architecture in He et al. (2016), we remove two batch normalization layers and one ReLU layer in each building block.
+
+We assume the data has $r$ labels, encoded as $r$ standard basis vectors in $\mathbb { R } ^ { r }$ , denoted by $e _ { 1 } , \ldots , e _ { r }$ . We have $n$ training examples $( x ^ { ( 1 ) } , y ^ { ( 1 ) } ) , \dots , ( x ^ { ( n ) } , y ^ { ( n ) } )$ , where $\boldsymbol { x } ^ { ( i ) } \in \mathbb { R } ^ { d }$ denotes the $i$ -th data and $y ^ { ( i ) } \in \{ e _ { 1 } , \ldots , e _ { r } \}$ denotes the $i$ -th label. Without loss of generality we assume the data are normalized so that $x ^ { ( i ) } = 1$ . We also make the mild assumption that no two data points are very close to each other.
+
+Assumption 3.1. We assume that for every $1 \leq i < j \leq n$ , we have $\| x ^ { ( i ) } - x ^ { ( j ) } \| ^ { 2 } \geq \rho$ for some absolute constant $\rho > 0$ .
+
+Images, for example, can always be imperceptibly perturbed in pixel space so as to satisfy this assumption for a small but constant $\rho$ .
+
+Under this mild assumption, we prove that residual networks have the power to express any possible labeling of the data as long as the number of parameters is a logarithmic factor larger than $n$ .
+
+Theorem 3.2. Suppose the training examples satisfy Assumption 3.1. Then, there exists a residual network $N$ (specified below) with ${ \bar { O } } ( n \log n + r ^ { 2 } )$ parameters that perfectly expresses the training data, i.e., for all $i \in \{ 1 , \ldots , n \}$ , the network $N$ maps $x ^ { ( i ) }$ to $\boldsymbol y ^ { ( i ) }$ .
+
+It is common in practice that $n > r ^ { 2 }$ , as is for example the case for the Imagenet data set where $n > 1 0 ^ { 6 }$ and $r = 1 0 0 0$ .
+
+We construct the following residual net using the building blocks of the form ${ \mathcal { T } } _ { U , V , s }$ as defined in equation (3.1). The network consists of $\ell + 1$ hidden layers $h _ { 0 } , \ldots , h _ { \ell }$ , and the output is denoted by $\hat { y } \in \mathbb { R } ^ { r }$ . The first layer of weights matrices $A _ { 0 }$ maps the $d$ -dimensional input to a $k$ -dimensional hidden variable $h _ { 0 }$ . Then we apply $\ell$ layers of building block $\tau$ with weight matrices $A _ { j } , B _ { j } \in \mathbb { R } ^ { k \times k }$ . Finally, we apply another layer to map the hidden variable $h _ { \ell }$ to the label $\hat { y }$ in $\mathbb { R } ^ { k }$ . Mathematically, we have
+
+$$
+\begin{array} { r l } & { h _ { 0 } = A _ { 0 } x , } \\ & { h _ { j } = h _ { j - 1 } + { \mathcal T } _ { A _ { j } , B _ { j } , b _ { j } } ( h _ { j - 1 } ) , \quad \forall j \in \{ 1 , \ldots , \ell \} } \\ & { \hat { y } = h _ { \ell } + { \mathcal T } _ { A _ { \ell + 1 } , B _ { \ell + 1 } , s _ { \ell + 1 } } ( h _ { \ell } ) . } \end{array}
+$$
+
+We note that here $A _ { \ell + 1 } \in \mathbb { R } ^ { k \times r }$ and $B _ { \ell + 1 } \in \mathbb { R } ^ { r \times r }$ so that the dimension is compatible. We assume the number of labels $r$ and the input dimension $d$ are both smaller than $n$ , which is safely true in practical applications.2 The hyperparameter $k$ will be chosen to be $O ( \log n )$ and the number of layers is chosen to be $\ell = [ \underline { { n } } / k ]$ . Thus, the first layer has $d k$ parameters, and each of the middle $\ell$ building blocks contains $2 k ^ { 2 }$ parameters and the final building block has $k r + r ^ { 2 }$ parameters. Hence, the total number of parameters is $O ( k d + \ell k ^ { 2 } + r k + r ^ { 2 } ) = O ( n \log n + r ^ { 2 } )$ .
+
+Towards constructing the network $N$ of the form above that fits the data, we first take a random matrix $A _ { 0 } \in \mathbb { R } ^ { k \times d }$ that maps all the data points $x ^ { ( i ) }$ to vectors $h _ { 0 } ^ { ( i ) } : = A _ { 0 } x ^ { ( i ) }$ . Here we will use $h _ { j } ^ { ( i ) }$ to denote the $j$ -th layer of hidden variable of the $i$ -th example. By Johnson-Lindenstrauss Theorem (Johnson $\&$ Lindenstrauss (1984), or see Wikipedia (2016)), with good probability, the resulting vectors $h _ { 0 } ^ { ( i ) }$ ’s remain to satisfy Assumption 3.1 (with slightly different scaling and larger constant $\rho )$ , that is, any two vectors $h _ { 0 } ^ { ( i ) }$ and $h _ { 0 } ^ { ( j ) }$ are not very correlated.
+
+Then we construct $\ell$ middle layers that maps $h _ { 0 } ^ { ( i ) }$ to $h _ { \ell } ^ { ( i ) }$ for every $i \in \{ 1 , \ldots , n \}$ . These vectors $h _ { \ell } ^ { ( i ) }$ will clustered into $r$ groups according to the labels, though they are in the $\mathbb { R } ^ { k }$ instead of in $\mathbb { R } ^ { r }$ as desired. Concretely, we design this cluster centers by picking $r$ random unit vectors $q _ { 1 } , \ldots , q _ { r }$ in $\mathbb { R } ^ { k }$ . We view them as the surrogate label vectors in dimension $k$ (note that $k$ is potentially much smaller than $r$ ). In high dimensions (technically, if $k > 4 \log r$ ) random unit vectors $q _ { 1 } , \ldots , q _ { r }$ are pair-wise uncorrelated with inner product less than $< 0 . 5$ . We associate the $i$ -th example with the target surrogate label vector $v ^ { ( i ) }$ defined as follows,
+
+$$
+\begin{array} { r } { \mathrm { ~ f ~ } y ^ { ( i ) } = e _ { j } , \operatorname { t h e n } v ^ { ( i ) } = q _ { j } . } \end{array}
+$$
+
+Then we will construct the matrices $( A _ { 1 } , B _ { 1 } ) , \ldots , ( A _ { \ell } , B _ { \ell } )$ such that the first $\ell$ layers of the network maps vector $h _ { 0 } ^ { ( i ) }$ to the surrogate label vector $v ^ { ( i ) }$ . Mathematically, we will construct $( A _ { 1 } , B _ { 1 } ) , \ldots , ( A _ { \ell } , B _ { \ell } )$ such that
+
+$$
+\forall i \in \{ 1 , \ldots , n \} , h _ { \ell } ^ { ( i ) } = v ^ { ( i ) } .
+$$
+
+Finally we will construct the last layer $\mathcal { T } _ { A _ { \ell + 1 } , B _ { \ell + 1 } , b _ { \ell + 1 } }$ so that it maps the vectors $q _ { 1 } , \ldots , q _ { r } \in \mathbb { R } ^ { k }$ to $e _ { 1 } , \ldots , e _ { r } \in \mathbb { R } ^ { r }$ ,
+
+$$
+\forall j \in \{ 1 , \ldots , r \} , q _ { j } + \mathcal { T } _ { A _ { \ell + 1 } , B _ { \ell + 1 } , b _ { \ell + 1 } } ( q _ { j } ) = e _ { j } .
+$$
+
+Putting these together, we have that by the definition (3.2) and equation (3.3), for every $i$ , if the label is $\boldsymbol y ^ { ( i ) }$ is $e _ { j }$ , then $h _ { \ell } ^ { ( i ) }$ will be $q _ { j }$ . Then by equation (3.4), we have that $\begin{array} { r l } { \hat { y } ^ { ( i ) } } & { { } = } \end{array}$ $q _ { j } + \mathcal { T } _ { A _ { \ell + 1 } , B _ { \ell + 1 } , b _ { \ell + 1 } } ( q _ { j } ) = e _ { j }$ . Hence we obtain that $\hat { y } ^ { ( i ) } = y ^ { ( i ) }$ .
+
+The key part of this plan is the construction of the middle $\ell$ layers of weight matrices so that $h _ { \ell } ^ { ( i ) } =$ $v ^ { ( i ) }$ . We encapsulate this into the following informal lemma. The formal statement and the full proof is deferred to Section B.
+
+Lemma 3.3 (Informal version of Lemma B.2). In the setting above, for (almost) arbitrary vectors $h _ { 0 } ^ { ( 1 ) } , \ldots , h _ { 0 } ^ { ( n ) }$ and $v ^ { ( 1 ) } , \ldots , v ^ { ( n ) } \in \{ q _ { 1 } , \ldots , q _ { r } \}$ , there exists weights matrices $( A _ { 1 } , B _ { 1 } ) , \ldots , ( \bar { A } _ { \ell } , B _ { \ell } )$ , such that,
+
+$$
+\forall i \in \{ 1 , \ldots , n \} , h _ { \ell } ^ { ( i ) } = v ^ { ( i ) } .
+$$
+
+We briefly sketch the proof of the Lemma to provide intuitions, and defer the full proof to Section B. The operation that each residual block applies to the hidden variable can be abstractly written as,
+
+$$
+\hat { h } \to h + { \mathcal { T } } _ { U , V , s } ( h ) .
+$$
+
+where $h$ corresponds to the hidden variable before the block and $\hat { h }$ corresponds to that after. We claim that for an (almost) arbitrary sequence of vectors ${ { h } ^ { ( 1 ) } } , \ldots , { { h } ^ { ( n ) } }$ , there exist $\mathcal { T } _ { U , V , s } ( \cdot )$ such that operation (3.5) transforms $k$ vectors of $\it { { h ^ { ( i ) } } }$ ’s to an arbitrary set of other $k$ vectors that we can freely choose, and maintain the value of the rest of $n - k$ vectors. Concretely, for any subset $S$ of size $k$ , and any desired vector $v ^ { ( i ) } ( i \in S )$ , there exist $U , V , s$ such that
+
+$$
+\begin{array} { r } { \boldsymbol { v } ^ { ( i ) } = \boldsymbol { h } ^ { ( i ) } + \mathcal { T } _ { U , V , s } ( \boldsymbol { h } ^ { ( i ) } ) ~ \forall i \in S } \\ { \boldsymbol { h } ^ { ( i ) } = \boldsymbol { h } ^ { ( i ) } + \mathcal { T } _ { U , V , s } ( \boldsymbol { h } ^ { ( i ) } ) ~ \forall i \notin S } \end{array}
+$$
+
+This claim is formalized in Lemma B.1. We can use it repeatedly to construct $\ell$ layers of building blocks, each of which transforms a subset of $k$ vectors in $\{ h _ { 0 } ^ { ( 1 ) } , \ldots , h _ { 0 } ^ { ( n ) } \}$ to the corresponding vectors in $\{ v ^ { ( 1 ) } , \ldots , v ^ { ( n ) } \}$ , and maintains the values of the others. Recall that we have $\ell = \lceil n / k \rceil$ layers and therefore after $\ell$ layers, all the vectors $h _ { 0 } ^ { ( i ) }$ ’s are transformed to $v ^ { ( i ) }$ ’s, which complete the proof sketch. □
+
+# 4 POWER OF ALL-CONVOLUTIONAL RESIDUAL NETWORKS
+
+Inspired by our theory, we experimented with all-convolutional residual networks on standard image classification benchmarks.
+
+# 4.1 CIFAR10 AND CIFAR100
+
+Our architectures for CIFAR10 and CIFAR100 are identical except for the final dimension corresponding to the number of classes 10 and 100, respectively. In Table 1, we outline our architecture. Each residual block has the form $x + C _ { 2 } ( \mathrm { R e L U } ( C _ { 1 } x ) )$ , where $C _ { 1 } , C _ { 2 }$ are convolutions of the specified dimension (kernel width, kernel height, number of input channels, number of output channels). The second convolution in each block always has stride 1, while the first may have stride 2 where indicated. In cases where transformation is not dimensionality-preserving, the original input $x$ is adjusted using averaging pooling and padding as is standard in residual layers.
+
+We trained our models with the Tensorflow framework, using a momentum optimizer with momen$\mathrm { t u m ~ 0 . 9 }$ , and batch size is 128. All convolutional weights are trained with weight decay 0.0001. The initial learning rate is 0.05, which drops by a factor 10 and 30000 and 50000 steps. The model reaches peak performance at around $5 0 k$ steps, which takes about $2 4 h$ on a single NVIDIA Tesla K40 GPU. Our code can be easily derived from an open source implementation3 by removing batch normalization, adjusting the residual components and model architecture. An important departure from the code is that we initialize a residual convolutional layer of kernel size $k \times k$ and $c$ output channels using a random normal initializer of standard deviation $\sigma = 1 / k ^ { 2 } c$ , rather than $1 / k { \sqrt { c } }$ used for standard convolutional layers. This substantially smaller weight initialization helped training, while not affecting representation.
+
+A notable difference from standard models is that the last layer is not trained, but simply a fixed random projection. On the one hand, this slightly improved test error (perhaps due to a regularizing effect). On the other hand, it means that the only trainable weights in our model are those of the convolutions, making our architecture “all-convolutional”.
+
+Table 1: Architecture for CIFAR10/100 (55 convolutions, 13.5M parameters)   
+
+<table><tr><td>variabledimensions</td><td>initial stride</td><td>description</td></tr><tr><td>3×3×3×16 3×3×16×64</td><td>1</td><td>1 standard conv</td></tr><tr><td rowspan="3">3×3×64×128 3×3×128×256</td><td>1</td><td>9 residual blocks</td></tr><tr><td>2</td><td>9 residual blocks</td></tr><tr><td>2</td><td>9 residual blocks</td></tr><tr><td rowspan="2">256 × num_classes</td><td></td><td>8 × 8 global average pool</td></tr><tr><td></td><td>random projection (not trained)</td></tr></table>
+
+![](images/03219ea76d1d037090c58d6ab6307932f31659ff790a6feac7618291024c18dc.jpg)  
+Figure 1: Convergence plots of best model for CIFAR10 (left) and CIFAR (100) right. One step is a gradient update with batch size 128.
+
+An interesting aspect of our model is that despite its massive size of 13.59 million trainable parameters, the model does not seem to overfit too quickly even though the data set size is 50000. In contrast, we found it difficult to train a model with batch normalization of this size without significant overfitting on CIFAR10.
+
+Table 2 summarizes the top-1 classification error of our models compared with a non-exhaustive list of previous works, restricted to the best previous all-convolutional result by Springenberg et al. (2014), the first residual results He et al. (2015), and state-of-the-art results on CIFAR by Huang et al. (2016). All results are with standard data augmentation.
+
+Table 2: Comparison of top-1 classification error on different benchmarks   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>CIFAR10</td><td rowspan=1 colspan=1>CIFAR100</td><td rowspan=1 colspan=1>ImageNet</td><td rowspan=1 colspan=1>remarks</td></tr><tr><td rowspan=1 colspan=1>All-CNN</td><td rowspan=1 colspan=1>7.25</td><td rowspan=1 colspan=1>32.39</td><td rowspan=1 colspan=1>41.2</td><td rowspan=1 colspan=1> all-convolutional, dropout, extra data processing</td></tr><tr><td rowspan=1 colspan=1>Ours</td><td rowspan=1 colspan=1>6.38</td><td rowspan=1 colspan=1>24.64</td><td rowspan=1 colspan=1>35.29</td><td rowspan=1 colspan=1>all-convolutional</td></tr><tr><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>6.43</td><td rowspan=1 colspan=1>25.16</td><td rowspan=1 colspan=1>19.38</td><td></td></tr><tr><td rowspan=1 colspan=1>DenseNet</td><td rowspan=1 colspan=1>3.74</td><td rowspan=1 colspan=1>19.25</td><td rowspan=1 colspan=1>N/A</td><td></td></tr></table>
+
+# 4.2 IMAGENET
+
+The ImageNet ILSVRC 2012 data set has 1, 281, 167 data points with 1000 classes. Each image is resized to $2 2 4 \times 2 2 4$ pixels with 3 channels. We experimented with an all-convolutional variant of the 34-layer network in He et al. (2015). The original model achieved $2 5 . 0 3 \%$ classification error. Our derived model has $3 5 . 7 M$ trainable parameters. We trained the model with a momentum optimizer (with momentum 0.9) and a learning rate schedule that decays by a factor of 0.94 every two epochs, starting from the initial learning rate 0.1. Training was distributed across 6 machines updating asynchronously. Each machine was equipped with 8 GPUs (NVIDIA Tesla K40) and used batch size 256 split across the 8 GPUs so that each GPU updated with batches of size 32.
+
+In contrast to the situation with CIFAR10 and CIFAR100, on ImageNet our all-convolutional model performed significantly worse than its original counterpart. Specifically, we experienced a significant amount of underfitting suggesting that a larger model would likely perform better.
+
+Despite this issue, our model still reached $3 5 . 2 9 \%$ top-1 classification error on the test set (50000 data points), and $1 4 . 1 7 \%$ top-5 test error after 700, 000 steps (about one week of training). While no longer state-of-the-art, this performance is significantly better than the $4 0 . 7 \%$ reported by Krizhevsky et al. (2012), as well as the best all-convolutional architecture by Springenberg et al. (2014). We believe it is quite likely that a better learning rate schedule and hyperparameter settings of our model could substantially improve on the preliminary performance reported here.
+
+# 5 CONCLUSION
+
+Our theory underlines the importance of identity parameterizations when training deep artificial neural networks. An outstanding open problem is to extend our optimization result to the non-linear case where each residual has a single ReLU activiation as in our expressivity result. We conjecture that a result analogous to Theorem 2.2 is true for the general non-linear case. Unlike with the standard parameterization, we see no fundamental obstacle for such a result.
+
+We hope our theory and experiments together help simplify the state of deep learning by aiming to explain its success with a few fundamental principles, rather than a multitude of tricks that need to be delicately combined. We believe that much of the advances in image recognition can be achieved with residual convolutional layers and ReLU activations alone. This could lead to extremely simple (albeit deep) architectures that match the state-of-the-art on all image classification benchmarks.
+
+# REFERENCES
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+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
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+J. T. Springenberg, A. Dosovitskiy, T. Brox, and M. Riedmiller. Striving for Simplicity: The All Convolutional Net. ArXiv e-prints, December 2014.
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+Eric W. Weisstein. Normal matrix, from mathworld–a wolfram web resource., 2016. URL http: //mathworld.wolfram.com/NormalMatrix.html.
+
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+
+# A MISSING PROOFS IN SECTION 2
+
+In this section, we give the complete proofs for Theorem 2.1 and Lemma 2.4, which are omitted in Section 2.
+
+# A.1 PROOF OF THEOREM 2.1
+
+It turns out the proof will be significantly easier if $R$ is assumed to be a symmetric positive semidefinite (PSD) matrix, or if we allow the variables to be complex matrices. Here we first give a proof sketch for the first special case. The readers can skip it and jumps to the full proof below. We will also prove stronger results, namely, $\| A ^ { \star } \| \le 3 \gamma / \ell$ , for the special case.
+
+When $R$ is PSD, it can be diagonalized by orthonormal matrix $U$ in the sense that $R = U Z U ^ { \top }$ , where $Z = \operatorname { d i a g } ( z _ { 1 } , \ldots , z _ { d } )$ is a diagonal matrix with non-negative diagonal entries $z _ { 1 } , \ldots , z _ { d }$ . Let $A _ { 1 } ^ { \star } = \dots = A _ { \ell } ^ { \star } = U \mathrm { d i a g } ( z _ { i } ^ { 1 / \ell } ) U ^ { \top } - \mathrm { I d }$ , then we have
+
+$$
+\begin{array} { r l } & { ( \mathrm { I d } + A _ { \ell } ^ { \star } ) \cdot \cdot \cdot ( \mathrm { I d } + A _ { 1 } ^ { \star } ) = ( U \mathrm { d i a g } ( z _ { i } ^ { 1 / \ell } ) U ^ { \top } ) ^ { \ell } = U \mathrm { d i a g } ( z _ { i } ^ { 1 / \ell } ) ^ { \ell } U \qquad \quad ( \mathrm { s i n c e ~ } U ^ { \top } U = \mathrm { I d } ) } \\ & { \qquad = U Z U ^ { \top } = R . } \end{array}
+$$
+
+We see that the network defined by $A ^ { \star }$ reconstruct the transformation $R$ , and therefore it’s a global minimum of the population risk (formally see Claim 2.3 below). Next, we verify that each of the $A _ { j } ^ { \star }$ has small spectral norm:
+
+$$
+\begin{array} { r l r } {  { \| A _ { j } ^ { \star } \| = \| \mathrm { I d } - U \mathrm { d i a g } ( z _ { i } ^ { 1 / \ell } ) U ^ { \top } ) \| = \| U ( \mathrm { I d } - \mathrm { d i a g } ( z _ { i } ) ^ { 1 / \ell } ) U ^ { \top } \| = \| \mathrm { I d } - \mathrm { d i a g } ( z _ { i } ) ^ { 1 / \ell } \| } } \\ & { } & { \mathrm { ( s i n c e ~ } U \mathrm { ~ i s ~ o r t h o n o n ~ } \mathrm { ~ } \quad } \\ & { } & { \mathrm { ~ } = \operatorname* { m a x } _ { i } | z _ { i } ^ { 1 / \ell } - 1 | . } \end{array}
+$$
+
+rmal)
+
+Since $\sigma _ { \operatorname* { m i n } } ( R ) \leq z _ { i } \leq \sigma _ { \operatorname* { m a x } } ( R )$ , we have $\ell \geq 3 \gamma \geq | \log z _ { i } |$ . It follows that
+
+$$
+| z _ { i } ^ { 1 / \ell } - 1 | = | e ^ { ( \log z _ { i } ) / \ell } - 1 | \leq 3 | ( \log z _ { i } ) / \ell | \leq 3 \gamma / \ell . \quad \mathrm { ( s i n c e ~ } | e ^ { x } - 1 | \leq 3 | x | \mathrm { ~ f ~ e ~ r ~ , ~ }
+$$
+
+Then using equation (A.1) and the equation above, we have that $\| A \| \leq \operatorname* { m a x } _ { j } \| A _ { j } ^ { \star } \| \leq 3 \gamma / \ell$ , which completes the proof for the special case.
+
+Next we give the formal full proof of Theorem 2.1.
+
+Proof of Theorem 2.1. We assume the dimension $d$ is an even number. The odd case has very similar proof and is left to the readers. Let $R = U K V ^ { \top }$ be its singular value decomposition, where $U , V$ are two orthonormal matrices and $K$ is a diagonal matrix. Since $U$ is a normal matrix (that is, $U$ satisfies that $U U ^ { \top } = U ^ { \top } U )$ , by Claim C.1, we have that $U$ can be block-diagnolaized by orthonormal matrix $S$ into $U = S D S ^ { - 1 }$ , where $D = \operatorname { d i a g } ( D _ { 1 } , \dots , D _ { d / 2 } )$ is a real block diagonal matrix with each block $D _ { i }$ being of size $2 \times 2$ .
+
+Since $U$ is orthonormal, $U$ has all its eigenvalues lying on the unit circle (in complex plane). Since $D$ and $U$ are unitarily similar to each other, $D$ also has eigenvalues lying on the unit circle, and so does each of the block $D _ { i }$ . This means that each $D _ { i }$ is a $2 \times 2$ dimensional rotation matrix. Each rotation matrix can be written as $T ( \theta ) = \left[ \sin \theta \quad - \sin \theta \right]$ . Suppose $D _ { i } = T ( \theta _ { i } )$ where $\theta _ { i } \in [ - \pi , \pi ]$ . Then we have that $D _ { i } = T ( \theta _ { i } / q ) ^ { q }$ for any integer $q$ (that is chosen later). Let $W = \mathrm { d i a g } ( T ( \theta _ { i } / q ) )$ . Therefore, it follows that $D = \mathrm { d i a g } ( D _ { i } ) = W ^ { q }$ . Moreover, we have $U = S D S ^ { - 1 } = ( \dot { S } W \dot { S } ^ { - \bar { 1 } } ) ^ { q }$ . Therefore, let $B _ { 1 } = B _ { 2 } = \cdot \cdot \cdot = B _ { q } = \operatorname { I d } - S W S ^ { - 1 }$ , then we have $U = ( \operatorname { I d } + B _ { q } ) \dots ( \operatorname { I d } + B _ { 1 } )$ . We verify the spectral norm of these matrices are indeed small,
+
+$$
+\begin{array} { r l r } {  { \| B _ { j } \| = \| \mathrm { I d } - S W S ^ { - 1 } \| = \| S ( \mathrm { I d } - W ) S ^ { - 1 } \| } } \\ & { } & { = \| \mathrm { I d } - W \| } \\ & { } & { = \underset { i \in [ d / 2 ] } { \operatorname* { m a x } } \| T ( 0 ) - T ( \theta _ { i } / q ) \| \qquad \mathrm { ( s i n c e ~ } W = \mathrm { d i a g } ( T ( \theta _ { i } / q ) \mathrm { ) } \mathrm { i } } \\ & { } & { = \operatorname* { m a x } | \mathrm { s i n } ( \theta _ { i } / q ) | \leq \pi / q . } \end{array}
+$$
+
+Similarly, we can choose $B _ { 1 } ^ { \prime } , \ldots , B _ { q } ^ { \prime }$ with $\| C _ { j } \| \le \pi / q$ so that $V ^ { \top } = \left( \operatorname { I d } + B _ { q } ^ { \prime } \right) \ldots \left( \operatorname { I d } + B _ { 1 } ^ { \prime } \right)$
+
+Last, we deal with the diagonal matrix $K$ . Let $\begin{array} { r c l } { K } & { = } & { \mathrm { d i a g } ( k _ { i } ) } \end{array}$ . We have $\begin{array} { r l } { \operatorname* { m i n } k _ { i } } & { { } = } \end{array}$ $\sigma _ { \operatorname* { m i n } } ( R ) , \operatorname* { m a x } k _ { i } = \sigma _ { \operatorname* { m a x } } ( R )$ . Then, we can write $K = ( K ^ { \prime } ) ^ { p }$ where $K ^ { \prime } = \mathrm { d i a g } ( k _ { i } ^ { 1 / p } )$ and $p$ is an integer to be chosen later. We have that $\lVert K ^ { \prime } - \mathbf { I d } \rVert \leq \operatorname* { m a x } | k _ { i } ^ { 1 / p } - 1 | \leq \operatorname* { m a x } | e ^ { \log k _ { i } \cdot 1 / p } - 1 | .$ . When $p \geq \gamma = \operatorname* { m a x } \{ \log \operatorname* { m a x } k _ { i } , - \log \operatorname* { m i n } k _ { i } \} = \operatorname* { m a x } \{ \log \sigma _ { \operatorname* { m a x } } ( R ) , - \log \sigma _ { \operatorname* { m i n } } ( R ) \}$ , we have that
+
+$$
+\begin{array} { r } { \| K ^ { \prime } - \mathbf { I d } \| \leq \operatorname* { m a x } \left| e ^ { \log k _ { i } \cdot 1 / p } - 1 \right| \leq 3 \operatorname* { m a x } \left| \log k _ { i } \cdot 1 / p \right| = 3 \gamma / p . } \end{array}
+$$
+
+Let $B _ { 1 } ^ { \prime \prime } = \cdot \cdot \cdot = B _ { p } ^ { \prime \prime } = K ^ { \prime } - \mathrm { I d }$ and then we have $K = ( { \mathrm { I d } } + B _ { p } ^ { \prime \prime } ) \cdot \cdot \cdot ( { \mathrm { I d } } + B _ { 1 } ^ { \prime \prime } )$ . Finally, we choose p = \` 3γ2(√π+√3γ) and $\begin{array} { r } { q = \frac { \ell \sqrt { \pi } } { \sqrt { \pi } + \sqrt { 3 \gamma } } } \end{array}$ , 4and let $A _ { 2 p + q } = B _ { q } , \cdot \cdot \cdot = A _ { p + q + 1 } = B _ { 1 } , A _ { p + q } =$ $B _ { p } ^ { \prime \prime } , \cdot \cdot \cdot , A _ { q + 1 } = B _ { 1 } ^ { \prime \prime } , A _ { q } = B _ { q } ^ { \prime } , \cdot \cdot \cdot , A _ { 1 } = B _ { 1 } ^ { \prime }$ . We have that $2 q + \ell = 1$ and
+
+$$
+R = U K V ^ { \top } = \left( \mathbf { I d } + A _ { \ell } \right) \ldots \left( \mathbf { I d } + A _ { 1 } \right) .
+$$
+
+Moreover, we have $\| A \| \le \operatorname* { m a x } \{ \| B _ { j } \| , \| B _ { j } ^ { \prime } \| . \| B _ { j } ^ { \prime \prime } \| \} \le \pi / q + 3 \gamma / p \le 2 ( \sqrt \pi + \sqrt { 3 \gamma } ) ^ { 2 } / \ell ,$ as desired.
+
+# A.2 PROOF OF LEMMA 2.4
+
+We compute the partial gradients by definition. Let $\Delta _ { j } \in \mathbb { R } ^ { d \times d }$ be an infinitesimal change to $A _ { j }$ . Using Claim 2.3, consider the Taylor expansion of $f ( \bar { A _ { 1 } } , \ldots , A _ { \ell } + \Delta _ { j } , \ldots , A _ { \ell } )$
+
+$$
+\begin{array} { r l } & { f ( A _ { 1 } , \ldots , A _ { \ell } + \Delta _ { j } , \ldots , A _ { \ell } ) } \\ & { = \left\| \big ( ( \mathbf { I d } + A _ { \ell } ) \cdots ( \mathbf { I d } + A _ { j } + \Delta _ { j } ) \ldots ( \mathbf { I d } + A _ { 1 } ) - R ) \Sigma ^ { 1 / 2 } \right\| _ { F } ^ { 2 } } \\ & { = \Big \| \big ( ( \mathbf { I d } + A _ { \ell } ) \cdot \cdot ( \mathbf { I d } + A _ { 1 } ) - R ) \Sigma ^ { 1 / 2 } + ( \mathbf { I d } + A _ { \ell } ) \cdot \cdot \cdot \Delta _ { j } \ldots ( \mathbf { I d } + A _ { 1 } ) \Sigma ^ { 1 / 2 } \Big \| _ { F } ^ { 2 } } \\ & { = \Big \| \big ( \mathbf { I d } + A _ { \ell } ) \cdot \cdot ( \mathbf { I d } + A _ { 1 } ) - R ) \Sigma ^ { 1 / 2 } \Big \| _ { F } ^ { 2 } + } \\ & { \quad 2 \langle ( ( \mathbf { I d } + A _ { \ell } ) \cdot \cdot ( \mathbf { I d } + A _ { 1 } ) - R ) \Sigma ^ { 1 / 2 } , ( \mathbf { I d } + A _ { \ell } ) \cdot \cdot \Delta _ { j } \ldots ( \mathbf { I d } + A _ { 1 } ) \Sigma ^ { 1 / 2 } \rangle + O ( \| \Delta _ { j } \| _ { F } ^ { 2 } ) } \\ & { = f ( A ) + 2 \langle ( \mathbf { I d } + A _ { \ell } ^ { \top } ) \ldots ( \mathbf { I d } + A _ { j + 1 } ^ { \top } ) E \Sigma ( \mathbf { I d } + A _ { j - 1 } ^ { \top } ) \ldots ( \mathbf { I d } + A _ { 1 } ^ { \top } ) , \Delta _ { j } \rangle + O ( \| \Delta _ { j } \| _ { F } ^ { 2 } ) . } \end{array}
+$$
+
+By definition, this means that the $\begin{array} { r } { \frac { \partial f } { \partial A _ { j } } \ = \ 2 \big ( \mathbf { I d } + A _ { \ell } ^ { \top } \big ) \ldots \big ( \mathbf { I d } + A _ { j + 1 } ^ { \top } \big ) E \Sigma \big ( \mathbf { I d } + A _ { j - 1 } ^ { \top } \big ) \ldots \big ( \mathbf { I d } + } \end{array}$ $A _ { 1 } ^ { \top }$ ).
+
+# B MISSING PROOFS IN SECTION 3
+
+In this section, we provide the full proof of Theorem 3.2. We start with the following Lemma that constructs a building block $\tau$ that transform $k$ vectors of an arbitrary sequence of $n$ vectors to any arbitrary set of vectors, and main the value of the others. For better abstraction we use $\alpha ^ { ( i ) } , \beta ^ { ( i ) }$ to denote the sequence of vectors.
+
+Lemma B.1. Let $S \subset [ n ]$ be of size $k$ . Suppose $\alpha ^ { ( 1 ) } , \ldots , \alpha ^ { ( n ) }$ is a sequences of $n$ vectors satisfying a) for every $1 \leq i \leq n$ , we have $1 - \rho ^ { \prime } \leq \| \alpha _ { i } \| ^ { 2 } \leq 1 + \rho ^ { \prime }$ , and $^ b$ ) if $\because j \neq j$ and $S$ contains at least one of $i , j$ , then $\| \alpha ^ { ( i ) } - \beta ^ { ( j ) } \| \ge 3 \rho ^ { \prime }$ . Let $\beta ^ { ( 1 ) } , \ldots , \beta ^ { ( n ) }$ be an arbitrary sequence of vectors. Then, there exists $U , V \in \mathbb { R } ^ { k \times k } , .$ s such that for every $i \in S$ , we have $\mathcal { T } _ { U , V , s } \big ( \alpha ^ { ( i ) } \big ) = \beta ^ { ( i ) } - \alpha ^ { ( i ) }$ , and moreover, for every $i \in [ n ] \backslash S$ we have $\mathcal { T } _ { U , V , s } ( \alpha ^ { ( i ) } ) = 0$ .
+
+We can see that the conclusion implies
+
+$$
+\begin{array} { r } { \beta ^ { ( i ) } = \alpha ^ { ( i ) } + \mathcal { T } _ { U , V , s } ( \alpha ^ { ( i ) } ) ~ \forall i \in S } \\ { \alpha ^ { ( i ) } = \alpha ^ { ( i ) } + \mathcal { T } _ { U , V , s } ( \alpha ^ { ( i ) } ) ~ \forall i \notin S } \end{array}
+$$
+
+which is a different way of writing equation (3.6).
+
+Proof of Lemma B.1. Without loss of generality, suppose $S = \{ 1 , \ldots , k \}$ . We construct $U , V , s$ as follows. Let the $i$ -th row of $U$ be $\alpha ^ { ( i ) }$ for $i \in [ k ]$ , and let $s = - ( 1 - 2 \rho ^ { \prime } ) \cdot { \bf 1 }$ where 1 denotes the all 1’s vector. Let the $i$ -column of $V$ be $\begin{array} { r } { \frac { 1 } { \| \alpha ^ { ( i ) } \| ^ { 2 } - ( 1 - 2 \rho ^ { \prime } ) } \big ( \beta ^ { ( i ) } - \alpha ^ { ( i ) } \big ) } \end{array}$ for $i \in [ k ]$ .
+
+Next we verify that the correctness of the construction. We first consider $1 \leq i \leq k$ . We have that $U \alpha ^ { ( i ) }$ is a a vector with $i$ -th coordinate equal to $\| \alpha ^ { ( i ) } \| ^ { 2 } \ge 1 - \rho ^ { \prime }$ . The $j$ -th coordinate of $U \alpha ^ { ( i ) }$ is equal to $\langle \alpha ^ { ( j ) } , \alpha ^ { ( i ) } \rangle$ , which can be upperbounded using the assumption of the Lemma by
+
+$$
+\langle \alpha ^ { ( j ) } , \alpha ^ { ( i ) } \rangle = \frac { 1 } { 2 } \left( \| \alpha ^ { ( i ) } \| ^ { 2 } + \| \alpha ^ { ( j ) } \| ^ { 2 } \right) - \| \alpha ^ { ( i ) } - \alpha ^ { ( j ) } \| ^ { 2 } \le 1 + \rho ^ { \prime } - 3 \rho ^ { \prime } \le 1 - 2 \rho ^ { \prime } .
+$$
+
+Therefore, this means $U \alpha ^ { ( i ) } \ : - \ : ( 1 \ : - \ : 2 \rho ^ { \prime } )$ · 1contains a single positive entry (with value at least $\| { \boldsymbol { \alpha } } ^ { ( i ) } \| ^ { 2 } - ( 1 - 2 \rho ^ { \prime } ) \geq \rho ^ { \prime } )$ , and all other entries being non-positive. This means that $\mathrm { R e L u } ( U \alpha ^ { ( i ) } + b ) = \left( \| \alpha ^ { ( i ) } \| ^ { 2 } - ( 1 - 2 \rho ^ { \prime } ) \right) e _ { i }$ where $e _ { i }$ is the $i$ -th natural basis vector. It follows that $V \mathrm { R e L u } ( U \alpha ^ { ( i ) } + b ) = ( \| \alpha ^ { ( i ) } \| ^ { 2 } - ( 1 - 2 \rho ^ { \prime } ) ) V e _ { i } = \beta ^ { ( i ) } - \alpha ^ { ( i ) } .$
+
+Finally, consider $n \geq i > k$ . Then similarly to the computation in equation (B.1), $U \alpha ^ { ( i ) }$ is a vector with all coordinates less than $1 - 2 \rho ^ { \prime }$ . Therefore $\boldsymbol { U } \boldsymbol { \alpha } ^ { ( i ) } + \boldsymbol { b }$ is a vector with negative entries. Hence we have $\mathtt { R e L u } ( U \alpha ^ { ( i ) } + b ) = 0$ , which implies $V \mathrm { R e L u } ( U \alpha ^ { ( i ) } + b ) = 0$ . □
+
+Now we are ready to state the formal version of Lemma 3.3.
+
+Lemma B.2. Suppose a sequence of $n$ vectors $z ^ { ( 1 ) } , \ldots , z ^ { ( n ) }$ satisfies a relaxed version of Assumption 3.1: a) for every i, $1 - \bar { \rho } ^ { \prime } \leq \| z ^ { ( i ) } \| ^ { 2 } \leq 1 + \rho ^ { \prime } b )$ for every $i \neq j$ , we have $\| z ^ { ( i ) } - z ^ { ( j ) } \| ^ { 2 } \geq \dot { \rho ^ { \prime } }$ ;. Let $v ^ { ( 1 ) } , \ldots , v ^ { ( n ) }$ be defined above. Then there exists weigh matrices $( A _ { 1 } , B _ { 1 } ) , \ldots , ( A _ { \ell } , B _ { \ell } )$ , such that given $\forall i , h _ { 0 } ^ { ( i ) } = z ^ { ( i ) }$ , we have,
+
+$$
+\forall i \in \{ 1 , \ldots , n \} , h _ { \ell } ^ { ( i ) } = v ^ { ( i ) } .
+$$
+
+We will use Lemma B.1 repeatedly to construct building blocks $\mathcal { T } _ { A _ { j } , B _ { k } , s _ { j } } ( \cdot )$ , and thus prove Lemma B.2. Each building block $\mathcal { T } _ { A _ { j } , B _ { k } , s _ { j } } \left( \cdot \right)$ takes a subset of $k$ vectors among $\{ z ^ { ( 1 ) } , \ldots , z ^ { ( n ) } \}$ and convert them to $v ^ { ( i ) }$ ’s, while maintaining all other vectors as fixed. Since they are totally $n / k$ layers, we finally maps all the $z ^ { ( i ) }$ ’s to the target vectors $v ^ { ( i ) }$ ’s.
+
+Proof of Lemma B.2. We use Lemma B.1 repeatedly. Let $S _ { 1 } = [ 1 , \ldots , k ]$ . Then using Lemma B.1 with $\alpha ^ { ( i ) } = z ^ { ( i ) }$ and $\beta ^ { ( i ) } = v ^ { ( i ) }$ for $i \in [ n ]$ , we obtain that there exists $A _ { 1 } , B _ { 1 } , b _ { 1 }$ such that for $i ~ \leq ~ k$ , it holds that $h _ { 1 } ^ { ( i ) } = z ^ { ( i ) } + \mathcal { T } _ { A _ { 1 } , B _ { 1 } , b _ { 1 } } \big ( z ^ { ( i ) } \big ) = v ^ { ( i ) }$ , and for $i \geq k$ , it holds that $h _ { 1 } ^ { ( i ) } =$ $z ^ { ( i ) } + \mathcal { T } _ { A _ { 1 } , B _ { 1 } , b _ { 1 } } ( z ^ { ( i ) } ) \bar { = } z ^ { ( i ) }$ .
+
+Now we construct the other layers inductively. We will construct the layers such that the hidden variable at layer $j$ satisfies $h _ { j } ^ { ( i ) } = v ^ { ( i ) }$ for every $1 \leq i \leq j k$ , and $h _ { j } ^ { ( i ) } = z ^ { ( i ) }$ for every $n \geq i >$ $j k$ . Assume that we have constructed the first $j$ layer and next we use Lemma B.1 to construct the $j + 1$ layer. Then we argue that the choice of $\alpha ^ { ( 1 ) } = v ^ { ( 1 ) } , \ldots , \alpha ^ { ( j k ) } = v ^ { ( j k ) } , \alpha ^ { ( j k + 1 ) } =$ $z ^ { ( j k + 1 ) } , \cdot \cdot \cdot , \dot { \alpha } ^ { ( n ) } = z ^ { ( n ) }$ , and $S = \{ j k + 1 , \ldots , ( j + 1 ) k \}$ satisfies the assumption of Lemma B.1. Indeed, because $q _ { i }$ ’s are chosen uniformly randomly, we have w.h.p for every $s$ and $i$ , $\langle q _ { s } , z ^ { ( i ) } \rangle \leq$ $1 - \rho ^ { \prime }$ . Thus, since $v ^ { ( i ) } \in \{ q _ { 1 } , \ldots , q _ { r } \}$ , we have that $v ^ { ( i ) }$ also doesn’t correlate with any of the $z ^ { ( i ) }$ . Then we apply Lemma B.1 and conclude that there exists $A _ { j + 1 } = U , B _ { j + 1 } = V , b _ { j + 1 } = s$ such that $\mathcal { T } _ { A _ { j + 1 } , b _ { j + 1 } , b _ { j + 1 } } ( v ^ { ( i ) } ) = 0$ for $i \leq j k , \mathcal { T } _ { A _ { j + 1 } , b _ { j + 1 } , b _ { j + 1 } } ( z ^ { ( i ) } ) = v ^ { ( i ) } - z ^ { ( i ) }$ for $j k < i \leq ( j + 1 ) k$ , and $\mathcal { T } _ { A _ { j + 1 } , b _ { j + 1 } , b _ { j + 1 } } ( z ^ { ( i ) } ) = 0$ for $n \geq i > ( j + 1 ) k$ . These imply that
+
+$$
+\begin{array} { r l } & { h _ { j + 1 } ^ { ( i ) } = h _ { j } ^ { ( i ) } + { \mathcal T } _ { A _ { j + 1 } , b _ { j + 1 } , b _ { j + 1 } } \big ( v ^ { ( i ) } \big ) = v ^ { ( i ) } \quad \forall 1 \leq i \leq j k } \\ & { h _ { j + 1 } ^ { ( i ) } = h _ { j } ^ { ( i ) } + { \mathcal T } _ { A _ { j + 1 } , b _ { j + 1 } , b _ { j + 1 } } \big ( z ^ { ( i ) } \big ) = v ^ { ( i ) } \quad \forall j k + 1 \leq i \leq ( j + 1 ) k } \\ & { h _ { j + 1 } ^ { ( i ) } = h _ { j } ^ { ( i ) } + { \mathcal T } _ { A _ { j + 1 } , b _ { j + 1 } , b _ { j + 1 } } \big ( z ^ { ( i ) } \big ) = z ^ { ( i ) } \quad \forall ( j + 1 ) k < i \leq n } \end{array}
+$$
+
+Therefore we constructed the $j + 1$ layers that meets the inductive hypothesis for layer $j + 1$ Therefore, by induction we get all the layers, and the last layer satisfies that $h _ { \ell } ^ { ( i ) } = v ^ { ( i ) }$ for every example $i$ . □
+
+Now we ready to prove Theorem 3.2, following the general plan sketched in Section 3.
+
+Proof of Theorem 3.2. We use formalize the intuition discussed below Theorem 3.2. First, take $k = \overset { \cdot } { c } ( \log n ) / \rho ^ { 2 }$ for sufficiently large absolute constant $c$ (for example, $c = 1 0$ works), by JohnsonLindenstrauss Theorem (Johnson $\&$ Lindenstrauss (1984), or see Wikipedia (2016)) we have that when $A _ { 0 }$ is a random matrix with standard normal entires, with high probability, all the pairwise distance between the the set of vectors $\{ 0 , x ^ { ( 1 ) } , \ldots , x ^ { ( n ) } \}$ are preserved up to $1 \pm \rho / 3$ factor. That is, we have that for every $i , 1 { - } \rho / 3 \le \| A _ { 0 } x ^ { ( i ) } \| \le 1 { + } \rho / 3$ , and for every $i \neq j$ , $\| A _ { 0 } x ^ { ( i ) } - A _ { 0 } x ^ { ( j ) } \| \ge$ $\rho ( 1 - \rho / 3 ) \ge 2 \rho / 3$ . Let $z ^ { ( i ) } = A _ { 0 } x ^ { ( i ) }$ and $\rho ^ { \prime } = \rho / 3$ . Then we have $z ^ { ( i ) }$ ’s satisfy the condition of Lemam B.2. We pick $r$ random vectors $q _ { 1 } , \ldots , q _ { r }$ in $\mathbb { R } ^ { k }$ . Let $v ^ { ( 1 ) } , \ldots , v ^ { ( n ) }$ be defined as in equation (3.2). Then by Lemma B.2, we can construct matrices $( A _ { 1 } , B _ { 1 } ) , \ldots , ( A _ { \ell } , B _ { \ell } )$ such that
+
+$$
+h _ { \ell } ^ { ( i ) } = v ^ { ( i ) } .
+$$
+
+Note that $v ^ { ( i ) } \in \{ q _ { 1 } , \ldots , q _ { r } \}$ , and $q _ { i }$ ’s are random unit vector. Therefore, the choice of $\alpha ^ { ( 1 ) } =$ $q _ { 1 } , \ldots , \alpha ^ { ( r ) } = q _ { r }$ , $\beta ^ { ( 1 ) } = e _ { 1 } , \ldots , \beta ^ { ( r ) } = e _ { r }$ , and satisfies the condition of Lemma B.1, and using Lemma B.1 we conclude that there exists $A _ { \ell + 1 } , B _ { \ell + 1 } , s _ { \ell + 1 }$ such that
+
+$$
+e _ { j } = v _ { j } + \mathcal { T } _ { A _ { \ell + 1 } , B _ { \ell + 1 } , b _ { \ell + 1 } } ( v _ { j } ) , \mathrm { ~ f o r ~ e v e r y ~ } j \in \{ 1 , \dots , r \} \dots
+$$
+
+By the definition of $v ^ { ( i ) }$ in equation (3.2) and equation (B.2), we conclude that $\hat { y } ^ { ( i ) } = h _ { \ell } ^ { ( i ) } +$ $\begin{array} { r } { \mathcal { T } _ { A _ { \ell + 1 } , B _ { \ell + 1 } , b _ { \ell + 1 } } ( h _ { \ell } ^ { ( i ) } ) = y ^ { ( i ) } . } \end{array}$ ., which complete the proof. □
+
+# C TOOLBOX
+
+In this section, we state two folklore linear algebra statements. The following Claim should be known, but we can’t find it in the literature. We provide the proof here for completeness.
+
+Claim C.1. Let $U \in \mathbb { R } ^ { d \times d }$ be a real normal matrix (that is, it satisfies $U U ^ { \top } = U ^ { \top } U$ ). Then, there exists an orthonormal matrix $S \in \mathbb { R } ^ { d \times d }$ such that
+
+$$
+U = S D S ^ { \top } ,
+$$
+
+where $D$ is a real block diagonal matrix that consists of blocks with size at most $2 \times 2$ . Moreover, if $d$ is even, then $D$ consists of blocks with size exactly $2 \times 2$ .
+
+Proof. Since $U$ is a normal matrix, it is unitarily diagonalizable (see Weisstein (2016) for backgrounds). Therefore, there exists unitary matrix $V$ in $\mathbb { C } ^ { d \times d }$ and diagonal matrix in $\mathbb { C } ^ { d \times \breve { d } }$ such that $U$ has eigen-decomposition $\begin{array} { r c l } { U } & { = } & { V \Lambda V ^ { * } } \end{array}$ . Since $U$ itself is a real matrix, we have that the eigenvalues (the diagonal entries of $\Lambda$ ) come as conjugate pairs, and so do the eigenvectors (which are the columns of $V$ ). That is, we can group the columns of $V$ into pairs $( v _ { 1 } , \bar { v } _ { 1 } ) , \ldots , ( v _ { s } , \bar { v _ { s } } ) , v _ { s + 1 } , \ldots , v _ { t } .$ , and let the corresponding eigenvalues be $\lambda _ { 1 } , \bar { \lambda } _ { 1 } , \dots , \bar { \lambda } _ { \lambda _ { s } } , \bar { \lambda } _ { s } , \lambda _ { s + 1 } , \dots , \bar { \lambda _ { t } }$ . Here $\lambda _ { s + 1 } , \ldots , \lambda _ { t } \in \mathbb { R }$ . Then we get that $U \ =$ $\begin{array} { r } { \sum _ { i = 1 } ^ { s } 2 \Re ( v _ { i } \lambda _ { i } v _ { i } ^ { * } ) + \sum _ { i = s + 1 } ^ { t } v _ { i } \lambda _ { i } v _ { i } ^ { \top } } \end{array}$ . Let $Q _ { i } = \Re ( v _ { i } \lambda _ { i } v _ { i } ^ { * } )$ , then we have that $Q _ { i }$ is a real matrix of rank-2. Let $S _ { i } \in \mathbb { R } ^ { d \times 2 }$ be a orthonormal basis of the column span of $Q _ { i }$ and then we have that $Q _ { i }$ can be written as $Q _ { i } = S _ { i } D _ { i } S _ { i } ^ { \top }$ where $D _ { i }$ is a $2 \times 2$ matrix. Finally, let $S = [ S _ { 1 } , \ldots , S _ { s } , v _ { s + 1 } , \ldots , v _ { t } ]$ , and $D = \mathrm { d i a g } ( D _ { 1 } , \dots , D _ { s } , \lambda _ { s + 1 } , \dots , \lambda _ { t } )$ we complete the proof. □
+
+The following Claim is used in the proof of Theorem 2.2. We provide a proof here for completeness. Claim C.2 (folklore). For any two matrices $A , B \in \mathbb { R } ^ { d \times d }$ , we have that
+
+$$
+\| A B \| _ { F } \geq \sigma _ { \operatorname* { m i n } } ( A ) \| B \| _ { F } .
+$$
+
+Proof. Since $\sigma _ { \mathrm { m i n } } ( A ) ^ { 2 }$ is the smallest eigenvalue of $A ^ { \top } A$ , we have that
+
+$$
+B ^ { \top } A ^ { \top } A B \succeq B ^ { \top } \cdot \sigma _ { \operatorname* { m i n } } ( A ) ^ { 2 } \mathrm { I d } \cdot B .
+$$
+
+Therefore, it follows that
+
+$$
+\begin{array} { r l } & { \| A B \| _ { F } ^ { 2 } = \mathrm { t r } ( B ^ { \top } A ^ { \top } A B ) \geq \mathrm { t r } ( B ^ { \top } \cdot \sigma _ { \operatorname* { m i n } } ( A ) ^ { 2 } \mathrm { I d } \cdot B ) } \\ & { \qquad = \sigma _ { \operatorname* { m i n } } ( A ) ^ { 2 } \mathrm { t r } ( B ^ { \top } B ) = \sigma _ { \operatorname* { m i n } } ( A ) ^ { 2 } \| B \| _ { F } ^ { 2 } . } \end{array}
+$$
+
+Taking square root of both sides completes the proof.