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+# SCALABLE UNBALANCED OPTIMAL TRANSPORT USING GENERATIVE ADVERSARIAL NETWORKS
+
+Karren D. Yang & Caroline Uhler Laboratory for Information & Decision Systems Institute for Data, Systems and Society Massachusetts Institute of Technology Cambridge, MA, USA {karren, cuhler}@mit.edu
+
+# ABSTRACT
+
+Generative adversarial networks (GANs) are an expressive class of neural generative models with tremendous success in modeling high-dimensional continuous measures. In this paper, we present a scalable method for unbalanced optimal transport (OT) based on the generativeadversarial framework. We formulate unbalanced OT as a problem of simultaneously learning a transport map and a scaling factor that push a source measure to a target measure in a cost-optimal manner. We provide theoretical justification for this formulation, showing that it is closely related to an existing static formulation by Liero et al. (2018). We then propose an algorithm for solving this problem based on stochastic alternating gradient updates, similar in practice to GANs, and perform numerical experiments demonstrating how this methodology can be applied to population modeling.
+
+# 1 INTRODUCTION
+
+We consider the problem of unbalanced optimal transport: given two measures, find a cost-optimal way to transform one measure to the other using a combination of mass variation and transport. Such problems arise, for example, when modeling the transformation of a source population into a target population (Figure 1a). In this setting, one needs to model mass transport to account for the features that are evolving, as well as local mass variations to allow sub-populations to become more or less prominent in the target population (Schiebinger et al., 2017).
+
+Classical optimal transport (OT) considers the problem of pushing a source to a target distribution in a way that is optimal with respect to some transport cost without allowing for mass variations. Modern approaches are based on the Kantorovich formulation (Kantorovich, 1942), which seeks the optimal probabilistic coupling between measures and can be solved using linear programming methods for discrete measures. Recently, Cuturi (2013) showed that regularizing the objective using an entropy term allows the dual problem to be solved more efficiently using the Sinkhorn algorithm. Stochastic methods based on the dual objective have been proposed for the continuous setting (Genevay et al., 2016; Seguy et al., 2017; Arjovsky et al., 2017). Optimal transport has been applied to many areas, such as computer graphics (Ferradans et al., 2014; Solomon et al., 2015) and domain adaptation (Courty et al., 2014; 2017).
+
+In many applications where a transport cost is not available, transport maps can also be learned using generative models such as generative adversarial networks (GANs) (Goodfellow et al., 2014), which push a source distribution to a target distribution by training against an adversary. Numerous transport problems in image translation (Mirza & Osindero, 2014; Zhu et al., 2017; Yi et al., 2017), natural language translation (He et al., 2016), domain adaptation (Bousmalis et al., 2017) and biological data integration (Amodio &
+
+![](images/77cead8f097106199f0fcac5724129b1bc02f84abf4e99c70b0a94a288141d93.jpg)  
+Figure 1: (a) Illustration of the problem of modeling the transformation of a source population $\mu$ to a target population $\nu$ . In this example, one sub-population is growing more rapidly than the others. (b) Schematic of Monge-like formulations of unbalanced optimal transport. The objective is to learn a transport map $T$ (for transporting mass) and scaling factor $\xi$ (for mass variation) to push the source $\mu$ to the target $\nu$ , using a deterministic transport map (top) (Chizat et al., 2015) or a stochastic transport map (bottom).
+
+Krishnaswamy, 2018) have been tackled using variants of GANs, with strategies such as conditioning or cycle-consistency employed to enforce correspondence between original and transported samples. However, all these methods conserve mass between the source and target and therefore cannot handle mass variation.
+
+Several formulations have been proposed for extending the theory of OT to the setting where the measures can have unbalanced masses (Chizat et al., 2015; 2018; Kondratyev et al., 2016; Liero et al., 2018; Frogner et al., 2015). In terms of numerical methods, a class of scaling algorithms (Chizat et al., 2016) that generalize the Sinkhorn algorithm for balanced OT have been developed for approximating the solution to optimal entropy-transport problems; this formulation of unbalanced OT by Liero et al. (2018) corresponds to the Kantorovich OT problem in which the hard marginal constraints are relaxed using divergences to allow for mass variation. In practice, these algorithms have been used to approximate unbalanced transport plans between discrete measures for applications such as computer graphics (Chizat et al., 2016), tumor growth modeling (Chizat & Di Marino, 2017) and computational biology (Schiebinger et al., 2017). However, while optimal entropy-transport allows mass variation, it cannot explicitly model it, and there are currently no methods that can perform unbalanced OT between continuous measures.
+
+Contributions. Inspired by the recent successes of GANs for high-dimensional transport problems, we present a novel framework for unbalanced optimal transport that directly models mass variation in addition to transport. Concretely, our contributions are the following:
+
+• We propose to solve a Monge-like formulation of unbalanced OT, in which the goal is to learn a stochastic transport map and scaling factor to push a source to a target measure in a cost-optimal manner. This generalizes the unbalanced Monge OT problem by Chizat et al. (2015).   
+• By relaxing this problem, we obtain an alternative form of the optimal entropy-transport problem by Liero et al. (2018), which confers desirable theoretical properties.   
+• We develop scalable methodology for solving the relaxed problem. Our derivation uses a convex conjugate representation of divergences, resulting in an alternating gradient descent method similar to GANs (Goodfellow et al., 2014).   
+• We demonstrate in practice how our methodology can be applied towards population modeling using the MNIST and USPS handwritten digits datasets, the CelebA dataset, and a recent single-cell RNA-seq dataset from zebrafish embrogenesis.
+
+In addition to these main contributions, for completeness we also propose a new scalable method (Algorithm 2) in the Appendix for solving the optimal-entropy transport problem by Liero et al. (2018) in the continuous setting. The algorithm extends the work of Seguy et al. (2017) to unbalanced OT and is a scalable alternative to the algorithm of Chizat et al. (2016) for very large or continuous datasets.
+
+# 2 PRELIMINARIES
+
+Notation. Let $\mathcal { X } , \mathcal { Y } \subseteq \mathbb { R } ^ { n }$ be topological spaces and let $\boldsymbol { B }$ denote the Borel $\sigma$ -algebra. Let $\mathcal { M } _ { + } ^ { 1 } ( \mathcal { X } ) , \mathcal { M } _ { + } ( \mathcal { X } )$ denote respectively the space of probability measures and finite non-negative measures over $\mathcal { X }$ . For a measurable function $T$ , let $T _ { \# }$ denote its pushforward operator: if $\mu$ is a measure, then $T _ { \# } \mu$ is the pushforward measure of $\mu$ under $T$ . Finally, let $\pi ^ { \mathcal { X } }$ , $\pi ^ { y }$ be functions that project onto $\mathcal { X }$ and $\mathcal { V }$ ; for a joint measure $\gamma \in \mathcal { M } _ { + } ( \mathcal { X } \times \mathcal { Y } )$ , $\pi _ { \# } ^ { \chi } \gamma$ and $\pi _ { \# } ^ { \dot { y } } \gamma$ are its marginals with respect to $\mathcal { X }$ and $\mathcal { V }$ respectively.
+
+Optimal transport (OT) addresses the problem of transporting between measures in a cost-optimal manner. Monge (1781) formulated this problem as a search over deterministic transport maps. Specifically, given $\mu \in \mathcal { M } _ { + } ^ { 1 } ( \mathcal { X } ) , \nu \in \mathcal { M } _ { + } ^ { 1 } ( \mathcal { Y } )$ and a cost function $c : \mathcal { X } \times \mathcal { Y }  \mathbb { R } ^ { + }$ , Monge OT seeks a measurable function $T : \mathcal { X } \overset { \cdot } {  } \mathcal { Y }$ minimizing
+
+$$
+\operatorname* { i n f } _ { T } \int _ { \mathcal { X } } c ( x , T ( x ) ) d \mu ( x )
+$$
+
+subject to the constraint $T _ { \# } \mu = \nu$ . While the optimal $T$ has an intuitive interpretation as an optimal transport map, the Monge problem is non-convex and not always feasible depending on the choices of $\mu$ and $\nu$ . The Kantorovich $o T$ problem is a convex relaxation of the Monge problem that formulates OT as a search over probabilistic transport plans. Given $\mu \in \mathcal { M } _ { + } ^ { 1 } ( \mathcal { X } ) , \nu \in \bar { \mathcal { M } } _ { + } ^ { 1 } ( \mathcal { V } )$ and a cost function $c : \mathcal { X } \times \mathcal { Y }  \mathbb { R } ^ { + }$ , Kantorovich OT seeks a joint measure $\gamma \in \mathcal { M } _ { + } ^ { 1 } ( \mathcal { X } \times \mathcal { Y } )$ subject to $\pi _ { \# } ^ { \chi } \gamma = \mu$ and $\pi _ { \# } ^ { y } \gamma = \nu$ minimizing
+
+$$
+W ( \mu , \nu ) : = \operatorname* { i n f } _ { \gamma } \int _ { \chi \times y } c ( x , y ) d \gamma ( x , y ) .
+$$
+
+Note that the conditional probability distributions $\gamma _ { y | x }$ specify stochastic maps from $\mathcal { X }$ to $\mathcal { V }$ and can be considered a “one-to-many” version of the deterministic map from the Monge problem. In terms of numerical methods, the relaxed problem is a linear program that is always feasible and can be solved in $O ( n ^ { 3 } )$ time for discrete $\mu , \nu$ . Cuturi (2013) recently showed that introducing entropic regularization results in a simpler dual optimization problem that can be solved efficiently using the Sinkhorn algorithm. Based on the entropyregularized dual problem, Genevay et al. (2016) and Seguy et al. (2017) proposed stochastic algorithms for computing transport plans that can handle continuous measures.
+
+Unbalanced OT. Several formulations that extend classical OT to handle mass variation have been proposed (Chizat et al., 2015; 2018; Kondratyev et al., 2016). Existing numerical methods are based on a Kantorovichlike formulation known as optimal-entropy transport (Liero et al., 2018). This formulation is obtained by relaxing the marginal constraints of (2) using divergences as follows: given two positive measures $\dot { \mu } \in \mathcal { M } _ { + } ( \bar { \mathcal { X } } )$ and $\nu \ \overset { \cdot } { \in } \ \mathcal { M } _ { + } ( \mathcal { V } )$ and a cost function $c : \bar { \mathcal { X } } \times \mathcal { Y }  \mathbb { R } ^ { + }$ , optimal entropy-transport finds a measure $\gamma \in \mathcal { M } _ { + } ( \mathcal { X } \times \mathcal { Y } )$ that minimizes
+
+$$
+W _ { u b } ( \mu , \nu ) : = \operatorname* { i n f } _ { \gamma } \int _ { \mathcal { X } \times \mathcal { Y } } c ( x , y ) d \gamma ( x , y ) + D _ { \psi _ { 1 } } ( \pi _ { \# } ^ { \mathcal { X } } \gamma | \mu ) + D _ { \psi _ { 2 } } ( \pi _ { \# } ^ { \mathcal { Y } } \gamma | \nu ) ,
+$$
+
+where $D _ { \psi _ { 1 } } , D _ { \psi _ { 2 } }$ are $\psi$ -divergences induced by $\psi _ { 1 } , \psi _ { 2 }$ . The $\psi$ -divergence between non-negative finite measures $P , Q$ over $\mathcal { T } \subseteq \mathbb { R } ^ { d }$ induced by a lower semi-continuous, convex entropy function $\psi : \mathbb { R } \to \mathbb { R } \cup \{ \infty \}$ is
+
+$$
+D _ { \psi } ( P | Q ) : = \psi _ { \infty } ^ { \prime } P ^ { \perp } ( { \cal T } ) + \int _ { \cal T } \psi \left( \frac { d P } { d Q } \right) d Q ,
+$$
+
+where $\begin{array} { r } { \psi _ { \infty } ^ { \prime } : = \operatorname* { l i m } _ { s  \infty } \frac { \psi ( s ) } { s } } \end{array}$ and $\frac { d P } { d Q } Q + P ^ { \perp }$ is the Lebesgue decomposition of $P$ with respect to $Q$ . Note that mass variation is allowed since the marginals of $\gamma$ are not constrained to be $\mu$ and $\nu$ . In terms of numerical
+
+![](images/aab4723e1bea6c7ab8ad1e6f35b8bcabd293305ddddaad2fc162ae81ea5f53a7.jpg)  
+Figure 2: Motivating examples for Unbalanced Monge OT.
+
+methods, the state-of-the-art in the discrete setting is a class of iterative scaling algorithms (Chizat et al., 2016) that generalize the Sinkhorn algorithm for computing regularized OT plans (Cuturi, 2013). There are no practical algorithms for unbalanced OT between continuous measures, especially in high-dimensional spaces.
+
+# 3 SCALABLE UNBALANCED OT USING GANS
+
+In this section, we propose the first algorithm for unbalanced OT that directly models mass variation and can be applied towards transport between high-dimensional continuous measures. The starting point of our development is the following Monge-like formulation of unbalanced OT, in which the goal is to learn a stochastic transport map and scaling factor to push a source to a target measure in a cost-optimal manner.
+
+Unbalanced Monge OT. Let $c _ { 1 } : \mathcal { X } \times \mathcal { Y }  \mathbb { R } ^ { + }$ be the cost of transport and $c _ { 2 } : \mathbb { R } ^ { + }  \mathbb { R } ^ { + }$ the cost of mass variation. Let the probability space $( { \mathcal { Z } } , B ( { \mathcal { Z } } ) , \lambda )$ be the source of randomness in the transport map $T$ . Given two positive measures $\mu \in \mathcal { M } _ { + } ( \mathcal { X } )$ and $\nu \in \mathcal { M } _ { + } ( \mathcal { V } )$ , we seek a transport map $T : \mathcal { X } \times \mathcal { Z }  \mathcal { Y }$ and a scaling factor $\xi : \mathcal { X } \to \mathbb { R } ^ { + }$ minimizing
+
+$$
+L ( \mu , \nu ) : = \operatorname* { i n f } _ { T , \xi } \int _ { \mathcal { X } } \int _ { \mathcal { Z } } c _ { 1 } ( x , T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) + \int _ { \mathcal { X } } c _ { 2 } ( \xi ( x ) ) d \mu ( x ) ,
+$$
+
+subject to the constraint $T _ { \# } ( \xi \mu \times \lambda ) = \nu$ . Concretely, the first and second terms of (5) penalize the cost of mass transport and variation respectively, and the equality constraint ensures that $( T , \xi )$ pushes $\mu$ to $\nu$ exactly. A special case of (5) is the unbalanced Monge OT problem by Chizat et al. (2015), which employs a deterministic transport map (Figure 1b). We consider the more general case of stochastic (i.e. one-to-many) maps because it is a more suitable model for many practical problems. For example, in cell biology, it is natural to think of one cell in a source population as potentially giving rise to multiple cells in a target population. In practice, one can take $\mathcal { Z } = \mathbb { R } ^ { n }$ and $\lambda$ to be the standard Gaussian measure if a stochastic map is desired; otherwise $\lambda$ can be set to a deterministic distribution. The following are examples of problems that can be modeled using unbalanced Monge OT.
+
+Example 3.1 (Figure 2a). Suppose the objective is to model the transformation from a source measure (Column 1) to the target measure (Column 2), which represent a population of interest at two distinct time points. The transport map $T$ models the transport/movement of points from the source to the target, while the scaling factor $\xi$ models the growth (replication) or shrinkage (death) of these points. Different models of transformation are optimal depending on the relative costs of mass transport and variation (Columns 3-6).
+
+Example 3.2 (Figure 2b). Suppose the objective is to transport points from a source measure (1st panel, color) to a target measure (1st panel, grey) in the presence of class imbalances. A pure transport map would muddle together points from different classes, while an unbalanced transport map with a scaling factor is able to ameliorate the class imbalance (2nd panel). In this case, the scaling factor tells us explicitly how to downweigh or upweigh samples in the source distribution to balance the classes with the target distribution (3rd panel).
+
+Relaxation. From an optimization standpoint, it is challenging to satisfy the constraint $T _ { \# } ( \xi \mu \times \lambda ) = \nu$ . We hence consider the following relaxation of (5) using a divergence penalty in place of the equality constraint:
+
+$$
+L _ { \psi } ( \mu , \nu ) : = \operatorname* { i n f } _ { T , \xi } \int _ { \mathcal { X } } \int _ { \mathcal { Z } } c _ { 1 } ( x , T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) + \int _ { \mathcal { X } } c _ { 2 } ( \xi ( x ) ) d \mu ( x ) + D _ { \psi } ( T _ { \# } ( \xi \mu \times \lambda ) | \nu ) ,
+$$
+
+using an appropriate choice of $\psi$ that satisfies the requirements of Lemma C.2 in the Appendix1.This relaxation is the Monge-like version of the optimal-entropy transport problem (3) by Liero et al. (2018). Specifically, $( T , \xi )$ specifies a joint measure $\gamma \in \mathcal { M } _ { + } ( \mathcal { X } \times \mathcal { Y } )$ given by
+
+$$
+\gamma ( C ) : = \int _ { \mathcal { X } } \int _ { \mathcal { Z } } \mathbb { 1 } _ { C } ( x , T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) , \quad \forall C \in \mathcal { B } ( \mathcal { X } ) \times \mathcal { B } ( \mathcal { Y } ) ,
+$$
+
+and by reformulating (6) in terms of $\gamma$ instead of $( T , \xi )$ , one obtains the objective function for optimal-entropy transport. The main difference between the formulations is their search space, since not all joint measures $\gamma \in \mathcal { M } _ { + } ( \mathcal { X } \times \mathcal { Y } )$ can be specified by some choice of $( T , \xi )$ . For example, if $T$ is a deterministic transport map, then $\gamma$ is necessarily restricted to the set of deterministic couplings. Even if $T$ is sufficiently random, it is generally not possible to specify all joint measures $\gamma \in \mathcal { M } _ { + } ( \mathcal { X } \times \mathcal { Y } )$ : in the asymmetric Monge formulation (6), all the mass transported to $\mathcal { V }$ must come from somewhere within the support of $\mu$ , since the scaling factor $\xi$ allows mass to grow but not to materialize outside of its original support. Therefore equivalence can be established in general only when restricting the support of $\gamma$ to $s u p p ( \mu ) \times \mathcal { Y }$ as described in the following lemma, whose proof is given in the Appendix.
+
+Lemma 3.3. Let $\mathcal { G }$ be the set of joint measures supported on supp $( \mu ) \times \mathcal { V }$ , and define
+
+$$
+\tilde { W } _ { c , \psi _ { 1 } , \psi _ { 2 } } ( \mu , \nu ) : = \operatorname* { i n f } _ { \gamma \in \mathcal { G } } \int c d \gamma + D _ { \psi _ { 1 } } ( \pi _ { \# } ^ { \mathcal { X } } \gamma | \mu ) + D _ { \psi _ { 2 } } ( \pi _ { \# } ^ { \mathcal { V } } \gamma | \nu ) .
+$$
+
+$I f ( \mathcal { Z } , B ( \mathcal { Z } ) , \lambda )$ is atomless and $c _ { 2 }$ is an entropy function, then $L _ { \psi } ( \mu , \nu ) = \tilde { W } _ { c _ { 1 } , c _ { 2 } , \psi } ( \mu , \nu )$ .
+
+Based on the relation between (3) and (6), several theoretical results for (6) follow from the analysis of optimal entropy-transport by Liero et al. (2018). Importantly, one can show the following theorem, namely that for an appropriate and sufficiently large choice of divergence penalty, solutions of the relaxed problem (6) converge to solutions of the original problem (5). The proof is given in the Appendix.
+
+Theorem 3.4. Suppose $c _ { 1 } , c _ { 2 } , \psi$ satisfy the existence assumptions of Proposition $B . I$ in the Appendix, and let $( { \mathcal { Z } } , B ( { \mathcal { Z } } ) , \lambda )$ be an atomless probability space. Furthermore, let $\psi$ be uniquely minimized at $\psi ( 1 ) = 0$ . Then for a sequence $0 < \zeta ^ { 1 } < \cdots < \zeta ^ { k } < \cdots$ diverging to $\infty$ indexed by $k$ , $\begin{array} { r } { \operatorname* { l i m } _ { k \to \infty } L _ { \zeta ^ { k } \psi } ( \mu , \nu ) = L ( \mu , \nu ) } \end{array}$ . Additionally, let $\gamma ^ { k }$ be the joint measure specified by a minimizer of $L _ { \zeta ^ { k } \psi } ( \mu , \nu )$ . If $L ( \mu , \nu ) < \infty$ , then up to extraction of a subsequence, $\gamma ^ { k }$ converges weakly to $\gamma ,$ , the joint measure specified by a minimizer of $L ( \mu , \nu )$ .
+
+Algorithm. Using the relaxation of unbalanced Monge OT in (6), we now show that the transport map and scaling factor can be learned by stochastic gradient methods. While the divergence term cannot easily be minimized using the definition in (4), we can write it as a penalty witnessed by an adversary function $f : \mathcal { V } \to ( - \mathrm { \bar { \infty } } , \psi _ { \infty } ^ { \prime } ]$ using the convex conjugate representation (see Lemma B.2):
+
+$$
+D _ { \psi } ( T _ { \# } ( \xi \mu \times \lambda ) | \nu ) = \operatorname* { s u p } _ { f } \int _ { \mathcal X } \int _ { \mathcal Z } f ( T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) - \int _ { \mathcal Y } \psi ^ { * } ( f ( y ) ) d \nu ( y ) ,
+$$
+
+where $\psi ^ { * }$ is the convex conjugate of $\psi$ . The objective in (6) can now be optimized using alternating stochastic gradient updates after parameterizing $T , \xi$ , and $f$ with neural networks; see Algorithm 1 2. The optimization procedure is similar to GAN training and can be interpreted as an adversarial game between $( T , \xi )$ and $f$ :
+
+• $T$ takes a point $x \sim \mu$ and transports it from $\mathcal { X }$ to $\mathcal { V }$ by generating $T ( x , z )$ where $z \sim \lambda$ .   
+• $\xi$ determines the importance weight of each transported point.   
+• Their shared objective is to minimize the divergence between transported samples and real samples from $\nu$ that is measured by the adversary $f$ .   
+• Additionally, cost functions $c _ { 1 }$ and $c _ { 2 }$ encourage $T , \xi$ to find the most cost-efficient strategy.
+
+# Algorithm 1 Generative-Adversarial Framework for Unbalanced Monge OT
+
+Input: Initial parameters $\theta$ , $\phi$ , $\omega$ ; step size $\eta$ ; normalized measures $\tilde { \mu } , \tilde { \nu }$ , constants $c _ { \mu } , c _ { \nu }$ .   
+Output: Updated parameters $\theta , \phi , \omega$ .   
+while $( \theta , \phi , \omega )$ not converged do Sample $x _ { 1 } , \cdots , x _ { n }$ from $\tilde { \mu }$ $, y _ { 1 } , \cdots , y _ { n }$ from $\tilde { \nu }$ , $z _ { 1 } , \cdots , z _ { n }$ from $\lambda$ ; $\begin{array} { r l r } {  { \operatorname { \mathrm {  ~ \Lambda ~ } } ^ { \mathrm { n o \to 1 , \Lambda ~ } } , \mathrm { \Lambda } ^ { \mathrm { \tiny ~ , ~ \times ~ n e a s e a ~ } , ~ p = 1 , \Lambda } , \mathrm { \Lambda } ^ { \mathrm { \tiny ~ , ~ \wedge ~ s o u s e a ~ } , ~ p = 1 , \Lambda } , } } \\ & { } & { \ell ( \theta , \phi , \omega ) : = \cfrac { 1 } { n } \sum _ { i = 1 } ^ { n } [ c _ { \mu } c _ { 1 } ( x _ { i } , T _ { \theta } ( x _ { i } , z _ { i } ) ) \xi _ { \phi } ( x _ { i } ) + c _ { \mu } c _ { 2 } ( \xi _ { \phi } ( x _ { i } ) ) } \\ & { } & { \mathrm + c _ { \mu } \xi _ { \phi } ( x _ { i } ) f _ { \omega } ( T _ { \theta } ( x _ { i } , z _ { i } ) ) - c _ { \nu } \psi ^ { * } ( f _ { \omega } ( y _ { i } ) ) . ] } \end{array}$ (8) Update $\omega$ by gradient descent on $- \ell ( \theta , \phi , \omega )$ . Update $\theta , \phi$ by gradient descent $\ell ( \theta , \phi , \omega )$ .   
+end while
+
+Table 1 in the Appendix provides some examples of divergences with corresponding entropy functions and convex conjugates that can be plugged into (7). Further practical considerations for implementation and training are discussed in Appendix C.
+
+Relation to other approaches. The probabilistic Monge-like formulation (6) is similar to the Kantorovichlike entropy-transport problem (3) in theory, but they result in quite different numerical methods in practice. Algorithm 1 solves the non-convex formulation (6) and learns a transport map $T$ and scaling factor $\xi$ parameterized by neural networks, enabling scalable optimization using stochastic gradient descent. The networks are immediately useful for many practical applications; for instance, it only requires a single forward pass to compute the transport and scaling of a point from the source domain to the target. Furthermore, the neural architectures of $T , \xi$ imbue their function classes with a particular structure, and when chosen appropriately, enable effective learning of these functions in high-dimensional settings. Due to the non-convexity of the optimization problem, however, Algorithm 1 is not guaranteed to find the global optimum. In contrast, the scaling algorithm of Chizat et al. (2016) based on (3) solves a convex optimization problem and is proven to converge, but is currently only practical for discrete problems and has limited scalability. For completeness, in Section A of the Appendix, we propose a new stochastic method based on the same dual objective as Chizat et al. (2016) that can handle transport between continuous measures (Algorithm 2 in the Appendix). This method generalizes the approach of Seguy et al. (2017) for handling transport between continuous measures and overcomes the scalability limitations of Chizat et al. (2016). However, the output is in the form of the dual solution, which is less interpretable for practical applications compared to the output of Algorithm 1. In particular, while one can compute a deterministic transport map known as a barycentric projection from the dual solution, it is unclear how best to obtain a scaling factor or a stochastic transport map that can generate samples outside of the target dataset. In the numerical experiments of Section 4, we show the advantage of directly learning a transport map and scaling factor using Algorithm 1.
+
+The problem of learning a scaling factor (or weighting factor) that “balances” measures $\mu$ and $\nu$ also arises in causal inference. Generally, $\mu$ is the distribution of covariates from a control population and $\nu$ is the distribution from a treated population. The goal is to scale the importance of different members from the control population based on how likely they are to be present in the treated population, in order to eliminate selection biases in the inference of treatment effects. Kallus (2018) proposed a generative-adversarial method for learning the scaling factor, but they do not consider transport.
+
+![](images/01c275c2927611da947034a1ab875f31c6606b0a5224cb91c59294fedd407472.jpg)  
+Figure 3: Learning weights on MNIST data using unbalanced OT.
+
+# 4 NUMERICAL EXPERIMENTS
+
+In this section, we illustrate in practice how Algorithm 1 performs unbalanced OT, with applications geared towards population modeling.
+
+MNIST-to-MNIST. We first apply Algorithm 1 to perform unbalanced optimal transport between two modified MNIST datasets. The source dataset consists of regular MNIST digits with the class distribution shown in column 1 of Figure 3a. The target dataset consists of either regular (for the experiment in Figure 3b) or dimmed (for the experiment in Figure 3c) MNIST digits with the class distribution shown in column 2 of Figure 3a. The class imbalance between the source and target datasets imitates a scenerio in which certain classes (digits 0-3) become more popular and others (6-9) become less popular in the target population, while the change in brightness is meant to reflect population drift. We evaluated Algorithm 1 on the problem of transporting the source distribution to the target distribution, enforcing a high cost of transport (w.r.t. Euclidean distance). In both cases, we found that the scaling factor over each of the digit classes roughly reflects its ratio of imbalance between the source and target distributions (Figure 3b-c). These experiments validate that the scaling factor learned by Algorithm 1 reflects the class imbalances and can be used to model growth or decline of different classes in a population. Figure 3d is a schematic illustrating the reweighting that occurs during unbalanced OT.
+
+MNIST-to-USPS. Next, we apply unbalanced OT from the MNIST dataset to the USPS dataset. As before, these two datasets are meant to imitate a population sampled at two different time points, this time with a large degree of evolution. We use Algorithm 1 to model the evolution of the MNIST distribution to the USPS distribution, taking as transport cost the Euclidean distance between the original and transported images. A summary of the unbalanced transport is visualized in Figure 4a. Each arrow originates from a real MNIST image and points towards the predicted appearance of this image in the USPS dataset. The size of the image reflects the scaling factor of the original MNIST image, i.e. whether it is relatively increasing or decreasing in prominence in the USPS dataset compared to the MNIST dataset according to the unbalanced OT model. Even though the Euclidean distance is not an ideal measure of correspondence between MNIST and USPS digits, many MNIST digits were able to preserve their likeness during the transport (Figure 4b). We analyzed which MNIST digits were considered as increasing or decreasing in prominence by the model. The MNIST digits with higher scaling factors were generally brighter (Figure 4c) and covered a larger area of pixels (Figure 4d) compared to the MNIST digits with lower scaling factors. These results are consistent with the observation that the target USPS digits are generally brighter and contain more pixels.
+
+![](images/f1945d66e3a4bbc47203129e1ec9fd61f23c5e65d98a2089910987d9bc1d453b.jpg)  
+Figure 4: Unbalanced Optimal Transport from MNIST to USPS digits.
+
+CelebA-Young-to-CelebA-Aged. We applied Algorithm 1 on the CelebA dataset to perform unbalanced OT from the population of young faces to the population of aged faces. This synthetic problem imitates a real application of interest, which is modeling the transformation of a population based on samples taken from two timepoints. Since the Euclidean distance between two faces is a poor measure of semantic similarity, we first train a variational autoencoder (VAE) (Kingma & Welling, 2013) on the CelebA dataset and encode all samples into the latent space. We then apply Algorithm 1 to perform unbalanced OT from the encoded young to the encoded aged faces, taking the transport cost to be the Euclidean distance in the latent space. A summary of the unbalanced transport is visualized in Figure 5a. Each arrow originates from a real face from the young population and points towards the predicted appearance of this face in the aged population. Generally, the transported faces retain the most salient features of the original faces (Figure 5b), although there are exceptions (e.g. gender swaps) which reflects that some features are not prominent components of the VAE encodings. Interestingly, the young faces with higher scaling factors were significantly enriched for males compared to young faces with lower scaling factors; $9 . 6 \%$ (9,913/103,287) of young female faces had a high scaling factor as compared to $1 8 . 5 \%$ (8,029/53,447) for young male faces (Figure 5c, top, $p = 0$ ). In other words, our model predicts growth in the prominence of male faces compared to female faces as the CelebA population evolves from young to aged. After observing this phenomenon, we confirmed based on checking the ground truth labels that there was indeed a strong gender imbalance between the young and aged populations: while the young population is predominantly female, the aged population is predominantly male (Figure 5c, bottom).
+
+Zebrafish embroygenesis. A problem of great interest in biology is lineage tracing of cells between different developmental stages or during disease progression. This is a natural application of transport in which the source and target distributions are unbalanced: some cells in the earlier stage are more poised to develop into cells seen in the later stage. To showcase the relevance of learning the scaling factor, we apply Algorithm 1 to recent single-cell gene expression data from two stages of zebrafish embryogenesis (Farrell et al., 2018). The source population is from a late stage of blastulation and the target population from an early stage of gastrulation (Figure 6a). The results of the transport are plotted in Figure 6b-c after dimensionality reduction by PCA and T-SNE (Maaten & Hinton, 2008). To assess the scaling factor, we extracted the cells from the blastula stage with higher scaling factors (i.e. over 90th percentile) and compared them to the remainder of the cells using differential gene expression analysis, producing a ranked list of upregulated genes. Using the GOrilla tool (Eden et al., 2009), we found that the cells with higher scaling factors were significantly enriched for genes associated with differentiation and development of the mesoderm (Figure 6d). This experiment shows that analysis of the scaling factor can be applied towards interesting and meaningful biological discovery.
+
+![](images/848903c9779b38a01fbefa40938a3c78753a90fbd247ce0e4db254269f5e5d7a.jpg)  
+Figure 5: Unbalanced Optimal Transport from Young to Aged CelebA Faces.
+
+![](images/b6efe530c64b1cb771f0ded266d3cfe324f357c7e0f7bca2209d1c92be27e400.jpg)  
+Figure 6: Unbalanced OT on Zebrafish Single-Cell Gene Expression Data. (a) Illustration of the blastula and gastrula stages of zebrafish embryogenesis. (b) T-SNE plot (Maaten & Hinton, 2008) of the unbalanced OT results for a subset of datapoints. The color of the transported points indicates the relative magnitude of the scaling factor $( \mathrm { b l a c k } = \mathrm { h i g h }$ , white $\mathbf { \tau } = 1 0 \mathbf { w }$ ). (c) Same plot as (b), where we have colored the source points instead of the transported points. (d) GOrilla output of significantly enriched processes (Eden et al., 2009) based on ranked list of enriched genes in cells with high scaling factors (black points from (c)) from a differential gene expression analysis. Processes in the graph are organized from more general (upstream) to more specific (downstream).
+
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+
+# APPENDIX
+
+# A DUAL STOCHASTIC METHOD
+
+In this section, we present a stochastic method for unbalanced OT based on the regularized dual formulation of (Chizat et al., 2015), which can be considered a natural generalization of Seguy et al. (2017). The dual formulation of (3) is given by
+
+$$
+\operatorname* { s u p } _ { u , v } - \int \psi _ { 1 } ^ { * } ( - u ) d \mu - \int \psi _ { 2 } ^ { * } ( - v ) d \nu
+$$
+
+subject to $u \oplus v \ \leq c$ , where the supremum is taken over functions $u : \mathcal { X }  [ - \psi _ { 1 \infty } ^ { \prime } , \infty ]$ and $v : \mathcal { V } $ $[ - \psi _ { 2 \infty } ^ { \prime } , \infty ]$ . This is a constrained optimization problem that is challenging to solve. A standard technique for making the dual problem unconstrained is to add a strongly convex regularization term to the primal objective (Blondel et al., 2017), such as an entropic regularization term (Cuturi, 2013):
+
+$$
+R _ { e } ( \gamma ) = \epsilon D _ { \psi _ { K L } } ( \gamma | \mu \otimes \nu )
+$$
+
+where $\epsilon > 0$ . Concretely, this term has a “smoothing” effect on the transport plan, in the sense that it encourages plans with high entropy. By the Fenchel-Rockafellar theorem, the dual of the regularized problem is given by,
+
+$$
+W _ { u b } ^ { \epsilon } ( \mu , \nu ) : = \operatorname* { s u p } _ { u , v } - \int \psi _ { 1 } ^ { * } ( - u ) d \mu - \int \psi _ { 2 } ^ { * } ( - v ) d \nu - \epsilon \int e ^ { ( u + v - c ) / \epsilon } d ( \mu \otimes \nu ) ,
+$$
+
+where the supremum is taken over functions $u : \mathcal { X }  [ - \psi _ { 1 \infty } ^ { \prime } , \infty ]$ and $v : \mathcal { V }  [ - \psi _ { 2 \infty } ^ { \prime } , \infty ]$ , and the relationship between the primal optimizer $\gamma ^ { * }$ and dual optimizer $( u ^ { * } , v ^ { * } )$ is given by
+
+$$
+d \gamma ^ { \ast } = e ^ { ( u ^ { \ast } + v ^ { \ast } - c ) / \epsilon } d ( \mu \otimes \nu ) .
+$$
+
+Next, we rewrite (9) in terms of expectations. We assume that one has access to samples from $\mu , \nu$ , and in the setting where $\mu , \nu$ are not normalized, then samples to the normalized measures $\tilde { \mu } , \tilde { \nu }$ as well as the normalization constants. Based on these assumptions, we have
+
+$$
+\begin{array} { r } { \mathcal { N } _ { u b } ^ { \epsilon } ( \mu , \nu ) = \underset { u , v } { \operatorname* { s u p } } - c _ { \mu } \mathbb { E } _ { x \sim \tilde { \mu } } \psi _ { 1 } ^ { * } ( - u ( x ) ) - c _ { \nu } \mathbb { E } _ { y \sim \tilde { \nu } } \psi _ { 2 } ^ { * } ( - v ( y ) ) - \epsilon c _ { \mu } c _ { \nu } \mathbb { E } _ { x \sim \tilde { \mu } , y \sim \tilde { \nu } } e ^ { ( u ( x ) + v ( y ) - c ( x , y ) ) / \epsilon } . } \end{array}
+$$
+
+If $\psi _ { 1 } ^ { * } , \psi _ { 2 } ^ { * }$ are differentiable, we can parameterize $u , v$ with neural networks $u _ { \theta } , v _ { \phi }$ and optimize $\theta , \phi$ using stochastic gradient descent. This is described in Algorithm 2. Note that this algorithm is a generalization of the algorithm of (Seguy et al., 2017) from classical OT to unbalanced OT. Indeed, taking $\psi _ { 1 } , \psi _ { 2 }$ to be equality constraints, (9) becomes
+
+$$
+\operatorname* { s u p } _ { u , v } \int u d \mu + \int v d \nu - \epsilon \int e ^ { ( u + v - c ) / \epsilon } d ( \mu \otimes \nu ) ,
+$$
+
+which is the dual of the entropy-regularized classical OT problem.
+
+# Algorithm 2 SGD for Unbalanced OT
+
+Input: Initial parameters $\theta$ , $\phi$ ; step size $\eta$ ; regularization parameter $\epsilon$ ; constants $c _ { \mu } , c _ { \nu }$ and normalized   
+measures $\tilde { \mu }$ , $\tilde { \nu }$   
+Output: Updated parameters $\theta$ , $\phi$   
+while $( \theta , \phi )$ not converged do Sample $( x _ { 1 } , y _ { 1 } ) , \cdot \cdot \cdot , ( x _ { n } , y _ { n } )$ from $\tilde { \mu } \otimes \tilde { \nu }$
+
+$$
+\ell ( \theta , \phi ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } [ c _ { \mu } \psi _ { 1 } ^ { * } ( - u ( x _ { i } ) ) + c _ { \nu } \psi _ { 2 } ^ { * } ( - v ( y _ { i } ) ) + \epsilon c _ { \mu } c _ { \nu } e ^ { ( u ( x _ { i } ) + v ( y _ { i } ) - c ( x _ { i } , y _ { i } ) ) / \epsilon } ]
+$$
+
+Update $\theta , \phi$ by gradient descent on $\ell ( \theta , \phi )$
+
+The dual solution $( u ^ { * } , v ^ { * } )$ learned from Algorithm 2 can be used to reconstruct the primal solution $\gamma ^ { * }$ based on the relation in (10). Concretely, $\gamma ^ { * }$ is a transport map that indicates the amount of mass transported between every pair of points in $\mathcal { X }$ and $\mathcal { V }$ . Note that the marginals of $\gamma ^ { * }$ with respect to $\mathcal { X }$ and $\mathcal { V }$ are not necessarily $\mu$ and $\nu$ , which is where mass variation is implicitly built into the problem. Given $\gamma ^ { * }$ , it is possible to also learn an “averaged” deterministic mapping from $\mathcal { X }$ to $\mathcal { V }$ . A standard approach is to take the barycentric projection $T : \mathcal { X }  \mathcal { Y }$ , defined as,
+
+$$
+T ( x ) = \operatorname* { m i n } _ { z \in \mathcal { V } } \mathbb { E } _ { y \sim \gamma ^ { * } ( \cdot | x ) } d ( z , y ) ,
+$$
+
+with respect to some distance $d : \mathcal { V } \times \mathcal { V }  \mathbb { R } ^ { + }$ . Seguy et al. (2017) proposed a stochastic algorithm for learning such a map from the dual solution, which we reproduce in Algorithm 3.
+
+# Algorithm 3 Learning Barycentric Projection
+
+<table><tr><td>Input:Learned functions u,U; initial Tθ; distance function d Output: Updated Te</td></tr><tr><td>while Tθ not converged do Sample (x1,yi),..,(xn,yn) from μ ⑧i</td></tr><tr><td></td></tr><tr><td>n</td></tr><tr><td>1 d(xi,Tθ(yi))e(u(xi)+u(yi)-c(xi,yi))/∈</td></tr><tr><td>l(0):= n</td></tr><tr><td>i=1</td></tr><tr><td>Update Tθ by gradient descent on l(0) end while</td></tr></table>
+
+# B SUPPLEMENT TO SECTION 3
+
+# B.1 RELAXATION TO OPTIMAL-ENTROPY TRANSPORT
+
+The objectives in (6) and (3) are equivalent if one reformulates (6) in terms of $\gamma$ instead of $( T , \xi )$ , where $\gamma \in \mathcal { M } _ { + } ( \mathcal { X } \times \mathcal { Y } )$ is a joint measure given by
+
+$$
+\gamma ( C ) : = \int _ { \mathcal { X } } \int _ { \mathcal { Z } } \mathbb { 1 } _ { C } ( x , T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) , \quad \forall C \in \mathcal { B } ( \mathcal { X } ) \times \mathcal { B } ( \mathcal { Y } ) .
+$$
+
+Furthermore, the formulations are equivalent if one restricts the search space of (3) to contain only those joint measures that can be specified by some $( T , \xi )$ . This relation between the formulations is formalized by Lemma 3.3.
+
+Proof of Lemma 3.3. First we show $L _ { \psi } ( \mu , \nu ) \ge \tilde { W } _ { c _ { 1 } , c _ { 2 } , \psi } ( \mu , \nu )$ . If $L _ { \psi } ( \mu , \nu ) = \infty$ , this is trivial, so assume $L _ { \psi } ( \mu , \nu ) < \infty$ . Let $( T , \xi )$ be any solution and define $\gamma$ by (12). Note by this definition that
+
+$$
+\pi _ { \# } ^ { \chi } \gamma ( A ) = \int _ { \chi } \mathbb { 1 } _ { A } ( x ) \xi ( x ) d \mu ( x ) , \quad \forall A \in B ( \mathcal { X } ) ,
+$$
+
+i.e. $\xi$ is the Radon-Nikodym derivative of $\pi _ { \# } ^ { \chi } \gamma$ with respect to $\mu$ . Also
+
+$$
+\pi _ { \# } ^ { \mathcal { Y } } \gamma ( B ) = \int _ { \mathcal { X } } \int _ { \mathcal { Z } } \mathbb { 1 } _ { B } ( T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) = T _ { \# } ( \xi \mu \times \lambda ) ( B ) , \quad \forall B \in B ( \mathcal { Y } ) .
+$$
+
+It follows that
+
+$$
+\begin{array} { l } { \displaystyle \int _ { \mathcal X } \left( \int _ { \mathcal z } c _ { 1 } ( x , T ( x , z ) ) d \lambda ( z ) \right) \xi ( x ) d \mu ( x ) + \int _ { \mathcal X } c _ { 2 } ( \xi ( x ) ) d \mu ( x ) + D _ { \psi } ( T _ { \# } ( \xi \mu \times \lambda ) | \nu ) } \\ { = \int _ { \mathcal X \times \mathcal Y } c _ { 1 } ( x , y ) d \gamma ( x , y ) + \int _ { \mathcal X } c _ { 2 } ( \xi ( x ) ) d \mu ( x ) + D _ { \psi } ( T _ { \# } ( \xi \mu \times \lambda ) | \nu ) } \end{array}
+$$
+
+$$
+= \int _ { \mathcal { X } \times \mathcal { Y } } c _ { 1 } ( x , y ) d \gamma ( x , y ) + \int _ { \mathcal { X } } c _ { 2 } ( \frac { d \pi _ { \# } ^ { \mathcal { X } } \gamma } { d \mu } ) d \mu ( x ) + D _ { \psi } ( T _ { \# } ( \xi \mu \times \lambda ) | \nu )
+$$
+
+$$
+\begin{array} { r l } & { = \displaystyle \int _ { \mathcal { X } \times \mathcal { Y } } c _ { 1 } ( x , y ) d \gamma ( x , y ) + D _ { c _ { 2 } } ( \pi _ { \# } ^ { \mathcal { X } } \gamma | \mu ) + D _ { \psi } ( T _ { \# } ( \xi \mu \times \lambda ) | \nu ) } \\ & { = \displaystyle \int _ { \mathcal { X } \times \mathcal { Y } } c _ { 1 } ( x , y ) d \gamma ( x , y ) + D _ { c _ { 2 } } ( \pi _ { \# } ^ { \mathcal { X } } \gamma | \mu ) + D _ { \psi } ( \pi _ { \# } ^ { \mathcal { Y } } \gamma | \nu ) } \\ & { \geq \tilde { W } _ { c , \psi _ { 1 } , \psi _ { 2 } } ( \mu , \nu ) . } \end{array}
+$$
+
+Since this inequality holds for any $( T , \xi )$ , taking the infimum over the left-hand side yields $L _ { \psi } ( \mu , \nu ) \ge \tilde { W } _ { c , \psi _ { 1 } , \psi _ { 2 } } \dot { ( } \mu , \nu )$ .
+
+To show $L _ { \psi } ( \mu , \nu ) \leq \tilde { W } _ { c , \psi _ { 1 } , \psi _ { 2 } } ( \mu , \nu )$ , assume $\tilde { W } _ { c , \psi _ { 1 } , \psi _ { 2 } } ( \mu , \nu ) < \infty$ and let $\gamma$ be any solution. By the disintegration theorem, there exists a family of probability measures $\{ \gamma _ { y | x } \} _ { x \in \mathcal { X } }$ in $\mathcal { M } _ { + } ^ { 1 } ( \mathcal { V } )$ such that
+
+$$
+\gamma ( C ) = \int _ { \mathcal { X } } \int _ { \mathcal { Y } } \mathbb { 1 } _ { C } ( x , y ) d \gamma _ { y \mid x } ( y ) d \pi _ { \# } ^ { \mathcal { X } } \gamma , \quad \forall C \in \mathcal { B } ( \mathcal { X } ) \times \mathcal { B } ( \mathcal { Y } ) .
+$$
+
+Since $( { \mathcal { Z } } , B ( { \mathcal { Z } } ) , \lambda )$ is atomless, it follows from Proposition 9.1.2 and Theorem 13.1.1 in (Dudley, 2018) that there exists a family of measurable functions $\{ T _ { x } : \mathcal { Z }  \mathcal { V } \} _ { x \in \mathcal { X } }$ such that $\gamma _ { y | x }$ is the pushforward measure of $\lambda$ under $T _ { x }$ for all $x \in \mathcal { X }$ . Denoting $T ( x , z ) : ( x , z ) \mapsto T _ { x } ( z )$ , then by a change of variables,
+
+$$
+\gamma ( C ) = \int _ { \mathcal { X } } \int _ { \mathcal { Z } } \mathbb { 1 } _ { C } ( x , T ( x , z ) ) d \lambda ( z ) d \pi _ { \# } ^ { \mathcal { X } } \gamma , \quad \forall C \in \mathcal { B } ( \mathcal { X } ) \times \mathcal { B } ( \mathcal { Y } ) .
+$$
+
+By hypothesis, $\pi _ { \# } ^ { \chi } \gamma$ is restricted to the support of $\mu$ , i.e. $\pi _ { \# } ^ { \chi } \gamma \ll \mu$ . Let $\xi$ be the Radon-Nikodym derivative $\frac { d \pi _ { \# } ^ { \chi } \gamma } { d \mu }$ . It follows from the Radon-Nikodym theorem that $( T , \xi )$ satisfy
+
+$$
+\gamma ( C ) = \int _ { \mathcal { X } } \int _ { \mathcal { Z } } \mathbb { 1 } _ { C } ( x , T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) , \quad \forall C \in \mathcal { B } ( \mathcal { X } ) \times \mathcal { B } ( \mathcal { Y } ) ,
+$$
+
+which is the same relation as in (12). Same as before,
+
+$$
+\pi _ { \# } ^ { \chi } \gamma ( A ) = \int _ { \chi } \mathbb { 1 } _ { A } ( x ) \xi ( x ) d \mu ( x ) , \quad \forall A \in B ( \mathcal { X } ) ,
+$$
+
+and
+
+$$
+\pi _ { \# } ^ { \mathcal { Y } } \gamma ( B ) = \int _ { \mathcal { X } } \int _ { \mathcal { Z } } \mathbb { 1 } _ { B } ( T ( x , z ) ) d \lambda ( z ) \xi ( x ) d \mu ( x ) = T _ { \# } ( \xi \mu \times \lambda ) ( B ) , \quad \forall B \in B ( \mathcal { Y } ) .
+$$
+
+It then follows that
+
+$$
+\begin{array} { l } { { \displaystyle \int _ { \mathbb { X } \times \mathbb { V } } c _ { 1 } d \gamma + D _ { c _ { 2 } } ( \pi _ { \# } ^ { \mathcal { X } } \gamma | \mu ) + D _ { \psi } ( \pi _ { \# } ^ { \mathcal { Y } } \gamma | \nu ) } } \\ { { \displaystyle = \int _ { \mathbb { X } } \left( \int _ { \mathbb { V } } c _ { 1 } ( x , y ) d \gamma _ { y | x } ( y ) \right) d \pi _ { \# } ^ { \mathcal { X } } \gamma + D _ { c _ { 2 } } ( \pi _ { \# } ^ { \mathcal { X } } \gamma | \mu ) + D _ { \psi } ( \pi _ { \# } ^ { \mathcal { Y } } \gamma | \nu ) } } \end{array}
+$$
+
+$$
+= \int _ { \mathcal { X } } \left( \int _ { \mathcal { Z } } c ( x , T ( x , z ) ) d \lambda ( z ) \right) d \pi _ { \# } ^ { \mathcal { X } } \gamma + D _ { c _ { 2 } } ( \pi _ { \# } ^ { \mathcal { X } } \gamma | \mu ) + D _ { \psi } ( \pi _ { \# } ^ { \mathcal { Y } } \gamma | \nu )
+$$
+
+$$
+\begin{array} { l l l } { { \displaystyle = \int _ { \mathcal X } \left( \int _ { \mathcal Z } c ( x , T ( x , z ) ) d \lambda ( z ) \right) d \pi _ { \# } ^ { \mathcal X } \gamma + \int c _ { 2 } ( \frac { d \pi _ { \# } ^ { \mathcal X } \gamma } { d \mu } ) d \mu ( x ) + D _ { \psi } ( \pi _ { \# } ^ { \mathcal Y } \gamma | \nu ) } }  \\ { { \displaystyle = \int _ { \mathcal X } \left( \int _ { \mathcal Z } c ( x , T ( x , z ) ) d \lambda ( z ) \right) \xi ( x ) d \mu ( x ) + \int c _ { 2 } ( \xi ( x ) ) d \mu ( x ) + D _ { \psi } ( \pi _ { \# } ^ { \mathcal Y } \gamma | \nu ) } } \end{array}
+$$
+
+$$
+\begin{array} { r l } & { = \displaystyle \int _ { \mathcal { X } } \left( \int _ { \mathcal { Z } } c ( x , T ( x , z ) ) d \lambda ( z ) \right) d \mu ( x ) + \int c _ { 2 } ( \xi ( x ) ) d \mu ( x ) + D _ { \psi } ( T _ { \# } ( \xi \mu \times \lambda ) | \nu ) } \\ & { \geq L _ { \psi } ( \mu , \nu ) . } \end{array}
+$$
+
+Since this inequality holds for any $\gamma$ , this implies that $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \psi } ( \mu , \nu ) \geq L _ { \psi } ( \mu , \nu )$ , which completes the proof.
+
+Due to the near equivalence of the formulations, several theoretical results for (6) follow from the analysis of optimal entropy-transport by Liero et al. (2018), such as the following existence and uniqueness result:
+
+Proposition B.1. Suppose $L _ { \psi } ( \mu , \nu ) < \infty$ , $c _ { 2 }$ is convex and lower semi-continuous, and $( \mathcal { Z } , B ( \mathcal { Z } ) , \lambda )$ is atomless. If $( i ) c _ { 1 }$ has compact sublevel sets in $\mathcal { X } \times \mathcal { V }$ and $c _ { 2 \infty } ^ { \prime } + \psi _ { \infty } ^ { \prime } > 0$ , or (ii) $c _ { 2 \infty } ^ { \prime } = \psi _ { \infty } ^ { \prime } = \infty$ then $a$ minimizer of $L _ { \psi } ( \mu , \nu )$ exists. If $c _ { 2 } , \psi$ are strictly convex, $\psi _ { \infty } ^ { \prime } = \infty$ and $c _ { 1 }$ satisfies Corollary 3.6 (Liero et al., 2018), then the joint measure $\gamma$ specified by any minimizer of $L _ { \psi } ( \mu , \nu )$ is unique.
+
+Proof of Proposition B.1. Note that $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \psi } ( \mu , \nu )$ is equivalent to $W _ { c _ { 1 } , c _ { 2 } , \psi } ( \mu , \nu )$ when $\mathcal { X }$ is restricted to the support of $\mu$ . If $c _ { 1 } , c _ { 2 } , \psi$ satisfy (i) or (ii), they also satisfy (i) and (ii) when $\mathcal { X }$ is restricted to the support of $\mu$ . By Theorem 3.3 of (Liero et al., 2018), $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \psi } ( \mu , \nu )$ has a minimizer. It follows from the construction of the proof of Lemma 3.3 that a minimizer of $L _ { \psi } ( \mu , \nu )$ also exists. For uniqueness, if $\psi _ { \infty } ^ { \prime } = \infty$ , then it follows from Lemma 3.5 of (Liero et al., 2018) and the fact that minimizers are restricted to $\mathcal { G }$ that the marginals $\pi _ { \# } ^ { \chi } \gamma$ $\pi _ { \# } ^ { y } \gamma$ are uniquely determined for any solution $\gamma$ of $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \psi } ( \mu , \nu )$ . The uniqueness of $\gamma$ then follows from the proof of Corollary 3.6 in (Liero et al., 2018). It follows from the construction of the proof of Lemma 3.3 that the product measure generated by the minimizers of $L _ { \psi } ( \mu , \nu )$ is unique, which completes the proof. □
+
+For certain cost functions and divergences, it can be shown that $L _ { \psi }$ defines a proper metric between positive measures $\mu$ and $\nu$ , i.e. taking $c _ { 2 } , \psi$ to be entropy functions corresponding to the KL-divergence and $c _ { 1 } = \log \cos _ { + } ^ { 2 } ( d ( x , y ) )$ , then $L _ { \psi } ( \mu , \nu )$ corresponds to the Hellinger-Kantorovich (Liero et al., 2018) or the Wasserstein-Fisher-Rao (Chizat et al., 2018) metric between positive measures $\mu$ and $\nu$ .
+
+Based on Lemma 3.3, the theoretical analysis of Liero et al. (2018), and standard results on constrained optimization, it can be shown that for an appropriate and sufficiently large choice of divergence penalty, solutions of the relaxed problem (6) converge to solutions of the original problem (5) (Theorem 3.4).
+
+Proof of Theorem 3.4. Since $\zeta ^ { k } \psi ( s )$ converges pointwise to the equality constraint $\iota _ { = } ( s )$ , which is 0 for $s = 1$ and $\infty$ otherwise, by Lemma 3.9 in (Liero et al., 2018), we have that $\begin{array} { r } { \operatorname* { l i m } \operatorname* { i n f } _ { k  \infty } \tilde { W } _ { c _ { 1 } , c _ { 2 } , \zeta ^ { k } \psi } ( \mu , \nu ) \geq } \end{array}$ $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { = } } ( \mu , \nu )$ . Additionally, $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \zeta ^ { k } \psi } ( \mu , \nu ) \leq \tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { = } } ( \mu , \nu )$ for any value of $k$ since for any minimizer $\gamma$ of $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { = } } ( \mu , \nu )$ , it holds that $\pi _ { \# } ^ { y } \gamma = \nu$ . Hence
+
+$$
+\tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { \mathbb { C } } } ( \mu , \nu ) = \int c _ { 1 } d \gamma + D _ { c _ { 2 } } ( \pi _ { \# } ^ { \chi } \gamma | \mu ) + D _ { \zeta ^ { k } \psi _ { 2 } } ( \pi _ { \# } ^ { \ y } \gamma | \nu ) \geq \tilde { W } _ { c _ { 1 } , c _ { 2 } , \zeta ^ { k } \psi } ( \mu , \nu ) ,
+$$
+
+for all $k$ . Therefore, $\begin{array} { r } { \operatorname* { l i m } _ { k \to \infty } \tilde { W } _ { c _ { 1 } , c _ { 2 } , \zeta ^ { k } \psi } ( \mu , \nu ) = \tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { = } } ( \mu , \nu ) } \end{array}$ , which then by Lemma 3.3 implies the first part of the proposition.
+
+For the second part, by the hypothesis we have that $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { = } } ( \mu , \nu ) = C < \infty$ and as a consequence $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \zeta ^ { k } \psi } ( \mu , \nu ) \leq C$ for all $k$ . Hence, by Proposition 2.10 in (Liero et al., 2018), the sequence of minimizers $\gamma ^ { k }$ is bounded. If the assumptions of Proposition B.1 are satisfied, then the sequence $\gamma ^ { k }$ is equally tight. For assumption (ii) this follows by Proposition 2.10 in (Liero et al., 2018) and for assumption (i) this follows by the Markov inequality: for any $\lambda > 0$ ,
+
+$$
+\gamma ^ { k } ( \{ ( x , y ) \in \mathcal { X } \times \mathcal { Y } | c _ { 1 } ( x , y ) > \lambda \} ) \le \frac { 1 } { \lambda } \int c _ { 1 } d \gamma ^ { k } \le \frac { C } { \lambda } .
+$$
+
+Since $\gamma ^ { k }$ are bounded and equally tight, by an extension of Prokhorov’s theorem (Theorem 2.2 of (Liero et al., 2018)), there exists a subsequence of $\gamma ^ { k }$ that is weakly convergent to some $\bar { \gamma }$ . Then by lower semicontinuity, we obtain that
+
+$$
+\int c _ { 1 } d \bar { \gamma } + D _ { c _ { 2 } } ( \pi _ { \# } ^ { \mathcal { X } } \bar { \gamma } | \mu ) + \operatorname* { l i m } _ { k \to \infty } D _ { \zeta ^ { k } \psi _ { 2 } } ( \pi _ { \# } ^ { \mathcal { Y } } \gamma ^ { k } | \nu ) \leq \tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { = } } ( \mu , \nu ) = C
+$$
+
+Since $D _ { \zeta ^ { k } \psi } ( \pi _ { \# } ^ { y } \gamma ^ { k } | \nu ) = \zeta ^ { k } D _ { \psi } ( \pi _ { \# } ^ { y } \gamma ^ { k } | \nu ) \geq 0$ and $\zeta ^ { k } \to \infty$ , for the left side to be finite, $D _ { \psi } ( \pi _ { \# } ^ { y } \gamma ^ { k } | \nu )$ must converge to $0$ , so $D _ { \psi } ( \pi _ { \# } ^ { y } \bar { \gamma } | \nu ) = 0$ by lower semicontinuity. Therefore, $\bar { \gamma }$ is a minimizer of $\tilde { W } _ { c _ { 1 } , c _ { 2 } , \iota _ { = } } ( \mu , \nu )$ . By construction of the proof of Lemma 3.3, $\gamma ^ { k }$ is equivalent to the product measure induced by minimizers of $L _ { \zeta ^ { k } \psi } ( \mu , \nu )$ , which implies the second part of the proposition. □
+
+# B.2 CONVEX CONJUGATE FORM OF DIVERGENCES
+
+In this section, we present the convex conjugate form of $\psi$ -divergence used to rewrite the main objective as a min-max problem.
+
+Lemma B.2. For non-negative finite measures $P , Q$ over $\mathcal { T } \subset \mathbb { R } ^ { d }$ , it holds that
+
+$$
+D _ { \psi } ( P | Q ) \geq \operatorname* { s u p } _ { f \in { \mathcal { F } } } \int f d P - \psi ^ { * } ( f ) d Q
+$$
+
+where $\mathcal { F }$ is a subset of measurable functions $\{ f : \mathcal { T } \to ( - \infty , \psi _ { \infty } ^ { \prime } ] \}$ . Equality holds if and only $i f \exists f \in { \mathcal { F } }$ such that (i) the restriction of $f$ to the support of $Q$ belongs to the subdifferential of $\psi ( { \frac { d P } { d Q } } )$ , i.e. the Radon-Nikodym derivative of $P$ with respect to $Q$ and (ii) $f = \psi _ { \infty } ^ { \prime }$ over the support of $P ^ { \perp }$ .
+
+We provide a simple proof of this result. A similar result under stronger assumptions was shown in Nguyen et al. (2008) and used by Nowozin et al. (2016) for generative modeling. A rigorous proof can be found in Liero et al. (2018).
+
+Proof of Lemma B.2. Note that
+
+$$
+\begin{array} { r l } & { D _ { \varphi } ( P | Q ) = \psi _ { \infty } ^ { \prime } P _ { \perp } ( T ) + \int _ { T } \psi \left( \frac { d P } { d Q } \right) d Q } \\ & { \quad \quad \quad = \psi _ { \infty } ^ { \prime } P _ { \perp } ( T ) + \int _ { T \in \cup \operatorname { i n f } } \{ \frac { d P } { d Q } - \psi ^ { \prime } ( \xi ) \} d Q } \\ & { \quad \quad \quad \quad ( \mathrm { b y ~ d e i n i o n ~ o f ~ c o m e x ~ c o n i n g ~ a t e } ) } \\ & { \quad \quad \quad = \psi _ { \infty } ^ { \prime } P _ { \perp } ( T ) + \int _ { T \in \{ c = - \infty , \psi _ { \infty } ^ { \prime } \} } \{ \xi \frac { d P } { d Q } - \psi ^ { \prime } ( \xi ) \} d Q } \\ & { \quad \quad \quad ( \mathrm { b y ~ L o m u a B . } ) \mathrm { b z ~ i s f o r ~ } } \\ & { \quad \quad \quad = \int _ { T \in \{ c = - \infty , \psi _ { \infty } ^ { \prime } \} } \{ \xi d P _ { \perp } + \xi \frac { d P } { d Q } d Q - \psi ^ { \prime } ( \xi ) d Q \} } \\ & { \quad \quad \quad = \operatorname* { s u p } _ { \xi \in \mathcal { S } } \int _ { T } d \phi - \psi ^ { \prime } ( f ) d \phi . } \end{array}
+$$
+
+By first-order optimality conditions, the optimal $f$ over the support of $Q$ is obtained when $\textstyle { \frac { d P } { d Q } }$ belongs to the subdifferential of $\psi ^ { * } ( f )$ , or equivalently when $f$ belongs to the subdifferential of $\psi ( { \frac { d P } { d Q } } )$ . It is straightforward to see that the optimal $f$ over the support of $P _ { \perp }$ is equal to $\psi _ { \infty } ^ { \prime }$ , which completes the proof. □
+
+Lemma B.3. If $\begin{array} { r } { \dot { \boldsymbol { \xi } } > \psi _ { \infty } ^ { \prime } : = \operatorname* { l i m } _ { s \to \infty } \frac { \psi ( s ) } { s } } \end{array}$ ψ(s) , then ψ∗(ξ) = ∞.
+
+Proof.
+
+$$
+\begin{array} { l } { \displaystyle \psi ^ { * } ( \xi ) = \underset { s \in \mathbb { R } } { \operatorname* { s u p } } s \xi - \psi ( s ) } \\ { \displaystyle \qquad \geq \underset { s  \infty } { \operatorname* { l i m } } s ( \xi - \frac { \psi ( s ) } { s } ) } \\ { \displaystyle = \infty \mathrm { ~ i f ~ } \xi > \psi _ { \infty } ^ { \prime } } \end{array}
+$$
+
+# C PRACTICAL CONSIDERATIONS FOR NUMERICAL EXPERIMENTS
+
+Choice of cost functions. Proposition B.1 gives sufficient conditions on $c _ { 1 } , c _ { 2 }$ for the problem to be wellposed. In practice, it is often convenient the cost of transport, $c _ { 1 }$ , to be some measurement of correspondence between $\mathcal { X }$ and $\mathcal { V }$ . For example, we can take $c _ { 1 } ( x , y )$ to be the Euclidean distance between $x$ and $y$ after mapping them to some common feature space. For the cost of mass adjustment, $c _ { 2 }$ , it is generally sensible to choose some convex function that vanishes at 1 (i.e. no mass adjustment) and such that $\begin{array} { r } { \operatorname* { l i m } _ { x  0 } c _ { 2 } ( x ) = } \end{array}$ $\begin{array} { r } { \operatorname* { l i m } _ { x  \infty } c _ { 2 } ( x ) = \infty } \end{array}$ to prevent $\xi$ from becoming too small or too large. Any of the entropy functions shown in Table 1 are reasonable choices.
+
+<table><tr><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>（s）</td><td rowspan=1 colspan=1>D(P|Q)</td><td rowspan=1 colspan=1>*（s）</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>Activation Layer</td></tr><tr><td rowspan=1 colspan=1>Kullback-Leibler (KL)</td><td rowspan=1 colspan=1>slogs-s+1</td><td rowspan=1 colspan=1>ʃlogddP-ʃdP+ʃdQ</td><td rowspan=1 colspan=1>e</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>Pearson x²</td><td rowspan=1 colspan=1>(s-1)²</td><td rowspan=1 colspan=1>器-1）2dQ</td><td rowspan=1 colspan=1>+s</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>Hellinger</td><td rowspan=1 colspan=1>(√s-1)²</td><td rowspan=1 colspan=1>品一1)²dQ</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1-es</td></tr><tr><td rowspan=1 colspan=1>Jensen-Shannon</td><td rowspan=1 colspan=1>slogs-(s+1)logs1</td><td rowspan=1 colspan=1>1DKL(P|PQ+DKL(QIP+Q)</td><td rowspan=1 colspan=1>-log(2-e)</td><td rowspan=1 colspan=1>log2</td><td rowspan=1 colspan=1>log(2)-log(1+e-))</td></tr></table>
+
+Table 1: Table of some common $\psi$ -divergences, associated entropy functions $\psi$ , and convex conjugates $\psi ^ { * }$ for Algorithm 1, partly adapted from (Nowozin et al., 2016).
+
+Choice of $\psi$ . In Nowozin et al. (2016), it was shown that any $\psi$ -divergence could be used to train generative models, i.e. to match a generated distribution $P$ to a true data distribution $Q$ . This is due to Jensen’s inequality: for any convex lower semi-continous entropy function $\psi$ , $D _ { \psi } ( P | Q )$ is uniquely minimized when $P = Q$ , where $P , Q$ are probability measures. However, this does not generally hold when $P , Q$ are not probability measures, as illustrated by the following example.
+
+Example C.1. In the original GAN paper, the discrminative objective,
+
+$$
+\operatorname* { s u p } _ { f } \int \log f ( x ) d P ( x ) - \int \log ( 1 - f ( x ) ) d Q ( x ) ,
+$$
+
+corresponds to $D _ { \psi } ( P | Q )$ with $\psi ( s ) = s \log s - ( s + 1 ) \log ( s + 1 )$ (Nowozin et al., 2016). If $P , Q$ are probability measures, this divergence is equivalent to the Jensen-Shannon divergence and is minimized when $P = Q$ . If $P , Q$ are non-negative measures with unconstrained total mass, the divergence is minimized when $P = \infty$ and $Q = 0$ .
+
+When $P , Q$ are not probability measures, we require an additional constraint on $\psi$ to ensure that divergence minimization matchces $P$ to $Q$ :
+
+Lemma C.2. Suppose $P , Q$ are non-negative finite measures over $T \subseteq \mathbb { R } ^ { n }$ . If $\psi ( s )$ attains a unique minimum at $s = 1$ with $\psi ( 1 ) = 0$ and $\psi _ { \infty } ^ { \prime } > 0$ , then ${ \dot { D } } _ { \psi } ( P | Q ) = 0 \Rightarrow P = Q$ . Otherwise, then $P \neq Q$ in general when $D _ { \psi } ( P | Q )$ is minimized.
+
+Proof. Suppose $\psi ( s )$ attains a unique minimum at $s = 1$ with $\psi ( 1 ) = 0$ , $\psi _ { \infty } ^ { \prime } > 0$ , and $P \neq Q$ over a region with positive measure. It is straightforward to see by the definition in (4) that $D _ { \psi } ( P | Q ) > 0$ , since at least one of the two terms will be strictly positive. Therefore, the first statement holds. For the second statement, suppose either $\psi ( s )$ does not attain a unique minimum at $s = 1$ or $\psi _ { \infty } ^ { \prime } \leq 0$ . If $\psi ( s )$ attains a minimum at some $s ^ { \prime } \neq 1$ , then taking $P = s ^ { \prime } Q$ results in a divergence that is equal to or less than $P = Q$ . If $\psi _ { \infty } ^ { \prime } \leq 0$ , then letting $P = Q + P _ { \perp }$ where $P _ { \perp }$ is a positive measure orthogonal to $Q$ results in a divergence that is equal to or less than $P = Q$ . □
+
+Table 1 provides some examples of $\psi$ corresponding to common divergences that can be used for unbalanced OT.
+
+Choice of $f$ . According to Lemma B.2, $f$ should belong to a class of functions that maps from $\mathcal { V }$ to $\left( - \infty , \psi _ { \infty } ^ { \prime } \right]$ . In practice, this can be enforced by parameterizing $f$ using a neural network with a final layer that maps to the correct range, also known as an output activation layer (Nowozin et al., 2016). Table 1 provides some examples of activation layers that can be used.
+
+Choice of neural architectures. For our experiments in Section 4, we used fully-connected feedforward networks with 3 hidden layers and ReLU activations. For $T$ , the output activation layer was a sigmoid function to map the final pixel brightness to the range $( 0 , 1 )$ . For $\xi$ , the output activation layer was a softplus function to map the scaling factor weight to the range $( 0 , \infty )$ .
+
+Gradient penalties. The training of GANs using alternating stochastic gradient descent is not guaranteed to converge locally (Mescheder et al., 2018). Stability can be improved using regularization in the form of added instance noise (Arjovsky & Bottou, 2017) or gradient penalties (Gulrajani et al., 2017). We observed that enforcing gradient penalities on $\xi$ and $f$ , while not necessary, improved the stability of Algorithm 1. A gradient penalty on $\xi$ effectively restricts the search to sufficiently smooth $\xi$ , which is reasonable in practice considering that similar regions of the source measure often have similar rates of growth or contraction. A gradient penalty on $f$ changes the nature of the relaxation from (5) to (6): the right-hand side of (7) is no longer equivalent to the $\psi$ -divergence, but is rather a lower-bound with a relation to bounded Lipschitz metrics (Gulrajani et al., 2017). In this case, while the problem formulation is not equivalent to optimal entropy-transport, it is still a valid relaxation of unbalanced Monge OT in (5).
+
+Choice of $\lambda$ . One can take $\lambda$ to be the standard Gaussian measure if a stochastic mapping is desired, similar to Almahairi et al. (2018). If a deterministic mapping is desired, then $\lambda$ is set to a deterministic distribution.
+
+Improved training dynamics. Recall that the objective function for our alternating gradient updates is
+
+$$
+\begin{array} { l } { \displaystyle \ell ( \theta , \phi , \omega ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } [ c _ { \mu } c _ { 1 } ( x _ { i } , T _ { \theta } ( x _ { i } , z _ { i } ) ) \xi _ { \phi } ( x _ { i } ) + c _ { \mu } c _ { 2 } ( \xi _ { \phi } ( x _ { i } ) ) } \\ { \displaystyle + c _ { \mu } \xi _ { \phi } ( x _ { i } ) f _ { \omega } ( T _ { \theta } ( x _ { i } , z _ { i } ) ) - c _ { \nu } \psi ^ { * } ( f _ { \omega } ( y _ { i } ) ) ] . } \end{array}
+$$
+
+Early in training, $\xi _ { \phi }$ can become very small for some $x _ { i }$ as none of the transported samples resemble samples from the target distribution. As a result, $T _ { \theta }$ may improve very slowly for some inputs $x _ { i }$ . One way to address this issue without changing the fixed point is to update $\xi _ { \phi } , f _ { \omega }$ using the above objective and update $T _ { \theta }$ using the following objective.
+
+$$
+\ell ( \theta , \phi , \omega ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } c _ { \mu } [ c _ { 1 } ( x _ { i } , T _ { \theta } ( x _ { i } , z _ { i } ) ) + f _ { \omega } ( T _ { \theta } ( x _ { i } , z _ { i } ) ) ]
+$$
+
+Note that we have omitted terms in the original objective that do not include $T _ { \theta }$ , and which therefore do not affect the gradient update. For the terms that remain, the difference is that we are rescaling the contribution of each sample $x _ { i }$ by $1 / \xi ( x _ { i } )$ . As long as $\xi ( x _ { i } ) > 0$ , this has the effect of rescaling the gradient update of the loss function with respect to each $T _ { \theta } ( x _ { i } , z _ { i } )$ without changing the direction of the update.

parse/train/HylVB3AqYm/HylVB3AqYm.md

@@ -0,0 +1,302 @@
+# PROXYLESSNAS: DIRECT NEURAL ARCHITECTURE SEARCH ON TARGET TASK AND HARDWARE
+
+Han Cai, Ligeng Zhu, Song Han Massachusetts Institute of Technology {hancai, ligeng, songhan}@mit.edu
+
+# ABSTRACT
+
+Neural architecture search (NAS) has a great impact by automatically designing effective neural network architectures. However, the prohibitive computational demand of conventional NAS algorithms (e.g. $1 0 ^ { 4 }$ GPU hours) makes it difficult to directly search the architectures on large-scale tasks (e.g. ImageNet). Differentiable NAS can reduce the cost of GPU hours via a continuous representation of network architecture but suffers from the high GPU memory consumption issue (grow linearly w.r.t. candidate set size). As a result, they need to utilize proxy tasks, such as training on a smaller dataset, or learning with only a few blocks, or training just for a few epochs. These architectures optimized on proxy tasks are not guaranteed to be optimal on the target task. In this paper, we present ProxylessNAS that can directly learn the architectures for large-scale target tasks and target hardware platforms. We address the high memory consumption issue of differentiable NAS and reduce the computational cost (GPU hours and GPU memory) to the same level of regular training while still allowing a large candidate set. Experiments on CIFAR-10 and ImageNet demonstrate the effectiveness of directness and specialization. On CIFAR-10, our model achieves $2 . 0 8 \%$ test error with only 5.7M parameters, better than the previous state-of-the-art architecture AmoebaNet-B, while using $6 \times$ fewer parameters. On ImageNet, our model achieves $3 . 1 \%$ better top-1 accuracy than MobileNetV2, while being $1 . 2 \times$ faster with measured GPU latency. We also apply ProxylessNAS to specialize neural architectures for hardware with direct hardware metrics (e.g. latency) and provide insights for efficient CNN architecture design.1
+
+# 1 INTRODUCTION
+
+Neural architecture search (NAS) has demonstrated much success in automating neural network architecture design for various deep learning tasks, such as image recognition (Zoph et al., 2018; Cai et al., 2018a; Liu et al., 2018a; Zhong et al., 2018) and language modeling (Zoph & Le, 2017). Despite the remarkable results, conventional NAS algorithms are prohibitively computation-intensive, requiring to train thousands of models on the target task in a single experiment. Therefore, directly applying NAS to a large-scale task (e.g. ImageNet) is computationally expensive or impossible, which makes it difficult for making practical industry impact. As a trade-off, Zoph et al. (2018) propose to search for building blocks on proxy tasks, such as training for fewer epochs, starting with a smaller dataset (e.g. CIFAR-10), or learning with fewer blocks. Then top-performing blocks are stacked and transferred to the large-scale target task. This paradigm has been widely adopted in subsequent NAS algorithms (Liu et al., 2018a;b; Real et al., 2018; Cai et al., 2018b; Liu et al., 2018c; Tan et al., 2018; Luo et al., 2018).
+
+However, these blocks optimized on proxy tasks are not guaranteed to be optimal on the target task, especially when taking hardware metrics such as latency into consideration. More importantly, to enable transferability, such methods need to search for only a few architectural motifs and then repeatedly stack the same pattern, which restricts the block diversity and thereby harms performance.
+
+In this work, we propose a simple and effective solution to the aforementioned limitations, called ProxylessNAS, which directly learns the architectures on the target task and hardware instead of with proxy (Figure 1). We also remove the restriction of repeating blocks in previous NAS works (Zoph et al., 2018; Liu et al., 2018c) and allow all of the blocks to be learned and specified. To achieve GPU Hours GPU Memory this, we reduce the computational cost (GPU hours and GPU memory) of architecture search to the same level of regular training in the following ways.
+
+![](images/4728660534263a050e47baba3f6f667425b2aaccf2f3e99f4a23fae5d67af64b.jpg)  
+Figure 1: ProxylessNAS directly optimizes neural network architectures on target task and hardware. Benefiting from the directness and specialization, ProxylessNAS can achieve remarkably better results than previous proxy-based approaches. On ImageNet, with only 200 GPU hours (200 $\times$ fewer than MnasNet (Tan et al., 2018)), our searched CNN model for mobile achieves the same level of top-1 accuracy as MobileNetV2 1.4 while being $1 . 8 \times$ faster.
+
+GPU hour-wise, inspired by recent works (Liu et al., 2018c; Bender et al., 2018), we formulate NAS as a path-level pruning process. Specifically, we directly train an over-parameterized network that contains all candidate paths (Figure 2). During training, we explicitly introduce architecture parameters to learn which paths are redundant, while these redundant paths are pruned at the end of training to get a compact optimized architecture. In this way, we only need to train a single network without any meta-controller (or hypernetwork) during architecture search.
+
+However, naively including all the candidate paths leads to GPU memory explosion (Liu et al., Normal Train NAS  DARTS & One-shot Proxyless (Ours)2018c; Bender et al., 2018), as the memory consumption grows linearly w.r.t. the number of choices. Need Proxy Need Proxy No ProxyThus, GPU memory-wise, we binarize the architecture parameters (1 or 0) and force only one path to be active at run-time, which reduces the required memory to the same level of training a compact model. We propose a gradient-based approach to train these binarized parameters based on BinaryConnect (Courbariaux et al., 2015). Furthermore, to handle non-differentiable hardware objectives (using latency as an example) for learning specialized network architectures on target hardware, we model network latency as a continuous function and optimize it as regularization loss. Additionally, we also present a REINFORCE-based (Williams, 1992) algorithm as an alternative strategy to handle hardware metrics.
+
+In our experiments on CIFAR-10 and ImageNet, benefiting from the directness and specialization, our method can achieve strong empirical results. On CIFAR-10, our model reaches $2 . 0 8 \%$ test error with only 5.7M parameters. On ImageNet, our model achieves $7 5 . 1 \%$ top-1 accuracy which is $3 . 1 \%$ higher than MobileNetV2 (Sandler et al., 2018) while being $1 . 2 \times$ faster. Our contributions can be summarized as follows:
+
+• ProxylessNAS is the first NAS algorithm that directly learns architectures on the largescale dataset (e.g. ImageNet) without any proxy while still allowing a large candidate set and removing the restriction of repeating blocks. It effectively enlarged the search space and achieved better performance.   
+• We provide a new path-level pruning perspective for NAS, showing a close connection between NAS and model compression (Han et al., 2016). We save memory consumption by one order of magnitude by using path-level binarization.   
+• We propose a novel gradient-based approach (latency regularization loss) for handling hardware objectives (e.g. latency). Given different hardware platforms: CPU/GPU/Mobile, ProxylessNAS enables hardware-aware neural network specialization that’s exactly optimized for the target hardware. To our best knowledge, it is the first work to study specialized neural network architectures for different hardware architectures. Extensive experiments showed the advantage of the directness property and the specialization property of ProxylessNAS. It achieved state-of-the-art accuracy performances on CIFAR-10 and ImageNet under latency constraints on different hardware platforms (GPU, CPU and mobile phone). We also analyze the insights of efficient CNN models specialized for different hardware platforms and raise the awareness that specialized neural network architecture is needed on different hardware architectures for efficient inference.
+
+# 2 RELATED WORK
+
+The use of machine learning techniques, such as reinforcement learning or neuro-evolution, to replace human experts in designing neural network architectures, usually referred to as neural architecture search, has drawn an increasing interest (Zoph & Le, 2017; Liu et al., 2018a;b;c; Cai et al., 2018a;b; Pham et al., 2018; Brock et al., 2018; Bender et al., 2018; Elsken et al., 2017; 2018b; Kamath et al., 2018). In NAS, architecture search is typically considered as a meta-learning process, and a meta-controller (e.g. a recurrent neural network (RNN)), is introduced to explore a given architecture space with training a network in the inner loop to get an evaluation for guiding exploration. Consequently, such methods are computationally expensive to run, especially on large-scale tasks, e.g. ImageNet.
+
+Some recent works (Brock et al., 2018; Pham et al., 2018) try to improve the efficiency of this meta-learning process by reducing the cost of getting an evaluation. In Brock et al. (2018), a hypernetwork is utilized to generate weights for each sampled network and hence can evaluate the architecture without training it. Similarly, Pham et al. (2018) propose to share weights among all sampled networks under the standard NAS framework (Zoph & Le, 2017). These methods speed up architecture search by orders of magnitude, however, they require a hypernetwork or an RNN controller and mainly focus on small-scale tasks (e.g. CIFAR) rather than large-scale tasks (e.g. ImageNet).
+
+Our work is most closely related to One-Shot (Bender et al., 2018) and DARTS (Liu et al., 2018c), both of which get rid of the meta-controller (or hypernetwork) by modeling NAS as a single training process of an over-parameterized network that comprises all candidate paths. Specifically, OneShot trains the over-parameterized network with DropPath (Zoph et al., 2018) that drops out each path with some fixed probability. Then they use the pre-trained over-parameterized network to evaluate architectures, which are sampled by randomly zeroing out paths. DARTS additionally introduces a real-valued architecture parameter for each path and jointly train weight parameters and architecture parameters via standard gradient descent. However, they suffer from the large GPU memory consumption issue and hence still need to utilize proxy tasks. In this work, we address the large memory issue in these two methods through path binarization.
+
+Another relevant topic is network pruning (Han et al., 2016) that aim to improve the efficiency of neural networks by removing insignificant neurons (Han et al., 2015) or channels (Liu et al., 2017). Similar to these works, we start with an over-parameterized network and then prune the redundant parts to derive the optimized architecture. The distinction is that they focus on layer-level pruning that only modifies the filter (or units) number of a layer but can not change the topology of the network, while we focus on learning effective network architectures through path-level pruning. We also allow both pruning and growing the number of layers.
+
+# 3 METHOD
+
+We first describe the construction of the over-parameterized network with all candidate paths, then introduce how we leverage binarized architecture parameters to reduce the memory consumption of training the over-parameterized network to the same level as regular training. We propose a gradient-based algorithm to train these binarized architecture parameters. Finally, we present two techniques to handle non-differentiable objectives (e.g. latency) for specializing neural networks on target hardware.
+
+# 3.1 CONSTRUCTION OF OVER-PARAMETERIZED NETWORK
+
+Denote a neural network as $\mathcal { N } ( e , \cdots , e _ { n } )$ where $e _ { i }$ represents a certain edge in the directed acyclic graph (DAG). Let $\mathcal { O } = \{ o _ { i } \}$ be the set of $N$ candidate primitive operations (e.g. convolution, pooling, identity, zero, etc). To construct the over-parameterized network that includes any architecture in the search space, instead of setting each edge to be a definite primitive operation, we set each edge to be a mixed operation that has $N$ parallel paths (Figure 2), denoted as $m _ { \mathcal { O } }$ . As such, the over-parameterized network can be expressed as $\mathcal { N } ( e = m _ { \ O } ^ { 1 } , \cdot \cdot \cdot , e _ { n } = m _ { \ O } ^ { n } )$ ).
+
+Given input $x$ , the output of a mixed operation $m _ { \mathcal { O } }$ is defined based on the outputs of its $N$ paths. In One-Shot, $m _ { \mathcal { O } } ( x )$ is the sum of $\{ o _ { i } ( x ) \}$ , while in DARTS, $m _ { \mathcal { O } } ( x )$ is weighted sum of $\{ o _ { i } ( x ) \}$ where the weights are calculated by applying softmax to $N$ real-valued architecture parameters $\left\{ \alpha _ { i } \right\}$
+
+![](images/91ba369eda3a18ef3afa416b394a07abeed74a7f6af9e90b5671b73469bfae98.jpg)  
+Figure 2: Learning both weight parameters and binarized architecture parameters.
+
+$$
+m _ { \mathcal { O } } ^ { \mathrm { { O n e - S h o t } } } ( x ) = \sum _ { i = 1 } ^ { N } o _ { i } ( x ) , \qquad m _ { \mathcal { O } } ^ { \mathrm { { D A R T S } } } ( x ) = \sum _ { i = 1 } ^ { N } p _ { i } o _ { i } ( x ) = \sum _ { i = 1 } ^ { N } \frac { \exp ( \alpha _ { i } ) } { \sum _ { j } \exp ( \alpha _ { j } ) } o _ { i } ( x ) .
+$$
+
+As shown in Eq. (1), the output feature maps of all $\mathbf { N }$ paths are calculated and stored in the memory, while training a compact model only involves one path. Therefore, One-Shot and DARTS roughly need $N$ Direct measurement:expensive and slow Latcheap, fatimes GPU memory and GPU hours compared to training a compact model. On largescale dataset, this can easily exceed the memory limits of hardware with large design space. In the following section, we solve this memory issue based on the idea of path binarization.
+
+# 3.2 LEARNING BINARIZED PATH
+
+To reduce memory footprint, we keep only one path when training the over-parameterized network. Unlike Courbariaux et al. (2015) which binarize individual weights, we binarize entire paths. We introduce $N$ real-valued architecture parameters $\{ \alpha _ { i } \}$ and then transforms the real-valued path weights to binary gates:
+
+$$
+g = \mathrm { b i n a r i z e } ( p _ { 1 } , \cdots , p _ { N } ) = \left\{ \begin{array} { l l } { { { \left[ 1 , 0 , \cdots ~ , 0 \right] } } } & { { \mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { 1 } , } } \\ { { \cdots } } & { { } } \\ { { \left[ 0 , 0 , \cdots ~ , 1 \right] } } & { { \mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { N } . } } \end{array} \right.
+$$
+
+Based on the binary gates $g$ i  5x53x3 , the output of the mixed operation is given as:
+
+$$
+m _ { \mathcal { O } } ^ { \mathrm { B i n a r y } } ( x ) = \sum _ { i = 1 } ^ { N } g _ { i } o _ { i } ( x ) = \left\{ { \begin{array} { l l } { o _ { 1 } ( x ) } & { \mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { 1 } } \\ { \cdots } & { } \\ { o _ { N } ( x ) } & { \mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { N } . } \end{array} } \right. .
+$$
+
+As illustrated in Eq. (3) and Figure 2, by using the binary gates rather than real-valued path weights (Liu et al., 2018c), only one path of activation is active in memory at run-time and the memory requirement of training the over-parameterized network is thus reduced to the same level of training a compact model. That’s more than an order of magnitude memory saving.
+
+# 3.2.1 TRAINING BINARIZED ARCHITECTURE PARAMETERS
+
+MIT Red-1Figure 2 illustrates the training procedure of the weight parameters and binarized architecture parameters in the over-parameterized network. When training weight parameters, we first freeze the architecture parameters and stochastically sample binary gates according to Eq. (2) for each batch of input data. Then the weight parameters of active paths are updated via standard gradient descent on the training set (Figure 2 left). When training architecture parameters, the weight parameters are frozen, then we reset the binary gates and update the architecture parameters on the validation set (Figure 2 right). These two update steps are performed in an alternative manner. Once the training of architecture parameters is finished, we can then derive the compact architecture by pruning redundant paths. In this work, we simply choose the path with the highest path weight.
+
+Unlike weight parameters, the architecture parameters are not directly involved in the computation graph and thereby cannot be updated using the standard gradient descent. In this section, we introduce a gradient-based approach to learn the architecture parameters.
+
+In BinaryConnect (Courbariaux et al., 2015), the real-valued weight is updated using the gradient w.r.t. its corresponding binary gate. In our case, analogously, the gradient w.r.t. architecture parameters can be approximately estimated using $\partial L / \partial g _ { i }$ in replace of $\bar { \partial } L / \partial p _ { i }$ :
+
+$$
+\frac { \partial L } { \partial \alpha _ { i } } = \sum _ { j = 1 } ^ { N } \frac { \partial L } { \partial p _ { j } } \frac { \partial p _ { j } } { \partial \alpha _ { i } } \approx \sum _ { j = 1 } ^ { N } \frac { \partial L } { \partial g _ { j } } \frac { \partial p _ { j } } { \partial \alpha _ { i } } = \sum _ { j = 1 } ^ { N } \frac { \partial L } { \partial g _ { j } } \frac { \partial \bigg ( \frac { \exp ( \alpha _ { j } ) } { \sum _ { k } \exp ( \alpha _ { k } ) } \bigg ) } { \partial \alpha _ { i } } = \sum _ { j = 1 } ^ { N } \frac { \partial L } { \partial g _ { j } } p _ { j } \big ( \delta _ { i j } - p _ { i } \big ) ,
+$$
+
+where $\delta _ { i j } = 1$ if $i = j$ and $\delta _ { i j } = 0$ if $i \neq j$ . Since the binary gates $g$ are involved in the computation graph, as shown in Eq. (3), $\partial L / \partial g _ { j }$ can be calculated through backpropagation. However, computing $\partial L / \partial g _ { j }$ requires to calculate and store $o _ { j } ( x )$ . Therefore, directly using Eq. (4) to update the architecture parameters would also require roughly $N$ times GPU memory compared to training a compact model.
+
+To address this issue, we consider factorizing the task of choosing one path out of $\mathbf { N }$ candidates into multiple binary selection tasks. The intuition is that if a path is the best choice at a particular position, it should be the better choice when solely compared to any other path.2
+
+Following this idea, within an update step of the architecture parameters, we first sample two paths according to the multinomial distribution $( p _ { 1 } , \cdots , p _ { N } )$ and mask all the other paths as if they do not exist. As such the number of candidates temporarily decrease from $N$ to 2, while the path weights $\{ p _ { i } \}$ and binary gates $\{ g _ { i } \}$ are reset accordingly. Then we update the architecture parameters of these two sampled paths using the gradients calculated via Eq. (4). Finally, as path weights are computed by applying softmax to the architecture parameters, we need to rescale the value of these two updated architecture parameters by multiplying a ratio to keep the path weights of unsampled paths unchanged. As such, in each update step, one of the sampled paths is enhanced (path weight increases) and the other sampled path is attenuated (path weight decreases) while all other paths keep unchanged. In this way, regardless of the value of $N$ , only two paths are involved in each update step of the architecture parameters, and thereby the memory requirement is reduced to the same level of training a compact model.
+
+# 3.3 HANDLING NON-DIFFERENTIABLE HARDWARE METRICS
+
+Besides accuracy, latency (not FLOPs) is another very important objective when designing efficient neural network architectures for hardware. Unfortunately, unlike accuracy that can be optimized using the gradient of the loss function, latency is non-differentiable. In this section, we present two algorithms to handle the non-differentiable objectives.
+
+# 3.3.1 MAKING LATENCY DIFFERENTIABLE
+
+To make latency differentiable, we model the latency of a network as a continuous function of the neural network dimensions 3. Consider a mixed operation with a candidate set $\{ o _ { j } \}$ and each $o _ { j }$ is associated with a path weight $p _ { j }$ which represents the probability of choosing $o _ { j }$ . As such, we have the expected latency of a mixed operation (i.e. a learnable block) as:
+
+$$
+\mathbb { E } [ \mathrm { l a t e n c y } _ { i } ] = \sum _ { j } p _ { j } ^ { i } \times F ( o _ { j } ^ { i } ) ,
+$$
+
+where $\mathbb { E } [ \mathrm { l a t e n c y } _ { i } ]$ is the expected latency of the $i ^ { t h }$ learnable block, $F ( \cdot )$ denotes the latency prediction model and $\bar { F } ( o _ { j } ^ { i } )$ is the predicted latency of $o _ { j } ^ { i }$ . The gradient of $\mathbb { E } [ \mathrm { l a t e n c y } _ { i } ]$ w.r.t. architecture parameters can thereby be given as: $\partial \mathbb { E } [ \mathrm { l a t e n c y } _ { i } ] / \partial p _ { j } ^ { i } = F ( o _ { j } ^ { i } ) .$ .
+
+For the whole network with a sequence of mixed operations (Figure 3 left), since these operations are executed sequentially during inference, the expected latency of the network can be expressed with the sum of these mixed operations’ expected latencies:
+
+$$
+\mathbb { E } [ \mathrm { l a t e n c y } ] = \sum _ { i } \mathbb { E } [ \mathrm { l a t e n c y } _ { i } ] ,
+$$
+
+![](images/408fcd3acff459c85e2d21839a193f855d92afc40095011c31df0b09827d215e.jpg)  
+Figure 3: Making latency differentiable by introducing latency regularization loss.
+
+We incorporate the expected latency of the network into the normal loss function by multiplying a scaling factor $\lambda _ { 2 } ( > 0 \bar { ) }$ which controls the trade-off between accuracy and latency. The final loss function is given as (also shown in Figure 3 right)
+
+$$
+L o s s = L o s s _ { C E } + \lambda _ { 1 } | | w | | _ { 2 } ^ { 2 } + \mathrm { ~ \qquad ~ } ,
+$$
+
+where $L o s s _ { C E }$ denotes the cross-entropy loss and $\lambda _ { 1 } | | w | | _ { 2 } ^ { 2 }$ is the weight decay term.
+
+# 3.3.2 REINFORCE-BASED APPROACH
+
+As an alternative to BinaryConnect, we can utilize REINFORCE to train binarized weights as well. Consider a network that has binarized parameters $\alpha$ , the goal of updating binarized parameters is to find the optimal binary gates $g$ that maximizes a certain reward, denoted as $R ( \cdot )$ . Here we assume the network only has one mixed operation for ease of illustration. Therefore, according to REINFORCE (Williams, 1992), we have the following updates for binarized parameters:
+
+$$
+\begin{array} { r l } & { \quad J ( \alpha ) = \mathbb { E } _ { g \sim \alpha } [ R ( \mathcal { N } _ { g } ) ] = \displaystyle \sum _ { i } p _ { i } R ( \mathcal { N } ( e = o _ { i } ) ) , } \\ & { \quad \nabla _ { \alpha } J ( \alpha ) = \displaystyle \sum _ { i } R ( \mathcal { N } ( e = o _ { i } ) ) \nabla _ { \alpha } p _ { i } = \displaystyle \sum _ { i } R ( \mathcal { N } ( e = o _ { i } ) ) p _ { i } \nabla _ { \alpha } \log ( p _ { i } ) , } \\ & { \quad \quad \quad = \mathbb { E } _ { g \sim \alpha } [ R ( \mathcal { N } _ { g } ) \nabla _ { \alpha } \log ( p ( g ) ) ] \approx \displaystyle \frac { 1 } { M } \displaystyle \sum _ { i = 1 } ^ { M } R ( \mathcal { N } _ { g ^ { i } } ) \nabla _ { \alpha } \log ( p ( g ^ { i } ) ) , } \end{array}
+$$
+
+where $g ^ { i }$ denotes the $i ^ { t h }$ sampled binary gates, $p ( g ^ { i } )$ denotes the probability of sampling $g ^ { i }$ according to Eq. (2) and $\mathcal { N } _ { g ^ { i } }$ is the compact network according to the binary gates $g ^ { i }$ . Since Eq. (8) does not require $R ( \mathcal { N } _ { g } )$ to be differentiable w.r.t. $g$ , it can thus handle non-differentiable objectives. An interesting observation is that Eq. (8) has a similar form to the standard NAS (Zoph & Le, 2017), while it is not a sequential decision-making process and no RNN meta-controller is used in our case. Furthermore, since both gradient-based updates and REINFORCE-based updates are essentially two different update rules to the same binarized architecture parameters, it is possible to combine them to form a new update rule for the architecture parameters.
+
+# 4 EXPERIMENTS AND RESULTS
+
+We demonstrate the effectiveness of our proposed method on two benchmark datasets (CIFAR-10 and ImageNet) for the image classification task. Unlike previous NAS works (Zoph et al., 2018; Liu et al., 2018c) that first learn CNN blocks on CIFAR-10 under small-scale setting (e.g. fewer blocks), then transfer the learned block to ImageNet or CIFAR-10 under large-scale setting by repeatedly stacking it, we directly learn the architectures on the target task (either CIFAR-10 or ImageNet) and target hardware (GPU, CPU and mobile phone) while allowing each block to be specified.
+
+# 4.1 EXPERIMENTS ON CIFAR-10
+
+Architecture Space. For CIFAR-10 experiments, we use the tree-structured architecture space that is introduced by Cai et al. (2018b) with PyramidNet (Han et al., 2017) as the backbone4. Specifically, we replace all $3 \times 3$ convolution layers in the residual blocks of a PyramidNet with tree-structured cells, each of which has a depth of 3 and the number of branches is set to be 2 at each node (except the leaf nodes). For further details about the tree-structured architecture space, we refer to the original paper (Cai et al., 2018b). Additionally, we use two hyperparameters to control the depth and width of a network in this architecture space, i.e. $B$ and $F$ , which respectively represents the number of blocks at each stage (totally 3 stages) and the number of output channels of the final block.
+
+Table 1: ProxylessNAS achieves state-of-the-art performance on CIFAR-10.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=1>Test error (%)</td></tr><tr><td rowspan=1 colspan=1>DenseNet-BC (Huang et al., 2017)PyramidNet (Han etal., 2017)Shake-Shake + c/o (DeVries &amp; Taylor, 2017)PyramidNet + SD(Yamada et al., 2018)</td><td rowspan=1 colspan=1>25.6M26.0M26.2M26.0M</td><td rowspan=1 colspan=1>3.463.312.562.31</td></tr><tr><td rowspan=1 colspan=1>ENAS+ c/o (Pham et al., 2018)DARTS + c/o (Liu et al., 2018c)NASNet-A + c/o (Zoph et al., 2018)PathLevel EAS + c/o (Cai et al., 2018b)AmoebaNet-B + c/o (Real et al., 2018)</td><td rowspan=1 colspan=1>4.6M3.4M27.6M14.3M34.9M</td><td rowspan=1 colspan=1>2.892.832.402.302.13</td></tr><tr><td rowspan=1 colspan=1>Proxyless-R+c/o (ours)Proxyless-G+c/o (ours)</td><td rowspan=1 colspan=1>5.8M5.7M</td><td rowspan=1 colspan=1>2.302.08</td></tr></table>
+
+Training Details. We randomly sample 5,000 images from the training set as a validation set for learning architecture parameters which are updated using the Adam optimizer with an initial learning rate of 0.006 for the gradient-based algorithm (Section 3.2.1) and 0.01 for the REINFORCEbased algorithm (Section 3.3.2). In the following discussions, we refer to these two algorithms as Proxyless-G (gradient) and Proxyless-R (REINFORCE) respectively.
+
+After the training process of the over-parameterized network completes, a compact network is derived according to the architecture parameters, as discussed in Section 3.2.1. Next, we train the compact network using the same training settings except that the number of training epochs increases from 200 to 300. Additionally, when the DropPath regularization (Zoph et al., 2018; Huang et al., 2016) is adopted, we further increase the number of training epochs to 600 (Zoph et al., 2018).
+
+Results. We apply the proposed method to learn architectures in the tree-structured architecture space with $B = 1 8$ and $F = 4 0 0$ . Since we do not repeat cells and each cell has 12 learnable edges, totally $1 2 \times 1 8 \times 3 = 6 4 8$ decisions are required to fully determine the architecture.
+
+The test error rate results of our proposed method and other state-of-the-art architectures on CIFAR10 are summarized in Table 1, where $\mathrm { { ^ { 6 6 } c / o ^ { 3 } } }$ indicates the use of Cutout (DeVries & Taylor, 2017). Compared to these state-of-the-art architectures, our proposed method can achieve not only lower test error rate but also better parameter efficiency. Specifically, Proxyless-G reaches a test error rate of $2 . 0 8 \%$ which is slightly better than AmoebaNet-B (Real et al., 2018) (the previous best architecture on CIFAR-10). Notably, AmoebaNet-B uses 34.9M parameters while our model only uses 5.7M parameters which is $6 \times$ fewer than AmoebaNet-B. Furthermore, compared with PathLevel EAS (Cai et al., 2018b) that also explores the tree-structured architecture space, both Proxyless-G and Proxyless-R achieves similar or lower test error rate results with half fewer parameters. The strong empirical results of our ProxylessNAS demonstrate the benefits of directly exploring a large architecture space instead of repeatedly stacking the same block.
+
+# 4.2 EXPERIMENTS ON IMAGENET
+
+For ImageNet experiments, we focus on learning efficient CNN architectures (Iandola et al., 2016; Howard et al., 2017; Sandler et al., 2018; Zhu et al., 2018) that have not only high accuracy but also low latency on specific hardware platforms. Therefore, it is a multi-objective NAS task (Hsu et al., 2018; Dong et al., 2018; Elsken et al., 2018a; He et al., 2018; Wang et al., 2018; Tan et al., 2018), where one of the objectives is non-differentiable (i.e. latency). We use three different hardware platforms, including mobile phone, GPU and CPU, in our experiments. The GPU latency is measured on V100 GPU with a batch size of 8 (single batch makes GPU severely under-utilized). The CPU latency is measured under batch size 1 on a server with two 2.40GHz Intel(R) Xeon(R)
+
+Table 2: ProxylessNAS achieves state-of-the art accuracy $( \% )$ on ImageNet (under mobile latency constraint $\leq 8 0 m s $ ) with $2 0 0 \times$ less search cost in GPU hours. “LL” indicates latency regularization loss. Details of MnasNet’s search cost are provided in appendix C.   
+
+<table><tr><td>Model</td><td>Top-1</td><td>Top-5</td><td>Mobile Latency</td><td>Hardware -aware</td><td>No Proxy</td><td>No Repeat</td><td>Search cost (GPU hours)</td></tr><tr><td>MobileNetV1[16] MobileNetV2 [30]</td><td>70.6 72.0</td><td>89.5 91.0</td><td>113ms 75ms</td><td>1</td><td>1 1</td><td>X ×</td><td>Manual Manual</td></tr><tr><td>NASNet-A [38]</td><td>74.0</td><td>91.3</td><td>183ms</td><td>1 X</td><td>X</td><td>X</td><td>48,000</td></tr><tr><td>AmoebaNet-A [29]</td><td>74.5</td><td>92.0</td><td>190ms</td><td>X</td><td>×</td><td>×</td><td>75,600</td></tr><tr><td>MnasNet [31]</td><td>74.0</td><td>91.8</td><td>76ms</td><td>√</td><td>×</td><td>×</td><td>40,000</td></tr><tr><td>MnasNet (our impl.)</td><td>74.0</td><td>91.8</td><td>79ms</td><td>√</td><td>X</td><td>X</td><td>40,000</td></tr><tr><td>Proxyless-G (mobile)</td><td>71.8</td><td>90.3</td><td>83ms</td><td>X</td><td>?</td><td>√</td><td>200</td></tr><tr><td>Proxyless-G + LL</td><td>74.2</td><td>91.7</td><td>79ms</td><td>√</td><td></td><td>√</td><td>200</td></tr><tr><td>Proxyless-R (mobile)</td><td>74.6</td><td>92.2</td><td>78ms</td><td>√</td><td>「</td><td>√</td><td>200</td></tr></table>
+
+![](images/5c66bb877d30a396515306d314713e4a93d5d485a7682f5ae667bce3ba3b91c3.jpg)  
+Figure 4: ProxylessNAS consistently outperforms MobileNetV2 under various latency settings.
+
+![](images/b69bf69ff8ddb72625298f86bd1b6e35c9ea60e28283d248c067bb3d7fb055e7.jpg)  
+Figure 5: Our mobile latency model is close to $y \ = \ x$ . The latency RMSE is $0 . 7 5 \mathrm { m s }$ .
+
+CPU E5-2640 v4. The mobile latency is measured on Google Pixel 1 phone with a batch size of 1. For Proxyless-R, we use $A C C ( m ) ^ { \cdot } \times [ L A T ( m ) / T ] ^ { w }$ as the optimization goal, where $A C C ( m )$ denotes the accuracy of model $m$ , $L A T ( m )$ denotes the latency of $m$ , $T$ is the target latency and $w$ is a hyperparameter for controlling the trade-off between accuracy and latency.
+
+Additionally, on mobile phone, we use the latency prediction model (Appendix B) during architecture search. As illustrated in Figure 5, we observe a strong correlation between the predicted latency and real measured latency on the test set, suggesting that the latency prediction model can be used to replace the expensive mobile farm infrastructure (Tan et al., 2018) with little error introduced.
+
+Architecture Space. We use MobileNetV2 (Sandler et al., 2018) as the backbone to build the architecture space. Specifically, rather than repeating the same mobile inverted bottleneck convolution (MBConv), we allow a set of MBConv layers with various kernel sizes $\{ 3 , 5 , 7 \}$ and expansion ratios $\{ 3 , 6 \}$ . To enable a direct trade-off between width and depth, we initiate a deeper over-parameterized network and allow a block with the residual connection to be skipped by adding the zero operation to the candidate set of its mixed operation. In this way, with a limited latency budget, the network can either choose to be shallower and wider by skipping more blocks and using larger MBConv layers or choose to be deeper and thinner by keeping more blocks and using smaller MBConv layers.
+
+Training Details. We randomly sample 50,000 images from the training set as a validation set during the architecture search. The settings for updating architecture parameters are the same as CIFAR-10 experiments except the initial learning rate is 0.001. The over-parameterized network is trained on the remaining training images with batch size 256.
+
+Table 3: ImageNet Accuracy $( \% )$ and GPU latency (Tesla V100) on ImageNet.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Top-1</td><td rowspan=1 colspan=1>Top-5</td><td rowspan=1 colspan=1>GPU latency</td></tr><tr><td rowspan=1 colspan=1>MobileNetV2 (Sandler et al., 2018)ShuffleNetV2 (1.5) (Ma et al., 2018)ResNet-34 (He et al., 2016)</td><td rowspan=1 colspan=1>72.072.673.3</td><td rowspan=1 colspan=1>91.0191.4</td><td rowspan=1 colspan=1>6.1ms7.3ms8.0ms</td></tr><tr><td rowspan=1 colspan=1>NASNet-A (Zoph et al., 2018)DARTS (Liu et al., 2018c)MnasNet (Tan et al., 2018)</td><td rowspan=1 colspan=1>74.073.174.0</td><td rowspan=1 colspan=1>91.391.091.8</td><td rowspan=1 colspan=1>38.3ms16.1ms</td></tr><tr><td rowspan=1 colspan=1>Proxyless (GPU)</td><td rowspan=1 colspan=1>75.1</td><td rowspan=1 colspan=1>92.5</td><td rowspan=1 colspan=1>5.1ms</td></tr></table>
+
+ImageNet Classification Results. We first apply our ProxylessNAS to learn specialized CNN models on the mobile phone. The summarized results are reported in Table 2. Compared to MobileNetV2, our model improves the top-1 accuracy by $2 . 6 \%$ while maintaining a similar latency on the mobile phone. Furthermore, by rescaling the width of the networks using a multiplier (Sandler et al., 2018; Tan et al., 2018), it is shown in Figure 4 that our model consistently outperforms MobileNetV2 by a significant margin under all latency settings. Specifically, to achieve the same level of top-1 accuracy performance (i.e. around $7 4 . 6 \%$ ), MobileNetV2 has 143ms latency while our model only needs 78ms $( { \bf 1 . 8 3 \times }$ faster). While compared with MnasNet (Tan et al., 2018), our model can achieve $0 . 6 \%$ higher top-1 accuracy with slightly lower mobile latency. More importantly, we are much more resource efficient: the GPU-hour is $2 0 0 \times$ fewer than MnasNet (Table 2).
+
+Additionally, we also observe that Proxyless-G has no incentive to choose computation-cheap operations if were not for the latency regularization loss. Its resulting architecture initially has 158ms latency on Pixel 1. After rescaling the network using the multiplier, its latency reduces to $8 3 \mathrm { m s }$ . However, this model can only achieve $7 1 . 8 \%$ top-1 accuracy on ImageNet, which is $2 . 4 \%$ lower than the result given by Proxyless-G with latency regularization loss. Therefore, we conclude that it is essential to take latency as a direct objective when learning efficient neural networks.
+
+Besides the mobile phone, we also apply our ProxylessNAS to learn specialized CNN models on GPU and CPU. Table 3 reports the results on GPU, where we find that our ProxylessNAS can still achieve superior performances compared to both human-designed and automatically searched architectures. Specifically, compared to MobileNetV2 and MnasNet, our model improves the top-1 accuracy by $3 . 1 \%$ and $1 . 1 \%$ respectively while being $1 . 2 \times$ faster. Table 4 shows the summarized results of our searched models on three different platforms. An interesting observation is that models optimized for GPU do not run fast on CPU and mobile phone, vice versa. Therefore, it is essential to learn specialized neural networks for different hardware architectures to achieve the best efficiency on different hardware.
+
+Specialized Models for Different Hardware. Figure 6 demonstrates the detailed architectures of our searched CNN models on three hardware platforms: GPU/CPU/Mobile. We notice that the architecture shows different preferences when targeting different platforms: (i) The GPU model is shallower and wider, especially in early stages where the feature map has higher resolution; (ii) The GPU model prefers large MBConv operations (e.g. $7 \times 7$ MBConv6), while the CPU model would go for smaller MBConv operations. This is because GPU has much higher parallelism than CPU so it can take advantage of large MBConv operations. Another interesting observation is that our searched models on all platforms prefer larger MBConv operations in the first block within each stage where the feature map is downsampled. We suppose it might because larger MBConv operations are beneficial for the network to preserve more information when downsampling. Notably, such kind of patterns cannot be captured in previous NAS methods as they force the blocks to share the same structure (Zoph et al., 2018; Liu et al., 2018a).
+
+# 5 CONCLUSION
+
+We introduced ProxylessNAS that can directly learn neural network architectures on the target task and target hardware without any proxy. We also reduced the search cost (GPU-hours and GPU memory) of NAS to the same level of normal training using path binarization. Benefiting from the direct search, we achieve strong empirical results on CIFAR-10 and ImageNet. Furthermore,
+
+![](images/9e0a1090b199cb164fd72d1ed54a3efec06e17ce9b325aae33e66b6340b57edf.jpg)  
+Figure 6: Efficient models optimized for different hardware. “MBConv3” and “MBConv6” denote mobile inverted bottleneck convolution layer with an expansion ratio of 3 and 6 respectively. Insights: GPU prefers shallow and wide model with early pooling; CPU prefers deep and narrow model with late pooling. Pooling layers prefer large and wide kernel. Early layers prefer small kernel. Late layers prefer large kernel.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Top-1 (%)</td><td rowspan=1 colspan=1>GPU latency</td><td rowspan=1 colspan=1>CPU latency</td><td rowspan=1 colspan=1>Mobile latency</td></tr><tr><td rowspan=1 colspan=1>Proxyless(GPU)</td><td rowspan=1 colspan=1>75.1</td><td rowspan=1 colspan=1>5.1ms</td><td rowspan=1 colspan=1>204.9ms</td><td rowspan=1 colspan=1>124ms</td></tr><tr><td rowspan=1 colspan=1>Proxyless(CPU)</td><td rowspan=1 colspan=1>75.3</td><td rowspan=1 colspan=1>7.4ms</td><td rowspan=1 colspan=1>138.7ms</td><td rowspan=1 colspan=1>116ms</td></tr><tr><td rowspan=1 colspan=1>Proxyless(mobile)</td><td rowspan=1 colspan=1>74.6</td><td rowspan=1 colspan=1>7.2ms</td><td rowspan=1 colspan=1>164.1ms</td><td rowspan=1 colspan=1>78ms</td></tr></table>
+
+Table 4: Hardware prefers specialized models. Models optimized for GPU does not run fast on CPU and mobile phone, vice versa. ProxylessNAS provides an efficient solution to search a specialized (3) Efficient GPU architecture found by ProxylessNAS. neural network architecture for a target hardware architecture, while cutting down the search cost by $2 0 0 \times$ compared with state-of-the-arts (Zoph & Le, 2017; Tan et al., 2018).
+
+we allow specializing network architectures for different platforms by directly incorporating the measured hardware latency into optimization objectives. We compared the optimized models on CPU/GPU/mobile and raised the awareness of the needs of specializing neural network architecture for different hardware architectures.
+
+# ACKNOWLEDGMENTS
+
+We thank MIT Quest for Intelligence, MIT-IBM Watson AI lab, SenseTime, Xilinx, Snap Research for supporting this work. We also thank AWS Cloud Credits for Research Program providing us the cloud computing resources.
+
+# REFERENCES
+
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+
+# A THE LIST OF CANDIDATE OPERATIONS USED ON CIFAR-10
+
+We adopt the following 7 operations in our CIFAR-10 experiments:
+
+• $3 \times 3$ dilated depthwise-separable convolution   
+• Identity   
+• $3 \times 3$ depthwise-separable convolution   
+• $5 \times 5$ depthwise-separable convolution   
+• $7 \times 7$ depthwise-separable convolution   
+• $3 \times 3$ average pooling   
+• $3 \times 3$ max pooling
+
+# B MOBILE LATENCY PREDICTION
+
+Measuring the latency on-device is accurate but not ideal for scalable neural architecture search. There are two reasons: (i) Slow. As suggested in TensorFlow-Lite, we need to average hundreds of runs to produce a precise measurement, approximately 20 seconds. This is far more slower than a single forward / backward execution. (ii) Expensive. A lot of mobile devices and software engineering work are required to build an automatic pipeline to gather the latency from a mobile farm. Instead of direct measurement, we build a model to estimate the latency. We need only 1 phone rather than a farm of phones, which has only $0 . 7 5 \mathrm { m s }$ latency RMSE. We use the latency model to search, and we use the measured latency to report the final model’s latency.
+
+We sampled $5 \mathrm { k }$ architectures from our candidate space, where $4 \mathrm { k \Omega }$ architectures are used to build the latency model and the rest are used for test. We measured the latency on Google Pixel 1 phone using TensorFlow-Lite. The features include (i) type of the operator (ii) input and output feature map size (iii) other attributes like kernel size, stride for convolution and expansion ratio.
+
+# C DETAILS OF MNASNET’S SEARCH COST
+
+Mnas (Tan et al., 2018) trains 8,000 mobile-sized models on ImageNet, each of which is trained for 5 epochs for learning architectures. If these models are trained on V100 GPUs, as done in our experiments, the search cost is roughly 40,000 GPU hours.
+
+# D IMPLEMENTAION OF THE GRADIENT-BASED ALGORITHM
+
+A naive implementation of the gradient-based algorithm (see Eq. (4)) is calculating and storing $o _ { j } ( x )$ in the forward step to later compute $\partial L / \partial g _ { j }$ in the backward step:
+
+$$
+\partial L / \partial g _ { j } = \mathrm { r e d u c e \_ s u m } ( \nabla _ { y } L \circ o _ { j } ( x ) ) ,
+$$
+
+where $\nabla _ { y } L$ denotes the gradient w.r.t. the output of the mixed operation $y$ , “◦” denotes the elementwise product, and “reduce $\_ \mathrm { s u m ( \cdot ) } ,$ ” denotes the sum of all elements.
+
+Notice that $o _ { j } ( x )$ is only used for calculating $\partial L / \partial g _ { j }$ when $j ^ { t h }$ path is not active (i.e. not involved in calculating $y$ ). So we do not need to actually allocate GPU memory to store $o _ { j } ( x )$ . Instead, we can calculate $o _ { j } ( x )$ after getting $\nabla _ { y } L$ in the backward step, use $o _ { j } ( x )$ to compute $\partial L / \partial g _ { j }$ following Eq. (9), then release the occupied GPU memory. In this way, without the approximation discussed in Section 3.2.1, we can reduce the GPU memory cost to the same level of training a compact model.

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+        "text": "PROXYLESSNAS: DIRECT NEURAL ARCHITECTURE SEARCH ON TARGET TASK AND HARDWARE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Han Cai, Ligeng Zhu, Song Han Massachusetts Institute of Technology {hancai, ligeng, songhan}@mit.edu ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Neural architecture search (NAS) has a great impact by automatically designing effective neural network architectures. However, the prohibitive computational demand of conventional NAS algorithms (e.g. $1 0 ^ { 4 }$ GPU hours) makes it difficult to directly search the architectures on large-scale tasks (e.g. ImageNet). Differentiable NAS can reduce the cost of GPU hours via a continuous representation of network architecture but suffers from the high GPU memory consumption issue (grow linearly w.r.t. candidate set size). As a result, they need to utilize proxy tasks, such as training on a smaller dataset, or learning with only a few blocks, or training just for a few epochs. These architectures optimized on proxy tasks are not guaranteed to be optimal on the target task. In this paper, we present ProxylessNAS that can directly learn the architectures for large-scale target tasks and target hardware platforms. We address the high memory consumption issue of differentiable NAS and reduce the computational cost (GPU hours and GPU memory) to the same level of regular training while still allowing a large candidate set. Experiments on CIFAR-10 and ImageNet demonstrate the effectiveness of directness and specialization. On CIFAR-10, our model achieves $2 . 0 8 \\%$ test error with only 5.7M parameters, better than the previous state-of-the-art architecture AmoebaNet-B, while using $6 \\times$ fewer parameters. On ImageNet, our model achieves $3 . 1 \\%$ better top-1 accuracy than MobileNetV2, while being $1 . 2 \\times$ faster with measured GPU latency. We also apply ProxylessNAS to specialize neural architectures for hardware with direct hardware metrics (e.g. latency) and provide insights for efficient CNN architecture design.1 ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "Neural architecture search (NAS) has demonstrated much success in automating neural network architecture design for various deep learning tasks, such as image recognition (Zoph et al., 2018; Cai et al., 2018a; Liu et al., 2018a; Zhong et al., 2018) and language modeling (Zoph & Le, 2017). Despite the remarkable results, conventional NAS algorithms are prohibitively computation-intensive, requiring to train thousands of models on the target task in a single experiment. Therefore, directly applying NAS to a large-scale task (e.g. ImageNet) is computationally expensive or impossible, which makes it difficult for making practical industry impact. As a trade-off, Zoph et al. (2018) propose to search for building blocks on proxy tasks, such as training for fewer epochs, starting with a smaller dataset (e.g. CIFAR-10), or learning with fewer blocks. Then top-performing blocks are stacked and transferred to the large-scale target task. This paradigm has been widely adopted in subsequent NAS algorithms (Liu et al., 2018a;b; Real et al., 2018; Cai et al., 2018b; Liu et al., 2018c; Tan et al., 2018; Luo et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "However, these blocks optimized on proxy tasks are not guaranteed to be optimal on the target task, especially when taking hardware metrics such as latency into consideration. More importantly, to enable transferability, such methods need to search for only a few architectural motifs and then repeatedly stack the same pattern, which restricts the block diversity and thereby harms performance. ",
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+    {
+        "type": "text",
+        "text": "In this work, we propose a simple and effective solution to the aforementioned limitations, called ProxylessNAS, which directly learns the architectures on the target task and hardware instead of with proxy (Figure 1). We also remove the restriction of repeating blocks in previous NAS works (Zoph et al., 2018; Liu et al., 2018c) and allow all of the blocks to be learned and specified. To achieve GPU Hours GPU Memory this, we reduce the computational cost (GPU hours and GPU memory) of architecture search to the same level of regular training in the following ways. ",
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+    {
+        "type": "image",
+        "img_path": "images/4728660534263a050e47baba3f6f667425b2aaccf2f3e99f4a23fae5d67af64b.jpg",
+        "image_caption": [
+            "Figure 1: ProxylessNAS directly optimizes neural network architectures on target task and hardware. Benefiting from the directness and specialization, ProxylessNAS can achieve remarkably better results than previous proxy-based approaches. On ImageNet, with only 200 GPU hours (200 $\\times$ fewer than MnasNet (Tan et al., 2018)), our searched CNN model for mobile achieves the same level of top-1 accuracy as MobileNetV2 1.4 while being $1 . 8 \\times$ faster. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "GPU hour-wise, inspired by recent works (Liu et al., 2018c; Bender et al., 2018), we formulate NAS as a path-level pruning process. Specifically, we directly train an over-parameterized network that contains all candidate paths (Figure 2). During training, we explicitly introduce architecture parameters to learn which paths are redundant, while these redundant paths are pruned at the end of training to get a compact optimized architecture. In this way, we only need to train a single network without any meta-controller (or hypernetwork) during architecture search. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "However, naively including all the candidate paths leads to GPU memory explosion (Liu et al., Normal Train NAS  DARTS & One-shot Proxyless (Ours)2018c; Bender et al., 2018), as the memory consumption grows linearly w.r.t. the number of choices. Need Proxy Need Proxy No ProxyThus, GPU memory-wise, we binarize the architecture parameters (1 or 0) and force only one path to be active at run-time, which reduces the required memory to the same level of training a compact model. We propose a gradient-based approach to train these binarized parameters based on BinaryConnect (Courbariaux et al., 2015). Furthermore, to handle non-differentiable hardware objectives (using latency as an example) for learning specialized network architectures on target hardware, we model network latency as a continuous function and optimize it as regularization loss. Additionally, we also present a REINFORCE-based (Williams, 1992) algorithm as an alternative strategy to handle hardware metrics. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In our experiments on CIFAR-10 and ImageNet, benefiting from the directness and specialization, our method can achieve strong empirical results. On CIFAR-10, our model reaches $2 . 0 8 \\%$ test error with only 5.7M parameters. On ImageNet, our model achieves $7 5 . 1 \\%$ top-1 accuracy which is $3 . 1 \\%$ higher than MobileNetV2 (Sandler et al., 2018) while being $1 . 2 \\times$ faster. Our contributions can be summarized as follows: ",
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• ProxylessNAS is the first NAS algorithm that directly learns architectures on the largescale dataset (e.g. ImageNet) without any proxy while still allowing a large candidate set and removing the restriction of repeating blocks. It effectively enlarged the search space and achieved better performance.   \n• We provide a new path-level pruning perspective for NAS, showing a close connection between NAS and model compression (Han et al., 2016). We save memory consumption by one order of magnitude by using path-level binarization.   \n• We propose a novel gradient-based approach (latency regularization loss) for handling hardware objectives (e.g. latency). Given different hardware platforms: CPU/GPU/Mobile, ProxylessNAS enables hardware-aware neural network specialization that’s exactly optimized for the target hardware. To our best knowledge, it is the first work to study specialized neural network architectures for different hardware architectures. Extensive experiments showed the advantage of the directness property and the specialization property of ProxylessNAS. It achieved state-of-the-art accuracy performances on CIFAR-10 and ImageNet under latency constraints on different hardware platforms (GPU, CPU and mobile phone). We also analyze the insights of efficient CNN models specialized for different hardware platforms and raise the awareness that specialized neural network architecture is needed on different hardware architectures for efficient inference. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "The use of machine learning techniques, such as reinforcement learning or neuro-evolution, to replace human experts in designing neural network architectures, usually referred to as neural architecture search, has drawn an increasing interest (Zoph & Le, 2017; Liu et al., 2018a;b;c; Cai et al., 2018a;b; Pham et al., 2018; Brock et al., 2018; Bender et al., 2018; Elsken et al., 2017; 2018b; Kamath et al., 2018). In NAS, architecture search is typically considered as a meta-learning process, and a meta-controller (e.g. a recurrent neural network (RNN)), is introduced to explore a given architecture space with training a network in the inner loop to get an evaluation for guiding exploration. Consequently, such methods are computationally expensive to run, especially on large-scale tasks, e.g. ImageNet. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Some recent works (Brock et al., 2018; Pham et al., 2018) try to improve the efficiency of this meta-learning process by reducing the cost of getting an evaluation. In Brock et al. (2018), a hypernetwork is utilized to generate weights for each sampled network and hence can evaluate the architecture without training it. Similarly, Pham et al. (2018) propose to share weights among all sampled networks under the standard NAS framework (Zoph & Le, 2017). These methods speed up architecture search by orders of magnitude, however, they require a hypernetwork or an RNN controller and mainly focus on small-scale tasks (e.g. CIFAR) rather than large-scale tasks (e.g. ImageNet). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our work is most closely related to One-Shot (Bender et al., 2018) and DARTS (Liu et al., 2018c), both of which get rid of the meta-controller (or hypernetwork) by modeling NAS as a single training process of an over-parameterized network that comprises all candidate paths. Specifically, OneShot trains the over-parameterized network with DropPath (Zoph et al., 2018) that drops out each path with some fixed probability. Then they use the pre-trained over-parameterized network to evaluate architectures, which are sampled by randomly zeroing out paths. DARTS additionally introduces a real-valued architecture parameter for each path and jointly train weight parameters and architecture parameters via standard gradient descent. However, they suffer from the large GPU memory consumption issue and hence still need to utilize proxy tasks. In this work, we address the large memory issue in these two methods through path binarization. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Another relevant topic is network pruning (Han et al., 2016) that aim to improve the efficiency of neural networks by removing insignificant neurons (Han et al., 2015) or channels (Liu et al., 2017). Similar to these works, we start with an over-parameterized network and then prune the redundant parts to derive the optimized architecture. The distinction is that they focus on layer-level pruning that only modifies the filter (or units) number of a layer but can not change the topology of the network, while we focus on learning effective network architectures through path-level pruning. We also allow both pruning and growing the number of layers. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We first describe the construction of the over-parameterized network with all candidate paths, then introduce how we leverage binarized architecture parameters to reduce the memory consumption of training the over-parameterized network to the same level as regular training. We propose a gradient-based algorithm to train these binarized architecture parameters. Finally, we present two techniques to handle non-differentiable objectives (e.g. latency) for specializing neural networks on target hardware. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 CONSTRUCTION OF OVER-PARAMETERIZED NETWORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Denote a neural network as $\\mathcal { N } ( e , \\cdots , e _ { n } )$ where $e _ { i }$ represents a certain edge in the directed acyclic graph (DAG). Let $\\mathcal { O } = \\{ o _ { i } \\}$ be the set of $N$ candidate primitive operations (e.g. convolution, pooling, identity, zero, etc). To construct the over-parameterized network that includes any architecture in the search space, instead of setting each edge to be a definite primitive operation, we set each edge to be a mixed operation that has $N$ parallel paths (Figure 2), denoted as $m _ { \\mathcal { O } }$ . As such, the over-parameterized network can be expressed as $\\mathcal { N } ( e = m _ { \\ O } ^ { 1 } , \\cdot \\cdot \\cdot , e _ { n } = m _ { \\ O } ^ { n } )$ ). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given input $x$ , the output of a mixed operation $m _ { \\mathcal { O } }$ is defined based on the outputs of its $N$ paths. In One-Shot, $m _ { \\mathcal { O } } ( x )$ is the sum of $\\{ o _ { i } ( x ) \\}$ , while in DARTS, $m _ { \\mathcal { O } } ( x )$ is weighted sum of $\\{ o _ { i } ( x ) \\}$ where the weights are calculated by applying softmax to $N$ real-valued architecture parameters $\\left\\{ \\alpha _ { i } \\right\\}$ ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/91ba369eda3a18ef3afa416b394a07abeed74a7f6af9e90b5671b73469bfae98.jpg",
+        "image_caption": [
+            "Figure 2: Learning both weight parameters and binarized architecture parameters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/752320c2dce8d94033d8ca87ee187276c1cfbe526d0cab8a3da2c6d112e55626.jpg",
+        "text": "$$\nm _ { \\mathcal { O } } ^ { \\mathrm { { O n e - S h o t } } } ( x ) = \\sum _ { i = 1 } ^ { N } o _ { i } ( x ) , \\qquad m _ { \\mathcal { O } } ^ { \\mathrm { { D A R T S } } } ( x ) = \\sum _ { i = 1 } ^ { N } p _ { i } o _ { i } ( x ) = \\sum _ { i = 1 } ^ { N } \\frac { \\exp ( \\alpha _ { i } ) } { \\sum _ { j } \\exp ( \\alpha _ { j } ) } o _ { i } ( x ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As shown in Eq. (1), the output feature maps of all $\\mathbf { N }$ paths are calculated and stored in the memory, while training a compact model only involves one path. Therefore, One-Shot and DARTS roughly need $N$ Direct measurement:expensive and slow Latcheap, fatimes GPU memory and GPU hours compared to training a compact model. On largescale dataset, this can easily exceed the memory limits of hardware with large design space. In the following section, we solve this memory issue based on the idea of path binarization. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 LEARNING BINARIZED PATH ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To reduce memory footprint, we keep only one path when training the over-parameterized network. Unlike Courbariaux et al. (2015) which binarize individual weights, we binarize entire paths. We introduce $N$ real-valued architecture parameters $\\{ \\alpha _ { i } \\}$ and then transforms the real-valued path weights to binary gates: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ccb87dcd4b61da2112088945b65560b78a3015d0522f4a4234b74539c5c960b.jpg",
+        "text": "$$\ng = \\mathrm { b i n a r i z e } ( p _ { 1 } , \\cdots , p _ { N } ) = \\left\\{ \\begin{array} { l l } { { { \\left[ 1 , 0 , \\cdots ~ , 0 \\right] } } } & { { \\mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { 1 } , } } \\\\ { { \\cdots } } & { { } } \\\\ { { \\left[ 0 , 0 , \\cdots ~ , 1 \\right] } } & { { \\mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { N } . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Based on the binary gates $g$ i  5x53x3 , the output of the mixed operation is given as: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36a9c7d57fa8f1fa449169325830b0e1efd4f7708e03c106a20069f4e8748443.jpg",
+        "text": "$$\nm _ { \\mathcal { O } } ^ { \\mathrm { B i n a r y } } ( x ) = \\sum _ { i = 1 } ^ { N } g _ { i } o _ { i } ( x ) = \\left\\{ { \\begin{array} { l l } { o _ { 1 } ( x ) } & { \\mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { 1 } } \\\\ { \\cdots } & { } \\\\ { o _ { N } ( x ) } & { \\mathrm { w i t h ~ p r o b a b i l i t y } ~ p _ { N } . } \\end{array} } \\right. .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Eq. (3) and Figure 2, by using the binary gates rather than real-valued path weights (Liu et al., 2018c), only one path of activation is active in memory at run-time and the memory requirement of training the over-parameterized network is thus reduced to the same level of training a compact model. That’s more than an order of magnitude memory saving. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 TRAINING BINARIZED ARCHITECTURE PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "MIT Red-1Figure 2 illustrates the training procedure of the weight parameters and binarized architecture parameters in the over-parameterized network. When training weight parameters, we first freeze the architecture parameters and stochastically sample binary gates according to Eq. (2) for each batch of input data. Then the weight parameters of active paths are updated via standard gradient descent on the training set (Figure 2 left). When training architecture parameters, the weight parameters are frozen, then we reset the binary gates and update the architecture parameters on the validation set (Figure 2 right). These two update steps are performed in an alternative manner. Once the training of architecture parameters is finished, we can then derive the compact architecture by pruning redundant paths. In this work, we simply choose the path with the highest path weight. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Unlike weight parameters, the architecture parameters are not directly involved in the computation graph and thereby cannot be updated using the standard gradient descent. In this section, we introduce a gradient-based approach to learn the architecture parameters. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In BinaryConnect (Courbariaux et al., 2015), the real-valued weight is updated using the gradient w.r.t. its corresponding binary gate. In our case, analogously, the gradient w.r.t. architecture parameters can be approximately estimated using $\\partial L / \\partial g _ { i }$ in replace of $\\bar { \\partial } L / \\partial p _ { i }$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2533f426ba3cd1df7255dc9f485683b08fab928842594e9439af3a6a573592b3.jpg",
+        "text": "$$\n\\frac { \\partial L } { \\partial \\alpha _ { i } } = \\sum _ { j = 1 } ^ { N } \\frac { \\partial L } { \\partial p _ { j } } \\frac { \\partial p _ { j } } { \\partial \\alpha _ { i } } \\approx \\sum _ { j = 1 } ^ { N } \\frac { \\partial L } { \\partial g _ { j } } \\frac { \\partial p _ { j } } { \\partial \\alpha _ { i } } = \\sum _ { j = 1 } ^ { N } \\frac { \\partial L } { \\partial g _ { j } } \\frac { \\partial \\bigg ( \\frac { \\exp ( \\alpha _ { j } ) } { \\sum _ { k } \\exp ( \\alpha _ { k } ) } \\bigg ) } { \\partial \\alpha _ { i } } = \\sum _ { j = 1 } ^ { N } \\frac { \\partial L } { \\partial g _ { j } } p _ { j } \\big ( \\delta _ { i j } - p _ { i } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            154,
+            782,
+            203
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\delta _ { i j } = 1$ if $i = j$ and $\\delta _ { i j } = 0$ if $i \\neq j$ . Since the binary gates $g$ are involved in the computation graph, as shown in Eq. (3), $\\partial L / \\partial g _ { j }$ can be calculated through backpropagation. However, computing $\\partial L / \\partial g _ { j }$ requires to calculate and store $o _ { j } ( x )$ . Therefore, directly using Eq. (4) to update the architecture parameters would also require roughly $N$ times GPU memory compared to training a compact model. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To address this issue, we consider factorizing the task of choosing one path out of $\\mathbf { N }$ candidates into multiple binary selection tasks. The intuition is that if a path is the best choice at a particular position, it should be the better choice when solely compared to any other path.2 ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following this idea, within an update step of the architecture parameters, we first sample two paths according to the multinomial distribution $( p _ { 1 } , \\cdots , p _ { N } )$ and mask all the other paths as if they do not exist. As such the number of candidates temporarily decrease from $N$ to 2, while the path weights $\\{ p _ { i } \\}$ and binary gates $\\{ g _ { i } \\}$ are reset accordingly. Then we update the architecture parameters of these two sampled paths using the gradients calculated via Eq. (4). Finally, as path weights are computed by applying softmax to the architecture parameters, we need to rescale the value of these two updated architecture parameters by multiplying a ratio to keep the path weights of unsampled paths unchanged. As such, in each update step, one of the sampled paths is enhanced (path weight increases) and the other sampled path is attenuated (path weight decreases) while all other paths keep unchanged. In this way, regardless of the value of $N$ , only two paths are involved in each update step of the architecture parameters, and thereby the memory requirement is reduced to the same level of training a compact model. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 HANDLING NON-DIFFERENTIABLE HARDWARE METRICS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Besides accuracy, latency (not FLOPs) is another very important objective when designing efficient neural network architectures for hardware. Unfortunately, unlike accuracy that can be optimized using the gradient of the loss function, latency is non-differentiable. In this section, we present two algorithms to handle the non-differentiable objectives. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 MAKING LATENCY DIFFERENTIABLE ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To make latency differentiable, we model the latency of a network as a continuous function of the neural network dimensions 3. Consider a mixed operation with a candidate set $\\{ o _ { j } \\}$ and each $o _ { j }$ is associated with a path weight $p _ { j }$ which represents the probability of choosing $o _ { j }$ . As such, we have the expected latency of a mixed operation (i.e. a learnable block) as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ddd320646022dfa97293492f7bd2d622e5692171115305cbbc65b17492b3cc0.jpg",
+        "text": "$$\n\\mathbb { E } [ \\mathrm { l a t e n c y } _ { i } ] = \\sum _ { j } p _ { j } ^ { i } \\times F ( o _ { j } ^ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            707,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { E } [ \\mathrm { l a t e n c y } _ { i } ]$ is the expected latency of the $i ^ { t h }$ learnable block, $F ( \\cdot )$ denotes the latency prediction model and $\\bar { F } ( o _ { j } ^ { i } )$ is the predicted latency of $o _ { j } ^ { i }$ . The gradient of $\\mathbb { E } [ \\mathrm { l a t e n c y } _ { i } ]$ w.r.t. architecture parameters can thereby be given as: $\\partial \\mathbb { E } [ \\mathrm { l a t e n c y } _ { i } ] / \\partial p _ { j } ^ { i } = F ( o _ { j } ^ { i } ) .$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the whole network with a sequence of mixed operations (Figure 3 left), since these operations are executed sequentially during inference, the expected latency of the network can be expressed with the sum of these mixed operations’ expected latencies: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ede1adcbadb6abf6e3305649d4e93f45be8ba402271deb561a86fb56f8750751.jpg",
+        "text": "$$\n\\mathbb { E } [ \\mathrm { l a t e n c y } ] = \\sum _ { i } \\mathbb { E } [ \\mathrm { l a t e n c y } _ { i } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/408fcd3acff459c85e2d21839a193f855d92afc40095011c31df0b09827d215e.jpg",
+        "image_caption": [
+            "Figure 3: Making latency differentiable by introducing latency regularization loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We incorporate the expected latency of the network into the normal loss function by multiplying a scaling factor $\\lambda _ { 2 } ( > 0 \\bar { ) }$ which controls the trade-off between accuracy and latency. The final loss function is given as (also shown in Figure 3 right) ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d2a86a0469946d62eb7a657b76d4f4ef773a4c2779fa52b989f870fcdbd86ac.jpg",
+        "text": "$$\nL o s s = L o s s _ { C E } + \\lambda _ { 1 } | | w | | _ { 2 } ^ { 2 } + \\mathrm { ~ \\qquad ~ } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $L o s s _ { C E }$ denotes the cross-entropy loss and $\\lambda _ { 1 } | | w | | _ { 2 } ^ { 2 }$ is the weight decay term. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3.2 REINFORCE-BASED APPROACH",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As an alternative to BinaryConnect, we can utilize REINFORCE to train binarized weights as well. Consider a network that has binarized parameters $\\alpha$ , the goal of updating binarized parameters is to find the optimal binary gates $g$ that maximizes a certain reward, denoted as $R ( \\cdot )$ . Here we assume the network only has one mixed operation for ease of illustration. Therefore, according to REINFORCE (Williams, 1992), we have the following updates for binarized parameters: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/c85bb6a07f47c1ff0ac4fbd975eb51c4c055218b87e83390d4bd285eb7822314.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad J ( \\alpha ) = \\mathbb { E } _ { g \\sim \\alpha } [ R ( \\mathcal { N } _ { g } ) ] = \\displaystyle \\sum _ { i } p _ { i } R ( \\mathcal { N } ( e = o _ { i } ) ) , } \\\\ & { \\quad \\nabla _ { \\alpha } J ( \\alpha ) = \\displaystyle \\sum _ { i } R ( \\mathcal { N } ( e = o _ { i } ) ) \\nabla _ { \\alpha } p _ { i } = \\displaystyle \\sum _ { i } R ( \\mathcal { N } ( e = o _ { i } ) ) p _ { i } \\nabla _ { \\alpha } \\log ( p _ { i } ) , } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { g \\sim \\alpha } [ R ( \\mathcal { N } _ { g } ) \\nabla _ { \\alpha } \\log ( p ( g ) ) ] \\approx \\displaystyle \\frac { 1 } { M } \\displaystyle \\sum _ { i = 1 } ^ { M } R ( \\mathcal { N } _ { g ^ { i } } ) \\nabla _ { \\alpha } \\log ( p ( g ^ { i } ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $g ^ { i }$ denotes the $i ^ { t h }$ sampled binary gates, $p ( g ^ { i } )$ denotes the probability of sampling $g ^ { i }$ according to Eq. (2) and $\\mathcal { N } _ { g ^ { i } }$ is the compact network according to the binary gates $g ^ { i }$ . Since Eq. (8) does not require $R ( \\mathcal { N } _ { g } )$ to be differentiable w.r.t. $g$ , it can thus handle non-differentiable objectives. An interesting observation is that Eq. (8) has a similar form to the standard NAS (Zoph & Le, 2017), while it is not a sequential decision-making process and no RNN meta-controller is used in our case. Furthermore, since both gradient-based updates and REINFORCE-based updates are essentially two different update rules to the same binarized architecture parameters, it is possible to combine them to form a new update rule for the architecture parameters. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We demonstrate the effectiveness of our proposed method on two benchmark datasets (CIFAR-10 and ImageNet) for the image classification task. Unlike previous NAS works (Zoph et al., 2018; Liu et al., 2018c) that first learn CNN blocks on CIFAR-10 under small-scale setting (e.g. fewer blocks), then transfer the learned block to ImageNet or CIFAR-10 under large-scale setting by repeatedly stacking it, we directly learn the architectures on the target task (either CIFAR-10 or ImageNet) and target hardware (GPU, CPU and mobile phone) while allowing each block to be specified. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTS ON CIFAR-10 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Architecture Space. For CIFAR-10 experiments, we use the tree-structured architecture space that is introduced by Cai et al. (2018b) with PyramidNet (Han et al., 2017) as the backbone4. Specifically, we replace all $3 \\times 3$ convolution layers in the residual blocks of a PyramidNet with tree-structured cells, each of which has a depth of 3 and the number of branches is set to be 2 at each node (except the leaf nodes). For further details about the tree-structured architecture space, we refer to the original paper (Cai et al., 2018b). Additionally, we use two hyperparameters to control the depth and width of a network in this architecture space, i.e. $B$ and $F$ , which respectively represents the number of blocks at each stage (totally 3 stages) and the number of output channels of the final block. ",
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+    {
+        "type": "table",
+        "img_path": "images/f0943c79bb848a7202938643d7029031a3d3c9988a0172ade7abe916611f1429.jpg",
+        "table_caption": [
+            "Table 1: ProxylessNAS achieves state-of-the-art performance on CIFAR-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=1>Test error (%)</td></tr><tr><td rowspan=1 colspan=1>DenseNet-BC (Huang et al., 2017)PyramidNet (Han etal., 2017)Shake-Shake + c/o (DeVries &amp; Taylor, 2017)PyramidNet + SD(Yamada et al., 2018)</td><td rowspan=1 colspan=1>25.6M26.0M26.2M26.0M</td><td rowspan=1 colspan=1>3.463.312.562.31</td></tr><tr><td rowspan=1 colspan=1>ENAS+ c/o (Pham et al., 2018)DARTS + c/o (Liu et al., 2018c)NASNet-A + c/o (Zoph et al., 2018)PathLevel EAS + c/o (Cai et al., 2018b)AmoebaNet-B + c/o (Real et al., 2018)</td><td rowspan=1 colspan=1>4.6M3.4M27.6M14.3M34.9M</td><td rowspan=1 colspan=1>2.892.832.402.302.13</td></tr><tr><td rowspan=1 colspan=1>Proxyless-R+c/o (ours)Proxyless-G+c/o (ours)</td><td rowspan=1 colspan=1>5.8M5.7M</td><td rowspan=1 colspan=1>2.302.08</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Training Details. We randomly sample 5,000 images from the training set as a validation set for learning architecture parameters which are updated using the Adam optimizer with an initial learning rate of 0.006 for the gradient-based algorithm (Section 3.2.1) and 0.01 for the REINFORCEbased algorithm (Section 3.3.2). In the following discussions, we refer to these two algorithms as Proxyless-G (gradient) and Proxyless-R (REINFORCE) respectively. ",
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+        "type": "text",
+        "text": "After the training process of the over-parameterized network completes, a compact network is derived according to the architecture parameters, as discussed in Section 3.2.1. Next, we train the compact network using the same training settings except that the number of training epochs increases from 200 to 300. Additionally, when the DropPath regularization (Zoph et al., 2018; Huang et al., 2016) is adopted, we further increase the number of training epochs to 600 (Zoph et al., 2018). ",
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+        "type": "text",
+        "text": "Results. We apply the proposed method to learn architectures in the tree-structured architecture space with $B = 1 8$ and $F = 4 0 0$ . Since we do not repeat cells and each cell has 12 learnable edges, totally $1 2 \\times 1 8 \\times 3 = 6 4 8$ decisions are required to fully determine the architecture. ",
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+    {
+        "type": "text",
+        "text": "The test error rate results of our proposed method and other state-of-the-art architectures on CIFAR10 are summarized in Table 1, where $\\mathrm { { ^ { 6 6 } c / o ^ { 3 } } }$ indicates the use of Cutout (DeVries & Taylor, 2017). Compared to these state-of-the-art architectures, our proposed method can achieve not only lower test error rate but also better parameter efficiency. Specifically, Proxyless-G reaches a test error rate of $2 . 0 8 \\%$ which is slightly better than AmoebaNet-B (Real et al., 2018) (the previous best architecture on CIFAR-10). Notably, AmoebaNet-B uses 34.9M parameters while our model only uses 5.7M parameters which is $6 \\times$ fewer than AmoebaNet-B. Furthermore, compared with PathLevel EAS (Cai et al., 2018b) that also explores the tree-structured architecture space, both Proxyless-G and Proxyless-R achieves similar or lower test error rate results with half fewer parameters. The strong empirical results of our ProxylessNAS demonstrate the benefits of directly exploring a large architecture space instead of repeatedly stacking the same block. ",
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTS ON IMAGENET ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For ImageNet experiments, we focus on learning efficient CNN architectures (Iandola et al., 2016; Howard et al., 2017; Sandler et al., 2018; Zhu et al., 2018) that have not only high accuracy but also low latency on specific hardware platforms. Therefore, it is a multi-objective NAS task (Hsu et al., 2018; Dong et al., 2018; Elsken et al., 2018a; He et al., 2018; Wang et al., 2018; Tan et al., 2018), where one of the objectives is non-differentiable (i.e. latency). We use three different hardware platforms, including mobile phone, GPU and CPU, in our experiments. The GPU latency is measured on V100 GPU with a batch size of 8 (single batch makes GPU severely under-utilized). The CPU latency is measured under batch size 1 on a server with two 2.40GHz Intel(R) Xeon(R) ",
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+        "table_caption": [
+            "Table 2: ProxylessNAS achieves state-of-the art accuracy $( \\% )$ on ImageNet (under mobile latency constraint $\\leq 8 0 m s $ ) with $2 0 0 \\times$ less search cost in GPU hours. “LL” indicates latency regularization loss. Details of MnasNet’s search cost are provided in appendix C. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Top-1</td><td>Top-5</td><td>Mobile Latency</td><td>Hardware -aware</td><td>No Proxy</td><td>No Repeat</td><td>Search cost (GPU hours)</td></tr><tr><td>MobileNetV1[16] MobileNetV2 [30]</td><td>70.6 72.0</td><td>89.5 91.0</td><td>113ms 75ms</td><td>1</td><td>1 1</td><td>X ×</td><td>Manual Manual</td></tr><tr><td>NASNet-A [38]</td><td>74.0</td><td>91.3</td><td>183ms</td><td>1 X</td><td>X</td><td>X</td><td>48,000</td></tr><tr><td>AmoebaNet-A [29]</td><td>74.5</td><td>92.0</td><td>190ms</td><td>X</td><td>×</td><td>×</td><td>75,600</td></tr><tr><td>MnasNet [31]</td><td>74.0</td><td>91.8</td><td>76ms</td><td>√</td><td>×</td><td>×</td><td>40,000</td></tr><tr><td>MnasNet (our impl.)</td><td>74.0</td><td>91.8</td><td>79ms</td><td>√</td><td>X</td><td>X</td><td>40,000</td></tr><tr><td>Proxyless-G (mobile)</td><td>71.8</td><td>90.3</td><td>83ms</td><td>X</td><td>?</td><td>√</td><td>200</td></tr><tr><td>Proxyless-G + LL</td><td>74.2</td><td>91.7</td><td>79ms</td><td>√</td><td></td><td>√</td><td>200</td></tr><tr><td>Proxyless-R (mobile)</td><td>74.6</td><td>92.2</td><td>78ms</td><td>√</td><td>「</td><td>√</td><td>200</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/5c66bb877d30a396515306d314713e4a93d5d485a7682f5ae667bce3ba3b91c3.jpg",
+        "image_caption": [
+            "Figure 4: ProxylessNAS consistently outperforms MobileNetV2 under various latency settings. "
+        ],
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+        "image_caption": [
+            "Figure 5: Our mobile latency model is close to $y \\ = \\ x$ . The latency RMSE is $0 . 7 5 \\mathrm { m s }$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "CPU E5-2640 v4. The mobile latency is measured on Google Pixel 1 phone with a batch size of 1. For Proxyless-R, we use $A C C ( m ) ^ { \\cdot } \\times [ L A T ( m ) / T ] ^ { w }$ as the optimization goal, where $A C C ( m )$ denotes the accuracy of model $m$ , $L A T ( m )$ denotes the latency of $m$ , $T$ is the target latency and $w$ is a hyperparameter for controlling the trade-off between accuracy and latency. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Additionally, on mobile phone, we use the latency prediction model (Appendix B) during architecture search. As illustrated in Figure 5, we observe a strong correlation between the predicted latency and real measured latency on the test set, suggesting that the latency prediction model can be used to replace the expensive mobile farm infrastructure (Tan et al., 2018) with little error introduced. ",
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+    {
+        "type": "text",
+        "text": "Architecture Space. We use MobileNetV2 (Sandler et al., 2018) as the backbone to build the architecture space. Specifically, rather than repeating the same mobile inverted bottleneck convolution (MBConv), we allow a set of MBConv layers with various kernel sizes $\\{ 3 , 5 , 7 \\}$ and expansion ratios $\\{ 3 , 6 \\}$ . To enable a direct trade-off between width and depth, we initiate a deeper over-parameterized network and allow a block with the residual connection to be skipped by adding the zero operation to the candidate set of its mixed operation. In this way, with a limited latency budget, the network can either choose to be shallower and wider by skipping more blocks and using larger MBConv layers or choose to be deeper and thinner by keeping more blocks and using smaller MBConv layers. ",
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+    {
+        "type": "text",
+        "text": "Training Details. We randomly sample 50,000 images from the training set as a validation set during the architecture search. The settings for updating architecture parameters are the same as CIFAR-10 experiments except the initial learning rate is 0.001. The over-parameterized network is trained on the remaining training images with batch size 256. ",
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+        "type": "table",
+        "img_path": "images/be27421d0f12a09b61053c8d06c13f0a54c38ba030ef8c1811619d4175ca550b.jpg",
+        "table_caption": [
+            "Table 3: ImageNet Accuracy $( \\% )$ and GPU latency (Tesla V100) on ImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Top-1</td><td rowspan=1 colspan=1>Top-5</td><td rowspan=1 colspan=1>GPU latency</td></tr><tr><td rowspan=1 colspan=1>MobileNetV2 (Sandler et al., 2018)ShuffleNetV2 (1.5) (Ma et al., 2018)ResNet-34 (He et al., 2016)</td><td rowspan=1 colspan=1>72.072.673.3</td><td rowspan=1 colspan=1>91.0191.4</td><td rowspan=1 colspan=1>6.1ms7.3ms8.0ms</td></tr><tr><td rowspan=1 colspan=1>NASNet-A (Zoph et al., 2018)DARTS (Liu et al., 2018c)MnasNet (Tan et al., 2018)</td><td rowspan=1 colspan=1>74.073.174.0</td><td rowspan=1 colspan=1>91.391.091.8</td><td rowspan=1 colspan=1>38.3ms16.1ms</td></tr><tr><td rowspan=1 colspan=1>Proxyless (GPU)</td><td rowspan=1 colspan=1>75.1</td><td rowspan=1 colspan=1>92.5</td><td rowspan=1 colspan=1>5.1ms</td></tr></table>",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "ImageNet Classification Results. We first apply our ProxylessNAS to learn specialized CNN models on the mobile phone. The summarized results are reported in Table 2. Compared to MobileNetV2, our model improves the top-1 accuracy by $2 . 6 \\%$ while maintaining a similar latency on the mobile phone. Furthermore, by rescaling the width of the networks using a multiplier (Sandler et al., 2018; Tan et al., 2018), it is shown in Figure 4 that our model consistently outperforms MobileNetV2 by a significant margin under all latency settings. Specifically, to achieve the same level of top-1 accuracy performance (i.e. around $7 4 . 6 \\%$ ), MobileNetV2 has 143ms latency while our model only needs 78ms $( { \\bf 1 . 8 3 \\times }$ faster). While compared with MnasNet (Tan et al., 2018), our model can achieve $0 . 6 \\%$ higher top-1 accuracy with slightly lower mobile latency. More importantly, we are much more resource efficient: the GPU-hour is $2 0 0 \\times$ fewer than MnasNet (Table 2). ",
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+    {
+        "type": "text",
+        "text": "Additionally, we also observe that Proxyless-G has no incentive to choose computation-cheap operations if were not for the latency regularization loss. Its resulting architecture initially has 158ms latency on Pixel 1. After rescaling the network using the multiplier, its latency reduces to $8 3 \\mathrm { m s }$ . However, this model can only achieve $7 1 . 8 \\%$ top-1 accuracy on ImageNet, which is $2 . 4 \\%$ lower than the result given by Proxyless-G with latency regularization loss. Therefore, we conclude that it is essential to take latency as a direct objective when learning efficient neural networks. ",
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+    {
+        "type": "text",
+        "text": "Besides the mobile phone, we also apply our ProxylessNAS to learn specialized CNN models on GPU and CPU. Table 3 reports the results on GPU, where we find that our ProxylessNAS can still achieve superior performances compared to both human-designed and automatically searched architectures. Specifically, compared to MobileNetV2 and MnasNet, our model improves the top-1 accuracy by $3 . 1 \\%$ and $1 . 1 \\%$ respectively while being $1 . 2 \\times$ faster. Table 4 shows the summarized results of our searched models on three different platforms. An interesting observation is that models optimized for GPU do not run fast on CPU and mobile phone, vice versa. Therefore, it is essential to learn specialized neural networks for different hardware architectures to achieve the best efficiency on different hardware. ",
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+    {
+        "type": "text",
+        "text": "Specialized Models for Different Hardware. Figure 6 demonstrates the detailed architectures of our searched CNN models on three hardware platforms: GPU/CPU/Mobile. We notice that the architecture shows different preferences when targeting different platforms: (i) The GPU model is shallower and wider, especially in early stages where the feature map has higher resolution; (ii) The GPU model prefers large MBConv operations (e.g. $7 \\times 7$ MBConv6), while the CPU model would go for smaller MBConv operations. This is because GPU has much higher parallelism than CPU so it can take advantage of large MBConv operations. Another interesting observation is that our searched models on all platforms prefer larger MBConv operations in the first block within each stage where the feature map is downsampled. We suppose it might because larger MBConv operations are beneficial for the network to preserve more information when downsampling. Notably, such kind of patterns cannot be captured in previous NAS methods as they force the blocks to share the same structure (Zoph et al., 2018; Liu et al., 2018a). ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We introduced ProxylessNAS that can directly learn neural network architectures on the target task and target hardware without any proxy. We also reduced the search cost (GPU-hours and GPU memory) of NAS to the same level of normal training using path binarization. Benefiting from the direct search, we achieve strong empirical results on CIFAR-10 and ImageNet. Furthermore, ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9e0a1090b199cb164fd72d1ed54a3efec06e17ce9b325aae33e66b6340b57edf.jpg",
+        "image_caption": [
+            "Figure 6: Efficient models optimized for different hardware. “MBConv3” and “MBConv6” denote mobile inverted bottleneck convolution layer with an expansion ratio of 3 and 6 respectively. Insights: GPU prefers shallow and wide model with early pooling; CPU prefers deep and narrow model with late pooling. Pooling layers prefer large and wide kernel. Early layers prefer small kernel. Late layers prefer large kernel. "
+        ],
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+    },
+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Top-1 (%)</td><td rowspan=1 colspan=1>GPU latency</td><td rowspan=1 colspan=1>CPU latency</td><td rowspan=1 colspan=1>Mobile latency</td></tr><tr><td rowspan=1 colspan=1>Proxyless(GPU)</td><td rowspan=1 colspan=1>75.1</td><td rowspan=1 colspan=1>5.1ms</td><td rowspan=1 colspan=1>204.9ms</td><td rowspan=1 colspan=1>124ms</td></tr><tr><td rowspan=1 colspan=1>Proxyless(CPU)</td><td rowspan=1 colspan=1>75.3</td><td rowspan=1 colspan=1>7.4ms</td><td rowspan=1 colspan=1>138.7ms</td><td rowspan=1 colspan=1>116ms</td></tr><tr><td rowspan=1 colspan=1>Proxyless(mobile)</td><td rowspan=1 colspan=1>74.6</td><td rowspan=1 colspan=1>7.2ms</td><td rowspan=1 colspan=1>164.1ms</td><td rowspan=1 colspan=1>78ms</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Hardware prefers specialized models. Models optimized for GPU does not run fast on CPU and mobile phone, vice versa. ProxylessNAS provides an efficient solution to search a specialized (3) Efficient GPU architecture found by ProxylessNAS. neural network architecture for a target hardware architecture, while cutting down the search cost by $2 0 0 \\times$ compared with state-of-the-arts (Zoph & Le, 2017; Tan et al., 2018). ",
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+    },
+    {
+        "type": "text",
+        "text": "we allow specializing network architectures for different platforms by directly incorporating the measured hardware latency into optimization objectives. We compared the optimized models on CPU/GPU/mobile and raised the awareness of the needs of specializing neural network architecture for different hardware architectures. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We thank MIT Quest for Intelligence, MIT-IBM Watson AI lab, SenseTime, Xilinx, Snap Research for supporting this work. We also thank AWS Cloud Credits for Research Program providing us the cloud computing resources. ",
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            797,
+            287,
+            813
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gabriel Bender, Pieter-Jan Kindermans, Barret Zoph, Vijay Vasudevan, and Quoc Le. Understanding and simplifying one-shot architecture search. In ICML, 2018. ",
+        "bbox": [
+            174,
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+            821,
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+        ],
+        "page_idx": 9
+    },
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+        "text": "Zhuang Liu, Jianguo Li, Zhiqiang Shen, Gao Huang, Shoumeng Yan, and Changshui Zhang. Learning efficient convolutional networks through network slimming. In ICCV, 2017.   \nRenqian Luo, Fei Tian, Tao Qin, and Tie-Yan Liu. Neural architecture optimization. arXiv preprint arXiv:1808.07233, 2018.   \nNingning Ma, Xiangyu Zhang, Hai-Tao Zheng, and Jian Sun. Shufflenet v2: Practical guidelines for efficient cnn architecture design. In ECCV, 2018.   \nHieu Pham, Melody Y Guan, Barret Zoph, Quoc V Le, and Jeff Dean. Efficient neural architecture search via parameter sharing. In ICML, 2018.   \nEsteban Real, Alok Aggarwal, Yanping Huang, and Quoc V Le. Regularized evolution for image classifier architecture search. arXiv preprint arXiv:1802.01548, 2018.   \nMark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In CVPR, 2018.   \nMingxing Tan, Bo Chen, Ruoming Pang, Vijay Vasudevan, and Quoc V Le. Mnasnet: Platformaware neural architecture search for mobile. arXiv preprint arXiv:1807.11626, 2018.   \nKuan Wang, Zhijian Liu, Yujun Lin, Ji Lin, and Song Han. Haq: Hardware-aware automated quantization. arXiv, 2018.   \nRonald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. In Reinforcement Learning. 1992.   \nYoshihiro Yamada, Masakazu Iwamura, and Koichi Kise. Shakedrop regularization. arXiv preprint arXiv:1802.02375, 2018.   \nZhao Zhong, Junjie Yan, Wei Wu, Jing Shao, and Cheng-Lin Liu. Practical block-wise neural network architecture generation. In CVPR, 2018.   \nLigeng Zhu, Ruizhi Deng, Michael Maire, Zhiwei Deng, Greg Mori, and Ping Tan. Sparsely aggregated convolutional networks. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 186–201, 2018.   \nBarret Zoph and Quoc V Le. Neural architecture search with reinforcement learning. In ICLR, 2017.   \nBarret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures for scalable image recognition. In CVPR, 2018. ",
+        "bbox": [
+            171,
+            98,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A THE LIST OF CANDIDATE OPERATIONS USED ON CIFAR-10",
+        "text_level": 1,
+        "bbox": [
+            173,
+            101,
+            712,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We adopt the following 7 operations in our CIFAR-10 experiments: ",
+        "bbox": [
+            173,
+            132,
+            616,
+            147
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• $3 \\times 3$ dilated depthwise-separable convolution   \n• Identity   \n• $3 \\times 3$ depthwise-separable convolution   \n• $5 \\times 5$ depthwise-separable convolution   \n• $7 \\times 7$ depthwise-separable convolution   \n• $3 \\times 3$ average pooling   \n• $3 \\times 3$ max pooling ",
+        "bbox": [
+            217,
+            157,
+            509,
+            263
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B MOBILE LATENCY PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            282,
+            472,
+            299
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Measuring the latency on-device is accurate but not ideal for scalable neural architecture search. There are two reasons: (i) Slow. As suggested in TensorFlow-Lite, we need to average hundreds of runs to produce a precise measurement, approximately 20 seconds. This is far more slower than a single forward / backward execution. (ii) Expensive. A lot of mobile devices and software engineering work are required to build an automatic pipeline to gather the latency from a mobile farm. Instead of direct measurement, we build a model to estimate the latency. We need only 1 phone rather than a farm of phones, which has only $0 . 7 5 \\mathrm { m s }$ latency RMSE. We use the latency model to search, and we use the measured latency to report the final model’s latency. ",
+        "bbox": [
+            173,
+            314,
+            825,
+            426
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We sampled $5 \\mathrm { k }$ architectures from our candidate space, where $4 \\mathrm { k \\Omega }$ architectures are used to build the latency model and the rest are used for test. We measured the latency on Google Pixel 1 phone using TensorFlow-Lite. The features include (i) type of the operator (ii) input and output feature map size (iii) other attributes like kernel size, stride for convolution and expansion ratio. ",
+        "bbox": [
+            174,
+            433,
+            825,
+            488
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C DETAILS OF MNASNET’S SEARCH COST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            508,
+            545,
+            525
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Mnas (Tan et al., 2018) trains 8,000 mobile-sized models on ImageNet, each of which is trained for 5 epochs for learning architectures. If these models are trained on V100 GPUs, as done in our experiments, the search cost is roughly 40,000 GPU hours. ",
+        "bbox": [
+            176,
+            540,
+            825,
+            583
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D IMPLEMENTAION OF THE GRADIENT-BASED ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            602,
+            687,
+            619
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A naive implementation of the gradient-based algorithm (see Eq. (4)) is calculating and storing $o _ { j } ( x )$ in the forward step to later compute $\\partial L / \\partial g _ { j }$ in the backward step: ",
+        "bbox": [
+            173,
+            633,
+            818,
+            662
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6efee5b3f95af63d4de66790215b4b0a4616e5e900f5aa884097fae507c093cc.jpg",
+        "text": "$$\n\\partial L / \\partial g _ { j } = \\mathrm { r e d u c e \\_ s u m } ( \\nabla _ { y } L \\circ o _ { j } ( x ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            670,
+            629,
+            688
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\nabla _ { y } L$ denotes the gradient w.r.t. the output of the mixed operation $y$ , “◦” denotes the elementwise product, and “reduce $\\_ \\mathrm { s u m ( \\cdot ) } ,$ ” denotes the sum of all elements. ",
+        "bbox": [
+            171,
+            694,
+            823,
+            722
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Notice that $o _ { j } ( x )$ is only used for calculating $\\partial L / \\partial g _ { j }$ when $j ^ { t h }$ path is not active (i.e. not involved in calculating $y$ ). So we do not need to actually allocate GPU memory to store $o _ { j } ( x )$ . Instead, we can calculate $o _ { j } ( x )$ after getting $\\nabla _ { y } L$ in the backward step, use $o _ { j } ( x )$ to compute $\\partial L / \\partial g _ { j }$ following Eq. (9), then release the occupied GPU memory. In this way, without the approximation discussed in Section 3.2.1, we can reduce the GPU memory cost to the same level of training a compact model. ",
+        "bbox": [
+            174,
+            728,
+            825,
+            799
+        ],
+        "page_idx": 12
+    }
+]

parse/train/rkYTTf-AZ/rkYTTf-AZ.md

@@ -0,0 +1,291 @@
+# UNSUPERVISED MACHINE TRANSLATION USING MONOLINGUAL CORPORA ONLY
+
+Guillaume Lample † ‡ , Alexis Conneau $\dagger$ , Ludovic Denoyer $^ \ddag$ , Marc’Aurelio Ranzato † † Facebook AI Research, ‡ Sorbonne Universites, UPMC Univ Paris 06, LIP6 UMR 7606, CNRS ´ {gl,aconneau,ranzato}@fb.com,ludovic.denoyer@lip6.fr
+
+# ABSTRACT
+
+Machine translation has recently achieved impressive performance thanks to recent advances in deep learning and the availability of large-scale parallel corpora. There have been numerous attempts to extend these successes to low-resource language pairs, yet requiring tens of thousands of parallel sentences. In this work, we take this research direction to the extreme and investigate whether it is possible to learn to translate even without any parallel data. We propose a model that takes sentences from monolingual corpora in two different languages and maps them into the same latent space. By learning to reconstruct in both languages from this shared feature space, the model effectively learns to translate without using any labeled data. We demonstrate our model on two widely used datasets and two language pairs, reporting BLEU scores of 32.8 and 15.1 on the Multi30k and WMT English-French datasets, without using even a single parallel sentence at training time.
+
+# 1 INTRODUCTION
+
+Thanks to recent advances in deep learning (Sutskever et al., 2014; Bahdanau et al., 2015) and the availability of large-scale parallel corpora, machine translation has now reached impressive performance on several language pairs (Wu et al., 2016). However, these models work very well only when provided with massive amounts of parallel data, in the order of millions of parallel sentences. Unfortunately, parallel corpora are costly to build as they require specialized expertise, and are often nonexistent for low-resource languages. Conversely, monolingual data is much easier to find, and many languages with limited parallel data still possess significant amounts of monolingual data.
+
+There have been several attempts at leveraging monolingual data to improve the quality of machine translation systems in a semi-supervised setting (Munteanu et al., 2004; Irvine, 2013; Irvine & Callison-Burch, 2015; Zheng et al., 2017). Most notably, Sennrich et al. (2015a) proposed a very effective data-augmentation scheme, dubbed “back-translation”, whereby an auxiliary translation system from the target language to the source language is first trained on the available parallel data, and then used to produce translations from a large monolingual corpus on the target side. The pairs composed of these translations with their corresponding ground truth targets are then used as additional training data for the original translation system.
+
+Another way to leverage monolingual data on the target side is to augment the decoder with a language model (Gulcehre et al., 2015). And finally, Cheng et al. (2016); He et al. (2016) have proposed to add an auxiliary auto-encoding task on monolingual data, which ensures that a translated sentence can be translated back to the original one. All these works still rely on several tens of thousands parallel sentences, however.
+
+Previous work on zero-resource machine translation has also relied on labeled information, not from the language pair of interest but from other related language pairs (Firat et al., 2016; Johnson et al., 2016; Chen et al., 2017) or from other modalities (Nakayama & Nishida, 2017; Lee et al., 2017). The only exception is the work by Ravi & Knight (2011); Pourdamghani & Knight (2017), where the machine translation problem is reduced to a deciphering problem. Unfortunately, their method is limited to rather short sentences and it has only been demonstrated on a very simplistic setting comprising of the most frequent short sentences, or very closely related languages.
+
+![](images/94c547e44b94e2f2ddd4b069149181cface3d24ae25686585f142de1dca80aed.jpg)  
+Figure 1: Toy illustration of the principles guiding the design of our objective function. Left (autoencoding): the model is trained to reconstruct a sentence from a noisy version of it. $x$ is the target, $C ( x )$ is the noisy input, $\hat { x }$ is the reconstruction. Right (translation): the model is trained to translate a sentence in the other domain. The input is a noisy translation (in this case, from source-to-target) produced by the model itself, $M$ , at the previous iteration $( t )$ , $y = M ^ { ( t ) } ( x )$ . The model is symmetric, and we repeat the same process in the other language. See text for more details.
+
+In this paper, we investigate whether it is possible to train a general machine translation system without any form of supervision whatsoever. The only assumption we make is that there exists a monolingual corpus on each language. This set up is interesting for a twofold reason. First, this is applicable whenever we encounter a new language pair for which we have no annotation. Second, it provides a strong lower bound performance on what any good semi-supervised approach is expected to yield.
+
+The key idea is to build a common latent space between the two languages (or domains) and to learn to translate by reconstructing in both domains according to two principles: (i) the model has to be able to reconstruct a sentence in a given language from a noisy version of it, as in standard denoising auto-encoders (Vincent et al., 2008). (ii) The model also learns to reconstruct any source sentence given a noisy translation of the same sentence in the target domain, and vice versa. For (ii), the translated sentence is obtained by using a back-translation procedure (Sennrich et al., 2015a), i.e. by using the learned model to translate the source sentence to the target domain. In addition to these reconstruction objectives, we constrain the source and target sentence latent representations to have the same distribution using an adversarial regularization term, whereby the model tries to fool a discriminator which is simultaneously trained to identify the language of a given latent sentence representation (Ganin et al., 2016). This procedure is then iteratively repeated, giving rise to translation models of increasing quality. To keep our approach fully unsupervised, we initialize our algorithm by using a na¨ıve unsupervised translation model based on a word by word translation of sentences with a bilingual lexicon derived from the same monolingual data (Conneau et al., 2017). As a result, and by only using monolingual data, we can encode sentences of both languages into the same feature space, and from there, we can also decode/translate in any of these languages; see Figure 1 for an illustration.
+
+While not being able to compete with supervised approaches using lots of parallel resources, we show in section 4 that our model is able to achieve remarkable performance. For instance, on the WMT dataset we can achieve the same translation quality of a similar machine translation system trained with full supervision on 100,000 sentence pairs. On the Multi30K-Task1 dataset we achieve a BLEU above 22 on all the language pairs, with up to 32.76 on English-French.
+
+Next, in section 2, we describe the model and the training algorithm. We then present experimental results in section 4. Finally, we further discuss related work in section 5 and summarize our findings in section 6.
+
+# 2 UNSUPERVISED NEURAL MACHINE TRANSLATION
+
+In this section, we first describe the architecture of the translation system, and then we explain how we train it.
+
+2.1 NEURAL MACHINE TRANSLATION MODEL
+
+The translation model we propose is composed of an encoder and a decoder, respectively responsible for encoding source and target sentences to a latent space, and to decode from that latent space to the source or the target domain. We use a single encoder and a single decoder for both domains (Johnson et al., 2016). The only difference when applying these modules to different languages is the choice of lookup tables.
+
+Let us denote by $\mathcal { W } _ { S }$ the set of words in the source domain associated with the (learned) words embeddings $\mathcal { Z } ^ { S } = ( z _ { 1 } ^ { s } , . . . . , z _ { | \mathcal { W } _ { S } | } ^ { s } )$ , and by $\mathcal { W } _ { T }$ the set of words in the target domain associated with the embeddings $\mathcal { Z } ^ { T } = ( z _ { 1 } ^ { t } , . . . , z _ { | \mathcal { W } _ { T } | } ^ { t } )$ , $\mathcal { Z }$ being the set of all the embeddings. Given an input sentence of $m$ words $\pmb { x } = ( x _ { 1 } , x _ { 2 } , . . . , x _ { m } )$ in a particular language \`, $\ell \in \{ s r c , t g t \}$ , an encoder $e _ { \theta _ { \mathrm { { e n c } } } , z } ( { \pmb x } , { \pmb \ell } )$ computes a sequence of $m$ hidden states $\boldsymbol { z } = ( z _ { 1 } , z _ { 2 } , . . . , z _ { m } )$ by using the corresponding word embeddings, i.e. ${ \mathcal { Z } } _ { S }$ if $\ell = s r c$ and ${ \mathcal { Z } } _ { T }$ if $\ell = t g t$ ; the other parameters $\theta _ { \mathrm { e n c } }$ are instead shared between the source and target languages. For the sake of simplicity, the encoder will be denoted as $e ( { \pmb x } , { \pmb \ell } )$ in the following. These hidden states are vectors in $\mathbb { R } ^ { n }$ , $n$ being the dimension of the latent space.
+
+A decoder $d _ { \theta _ { \mathrm { d e c } } , \mathcal { Z } } ( z , \ell )$ takes as input $_ z$ and a language $\ell$ , and generates an output sequence ${ \textbf { \em y } } =$ $\left( y _ { 1 } , y _ { 2 } , . . . , y _ { k } \right)$ , where each word $y _ { i }$ is in the corresponding vocabulary $\mathcal { W } ^ { \ell }$ . This decoder makes use of the corresponding word embeddings, and it is otherwise parameterized by a vector $\theta _ { \mathrm { d e c } }$ that does not depend on the output language. It will thus be denoted $d ( z , \ell )$ in the following. To generate an output word $y _ { i }$ , the decoder iteratively takes as input the previously generated word $y _ { i - 1 }$ $y _ { 0 }$ being a start symbol which is language dependent), updates its internal state, and returns the word that has the highest probability of being the next one. The process is repeated until the decoder generates a stop symbol indicating the end of the sequence.
+
+In this article, we use a sequence-to-sequence model with attention (Bahdanau et al., 2015), without input-feeding. The encoder is a bidirectional-LSTM which returns a sequence of hidden states $\boldsymbol { z } = ( z _ { 1 } , z _ { 2 } , . . . , z _ { m } )$ . At each step, the decoder, which is also an LSTM, takes as input the previous hidden state, the current word and a context vector given by a weighted sum over the encoder states. In all the experiments we consider, both encoder and decoder have 3 layers. The LSTM layers are shared between the source and target encoder, as well as between the source and target decoder. We also share the attention weights between the source and target decoder. The embedding and LSTM hidden state dimensions are all set to 300. Sentences are generated using greedy decoding.
+
+# 2.2 OVERVIEW OF THE METHOD
+
+We consider a dataset of sentences in the source domain, denoted by $\mathcal { D } _ { s r c }$ , and another dataset in the target domain, denoted by $\mathcal { D } _ { t g t }$ . These datasets do not correspond to each other, in general. We train the encoder and decoder by reconstructing a sentence in a particular domain, given a noisy version of the same sentence in the same or in the other domain.
+
+At a high level, the model starts with an unsupervised na¨ıve translation model obtained by making word-by-word translation of sentences using a parallel dictionary learned in an unsupervised way (Conneau et al., 2017). Then, at each iteration, the encoder and decoder are trained by minimizing an objective function that measures their ability to both reconstruct and translate from a noisy version of an input training sentence. This noisy input is obtained by dropping and swapping words in the case of the auto-encoding task, while it is the result of a translation with the model at the previous iteration in the case of the translation task. In order to promote alignment of the latent distribution of sentences in the source and the target domains, our approach also simultaneously learns a discriminator in an adversarial setting. The newly learned encoder/decoder are then used at the next iteration to generate new translations, until convergence of the algorithm. At test time and despite the lack of parallel data at training time, the encoder and decoder can be composed into a standard machine translation system.
+
+# 2.3 DENOISING AUTO-ENCODING
+
+Training an autoencoder of sentences is a trivial task, if the sequence-to-sequence model is provided with an attention mechanism like in our work 1. Without any constraint, the auto-encoder very quickly learns to merely copy every input word one by one. Such a model would also perfectly copy sequences of random words, suggesting that the model does not learn any useful structure in the data. To address this issue, we adopt the same strategy of Denoising Auto-encoders (DAE) (Vincent et al., 2008)), and add noise to the input sentences (see Figure 1-left), similarly to Hill et al. (2016). Considering a domain $\ell = s r c$ or $\ell = t g t$ , and a stochastic noise model denoted by $C$ which operates on sentences, we define the following objective function:
+
+$$
+\mathcal { L } _ { a u t o } ( \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } , \ell ) = \mathbb { E } _ { x \sim \mathcal { D } _ { \ell } , \hat { x } \sim d ( e ( C ( x ) , \ell ) , \ell ) } \left[ \Delta ( \hat { x } , x ) \right]
+$$
+
+where $\hat { x } \sim d ( e ( C ( x ) , \ell ) , \ell )$ means that $\hat { x }$ is a reconstruction of the corrupted version of $x$ , with $x$ sampled from the monolingual dataset $\mathcal { D } _ { \ell }$ . In this equation, $\Delta$ is a measure of discrepancy between the two sequences, the sum of token-level cross-entropy losses in our case.
+
+Noise model $C ( x )$ is a randomly sampled noisy version of sentence $x$ . In particular, we add two different types of noise to the input sentence. First, we drop every word in the input sentence with a probability $p _ { w d }$ . Second, we slightly shuffle the input sentence. To do so, we apply a random permutation $\sigma$ to the input sentence, verifying the condition $\forall i \in \{ 1 , n \} , | \sigma ( i ) - i | \tilde { \leq k }$ where $n$ is the length of the input sentence, and $k$ is a tunable parameter.
+
+To generate a random permutation verifying the above condition for a sentence of size $n$ , we generate a random vector $q$ of size $n$ , where $q _ { i } = i + U ( 0 , \alpha )$ , and $U$ is a draw from the uniform distribution in the specified range. Then, we define $\sigma$ to be the permutation that sorts the array $q$ . In particular, $\alpha < 1$ will return the identity, $\alpha = + \infty$ can return any permutation, and $\alpha = k + 1$ will return permutations $\sigma$ verifying $\forall i \ \in \ \{ 1 , n \} , | \sigma ( i ) - i | \ \leq \ k$ . Although biased, this method generates permutations similar to the noise observed with word-by-word translation.
+
+In our experiments, both the word dropout and the input shuffling strategies turned out to have a critical impact on the results, see also section 4.5, and using both strategies at the same time gave us the best performance. In practice, we found $p _ { w d } = 0 . 1$ and $k = 3$ to be good parameters.
+
+# 2.4 CROSS DOMAIN TRAINING
+
+The second objective of our approach is to constrain the model to be able to map an input sentence from a the source/target domain $\ell _ { 1 }$ to the target/source domain $\ell _ { 2 }$ , which is what we are ultimately interested in at test time. The principle here is to sample a sentence $x \in \mathcal { D } _ { \ell _ { 1 } }$ , and to generate a corrupted translation of this sentence in $\ell _ { 2 }$ . This corrupted version is generated by applying the current translation model denoted $M$ to $x$ such that $y = M ( x )$ . Then a corrupted version $C ( y )$ is sampled (see Figure 1-right). The objective is thus to learn the encoder and the decoder such that they can reconstruct $x$ from $C ( y )$ . The cross-domain loss can be written as:
+
+$$
+\mathcal { L } _ { c d } \left( \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } , \ell _ { 1 } , \ell _ { 2 } \right) = \mathbb { E } _ { x \sim \mathcal { D } _ { \ell _ { 1 } } , \hat { x } \sim d \left( e \left( C \left( M \left( x \right) \right) , \ell _ { 2 } \right) , \ell _ { 1 } \right) } \left[ \Delta \left( \hat { x } , x \right) \right]
+$$
+
+where $\Delta$ is again the sum of token-level cross-entropy losses.
+
+# 2.5 ADVERSARIAL TRAINING
+
+Intuitively, the decoder of a neural machine translation system works well only when its input is produced by the encoder it was trained with, or at the very least, when that input comes from a distribution very close to the one induced by its encoder. Therefore, we would like our encoder to output features in the same space regardless of the actual language of the input sentence. If such condition is satisfied, our decoder may be able to decode in a certain language regardless of the language of the encoder input sentence.
+
+Note however that the decoder could still produce a bad translation while yielding a valid sentence in the target domain, as constraining the encoder to map two languages in the same feature space does not imply a strict correspondence between sentences. Fortunately, the previously introduced loss for cross-domain training in equation 2 mitigates this concern. Also, recent work on bilingual lexical induction has shown that such a constraint is very effective at the word level, suggesting that it may also work at the sentence level, as long as the two latent representations exhibit strong structure in feature space.
+
+In order to add such a constraint, we train a neural network, which we will refer to as the discriminator, to classify between the encoding of source sentences and the encoding of target sentences (Ganin et al., 2016). The discriminator operates on the output of the encoder, which is a sequence of latent vectors $\left( z _ { 1 } , . . . , z _ { m } \right)$ , with $z _ { i } \in \mathbb { R } ^ { n }$ , and produces a binary prediction about the language of the encoder input sentence: $p _ { D } ( l | z _ { 1 } , . . . , z _ { m } ) \propto \prod _ { j = 1 } ^ { m } p _ { D } ( \ell | z _ { j } )$ , with $p _ { D } : \mathbb { R } ^ { n }  [ 0 ; 1 ]$ , where 0 corresponds to the source domain, and 1 to the target domain.
+
+The discriminator is trained to predict the language by minimizing the following cross-entropy loss: $\mathcal { L } _ { \mathcal { D } } ( \theta _ { D } | \theta , \mathcal { Z } ) = - \mathbb { E } _ { ( x _ { i } , \ell _ { i } ) } [ \log p _ { D } ( \ell _ { i } | e ( x _ { i } , \ell _ { i } ) ) ]$ , where $( x _ { i } , \ell _ { i } )$ corresponds to sentence and language id pairs uniformly sampled from the two monolingual datasets, $\theta _ { D }$ are the parameters of the discriminator, $\theta _ { \mathrm { e n c } }$ are the parameters of the encoder, and $\mathcal { Z }$ are the encoder word embeddings.
+
+The encoder is trained instead to fool the discriminator:
+
+$$
+\mathcal { L } _ { a d v } ( \theta _ { \mathrm { e n c } } , \mathcal { Z } | \theta _ { D } ) = - \mathbb { E } _ { ( x _ { i } , \ell _ { i } ) } [ \log p _ { D } ( \ell _ { j } | e ( x _ { i } , \ell _ { i } ) ) ]
+$$
+
+with $\ell _ { j } = \ell _ { 1 }$ if $\ell _ { i } = \ell _ { 2 }$ , and vice versa.
+
+Final Objective function The final objective function at one iteration of our learning algorithm is thus:
+
+$$
+\begin{array} { r l } & { \mathcal { L } ( \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } ) = \lambda _ { a u t o } [ \mathcal { L } _ { a u t o } ( \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } , s r c ) + \mathcal { L } _ { a u t o } ( \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } , t g t ) ] + } \\ & { \qquad \lambda _ { c d } [ \mathcal { L } _ { c d } ( \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } , s r c , t g t ) + \mathcal { L } _ { c d } ( \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } , t g t , s r c ) ] + } \\ & { \qquad \lambda _ { a d v } \mathcal { L } _ { a d v } ( \theta _ { \mathrm { e n c } } , \mathcal { Z } | \theta _ { D } ) } \end{array}
+$$
+
+where $\lambda _ { a u t o }$ , $\lambda _ { c d }$ , and $\lambda _ { a d v }$ are hyper-parameters weighting the importance of the auto-encoding, cross-domain and adversarial loss. In parallel, the discriminator loss $\mathcal { L } _ { D }$ is minimized to update the discriminator.
+
+# 3 TRAINING
+
+In this section we describe the overall training algorithm and the unsupervised criterion we used to select hyper-parameters.
+
+# 3.1 ITERATIVE TRAINING
+
+The final learning algorithm is described in Algorithm 1 and the general architecture of the model is shown in Figure 2. As explained previously, our model relies on an iterative algorithm which starts from an initial translation model $M ^ { ( 1 ) }$ (line 3). This is used to translate the available monolingual data, as needed by the cross-domain loss function of Equation 2. At each iteration, a new encoder and decoder are trained by minimizing the loss of Equation 4 – line 7 of the algorithm. Then, a new translation model $M ^ { ( t + 1 ) }$ is created by composing the resulting encoder and decoder, and the process repeats.
+
+To jump start the process, $M ^ { ( 1 ) }$ simply makes a word-by-word translation of each sentence using a parallel dictionary learned using the unsupervised method proposed by Conneau et al. (2017), which only leverages monolingual data.
+
+The intuition behind our algorithm is that as long as the initial translation model $M ^ { ( 1 ) }$ retains at least some information of the input sentence, the encoder will map such translation into a representation in feature space that also corresponds to a cleaner version of the input, since the encoder is trained to denoise. At the same time, the decoder is trained to predict noiseless outputs, conditioned on noisy features. Putting these two pieces together will produce less noisy translations, which will enable better back-translations at the next iteration, and so on so forth.
+
+![](images/a6ebd6e78ac84bb7951bc3db503a594c1ee86aff87bf655cf571909182f31264.jpg)  
+Figure 2: Illustration of the proposed architecture and training objectives. The architecture is a sequence to sequence model, with both encoder and decoder operating on two languages depending on an input language identifier that swaps lookup tables. Top (auto-encoding): the model learns to denoise sentences in each domain. Bottom (translation): like before, except that we encode from another language, using as input the translation produced by the model at the previous iteration (light blue box). The green ellipses indicate terms in the loss function.
+
+# Algorithm 1 Unsupervised Training for Machine Translation
+
+1: procedure TRAINING $\mathcal { D } _ { s r c }$ , $\mathcal { D } _ { t g t } , T )$   
+2: Infer bilingual dictionary using monolingual data (Conneau et al., 2017)   
+3: $M ^ { ( 1 ) } \gets$ unsupervised word-by-word translation model using the inferred dictionary   
+4: for $t = 1 , T$ do   
+5: using ${ \bf \nabla } _ { M } ( t )$ , translate each monolingual dataset   
+6: $/ /$ discriminator training $\&$ model training as in eq. 4   
+7: $\theta _ { \mathrm { d i s c r } }  \arg \operatorname* { m i n } \mathcal { L } _ { D }$ , $\theta _ { \mathrm { e n c } } , \theta _ { \mathrm { d e c } } , \mathcal { Z } \gets \arg \operatorname* { m i n } \mathcal { L }$   
+8: $M ^ { ( t + 1 ) } \gets \breve { e } ^ { ( t ) } \circ d ^ { ( t ) } / /$ update MT model   
+9: end for   
+10: return M (T +1)   
+11: end procedure
+
+# 3.2 UNSUPERVISED MODEL SELECTION CRITERION
+
+In order to select hyper-parameters, we wish to have a criterion correlated with the translation quality. However, we do not have access to parallel sentences to judge how well our model translates, not even at validation time. Therefore, we propose the surrogate criterion which we show correlates well with BLEU (Papineni et al., 2002), the metric we care about at test time.
+
+For all sentences $x$ in a domain $\ell _ { 1 }$ , we translate these sentences to the other domain $\ell _ { 2 }$ , and then translate the resulting sentences back to $\ell _ { 1 }$ . The quality of the model is then evaluated by computing the BLEU score over the original inputs and their reconstructions via this two-step translation process. The performance is then averaged over the two directions, and the selected model is the one with the highest average score.
+
+Given an encoder $e$ , a decoder $d$ and two non-parallel datasets $\mathcal { D } _ { s r c }$ and $\mathcal { D } _ { t g t }$ , we denote $M _ { s r c  t g t } ( x ) = d ( e ( x , s r c ) , t g t )$ the translation model from src to tgt, and $M _ { t g t  s r c }$ the model in the opposite direction. Our model selection criterion $M S ( e , d , \mathcal { D } _ { s r c } , \mathcal { D } _ { t g t } )$ is:
+
+$$
+\begin{array} { r c l } { { M S ( e , d , \mathcal { D } _ { s r c } , \mathcal { D } _ { t g t } ) } } & { { = } } & { { \displaystyle \frac { 1 } { 2 } \mathbb { E } _ { x \sim \mathcal { D } _ { s r c } } [ \mathrm { B L E U } ( x , M _ { s r c  t g t } \circ M _ { t g t  s r c } ( x ) ) ] + } } \\ { { } } & { { } } & { { \displaystyle \frac { 1 } { 2 } \mathbb { E } _ { x \sim \mathcal { D } _ { t g t } } [ \mathrm { B L E U } ( x , M _ { t g t  s r c } \circ M _ { s r c  t g t } ( x ) ) ] } } \end{array}
+$$
+
+Figure 3 shows a typical example of the correlation between this measure and the final translation model performance (evaluated here using a parallel dataset).
+
+The unsupervised model selection criterion is used both to a) determine when to stop training and b) to select the best hyper-parameter setting across different experiments. In the former case, the
+
+![](images/7d5c691d1839c8ac8f8acfc1059b5b6a7784330639f26875fb7813bd2392df39.jpg)  
+Figure 3: Unsupervised model selection. BLEU score of the source to target and target to source models on the Multi30k-Task1 English-French dataset as a function of the number of passes through the dataset at iteration $\mathbf { \Omega } ( t ) \ = \ 1$ of the algorithm (training $M ( 2 )$ given $M ( 1 ) ,$ ). BLEU correlates very well with the proposed model selection criterion, see Equation 5.
+
+Spearman correlation coefficient between the proposed criterion and BLEU on the test set is 0.95 in average. In the latter case, the coefficient is in average 0.75, which is fine but not nearly as good. For instance, the BLEU score on the test set of models selected with the unsupervised criterion are sometimes up to 1 or 2 BLEU points below the score of models selected using a small validation set of 500 parallel sentences.
+
+# 4 EXPERIMENTS
+
+In this section, we first describe the datasets and the pre-processing we used, then we introduce the baselines we considered, and finally we report the extensive empirical validation proving the effectiveness of our method. We will release the code to the public once the revision process is over.
+
+# 4.1 DATASETS
+
+In our experiments, we consider the English-French and English-German language pairs, on three different datasets.
+
+WMT’14 English-French We use the full training set of 36 million pairs, we lower-case them and remove sentences longer than 50 words, as well as pairs with a source/target length ratio above 1.5, resulting in a parallel corpus of about 30 million sentences. Next, we build monolingual corpora by selecting the English sentences from 15 million random pairs, and selecting the French sentences from the complementary set. The former set constitutes our English monolingual dataset. The latter set is our French monolingual dataset. The lack of overlap between the two sets ensures that there is not exact correspondence between examples in the two datasets.
+
+The validation set is comprised of 3,000 English and French sentences extracted from our monolingual training corpora described above. These sentences are not the translation of each other, and they will be used by our unsupervised model selection criterion, as explained in 3.2. Finally, we report results on the full newstest2014 dataset.
+
+WMT’16 English-German We follow the same procedure as above to create monolingual training and validation corpora in English and German, which results in two monolingual training corpora of 1.8 million sentences each. We test our model on the newstest2016 dataset.
+
+Multi30k-Task1 The task 1 of the Multi30k dataset (Elliott et al., 2016) has 30,000 images, with annotations in English, French and German, that are translations of each other. We consider the English-French and English-German pairs. We disregard the images and only consider the parallel annotations, with the provided training, validation and test sets, composed of 29,000, 1,000 and 1,000 pairs of sentences respectively. For both pairs of languages and similarly to the WMT datasets above, we split the training and validation sets into monolingual corpora, resulting in 14,500 monolingual source and target sentences in the training set, and 500 sentences in the validation set.
+
+Table 1: Multi30k-Task1 and WMT datasets statistics. To limit the vocabulary size in the WMT en-fr and WMT de-en datasets, we only considered words with more than 100 and 25 occurrences, respectively.   
+
+<table><tr><td></td><td>MMT1 en-fr</td><td>MMT1 de-en</td><td>WMT en-fr</td><td>WMT de-en</td></tr><tr><td>Monolingual sentences</td><td>14.5k</td><td>14.5k</td><td>15M</td><td>1.8M</td></tr><tr><td>Vocabulary size</td><td>10k/11k</td><td>19k/10k</td><td>67k/78k</td><td>80k/46k</td></tr></table>
+
+Table 1 summarizes the number of monolingual sentences in each dataset, along with the vocabulary size. To limit the vocabulary size on the WMT en-fr and WMT de-en datasets, we only considered words with more than 100 and 25 occurrences, respectively.
+
+# 4.2 BASELINES
+
+Word-by-word translation (WBW) The first baseline is a system that performs word-by-word translations of the input sentences using the inferred bilingual dictionary (Conneau et al., 2017). This baseline provides surprisingly good results for related language pairs, like English-French, where the word order is similar, but performs rather poorly on more distant pairs like English-German, as can be seen in Table 2.
+
+Word reordering (WR) After translating word-by-word as in WBW, here we reorder words using an LSTM-based language model trained on the target side. Since we cannot exhaustively score every possible word permutation (some sentences have about 100 words), we consider all pairwise swaps of neighboring words, we select the best swap, and iterate ten times. We use this baseline only on the WMT dataset that has a large enough monolingual data to train a language model.
+
+Oracle Word Reordering (OWR) Using the reference, we produce the best possible generation using only the words given by WBW. The performance of this method is an upper-bound of what any model could do without replacing words.
+
+Supervised Learning We finally consider exactly the same model as ours, but trained with supervision, using the standard cross-entropy loss on the original parallel sentences.
+
+# 4.3 UNSUPERVISED DICTIONARY LEARNING
+
+To implement our baseline and also to initialize the embeddings $\mathcal { Z }$ of our model, we first train word embeddings on the source and target monolingual corpora using fastText (Bojanowski et al., 2017), and then we apply the unsupervised method proposed by Conneau et al. (2017) to infer a bilingual dictionary which can be use for word-by-word translation.
+
+Since WMT yields a very large-scale monolingual dataset, we obtain very high-quality embeddings and dictionaries, with an accuracy of $8 4 . { \bar { 4 } } 8 \%$ and $7 7 . 2 9 \%$ on French-English and GermanEnglish, which is on par with what could be obtained using a state-of-the-art supervised alignment method (Conneau et al., 2017).
+
+On the Multi30k datasets instead, the monolingual training corpora are too small to train good word embeddings (more than two order of magnitude smaller than WMT). We therefore learn word vectors on Wikipedia using fastText2.
+
+Table 2: BLEU score on the Multi30k-Task1 and WMT datasets using greedy decoding.   
+
+<table><tr><td></td><td colspan="4">Multi30k-Task1</td><td colspan="4">WMT</td></tr><tr><td></td><td>en-fr</td><td>fr-en</td><td>de-en</td><td>en-de</td><td>en-fr</td><td>fr-en</td><td>de-en</td><td>en-de</td></tr><tr><td>Supervised</td><td>56.83</td><td>50.77</td><td>38.38</td><td>35.16</td><td>27.97</td><td>26.13</td><td>25.61</td><td>21.33</td></tr><tr><td>word-by-word</td><td>8.54</td><td>16.77</td><td>15.72</td><td>5.39</td><td>6.28</td><td>10.09</td><td>10.77</td><td>7.06</td></tr><tr><td>word reordering</td><td>-</td><td>-</td><td>1</td><td>-</td><td>6.68</td><td>11.69</td><td>10.84</td><td>6.70</td></tr><tr><td>oracle word reordering</td><td>11.62</td><td>24.88</td><td>18.27</td><td>6.79</td><td>10.12</td><td>20.64</td><td>19.42</td><td>11.57</td></tr><tr><td>Our model: 1st iteration</td><td>27.48</td><td>28.07</td><td>23.69</td><td>19.32</td><td>12.10</td><td>11.79</td><td>11.10</td><td>8.86</td></tr><tr><td>Our model: 2nd iteration</td><td>31.72</td><td>30.49</td><td>24.73</td><td>21.16</td><td>14.42</td><td>13.49</td><td>13.25</td><td>9.75</td></tr><tr><td>Our model: 3rd iteration</td><td>32.76</td><td>32.07</td><td>26.26</td><td>22.74</td><td>15.05</td><td>14.31</td><td>13.33</td><td>9.64</td></tr></table>
+
+# 4.4 EXPERIMENTAL DETAILS
+
+Discriminator Architecture The discriminator is a multilayer perceptron with three hidden layers of size 1024, Leaky-ReLU activation functions and an output logistic unit. Following Goodfellow (2016), we include a smoothing coefficient $s = 0 . 1$ in the discriminator predictions.
+
+Training Details The encoder and the decoder are trained using Adam (Kingma & Ba, 2014), with a learning rate of 0.0003, $\beta _ { 1 } = 0 . 5$ , and a mini-batch size of 32. The discriminator is trained using RMSProp (Tieleman & Hinton, 2012) with a learning rate of 0.0005. We evenly alternate between one encoder-decoder and one discriminator update. We set $\lambda _ { a u t o } = \lambda _ { c d } = \lambda _ { a d v } = 1$ .
+
+# 4.5 EXPERIMENTAL RESULTS
+
+Table 2 shows the BLEU scores achieved by our model and the baselines we considered. First, we observe that word-by-word translation is surprisingly effective when translating into English, obtaining a BLEU score of 16.77 and 10.09 for $f r$ -en on respectively Multi30k-Task1 and WMT datasets. Word-reordering only slightly improves upon word-by-word translation. Our model instead, clearly outperforms these baselines, even on the WMT dataset which has more diversity of topics and sentences with much more complicated structure. After just one iteration, we obtain a BLEU score of 27.48 and 12.10 for the en-fr task. Interestingly, we do even better than oracle reordering on some language pairs, suggesting that our model not only reorders but also correctly substitutes some words. After a few iterations, our model obtains BLEU of 32.76 and 15.05 on Multi30k-Task1 and WMT datasets for the English to French task, which is rather remarkable.
+
+Comparison with supervised approaches Here, we assess how much labeled data are worth our two large monolingual corpora. On WMT, we trained the very same NMT architecture on both language pairs, but with supervision using various amounts of parallel data. Figure 4-right shows the resulting performance. Our unsupervised approach obtains the same performance than a supervised NMT model trained on about 100,000 parallel sentences, which is impressive. Of course, adding more parallel examples allows the supervised approach to outperform our method, but the good performance of our unsupervised method suggests that it could be very effective for low-resources languages where no parallel data are available. Moreover, these results open the door to the development of semi-supervised translation models, which will be the focus of future investigation. With a phrase-based machine translation system, we obtain 21.6 and 22.4 BLEU on WMT en- ${ \mathcal { f } } { \mathit { r } }$ and $f r { - } e n$ , which is better than the supervised NMT baseline we report for that same amount of parallel sentences, which is 16.8 and 16.4 respectively. However, if we train the same supervised NMT model with BPE (Sennrich et al., 2015b), we obtain 22.6 BLEU for en-fr, suggesting that our results on unsupervised machine translation could also be improved by using BPE, as this removes unknown words (about $9 \%$ of the words in de-en are replaced by the unknown token otherwise).
+
+Iterative Learning Figure 4-left illustrates the quality of the learned model after each iteration of the learning process in the language pairs of Multi30k-Task1 dataset, other results being provided in Table 2. One can see that the quality of the obtained model is high just after the first iteration of the process. Subsequent iterations yield significant gains although with diminishing returns. At iteration 3, the performance gains are marginal, showing that our approach quickly converges.
+
+![](images/883b63a0e414781e577adbbaf839014448b02037b6e20d1cafd7034972cea13d.jpg)  
+Figure 4: Left: BLEU as a function of the number of iterations of our algorithm on the Multi30kTask1 datasets. Right: The curves show BLEU as a function of the amount of parallel data on WMT datasets. The unsupervised method which leverages about 15 million monolingual sentences in each language, achieves performance (see horizontal lines) close to what we would obtain by employing 100,000 parallel sentences.
+
+Table 3: Unsupervised translations. Examples of translations on the French-English pair of the Multi30k-Task1 dataset. Iteration 0 corresponds to word-by-word translation. After 3 iterations, the model generates very good translations.   
+
+<table><tr><td>Source Iteration O Iteration 1 Iteration 2 Iteration 3 Reference</td><td>un homme est debout pres d&#x27;une série de jeux vidéo dans un bar . a man is seated neara series of games video in a bar. a man is standing nearacloseup of other games in a bar . a man is standing near a bunch of video video game in a bar . a man is standing neara bunch of video games ina bar. a man is standing by a group of video games in a bar.</td></tr><tr><td>Source Iteration O</td><td>une femme aux cheveux roses habillée en noir parle ä un homme . a woman at hair roses dressed in black speaks to a man .</td></tr><tr><td>Iteration 1</td><td>a woman at glasses dressed in black talking to a man.</td></tr><tr><td>Iteration 2</td><td>a woman at pink hair dressed in black speaks to a man.</td></tr><tr><td>Iteration 3 Reference</td><td>a woman with pink hair dressed in black is talking to a man.</td></tr><tr><td></td><td>a woman with pink hair dressed in black talks to a man.</td></tr><tr><td>Source</td><td>une photo d&#x27;une rue bondée en ville .</td></tr><tr><td>Iteration O</td><td>a photo a street crowded in city .</td></tr><tr><td>Iteration 1</td><td>a picture of a street crowded in a city .</td></tr><tr><td>Iteration 2</td><td></td></tr><tr><td>Iteration 3</td><td>a picture of a crowded city street.</td></tr><tr><td></td><td>a picture of a crowded street in a city .</td></tr><tr><td>Reference</td><td>aview of acrowded citystreet.</td></tr><tr><td></td><td></td></tr></table>
+
+Table 3 shows examples of translations of three sentences on the Multi30k dataset, as we iterate. Iteration 0 corresponds to the word-by-word translation obtained with our cross-lingual dictionary, which clearly suffers from word order issues. We can observe that the quality of the translations increases at every iteration.
+
+Ablation Study We perform an ablation study to understand the importance of the different components of our system. To this end, we have trained multiple versions of our model with some missing components: the discriminator, the cross-domain loss, the auto-encoding loss, etc. Table 4 shows that the best performance is obtained with the simultaneous use of all the described elements.
+
+Table 4: Ablation study on the Multi30k-Task1 dataset.   
+
+<table><tr><td></td><td>en-fr</td><td>fr-en</td><td>de-en</td><td>en-de</td></tr><tr><td>Xcd=0</td><td>25.44</td><td>27.14</td><td>20.56</td><td>14.42</td></tr><tr><td>Without pretraining</td><td>25.29</td><td>26.10</td><td>21.44</td><td>17.23</td></tr><tr><td>Without pretraining, Xcd = 0</td><td>8.78</td><td>9.15</td><td>7.52</td><td>6.24</td></tr><tr><td>Without noise,C(x) = x</td><td>16.76</td><td>16.85</td><td>16.85</td><td>14.61</td></tr><tr><td>Xauto=0</td><td>24.32</td><td>20.02</td><td>19.10</td><td>14.74</td></tr><tr><td>Aadu =0</td><td>24.12</td><td>22.74</td><td>19.87</td><td>15.13</td></tr><tr><td>Full</td><td>27.48</td><td>28.07</td><td>23.69</td><td>19.32</td></tr></table>
+
+The most critical component is the unsupervised word alignment technique, either in the form of a back-translation dataset generated using word-by-word translation, or in the form of pretrained embeddings which enable to map sentences of different languages in the same latent space.
+
+On the English-French pair of Multi30k-Task1, with a back-translation dataset but without pretrained embeddings, our model obtains a BLEU score of 25.29 and 26.10, which is only a few points below the model using all components. Similarly, when the model uses pretrained embeddings but no back-translation dataset (when $\lambda _ { c d } = 0$ ), it obtains 25.44 and 27.14. On the other hand, a model that does not use any of these components only reaches 8.78 and 9.15 BLEU.
+
+The adversarial component also significantly improves the performance of our system, with a difference of up to 5.33 BLEU in the French-English pair of Multi30k-Task1. This confirms our intuition that, to really benefit from the cross-domain loss, one has to ensure that the distribution of latent sentence representations is similar across the two languages. Without the auto-encoding loss (when $\lambda _ { a u t o } = 0$ ), the model only obtains 20.02, which is 8.05 BLEU points below the method using all components. Finally, performance is greatly degraded also when the corruption process of the input sentences is removed, as the model has much harder time learning useful regularities and merely learns to copy input data.
+
+# 5 RELATED WORK
+
+A similar work to ours is the style transfer method with non-parallel text by Shen et al. (2017). The authors consider a sequence-to-sequence model, where the latent state given to the decoder is also fed to a discriminator. The encoder is trained with the decoder to reconstruct the input, but also to fool the discriminator. The authors also found it beneficial to train two discriminators, one for the source and one for the target domain. Then, they trained the decoder so that the recurrent hidden states during the decoding process of a sentence in a particular domain are not distinguishable according to the respective discriminator. This algorithm, called Professor forcing, was initially introduced by Lamb et al. (2016) to encourage the dynamics of the decoder observed during inference to be similar to the ones observed at training time.
+
+Similarly, Xie et al. (2017) also propose to use an adversarial training approach to learn representations invariant to specific attributes. In particular, they train an encoder to map the observed data to a latent feature space, and a model to make predictions based on the encoder output. To remove bias existing in the data from the latent codes, a discriminator is also trained on the encoder outputs to predict specific attributes, while the encoder is jointly trained to fool the discriminator. They show that the obtained invariant representations lead to better generalization on classification and generation tasks.
+
+Before that, Hu et al. (2017) trained a variational autoencoder (Kingma & Welling, 2013) where the decoder input is the concatenation of an unstructured latent vector, and a structured code representing the attribute of the sentence to generate. A discriminator is trained on top of the decoder to classify the labels of generated sentences, while the decoder is trained to satisfy this discriminator. Because of the non-differentiability of the decoding process, at each step, their decoder takes as input the probability vector predicted at the previous step.
+
+Perhaps, the most relevant prior work is by He et al. (2016), who essentially optimizes directly for the model selection metric we propose in section 3.2. One drawback of their approach, which has not been applied to the fully unsupervised setting, is that it requires to back-propagate through the sequence of discrete predictions using reinforcement learning-based approaches which are notoriously inefficient. In this work, we instead propose to a) use a symmetric architecture, and b) freeze the translator from source to target when training the translator from target to source, and vice versa. By alternating this process we operate with a fully differentiable model and we efficiently converge.
+
+In the vision domain, several studies tackle the unsupervised image translation problem, where the task consists in mapping two image domains A and B, without paired supervision. For instance, in the CoGAN architecture (Liu & Tuzel, 2016), two generators are trained to learn a common representation space between two domains, by sharing some of their convolutional layers. This is similar to our strategy of sharing the LSTM weights across the source and target encoders and decoders. Liu et al. (2017) propose a similar approach, based on variational autoencoders, and generative adversarial networks (Goodfellow et al., 2014). Taigman et al. (2016) use similar approaches for emoji generation, and apply a regularization term to the generator so that it behaves like an identity mapping when provided with input images from the target domain. Zhu et al. (2017) introduced a cycle consistency loss, to capture the intuition that if an image is mapped from A to B, then from B to A, then the resulting image should be identical to the input one.
+
+Our approach is also reminiscent of the Fader Networks architecture (Lample et al., 2017), where a discriminator is used to remove the information related to specific attributes from the latent states of an autoencoder of images. The attribute values are then given as input to the decoder. The decoder is trained with real attributes, but at inference, it can be fed with any attribute values to generate variations of the input images. The model presented in this paper can be seen as an extension to the text domain of the Fader Networks, where the attribute is the language itself.
+
+# 6 CONCLUSION
+
+We presented a new approach to neural machine translation where a translation model is learned using monolingual datasets only, without any alignment between sentences or documents. The principle of our approach is to start from a simple unsupervised word-by-word translation model, and to iteratively improve this model based on a reconstruction loss, and using a discriminator to align latent distributions of both the source and the target languages. Our experiments demonstrate that our approach is able to learn effective translation models without any supervision of any sort.
+
+# REFERENCES
+
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parse/train/rkYTTf-AZ/rkYTTf-AZ_content_list.json

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+    {
+        "type": "text",
+        "text": "UNSUPERVISED MACHINE TRANSLATION USING MONOLINGUAL CORPORA ONLY ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Guillaume Lample † ‡ , Alexis Conneau $\\dagger$ , Ludovic Denoyer $^ \\ddag$ , Marc’Aurelio Ranzato † † Facebook AI Research, ‡ Sorbonne Universites, UPMC Univ Paris 06, LIP6 UMR 7606, CNRS ´ {gl,aconneau,ranzato}@fb.com,ludovic.denoyer@lip6.fr ",
+        "bbox": [
+            184,
+            169,
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+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine translation has recently achieved impressive performance thanks to recent advances in deep learning and the availability of large-scale parallel corpora. There have been numerous attempts to extend these successes to low-resource language pairs, yet requiring tens of thousands of parallel sentences. In this work, we take this research direction to the extreme and investigate whether it is possible to learn to translate even without any parallel data. We propose a model that takes sentences from monolingual corpora in two different languages and maps them into the same latent space. By learning to reconstruct in both languages from this shared feature space, the model effectively learns to translate without using any labeled data. We demonstrate our model on two widely used datasets and two language pairs, reporting BLEU scores of 32.8 and 15.1 on the Multi30k and WMT English-French datasets, without using even a single parallel sentence at training time. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            496,
+            336,
+            512
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Thanks to recent advances in deep learning (Sutskever et al., 2014; Bahdanau et al., 2015) and the availability of large-scale parallel corpora, machine translation has now reached impressive performance on several language pairs (Wu et al., 2016). However, these models work very well only when provided with massive amounts of parallel data, in the order of millions of parallel sentences. Unfortunately, parallel corpora are costly to build as they require specialized expertise, and are often nonexistent for low-resource languages. Conversely, monolingual data is much easier to find, and many languages with limited parallel data still possess significant amounts of monolingual data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "There have been several attempts at leveraging monolingual data to improve the quality of machine translation systems in a semi-supervised setting (Munteanu et al., 2004; Irvine, 2013; Irvine & Callison-Burch, 2015; Zheng et al., 2017). Most notably, Sennrich et al. (2015a) proposed a very effective data-augmentation scheme, dubbed “back-translation”, whereby an auxiliary translation system from the target language to the source language is first trained on the available parallel data, and then used to produce translations from a large monolingual corpus on the target side. The pairs composed of these translations with their corresponding ground truth targets are then used as additional training data for the original translation system. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Another way to leverage monolingual data on the target side is to augment the decoder with a language model (Gulcehre et al., 2015). And finally, Cheng et al. (2016); He et al. (2016) have proposed to add an auxiliary auto-encoding task on monolingual data, which ensures that a translated sentence can be translated back to the original one. All these works still rely on several tens of thousands parallel sentences, however. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Previous work on zero-resource machine translation has also relied on labeled information, not from the language pair of interest but from other related language pairs (Firat et al., 2016; Johnson et al., 2016; Chen et al., 2017) or from other modalities (Nakayama & Nishida, 2017; Lee et al., 2017). The only exception is the work by Ravi & Knight (2011); Pourdamghani & Knight (2017), where the machine translation problem is reduced to a deciphering problem. Unfortunately, their method is limited to rather short sentences and it has only been demonstrated on a very simplistic setting comprising of the most frequent short sentences, or very closely related languages. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/94c547e44b94e2f2ddd4b069149181cface3d24ae25686585f142de1dca80aed.jpg",
+        "image_caption": [
+            "Figure 1: Toy illustration of the principles guiding the design of our objective function. Left (autoencoding): the model is trained to reconstruct a sentence from a noisy version of it. $x$ is the target, $C ( x )$ is the noisy input, $\\hat { x }$ is the reconstruction. Right (translation): the model is trained to translate a sentence in the other domain. The input is a noisy translation (in this case, from source-to-target) produced by the model itself, $M$ , at the previous iteration $( t )$ , $y = M ^ { ( t ) } ( x )$ . The model is symmetric, and we repeat the same process in the other language. See text for more details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we investigate whether it is possible to train a general machine translation system without any form of supervision whatsoever. The only assumption we make is that there exists a monolingual corpus on each language. This set up is interesting for a twofold reason. First, this is applicable whenever we encounter a new language pair for which we have no annotation. Second, it provides a strong lower bound performance on what any good semi-supervised approach is expected to yield. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The key idea is to build a common latent space between the two languages (or domains) and to learn to translate by reconstructing in both domains according to two principles: (i) the model has to be able to reconstruct a sentence in a given language from a noisy version of it, as in standard denoising auto-encoders (Vincent et al., 2008). (ii) The model also learns to reconstruct any source sentence given a noisy translation of the same sentence in the target domain, and vice versa. For (ii), the translated sentence is obtained by using a back-translation procedure (Sennrich et al., 2015a), i.e. by using the learned model to translate the source sentence to the target domain. In addition to these reconstruction objectives, we constrain the source and target sentence latent representations to have the same distribution using an adversarial regularization term, whereby the model tries to fool a discriminator which is simultaneously trained to identify the language of a given latent sentence representation (Ganin et al., 2016). This procedure is then iteratively repeated, giving rise to translation models of increasing quality. To keep our approach fully unsupervised, we initialize our algorithm by using a na¨ıve unsupervised translation model based on a word by word translation of sentences with a bilingual lexicon derived from the same monolingual data (Conneau et al., 2017). As a result, and by only using monolingual data, we can encode sentences of both languages into the same feature space, and from there, we can also decode/translate in any of these languages; see Figure 1 for an illustration. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While not being able to compete with supervised approaches using lots of parallel resources, we show in section 4 that our model is able to achieve remarkable performance. For instance, on the WMT dataset we can achieve the same translation quality of a similar machine translation system trained with full supervision on 100,000 sentence pairs. On the Multi30K-Task1 dataset we achieve a BLEU above 22 on all the language pairs, with up to 32.76 on English-French. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Next, in section 2, we describe the model and the training algorithm. We then present experimental results in section 4. Finally, we further discuss related work in section 5 and summarize our findings in section 6. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 UNSUPERVISED NEURAL MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this section, we first describe the architecture of the translation system, and then we explain how we train it. ",
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+    {
+        "type": "text",
+        "text": "2.1 NEURAL MACHINE TRANSLATION MODEL ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The translation model we propose is composed of an encoder and a decoder, respectively responsible for encoding source and target sentences to a latent space, and to decode from that latent space to the source or the target domain. We use a single encoder and a single decoder for both domains (Johnson et al., 2016). The only difference when applying these modules to different languages is the choice of lookup tables. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let us denote by $\\mathcal { W } _ { S }$ the set of words in the source domain associated with the (learned) words embeddings $\\mathcal { Z } ^ { S } = ( z _ { 1 } ^ { s } , . . . . , z _ { | \\mathcal { W } _ { S } | } ^ { s } )$ , and by $\\mathcal { W } _ { T }$ the set of words in the target domain associated with the embeddings $\\mathcal { Z } ^ { T } = ( z _ { 1 } ^ { t } , . . . , z _ { | \\mathcal { W } _ { T } | } ^ { t } )$ , $\\mathcal { Z }$ being the set of all the embeddings. Given an input sentence of $m$ words $\\pmb { x } = ( x _ { 1 } , x _ { 2 } , . . . , x _ { m } )$ in a particular language \\`, $\\ell \\in \\{ s r c , t g t \\}$ , an encoder $e _ { \\theta _ { \\mathrm { { e n c } } } , z } ( { \\pmb x } , { \\pmb \\ell } )$ computes a sequence of $m$ hidden states $\\boldsymbol { z } = ( z _ { 1 } , z _ { 2 } , . . . , z _ { m } )$ by using the corresponding word embeddings, i.e. ${ \\mathcal { Z } } _ { S }$ if $\\ell = s r c$ and ${ \\mathcal { Z } } _ { T }$ if $\\ell = t g t$ ; the other parameters $\\theta _ { \\mathrm { e n c } }$ are instead shared between the source and target languages. For the sake of simplicity, the encoder will be denoted as $e ( { \\pmb x } , { \\pmb \\ell } )$ in the following. These hidden states are vectors in $\\mathbb { R } ^ { n }$ , $n$ being the dimension of the latent space. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A decoder $d _ { \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } } ( z , \\ell )$ takes as input $_ z$ and a language $\\ell$ , and generates an output sequence ${ \\textbf { \\em y } } =$ $\\left( y _ { 1 } , y _ { 2 } , . . . , y _ { k } \\right)$ , where each word $y _ { i }$ is in the corresponding vocabulary $\\mathcal { W } ^ { \\ell }$ . This decoder makes use of the corresponding word embeddings, and it is otherwise parameterized by a vector $\\theta _ { \\mathrm { d e c } }$ that does not depend on the output language. It will thus be denoted $d ( z , \\ell )$ in the following. To generate an output word $y _ { i }$ , the decoder iteratively takes as input the previously generated word $y _ { i - 1 }$ $y _ { 0 }$ being a start symbol which is language dependent), updates its internal state, and returns the word that has the highest probability of being the next one. The process is repeated until the decoder generates a stop symbol indicating the end of the sequence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this article, we use a sequence-to-sequence model with attention (Bahdanau et al., 2015), without input-feeding. The encoder is a bidirectional-LSTM which returns a sequence of hidden states $\\boldsymbol { z } = ( z _ { 1 } , z _ { 2 } , . . . , z _ { m } )$ . At each step, the decoder, which is also an LSTM, takes as input the previous hidden state, the current word and a context vector given by a weighted sum over the encoder states. In all the experiments we consider, both encoder and decoder have 3 layers. The LSTM layers are shared between the source and target encoder, as well as between the source and target decoder. We also share the attention weights between the source and target decoder. The embedding and LSTM hidden state dimensions are all set to 300. Sentences are generated using greedy decoding. ",
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+    {
+        "type": "text",
+        "text": "2.2 OVERVIEW OF THE METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider a dataset of sentences in the source domain, denoted by $\\mathcal { D } _ { s r c }$ , and another dataset in the target domain, denoted by $\\mathcal { D } _ { t g t }$ . These datasets do not correspond to each other, in general. We train the encoder and decoder by reconstructing a sentence in a particular domain, given a noisy version of the same sentence in the same or in the other domain. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "At a high level, the model starts with an unsupervised na¨ıve translation model obtained by making word-by-word translation of sentences using a parallel dictionary learned in an unsupervised way (Conneau et al., 2017). Then, at each iteration, the encoder and decoder are trained by minimizing an objective function that measures their ability to both reconstruct and translate from a noisy version of an input training sentence. This noisy input is obtained by dropping and swapping words in the case of the auto-encoding task, while it is the result of a translation with the model at the previous iteration in the case of the translation task. In order to promote alignment of the latent distribution of sentences in the source and the target domains, our approach also simultaneously learns a discriminator in an adversarial setting. The newly learned encoder/decoder are then used at the next iteration to generate new translations, until convergence of the algorithm. At test time and despite the lack of parallel data at training time, the encoder and decoder can be composed into a standard machine translation system. ",
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+    {
+        "type": "text",
+        "text": "2.3 DENOISING AUTO-ENCODING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training an autoencoder of sentences is a trivial task, if the sequence-to-sequence model is provided with an attention mechanism like in our work 1. Without any constraint, the auto-encoder very quickly learns to merely copy every input word one by one. Such a model would also perfectly copy sequences of random words, suggesting that the model does not learn any useful structure in the data. To address this issue, we adopt the same strategy of Denoising Auto-encoders (DAE) (Vincent et al., 2008)), and add noise to the input sentences (see Figure 1-left), similarly to Hill et al. (2016). Considering a domain $\\ell = s r c$ or $\\ell = t g t$ , and a stochastic noise model denoted by $C$ which operates on sentences, we define the following objective function: ",
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+    {
+        "type": "equation",
+        "img_path": "images/b2895f3393e737b2eed9d8fb264f12d9f0ede644bb9e90fb0a338cde627f890e.jpg",
+        "text": "$$\n\\mathcal { L } _ { a u t o } ( \\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } , \\ell ) = \\mathbb { E } _ { x \\sim \\mathcal { D } _ { \\ell } , \\hat { x } \\sim d ( e ( C ( x ) , \\ell ) , \\ell ) } \\left[ \\Delta ( \\hat { x } , x ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { x } \\sim d ( e ( C ( x ) , \\ell ) , \\ell )$ means that $\\hat { x }$ is a reconstruction of the corrupted version of $x$ , with $x$ sampled from the monolingual dataset $\\mathcal { D } _ { \\ell }$ . In this equation, $\\Delta$ is a measure of discrepancy between the two sequences, the sum of token-level cross-entropy losses in our case. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Noise model $C ( x )$ is a randomly sampled noisy version of sentence $x$ . In particular, we add two different types of noise to the input sentence. First, we drop every word in the input sentence with a probability $p _ { w d }$ . Second, we slightly shuffle the input sentence. To do so, we apply a random permutation $\\sigma$ to the input sentence, verifying the condition $\\forall i \\in \\{ 1 , n \\} , | \\sigma ( i ) - i | \\tilde { \\leq k }$ where $n$ is the length of the input sentence, and $k$ is a tunable parameter. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To generate a random permutation verifying the above condition for a sentence of size $n$ , we generate a random vector $q$ of size $n$ , where $q _ { i } = i + U ( 0 , \\alpha )$ , and $U$ is a draw from the uniform distribution in the specified range. Then, we define $\\sigma$ to be the permutation that sorts the array $q$ . In particular, $\\alpha < 1$ will return the identity, $\\alpha = + \\infty$ can return any permutation, and $\\alpha = k + 1$ will return permutations $\\sigma$ verifying $\\forall i \\ \\in \\ \\{ 1 , n \\} , | \\sigma ( i ) - i | \\ \\leq \\ k$ . Although biased, this method generates permutations similar to the noise observed with word-by-word translation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, both the word dropout and the input shuffling strategies turned out to have a critical impact on the results, see also section 4.5, and using both strategies at the same time gave us the best performance. In practice, we found $p _ { w d } = 0 . 1$ and $k = 3$ to be good parameters. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 CROSS DOMAIN TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The second objective of our approach is to constrain the model to be able to map an input sentence from a the source/target domain $\\ell _ { 1 }$ to the target/source domain $\\ell _ { 2 }$ , which is what we are ultimately interested in at test time. The principle here is to sample a sentence $x \\in \\mathcal { D } _ { \\ell _ { 1 } }$ , and to generate a corrupted translation of this sentence in $\\ell _ { 2 }$ . This corrupted version is generated by applying the current translation model denoted $M$ to $x$ such that $y = M ( x )$ . Then a corrupted version $C ( y )$ is sampled (see Figure 1-right). The objective is thus to learn the encoder and the decoder such that they can reconstruct $x$ from $C ( y )$ . The cross-domain loss can be written as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cafdb6b4349da84733faf45700d20334d58ddac55bdc6a4e3cbe709e5aba9d72.jpg",
+        "text": "$$\n\\mathcal { L } _ { c d } \\left( \\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } , \\ell _ { 1 } , \\ell _ { 2 } \\right) = \\mathbb { E } _ { x \\sim \\mathcal { D } _ { \\ell _ { 1 } } , \\hat { x } \\sim d \\left( e \\left( C \\left( M \\left( x \\right) \\right) , \\ell _ { 2 } \\right) , \\ell _ { 1 } \\right) } \\left[ \\Delta \\left( \\hat { x } , x \\right) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            678,
+            720,
+            696
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\Delta$ is again the sum of token-level cross-entropy losses. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.5 ADVERSARIAL TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, the decoder of a neural machine translation system works well only when its input is produced by the encoder it was trained with, or at the very least, when that input comes from a distribution very close to the one induced by its encoder. Therefore, we would like our encoder to output features in the same space regardless of the actual language of the input sentence. If such condition is satisfied, our decoder may be able to decode in a certain language regardless of the language of the encoder input sentence. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note however that the decoder could still produce a bad translation while yielding a valid sentence in the target domain, as constraining the encoder to map two languages in the same feature space does not imply a strict correspondence between sentences. Fortunately, the previously introduced loss for cross-domain training in equation 2 mitigates this concern. Also, recent work on bilingual lexical induction has shown that such a constraint is very effective at the word level, suggesting that it may also work at the sentence level, as long as the two latent representations exhibit strong structure in feature space. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In order to add such a constraint, we train a neural network, which we will refer to as the discriminator, to classify between the encoding of source sentences and the encoding of target sentences (Ganin et al., 2016). The discriminator operates on the output of the encoder, which is a sequence of latent vectors $\\left( z _ { 1 } , . . . , z _ { m } \\right)$ , with $z _ { i } \\in \\mathbb { R } ^ { n }$ , and produces a binary prediction about the language of the encoder input sentence: $p _ { D } ( l | z _ { 1 } , . . . , z _ { m } ) \\propto \\prod _ { j = 1 } ^ { m } p _ { D } ( \\ell | z _ { j } )$ , with $p _ { D } : \\mathbb { R } ^ { n }  [ 0 ; 1 ]$ , where 0 corresponds to the source domain, and 1 to the target domain. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The discriminator is trained to predict the language by minimizing the following cross-entropy loss: $\\mathcal { L } _ { \\mathcal { D } } ( \\theta _ { D } | \\theta , \\mathcal { Z } ) = - \\mathbb { E } _ { ( x _ { i } , \\ell _ { i } ) } [ \\log p _ { D } ( \\ell _ { i } | e ( x _ { i } , \\ell _ { i } ) ) ]$ , where $( x _ { i } , \\ell _ { i } )$ corresponds to sentence and language id pairs uniformly sampled from the two monolingual datasets, $\\theta _ { D }$ are the parameters of the discriminator, $\\theta _ { \\mathrm { e n c } }$ are the parameters of the encoder, and $\\mathcal { Z }$ are the encoder word embeddings. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The encoder is trained instead to fool the discriminator: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d872c755d36260a3426f62659daefbea006e5c77ab341406a7f0cda8f0413d19.jpg",
+        "text": "$$\n\\mathcal { L } _ { a d v } ( \\theta _ { \\mathrm { e n c } } , \\mathcal { Z } | \\theta _ { D } ) = - \\mathbb { E } _ { ( x _ { i } , \\ell _ { i } ) } [ \\log p _ { D } ( \\ell _ { j } | e ( x _ { i } , \\ell _ { i } ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            372,
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+            391
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $\\ell _ { j } = \\ell _ { 1 }$ if $\\ell _ { i } = \\ell _ { 2 }$ , and vice versa. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Final Objective function The final objective function at one iteration of our learning algorithm is thus: ",
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3ed7c6327b6b1937c822b7f1569d4d024317097dab8ceaad56a15bd65e0294f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ( \\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } ) = \\lambda _ { a u t o } [ \\mathcal { L } _ { a u t o } ( \\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } , s r c ) + \\mathcal { L } _ { a u t o } ( \\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } , t g t ) ] + } \\\\ & { \\qquad \\lambda _ { c d } [ \\mathcal { L } _ { c d } ( \\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } , s r c , t g t ) + \\mathcal { L } _ { c d } ( \\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } , t g t , s r c ) ] + } \\\\ & { \\qquad \\lambda _ { a d v } \\mathcal { L } _ { a d v } ( \\theta _ { \\mathrm { e n c } } , \\mathcal { Z } | \\theta _ { D } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda _ { a u t o }$ , $\\lambda _ { c d }$ , and $\\lambda _ { a d v }$ are hyper-parameters weighting the importance of the auto-encoding, cross-domain and adversarial loss. In parallel, the discriminator loss $\\mathcal { L } _ { D }$ is minimized to update the discriminator. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we describe the overall training algorithm and the unsupervised criterion we used to select hyper-parameters. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 ITERATIVE TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The final learning algorithm is described in Algorithm 1 and the general architecture of the model is shown in Figure 2. As explained previously, our model relies on an iterative algorithm which starts from an initial translation model $M ^ { ( 1 ) }$ (line 3). This is used to translate the available monolingual data, as needed by the cross-domain loss function of Equation 2. At each iteration, a new encoder and decoder are trained by minimizing the loss of Equation 4 – line 7 of the algorithm. Then, a new translation model $M ^ { ( t + 1 ) }$ is created by composing the resulting encoder and decoder, and the process repeats. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To jump start the process, $M ^ { ( 1 ) }$ simply makes a word-by-word translation of each sentence using a parallel dictionary learned using the unsupervised method proposed by Conneau et al. (2017), which only leverages monolingual data. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The intuition behind our algorithm is that as long as the initial translation model $M ^ { ( 1 ) }$ retains at least some information of the input sentence, the encoder will map such translation into a representation in feature space that also corresponds to a cleaner version of the input, since the encoder is trained to denoise. At the same time, the decoder is trained to predict noiseless outputs, conditioned on noisy features. Putting these two pieces together will produce less noisy translations, which will enable better back-translations at the next iteration, and so on so forth. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/a6ebd6e78ac84bb7951bc3db503a594c1ee86aff87bf655cf571909182f31264.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the proposed architecture and training objectives. The architecture is a sequence to sequence model, with both encoder and decoder operating on two languages depending on an input language identifier that swaps lookup tables. Top (auto-encoding): the model learns to denoise sentences in each domain. Bottom (translation): like before, except that we encode from another language, using as input the translation produced by the model at the previous iteration (light blue box). The green ellipses indicate terms in the loss function. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Unsupervised Training for Machine Translation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: procedure TRAINING $\\mathcal { D } _ { s r c }$ , $\\mathcal { D } _ { t g t } , T )$   \n2: Infer bilingual dictionary using monolingual data (Conneau et al., 2017)   \n3: $M ^ { ( 1 ) } \\gets$ unsupervised word-by-word translation model using the inferred dictionary   \n4: for $t = 1 , T$ do   \n5: using ${ \\bf \\nabla } _ { M } ( t )$ , translate each monolingual dataset   \n6: $/ /$ discriminator training $\\&$ model training as in eq. 4   \n7: $\\theta _ { \\mathrm { d i s c r } }  \\arg \\operatorname* { m i n } \\mathcal { L } _ { D }$ , $\\theta _ { \\mathrm { e n c } } , \\theta _ { \\mathrm { d e c } } , \\mathcal { Z } \\gets \\arg \\operatorname* { m i n } \\mathcal { L }$   \n8: $M ^ { ( t + 1 ) } \\gets \\breve { e } ^ { ( t ) } \\circ d ^ { ( t ) } / /$ update MT model   \n9: end for   \n10: return M (T +1)   \n11: end procedure ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 UNSUPERVISED MODEL SELECTION CRITERION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to select hyper-parameters, we wish to have a criterion correlated with the translation quality. However, we do not have access to parallel sentences to judge how well our model translates, not even at validation time. Therefore, we propose the surrogate criterion which we show correlates well with BLEU (Papineni et al., 2002), the metric we care about at test time. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For all sentences $x$ in a domain $\\ell _ { 1 }$ , we translate these sentences to the other domain $\\ell _ { 2 }$ , and then translate the resulting sentences back to $\\ell _ { 1 }$ . The quality of the model is then evaluated by computing the BLEU score over the original inputs and their reconstructions via this two-step translation process. The performance is then averaged over the two directions, and the selected model is the one with the highest average score. ",
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+    {
+        "type": "text",
+        "text": "Given an encoder $e$ , a decoder $d$ and two non-parallel datasets $\\mathcal { D } _ { s r c }$ and $\\mathcal { D } _ { t g t }$ , we denote $M _ { s r c  t g t } ( x ) = d ( e ( x , s r c ) , t g t )$ the translation model from src to tgt, and $M _ { t g t  s r c }$ the model in the opposite direction. Our model selection criterion $M S ( e , d , \\mathcal { D } _ { s r c } , \\mathcal { D } _ { t g t } )$ is: ",
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+        "type": "equation",
+        "img_path": "images/ba1640b6a6096fd61c218dc9afe77c35d5e73d0b64cc23b1e4a122ec8216eade.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { M S ( e , d , \\mathcal { D } _ { s r c } , \\mathcal { D } _ { t g t } ) } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\mathbb { E } _ { x \\sim \\mathcal { D } _ { s r c } } [ \\mathrm { B L E U } ( x , M _ { s r c  t g t } \\circ M _ { t g t  s r c } ( x ) ) ] + } } \\\\ { { } } & { { } } & { { \\displaystyle \\frac { 1 } { 2 } \\mathbb { E } _ { x \\sim \\mathcal { D } _ { t g t } } [ \\mathrm { B L E U } ( x , M _ { t g t  s r c } \\circ M _ { s r c  t g t } ( x ) ) ] } } \\end{array}\n$$",
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+        "type": "text",
+        "text": "Figure 3 shows a typical example of the correlation between this measure and the final translation model performance (evaluated here using a parallel dataset). ",
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+        "type": "text",
+        "text": "The unsupervised model selection criterion is used both to a) determine when to stop training and b) to select the best hyper-parameter setting across different experiments. In the former case, the ",
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+        "img_path": "images/7d5c691d1839c8ac8f8acfc1059b5b6a7784330639f26875fb7813bd2392df39.jpg",
+        "image_caption": [
+            "Figure 3: Unsupervised model selection. BLEU score of the source to target and target to source models on the Multi30k-Task1 English-French dataset as a function of the number of passes through the dataset at iteration $\\mathbf { \\Omega } ( t ) \\ = \\ 1$ of the algorithm (training $M ( 2 )$ given $M ( 1 ) ,$ ). BLEU correlates very well with the proposed model selection criterion, see Equation 5. "
+        ],
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+        "type": "text",
+        "text": "Spearman correlation coefficient between the proposed criterion and BLEU on the test set is 0.95 in average. In the latter case, the coefficient is in average 0.75, which is fine but not nearly as good. For instance, the BLEU score on the test set of models selected with the unsupervised criterion are sometimes up to 1 or 2 BLEU points below the score of models selected using a small validation set of 500 parallel sentences. ",
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+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we first describe the datasets and the pre-processing we used, then we introduce the baselines we considered, and finally we report the extensive empirical validation proving the effectiveness of our method. We will release the code to the public once the revision process is over. ",
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+    {
+        "type": "text",
+        "text": "4.1 DATASETS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In our experiments, we consider the English-French and English-German language pairs, on three different datasets. ",
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+        "type": "text",
+        "text": "WMT’14 English-French We use the full training set of 36 million pairs, we lower-case them and remove sentences longer than 50 words, as well as pairs with a source/target length ratio above 1.5, resulting in a parallel corpus of about 30 million sentences. Next, we build monolingual corpora by selecting the English sentences from 15 million random pairs, and selecting the French sentences from the complementary set. The former set constitutes our English monolingual dataset. The latter set is our French monolingual dataset. The lack of overlap between the two sets ensures that there is not exact correspondence between examples in the two datasets. ",
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+        "type": "text",
+        "text": "The validation set is comprised of 3,000 English and French sentences extracted from our monolingual training corpora described above. These sentences are not the translation of each other, and they will be used by our unsupervised model selection criterion, as explained in 3.2. Finally, we report results on the full newstest2014 dataset. ",
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+        "type": "text",
+        "text": "WMT’16 English-German We follow the same procedure as above to create monolingual training and validation corpora in English and German, which results in two monolingual training corpora of 1.8 million sentences each. We test our model on the newstest2016 dataset. ",
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+    {
+        "type": "text",
+        "text": "Multi30k-Task1 The task 1 of the Multi30k dataset (Elliott et al., 2016) has 30,000 images, with annotations in English, French and German, that are translations of each other. We consider the English-French and English-German pairs. We disregard the images and only consider the parallel annotations, with the provided training, validation and test sets, composed of 29,000, 1,000 and 1,000 pairs of sentences respectively. For both pairs of languages and similarly to the WMT datasets above, we split the training and validation sets into monolingual corpora, resulting in 14,500 monolingual source and target sentences in the training set, and 500 sentences in the validation set. ",
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+    {
+        "type": "table",
+        "img_path": "images/41dee63c73cfcccc7d83e6c3356a665d98bc4ae0e157b451b323f47745269bb0.jpg",
+        "table_caption": [
+            "Table 1: Multi30k-Task1 and WMT datasets statistics. To limit the vocabulary size in the WMT en-fr and WMT de-en datasets, we only considered words with more than 100 and 25 occurrences, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MMT1 en-fr</td><td>MMT1 de-en</td><td>WMT en-fr</td><td>WMT de-en</td></tr><tr><td>Monolingual sentences</td><td>14.5k</td><td>14.5k</td><td>15M</td><td>1.8M</td></tr><tr><td>Vocabulary size</td><td>10k/11k</td><td>19k/10k</td><td>67k/78k</td><td>80k/46k</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Table 1 summarizes the number of monolingual sentences in each dataset, along with the vocabulary size. To limit the vocabulary size on the WMT en-fr and WMT de-en datasets, we only considered words with more than 100 and 25 occurrences, respectively. ",
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+    {
+        "type": "text",
+        "text": "4.2 BASELINES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Word-by-word translation (WBW) The first baseline is a system that performs word-by-word translations of the input sentences using the inferred bilingual dictionary (Conneau et al., 2017). This baseline provides surprisingly good results for related language pairs, like English-French, where the word order is similar, but performs rather poorly on more distant pairs like English-German, as can be seen in Table 2. ",
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+        "type": "text",
+        "text": "Word reordering (WR) After translating word-by-word as in WBW, here we reorder words using an LSTM-based language model trained on the target side. Since we cannot exhaustively score every possible word permutation (some sentences have about 100 words), we consider all pairwise swaps of neighboring words, we select the best swap, and iterate ten times. We use this baseline only on the WMT dataset that has a large enough monolingual data to train a language model. ",
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+        "type": "text",
+        "text": "Oracle Word Reordering (OWR) Using the reference, we produce the best possible generation using only the words given by WBW. The performance of this method is an upper-bound of what any model could do without replacing words. ",
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+        "type": "text",
+        "text": "Supervised Learning We finally consider exactly the same model as ours, but trained with supervision, using the standard cross-entropy loss on the original parallel sentences. ",
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+    {
+        "type": "text",
+        "text": "4.3 UNSUPERVISED DICTIONARY LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To implement our baseline and also to initialize the embeddings $\\mathcal { Z }$ of our model, we first train word embeddings on the source and target monolingual corpora using fastText (Bojanowski et al., 2017), and then we apply the unsupervised method proposed by Conneau et al. (2017) to infer a bilingual dictionary which can be use for word-by-word translation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since WMT yields a very large-scale monolingual dataset, we obtain very high-quality embeddings and dictionaries, with an accuracy of $8 4 . { \\bar { 4 } } 8 \\%$ and $7 7 . 2 9 \\%$ on French-English and GermanEnglish, which is on par with what could be obtained using a state-of-the-art supervised alignment method (Conneau et al., 2017). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "On the Multi30k datasets instead, the monolingual training corpora are too small to train good word embeddings (more than two order of magnitude smaller than WMT). We therefore learn word vectors on Wikipedia using fastText2. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/21f3eca102928cae7b0b4699f0a4e2f7bfe13b2a6f20035b8f42e9d4ff6628ac.jpg",
+        "table_caption": [
+            "Table 2: BLEU score on the Multi30k-Task1 and WMT datasets using greedy decoding. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Multi30k-Task1</td><td colspan=\"4\">WMT</td></tr><tr><td></td><td>en-fr</td><td>fr-en</td><td>de-en</td><td>en-de</td><td>en-fr</td><td>fr-en</td><td>de-en</td><td>en-de</td></tr><tr><td>Supervised</td><td>56.83</td><td>50.77</td><td>38.38</td><td>35.16</td><td>27.97</td><td>26.13</td><td>25.61</td><td>21.33</td></tr><tr><td>word-by-word</td><td>8.54</td><td>16.77</td><td>15.72</td><td>5.39</td><td>6.28</td><td>10.09</td><td>10.77</td><td>7.06</td></tr><tr><td>word reordering</td><td>-</td><td>-</td><td>1</td><td>-</td><td>6.68</td><td>11.69</td><td>10.84</td><td>6.70</td></tr><tr><td>oracle word reordering</td><td>11.62</td><td>24.88</td><td>18.27</td><td>6.79</td><td>10.12</td><td>20.64</td><td>19.42</td><td>11.57</td></tr><tr><td>Our model: 1st iteration</td><td>27.48</td><td>28.07</td><td>23.69</td><td>19.32</td><td>12.10</td><td>11.79</td><td>11.10</td><td>8.86</td></tr><tr><td>Our model: 2nd iteration</td><td>31.72</td><td>30.49</td><td>24.73</td><td>21.16</td><td>14.42</td><td>13.49</td><td>13.25</td><td>9.75</td></tr><tr><td>Our model: 3rd iteration</td><td>32.76</td><td>32.07</td><td>26.26</td><td>22.74</td><td>15.05</td><td>14.31</td><td>13.33</td><td>9.64</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.4 EXPERIMENTAL DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Discriminator Architecture The discriminator is a multilayer perceptron with three hidden layers of size 1024, Leaky-ReLU activation functions and an output logistic unit. Following Goodfellow (2016), we include a smoothing coefficient $s = 0 . 1$ in the discriminator predictions. ",
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+        "type": "text",
+        "text": "Training Details The encoder and the decoder are trained using Adam (Kingma & Ba, 2014), with a learning rate of 0.0003, $\\beta _ { 1 } = 0 . 5$ , and a mini-batch size of 32. The discriminator is trained using RMSProp (Tieleman & Hinton, 2012) with a learning rate of 0.0005. We evenly alternate between one encoder-decoder and one discriminator update. We set $\\lambda _ { a u t o } = \\lambda _ { c d } = \\lambda _ { a d v } = 1$ . ",
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+    {
+        "type": "text",
+        "text": "4.5 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 2 shows the BLEU scores achieved by our model and the baselines we considered. First, we observe that word-by-word translation is surprisingly effective when translating into English, obtaining a BLEU score of 16.77 and 10.09 for $f r$ -en on respectively Multi30k-Task1 and WMT datasets. Word-reordering only slightly improves upon word-by-word translation. Our model instead, clearly outperforms these baselines, even on the WMT dataset which has more diversity of topics and sentences with much more complicated structure. After just one iteration, we obtain a BLEU score of 27.48 and 12.10 for the en-fr task. Interestingly, we do even better than oracle reordering on some language pairs, suggesting that our model not only reorders but also correctly substitutes some words. After a few iterations, our model obtains BLEU of 32.76 and 15.05 on Multi30k-Task1 and WMT datasets for the English to French task, which is rather remarkable. ",
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+    {
+        "type": "text",
+        "text": "Comparison with supervised approaches Here, we assess how much labeled data are worth our two large monolingual corpora. On WMT, we trained the very same NMT architecture on both language pairs, but with supervision using various amounts of parallel data. Figure 4-right shows the resulting performance. Our unsupervised approach obtains the same performance than a supervised NMT model trained on about 100,000 parallel sentences, which is impressive. Of course, adding more parallel examples allows the supervised approach to outperform our method, but the good performance of our unsupervised method suggests that it could be very effective for low-resources languages where no parallel data are available. Moreover, these results open the door to the development of semi-supervised translation models, which will be the focus of future investigation. With a phrase-based machine translation system, we obtain 21.6 and 22.4 BLEU on WMT en- ${ \\mathcal { f } } { \\mathit { r } }$ and $f r { - } e n$ , which is better than the supervised NMT baseline we report for that same amount of parallel sentences, which is 16.8 and 16.4 respectively. However, if we train the same supervised NMT model with BPE (Sennrich et al., 2015b), we obtain 22.6 BLEU for en-fr, suggesting that our results on unsupervised machine translation could also be improved by using BPE, as this removes unknown words (about $9 \\%$ of the words in de-en are replaced by the unknown token otherwise). ",
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+        "type": "text",
+        "text": "Iterative Learning Figure 4-left illustrates the quality of the learned model after each iteration of the learning process in the language pairs of Multi30k-Task1 dataset, other results being provided in Table 2. One can see that the quality of the obtained model is high just after the first iteration of the process. Subsequent iterations yield significant gains although with diminishing returns. At iteration 3, the performance gains are marginal, showing that our approach quickly converges. ",
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+    {
+        "type": "image",
+        "img_path": "images/883b63a0e414781e577adbbaf839014448b02037b6e20d1cafd7034972cea13d.jpg",
+        "image_caption": [
+            "Figure 4: Left: BLEU as a function of the number of iterations of our algorithm on the Multi30kTask1 datasets. Right: The curves show BLEU as a function of the amount of parallel data on WMT datasets. The unsupervised method which leverages about 15 million monolingual sentences in each language, achieves performance (see horizontal lines) close to what we would obtain by employing 100,000 parallel sentences. "
+        ],
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+        "table_caption": [
+            "Table 3: Unsupervised translations. Examples of translations on the French-English pair of the Multi30k-Task1 dataset. Iteration 0 corresponds to word-by-word translation. After 3 iterations, the model generates very good translations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source Iteration O Iteration 1 Iteration 2 Iteration 3 Reference</td><td>un homme est debout pres d&#x27;une série de jeux vidéo dans un bar . a man is seated neara series of games video in a bar. a man is standing nearacloseup of other games in a bar . a man is standing near a bunch of video video game in a bar . a man is standing neara bunch of video games ina bar. a man is standing by a group of video games in a bar.</td></tr><tr><td>Source Iteration O</td><td>une femme aux cheveux roses habillée en noir parle ä un homme . a woman at hair roses dressed in black speaks to a man .</td></tr><tr><td>Iteration 1</td><td>a woman at glasses dressed in black talking to a man.</td></tr><tr><td>Iteration 2</td><td>a woman at pink hair dressed in black speaks to a man.</td></tr><tr><td>Iteration 3 Reference</td><td>a woman with pink hair dressed in black is talking to a man.</td></tr><tr><td></td><td>a woman with pink hair dressed in black talks to a man.</td></tr><tr><td>Source</td><td>une photo d&#x27;une rue bondée en ville .</td></tr><tr><td>Iteration O</td><td>a photo a street crowded in city .</td></tr><tr><td>Iteration 1</td><td>a picture of a street crowded in a city .</td></tr><tr><td>Iteration 2</td><td></td></tr><tr><td>Iteration 3</td><td>a picture of a crowded city street.</td></tr><tr><td></td><td>a picture of a crowded street in a city .</td></tr><tr><td>Reference</td><td>aview of acrowded citystreet.</td></tr><tr><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Table 3 shows examples of translations of three sentences on the Multi30k dataset, as we iterate. Iteration 0 corresponds to the word-by-word translation obtained with our cross-lingual dictionary, which clearly suffers from word order issues. We can observe that the quality of the translations increases at every iteration. ",
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+    {
+        "type": "text",
+        "text": "Ablation Study We perform an ablation study to understand the importance of the different components of our system. To this end, we have trained multiple versions of our model with some missing components: the discriminator, the cross-domain loss, the auto-encoding loss, etc. Table 4 shows that the best performance is obtained with the simultaneous use of all the described elements. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/37649b99ee66575888d8040b5bc3f90bf4c277a6442be932afce81e0be2462ed.jpg",
+        "table_caption": [
+            "Table 4: Ablation study on the Multi30k-Task1 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>en-fr</td><td>fr-en</td><td>de-en</td><td>en-de</td></tr><tr><td>Xcd=0</td><td>25.44</td><td>27.14</td><td>20.56</td><td>14.42</td></tr><tr><td>Without pretraining</td><td>25.29</td><td>26.10</td><td>21.44</td><td>17.23</td></tr><tr><td>Without pretraining, Xcd = 0</td><td>8.78</td><td>9.15</td><td>7.52</td><td>6.24</td></tr><tr><td>Without noise,C(x) = x</td><td>16.76</td><td>16.85</td><td>16.85</td><td>14.61</td></tr><tr><td>Xauto=0</td><td>24.32</td><td>20.02</td><td>19.10</td><td>14.74</td></tr><tr><td>Aadu =0</td><td>24.12</td><td>22.74</td><td>19.87</td><td>15.13</td></tr><tr><td>Full</td><td>27.48</td><td>28.07</td><td>23.69</td><td>19.32</td></tr></table>",
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The most critical component is the unsupervised word alignment technique, either in the form of a back-translation dataset generated using word-by-word translation, or in the form of pretrained embeddings which enable to map sentences of different languages in the same latent space. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "On the English-French pair of Multi30k-Task1, with a back-translation dataset but without pretrained embeddings, our model obtains a BLEU score of 25.29 and 26.10, which is only a few points below the model using all components. Similarly, when the model uses pretrained embeddings but no back-translation dataset (when $\\lambda _ { c d } = 0$ ), it obtains 25.44 and 27.14. On the other hand, a model that does not use any of these components only reaches 8.78 and 9.15 BLEU. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The adversarial component also significantly improves the performance of our system, with a difference of up to 5.33 BLEU in the French-English pair of Multi30k-Task1. This confirms our intuition that, to really benefit from the cross-domain loss, one has to ensure that the distribution of latent sentence representations is similar across the two languages. Without the auto-encoding loss (when $\\lambda _ { a u t o } = 0$ ), the model only obtains 20.02, which is 8.05 BLEU points below the method using all components. Finally, performance is greatly degraded also when the corruption process of the input sentences is removed, as the model has much harder time learning useful regularities and merely learns to copy input data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A similar work to ours is the style transfer method with non-parallel text by Shen et al. (2017). The authors consider a sequence-to-sequence model, where the latent state given to the decoder is also fed to a discriminator. The encoder is trained with the decoder to reconstruct the input, but also to fool the discriminator. The authors also found it beneficial to train two discriminators, one for the source and one for the target domain. Then, they trained the decoder so that the recurrent hidden states during the decoding process of a sentence in a particular domain are not distinguishable according to the respective discriminator. This algorithm, called Professor forcing, was initially introduced by Lamb et al. (2016) to encourage the dynamics of the decoder observed during inference to be similar to the ones observed at training time. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Similarly, Xie et al. (2017) also propose to use an adversarial training approach to learn representations invariant to specific attributes. In particular, they train an encoder to map the observed data to a latent feature space, and a model to make predictions based on the encoder output. To remove bias existing in the data from the latent codes, a discriminator is also trained on the encoder outputs to predict specific attributes, while the encoder is jointly trained to fool the discriminator. They show that the obtained invariant representations lead to better generalization on classification and generation tasks. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Before that, Hu et al. (2017) trained a variational autoencoder (Kingma & Welling, 2013) where the decoder input is the concatenation of an unstructured latent vector, and a structured code representing the attribute of the sentence to generate. A discriminator is trained on top of the decoder to classify the labels of generated sentences, while the decoder is trained to satisfy this discriminator. Because of the non-differentiability of the decoding process, at each step, their decoder takes as input the probability vector predicted at the previous step. ",
+        "bbox": [
+            174,
+            804,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Perhaps, the most relevant prior work is by He et al. (2016), who essentially optimizes directly for the model selection metric we propose in section 3.2. One drawback of their approach, which has not been applied to the fully unsupervised setting, is that it requires to back-propagate through the sequence of discrete predictions using reinforcement learning-based approaches which are notoriously inefficient. In this work, we instead propose to a) use a symmetric architecture, and b) freeze the translator from source to target when training the translator from target to source, and vice versa. By alternating this process we operate with a fully differentiable model and we efficiently converge. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the vision domain, several studies tackle the unsupervised image translation problem, where the task consists in mapping two image domains A and B, without paired supervision. For instance, in the CoGAN architecture (Liu & Tuzel, 2016), two generators are trained to learn a common representation space between two domains, by sharing some of their convolutional layers. This is similar to our strategy of sharing the LSTM weights across the source and target encoders and decoders. Liu et al. (2017) propose a similar approach, based on variational autoencoders, and generative adversarial networks (Goodfellow et al., 2014). Taigman et al. (2016) use similar approaches for emoji generation, and apply a regularization term to the generator so that it behaves like an identity mapping when provided with input images from the target domain. Zhu et al. (2017) introduced a cycle consistency loss, to capture the intuition that if an image is mapped from A to B, then from B to A, then the resulting image should be identical to the input one. ",
+        "bbox": [
+            174,
+            180,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Our approach is also reminiscent of the Fader Networks architecture (Lample et al., 2017), where a discriminator is used to remove the information related to specific attributes from the latent states of an autoencoder of images. The attribute values are then given as input to the decoder. The decoder is trained with real attributes, but at inference, it can be fed with any attribute values to generate variations of the input images. The model presented in this paper can be seen as an extension to the text domain of the Fader Networks, where the attribute is the language itself. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            444,
+            318,
+            460
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We presented a new approach to neural machine translation where a translation model is learned using monolingual datasets only, without any alignment between sentences or documents. The principle of our approach is to start from a simple unsupervised word-by-word translation model, and to iteratively improve this model based on a reconstruction loss, and using a discriminator to align latent distributions of both the source and the target languages. Our experiments demonstrate that our approach is able to learn effective translation models without any supervision of any sort. ",
+        "bbox": [
+            174,
+            474,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    },
+    {
+        "type": "text",
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Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning, pp. 1096–1103. ACM, 2008.   \nYonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. Google’s neural machine translation system: Bridging the gap between human and machine translation. arXiv preprint arXiv:1609.08144, 2016.   \nQizhe Xie, Zihang Dai, Yulun Du, Eduard Hovy, and Graham Neubig. Controllable invariance through adversarial feature learning. arXiv preprint arXiv:1705.11122, 2017.   \nH. Zheng, Y. Cheng, and Y. Liu. Maximum expected likelihood estimation for zero-resource neural machine translation. In IJCAI, 2017.   \nJun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017. ",
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parse/train/uSQQH7Fj5U/uSQQH7Fj5U.md

@@ -0,0 +1,366 @@
+# Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov space
+
+Taiji Suzuki Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan RIKEN Center for Advanced Intelligence Project, Tokyo, Japan taiji@mist.i.u-tokyo.ac.jp
+
+Atsushi Nitanda Kyushu Institute of Technology, Fukuoka, Japan RIKEN Center for Advanced Intelligence Project, Tokyo, Japan nitanda@ai.kyutech.ac.jp
+
+# Abstract
+
+Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far. We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse of dimensionality can be avoided if the smoothness of the target function is highly anisotropic. Unlike existing studies, our analysis does not require a low-dimensional structure of the input data. We also investigate the minimax optimality of deep learning and compare its performance with that of the kernel method (more generally, linear estimators). The results show that deep learning has better dependence on the input dimensionality if the target function possesses anisotropic smoothness, and it achieves an adaptive rate for functions with spatially inhomogeneous smoothness.
+
+# 1 Introduction
+
+Based on the recent literature pertaining to machine learning, deep learning has exhibited superior performance in several tasks such as image recognition (Krizhevsky et al., 2012), natural language processing (Devlin et al., 2018), and image synthesis (Radford et al., 2015). In particular, its superiority is remarkable for complicated and high-dimensional data like images. This is mainly due to its high flexibility and superior feature-extraction ability for effectively extracting the intrinsic structure of data. Its theoretical analysis also has been extensively developed considering several aspects such as expressive ability, optimization, and generalization error.
+
+Amongst representation ability analysis of deep neural networks such as universal approximation ability (Cybenko, 1989; Hornik, 1991; Sonoda & Murata, 2017), approximation theory of deep neural networks on typical function classes such as Holder, Sobolev, and Besov spaces have been¨ extensively studied. In particular, analyses of deep neural networks with the ReLU activation (Nair & Hinton, 2010; Glorot et al., 2011) have been recently developed. Schmidt-Hieber (2020) showed that the deep learning with ReLU activations can achieve the minimax optimal estimation accuracy to estimate composite functions in Holder spaces by using the approximation theory of ¨ Yarotsky (2017). Suzuki (2019) generalized this analysis to those on the Besov space and the mixed smooth
+
+Table 1: Relationship between existing research and our work. $\beta$ indicates the smoothness of the target function, $d$ is the dimensionality of input $x$ , $D$ is the dimensionality of a low-dimensional structure on which the data are distributed, and $\widetilde { \beta }$ is the average smoothness of an anisotropic Besov space (Eq. (1)).   
+
+<table><tr><td rowspan=1 colspan=1>Functionclass</td><td rowspan=1 colspan=1>Holder</td><td rowspan=1 colspan=1>Besov</td><td rowspan=1 colspan=1>mixed  smoothBesov</td><td rowspan=1 colspan=1>Holder onalow-dimensional set</td><td rowspan=1 colspan=1>anisotropicBesov</td></tr><tr><td rowspan=1 colspan=1>Author</td><td rowspan=1 colspan=1>Schmidt-Hieber(2020)</td><td rowspan=1 colspan=1>Suzuki(2019)</td><td rowspan=1 colspan=1>Suzuki (2019)</td><td rowspan=1 colspan=1>Nakada &amp; Imaizumi(2020);Schmidt-Hieber(2019); Chen et al. (2019)</td><td rowspan=1 colspan=1>This work</td></tr><tr><td rowspan=1 colspan=1>Estimationerror</td><td rowspan=1 colspan=1>O(n-2a）2β</td><td rowspan=1 colspan=1>O(n-2)2β</td><td rowspan=1 colspan=1>o(n-2β2+1×2(d-1)（u+β)log(n)  1+2β</td><td rowspan=1 colspan=1>O(n-2D)2β</td><td rowspan=1 colspan=1>2β0（n-2+1）</td></tr></table>
+
+Besov space by utilizing the techniques developed in approximation theories (Temlyakov, 1993; DeVore, 1998). It was shown that deep learning can achieve an adaptive approximation error rate that is faster than that of (non-adaptive) linear approximation methods (DeVore & Popov, 1988; DeVore et al., 1993; Dung ˜ , 2011), and it outperforms any linear estimators (including kernel ridge regression) in terms of the minimax optimal rate.
+
+From these analyses, one can see that the approximation errors and estimation errors are strongly influenced by two factors, i.e., the smoothness of the target function and the dimensionality of the input (see Table 1). In particular, they suffer from the curse of dimensionality, which is unavoidable. However, these analyses are about the worst case errors and do not exploit specific intrinsic properties of the true distributions. For example, practically encountered data usually possess low intrinsic dimensionality, i.e., data are distributed on a low dimensional sub-manifold of the input space (Tenenbaum et al., 2000; Belkin & Niyogi, 2003). Recently, Nakada & Imaizumi (2020); Schmidt-Hieber (2019); Chen et al. (2019); Chen et al. (2019) have shown that deep ReLU network has adaptivity to the intrinsic dimensionality of data and can avoid curse of dimensionality if the intrinsic dimensionality is small. However, one drawback is that they assumed exact low dimensionality of the input data. This could be a strong assumption because practically observed data are always noisy, and injecting noise immediately destroys the low-dimensional structure. Therefore, we consider another direction in this paper. In terms of curse of dimensionality, Suzuki (2019) showed that deep learning can alleviate the curse of dimensionality to estimate functions in a so called mixed smooth Besov space (m-Besov). However, m-Besov space assumes strong smoothness toward all directions uniformly and does not include the ordinary Besov space as a special case. Moreover, the convergence rate includes heavy poly-log term which is not negligible (see Table 1).
+
+In practice, one of the typically expected properties of a true function on high-dimensional data is that it is invariant against perturbations of an input in some specific directions (Figure 1). For example, in image-recognition tasks, the target function must be invariant against the spatial shift of an input image, which is utilized by data-augmentation techniques (Simard et al., 2003; Krizhevsky et al., 2012). In this paper, we investigate the approximation and estimation abilities of deep learning on anisotropic Besov spaces (Nikol’skii, 1975; Vybiral, 2006; Triebel, 2011) (also called dominated mixed-smooth Besov spaces). An anisotropic Besov space is a set of functions that have “direction-dependent” smoothness, whereas ordinary function spaces such as Holder, Sobolev, and ¨ Besov spaces assume isotropic smoothness that is uniform in all directions. We consider a composition of functions included in an anisotropic Besov space, including several existing settings as special cases; it includes analyses of the Holder space ¨ Schmidt-Hieber (2020) and Besov space Suzuki (2019), as well as the low-dimensional sub-manifold setting (Nakada & Imaizumi, 2020; Schmidt-Hieber, 2019; Chen et al., 2019; Chen et al., 2019)1. By considering such a space, we can show that deep learning can alleviate curse of dimensionality if the smoothness in each direction is highly anisotropic. Interestingly, any linear estimator (including kernel ridge regression) has worse dependence on the dimensionality than deep learning. Our contributions can be summarized as follows:
+
+• We consider a situation in which the target function is included in a class of anisotropic Besov spaces and show that deep learning can avoid the curse of dimensionality even if the input data do not lie on a low-dimensional manifold. Moreover, deep learning can achieve the optimal adaptive approximation error rate and minimax optimal estimation error rate.
+
+• We compare deep learning with general linear estimators (including kernel methods) and show that deep learning has better dependence on the input dimensionality than linear estimators.
+
+# 2 Problem setting and the model
+
+In this section, we describe the problem setting considered in this work. We consider the following nonparametric regression model:
+
+$$
+y _ { i } = f ^ { \mathrm { o } } ( x _ { i } ) + \xi _ { i } \quad ( i = 1 , \ldots , n ) ,
+$$
+
+where $x _ { i }$ is generated from a probability distribution $P _ { X }$ on $[ 0 , 1 ] ^ { d } , \ \xi _ { i } \ \sim \ N ( 0 , \sigma ^ { 2 } )$ , and the data $D _ { n } \ = \ ( x _ { i } , y _ { i } ) _ { i = 1 } ^ { n }$ are independently identically distributed. $f ^ { \mathrm { o } }$ is the true function that we want to estimate. We are interested in the mean squared estimation error of an estimator $\hat { f }$ : $\mathrm { E } _ { D _ { n } } [ \| \widehat { f } - f ^ { \mathrm { o } } \| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ]$ , where $\operatorname { E } _ { D _ { n } } [ . ]$ indicates the expectation with respect to the training data $D _ { n }$ . We consider a least-squares estimator in the deep neural network model as $\widehat { f }$ (see Eq. (5)) and discuss its optimality. More specifically, we investigate how the “intrinsic dimensionality” of data affects the estimation accuracy of deep learning. For this purpose, we consider an anisotropic Besov space as a model of the target function.
+
+# 2.1 Anisotropic Besov space
+
+In this section, we introduce the anisotropic Besov which was investigated as the model of the true function in this paper. Throughout this paper, we set the domain of the input to $\Omega = [ 0 , 1 ] ^ { d }$ . For a function $f : \Omega \to \mathbb { R }$ , let $\begin{array} { r } { \| f \| _ { p } : = \| f \| _ { L ^ { p } ( \Omega ) } : = ( \int _ { \Omega } | f | ^ { p } \mathrm { d } x ) ^ { 1 / p } } \end{array}$ for $0 < p < \infty$ . For $p = \infty$ , we define $\| f \| _ { \infty } : = \| f \| _ { L ^ { \infty } ( \Omega ) } : = \operatorname* { s u p } _ { x \in \Omega } | f ( x ) |$ . For $\beta \in \mathbb { R } _ { + + } ^ { d }$ , let $\begin{array} { r } { | \beta | = \sum _ { j = 1 } ^ { d } | \beta _ { j } | ^ { 2 } } \end{array}$ .
+
+For a function $f :  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ , we define the $r$ th difference of $f$ in the direction $h \in \mathbb { R } ^ { d }$ as
+
+$$
+\Delta _ { h } ^ { r } ( f ) ( x ) : = \Delta _ { h } ^ { r - 1 } ( f ) ( x + h ) - \Delta _ { h } ^ { r - 1 } ( f ) ( x ) , \Delta _ { h } ^ { 0 } ( f ) ( x ) : = f ( x ) ,
+$$
+
+for $x \in \Omega$ with $x + r h \in \Omega$ , otherwise, let $\Delta _ { h } ^ { r } ( f ) ( x ) = 0$ .
+
+Definition 1. For a function $f \in L ^ { p } ( \Omega )$ where $p \in ( 0 , \infty ]$ , the $r$ -th modulus of smoothness of $f$ is defined by $\begin{array} { r } { w _ { r , p } ( f , t ) = \operatorname* { s u p } _ { h \in \mathbb { R } ^ { d } : | h _ { i } | \leq t _ { i } } \| \Delta _ { h } ^ { r } ( f ) \| _ { p } , f o r t = ( t _ { 1 } , \ldots , t _ { d } ) , t _ { i } > 0 . } \end{array}$
+
+With this modulus of smoothness, we define the anisotropic Besov space $B _ { p , q } ^ { \beta } ( \Omega )$ for $\beta \ =$ $( \beta _ { 1 } , \ldots , \beta _ { d } ) ^ { \top } \in \mathbb { R } _ { + + } ^ { d }$ as follows.
+
+Definition 2 (Anisotropic Besov space $( B _ { p , q } ^ { \beta } ( \Omega ) ) )$ . For $0 < p , q \le \infty$ , $\beta = ( \beta _ { 1 } , \ldots , \beta _ { d } ) ^ { \top } \in \mathbb { R } _ { + + } ^ { d } ,$ r := maxibβic + 1, let the seminorm | · |Bαp,q be
+
+$$
+| f | _ { B _ { p , q } ^ { \beta } } : = \left\{ \begin{array} { l l } { \left( \sum _ { k = 0 } ^ { \infty } [ 2 ^ { k } w _ { r , p } ( f , ( 2 ^ { - k / \beta _ { 1 } } , \ldots , 2 ^ { - k / \beta _ { d } } ) ) ] ^ { q } \right) ^ { 1 / q } } & { ( q < \infty ) , } \\ { \operatorname* { s u p } _ { k \geq 0 } 2 ^ { k } w _ { r , p } ( f , ( 2 ^ { - k / \beta _ { 1 } } , \ldots , 2 ^ { - k / \beta _ { d } } ) ) } & { ( q = \infty ) . } \end{array} \right.
+$$
+
+The norm of the anisotropic Besov space $B _ { p , q } ^ { \beta } ( \Omega )$ is defined by $\lVert f \rVert _ { B _ { p , q } ^ { \beta } } : = \lVert f \rVert _ { p } + \lvert f \rvert _ { B _ { p , q } ^ { \beta } }$ , and $B _ { p , q } ^ { \beta } ( \Omega ) = \{ f \in L ^ { p } ( \Omega ) \mid \| f \| _ { B _ { p , q } ^ { \beta } } < \infty \} .$ .
+
+Roughly speaking $\beta$ represents the smoothness in each direction. If $\beta _ { i }$ is large, then a function in $B _ { p , q } ^ { \beta }$ is smooth to the $i$ th coordinate direction, otherwise, it is non-smooth to that direction. $p$ is also an important quantity that controls the spatial inhomogeneity of the smoothness. If $\beta _ { 1 } = \beta _ { 2 } = \cdot \cdot \cdot =$ $\beta _ { d }$ , then the definition is equivalent to the usual Besov space (DeVore & Popov, 1988; DeVore et al., 1993). Suzuki (2019) analyzed curse of dimensionality of deep learning through a so-called mixed smooth Besov (m-Besov) space which imposes a stronger condition toward all directions uniformly.
+
+Particularly, it imposes stronger smoothness toward non-coordinate axis directions. Moreover, mBesov space does not include the vanilla Besov space as a special case and thus cannot capture the situation that we consider in this paper.
+
+Throughout this paper, for given $\beta = ( \beta _ { 1 } , \ldots , \beta _ { d } ) ^ { \top } \in \mathbb { R } _ { + + } ^ { d }$ , we write $\underline { { \beta } } : = \operatorname* { m i n } _ { i } \beta _ { i }$ (smallest smoothness) and $\overline { { \beta } } : = \operatorname* { m a x } _ { i } \beta _ { i }$ (largest smoothness). The approximation error of a function in anisotropic Besov spaces is characterized by the harmonic mean of $( \beta _ { j } ) _ { j = 1 } ^ { d }$ , which corresponds to the average smoothness, and thus we define
+
+$$
+\begin{array} { r } { \widetilde { \beta } : = \left( \sum _ { j = 1 } ^ { d } 1 / \beta _ { j } \right) ^ { - 1 } . } \end{array}
+$$
+
+The Besov space is closely related to other function spaces such as Holder space. Let ¨ $\partial ^ { \alpha } f ( x ) =$ $\frac { \partial ^ { | \alpha | } f } { \partial ^ { \alpha _ { 1 } } x _ { 1 } . . . \partial ^ { \alpha _ { d } } x _ { d } } ( x )$ .
+
+Definition 3 (Holder space¨ $( \mathscr { C } ^ { \beta } ( \Omega ) ) )$ . For a smoothness paraemter $\beta \in \mathbb { R } _ { + + }$ with $\beta \notin \mathbb { N }$ , consider an $m$ times differentiable function $f ~ : ~ \mathbb { R } ^ { d } ~ \to ~ \mathbb { R }$ where $m ~ = ~ \left\lfloor \beta \right\rfloor$ (the largest integer less than $\beta$ ), and let the norm of the Holder space ¨ $\mathscr { C } ^ { \beta } ( \Omega )$ be $\| f \| _ { \mathcal { C } ^ { \beta } } : = \operatorname* { m a x } _ { | \alpha | \leq m } \| \partial ^ { \alpha } f \| _ { \infty } +$ $\begin{array} { r l } & { \operatorname* { m a x } _ { | \alpha | = m } \operatorname* { s u p } _ { x , y \in \Omega } \frac { | \partial ^ { \alpha } f ( x ) - \partial ^ { \alpha } f ( y ) | } { \| x - y \| ^ { \beta - m } } } \end{array}$ |∂αf(x)−∂αf(y)|kx−ykβ−m . Then, (β-)Holder space ¨ Cβ(Ω) is defined as Cβ(Ω) = {f | $\| f \| _ { C ^ { \beta } } < \infty \}$ .
+
+Let ${ \mathcal { C } } ^ { 0 } ( \Omega )$ be the set of continuous functions equipped with $L ^ { \infty }$ -norm: $\mathcal { C } ^ { 0 } ( \Omega ) : = \{ f : \Omega \to \mathbb { R } \ |$ $f$ is continuous and $\| f \| _ { \infty } < \infty \}$ . These function spaces are closely related to each other.
+
+Proposition 1 (Triebel (2011)). There exist the following relations between the spaces:
+
+1. For $\beta = ( \beta _ { 0 } , \ldots , \beta _ { 0 } ) ^ { \intercal } \in \mathbb { R } ^ { d }$ with $\beta _ { 0 } \notin \mathbb { N } ,$ , it holds that $\mathcal { C } ^ { \beta _ { 0 } } ( \Omega ) = B _ { \infty , \infty } ^ { \beta } ( \Omega )$ . 2. For $0 < p _ { 1 } , p _ { 2 } , q \le \infty$ , $p _ { 1 } \leq p _ { 2 }$ and $\beta \in \mathbb { R } _ { + + } ^ { d }$ with $\widetilde { \beta } > ( 1 / p _ { 1 } - 1 / p _ { 2 } ) _ { + } { } ^ { 3 }$ , it holds that4 $B _ { p _ { 1 } , q } ^ { \beta } ( \Omega ) \hookrightarrow B _ { p _ { 2 } , q } ^ { \gamma \beta } ( \Omega ) f o r \gamma = 1 - ( 1 / p _ { 1 } - 1 / p _ { 2 } ) _ { + } / \tilde { \beta }$ . 13. For $0 < p , q _ { 1 } , q _ { 2 } \leq \infty$ , $q _ { 1 } < q _ { 2 }$ , and $\beta \in \mathbb { R } _ { + + } ^ { d }$ , it holds that $B _ { p , q _ { 1 } } ^ { \beta } \hookrightarrow B _ { p , q _ { 2 } } ^ { \beta }$ . In particular, with properties $^ { l }$ and 2, $i f \widetilde { \beta } > 1 / p ,$ , it holds that $B _ { p , q } ^ { \beta } ( \Omega ) \hookrightarrow \mathcal { C } ^ { \gamma \beta } ( \Omega )$ where $\gamma = 1 - 1 / ( \widetilde { \beta } p )$ . 4. For $0 < p , q \le \infty$ and $\beta \in \mathbb { R } _ { + + } ^ { d }$ , $i f { \widetilde \beta } > 1 / p$ , then $B _ { p , q } ^ { \beta } ( \Omega ) \hookrightarrow \mathcal { C } ^ { 0 } ( \Omega )$ .
+
+This result is basically proven by Triebel (2011). For completeness, we provide its derivation in Appendix D. If the average smoothness $\widetilde { \beta }$ is sufficiently large $( \widetilde { \beta } > 1 / \bar { p } )$ , then the functions in $B _ { p , q } ^ { \beta }$ are continuous; however, if it is small $( { \widetilde \beta } < 1 / p )$ , then they are no longer continuous. Small $p$ indicates spatially inhomogeneous smoothness; thus, spikes and jumps appear (see Donoho & Johnstone (1998) for this perspective, from the viewpoint of wavelet analysis).
+
+# 2.2 Model of the true function
+
+As a model of the true function $f ^ { \mathrm { o } }$ , we consider two types of models: Affien composition model and deep composition model. For a Banach space $\mathcal { H }$ , we let $U ( \mathcal { H } )$ be the unit ball of $\mathcal { H }$ .
+
+(a) Affine composition model: The first model we introduced is a very naive model which is just a composition of an affine transformation and a function in the anisotropic Besov space:
+
+$$
+\begin{array} { r l } & { \mathcal { H } _ { \mathrm { a f f } } : = \{ h ( A x + b ) ~ \vert ~ h \in U ( B _ { p , q } ^ { \beta } ( [ 0 , 1 ] ^ { \bar { d } } ) ) , ~ A \in \mathbb { R } ^ { \bar { d } \times d } , ~ b \in \mathbb { R } ^ { b } \mathrm { ~ s . t . ~ } A x + b \in [ 0 , 1 ] ^ { \bar { d } } \left( \forall x \in \Omega \right) \} , } \end{array}
+$$
+
+where we assume $\tilde { d } \leq d$ . Here, we assumed that the affine transformation has an appropriate scaling such that $A x + b$ is included in the domain of $h$ for all $x \in \Omega$ . This is a quite naive model but provides an instructive example to understand how the estimation error of deep learning behaves under the anisotropic setting.
+
+(b) Deep composition model: The deep composition model generalizes the affine composition model to a composition of nonlinear functions. Let $m _ { 1 } = d , m _ { L + 1 } = 1 , m _ { \ell }$ be the dimension of the \`th layer, and let $\beta ^ { ( \ell ) } \in \mathbb { R } _ { + + } ^ { m _ { \ell } }$ be the smoothness parameter in the \`th layer. The deep composition model is defined as
+
+![](images/4d5b1a3f7398cfbdef60e024a372562d2899f3a8e5c91cb51a1858004d8d82b9.jpg)  
+Figure 1: Near low dimensional data distribution with anisotropic smoothness of the target function. The target function has less smoothness $( s _ { 1 } , s _ { 2 } )$ toward the first two coordinates on the manifold while it is almost constant toward the third coordinate (large $s _ { 3 }$ ).
+
+$$
+H _ { \mathrm { d e e p } } : = \{ h _ { H } \circ \cdot \cdot \circ h _ { 1 } ( x ) \mid h _ { \ell } : [ 0 , 1 ] ^ { m _ { \ell } } \} [ 0 , 1 ] ^ { m _ { \ell + 1 } } , h _ { \ell , k } \in U ( B _ { p , q } ^ { \beta ^ { ( \ell ) } } ( [ 0 , 1 ] ^ { m _ { \ell } } ) ) \left( \forall k \in [ m _ { \ell + 1 } ] \right) \} .
+$$
+
+Here, the interval ference can be abs $[ 0 , 1 ]$ can be replaced by another compact interval, by changing a scaling factor. The assumption $\| h _ { \ell , k } \| _ { B _ { p , q } ^ { \beta ^ { ( \ell ) } } } \leq 1$ $[ a _ { \ell } , b _ { \ell } ]$ but this dif-can also be affine composition model as a special case. However, it requires a stronger assumption to properly evaluate the estimation error on this model.
+
+Examples The model we have introduced includes some instructive examples as listed below:
+
+(a) Linear projection Schmidt-Hieber (2020) analyzed estimation of the following model by deep learning: $\bar { f ^ { \mathrm { o } } } ( \bar { x } ) = g ( w ^ { \top } x )$ where $g \in \mathcal { C } ^ { \beta } ( [ 0 , 1 ] )$ and $w \in \mathbb { R } ^ { d }$ . In this example, the function $f ^ { \mathrm { o } }$ varies along only one direction, $w$ . Apparently, this is an example of the affine composition model.
+
+(b) Distribution on low dimensional smooth manifold Assume that the input $x$ is distributed on a low-dimensional smooth manifold embedded in $\Omega$ , and the smoothness of the true function $f ^ { \mathrm { o } }$ is anisotropic along a coordinate direction on the manifold. We suppose that the low dimensional manifold is $\tilde { d }$ -dimensional and $\tilde { d } \ll d$ . In this situation, the true function can be written as $f ^ { \mathrm { o } } ( x ) = h ( \phi ( x ) )$ where $\phi : \mathbb { R } ^ { d }  \mathbb { R } ^ { \tilde { d } }$ is a map that returns the coordinate of $x$ on the manifold and $h$ is an element in an anisotropic Besov space on $\mathbb { R } ^ { \tilde { d } }$ . This situation appears if data is distributed on a low-dimensional sub-manifold of $\Omega$ and the target function is invariant against noise injection to some direction on the manifold at each input point $x$ (Figure 1 illustrates this situation). One typical example of this situation is a function invariant with data augmentation (Simard et al., 2003; Krizhevsky et al., 2012). Even if the noise injection destroys low dimensionality of the data distribution (i.e., $\bar { \tilde { d } } = d _ { \ast }$ , an anisotropic smoothness of the target function eases the curse of dimensionality as analyzed below, which is quite different from existing works (Yang & Dunson, 2016; Bickel & Li, 2007; Nakada & Imaizumi, 2020; Schmidt-Hieber, 2019; Chen et al., 2019; Chen et al., 2019).
+
+Related work Here, we introduce some more related work and discuss their relation to our analysis. The statistical analysis on an anisotropic Besov space can be back to Ibragimov & Khas’minskii (1984) who considered density estimation, where the density is assumed to be included in an anisotropic Sobolev space with $p \geq 2$ , and derived the minimax optimal rate $n ^ { - r \widetilde { \beta } / ( 2 \widetilde { \beta } + 1 ) }$ with respect to $L ^ { r }$ -norm. Nyssbaum (1983, 1987) analyzed a nonparametric regression problem on an anisotropic Besov space. Following these results, several studied have been conducted in the literature pertaining to nonparametric statistics, such as nonlinear kernel estimator Kerkyacharian et al. (2001), adaptive confidence band construction Hoffman & Lepski (2002), optimal aggregation Gaiffas & Lecue (2011), Gaussian process estimator Bhattacharya et al. (2011, 2014), and kernel ridge regression Hang & Steinwart (2018). Basically, these studies investigated estimation problems in which the target function is in anisotropic Besov spaces, but the composition models considered in this paper have not been analyzed. Hoffman & Lepski (2002); Bhattacharya et al. (2011) considered a dimension reduction model; that is, the target function is dependent on only a few variables of $x$ , but they did not deal with more general models, such as the affine/deep composition models. The nonparametric regression problems where the input data are distributed on a low-dimensional smooth manifold has been studied as a “manifold regression” Yang & Dunson (2016); Bickel & Li (2007); Yang & Tokdar (2015). Such a model can be considered as a specific example of the deep composition model. In this sense, our analysis is a significant extension of these analyses.
+
+# 3 Approximation error analysis
+
+Here, we consider approximating the true function $f ^ { \mathrm { o } }$ via a deep neural network and derive the approximation error. As the activation function, we consider the ReLU activation denoted by $\eta ( x ) = \operatorname* { m a x } \{ x , 0 \}$ $( x \in \mathbb { R }$ ). For a vector $x$ , $\eta ( x )$ is operated in an element-wise manner. The model of neural networks with height $L$ , width $W$ , sparsity constraint $S$ , and norm constraint $B$ as $\Phi ( L , W , S , B ) : = \{ ( \mathcal { W } ^ { ( L ) } \eta ( \cdot ) + \bar { b ^ { ( L ) } } ) \circ \cdot \cdot \cdot \circ ( \mathcal { W } ^ { ( 1 ) } \bar { x } + b ^ { ( 1 ) } ) \ | \ \mathcal { W } ^ { ( L ) } \in \mathbb { R } ^ { 1 \times W } .$ , $b ^ { ( L ) } \in \mathbb { R }$ , $\mathcal { W } ^ { ( 1 ) } \in$ $\mathbb { R } ^ { W \times d }$ , $b ^ { ( 1 ) } \in \mathbb { R } ^ { W } , \mathcal { W } ^ { ( \ell ) } \in \mathbb { R } ^ { W \times W }$ , $b ^ { ( \ell ) } \in \mathbb { R } ^ { W } ( 1 < \ell < L ) , \sum _ { \ell = 1 } ^ { L } ( \| \mathcal { W } ^ { ( \ell ) } \| _ { 0 } + \| b ^ { ( \ell ) } \| _ { 0 } ) \le$ $S , \operatorname* { m a x } _ { \ell } \| \mathcal W ^ { ( \ell ) } \| _ { \infty } \vee \| b ^ { ( \ell ) } \| _ { \infty } \le B \}$ , where $\| \cdot \| _ { 0 }$ is the $\ell _ { 0 }$ -norm of the matrix (the number of nonzero elements of the matrix), and $\| \cdot \| _ { \infty }$ is the $\ell _ { \infty }$ -norm of the matrix (maximum of the absolute values of the elements). The sparsity constraint and norm bounds are required to obtain the nearoptimal rate of the estimation error. To evaluate the accuracy of the deep neural network model in approximating target functions, we define the worst-case approximation error as
+
+$$
+\begin{array} { r } { R _ { r } ( \mathcal { F } , \mathcal { H } ) : = \operatorname* { s u p } _ { f ^ { * } \in \mathcal { H } } \operatorname* { i n f } _ { f \in \mathcal { F } } \| f ^ { * } - f \| _ { L ^ { r } ( \Omega ) } , } \end{array}
+$$
+
+where $\mathcal { F }$ is the set of functions used for approximation, and $\mathcal { H }$ is the set of target functions.
+
+Proposition 2 (Approximation ability for anisotropic Besov space). Suppose that $0 < p , q , r \leq \infty$ and $\beta \in \mathbb { R } _ { + + } ^ { d }$ satisfy the following condition: $\widetilde { \beta } > ( 1 / p - 1 / r ) _ { + }$ . Assume that $m \in \mathbb { N }$ satisfies $0 < \overline { { \beta } } < \operatorname* { m i n } ( m , m - 1 + 1 / p )$ . Let $\delta = ( 1 / p - 1 / r ) _ { + }$ , $\nu = ( \widetilde { \beta } - \delta ) / ( 2 \delta )$ and $W _ { 0 } ( d ) : =$ $6 d m ( m + 2 ) + 2 d$ . Then, for $N \in \mathbb N$ , we can bound the approximation error as
+
+$$
+R _ { r } ( \Phi ( L _ { 1 } , W _ { 1 } , S _ { 1 } , B _ { 1 } ) , U ( B _ { p , q } ^ { \beta } ( \Omega ) ) ) \lesssim N ^ { - \widetilde { \beta } } ,
+$$
+
+by setting
+
+$$
+\begin{array} { r l } & { L _ { 1 } ( d ) : = 3 + 2 \lceil \log _ { 2 } \left( \frac { 3 ^ { d \vee m } } { \epsilon c _ { ( d , m ) } } \right) + 5 \rceil \lceil \log _ { 2 } ( d \vee m ) \rceil , W _ { 1 } ( d ) : = N W _ { 0 } , } \\ & { S _ { 1 } ( d ) : = \lbrack ( L - 1 ) W _ { 0 } ^ { 2 } + 1 \rbrack N , B _ { 1 } ( d ) : = O ( N ^ { d ( 1 + \nu ^ { - 1 } ) ( 1 / p - \widetilde { \beta } ) _ { + } } ) , } \end{array}
+$$
+
+for $\epsilon = N ^ { - \widetilde { \beta } } \log ( N ) ^ { - 1 }$ and a constant $c _ { ( d , m ) }$ depending only on $d$ and $m$
+
+The proof of this proposition is provided in Appendix B. The rate $N ^ { - \widetilde { \beta } }$ is the optimal adaptive approximation error rate that can be achieved by a model with $N$ parameters (the difference between adaptive and non-adaptive methods is explained in the discussion below). Note that this is an approximation error in an oracle setting and no sample complexity appears here. We notice that we can avoid the curse of dimensionality if the average smoothness $\widetilde { \beta }$ is small. This means that if the target function is non-smooth in only a few directions and smooth in other directions, we can avoid the curse of dimensionality. In contrast, if we consider an isotropic Besov space where $\beta _ { 1 } = \cdots = \beta _ { d } ( = \underline { { \beta } } )$ , then $\widetilde { \beta } = \underline { { \beta } } / d$ , which directly depends on the dimensionality $d$ , and we need an exponentially large number of parameters in this situation to achieve $\epsilon$ -accuracy. Therefore, the anisotropic smoothness has a significant impact on the approximation error rate. The assumption $\widetilde { \beta } > ( 1 / p - 1 / r ) _ { + }$ ensures the $L _ { r }$ -integrability of the target function, and the inequality (without equality) admits a near-optimal wavelet approximation of the target function in terms of $L _ { r }$ -norm.
+
+Using this evaluation as a basic tool, we can obtain the approximation error for the deep composition models. We can also obtain the approximation error for the affine composition models, but it is almost identical to Proposition 2. Therefore, we defer it to Appendix A.
+
+Theorem 1 (Deep composition model). Assume that $\widetilde { \beta } ^ { ( \ell ) } > 1 / p$ for all $\ell = 1 , \ldots , H$ . Then, the estimation error on the deep composition model is bounded as
+
+$$
+R _ { \infty } ( \Phi ( L , W , S , B ) , \mathcal { H } _ { \mathrm { d e e p } } ) \lesssim \operatorname* { m a x } _ { \ell \in [ H ] } N ^ { - \widetilde { \beta } ^ { * ( \ell ) } } ,
+$$
+
+$$
+\begin{array} { r l } & { v h e r e ~ \widetilde { \beta } ^ { * } ^ { ( \ell ) } = \widetilde { \beta } ^ { ( \ell ) } \prod _ { k = \ell + 1 } ^ { H } [ ( \underline { { \beta } } ^ { ( k ) } - 1 / p ) \wedge 1 ] , a n d \ L = \sum _ { \ell = 1 } ^ { H } ( L _ { 1 } ( m _ { \ell } ) + 1 ) , W = \operatorname* { m a x } _ { \ell } ( W _ { 1 } ( m _ { \ell } ) \vee \ell ) } \\ & { n _ { \ell + 1 } ) , S = \sum _ { \ell = 1 } ^ { H } ( S _ { 1 } ( m _ { \ell } ) + 3 m _ { \ell + 1 } ) , B = \operatorname* { m a x } _ { \ell } B _ { 1 } ( m _ { \ell } ) . } \end{array}
+$$
+
+The proof can be found in Appendix B.1. Since the model is more general than the vanilla anisotropic Besov space, we require a stronger assumption $\widetilde { \beta } ^ { ( \ell ) } > 1 / p$ on $\widetilde { \beta } ^ { \left( \ell \right) }$ than the condition in Proposition 2. This is because we need to bound the Holder smoothness of the remaining ¨ layers to bound the influence of the approximation error in the internal layers to the entire function. Holder smoothness is ensured according to the embedding property under this condition (Proposi- ¨ tion 1). This Holder smoothness assumption affects the approximation error rate. The convergence ¨ rate ${ \widetilde { \beta } } ^ { * ( \ell ) }$ in Eq. (4) is different from that in Eq. (8). This is because the approximation error in the internal layers are propagated through the remaining layers with Holder smoothness and its ¨ amplitude is controlled by the Holder smoothness. ¨
+
+Approximation error by non-adaptive method The approximation error obtained in the previous section is called an adaptive error rate in the literature regarding approximation theory (DeVore, 1998). If we fix $N$ bases beforehand and approximate the target function by a linear combination of the $N$ bases (which is called the non-adaptive method), then we cannot achieve the adaptive error rate obtained in the previous section. Roughly speaking, the approximation error of non-adaptive methods is lower bounded by $\begin{array} { r } { N ^ { - \left( \widetilde { \beta } - \left( \frac { 1 } { p } - \frac { 1 } { \operatorname* { m i n } \left\{ 2 , r \right\} } \right) + \right) } } \end{array}$ (Myronyuk, 2015, 2016, 2017), which is slower than the approximation error rate of deep neural networks especially for small $p$ .
+
+# 4 Estimation error analysis
+
+In this section, we analyze the accuracy of deep learning in estimating a function in compositions of anisotropic Besov spaces. We consider a least-squares estimator in the deep neural network model:
+
+$$
+\begin{array} { r } { \widehat { f } = \operatorname * { a r g m i n } _ { \bar { f } : f \in \Phi ( L , W , S , B ) } \sum _ { i = 1 } ^ { n } ( y _ { i } - \bar { f } ( x _ { i } ) ) ^ { 2 } } \end{array}
+$$
+
+where $\bar { f }$ is the clipping of $f$ defined by $\bar { f } = \operatorname* { m i n } \{ \operatorname* { m a x } \{ f , - F \} , F \}$ for a constant $F > 0$ which is realized by ReLU units. The network parameters $( L , W , S , B )$ should be specified appropriately as indicated in Theorems 2 and 3. In practice, these parameters can be specified by cross validation. Indeed, we can theoretically show that cross validation can provide the appropriate choice of these parameters in compensation to an additional $\log ( n )$ -factor in the estimation error bound. This estimator can be seen as a sparsely regularized estimator because there are constraints on $S$ . In terms of optimization, this requires a combinatorial optimization, but we do not pursue the computational aspect. The estimation error that we derive in this section can involve the optimization error, but for simplicity, we only demonstrate the estimation error of the ideal situation where there is no optimization error.
+
+Affine composition model The following theorem provides an upper bound of the estimation error for the affine composition model.
+
+Theorem 2. Assume the same condition as in Theorem $6$ ; in particular, suppose $0 < p , q \le \infty$ and $\widetilde { \beta } > ( 1 / p - 1 / 2 ) _ { + }$ for $\widetilde { \beta } \in \mathbb { R } _ { + + } ^ { \tilde { d } }$ . Moreover, we assume that the distribution $P _ { X }$ has a density $p _ { X }$ such that $\| p _ { X } \| _ { \infty } \leq R$ for a constant $R > 0$ . If $f ^ { \mathrm { o } } \in \mathcal { H } _ { \mathrm { a f f } } \cap L ^ { \infty } ( \Omega )$ , and $\| f ^ { \mathrm { o } } \| _ { \infty } \leq F$ for $F \geq 1$ ; then, letting $( W , L , S , B ) = \big ( L _ { 1 } ( \tilde { d } ) , W _ { 1 } ( \tilde { d } ) , S _ { 1 } ( \tilde { d } ) , ( \tilde { d } C + 1 ) B _ { 1 } ( \tilde { d } ) \big )$ as in Theorem $6$ with $N \asymp n ^ { \frac { 1 } { 2 \tilde { \beta } + 1 } }$ , we obtain
+
+$$
+\mathrm { E } _ { D _ { n } } [ \| f ^ { \mathrm { o } } - \widehat { f } \| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] \lesssim n ^ { - \frac { 2 \widetilde { \beta } } { 2 \widetilde { \beta } + 1 } } \log ( n ) ^ { 3 } ,
+$$
+
+where $\operatorname { E } _ { D _ { n } } [ \cdot ]$ indicates the expectation with respect to the training data $D _ { n }$
+
+The proof is provided in Appendix C. We will show that the convergence rate $n ^ { - 2 \widetilde { \beta } / ( 2 \widetilde { \beta } + 1 ) }$ is minimax optimal in Section 5 (see also Kerkyacharian & Picard (1992); Donoho et al. (1996); Donoho & Johnstone (1998); Gine & Nickl ´ (2015) for ordinary Besov spaces). The $L ^ { \infty }$ -norm constraint $\| f ^ { \mathrm { o } } \| _ { \infty } \leq F$ is used to derive a uniform bound on the discrepancy between the population and the empirical $L ^ { 2 }$ -norm. Without this condition, the convergence rate could be slower.
+
+Deep composition model For the deep composition model, we obtain the following convergence rate. This is an extension of Theorem 2 but requires a stronger assumption on the smoothness.
+
+Theorem 3. Suppose that $0 < p , q \le \infty$ and $\widetilde { \beta } ^ { ( \ell ) } > 1 / p$ for all $\ell \in [ H ]$ . If $f ^ { \mathrm { o } } \in \mathcal { H } _ { \mathrm { d e e p } } \cap L ^ { \infty } ( \Omega )$ and $\| f \| _ { \infty } \leq F$ for $F \geq 1$ , then we obtain
+
+$$
+\begin{array} { r } { { \mathrm { E } } _ { D _ { n } } [ \| f ^ { \mathrm { o } } - \widehat f \| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] \lesssim \operatorname* { m a x } _ { \ell \in [ H ] } n ^ { - 2 \widetilde { \beta } ^ { * ( \ell ) } / ( 2 \widetilde { \beta } ^ { * ( \ell ) } + 1 ) } \log ( n ) ^ { 3 } , } \end{array}
+$$
+
+where ${ \widetilde { \beta } } ^ { * ( \ell ) }$ is defined in Theorem 1, and $( L , W , S , B )$ is as given in Theorem 1 with $N \ \asymp$ $\begin{array} { r } { \operatorname* { m a x } _ { \ell \in [ L ] } n ^ { \frac { 1 } { 2 \widetilde { \beta } ^ { * ( \ell ) } + 1 } } } \end{array}$ .
+
+The proof is provided in Appendix C. We will show that this is also minimax optimal in Theorem 4. Because of the Holder continuity, the convergence rate becomes slower than the affine composition ¨ model (that is, $\widetilde { \beta } ^ { * ( \ell ) } \le \widetilde { \beta } ^ { ( \ell ) } )$ . However, this slower rate is unavoidable in terms of the minimax optimal rate. Schmidt-Hieber (2020) analyzed the same situation for the Holder class which corre- ¨ sponds to $\beta _ { 1 } ^ { ( \ell ) } = \cdot \cdot \cdot = \beta _ { d } ^ { ( \ell ) }$ $( \forall \ell )$ and $p = q = \infty$ . Our analysis far extends their analysis to the setting of anisotropic Besov spaces in which the parameters $\beta ^ { ( \ell ) } , p , q$ have much more freedom.
+
+From these two bounds (Theorems 2 and 3), we can see that as the smoothness $\widetilde { \beta }$ becomes large, the convergence rates faster. If the target function is included in the isotropic Besov space with smoothness $\beta _ { 1 } = \cdots = \beta _ { d } ( = \underline { { \beta } } )$ , then the estimation error becomes
+
+# (Isotropic Besov)
+
+$$
+n ^ { - 2 \underline { { \beta } } / ( 2 \underline { { \beta } } + d ) } .
+$$
+
+In the exponent, the dimensionality $d$ appears, which causes the curse of dimensionality. In contrast, if the target function is in the anisotropic Besov space, and the smoothness in each direction is sufficiently imbalanced such that $\widetilde { \beta }$ does not depend on $d$ , our obtained rate
+
+# (Anisotropic Besov)
+
+$$
+n ^ { - 2 \widetilde { \beta } / ( 2 \widetilde { \beta } + 1 ) }
+$$
+
+avoids the curse of dimensionality. For high-dimensional settings, there would be several redundant directions in which the true function does not change. Deep learning is adaptive to this redundancy and achieves a better estimation error. However, in Section 6, we prove that linear estimators are affected by the dimensionality more strongly than deep learning. This indicates the superiority of deep learning.
+
+# 5 Minimax optimal rate
+
+Here, we show that the estimation error rate, that we have presented, of deep learning achieves the minimax optimal rate. Roughly speaking the minimax optimal risk on a model ${ \mathcal { F } } ^ { \circ }$ of the true function is the smallest worst case error over all estimators:
+
+$$
+\begin{array} { r } { R _ { * } ( \mathcal { F } ^ { \circ } ) : = \operatorname* { i n f } _ { \widehat f } { \operatorname* { s u p } _ { f ^ { \circ } \in \mathcal { F } ^ { \circ } } { \operatorname E } _ { D _ { n } } [ \| \widehat f - f ^ { \circ } \| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] } , } \end{array}
+$$
+
+where $\widehat { f }$ runs over all estimators. The convergence rate of the minimax optimal risk is referred to as minimax optimal rate. We obtain the following minimax optimal rate for anisotropic Besov spaces.
+
+Theorem 4. (a) Affine composition model: For $0 < p , q \le \infty$ and $\beta \in \mathbb { R } _ { + + } ^ { d }$ , assume that $\widetilde { \beta } > \operatorname* { m a x } { \{ 1 / p - 1 / 2 , 1 / p - 1 , 0 \} }$ . Then, the minimax optimal risk of the affine composition model is lower bounded as $R _ { * } ( \mathcal { H } _ { \mathrm { a f f } } ) \gtrsim n ^ { - \frac { 2 \widetilde { \beta } } { 2 \widetilde { \beta } + 1 } }$ . (b) Deep composition model: For $0 < p , q \le \infty$ and $\beta ^ { ( \ell ) } \in \mathbb { R } _ { + + } ^ { d } \ ( \ell = 1 , \dots , H )$ , assume that $\widetilde { \beta } ^ { ( \ell ) } > 1 / p$ . Let $\epsilon > 0$ be arbitrarily small for $q < \infty$ , and let $\epsilon = 0 f o r q = 0$ . Let $\begin{array} { r } { \widetilde { \beta } ^ { * ( \ell ) } = \widetilde { \beta } ^ { ( \ell ) } \prod _ { k = \ell + 1 } ^ { H } [ ( \underline { { \beta } } ^ { ( k ) } - 1 / p + \epsilon ) \wedge 1 ] } \end{array}$ , and $\begin{array} { r } { \widetilde { \beta } ^ { * * } : = \operatorname* { m i n } _ { \ell } \widetilde { \beta } ^ { * ( \ell ) } } \end{array}$ . Then, the minimax optimal risk of the deep composition model is lower bounded as $R _ { * } ( \mathcal { H } _ { \mathrm { d e e p } } ) \gtrsim n ^ { - \frac { 2 \widetilde { \beta } ^ { * * } } { 2 \widetilde { \beta } ^ { * * } + 1 } }$ .
+
+The proof is provided in Appendix E (see also Ibragimov & Khas’minskii (1984); Nyssbaum (1987)). From this theorem, we can see that the estimation error of deep learning shown in Theorems 2 and 3 indeed achieve the minimax optimal rate up to a poly- $\log ( n )$ factor.
+
+# 6 Suboptimality of linear estimators
+
+In this section, we give the minimax optimal rate in the class of linear estimators. The linear estimator is a class of estimators that can be written as
+
+$$
+\begin{array} { r } { \widehat { f } ( x ) = \sum _ { i = 1 } ^ { n } y _ { i } \varphi _ { i } ( x ; X ^ { n } ) , } \end{array}
+$$
+
+where $X ^ { n } \ = \ ( x _ { 1 } , \ldots , x _ { n } )$ and $\varphi _ { i } ( x ; X ^ { n } ) ~ ( i = 1 , \dots , n )$ are (measurable) functions that only depend on $x$ and $X ^ { n }$ . This is linearly dependent on $Y ^ { n } = ( y _ { 1 } , \dots , y _ { n } )$ . We notice that the kernel ridge regression is included in this class because it can be written as $\widehat { f } ( x ) = k _ { x , X ^ { n } } ( k _ { X ^ { n } , X ^ { n } } +$ $\lambda \mathrm { I } ) ^ { - 1 } Y ^ { n }$ , which linearly depends on $Y ^ { n }$ . This class includes other important estimators, such as the Nadaraya–Watson estimator, the $k$ -nearest neighbor estimator, and the sieve estimator. We compare deep learning with the linear estimators in terms of minimax risk. For this purpose, we define the minimax risk of the class of linear estimators:
+
+$$
+R _ { * } ^ { ( \mathrm { l i n } ) } ( \mathcal { F } ^ { \circ } ) : = \operatorname* { i n f } _ { \widehat { f } : \mathrm { l i n e a r } f ^ { \circ } \in \mathcal { F } ^ { \circ } } \mathrm { E } _ { D _ { n } } [ \| f ^ { \mathrm { o } } - \widehat { f } \| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] ,
+$$
+
+where $\widehat { f }$ runs over all linear estimators. We can see that linear estimators suffer from the sub-optimal rate because of the following two points: (i) they do not have adaptivity, and (ii) they significantly suffer from the curse of dimensionality.
+
+Theorem 5. (i) Suppose that the input distribution $P _ { X }$ is the uniform distribution on $\Omega = [ 0 , 1 ] ^ { d }$ and assume that ${ \widetilde \beta } > 1 / p$ and $1 \leq p , q \leq \infty$ . Then, the minimax optimal rate of the linear estimators is lower bounded as
+
+$$
+R _ { * } ^ { ( \mathrm { l i n } ) } ( U ( B _ { p , q } ^ { \beta } ) ) \gtrsim n ^ { - \frac { 2 \widetilde { \beta } - v } { 2 \widetilde { \beta } + 1 - v } } ,
+$$
+
+where $v = 2 ( 1 / p - 1 / 2 ) _ { + }$
+
+(ii) In addition to the above conditions, we assume that the true function is included in the affine composition model with $\tilde { d } \leq d , \underline { { \beta } } = \beta _ { 1 } = \cdot \cdot \cdot = \beta _ { \tilde { d } }$ and $0 < p \le 2$ . Let $a _ { d } = 1 + \kappa$ with arbitrary small $\kappa > 0$ when $\tilde { d } < d / 2$ , and let $a _ { d } = 0$ when $\tilde { d } \ge d / 2$ . Then, the minimax rate of the linear estimators on the affine composition model is lower bounded by
+
+$$
+\begin{array} { r } { R _ { * } ^ { ( \mathrm { l i n } ) } ( \mathcal { H } _ { \mathrm { a f f } } ) \gtrsim n ^ { - \frac { 2 ( \underline { { \beta } } - \tilde { d } / p + d / 2 + a _ { d } ) } { 2 ( \underline { { \beta } } - \tilde { d } / p + d / 2 + a _ { d } ) + d } } . } \end{array}
+$$
+
+The proof is provided in Appendix F. (i) The lower bound (7) reveals the suboptimality of linear estimators in terms of input dimensionality. Actually, if we consider a particular case where $\tilde { d } = 1$ , $p = 1$ and $d \gg \tilde { d }$ , then the obtained minimax rate of linear estimators and the estimation error rate of deep learning can be summarized as
+
+$$
+\mathrm { l i n e a r : } \ n ^ { - \frac { 2 \beta + d } { 2 \beta + 2 d } } , \ \qquad \mathrm { d e e p : } \ n ^ { - \frac { 2 \beta } { 2 \beta + 1 } } ,
+$$
+
+by Theorem 2 when $\underline { { \beta } } > 1$ (which can be checked by noticing $\widetilde { d } = \underline { { \beta } } / \widetilde { \beta } = 1$ in this situation). We can see that the dependence on the dimensionality of linear estimators is significantly worse than that of deep leaning. This indicates poor adaptivity of linear estimators to the intrinsic dimensionality√ of data. Actually, as $d$ becomes large, the rate for the linear estimator approaches to $1 / { \sqrt { n } }$ but that for the deep learning is not affected by $d$ and still faster than $1 / \sqrt { n }$ . To show the theorem, we used the “convex-hull argument” developed by Hayakawa & Suzuki (2019); Donoho $\&$ Johnstone (1998). We combined this technique with the so-called Irie-Miyake’s integral representation (Irie & Miyake, 1988; Hornik et al., 1990). Note that this difference appears because there is an affine transformation in the first layer of the affine composition model. Deep learning is flexible against such a coordinate transform so that it can find directions to which the target function is smooth. In contrast, kernel methods do not have such adaptivity because there is no feature extraction layer. (ii) The lower bound (6) states that when $p < 2$ (that is, $v > 0$ ), the minimax rate of the linear estimators is outperformed by that of deep learning (Theorem 2). This is due to the “adaptivity” of deep leaning. When $p$ is small, the smoothness of the target function is less homogeneous, and it requires an adaptive approximation scheme to achieve the best estimation error. Linear estimators do not have adaptivity and thus fail to achieve the minimax optimal rate. Our bound (6) extends the result by Zhang et al. (2002) to a multivariate anisotropic Besov space while Zhang et al. (2002) investigated the univariate space $\langle d = 1 \rangle$ ).
+
+# 7 Conclusion
+
+We investigated the approximation error and estimation error of deep learning in the anisotropic Besov spaces. It was proved that the convergence rate is determined by the average of the anisotropic smoothness, which results in milder dependence on the input dimensionality. If the smoothness is highly anisotropic, deep learning can avoid overfitting. We also compared the error rate of deep learning with that of linear estimators and showed that deep learning has better dependence on the input dimensionality. Moreover, it was shown that deep learning can achieve the adaptive rate and outperform non-adaptive approximation methods and linear estimators if the homogeneity $p$ of smoothness is small. These analyses strongly support the practical success of deep learning from a theoretical perspective.
+
+Limitations of this work Our work does not cover the optimization aspect of deep learning. It is assumed that the regularized least squares (5) can be executed. It would be nice to combine our study with recent developments of non-convex optimization techniques (Vempala & Wibisono, 2019; Suzuki & Akiyama, 2021).
+
+Potential negative societal impact Since this is purely theoretical result, it is not expected that there is a direct negative societal impact. However, revealing detailed properties of the deep learning could promote an opportunity to pervert deep learning.
+
+# Acknowledgment
+
+TS was partially supported by JSPS KAKENHI (18H03201), Japan Digital Design and JST CREST.   
+AN was partially supported by JSPS Kakenhi (19K20337) and JST-PRESTO.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] Each theorem explicitly describes assumptions on which the theorem is established.   
+(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix in the supplementary material. All proofs are included there.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/uSQQH7Fj5U/uSQQH7Fj5U_content_list.json

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+    {
+        "type": "text",
+        "text": "Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov space ",
+        "text_level": 1,
+        "bbox": [
+            186,
+            122,
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+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Taiji Suzuki Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan RIKEN Center for Advanced Intelligence Project, Tokyo, Japan taiji@mist.i.u-tokyo.ac.jp ",
+        "bbox": [
+            232,
+            224,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Atsushi Nitanda Kyushu Institute of Technology, Fukuoka, Japan RIKEN Center for Advanced Intelligence Project, Tokyo, Japan nitanda@ai.kyutech.ac.jp ",
+        "bbox": [
+            289,
+            303,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            393,
+            535,
+            411
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far. We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse of dimensionality can be avoided if the smoothness of the target function is highly anisotropic. Unlike existing studies, our analysis does not require a low-dimensional structure of the input data. We also investigate the minimax optimality of deep learning and compare its performance with that of the kernel method (more generally, linear estimators). The results show that deep learning has better dependence on the input dimensionality if the target function possesses anisotropic smoothness, and it achieves an adaptive rate for functions with spatially inhomogeneous smoothness. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            656,
+            310,
+            674
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Based on the recent literature pertaining to machine learning, deep learning has exhibited superior performance in several tasks such as image recognition (Krizhevsky et al., 2012), natural language processing (Devlin et al., 2018), and image synthesis (Radford et al., 2015). In particular, its superiority is remarkable for complicated and high-dimensional data like images. This is mainly due to its high flexibility and superior feature-extraction ability for effectively extracting the intrinsic structure of data. Its theoretical analysis also has been extensively developed considering several aspects such as expressive ability, optimization, and generalization error. ",
+        "bbox": [
+            174,
+            688,
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+            785
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Amongst representation ability analysis of deep neural networks such as universal approximation ability (Cybenko, 1989; Hornik, 1991; Sonoda & Murata, 2017), approximation theory of deep neural networks on typical function classes such as Holder, Sobolev, and Besov spaces have been¨ extensively studied. In particular, analyses of deep neural networks with the ReLU activation (Nair & Hinton, 2010; Glorot et al., 2011) have been recently developed. Schmidt-Hieber (2020) showed that the deep learning with ReLU activations can achieve the minimax optimal estimation accuracy to estimate composite functions in Holder spaces by using the approximation theory of ¨ Yarotsky (2017). Suzuki (2019) generalized this analysis to those on the Besov space and the mixed smooth ",
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+            174,
+            791,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/f4e024592c440ca271f62c9da45aec45576d725beadad11c2cc6573e265dedb1.jpg",
+        "table_caption": [
+            "Table 1: Relationship between existing research and our work. $\\beta$ indicates the smoothness of the target function, $d$ is the dimensionality of input $x$ , $D$ is the dimensionality of a low-dimensional structure on which the data are distributed, and $\\widetilde { \\beta }$ is the average smoothness of an anisotropic Besov space (Eq. (1)). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Functionclass</td><td rowspan=1 colspan=1>Holder</td><td rowspan=1 colspan=1>Besov</td><td rowspan=1 colspan=1>mixed  smoothBesov</td><td rowspan=1 colspan=1>Holder onalow-dimensional set</td><td rowspan=1 colspan=1>anisotropicBesov</td></tr><tr><td rowspan=1 colspan=1>Author</td><td rowspan=1 colspan=1>Schmidt-Hieber(2020)</td><td rowspan=1 colspan=1>Suzuki(2019)</td><td rowspan=1 colspan=1>Suzuki (2019)</td><td rowspan=1 colspan=1>Nakada &amp; Imaizumi(2020);Schmidt-Hieber(2019); Chen et al. (2019)</td><td rowspan=1 colspan=1>This work</td></tr><tr><td rowspan=1 colspan=1>Estimationerror</td><td rowspan=1 colspan=1>O(n-2a）2β</td><td rowspan=1 colspan=1>O(n-2)2β</td><td rowspan=1 colspan=1>o(n-2β2+1×2(d-1)（u+β)log(n)  1+2β</td><td rowspan=1 colspan=1>O(n-2D)2β</td><td rowspan=1 colspan=1>2β0（n-2+1）</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Besov space by utilizing the techniques developed in approximation theories (Temlyakov, 1993; DeVore, 1998). It was shown that deep learning can achieve an adaptive approximation error rate that is faster than that of (non-adaptive) linear approximation methods (DeVore & Popov, 1988; DeVore et al., 1993; Dung ˜ , 2011), and it outperforms any linear estimators (including kernel ridge regression) in terms of the minimax optimal rate. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "From these analyses, one can see that the approximation errors and estimation errors are strongly influenced by two factors, i.e., the smoothness of the target function and the dimensionality of the input (see Table 1). In particular, they suffer from the curse of dimensionality, which is unavoidable. However, these analyses are about the worst case errors and do not exploit specific intrinsic properties of the true distributions. For example, practically encountered data usually possess low intrinsic dimensionality, i.e., data are distributed on a low dimensional sub-manifold of the input space (Tenenbaum et al., 2000; Belkin & Niyogi, 2003). Recently, Nakada & Imaizumi (2020); Schmidt-Hieber (2019); Chen et al. (2019); Chen et al. (2019) have shown that deep ReLU network has adaptivity to the intrinsic dimensionality of data and can avoid curse of dimensionality if the intrinsic dimensionality is small. However, one drawback is that they assumed exact low dimensionality of the input data. This could be a strong assumption because practically observed data are always noisy, and injecting noise immediately destroys the low-dimensional structure. Therefore, we consider another direction in this paper. In terms of curse of dimensionality, Suzuki (2019) showed that deep learning can alleviate the curse of dimensionality to estimate functions in a so called mixed smooth Besov space (m-Besov). However, m-Besov space assumes strong smoothness toward all directions uniformly and does not include the ordinary Besov space as a special case. Moreover, the convergence rate includes heavy poly-log term which is not negligible (see Table 1). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In practice, one of the typically expected properties of a true function on high-dimensional data is that it is invariant against perturbations of an input in some specific directions (Figure 1). For example, in image-recognition tasks, the target function must be invariant against the spatial shift of an input image, which is utilized by data-augmentation techniques (Simard et al., 2003; Krizhevsky et al., 2012). In this paper, we investigate the approximation and estimation abilities of deep learning on anisotropic Besov spaces (Nikol’skii, 1975; Vybiral, 2006; Triebel, 2011) (also called dominated mixed-smooth Besov spaces). An anisotropic Besov space is a set of functions that have “direction-dependent” smoothness, whereas ordinary function spaces such as Holder, Sobolev, and ¨ Besov spaces assume isotropic smoothness that is uniform in all directions. We consider a composition of functions included in an anisotropic Besov space, including several existing settings as special cases; it includes analyses of the Holder space ¨ Schmidt-Hieber (2020) and Besov space Suzuki (2019), as well as the low-dimensional sub-manifold setting (Nakada & Imaizumi, 2020; Schmidt-Hieber, 2019; Chen et al., 2019; Chen et al., 2019)1. By considering such a space, we can show that deep learning can alleviate curse of dimensionality if the smoothness in each direction is highly anisotropic. Interestingly, any linear estimator (including kernel ridge regression) has worse dependence on the dimensionality than deep learning. Our contributions can be summarized as follows: ",
+        "bbox": [
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+        "type": "text",
+        "text": "• We consider a situation in which the target function is included in a class of anisotropic Besov spaces and show that deep learning can avoid the curse of dimensionality even if the input data do not lie on a low-dimensional manifold. Moreover, deep learning can achieve the optimal adaptive approximation error rate and minimax optimal estimation error rate. ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "text",
+        "text": "• We compare deep learning with general linear estimators (including kernel methods) and show that deep learning has better dependence on the input dimensionality than linear estimators. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Problem setting and the model ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In this section, we describe the problem setting considered in this work. We consider the following nonparametric regression model: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3c7ac3f1c4062f7e3bf50cd41ba39b64148c5a16f7f9d50b5f88e36fe22d799.jpg",
+        "text": "$$\ny _ { i } = f ^ { \\mathrm { o } } ( x _ { i } ) + \\xi _ { i } \\quad ( i = 1 , \\ldots , n ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $x _ { i }$ is generated from a probability distribution $P _ { X }$ on $[ 0 , 1 ] ^ { d } , \\ \\xi _ { i } \\ \\sim \\ N ( 0 , \\sigma ^ { 2 } )$ , and the data $D _ { n } \\ = \\ ( x _ { i } , y _ { i } ) _ { i = 1 } ^ { n }$ are independently identically distributed. $f ^ { \\mathrm { o } }$ is the true function that we want to estimate. We are interested in the mean squared estimation error of an estimator $\\hat { f }$ : $\\mathrm { E } _ { D _ { n } } [ \\| \\widehat { f } - f ^ { \\mathrm { o } } \\| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ]$ , where $\\operatorname { E } _ { D _ { n } } [ . ]$ indicates the expectation with respect to the training data $D _ { n }$ . We consider a least-squares estimator in the deep neural network model as $\\widehat { f }$ (see Eq. (5)) and discuss its optimality. More specifically, we investigate how the “intrinsic dimensionality” of data affects the estimation accuracy of deep learning. For this purpose, we consider an anisotropic Besov space as a model of the target function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 Anisotropic Besov space ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we introduce the anisotropic Besov which was investigated as the model of the true function in this paper. Throughout this paper, we set the domain of the input to $\\Omega = [ 0 , 1 ] ^ { d }$ . For a function $f : \\Omega \\to \\mathbb { R }$ , let $\\begin{array} { r } { \\| f \\| _ { p } : = \\| f \\| _ { L ^ { p } ( \\Omega ) } : = ( \\int _ { \\Omega } | f | ^ { p } \\mathrm { d } x ) ^ { 1 / p } } \\end{array}$ for $0 < p < \\infty$ . For $p = \\infty$ , we define $\\| f \\| _ { \\infty } : = \\| f \\| _ { L ^ { \\infty } ( \\Omega ) } : = \\operatorname* { s u p } _ { x \\in \\Omega } | f ( x ) |$ . For $\\beta \\in \\mathbb { R } _ { + + } ^ { d }$ , let $\\begin{array} { r } { | \\beta | = \\sum _ { j = 1 } ^ { d } | \\beta _ { j } | ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For a function $f :  { \\mathbb { R } ^ { d } } \\to  { \\mathbb { R } }$ , we define the $r$ th difference of $f$ in the direction $h \\in \\mathbb { R } ^ { d }$ as ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/24a639c049482bcf8b5ce5952ff47d7d3f7263b93093558e8887b9df2623368c.jpg",
+        "text": "$$\n\\Delta _ { h } ^ { r } ( f ) ( x ) : = \\Delta _ { h } ^ { r - 1 } ( f ) ( x + h ) - \\Delta _ { h } ^ { r - 1 } ( f ) ( x ) , \\Delta _ { h } ^ { 0 } ( f ) ( x ) : = f ( x ) ,\n$$",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "for $x \\in \\Omega$ with $x + r h \\in \\Omega$ , otherwise, let $\\Delta _ { h } ^ { r } ( f ) ( x ) = 0$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 1. For a function $f \\in L ^ { p } ( \\Omega )$ where $p \\in ( 0 , \\infty ]$ , the $r$ -th modulus of smoothness of $f$ is defined by $\\begin{array} { r } { w _ { r , p } ( f , t ) = \\operatorname* { s u p } _ { h \\in \\mathbb { R } ^ { d } : | h _ { i } | \\leq t _ { i } } \\| \\Delta _ { h } ^ { r } ( f ) \\| _ { p } , f o r t = ( t _ { 1 } , \\ldots , t _ { d } ) , t _ { i } > 0 . } \\end{array}$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With this modulus of smoothness, we define the anisotropic Besov space $B _ { p , q } ^ { \\beta } ( \\Omega )$ for $\\beta \\ =$ $( \\beta _ { 1 } , \\ldots , \\beta _ { d } ) ^ { \\top } \\in \\mathbb { R } _ { + + } ^ { d }$ as follows. ",
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+    {
+        "type": "text",
+        "text": "Definition 2 (Anisotropic Besov space $( B _ { p , q } ^ { \\beta } ( \\Omega ) ) )$ . For $0 < p , q \\le \\infty$ , $\\beta = ( \\beta _ { 1 } , \\ldots , \\beta _ { d } ) ^ { \\top } \\in \\mathbb { R } _ { + + } ^ { d } ,$ r := maxibβic + 1, let the seminorm | · |Bαp,q be ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64da6b2ad1a24e4d504813d6024f1aa3a78b449c32f6abd00bd1a78b27796d16.jpg",
+        "text": "$$\n| f | _ { B _ { p , q } ^ { \\beta } } : = \\left\\{ \\begin{array} { l l } { \\left( \\sum _ { k = 0 } ^ { \\infty } [ 2 ^ { k } w _ { r , p } ( f , ( 2 ^ { - k / \\beta _ { 1 } } , \\ldots , 2 ^ { - k / \\beta _ { d } } ) ) ] ^ { q } \\right) ^ { 1 / q } } & { ( q < \\infty ) , } \\\\ { \\operatorname* { s u p } _ { k \\geq 0 } 2 ^ { k } w _ { r , p } ( f , ( 2 ^ { - k / \\beta _ { 1 } } , \\ldots , 2 ^ { - k / \\beta _ { d } } ) ) } & { ( q = \\infty ) . } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The norm of the anisotropic Besov space $B _ { p , q } ^ { \\beta } ( \\Omega )$ is defined by $\\lVert f \\rVert _ { B _ { p , q } ^ { \\beta } } : = \\lVert f \\rVert _ { p } + \\lvert f \\rvert _ { B _ { p , q } ^ { \\beta } }$ , and $B _ { p , q } ^ { \\beta } ( \\Omega ) = \\{ f \\in L ^ { p } ( \\Omega ) \\mid \\| f \\| _ { B _ { p , q } ^ { \\beta } } < \\infty \\} .$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Roughly speaking $\\beta$ represents the smoothness in each direction. If $\\beta _ { i }$ is large, then a function in $B _ { p , q } ^ { \\beta }$ is smooth to the $i$ th coordinate direction, otherwise, it is non-smooth to that direction. $p$ is also an important quantity that controls the spatial inhomogeneity of the smoothness. If $\\beta _ { 1 } = \\beta _ { 2 } = \\cdot \\cdot \\cdot =$ $\\beta _ { d }$ , then the definition is equivalent to the usual Besov space (DeVore & Popov, 1988; DeVore et al., 1993). Suzuki (2019) analyzed curse of dimensionality of deep learning through a so-called mixed smooth Besov (m-Besov) space which imposes a stronger condition toward all directions uniformly. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Particularly, it imposes stronger smoothness toward non-coordinate axis directions. Moreover, mBesov space does not include the vanilla Besov space as a special case and thus cannot capture the situation that we consider in this paper. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Throughout this paper, for given $\\beta = ( \\beta _ { 1 } , \\ldots , \\beta _ { d } ) ^ { \\top } \\in \\mathbb { R } _ { + + } ^ { d }$ , we write $\\underline { { \\beta } } : = \\operatorname* { m i n } _ { i } \\beta _ { i }$ (smallest smoothness) and $\\overline { { \\beta } } : = \\operatorname* { m a x } _ { i } \\beta _ { i }$ (largest smoothness). The approximation error of a function in anisotropic Besov spaces is characterized by the harmonic mean of $( \\beta _ { j } ) _ { j = 1 } ^ { d }$ , which corresponds to the average smoothness, and thus we define ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b2c5a9250643b94e870dfde41d477ccbdfe5c3724c3d0f2b8e144fdbc5ec1d28.jpg",
+        "text": "$$\n\\begin{array} { r } { \\widetilde { \\beta } : = \\left( \\sum _ { j = 1 } ^ { d } 1 / \\beta _ { j } \\right) ^ { - 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The Besov space is closely related to other function spaces such as Holder space. Let ¨ $\\partial ^ { \\alpha } f ( x ) =$ $\\frac { \\partial ^ { | \\alpha | } f } { \\partial ^ { \\alpha _ { 1 } } x _ { 1 } . . . \\partial ^ { \\alpha _ { d } } x _ { d } } ( x )$ . ",
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+    {
+        "type": "text",
+        "text": "Definition 3 (Holder space¨ $( \\mathscr { C } ^ { \\beta } ( \\Omega ) ) )$ . For a smoothness paraemter $\\beta \\in \\mathbb { R } _ { + + }$ with $\\beta \\notin \\mathbb { N }$ , consider an $m$ times differentiable function $f ~ : ~ \\mathbb { R } ^ { d } ~ \\to ~ \\mathbb { R }$ where $m ~ = ~ \\left\\lfloor \\beta \\right\\rfloor$ (the largest integer less than $\\beta$ ), and let the norm of the Holder space ¨ $\\mathscr { C } ^ { \\beta } ( \\Omega )$ be $\\| f \\| _ { \\mathcal { C } ^ { \\beta } } : = \\operatorname* { m a x } _ { | \\alpha | \\leq m } \\| \\partial ^ { \\alpha } f \\| _ { \\infty } +$ $\\begin{array} { r l } & { \\operatorname* { m a x } _ { | \\alpha | = m } \\operatorname* { s u p } _ { x , y \\in \\Omega } \\frac { | \\partial ^ { \\alpha } f ( x ) - \\partial ^ { \\alpha } f ( y ) | } { \\| x - y \\| ^ { \\beta - m } } } \\end{array}$ |∂αf(x)−∂αf(y)|kx−ykβ−m . Then, (β-)Holder space ¨ Cβ(Ω) is defined as Cβ(Ω) = {f | $\\| f \\| _ { C ^ { \\beta } } < \\infty \\}$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Let ${ \\mathcal { C } } ^ { 0 } ( \\Omega )$ be the set of continuous functions equipped with $L ^ { \\infty }$ -norm: $\\mathcal { C } ^ { 0 } ( \\Omega ) : = \\{ f : \\Omega \\to \\mathbb { R } \\ |$ $f$ is continuous and $\\| f \\| _ { \\infty } < \\infty \\}$ . These function spaces are closely related to each other. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 (Triebel (2011)). There exist the following relations between the spaces: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. For $\\beta = ( \\beta _ { 0 } , \\ldots , \\beta _ { 0 } ) ^ { \\intercal } \\in \\mathbb { R } ^ { d }$ with $\\beta _ { 0 } \\notin \\mathbb { N } ,$ , it holds that $\\mathcal { C } ^ { \\beta _ { 0 } } ( \\Omega ) = B _ { \\infty , \\infty } ^ { \\beta } ( \\Omega )$ . 2. For $0 < p _ { 1 } , p _ { 2 } , q \\le \\infty$ , $p _ { 1 } \\leq p _ { 2 }$ and $\\beta \\in \\mathbb { R } _ { + + } ^ { d }$ with $\\widetilde { \\beta } > ( 1 / p _ { 1 } - 1 / p _ { 2 } ) _ { + } { } ^ { 3 }$ , it holds that4 $B _ { p _ { 1 } , q } ^ { \\beta } ( \\Omega ) \\hookrightarrow B _ { p _ { 2 } , q } ^ { \\gamma \\beta } ( \\Omega ) f o r \\gamma = 1 - ( 1 / p _ { 1 } - 1 / p _ { 2 } ) _ { + } / \\tilde { \\beta }$ . 13. For $0 < p , q _ { 1 } , q _ { 2 } \\leq \\infty$ , $q _ { 1 } < q _ { 2 }$ , and $\\beta \\in \\mathbb { R } _ { + + } ^ { d }$ , it holds that $B _ { p , q _ { 1 } } ^ { \\beta } \\hookrightarrow B _ { p , q _ { 2 } } ^ { \\beta }$ . In particular, with properties $^ { l }$ and 2, $i f \\widetilde { \\beta } > 1 / p ,$ , it holds that $B _ { p , q } ^ { \\beta } ( \\Omega ) \\hookrightarrow \\mathcal { C } ^ { \\gamma \\beta } ( \\Omega )$ where $\\gamma = 1 - 1 / ( \\widetilde { \\beta } p )$ . 4. For $0 < p , q \\le \\infty$ and $\\beta \\in \\mathbb { R } _ { + + } ^ { d }$ , $i f { \\widetilde \\beta } > 1 / p$ , then $B _ { p , q } ^ { \\beta } ( \\Omega ) \\hookrightarrow \\mathcal { C } ^ { 0 } ( \\Omega )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This result is basically proven by Triebel (2011). For completeness, we provide its derivation in Appendix D. If the average smoothness $\\widetilde { \\beta }$ is sufficiently large $( \\widetilde { \\beta } > 1 / \\bar { p } )$ , then the functions in $B _ { p , q } ^ { \\beta }$ are continuous; however, if it is small $( { \\widetilde \\beta } < 1 / p )$ , then they are no longer continuous. Small $p$ indicates spatially inhomogeneous smoothness; thus, spikes and jumps appear (see Donoho & Johnstone (1998) for this perspective, from the viewpoint of wavelet analysis). ",
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+    {
+        "type": "text",
+        "text": "2.2 Model of the true function ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "As a model of the true function $f ^ { \\mathrm { o } }$ , we consider two types of models: Affien composition model and deep composition model. For a Banach space $\\mathcal { H }$ , we let $U ( \\mathcal { H } )$ be the unit ball of $\\mathcal { H }$ . ",
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+    {
+        "type": "text",
+        "text": "(a) Affine composition model: The first model we introduced is a very naive model which is just a composition of an affine transformation and a function in the anisotropic Besov space: ",
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+    {
+        "type": "equation",
+        "img_path": "images/618a746c11f4cdb851c6c5407df4e10d519a12906c5a7b424cba5f5958942647.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { H } _ { \\mathrm { a f f } } : = \\{ h ( A x + b ) ~ \\vert ~ h \\in U ( B _ { p , q } ^ { \\beta } ( [ 0 , 1 ] ^ { \\bar { d } } ) ) , ~ A \\in \\mathbb { R } ^ { \\bar { d } \\times d } , ~ b \\in \\mathbb { R } ^ { b } \\mathrm { ~ s . t . ~ } A x + b \\in [ 0 , 1 ] ^ { \\bar { d } } \\left( \\forall x \\in \\Omega \\right) \\} , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where we assume $\\tilde { d } \\leq d$ . Here, we assumed that the affine transformation has an appropriate scaling such that $A x + b$ is included in the domain of $h$ for all $x \\in \\Omega$ . This is a quite naive model but provides an instructive example to understand how the estimation error of deep learning behaves under the anisotropic setting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(b) Deep composition model: The deep composition model generalizes the affine composition model to a composition of nonlinear functions. Let $m _ { 1 } = d , m _ { L + 1 } = 1 , m _ { \\ell }$ be the dimension of the \\`th layer, and let $\\beta ^ { ( \\ell ) } \\in \\mathbb { R } _ { + + } ^ { m _ { \\ell } }$ be the smoothness parameter in the \\`th layer. The deep composition model is defined as ",
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+        "type": "image",
+        "img_path": "images/4d5b1a3f7398cfbdef60e024a372562d2899f3a8e5c91cb51a1858004d8d82b9.jpg",
+        "image_caption": [
+            "Figure 1: Near low dimensional data distribution with anisotropic smoothness of the target function. The target function has less smoothness $( s _ { 1 } , s _ { 2 } )$ toward the first two coordinates on the manifold while it is almost constant toward the third coordinate (large $s _ { 3 }$ ). "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/86020b32102e67fa6412824ae8b025fd951868bac7e8dd87e45baa4c7f96afe7.jpg",
+        "text": "$$\nH _ { \\mathrm { d e e p } } : = \\{ h _ { H } \\circ \\cdot \\cdot \\circ h _ { 1 } ( x ) \\mid h _ { \\ell } : [ 0 , 1 ] ^ { m _ { \\ell } } \\} [ 0 , 1 ] ^ { m _ { \\ell + 1 } } , h _ { \\ell , k } \\in U ( B _ { p , q } ^ { \\beta ^ { ( \\ell ) } } ( [ 0 , 1 ] ^ { m _ { \\ell } } ) ) \\left( \\forall k \\in [ m _ { \\ell + 1 } ] \\right) \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, the interval ference can be abs $[ 0 , 1 ]$ can be replaced by another compact interval, by changing a scaling factor. The assumption $\\| h _ { \\ell , k } \\| _ { B _ { p , q } ^ { \\beta ^ { ( \\ell ) } } } \\leq 1$ $[ a _ { \\ell } , b _ { \\ell } ]$ but this dif-can also be affine composition model as a special case. However, it requires a stronger assumption to properly evaluate the estimation error on this model. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Examples The model we have introduced includes some instructive examples as listed below: ",
+        "bbox": [
+            178,
+            414,
+            800,
+            429
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(a) Linear projection Schmidt-Hieber (2020) analyzed estimation of the following model by deep learning: $\\bar { f ^ { \\mathrm { o } } } ( \\bar { x } ) = g ( w ^ { \\top } x )$ where $g \\in \\mathcal { C } ^ { \\beta } ( [ 0 , 1 ] )$ and $w \\in \\mathbb { R } ^ { d }$ . In this example, the function $f ^ { \\mathrm { o } }$ varies along only one direction, $w$ . Apparently, this is an example of the affine composition model. ",
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(b) Distribution on low dimensional smooth manifold Assume that the input $x$ is distributed on a low-dimensional smooth manifold embedded in $\\Omega$ , and the smoothness of the true function $f ^ { \\mathrm { o } }$ is anisotropic along a coordinate direction on the manifold. We suppose that the low dimensional manifold is $\\tilde { d }$ -dimensional and $\\tilde { d } \\ll d$ . In this situation, the true function can be written as $f ^ { \\mathrm { o } } ( x ) = h ( \\phi ( x ) )$ where $\\phi : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { \\tilde { d } }$ is a map that returns the coordinate of $x$ on the manifold and $h$ is an element in an anisotropic Besov space on $\\mathbb { R } ^ { \\tilde { d } }$ . This situation appears if data is distributed on a low-dimensional sub-manifold of $\\Omega$ and the target function is invariant against noise injection to some direction on the manifold at each input point $x$ (Figure 1 illustrates this situation). One typical example of this situation is a function invariant with data augmentation (Simard et al., 2003; Krizhevsky et al., 2012). Even if the noise injection destroys low dimensionality of the data distribution (i.e., $\\bar { \\tilde { d } } = d _ { \\ast }$ , an anisotropic smoothness of the target function eases the curse of dimensionality as analyzed below, which is quite different from existing works (Yang & Dunson, 2016; Bickel & Li, 2007; Nakada & Imaizumi, 2020; Schmidt-Hieber, 2019; Chen et al., 2019; Chen et al., 2019). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Related work Here, we introduce some more related work and discuss their relation to our analysis. The statistical analysis on an anisotropic Besov space can be back to Ibragimov & Khas’minskii (1984) who considered density estimation, where the density is assumed to be included in an anisotropic Sobolev space with $p \\geq 2$ , and derived the minimax optimal rate $n ^ { - r \\widetilde { \\beta } / ( 2 \\widetilde { \\beta } + 1 ) }$ with respect to $L ^ { r }$ -norm. Nyssbaum (1983, 1987) analyzed a nonparametric regression problem on an anisotropic Besov space. Following these results, several studied have been conducted in the literature pertaining to nonparametric statistics, such as nonlinear kernel estimator Kerkyacharian et al. (2001), adaptive confidence band construction Hoffman & Lepski (2002), optimal aggregation Gaiffas & Lecue (2011), Gaussian process estimator Bhattacharya et al. (2011, 2014), and kernel ridge regression Hang & Steinwart (2018). Basically, these studies investigated estimation problems in which the target function is in anisotropic Besov spaces, but the composition models considered in this paper have not been analyzed. Hoffman & Lepski (2002); Bhattacharya et al. (2011) considered a dimension reduction model; that is, the target function is dependent on only a few variables of $x$ , but they did not deal with more general models, such as the affine/deep composition models. The nonparametric regression problems where the input data are distributed on a low-dimensional smooth manifold has been studied as a “manifold regression” Yang & Dunson (2016); Bickel & Li (2007); Yang & Tokdar (2015). Such a model can be considered as a specific example of the deep composition model. In this sense, our analysis is a significant extension of these analyses. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 Approximation error analysis ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here, we consider approximating the true function $f ^ { \\mathrm { o } }$ via a deep neural network and derive the approximation error. As the activation function, we consider the ReLU activation denoted by $\\eta ( x ) = \\operatorname* { m a x } \\{ x , 0 \\}$ $( x \\in \\mathbb { R }$ ). For a vector $x$ , $\\eta ( x )$ is operated in an element-wise manner. The model of neural networks with height $L$ , width $W$ , sparsity constraint $S$ , and norm constraint $B$ as $\\Phi ( L , W , S , B ) : = \\{ ( \\mathcal { W } ^ { ( L ) } \\eta ( \\cdot ) + \\bar { b ^ { ( L ) } } ) \\circ \\cdot \\cdot \\cdot \\circ ( \\mathcal { W } ^ { ( 1 ) } \\bar { x } + b ^ { ( 1 ) } ) \\ | \\ \\mathcal { W } ^ { ( L ) } \\in \\mathbb { R } ^ { 1 \\times W } .$ , $b ^ { ( L ) } \\in \\mathbb { R }$ , $\\mathcal { W } ^ { ( 1 ) } \\in$ $\\mathbb { R } ^ { W \\times d }$ , $b ^ { ( 1 ) } \\in \\mathbb { R } ^ { W } , \\mathcal { W } ^ { ( \\ell ) } \\in \\mathbb { R } ^ { W \\times W }$ , $b ^ { ( \\ell ) } \\in \\mathbb { R } ^ { W } ( 1 < \\ell < L ) , \\sum _ { \\ell = 1 } ^ { L } ( \\| \\mathcal { W } ^ { ( \\ell ) } \\| _ { 0 } + \\| b ^ { ( \\ell ) } \\| _ { 0 } ) \\le$ $S , \\operatorname* { m a x } _ { \\ell } \\| \\mathcal W ^ { ( \\ell ) } \\| _ { \\infty } \\vee \\| b ^ { ( \\ell ) } \\| _ { \\infty } \\le B \\}$ , where $\\| \\cdot \\| _ { 0 }$ is the $\\ell _ { 0 }$ -norm of the matrix (the number of nonzero elements of the matrix), and $\\| \\cdot \\| _ { \\infty }$ is the $\\ell _ { \\infty }$ -norm of the matrix (maximum of the absolute values of the elements). The sparsity constraint and norm bounds are required to obtain the nearoptimal rate of the estimation error. To evaluate the accuracy of the deep neural network model in approximating target functions, we define the worst-case approximation error as ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/8f7f8af151b4a7ac01583672c920c07e02650ecbc0b5a289244e5450f9959d7f.jpg",
+        "text": "$$\n\\begin{array} { r } { R _ { r } ( \\mathcal { F } , \\mathcal { H } ) : = \\operatorname* { s u p } _ { f ^ { * } \\in \\mathcal { H } } \\operatorname* { i n f } _ { f \\in \\mathcal { F } } \\| f ^ { * } - f \\| _ { L ^ { r } ( \\Omega ) } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            334,
+            653,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { F }$ is the set of functions used for approximation, and $\\mathcal { H }$ is the set of target functions. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2 (Approximation ability for anisotropic Besov space). Suppose that $0 < p , q , r \\leq \\infty$ and $\\beta \\in \\mathbb { R } _ { + + } ^ { d }$ satisfy the following condition: $\\widetilde { \\beta } > ( 1 / p - 1 / r ) _ { + }$ . Assume that $m \\in \\mathbb { N }$ satisfies $0 < \\overline { { \\beta } } < \\operatorname* { m i n } ( m , m - 1 + 1 / p )$ . Let $\\delta = ( 1 / p - 1 / r ) _ { + }$ , $\\nu = ( \\widetilde { \\beta } - \\delta ) / ( 2 \\delta )$ and $W _ { 0 } ( d ) : =$ $6 d m ( m + 2 ) + 2 d$ . Then, for $N \\in \\mathbb N$ , we can bound the approximation error as ",
+        "bbox": [
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+            375,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e481ea49d7cc808d7f549c8b3e243fb4d0ce3b51319994c8c1958ba17db8ed6.jpg",
+        "text": "$$\nR _ { r } ( \\Phi ( L _ { 1 } , W _ { 1 } , S _ { 1 } , B _ { 1 } ) , U ( B _ { p , q } ^ { \\beta } ( \\Omega ) ) ) \\lesssim N ^ { - \\widetilde { \\beta } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            443,
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+            465
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "by setting ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/38c6c633e8c1094e26a42810a13555ce25dcf183e5f9de71bf8e1a3208973091.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { 1 } ( d ) : = 3 + 2 \\lceil \\log _ { 2 } \\left( \\frac { 3 ^ { d \\vee m } } { \\epsilon c _ { ( d , m ) } } \\right) + 5 \\rceil \\lceil \\log _ { 2 } ( d \\vee m ) \\rceil , W _ { 1 } ( d ) : = N W _ { 0 } , } \\\\ & { S _ { 1 } ( d ) : = \\lbrack ( L - 1 ) W _ { 0 } ^ { 2 } + 1 \\rbrack N , B _ { 1 } ( d ) : = O ( N ^ { d ( 1 + \\nu ^ { - 1 } ) ( 1 / p - \\widetilde { \\beta } ) _ { + } } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for $\\epsilon = N ^ { - \\widetilde { \\beta } } \\log ( N ) ^ { - 1 }$ and a constant $c _ { ( d , m ) }$ depending only on $d$ and $m$ ",
+        "bbox": [
+            173,
+            546,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The proof of this proposition is provided in Appendix B. The rate $N ^ { - \\widetilde { \\beta } }$ is the optimal adaptive approximation error rate that can be achieved by a model with $N$ parameters (the difference between adaptive and non-adaptive methods is explained in the discussion below). Note that this is an approximation error in an oracle setting and no sample complexity appears here. We notice that we can avoid the curse of dimensionality if the average smoothness $\\widetilde { \\beta }$ is small. This means that if the target function is non-smooth in only a few directions and smooth in other directions, we can avoid the curse of dimensionality. In contrast, if we consider an isotropic Besov space where $\\beta _ { 1 } = \\cdots = \\beta _ { d } ( = \\underline { { \\beta } } )$ , then $\\widetilde { \\beta } = \\underline { { \\beta } } / d$ , which directly depends on the dimensionality $d$ , and we need an exponentially large number of parameters in this situation to achieve $\\epsilon$ -accuracy. Therefore, the anisotropic smoothness has a significant impact on the approximation error rate. The assumption $\\widetilde { \\beta } > ( 1 / p - 1 / r ) _ { + }$ ensures the $L _ { r }$ -integrability of the target function, and the inequality (without equality) admits a near-optimal wavelet approximation of the target function in terms of $L _ { r }$ -norm. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using this evaluation as a basic tool, we can obtain the approximation error for the deep composition models. We can also obtain the approximation error for the affine composition models, but it is almost identical to Proposition 2. Therefore, we defer it to Appendix A. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (Deep composition model). Assume that $\\widetilde { \\beta } ^ { ( \\ell ) } > 1 / p$ for all $\\ell = 1 , \\ldots , H$ . Then, the estimation error on the deep composition model is bounded as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5161db53d40e445b0dc21893f9c3be131920c2584072d6e95abc150f8a9acb5.jpg",
+        "text": "$$\nR _ { \\infty } ( \\Phi ( L , W , S , B ) , \\mathcal { H } _ { \\mathrm { d e e p } } ) \\lesssim \\operatorname* { m a x } _ { \\ell \\in [ H ] } N ^ { - \\widetilde { \\beta } ^ { * ( \\ell ) } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71079c598f06838e3e6dcc89dda3b7328b9328286a2da91a96b6a0380a1db367.jpg",
+        "text": "$$\n\\begin{array} { r l } & { v h e r e ~ \\widetilde { \\beta } ^ { * } ^ { ( \\ell ) } = \\widetilde { \\beta } ^ { ( \\ell ) } \\prod _ { k = \\ell + 1 } ^ { H } [ ( \\underline { { \\beta } } ^ { ( k ) } - 1 / p ) \\wedge 1 ] , a n d \\ L = \\sum _ { \\ell = 1 } ^ { H } ( L _ { 1 } ( m _ { \\ell } ) + 1 ) , W = \\operatorname* { m a x } _ { \\ell } ( W _ { 1 } ( m _ { \\ell } ) \\vee \\ell ) } \\\\ & { n _ { \\ell + 1 } ) , S = \\sum _ { \\ell = 1 } ^ { H } ( S _ { 1 } ( m _ { \\ell } ) + 3 m _ { \\ell + 1 } ) , B = \\operatorname* { m a x } _ { \\ell } B _ { 1 } ( m _ { \\ell } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The proof can be found in Appendix B.1. Since the model is more general than the vanilla anisotropic Besov space, we require a stronger assumption $\\widetilde { \\beta } ^ { ( \\ell ) } > 1 / p$ on $\\widetilde { \\beta } ^ { \\left( \\ell \\right) }$ than the condition in Proposition 2. This is because we need to bound the Holder smoothness of the remaining ¨ layers to bound the influence of the approximation error in the internal layers to the entire function. Holder smoothness is ensured according to the embedding property under this condition (Proposi- ¨ tion 1). This Holder smoothness assumption affects the approximation error rate. The convergence ¨ rate ${ \\widetilde { \\beta } } ^ { * ( \\ell ) }$ in Eq. (4) is different from that in Eq. (8). This is because the approximation error in the internal layers are propagated through the remaining layers with Holder smoothness and its ¨ amplitude is controlled by the Holder smoothness. ¨ ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Approximation error by non-adaptive method The approximation error obtained in the previous section is called an adaptive error rate in the literature regarding approximation theory (DeVore, 1998). If we fix $N$ bases beforehand and approximate the target function by a linear combination of the $N$ bases (which is called the non-adaptive method), then we cannot achieve the adaptive error rate obtained in the previous section. Roughly speaking, the approximation error of non-adaptive methods is lower bounded by $\\begin{array} { r } { N ^ { - \\left( \\widetilde { \\beta } - \\left( \\frac { 1 } { p } - \\frac { 1 } { \\operatorname* { m i n } \\left\\{ 2 , r \\right\\} } \\right) + \\right) } } \\end{array}$ (Myronyuk, 2015, 2016, 2017), which is slower than the approximation error rate of deep neural networks especially for small $p$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Estimation error analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            342,
+            416,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we analyze the accuracy of deep learning in estimating a function in compositions of anisotropic Besov spaces. We consider a least-squares estimator in the deep neural network model: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/32caaa108a57a85135990b28d2ba15622501f02a26b64b74f143e78d9a0ad0af.jpg",
+        "text": "$$\n\\begin{array} { r } { \\widehat { f } = \\operatorname * { a r g m i n } _ { \\bar { f } : f \\in \\Phi ( L , W , S , B ) } \\sum _ { i = 1 } ^ { n } ( y _ { i } - \\bar { f } ( x _ { i } ) ) ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            404,
+            656,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\bar { f }$ is the clipping of $f$ defined by $\\bar { f } = \\operatorname* { m i n } \\{ \\operatorname* { m a x } \\{ f , - F \\} , F \\}$ for a constant $F > 0$ which is realized by ReLU units. The network parameters $( L , W , S , B )$ should be specified appropriately as indicated in Theorems 2 and 3. In practice, these parameters can be specified by cross validation. Indeed, we can theoretically show that cross validation can provide the appropriate choice of these parameters in compensation to an additional $\\log ( n )$ -factor in the estimation error bound. This estimator can be seen as a sparsely regularized estimator because there are constraints on $S$ . In terms of optimization, this requires a combinatorial optimization, but we do not pursue the computational aspect. The estimation error that we derive in this section can involve the optimization error, but for simplicity, we only demonstrate the estimation error of the ideal situation where there is no optimization error. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Affine composition model The following theorem provides an upper bound of the estimation error for the affine composition model. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Assume the same condition as in Theorem $6$ ; in particular, suppose $0 < p , q \\le \\infty$ and $\\widetilde { \\beta } > ( 1 / p - 1 / 2 ) _ { + }$ for $\\widetilde { \\beta } \\in \\mathbb { R } _ { + + } ^ { \\tilde { d } }$ . Moreover, we assume that the distribution $P _ { X }$ has a density $p _ { X }$ such that $\\| p _ { X } \\| _ { \\infty } \\leq R$ for a constant $R > 0$ . If $f ^ { \\mathrm { o } } \\in \\mathcal { H } _ { \\mathrm { a f f } } \\cap L ^ { \\infty } ( \\Omega )$ , and $\\| f ^ { \\mathrm { o } } \\| _ { \\infty } \\leq F$ for $F \\geq 1$ ; then, letting $( W , L , S , B ) = \\big ( L _ { 1 } ( \\tilde { d } ) , W _ { 1 } ( \\tilde { d } ) , S _ { 1 } ( \\tilde { d } ) , ( \\tilde { d } C + 1 ) B _ { 1 } ( \\tilde { d } ) \\big )$ as in Theorem $6$ with $N \\asymp n ^ { \\frac { 1 } { 2 \\tilde { \\beta } + 1 } }$ , we obtain ",
+        "bbox": [
+            173,
+            604,
+            825,
+            685
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f5fc2477ff49a8851d79c97e4753d7c845c1bc3086a4809a2f6ec1afba89b6b.jpg",
+        "text": "$$\n\\mathrm { E } _ { D _ { n } } [ \\| f ^ { \\mathrm { o } } - \\widehat { f } \\| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] \\lesssim n ^ { - \\frac { 2 \\widetilde { \\beta } } { 2 \\widetilde { \\beta } + 1 } } \\log ( n ) ^ { 3 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            689,
+            637,
+            717
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { E } _ { D _ { n } } [ \\cdot ]$ indicates the expectation with respect to the training data $D _ { n }$ ",
+        "bbox": [
+            176,
+            722,
+            669,
+            737
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The proof is provided in Appendix C. We will show that the convergence rate $n ^ { - 2 \\widetilde { \\beta } / ( 2 \\widetilde { \\beta } + 1 ) }$ is minimax optimal in Section 5 (see also Kerkyacharian & Picard (1992); Donoho et al. (1996); Donoho & Johnstone (1998); Gine & Nickl ´ (2015) for ordinary Besov spaces). The $L ^ { \\infty }$ -norm constraint $\\| f ^ { \\mathrm { o } } \\| _ { \\infty } \\leq F$ is used to derive a uniform bound on the discrepancy between the population and the empirical $L ^ { 2 }$ -norm. Without this condition, the convergence rate could be slower. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Deep composition model For the deep composition model, we obtain the following convergence rate. This is an extension of Theorem 2 but requires a stronger assumption on the smoothness. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Theorem 3. Suppose that $0 < p , q \\le \\infty$ and $\\widetilde { \\beta } ^ { ( \\ell ) } > 1 / p$ for all $\\ell \\in [ H ]$ . If $f ^ { \\mathrm { o } } \\in \\mathcal { H } _ { \\mathrm { d e e p } } \\cap L ^ { \\infty } ( \\Omega )$ and $\\| f \\| _ { \\infty } \\leq F$ for $F \\geq 1$ , then we obtain ",
+        "bbox": [
+            173,
+            856,
+            821,
+            887
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6ad9646f1ce7beb012495f28d79c0067485a0fe0c2f0745f1b8547ce69d31fbb.jpg",
+        "text": "$$\n\\begin{array} { r } { { \\mathrm { E } } _ { D _ { n } } [ \\| f ^ { \\mathrm { o } } - \\widehat f \\| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] \\lesssim \\operatorname* { m a x } _ { \\ell \\in [ H ] } n ^ { - 2 \\widetilde { \\beta } ^ { * ( \\ell ) } / ( 2 \\widetilde { \\beta } ^ { * ( \\ell ) } + 1 ) } \\log ( n ) ^ { 3 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            892,
+            707,
+            916
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\widetilde { \\beta } } ^ { * ( \\ell ) }$ is defined in Theorem 1, and $( L , W , S , B )$ is as given in Theorem 1 with $N \\ \\asymp$ $\\begin{array} { r } { \\operatorname* { m a x } _ { \\ell \\in [ L ] } n ^ { \\frac { 1 } { 2 \\widetilde { \\beta } ^ { * ( \\ell ) } + 1 } } } \\end{array}$ . ",
+        "bbox": [
+            174,
+            88,
+            825,
+            128
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The proof is provided in Appendix C. We will show that this is also minimax optimal in Theorem 4. Because of the Holder continuity, the convergence rate becomes slower than the affine composition ¨ model (that is, $\\widetilde { \\beta } ^ { * ( \\ell ) } \\le \\widetilde { \\beta } ^ { ( \\ell ) } )$ . However, this slower rate is unavoidable in terms of the minimax optimal rate. Schmidt-Hieber (2020) analyzed the same situation for the Holder class which corre- ¨ sponds to $\\beta _ { 1 } ^ { ( \\ell ) } = \\cdot \\cdot \\cdot = \\beta _ { d } ^ { ( \\ell ) }$ $( \\forall \\ell )$ and $p = q = \\infty$ . Our analysis far extends their analysis to the setting of anisotropic Besov spaces in which the parameters $\\beta ^ { ( \\ell ) } , p , q$ have much more freedom. ",
+        "bbox": [
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+            135,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "From these two bounds (Theorems 2 and 3), we can see that as the smoothness $\\widetilde { \\beta }$ becomes large, the convergence rates faster. If the target function is included in the isotropic Besov space with smoothness $\\beta _ { 1 } = \\cdots = \\beta _ { d } ( = \\underline { { \\beta } } )$ , then the estimation error becomes ",
+        "bbox": [
+            174,
+            236,
+            825,
+            280
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(Isotropic Besov) ",
+        "text_level": 1,
+        "bbox": [
+            189,
+            290,
+            312,
+            305
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8eca2afa859c45f86b698771505b0865d632b707f43dee9c68ce40c3871f6e7.jpg",
+        "text": "$$\nn ^ { - 2 \\underline { { \\beta } } / ( 2 \\underline { { \\beta } } + d ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            486,
+            286,
+            578,
+            304
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the exponent, the dimensionality $d$ appears, which causes the curse of dimensionality. In contrast, if the target function is in the anisotropic Besov space, and the smoothness in each direction is sufficiently imbalanced such that $\\widetilde { \\beta }$ does not depend on $d$ , our obtained rate ",
+        "bbox": [
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+            311,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(Anisotropic Besov) ",
+        "text_level": 1,
+        "bbox": [
+            189,
+            367,
+            330,
+            382
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71c38eb4d655660df45722ded210463104c9d3d12c6652131f2520d27c8096f2.jpg",
+        "text": "$$\nn ^ { - 2 \\widetilde { \\beta } / ( 2 \\widetilde { \\beta } + 1 ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            485,
+            362,
+            570,
+            380
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "avoids the curse of dimensionality. For high-dimensional settings, there would be several redundant directions in which the true function does not change. Deep learning is adaptive to this redundancy and achieves a better estimation error. However, in Section 6, we prove that linear estimators are affected by the dimensionality more strongly than deep learning. This indicates the superiority of deep learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Minimax optimal rate ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            473,
+            390,
+            491
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here, we show that the estimation error rate, that we have presented, of deep learning achieves the minimax optimal rate. Roughly speaking the minimax optimal risk on a model ${ \\mathcal { F } } ^ { \\circ }$ of the true function is the smallest worst case error over all estimators: ",
+        "bbox": [
+            173,
+            501,
+            825,
+            542
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27e738b117997f31c68276ac61901a67508213a0b151461f9c8395b581205e24.jpg",
+        "text": "$$\n\\begin{array} { r } { R _ { * } ( \\mathcal { F } ^ { \\circ } ) : = \\operatorname* { i n f } _ { \\widehat f } { \\operatorname* { s u p } _ { f ^ { \\circ } \\in \\mathcal { F } ^ { \\circ } } { \\operatorname E } _ { D _ { n } } [ \\| \\widehat f - f ^ { \\circ } \\| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            546,
+            665,
+            569
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\widehat { f }$ runs over all estimators. The convergence rate of the minimax optimal risk is referred to as minimax optimal rate. We obtain the following minimax optimal rate for anisotropic Besov spaces. ",
+        "bbox": [
+            173,
+            578,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4. (a) Affine composition model: For $0 < p , q \\le \\infty$ and $\\beta \\in \\mathbb { R } _ { + + } ^ { d }$ , assume that $\\widetilde { \\beta } > \\operatorname* { m a x } { \\{ 1 / p - 1 / 2 , 1 / p - 1 , 0 \\} }$ . Then, the minimax optimal risk of the affine composition model is lower bounded as $R _ { * } ( \\mathcal { H } _ { \\mathrm { a f f } } ) \\gtrsim n ^ { - \\frac { 2 \\widetilde { \\beta } } { 2 \\widetilde { \\beta } + 1 } }$ . (b) Deep composition model: For $0 < p , q \\le \\infty$ and $\\beta ^ { ( \\ell ) } \\in \\mathbb { R } _ { + + } ^ { d } \\ ( \\ell = 1 , \\dots , H )$ , assume that $\\widetilde { \\beta } ^ { ( \\ell ) } > 1 / p$ . Let $\\epsilon > 0$ be arbitrarily small for $q < \\infty$ , and let $\\epsilon = 0 f o r q = 0$ . Let $\\begin{array} { r } { \\widetilde { \\beta } ^ { * ( \\ell ) } = \\widetilde { \\beta } ^ { ( \\ell ) } \\prod _ { k = \\ell + 1 } ^ { H } [ ( \\underline { { \\beta } } ^ { ( k ) } - 1 / p + \\epsilon ) \\wedge 1 ] } \\end{array}$ , and $\\begin{array} { r } { \\widetilde { \\beta } ^ { * * } : = \\operatorname* { m i n } _ { \\ell } \\widetilde { \\beta } ^ { * ( \\ell ) } } \\end{array}$ . Then, the minimax optimal risk of the deep composition model is lower bounded as $R _ { * } ( \\mathcal { H } _ { \\mathrm { d e e p } } ) \\gtrsim n ^ { - \\frac { 2 \\widetilde { \\beta } ^ { * * } } { 2 \\widetilde { \\beta } ^ { * * } + 1 } }$ . ",
+        "bbox": [
+            173,
+            609,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The proof is provided in Appendix E (see also Ibragimov & Khas’minskii (1984); Nyssbaum (1987)). From this theorem, we can see that the estimation error of deep learning shown in Theorems 2 and 3 indeed achieve the minimax optimal rate up to a poly- $\\log ( n )$ factor. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 Suboptimality of linear estimators ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            795,
+            491,
+            813
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we give the minimax optimal rate in the class of linear estimators. The linear estimator is a class of estimators that can be written as ",
+        "bbox": [
+            173,
+            825,
+            825,
+            854
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23fe413c691e68ff7a22c93ec39c9394f609bc23185cbcd7642fcc6acd3fd1bc.jpg",
+        "text": "$$\n\\begin{array} { r } { \\widehat { f } ( x ) = \\sum _ { i = 1 } ^ { n } y _ { i } \\varphi _ { i } ( x ; X ^ { n } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            858,
+            591,
+            878
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $X ^ { n } \\ = \\ ( x _ { 1 } , \\ldots , x _ { n } )$ and $\\varphi _ { i } ( x ; X ^ { n } ) ~ ( i = 1 , \\dots , n )$ are (measurable) functions that only depend on $x$ and $X ^ { n }$ . This is linearly dependent on $Y ^ { n } = ( y _ { 1 } , \\dots , y _ { n } )$ . We notice that the kernel ridge regression is included in this class because it can be written as $\\widehat { f } ( x ) = k _ { x , X ^ { n } } ( k _ { X ^ { n } , X ^ { n } } +$ $\\lambda \\mathrm { I } ) ^ { - 1 } Y ^ { n }$ , which linearly depends on $Y ^ { n }$ . This class includes other important estimators, such as the Nadaraya–Watson estimator, the $k$ -nearest neighbor estimator, and the sieve estimator. We compare deep learning with the linear estimators in terms of minimax risk. For this purpose, we define the minimax risk of the class of linear estimators: ",
+        "bbox": [
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+            882,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            88,
+            825,
+            162
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de626e72ddf7bd453c6baee831ee4da6b42dabcdaf25072a96b9a4432d234f2e.jpg",
+        "text": "$$\nR _ { * } ^ { ( \\mathrm { l i n } ) } ( \\mathcal { F } ^ { \\circ } ) : = \\operatorname* { i n f } _ { \\widehat { f } : \\mathrm { l i n e a r } f ^ { \\circ } \\in \\mathcal { F } ^ { \\circ } } \\mathrm { E } _ { D _ { n } } [ \\| f ^ { \\mathrm { o } } - \\widehat { f } \\| _ { L ^ { 2 } ( P _ { X } ) } ^ { 2 } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            166,
+            666,
+            196
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\widehat { f }$ runs over all linear estimators. We can see that linear estimators suffer from the sub-optimal rate because of the following two points: (i) they do not have adaptivity, and (ii) they significantly suffer from the curse of dimensionality. ",
+        "bbox": [
+            174,
+            205,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5. (i) Suppose that the input distribution $P _ { X }$ is the uniform distribution on $\\Omega = [ 0 , 1 ] ^ { d }$ and assume that ${ \\widetilde \\beta } > 1 / p$ and $1 \\leq p , q \\leq \\infty$ . Then, the minimax optimal rate of the linear estimators is lower bounded as ",
+        "bbox": [
+            174,
+            253,
+            825,
+            299
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/abda58ff5778539c70c9a1703d9dd15510ff0d242b34a6a1f8166b07394bb7d8.jpg",
+        "text": "$$\nR _ { * } ^ { ( \\mathrm { l i n } ) } ( U ( B _ { p , q } ^ { \\beta } ) ) \\gtrsim n ^ { - \\frac { 2 \\widetilde { \\beta } - v } { 2 \\widetilde { \\beta } + 1 - v } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            303,
+            599,
+            330
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $v = 2 ( 1 / p - 1 / 2 ) _ { + }$ ",
+        "bbox": [
+            174,
+            337,
+            352,
+            353
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(ii) In addition to the above conditions, we assume that the true function is included in the affine composition model with $\\tilde { d } \\leq d , \\underline { { \\beta } } = \\beta _ { 1 } = \\cdot \\cdot \\cdot = \\beta _ { \\tilde { d } }$ and $0 < p \\le 2$ . Let $a _ { d } = 1 + \\kappa$ with arbitrary small $\\kappa > 0$ when $\\tilde { d } < d / 2$ , and let $a _ { d } = 0$ when $\\tilde { d } \\ge d / 2$ . Then, the minimax rate of the linear estimators on the affine composition model is lower bounded by ",
+        "bbox": [
+            173,
+            358,
+            825,
+            420
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/381c1da2d911a7dd5839b0b139c470a41f21693a02715c1916d610e63841857d.jpg",
+        "text": "$$\n\\begin{array} { r } { R _ { * } ^ { ( \\mathrm { l i n } ) } ( \\mathcal { H } _ { \\mathrm { a f f } } ) \\gtrsim n ^ { - \\frac { 2 ( \\underline { { \\beta } } - \\tilde { d } / p + d / 2 + a _ { d } ) } { 2 ( \\underline { { \\beta } } - \\tilde { d } / p + d / 2 + a _ { d } ) + d } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            428,
+            620,
+            454
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The proof is provided in Appendix F. (i) The lower bound (7) reveals the suboptimality of linear estimators in terms of input dimensionality. Actually, if we consider a particular case where $\\tilde { d } = 1$ , $p = 1$ and $d \\gg \\tilde { d }$ , then the obtained minimax rate of linear estimators and the estimation error rate of deep learning can be summarized as ",
+        "bbox": [
+            173,
+            467,
+            825,
+            529
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cafe8e83a3b54fdefa312aa0e2811ba8605a37db0727165b22d0205773fe716d.jpg",
+        "text": "$$\n\\mathrm { l i n e a r : } \\ n ^ { - \\frac { 2 \\beta + d } { 2 \\beta + 2 d } } , \\ \\qquad \\mathrm { d e e p : } \\ n ^ { - \\frac { 2 \\beta } { 2 \\beta + 1 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            535,
+            635,
+            559
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "by Theorem 2 when $\\underline { { \\beta } } > 1$ (which can be checked by noticing $\\widetilde { d } = \\underline { { \\beta } } / \\widetilde { \\beta } = 1$ in this situation). We can see that the dependence on the dimensionality of linear estimators is significantly worse than that of deep leaning. This indicates poor adaptivity of linear estimators to the intrinsic dimensionality√ of data. Actually, as $d$ becomes large, the rate for the linear estimator approaches to $1 / { \\sqrt { n } }$ but that for the deep learning is not affected by $d$ and still faster than $1 / \\sqrt { n }$ . To show the theorem, we used the “convex-hull argument” developed by Hayakawa & Suzuki (2019); Donoho $\\&$ Johnstone (1998). We combined this technique with the so-called Irie-Miyake’s integral representation (Irie & Miyake, 1988; Hornik et al., 1990). Note that this difference appears because there is an affine transformation in the first layer of the affine composition model. Deep learning is flexible against such a coordinate transform so that it can find directions to which the target function is smooth. In contrast, kernel methods do not have such adaptivity because there is no feature extraction layer. (ii) The lower bound (6) states that when $p < 2$ (that is, $v > 0$ ), the minimax rate of the linear estimators is outperformed by that of deep learning (Theorem 2). This is due to the “adaptivity” of deep leaning. When $p$ is small, the smoothness of the target function is less homogeneous, and it requires an adaptive approximation scheme to achieve the best estimation error. Linear estimators do not have adaptivity and thus fail to achieve the minimax optimal rate. Our bound (6) extends the result by Zhang et al. (2002) to a multivariate anisotropic Besov space while Zhang et al. (2002) investigated the univariate space $\\langle d = 1 \\rangle$ ). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            837,
+            299,
+            854
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We investigated the approximation error and estimation error of deep learning in the anisotropic Besov spaces. It was proved that the convergence rate is determined by the average of the anisotropic smoothness, which results in milder dependence on the input dimensionality. If the smoothness is highly anisotropic, deep learning can avoid overfitting. We also compared the error rate of deep learning with that of linear estimators and showed that deep learning has better dependence on the input dimensionality. Moreover, it was shown that deep learning can achieve the adaptive rate and outperform non-adaptive approximation methods and linear estimators if the homogeneity $p$ of smoothness is small. These analyses strongly support the practical success of deep learning from a theoretical perspective. ",
+        "bbox": [
+            176,
+            869,
+            823,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Limitations of this work Our work does not cover the optimization aspect of deep learning. It is assumed that the regularized least squares (5) can be executed. It would be nice to combine our study with recent developments of non-convex optimization techniques (Vempala & Wibisono, 2019; Suzuki & Akiyama, 2021). ",
+        "bbox": [
+            174,
+            183,
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+            238
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Potential negative societal impact Since this is purely theoretical result, it is not expected that there is a direct negative societal impact. However, revealing detailed properties of the deep learning could promote an opportunity to pervert deep learning. ",
+        "bbox": [
+            174,
+            247,
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+            289
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgment ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            309,
+            321,
+            327
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "TS was partially supported by JSPS KAKENHI (18H03201), Japan Digital Design and JST CREST.   \nAN was partially supported by JSPS Kakenhi (19K20337) and JST-PRESTO. ",
+        "bbox": [
+            171,
+            340,
+            823,
+            369
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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