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+# Fast Policy Extragradient Methods for Competitive Games with Entropy Regularization
+
+Shicong Cen Carnegie Mellon University shicongc@andrew.cmu.edu
+
+Yuting Wei University of Pennsylvania ytwei@wharton.upenn.edu
+
+Yuejie Chi Carnegie Mellon University yuejiechi@cmu.edu
+
+# Abstract
+
+This paper investigates the problem of computing the equilibrium of competitive games, which is often modeled as a constrained saddle-point optimization problem with probability simplex constraints. Despite recent efforts in understanding the last-iterate convergence of extragradient methods in the unconstrained setting, the theoretical underpinnings of these methods in the constrained settings, especially those using multiplicative updates, remain highly inadequate, even when the objective function is bilinear. Motivated by the algorithmic role of entropy regularization in single-agent reinforcement learning and game theory, we develop provably efficient extragradient methods to find the quantal response equilibrium (QRE)—which are solutions to zero-sum two-player matrix games with entropy regularization—at a linear rate. The proposed algorithms can be implemented in a decentralized manner, where each player executes symmetric and multiplicative updates iteratively using its own payoff without observing the opponent’s actions directly. In addition, by controlling the knob of entropy regularization, the proposed algorithms can locate an approximate Nash equilibrium of the unregularized matrix game at a sublinear rate without assuming the Nash equilibrium to be unique. Our methods also lead to efficient policy extragradient algorithms for solving entropy-regularized zero-sum Markov games at a linear rate. All of our convergence rates are nearly dimension-free, which are independent of the size of the state and action spaces up to logarithm factors, highlighting the positive role of entropy regularization for accelerating convergence.
+
+# 1 Introduction
+
+Finding the equilibrium of competitive games, which can be viewed as constrained saddle-point optimization problems with probability simplex constraints, lies at the heart of modern machine learning and decision making paradigms such as Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), competitive reinforcement learning (RL) (Littman, 1994), game theory (Shapley, 1953), adversarial training (Mertikopoulos et al., 2018b), to name a few.
+
+In this paper, we study one of the most basic forms of competitive games, namely two-player zero-sum games, in both the matrix setting and the Markov setting. Our goal is to find the equilibrium policies of both players in an independent and decentralized manner (Daskalakis et al., 2020; Wei et al., 2021a) with guaranteed last-iterate convergence. Namely, each player will execute symmetric and independent updates iteratively using its own payoff without observing the opponent’s actions directly, and the final policies of the iterative process should be a close approximation to the equilibrium up to any prescribed precision. This kind of algorithms is more advantageous and versatile especially in federated environments, as it requires neither prior coordination between the players like twotimescale algorithms, nor a central controller to collect and disseminate the policies of all the players, which are often unavailable due to privacy constraints.
+
+# 1.1 Last-iterate convergence in competitive games
+
+In recent years, there have been significant progresses in understanding the last-iterate convergence of simple iterative algorithms for unconstrained saddle-point optimization, where one is interested in bounding the sub-optimality of the last iterate of the algorithm, rather than say, the ergodic iterate — which is the average of all the iterations — that are commonly studied in the earlier literature. This shift of focus is motivated, for example, by the infeasibility of averaging large machine learning models in training GANs (Goodfellow et al., 2014). While vanilla Gradient Descent / Ascent (GDA) may diverge or cycle even for bilinear matrix games (Daskalakis et al., 2018), quite remarkably, small modifications lead to guaranteed last-iterate convergence to the equilibrium in a non-asymptotic fashion. A flurry of algorithms is proposed, including Optimistic Gradient Descent Ascent (OGDA) (Rakhlin and Sridharan, 2013; Daskalakis and Panageas, 2018b; Wei et al., 2021b), predictive updates (Yadav et al., 2017), implicit updates (Liang and Stokes, 2019), and more. Several unified analyses of these algorithms have been carried out (see, e.g. Mokhtari et al. (2020a); Liang and Stokes (2019) and references therein), where these methods in principle all make clever extrapolation of the local curvature in a predictive manner to accelerate convergence. With slight abuse of terminology, in this paper, we refer to this ensemble of algorithms as extragradient methods (Korpelevich, 1976; Tseng, 1995; Mertikopoulos et al., 2018a; Harker and Pang, 1990).
+
+However, saddle-point optimization in the constrained setting, which includes competitive games as a special case, remains largely under-explored even for bilinear matrix games. While it is possible to reformulate constrained bilinear games to unconstrained ones using softmax parameterization of the probability simplex, this approach falls short of preserving the bilinear structure and convexconcave properties in the original problem, which are crucial to the convergence of gradient methods. Therefore, there is a strong necessity of understanding and developing improved extragradient methods in the constrained setting. Daskalakis and Panageas (2018a) proposed the optimistic variant of the multiplicative weight updates (MWU) method (Arora et al., 2012) – which is extremely natural and popular for optimizing over probability simplexes – called Optimistic Multiplicative Weight Updates (OMWU), and established the asymptotic last-iterate convergence of OMWU for matrix games. Very recently, Wei et al. (2021b) established non-asymptotic last-iterate convergences of OMWU. However, these last-iterate convergence results require the Nash equilibrium to be unique, and cannot be applied to problems with multiple Nash equilibria.
+
+# 1.2 Our contributions
+
+Motivated by the algorithmic role of entropy regularization in single-agent RL (Neu et al., 2017; Geist et al., 2019; Cen et al., 2020) as well as its wide use in game theory to account for imperfect and noisy information (McKelvey and Palfrey, 1995; Savas et al., 2019), we initiate the design and analysis of extragradient algorithms using multiplicative updates for finding the quantal response equilibrium (QRE), which are solutions to competitive games with entropy regularization (McKelvey and Palfrey, 1995). While finding QRE is of interest in its own right, by controlling the knob of entropy regularization, the QRE provides a close approximation to the Nash equilibrium (NE), and in turn acts as a smoothing scheme for finding the NE. Our contributions are summarized below.
+
+• Near dimension-free last-iterate convergence to QRE of entropy-regularized matrix games. We propose two policy extragradient algorithms to solve entropy-regularized matrix games, namely the Predictive Update (PU) and OMWU methods, where both players execute symmetric and multiplicative updates without knowing the entire payoff matrix nor the opponent’s actions. Encouragingly, we show that the last iterate of the proposed algorithms converges to the unique QRE at a linear rate that is almost independent of the size of the action spaces. Roughly speaking, to find an $\epsilon$ -optimal QRE in terms of Kullback-Leibler (KL) divergence, it takes no more than $\begin{array} { r } { \widetilde O \left( \frac { 1 } { \eta \tau } \log \left( \frac { 1 } { \epsilon } \right) \right) } \end{array}$ iterations, where ${ \widetilde { O } } ( \cdot )$ hides logarithmic dependencies. Here, $\tau$ is the regularization parameter, and $\eta$ is the learning rate of both players. Maximizing the learning rate, the iteration complexity is bounded by $\widetilde { O } \left( ( 1 + \| A \| _ { \infty } / \tau ) \log ( 1 / \epsilon ) \right)$ , where $\left\| A \right\| _ { \infty } = \operatorname* { m a x } _ { i , j } \left| A _ { i , j } \right|$ is the $\ell _ { \infty }$ norm of the payoff matrix $A$ .
+
+• Last-iterate convergence to $\epsilon$ -NE of unregularized matrix games without uniqueness assumption. The QRE provides an accurate approximation to the NE by setting the entropy regularization $\tau$ sufficiently small, therefore our result directly translates to finding a NE with last-iterate convergence guarantee. Roughly speaking, to find an $\epsilon$ -NE (Zhang et al., 2020, Definition 2.1), it takes no more than $\begin{array} { r } { \widetilde { O } \left( 1 + \frac { \| A \| _ { \infty } } { \epsilon } \right) } \end{array}$ iterations with optimized learning rates, which is again independent of the size of the action spaces up to logarithmic factors. Unlike prior literature (Daskalakis and Panageas, 2018a; Wei et al., 2021b), our last-iterate convergence guarantee does not require the NE to be unique.
+
+Table 1: Comparisons of last-iterate convergence of the proposed entropy-regularized PU and OMWU methods with prior results for finding $\epsilon$ -QRE or $\mathrm { \epsilon - N E }$ of competitive matrix games. We note that the convergence rates of unregularized OMWU established in Wei et al. (2021b) are problemdependent, and scale at least polynomially on the size of the action spaces. Desirable features in the last two columns are highlighted in blue.   
+
+<table><tr><td rowspan=1 colspan=1>Equilibriumtype</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Convergence rate</td><td rowspan=1 colspan=1>Dimension-free</td><td rowspan=1 colspan=1>Requireunique NE</td></tr><tr><td rowspan=1 colspan=1>E-QRE</td><td rowspan=1 colspan=1>PU&amp;OMWU(this work)</td><td rowspan=1 colspan=1>linear</td><td rowspan=1 colspan=1>yes</td><td rowspan=1 colspan=1>n/a</td></tr><tr><td rowspan=3 colspan=1>e-NE</td><td rowspan=1 colspan=1>OMWU(Daskalakis and Panageas, 2018a)</td><td rowspan=1 colspan=1>asymptotic</td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>yes</td></tr><tr><td rowspan=1 colspan=1>OMWU(Wei et al., 2021b)</td><td rowspan=1 colspan=1>sublinear + linear</td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>yes</td></tr><tr><td rowspan=1 colspan=1>PU&amp;OMWU(this work)</td><td rowspan=1 colspan=1>sublinear</td><td rowspan=1 colspan=1>yes</td><td rowspan=1 colspan=1>no</td></tr></table>
+
+• Extensions to two-player zero-sum Markov games. By connecting value iteration with matrix games, we propose a policy extragradient method for solving infinite-horizon discounted entropyregularized zero-sum Markov games, which finds an $\epsilon$ -optimal minimax soft Q-function—in terms of $\ell _ { \infty }$ error—in at most $\begin{array} { r } { \widetilde O \left( \frac { 1 } { \tau ( 1 - \gamma ) ^ { 2 } } \log ^ { 2 } \left( \frac { 1 } { \epsilon } \right) \right) } \end{array}$ iterations, where $\gamma \in ( 0 , 1 )$ is the discount factor.
+
+To the best of our knowledge, our paper is the first that develops policy extragradient algorithms for solving entropy-regularized competitive games with multiplicative updates and dimension-free linear last-iterate convergence, and demonstrates entropy regularization as a smoothing technique to find $\mathrm { \epsilon - N E }$ without the uniqueness assumption. Table 1 provides detailed comparisons of the proposed methods with prior arts for solving matrix games. Our results highlight the positive role of entropy regularization for accelerating convergence and safeguarding against imperfect information in competitive games. We defer the complete proof of our results to Cen et al. (2021).
+
+# 1.3 Related works
+
+Our work lies at the intersection of saddle-point optimization, game theory, and reinforcement learning. In what follows, we discuss a few topics that are closely related to ours.
+
+Unregularized matrix game. Freund and Schapire (1999) showed that many standard methods such as GDA and MWU have a converging average duality gap at the rate of $O ( 1 / \sqrt { T } )$ , which is improved to $O ( 1 / T )$ by considering optimistic variants of these methods, such as OGDA and OMWU (Rakhlin and Sridharan, 2013; Daskalakis et al., 2011; Syrgkanis et al., 2015). However, the last-iterate convergence of these methods are less understood until recently (Daskalakis and Panageas, 2018a; Wei et al., 2021b). In particular, under the assumption that the NE is unique for the unregularized matrix game, Daskalakis and Panageas (2018a) showed the asymptotic convergence of the last iterate of OMWU to the unique equilibrium, and Wei et al. (2021b) showed the last iterate of OMWU achieves a linear rate of convergence after an initial phase of sublinear convergence, however the rates therein can be highly pessimistic in terms of the problem dimension, while our rate for entropy-regularized OMWU is dimension-free up to logarithmic factors.
+
+Saddle-point optimization. Considerable progress has been made towards understanding OGDA and extragradient (EG) methods in the unconstrained convex-concave saddle-point optimization with general objective functions (Mokhtari et al., 2020a,b; Nemirovski, 2004; Liang and Stokes, 2019). However, the last-iterate convergence of constrained convex-concave saddle-point optimization still lacks theoretical understanding in general and most works fall short of characterizing a finite-time convergence result. In particular, Mertikopoulos et al. (2018a) demonstrated the asymptotic lastiterate convergence of EG, and Hsieh et al. (2019) investigated similar questions for single-call EG algorithms. Lei et al. (2021) showed that OMWU converges to the equilibrium locally without an explicit rate. Wei et al. (2021b) showed that the last-iterate of OGDA converges linearly for strongly-convex strongly-concave constrained saddle-point optimization with an explicit rate.
+
+Entropy regularization in RL and games. In single-agent RL, the role of entropy regularization as an algorithmic mechanism to encourage exploration and accelerate convergence has been investigated extensively (Neu et al., 2017; Geist et al., 2019; Mei et al., 2020; Cen et al., 2020; Lan, 2021; Zhan et al., 2021). Turning to the game setting, entropy regularization is used to account for imperfect information in the seminal work of McKelvey and Palfrey (1995) that introduced the QRE, and a few representative works on entropy and more general regularizations in games include Savas et al. (2019); Hofbauer and Sandholm (2002); Mertikopoulos and Sandholm (2016).
+
+Zero-sum Markov games. There have been a significant recent interest in developing provably efficient self-play algorithms for Markov games, including model-based algorithms (Perolat et al., 2015; Zhang et al., 2020), value-based algorithms (Bai and Jin, 2020; Xie et al., 2020), and policybased algorithms (Daskalakis et al., 2020; Wei et al., 2021a; Zhao et al., 2021). The iteration complexities in prior works (Perolat et al., 2015; Daskalakis et al., 2020; Wei et al., 2021a; Zhao et al., 2021) depend on various notions of concentrability coefficient and therefore can scale quite pessimistically with the problem dimension. Our approach can be regarded as a policy-based algorithm to approximate value iteration, which can be implemented in a decentralized manner with symmetric and multiplicative updates from both players, and the iteration complexity is almost independent of the size of the state-action space.
+
+Notation. We denote by $\Delta ( \mathcal { A } )$ the probability simplex over the set $\mathcal { A }$ . We overload the functions such as $\log ( \cdot )$ and $\exp ( \cdot )$ to take vector inputs with the understanding that the function is applied in an entrywise manner. For instance, given any vector $z = [ z _ { i } ] _ { 1 \leq i \leq n } \in \mathbb { R } ^ { n }$ , the notation $\exp ( z )$ denotes $\exp ( z ) : = [ \exp ( z _ { i } ) ] _ { 1 \leq i \leq n }$ ; other functions are defined analogously. Given two probability distributions $\mu$ and $\mu ^ { \prime }$ over $\mathcal { A }$ , the KL divergence from $\mu ^ { \prime }$ to $\mu$ is defined by $\begin{array} { r } { \mathsf { K L } ( \mu \parallel \mu ^ { \prime } ) : = \sum _ { a \in \mathcal { A } } \mu ( a ) \log \frac { \mu ( a ) } { \mu ^ { \prime } ( a ) } } \end{array}$ . Given a matrix $A$ , $\| A \| _ { \infty }$ is used to denote entrywise maximum norm, namely, $\left\| A \right\| _ { \infty } = \operatorname* { m a x } _ { i , j } \left| A _ { i , j } \right|$ . The all-one vector is denoted as 1.
+
+# 2 Zero-sum matrix games with entropy regularization
+
+We first consider a two-player zero-sum game with bilinear objective and probability simplex constraints, and demonstrate the positive role of entropy regularization in solving this problem. Throughout this paper, let $\mathcal { A } = \{ 1 , \dotsc , m \}$ and $\boldsymbol { B } = \{ 1 , \ldots , \bar { n } \}$ be the action spaces of each player.
+
+# 2.1 Background and problem formulation
+
+Zero-sum two-player matrix game. The focal point of this subsection is a constrained two-player zero-sum matrix game, which can be formulated as the following min-max problem (or saddle point optimization problem):
+
+$$
+\operatorname* { m a x } _ { \mu \in \Delta ( \mathcal { A } ) } \operatorname* { m i n } _ { \nu \in \Delta ( \mathcal { B } ) } f ( \mu , \nu ) : = \mu ^ { \top } A \nu ,
+$$
+
+where $\textbf { \textit { A } } \in \mathbb { R } ^ { m \times n }$ denotes the payoff matrix, $\mu ~ \in ~ \Delta ( { \mathcal { A } } )$ and $\nu ~ \in ~ \Delta ( B )$ stand for the mixed/randomized policies of each player, defined respectively as distributions over the probability simplex $\Delta ( \mathcal { A } )$ and $\Delta ( B )$ . A pair of policies $( \mu ^ { \star } , \nu ^ { \star } )$ is said to be a Nash equilibrium (NE) of (1) if $f ( \mu ^ { \star } , \nu ) \geq f ( \mu ^ { \star } , \nu ^ { \star } ) \geq f ( \mu , \nu ^ { \star } )$ for all $( \mu , \nu ) \in \Delta ( \mathcal { A } ) \times \Delta ( \mathcal { B } )$ .
+
+Entropy-regularized zero-sum two-player matrix game. There is no shortage of scenarios where the payoff matrix $A$ might not be known perfectly. In an attempt to accommodate imperfect knowledge of $A$ , McKelvey and Palfrey (1995) proposed a seminal extension to the Nash equilibrium called the quantal response equilibrium $( Q R E )$ when the payoffs are perturbed by Gumbel-distributed noise. Formally, this amounts to solving the following matrix game with entropy regularization (Mertikopoulos and Sandholm, 2016):
+
+$$
+\operatorname* { m a x } _ { \mu \in \Delta ( \mathcal { A } ) } \operatorname* { m i n } _ { \nu \in \Delta ( \mathcal { B } ) } f _ { \tau } ( \mu , \nu ) : = \mu ^ { \top } A \nu + \tau \mathcal { H } ( \mu ) - \tau \mathcal { H } ( \nu ) ,
+$$
+
+where $\begin{array} { r } { \mathcal { H } ( \pi ) = - \sum _ { i } \pi _ { i } \log ( \pi _ { i } ) } \end{array}$ denotes the Shannon entropy of a distribution $\pi$ , and $\tau \geq 0$ is the regularization parameter. As is well known, the optimal solution $( \mu _ { \tau } ^ { \star } , \nu _ { \tau } ^ { \star } )$ to (2), dubbed as the QRE,
+
+is unique whenever $\tau > 0$ (due to the presence of strong concavity/convexity), which satisfies the following fixed point equations:
+
+$$
+\begin{array} { r } { \biggr \{ \mu _ { \tau } ^ { \star } ( a ) = \frac { \exp ( [ A \nu _ { \tau } ^ { \star } ] a / \tau ) } { \sum _ { a = 1 } ^ { m } \exp ( [ A \nu _ { \tau } ^ { \star } ] a / \tau ) } \propto \exp ( [ A \nu _ { \tau } ^ { \star } ] a / \tau ) , \qquad \mathrm { f o r ~ a l l } \ a \in \mathcal { A } , } \\ { \nu _ { \tau } ^ { \star } ( b ) = \frac { \exp ( - [ A ^ { \top } \mu _ { \tau } ^ { \star } ] b / \tau ) } { \sum _ { b = 1 } ^ { n } \exp ( - [ A ^ { \top } \mu _ { \tau } ^ { \star } ] b / \tau ) } \propto \exp ( - [ A ^ { \top } \mu _ { \tau } ^ { \star } ] b / \tau ) , \quad \mathrm { f o r ~ a l l } \ b \in \mathcal { B } . } \end{array}
+$$
+
+Goal. We aim to efficiently compute the QRE of the entropy-regularized matrix game in a decentralized manner, and investigate how an efficient solver of QRE can be leveraged to find a NE of the unregularized matrix game (1). Namely, we only assume access to “first-order information” as opposed to full knowledge of the payoff matrix $A$ or the actions of the opponent. The information received by each player is formally described in the following sampling oracle.
+
+Definition 1 (Sampling oracle for matrix games). For any policy pair $( \mu , \nu )$ and payoff matrix $A$ the sampling oracle returns the exact values of $\mu ^ { \top } A$ and $A \nu$ .
+
+Additional notation. For notational convenience, we let $\zeta$ represent the concatenation of $\mu \in \mathbb { R } ^ { | \mathcal { A } | }$ and $\nu \in \mathbb { R } ^ { | B | }$ , namely, $\zeta = ( \mu , \nu )$ . The solution to (2), which is specified in (3), is denoted by $\zeta _ { \tau } ^ { \star } = ( \mu _ { \tau } ^ { \star } , \nu _ { \tau } ^ { \star } )$ . For any $\zeta = ( \mu , \nu )$ and $\zeta ^ { \prime } = ( \mu ^ { \prime } , \nu ^ { \prime } )$ , we shall often abuse the notation and let $\mathsf { K L } \big ( \zeta \| \zeta ^ { \prime } \big ) = \mathsf { K L } \big ( \mu \| \mu ^ { \prime } \big ) + \mathsf { K L } \big ( \nu \| \nu ^ { \prime } \big )$ . The duality gap of the entropy-regularized matrix game (2) at $\zeta = ( \mu , \nu )$ is defined as $\begin{array} { r } { \mathsf { D u a l G a p } _ { \tau } ( \zeta ) = \operatorname* { m a x } _ { \mu ^ { \prime } \in \Delta ( A ) } f _ { \tau } ( \mu ^ { \prime } , \nu ) - \operatorname* { m i n } _ { \nu ^ { \prime } \in \Delta ( B ) } f _ { \tau } ( \mu , \nu ^ { \prime } ) } \end{array}$ which is clearly nonnegative and $\mathsf { D u a l G a p } _ { \tau } ( \boldsymbol { \zeta } _ { \tau } ^ { \star } ) = 0$ . Similarly, let the optimality gap of the entropyregularized matrix game (2) at $\zeta = ( \mu , \nu )$ be $\mathsf { O p t G a p } ( \zeta ) = \left| f _ { \tau } ( \mu , \nu ) - f _ { \tau } ( \mu _ { \tau } ^ { \star } , \nu _ { \tau } ^ { \star } ) \right|$ .
+
+# 2.2 Proposed extragradient methods: PU and OMWU
+
+To begin, assume we are given a pair of policies $z _ { 1 } \in \Delta ( \mathcal { A } )$ , $z _ { 2 } \in \Delta ( B )$ employed by each player respectively. If we proceed with fictitious play, i.e. player 1 (resp. player 2) aims to optimize its own policy by assuming the opponent’s policy is fixed as $z _ { 2 }$ (resp. $z _ { 1 }$ ), the saddle-point optimization problem (2) is then decoupled into two independent min/max optimization problems:
+
+$$
+\operatorname* { m a x } _ { \mu \in \Delta ( A ) } \mu ^ { \top } A z _ { 2 } + \tau \mathcal { H } ( \mu ) - \tau \mathcal { H } ( z _ { 2 } ) \qquad \mathrm { a n d } \qquad \operatorname* { m i n } _ { \nu \in \Delta ( B ) } z _ { 1 } ^ { \top } A \nu + \tau \mathcal { H } ( z _ { 1 } ) - \tau \mathcal { H } ( \nu ) ,
+$$
+
+which are naturally solved via mirror descent / ascent with KL divergence. Specifically, one step of mirror descent / ascent takes the form
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \mu ^ { ( t + 1 ) } ( a ) \propto \mu ^ { ( t ) } ( a ) ^ { 1 - \eta \tau } \exp ( \eta [ A z _ { 2 } ] _ { a } ) , } & { \mathrm { f o r ~ a l l ~ } a \in \mathcal { A } , } \\ { \nu ^ { ( t + 1 ) } ( b ) \propto \nu ^ { ( t ) } ( b ) ^ { 1 - \eta \tau } \exp ( - \eta [ A ^ { \top } z _ { 1 } ] _ { b } ) , } & { \mathrm { f o r ~ a l l ~ } b \in \mathcal { B } , } \end{array} \right. } \end{array}
+$$
+
+where $\eta$ is the learning rate. The above update rule forms the basis of our algorithm design.
+
+Motivation: a form of implicit updates with linear convergence. It turns out, if we could select the policy pair $( z _ { 1 } , z _ { 2 } ) = \hat { \zeta } ^ { ( t + 1 ) } : \stackrel { \bullet } { = } ( \mu ^ { ( t + 1 ) } , \nu ^ { ( t + 1 ) } )$ as the ones to be taken in the future, and call the resulting update rule as the Implicit Update (IU) method:
+
+$$
+\begin{array} { r } { \left\{ \mu ^ { ( t + 1 ) } ( a ) \propto \mu ^ { ( t ) } ( a ) ^ { 1 - \eta \tau } \exp ( \eta [ A \nu ^ { ( t + 1 ) } ] _ { a } ) , \right. \left. \mathrm { f o r ~ a l l } \ : a \in \mathcal { A } , \right. } \\ { \nu ^ { ( t + 1 ) } ( b ) \propto \nu ^ { ( t ) } ( b ) ^ { 1 - \eta \tau } \exp ( - \eta [ A ^ { \top } \mu ^ { ( t + 1 ) } ] _ { b } ) , \left. \mathrm { f o r ~ a l l } \ : b \in \mathcal { B } . \right. } \end{array}
+$$
+
+Though unrealistic — since it uses the future updates — it leads to a one-step convergence to the QRE when $\eta = 1 / \tau$ (see the optimality condition in (3)). Encouragingly, we have the following linear convergence guarantee of IU when adopting a general learning rate.
+
+Proposition 1 (Linear convergence of IU). Assume $0 < \eta \leq 1 / \tau$ , then for all $t \geq 0$ , the iterates $\zeta ^ { ( t ) } : = ( \mu ^ { ( t ) } , \nu ^ { ( t ) } )$ of the $I U$ method in (5) satisfy $\mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \parallel \zeta ^ { ( t ) } \big ) \leq ( 1 - \eta \tau ) ^ { t } \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \parallel \zeta ^ { ( 0 ) } \big )$ .
+
+In words, the IU method achieves an appealing linear rate of convergence that is independent of the problem dimension. Motivated by this observation, we seek to design algorithms where the policies $( z _ { 1 } , z _ { 2 } )$ employed in (4) serve as good predictions of $( \mu ^ { ( t + 1 ) } , \nu ^ { ( t + \bar { 1 ) } } )$ , such that the resulting algorithms are both practical and retain the appealing convergence rate of IU.
+
+Proposed algorithms. We propose two extragradient algorithms for solving the entropy-regularized matrix game, namely the Predictive Update $( P U )$ method and the Optimistic Multiplicative Weights
+
+# Algorithm 1: The PU method
+
+# Algorithm 2: The OMWU method
+
+1 initialization: $\mu ^ { ( 0 ) }$ , $\nu ^ { ( 0 ) }$
+
+2 for $t = 0 , 1 , 2 , \cdots$ do
+
+2 for $t = 0 , 1 , 2 , \cdots$ do
+
+3 Update $\bar { \mu }$ and $\bar { \nu }$ according to
+
+3
+
+Update $\bar { \mu }$ and $\bar { \nu }$ according to
+
+$$
+\begin{array} { r } { \left\{ \bar { \mu } ^ { ( t + 1 ) } ( a ) \propto { \mu } ^ { ( t ) } ( a ) ^ { 1 - \eta \tau } \exp ( \eta [ A \nu ^ { ( t ) } ] _ { a } ) , \right. \qquad } \\ { \left. \bar { \nu } ^ { ( t + 1 ) } ( b ) \propto \nu ^ { ( t ) } ( b ) ^ { 1 - \eta \tau } \exp ( - \eta [ A ^ { \top } { \mu } ^ { ( t ) } ] _ { b } ) . \right. } \end{array}
+$$
+
+$$
+\begin{array} { r } { \left\{ \bar { \mu } ^ { ( t + 1 ) } ( a ) \propto { \mu } ^ { ( t ) } ( a ) ^ { 1 - \eta \tau } \exp ( \eta [ A \bar { \nu } ^ { ( t ) } ] _ { a } ) , \right. \mathrm { ~ } } \\ { \left. \bar { \nu } ^ { ( t + 1 ) } ( b ) \propto \nu ^ { ( t ) } ( b ) ^ { 1 - \eta \tau } \exp ( - \eta [ A ^ { \top } \bar { \mu } ^ { ( t ) } ] _ { b } ) . \right. } \end{array}
+$$
+
+4 Update $\mu$ and $\nu$ according to
+
+Update $\mu$ and $\nu$ according to
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \mu ^ { ( t + 1 ) } ( a ) \propto \mu ^ { ( t ) } ( a ) ^ { 1 - \eta \tau } \exp ( \eta [ A \bar { \nu } ^ { ( t + 1 ) } ] _ { a } ) , } \\ { \nu ^ { ( t + 1 ) } ( b ) \propto \nu ^ { ( t ) } ( b ) ^ { 1 - \eta \tau } \exp ( - \eta [ A ^ { \top } \bar { \mu } ^ { ( t + 1 ) } ] _ { b } ) . } \end{array} \right. } \end{array}
+$$
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \mu ^ { ( t + 1 ) } ( a ) \propto \mu ^ { ( t ) } ( a ) ^ { 1 - \eta \tau } \exp ( \eta [ A \bar { \nu } ^ { ( t + 1 ) } ] _ { a } ) , } \\ { \nu ^ { ( t + 1 ) } ( b ) \propto \nu ^ { ( t ) } ( b ) ^ { 1 - \eta \tau } \exp ( - \eta [ A ^ { \top } \bar { \mu } ^ { ( t + 1 ) } ] _ { b } ) . } \end{array} \right. } \end{array}
+$$
+
+Update (OMWU) method, the latter adapted from Rakhlin and Sridharan (2013); Daskalakis et al. (2011). Detailed procedures can be found in Algorithm 1 and Algorithm 2, respectively. On a high level, both algorithms maintain two intertwined sequences $\{ ( \mu ^ { ( \bar { t } ) } , \nu ^ { ( t ) } ) \} _ { t \geq 0 }$ and $\{ ( \bar { \mu } ^ { ( \dot { t } ) } , \bar { \nu } ^ { ( t ) } ) \} _ { t \geq 0 }$ , and in each iteration $t = 0 , 1 , \ldots$ , proceed in two steps:
+
+• The midpoint $( \bar { \mu } ^ { ( t + 1 ) } , \bar { \nu } ^ { ( t + 1 ) } )$ serves as a prediction of $( \mu ^ { ( t + 1 ) } , \nu ^ { ( t + 1 ) } )$ by running one step of mirror descent / ascent (cf. (4)) from either $( z _ { 1 } , z _ { 2 } ) = ( \mu ^ { ( t ) } , \nu ^ { ( t ) } )$ (for PU) or $( z _ { 1 } , z _ { 2 } ) = ( \bar { \mu } ^ { ( t ) } , \bar { \nu } ^ { ( t ) } )$ (for OMWU).
+
+• The update of $( \boldsymbol { \mu } ^ { ( t + 1 ) } , \boldsymbol { \nu } ^ { ( t + 1 ) } )$ then mimics the implicit update (5) using the prediction $( \bar { \mu } ^ { ( t + 1 ) } , \bar { \nu } ^ { ( t + 1 ) } )$ obtained above.
+
+When the proposed algorithms converge, both $( \mu ^ { ( t ) } , \nu ^ { ( t ) } )$ and $( \bar { \mu } ^ { ( t ) } , \bar { \nu } ^ { ( t ) } )$ converge to the same point. The two players are completely symmetric and adopt the same learning rate, and require only first-order information provided by the sampling oracle. While the two algorithms resemble each other in many aspects, a key difference lies in the query and use of the sampling oracle: in each iteration, OMWU makes a single call to the sampling oracle for gradient evaluation, while PU calls the sampling oracle twice. It is worth noting that, when $\tau = 0$ (i.e., no entropy regularization is enforced), the OMWU method in Algorithm 2 reduces to the method analyzed in Rakhlin and Sridharan (2013); Daskalakis and Panageas (2018a); Wei et al. (2021b) without entropy regularization.
+
+Remark 1. It is worth highlighting that the proposed algorithms are different from Mertikopoulos et al. (2018a), as the extragradient is only applied to the bilinear term but not the entropy regularization term. This seemingly small, but important, difference leads to a more concise closed-form update rule and a cleaner analysis, as shall be seen momentarily.
+
+# 2.3 Performance guarantees
+
+We are now positioned to present our main theorem concerning the last-iterate convergence of PU and OMWU for solving (2).
+
+Theorem 1 (Last-iterate convergence of PU and OMWU). Suppose that the learning rates $\eta = \eta _ { \mathsf { P U } }$ of $P U$ in Algorithm $I$ and $\eta = \eta$ OMWU of OMWU in Algorithm 2 satisfy
+
+$$
+0 < \eta _ { \mathsf { P U } } \leq \frac { 1 } { \tau + 2 \left\| A \right\| _ { \infty } } , a n d 0 < \eta _ { \mathsf { O M W U } } \leq \operatorname* { m i n } \left\{ \frac { 1 } { 2 \tau + 2 \left\| A \right\| _ { \infty } } , \frac { 1 } { 4 \left\| A \right\| _ { \infty } } \right\} .
+$$
+
+Then for any $t \geq 0$ , the iterates $\zeta ^ { ( t ) } = ( \mu ^ { ( t ) } , \nu ^ { ( t ) } )$ and $\bar { \zeta } ^ { ( t ) } = ( \bar { \mu } ^ { ( t ) } , \bar { \nu } ^ { ( t ) } )$ of $P U$ and OMWU achieve
+
+# • Linear convergence of policies in KL divergence and entrywise log-ratios:
+
+$$
+\begin{array} { r l } & { \operatorname* { m a x } \left\{ \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \| \zeta ^ { ( t ) } \big ) , \frac { 1 } { 2 } \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \| \bar { \zeta } ^ { ( t + 1 ) } \big ) \right\} \leq ( 1 - \eta \tau ) ^ { t } \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \| \zeta ^ { ( 0 ) } \big ) , } \\ & { \left\| \log \frac { \zeta ^ { ( t ) } } { \zeta _ { \tau } ^ { \star } } \right\| _ { \infty } \leq 2 ( 1 - \eta \tau ) ^ { t } \left\| \log \frac { \zeta ^ { ( 0 ) } } { \zeta _ { \tau } ^ { \star } } \right\| _ { \infty } + \frac { 8 \| A \| _ { \infty } } { \tau } ( 1 - \eta \tau ) ^ { t / 2 } \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \| \zeta ^ { ( 0 ) } \big ) ^ { 1 / 2 } . } \end{array}
+$$
+
+# • Linear convergence of values in optimality and duality gaps:
+
+$$
+\begin{array} { r l } & { \mathsf { O p t G a p } _ { \tau } ( \bar { \zeta } ^ { ( t ) } ) \leq \eta ^ { - 1 } \cdot \frac { 1 } { 1 - ( \tau + \| A \| _ { \infty } ) \eta } \cdot \frac { ( 1 - \eta \tau ) ^ { t } } { 1 - ( 1 - \eta \tau ) ^ { t } } \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \| \zeta ^ { ( 0 ) } \big ) , } \\ & { \mathsf { D u a l G a p } _ { \tau } ( \bar { \zeta } ^ { ( t ) } ) \leq \left( \eta ^ { - 1 } + 2 \tau ^ { - 1 } \| A \| _ { \infty } ^ { 2 } \right) ( 1 - \eta \tau ) ^ { t - 1 } \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \| \zeta ^ { ( 0 ) } \big ) . } \end{array}
+$$
+
+Remark 2. Setting $\mu ^ { ( 0 ) }$ and $\nu ^ { ( 0 ) }$ to be uniform policies leads to a universal bound
+
+$$
+\begin{array} { r } { \mathsf { K L } \big ( \zeta _ { \tau } ^ { \star } \| \zeta ^ { ( 0 ) } \big ) = \log | \mathcal { A } | + \log | \mathcal { B } | - \mathcal { H } ( \mu _ { \tau } ^ { \star } ) - \mathcal { H } ( \nu _ { \tau } ^ { \star } ) \leq \log | \mathcal { A } | + \log | \mathcal { B } | . } \end{array}
+$$
+
+Remark 3. Similar results continue to hold even when the two players use different regularization parameters $\tau _ { \mu } , \tau _ { \nu } > 0$ in (2), as long as the regularization parameter $\tau$ is replaced by max $\{ \tau _ { \mu } , \tau _ { \nu } \}$ in the upper bounds of the learning rate, and the contraction parameter is replaced by $1 - \operatorname* { m i n } \{ \tau _ { \mu } , \tau _ { \nu } \} \eta$ .
+
+Theorem 1 characterizes the convergence of the last-iterates $\zeta ^ { ( t ) }$ and $\bar { \zeta } ^ { ( t ) }$ of PU and OMWU as long as the learning rate lies within the specified ranges. While PU doubles the number of calls to the sampling oracle, it also allows roughly as large as twice the learning rate compared with OMWU (cf. (6)). Compared with the vast literature analyzing the average-iterate performance of variants of extragradient methods, our results contribute towards characterizing the last-iterate convergence of multiplicative update methods in the presence of entropy regularization and simplex constraints, which to the best of our knowledge, are the first of its kind. Several remarks are in order.
+
+Linear convergence to QRE. To achieve an $\epsilon$ -accurate estimate of the QRE in terms of the KL divergence, the bound (7a) tells that it is sufficient to take
+
+$$
+\frac { 1 } { \eta \tau } \log \left( \frac { \log | \cal { A } | + \log | \cal { B } | } { \epsilon } \right)
+$$
+
+iterations using either PU or OMWU. Notably, this iteration complexity does not depend on any hidden constants and only depends double logarithmically on the cardinality of action spaces, which is almost dimension-free. Maximizing the learning rate, the iteration complexity is bounded by $( 1 + \| A \| _ { \infty } / \tau ) \log ( 1 / \epsilon )$ (modulo log factors), which only depends on the ratio $\| A \| _ { \infty } / \tau$ .
+
+Entrywise error of the policy log-ratios. Both PU and OMWU enjoy strong entrywise guarantees in the sense we can guarantee the convergence of the $\ell _ { \infty }$ norm of the log-ratios between the learned policy pair and the QRE at the same dimension-free linear rate (cf. (7b)), which suggests the policy pair converges in a somewhat uniform manner across the entire action space.
+
+Linear convergence of optimality and duality gaps. Our theorem also establishes the last-iterate convergence of the game values in terms of the optimality gap (cf. (7c)) and the duality gap (cf. (7d)) for both PU and OMWU. In particular, as will be seen, bounding the optimality gap of matrix games turns out to be the key enabler for generalizing our algorithms to Markov games, and bounding the duality gap allows to directly translate our results to finding a NE of unregularized matrix games.
+
+Last-iterate convergence to approximate NE. The entropy-regularized matrix game can be thought as a smooth surrogate of the unregularized matrix game (1); in particular, it is possible to find an $\epsilon$ -NE by setting $\tau$ sufficiently small in (2). According to (Zhang et al., 2020, Definition 2.1), a policy pair $\bar { \zeta } = ( \mu , \bar { \nu } )$ is an $\epsilon$ -NE if it satisfies $\begin{array} { r } { \mathsf { D u a l G a p } ( \zeta ) : = \operatorname* { m a x } _ { \mu ^ { \prime } \in \Delta ( A ) } f ( \mu ^ { \prime } , \nu ) - \operatorname* { m i n } _ { \nu ^ { \prime } \in \Delta ( B ) } f ( \mu , \nu ^ { \prime } ) \leq \epsilon } \end{array}$ .
+
+Observe that setting $\begin{array} { r } { \tau = \frac { \epsilon / 4 } { \log | \mathcal { A } | + \log | \mathcal { B } | } } \end{array}$ guarantees that $| f _ { \tau } ( \mu , \nu ) - f ( \mu , \nu ) | < \epsilon / 4$ uniformly over $( \mu , \nu ) \in \Delta ( \mathcal { A } ) \times \Delta ( \mathcal { B } )$ in view of the boundedness of the Shannon entropy $\mathcal { H } ( \cdot )$ . Theorem 7 (cf. (7d)) also ensures that our proposed algorithms find an approximate QRE $\bar { \zeta } ^ { ( T ) }$ such that $\mathsf { D u a l G a p } _ { \tau } \bigl ( \bar { \zeta } ^ { ( T ) } \bigr ) \leq$ $\epsilon / 2$ after taking $\begin{array} { r } { T = \widetilde { O } \left( \frac { 1 } { \eta \epsilon } \right) } \end{array}$ iterations, which is no more than $\begin{array} { r } { \widetilde { O } \left( 1 + \frac { \| A \| _ { \infty } } { \epsilon } \right) } \end{array}$ iterations with optimized learning rates. It follows immediately that
+
+[ $\mathtt { \mathtt { M a l G a p } } ( \bar { \zeta } ^ { ( T ) } ) \le \mathtt { D u a l G a p } _ { \tau } ( \bar { \zeta } ^ { ( T ) } ) + \operatorname* { m a x } _ { \mu ^ { \prime } , \nu ^ { \prime } } \Big | f _ { \tau } ( \mu ^ { \prime } , \bar { \nu } ^ { ( T ) } ) - f _ { \tau } \big ( \bar { \mu } ^ { ( T ) } , \nu ^ { \prime } \big ) - \big ( f ( \mu ^ { \prime } , \bar { \nu } ^ { ( T ) } ) - f ( \bar { \mu } ^ { ( T ) } , \nu ^ { \prime } ) \big ) \Big | \le \epsilon ,$ and therefore $\bar { \zeta } ^ { ( T ) }$ is an $\mathrm { \epsilon - N E }$ . Intriguingly, unlike prior work (Daskalakis and Panageas, 2018a; Wei et al., 2021b) that analyzed the last-iterate convergence of OMWU in the unregularized setting $\mathit { \Pi } _ { \mathcal { T } } = 0 \mathit { \Pi } _ { \mathcal { c } }$ ), our last-iterate convergence does not require the NE of (1) to be unique.
+
+Rationality. Another attractive feature of the algorithms developed above is being rational (as introduced in Bowling and Veloso (2001)) in the sense that the algorithm returns the best-response policy of one player when the opponent takes any fixed stationary policy. More specially, in terms of matrix games, when player 2 sticks to a stationary policy $\nu$ , the update of player 1 reduces to $\mu ^ { ( t + 1 ) } ( a ) \bar { \propto } \mu ^ { ( t ) } ( a ) ^ { 1 - \bar { \eta tau } } \exp ( \eta [ A \nu ] _ { a } )$ . In this case, Theorem 1 can be established in exactly the same fashion by restricting attention only to the updates of $\mu ^ { ( t ) }$ .
+
+![](images/4e4a6167f1bf0473e8d7a714c8b874f38c7ea875b79a1295462ea5a711766939.jpg)  
+Figure 1: Performance illustration of the PU and OMWU methods for solving entropy-regularized matrix games with $| \mathcal { A } | = | \mathcal { B } | = 1 0 0$ , where the entries of the payoff matrix $A$ is generated independently from the uniform distribution on $[ - 1 , 1 ]$ . The learning rates are fixed as $\eta = 0 . 1$ . The left panel plots various error metrics of convergence w.r.t. the iteration count with $\tau = 0 . 0 1$ , while the right panel plots these error metrics at 1000-th iteration with different choices of $\tau$ .
+
+No-regret learning of OMWU. Besides convergence to equilibria, in game-theoretical settings, it is often desirable to design and implement no-regret algorithms, which are capable of providing black-box guarantees over arbitrary sequences played by the opponent (Cesa-Bianchi and Lugosi, 2006; Rakhlin and Sridharan, 2013). Fortunately, it turns out that entropy regularization not only accelerates the convergence, but also enables no-regret learning somewhat “for free”: it encourages exploration by putting a positive mass on every action, therefore guards against adversaries. By using a properly chosen learning rate schedule, the proposed OMWU (Algorithm 2) can be further established as a no-regret algorithm; the details can be found in (Cen et al., 2021).
+
+# 3 Zero-sum Markov games with entropy regularization
+
+Leveraging the success of PU and OMWU in solving the entropy-regularized matrix games, this section extends our current analysis to solve the zero-sum two-player Markov game with entropy regularization, which is again formulated as finding the equilibrium of a saddle-point optimization problem. We start by introducing its basic setup, which will be followed by the proposed policy extragradient method with its theoretical guarantees.
+
+# 3.1 Background and problem formulation
+
+We consider a discounted Markov Game (MG) which is defined as $\mathcal { M } = \{ { S , A , B , P , r , \gamma } \}$ , with discrete state space $s$ , action spaces of two players $\mathcal { A }$ and $\boldsymbol { B }$ , transition probability $P$ , reward function $r : \mathcal { S } \times \mathcal { A } \times \mathcal { B }  [ 0 , 1 ]$ and discount factor $\gamma \in [ 0 , 1 )$ . A policy $\mu : { \mathcal { S } }  \Delta ( { \mathcal { A } } )$ (resp. $\nu : S \to \Delta ( B ) )$ defines how player 1 (resp. player 2) reacts to a given state $s$ , where the probability of taking action $a \in { \mathcal { A } }$ (resp. $b \in B ,$ ) is $\mu ( a | s )$ (resp. $\nu ( b | s ) )$ . The transition probability kernel $P : \mathcal { S } \times \mathcal { A } \times \mathcal { B }  \Delta ( \mathcal { S } )$ defines the dynamics of the Markov game, where $P ( s ^ { \prime } | s , a , b )$ specifies the probability of transiting to state $s ^ { \prime }$ from state $s$ when the players take actions $a$ and $b$ respectively.
+
+Motivated by entropy regularization in Markov decision processes (MDP) (Geist et al., 2019), we consider an entropy-regularized variant of MG, where the value function is defined as
+
+$$
+V _ { \tau } ^ { \mu , \nu } ( s ) : = \mathbb { E } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \left( r ( s _ { t } , a _ { t } , b _ { t } ) - \tau \log \mu ( a _ { t } | s _ { t } ) + \tau \log \nu ( b _ { t } | s _ { t } ) \right) ~ \middle | s _ { 0 } = s \right] ,
+$$
+
+where the quantity $\tau \geq 0$ denotes the regularization parameter, and the expectation is evaluated over the randomness of the transition kernel as well as the policies. The regularized Q-function $Q _ { \tau } ^ { \mu , \nu }$ of a
+
+policy pair $( \mu , \nu )$ is related to $V _ { \tau } ^ { \mu , \nu }$ as
+
+$$
+\begin{array} { r } { Q _ { \tau } ^ { \mu , \nu } ( s , a , b ) = r ( s , a , b ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim P ( \cdot \mid s , a , b ) } \bigl [ V _ { \tau } ^ { \mu , \nu } \bigl ( s ^ { \prime } \bigr ) \bigr ] . } \end{array}
+$$
+
+We will call $V _ { \tau } ^ { \mu , \nu }$ and $Q _ { \tau } ^ { \mu , \nu }$ the soft value function and soft $Q$ -function, respectively. A policy pair $( \mu _ { \tau } ^ { \star } , \nu _ { \tau } ^ { \star } )$ is said to be the quantal response equilibrium (QRE) of the entropy-regularized MG, if its value attains the minimax value of the entropy-regularized MG over all states $s \in S$ , i.e.
+
+$$
+V _ { \tau } ^ { \star } ( s ) = \operatorname* { m a x } _ { \mu } \operatorname* { m i n } _ { \nu } V _ { \tau } ^ { \mu , \nu } ( s ) = \operatorname* { m i n } _ { \nu } \operatorname* { m a x } _ { \mu } V _ { \tau } ^ { \mu , \nu } ( s ) : = V _ { \tau } ^ { \mu _ { \tau } ^ { \star } , \nu _ { \tau } ^ { \star } } ( s ) ,
+$$
+
+where ${ \cal V } _ { \tau } ^ { \star }$ is called the optimal minimax soft value function, and similarly $Q _ { \tau } ^ { \star } : = Q _ { \tau } ^ { \mu _ { \tau } ^ { \star } , \nu _ { \tau } ^ { \star } }$ is called the optimal minimax soft Q-function.
+
+Goal. Our goal is to find the QRE of the entropy-regularized MG in a decentralized manner where the players only observe its own reward without accessing the opponent’s actions. By setting the regularization parameter sufficiently small $\tau$ , this also allows us to find an approximate NE of the unregularized MG.
+
+# 3.2 From value iteration to policy extragradient methods
+
+Entropy-regularized value iteration. It is known that classical dynamic programming approaches such as value iteration can be extended to solve MG (Perolat et al., 2015), where each iteration amounts to solving a series of matrix games for each state. Similar to the single-agent case (Cen et al., 2020), we can extend these approaches to solve the entropy-regularized MG. Setting the stage, let us introduce the per-state $\mathbf { Q }$ -value matrix $Q ( s ) : = Q ( s , \cdot , \cdot ) \bar { \in \mathbb { R } ^ { | \mathcal { A } | \times | \mathcal { B } | } }$ for every $s \in S$ , where the element indexed by the action pair $( a , b )$ is $Q ( s , a , b )$ . Similarly, we define the per-state policies $\mu ( s ) : = \mu ( \cdot | s ) \in \Delta ( { \dot { A } } )$ and $\nu ( s ) : = \nu ( \cdot | s ) \in \Delta ( B )$ for both players.
+
+In parallel to the original Bellman operator, we denote the soft Bellman operator $\mathcal { T } _ { \tau }$ as
+
+$$
+\mathcal { T } _ { \tau } ( Q ) ( s , a , b ) : = r ( s , a , b ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim P ( \cdot \vert s , a , b ) } \left[ \operatorname* { m a x } _ { \mu ( s ^ { \prime } ) \in \Delta ( A ) } \operatorname* { m i n } _ { \nu ( s ^ { \prime } ) \in \Delta ( B ) } f _ { \tau } \left( Q ( s ^ { \prime } ) ; \mu ( s ^ { \prime } ) , \nu ( s ^ { \prime } ) \right) \right] ,
+$$
+
+where for each per-state Q-value matrix $Q ( s )$ , we introduce an entropy-regularized matrix game in the form of
+
+$$
+\displaystyle \operatorname* { m a x } _ { \mu \in \Delta ( A ) } \operatorname* { m i n } _ { \nu \in \Delta ( B ) } f _ { \tau } \big ( Q ( s ) ; \mu ( s ) , \nu ( s ) \big ) : = \mu ( s ) ^ { \top } Q ( s ) \nu ( s ) - \tau \mathcal { H } ( \mu ( s ) ) + \tau \mathcal { H } ( \nu ( s ) ) .
+$$
+
+The entropy-regularized value iteration then proceeds as
+
+$$
+Q ^ { ( t + 1 ) } = T _ { \tau } ( Q ^ { ( t ) } ) ,
+$$
+
+where $Q ^ { ( 0 ) }$ is an initialization. By definition, the optimal minimax soft Q-function obeys ${ \cal T } _ { \tau } ( Q _ { \tau } ^ { \star } ) =$ $Q _ { \tau } ^ { \star }$ and therefore corresponds to the fix point of the soft Bellman operator. Given the above entropyregularized value iteration, the following lemma states its iterates contract linearly to the optimal minimax soft Q-function at a rate of the discount factor $\gamma$ .
+
+Proposition 2. The entropy-regularized value iteration (10) converges at a linear rate, i.e. $\parallel Q ^ { ( t ) } -$   
+$Q _ { \tau } ^ { \star } \| _ { \infty } \leq \gamma ^ { t } \| Q ^ { ( 0 ) } - Q _ { \tau } ^ { \star } \| _ { \infty }$ .
+
+Approximate value iteration via policy extragradient methods. Proposition 2 suggests that the optimal minimax soft Q-function of the entropy-regularized MG can be found by solving a series of entropy-regularized matrix games induced by $\bar { \{ Q ^ { ( t ) } \} } _ { t \geq 0 }$ in (10), a task that can be accomplished by adopting the fast extragradient methods developed earlier. To proceed, we first define the following sampling oracle, which makes it rigorous that the proposed algorithm does not require access to the Q-function of the entire MG, but only its own single-agent Q-function when playing against the opponent’s policy.
+
+Definition 2 (Sampling oracle for Markov games). Given any policy pair $\mu ( s ) , \nu ( s )$ and $Q$ -value matrix $Q ( s )$ for any $s \in S$ , the sampling oracle returns
+
+$$
+[ Q ( s ) \nu ( s ) ] _ { a } = \mathbb { E } _ { b \sim \nu ( s ) } \left[ Q ( s , a , b ) \right] , \qquad a n d \qquad [ Q ( s ) ^ { \top } \mu ( s ) ] _ { b } = \mathbb { E } _ { a \sim \mu ( s ) } \left[ Q ( s , a , b ) \right]
+$$
+
+for any $a \in { \mathcal { A } }$ and $b \in B$ .
+
+# Algorithm 3: Policy Extragradient Method for Entropy-regularized Markov Game
+
+1 initialization: $Q ^ { ( 0 ) } = 0$ .   
+2 for $t = 0 , 1 , 2 , \cdots , T _ { \mathrm { m a i n } } \mathrm { { \bf d o } }$   
+3 Let $Q ^ { ( t ) }$ denote $\begin{array} { r } { Q ^ { ( t ) } ( s , a , b ) = r ( s , a , b ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim P ( \cdot | s , a , b ) } V ^ { ( t ) } ( s ^ { \prime } ) . } \end{array}$ (11)   
+4 Invoke PU (Algorithm 1) or OMWU (Algorithm 2) for $T _ { \mathrm { s u b } }$ iterations to solve the following entropy-regularized matrix game for every state $s$ , where the initialization is set as uniform distributions: $\operatorname* { m a x } _ { \mu ( s ) \in \Delta ( A ) } \operatorname* { m i n } _ { \nu ( s ) \in \Delta ( B ) } f _ { \tau } \big ( Q ^ { ( t ) } ( s ) ; \mu ( s ) , \nu ( s ) \big ) .$ Return the last iterate $\bar { \mu } ^ { ( t , T _ { \mathrm { s u b } } ) } ( s ) , \bar { \nu } ^ { ( t , T _ { \mathrm { s u b } } ) } ( s )$ .   
+5 Set $V ^ { ( t + 1 ) } ( s ) = f _ { \tau } \left( Q ^ { ( t ) } ( s ) ; \bar { \mu } ^ { ( t , \bar { T } _ { \mathrm { s u b } } ) } ( s ) , \bar { \nu } ^ { ( t , \bar { T } _ { \mathrm { s u b } } ) } ( s ) \right)$ .
+
+Encouragingly, by judiciously setting the number of iterations in both the outer loop (for updating the Q-value matrices) and the inner loop (for updating the QRE of the corresponding Q-value matrix), we are guaranteed to find the QRE of the entropy-regularized MG in a small number of iterations without solving the iteration-varying matrix games exactly, as dictated by the following theorem.
+
+Theorem 2. Assume $| { \mathcal { A } } | \geq | { \mathcal { B } } |$ and $\tau \leq 1$ . Setting $\begin{array} { r } { \eta = \frac { 1 - \gamma } { 2 ( 1 + \tau ( \log | \mathcal { A } | + 1 - \gamma ) ) } } \end{array}$ , the total iterations (namely, the product $T _ { \mathrm { m a i n } } \cdot T _ { \mathrm { s u b , } }$ ) required for Algorithm 3 to achieve $\left\| Q ^ { ( { \vec { T } _ { \operatorname* { m i n } } } ) } - Q _ { \tau } ^ { \star } \right\| _ { \infty } \leq \epsilon$ is at most $\begin{array} { r } { O \left( \frac { ( \log | \cal { A } | + 1 / \tau ) } { ( 1 - \gamma ) ^ { 2 } } \left( \log \frac { \log | \cal { A } | } { ( 1 - \gamma ) \epsilon } \right) ^ { 2 } \right) } \end{array}$ .
+
+Theorem 2 ensures that within $\begin{array} { r } { \widetilde O \left( \frac { 1 } { \tau ( 1 - \gamma ) ^ { 2 } } \log ^ { 2 } \left( \frac { 1 } { \epsilon } \right) \right) } \end{array}$ iterations, Algorithm 3 finds a pair of policies whose value is close to the optimal minimax soft Q-function $Q _ { \tau } ^ { \star }$ in an entrywise manner to a prescribed accuracy $\epsilon$ . Remarkably, the iteration complexity is independent of the dimensions of the state space and the action space (up to log factors).
+
+Remark 4 (Duality gap and solving the unregularized MG). Solving the entropy-regularized MG provides a viable strategy to find an $\epsilon$ -approximate NE of the unregularized MG, where the optimality of a policy pair is typically gauged by the duality gap. Fortunately, this can be achieved similar to the case of matrix games, and we refer interested readers to Cen et al. (2021) for details.
+
+# 4 Conclusions
+
+This paper develops provably efficient policy extragradient methods (PU and OMWU) for entropyregularized matrix games and Markov games, whose last iterates are guaranteed to converge linearly to the quantal response equilibrium at a linear rate. Encouragingly, the rate of convergence is independent of the dimension of the problem, i.e. the sizes of the space space and the action space. In addition, the last iterates of the proposed algorithms can also be used to locate Nash equilibria for the unregularized competitive games without assuming the uniqueness of the Nash equilibria by judiciously tuning the amount of regularization. This work opens up interesting opportunities for further investigations of policy extragradient methods for solving competitive games. For example, can we develop a two-time-scale policy extragradient algorithms for Markov games where the Qfunction is updated simultaneously with the policy but potentially at a different time scale, using samples, such as in an actor-critic algorithm (Konda and Tsitsiklis, 2000)?
+
+# Acknowledgments and Disclosure of Funding
+
+S. Cen and Y. Chi are supported in part by the grants ONR N00014-18-1-2142 and N00014-19-1- 2404, ARO W911NF-18-1-0303, NSF CCF-1901199, CCF-2007911 and CCF-2106778. Y. Wei is supported in part by the NSF grants CCF-2007911, DMS-2147546/2015447 and CCF-2106778.
+
+References   
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parse/train/74RmfBweB60/74RmfBweB60_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "Fast Policy Extragradient Methods for Competitive Games with Entropy Regularization ",
+        "text_level": 1,
+        "bbox": [
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+            122,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Shicong Cen Carnegie Mellon University shicongc@andrew.cmu.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yuting Wei University of Pennsylvania ytwei@wharton.upenn.edu ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yuejie Chi Carnegie Mellon University yuejiechi@cmu.edu ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            304,
+            535,
+            320
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper investigates the problem of computing the equilibrium of competitive games, which is often modeled as a constrained saddle-point optimization problem with probability simplex constraints. Despite recent efforts in understanding the last-iterate convergence of extragradient methods in the unconstrained setting, the theoretical underpinnings of these methods in the constrained settings, especially those using multiplicative updates, remain highly inadequate, even when the objective function is bilinear. Motivated by the algorithmic role of entropy regularization in single-agent reinforcement learning and game theory, we develop provably efficient extragradient methods to find the quantal response equilibrium (QRE)—which are solutions to zero-sum two-player matrix games with entropy regularization—at a linear rate. The proposed algorithms can be implemented in a decentralized manner, where each player executes symmetric and multiplicative updates iteratively using its own payoff without observing the opponent’s actions directly. In addition, by controlling the knob of entropy regularization, the proposed algorithms can locate an approximate Nash equilibrium of the unregularized matrix game at a sublinear rate without assuming the Nash equilibrium to be unique. Our methods also lead to efficient policy extragradient algorithms for solving entropy-regularized zero-sum Markov games at a linear rate. All of our convergence rates are nearly dimension-free, which are independent of the size of the state and action spaces up to logarithm factors, highlighting the positive role of entropy regularization for accelerating convergence. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            660,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Finding the equilibrium of competitive games, which can be viewed as constrained saddle-point optimization problems with probability simplex constraints, lies at the heart of modern machine learning and decision making paradigms such as Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), competitive reinforcement learning (RL) (Littman, 1994), game theory (Shapley, 1953), adversarial training (Mertikopoulos et al., 2018b), to name a few. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we study one of the most basic forms of competitive games, namely two-player zero-sum games, in both the matrix setting and the Markov setting. Our goal is to find the equilibrium policies of both players in an independent and decentralized manner (Daskalakis et al., 2020; Wei et al., 2021a) with guaranteed last-iterate convergence. Namely, each player will execute symmetric and independent updates iteratively using its own payoff without observing the opponent’s actions directly, and the final policies of the iterative process should be a close approximation to the equilibrium up to any prescribed precision. This kind of algorithms is more advantageous and versatile especially in federated environments, as it requires neither prior coordination between the players like twotimescale algorithms, nor a central controller to collect and disseminate the policies of all the players, which are often unavailable due to privacy constraints. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1.1 Last-iterate convergence in competitive games ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            92,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In recent years, there have been significant progresses in understanding the last-iterate convergence of simple iterative algorithms for unconstrained saddle-point optimization, where one is interested in bounding the sub-optimality of the last iterate of the algorithm, rather than say, the ergodic iterate — which is the average of all the iterations — that are commonly studied in the earlier literature. This shift of focus is motivated, for example, by the infeasibility of averaging large machine learning models in training GANs (Goodfellow et al., 2014). While vanilla Gradient Descent / Ascent (GDA) may diverge or cycle even for bilinear matrix games (Daskalakis et al., 2018), quite remarkably, small modifications lead to guaranteed last-iterate convergence to the equilibrium in a non-asymptotic fashion. A flurry of algorithms is proposed, including Optimistic Gradient Descent Ascent (OGDA) (Rakhlin and Sridharan, 2013; Daskalakis and Panageas, 2018b; Wei et al., 2021b), predictive updates (Yadav et al., 2017), implicit updates (Liang and Stokes, 2019), and more. Several unified analyses of these algorithms have been carried out (see, e.g. Mokhtari et al. (2020a); Liang and Stokes (2019) and references therein), where these methods in principle all make clever extrapolation of the local curvature in a predictive manner to accelerate convergence. With slight abuse of terminology, in this paper, we refer to this ensemble of algorithms as extragradient methods (Korpelevich, 1976; Tseng, 1995; Mertikopoulos et al., 2018a; Harker and Pang, 1990). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, saddle-point optimization in the constrained setting, which includes competitive games as a special case, remains largely under-explored even for bilinear matrix games. While it is possible to reformulate constrained bilinear games to unconstrained ones using softmax parameterization of the probability simplex, this approach falls short of preserving the bilinear structure and convexconcave properties in the original problem, which are crucial to the convergence of gradient methods. Therefore, there is a strong necessity of understanding and developing improved extragradient methods in the constrained setting. Daskalakis and Panageas (2018a) proposed the optimistic variant of the multiplicative weight updates (MWU) method (Arora et al., 2012) – which is extremely natural and popular for optimizing over probability simplexes – called Optimistic Multiplicative Weight Updates (OMWU), and established the asymptotic last-iterate convergence of OMWU for matrix games. Very recently, Wei et al. (2021b) established non-asymptotic last-iterate convergences of OMWU. However, these last-iterate convergence results require the Nash equilibrium to be unique, and cannot be applied to problems with multiple Nash equilibria. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.2 Our contributions ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            536,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the algorithmic role of entropy regularization in single-agent RL (Neu et al., 2017; Geist et al., 2019; Cen et al., 2020) as well as its wide use in game theory to account for imperfect and noisy information (McKelvey and Palfrey, 1995; Savas et al., 2019), we initiate the design and analysis of extragradient algorithms using multiplicative updates for finding the quantal response equilibrium (QRE), which are solutions to competitive games with entropy regularization (McKelvey and Palfrey, 1995). While finding QRE is of interest in its own right, by controlling the knob of entropy regularization, the QRE provides a close approximation to the Nash equilibrium (NE), and in turn acts as a smoothing scheme for finding the NE. Our contributions are summarized below. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Near dimension-free last-iterate convergence to QRE of entropy-regularized matrix games. We propose two policy extragradient algorithms to solve entropy-regularized matrix games, namely the Predictive Update (PU) and OMWU methods, where both players execute symmetric and multiplicative updates without knowing the entire payoff matrix nor the opponent’s actions. Encouragingly, we show that the last iterate of the proposed algorithms converges to the unique QRE at a linear rate that is almost independent of the size of the action spaces. Roughly speaking, to find an $\\epsilon$ -optimal QRE in terms of Kullback-Leibler (KL) divergence, it takes no more than $\\begin{array} { r } { \\widetilde O \\left( \\frac { 1 } { \\eta \\tau } \\log \\left( \\frac { 1 } { \\epsilon } \\right) \\right) } \\end{array}$ iterations, where ${ \\widetilde { O } } ( \\cdot )$ hides logarithmic dependencies. Here, $\\tau$ is the regularization parameter, and $\\eta$ is the learning rate of both players. Maximizing the learning rate, the iteration complexity is bounded by $\\widetilde { O } \\left( ( 1 + \\| A \\| _ { \\infty } / \\tau ) \\log ( 1 / \\epsilon ) \\right)$ , where $\\left\\| A \\right\\| _ { \\infty } = \\operatorname* { m a x } _ { i , j } \\left| A _ { i , j } \\right|$ is the $\\ell _ { \\infty }$ norm of the payoff matrix $A$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Last-iterate convergence to $\\epsilon$ -NE of unregularized matrix games without uniqueness assumption. The QRE provides an accurate approximation to the NE by setting the entropy regularization $\\tau$ sufficiently small, therefore our result directly translates to finding a NE with last-iterate convergence guarantee. Roughly speaking, to find an $\\epsilon$ -NE (Zhang et al., 2020, Definition 2.1), it takes no more than $\\begin{array} { r } { \\widetilde { O } \\left( 1 + \\frac { \\| A \\| _ { \\infty } } { \\epsilon } \\right) } \\end{array}$ iterations with optimized learning rates, which is again independent of the size of the action spaces up to logarithmic factors. Unlike prior literature (Daskalakis and Panageas, 2018a; Wei et al., 2021b), our last-iterate convergence guarantee does not require the NE to be unique. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/7cd63fa4d3a31b21549b1c3cba203d1cfde83dc305268984dcc577e9d1c629f3.jpg",
+        "table_caption": [
+            "Table 1: Comparisons of last-iterate convergence of the proposed entropy-regularized PU and OMWU methods with prior results for finding $\\epsilon$ -QRE or $\\mathrm { \\epsilon - N E }$ of competitive matrix games. We note that the convergence rates of unregularized OMWU established in Wei et al. (2021b) are problemdependent, and scale at least polynomially on the size of the action spaces. Desirable features in the last two columns are highlighted in blue. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Equilibriumtype</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Convergence rate</td><td rowspan=1 colspan=1>Dimension-free</td><td rowspan=1 colspan=1>Requireunique NE</td></tr><tr><td rowspan=1 colspan=1>E-QRE</td><td rowspan=1 colspan=1>PU&amp;OMWU(this work)</td><td rowspan=1 colspan=1>linear</td><td rowspan=1 colspan=1>yes</td><td rowspan=1 colspan=1>n/a</td></tr><tr><td rowspan=3 colspan=1>e-NE</td><td rowspan=1 colspan=1>OMWU(Daskalakis and Panageas, 2018a)</td><td rowspan=1 colspan=1>asymptotic</td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>yes</td></tr><tr><td rowspan=1 colspan=1>OMWU(Wei et al., 2021b)</td><td rowspan=1 colspan=1>sublinear + linear</td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>yes</td></tr><tr><td rowspan=1 colspan=1>PU&amp;OMWU(this work)</td><td rowspan=1 colspan=1>sublinear</td><td rowspan=1 colspan=1>yes</td><td rowspan=1 colspan=1>no</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Extensions to two-player zero-sum Markov games. By connecting value iteration with matrix games, we propose a policy extragradient method for solving infinite-horizon discounted entropyregularized zero-sum Markov games, which finds an $\\epsilon$ -optimal minimax soft Q-function—in terms of $\\ell _ { \\infty }$ error—in at most $\\begin{array} { r } { \\widetilde O \\left( \\frac { 1 } { \\tau ( 1 - \\gamma ) ^ { 2 } } \\log ^ { 2 } \\left( \\frac { 1 } { \\epsilon } \\right) \\right) } \\end{array}$ iterations, where $\\gamma \\in ( 0 , 1 )$ is the discount factor. ",
+        "bbox": [
+            174,
+            392,
+            825,
+            457
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, our paper is the first that develops policy extragradient algorithms for solving entropy-regularized competitive games with multiplicative updates and dimension-free linear last-iterate convergence, and demonstrates entropy regularization as a smoothing technique to find $\\mathrm { \\epsilon - N E }$ without the uniqueness assumption. Table 1 provides detailed comparisons of the proposed methods with prior arts for solving matrix games. Our results highlight the positive role of entropy regularization for accelerating convergence and safeguarding against imperfect information in competitive games. We defer the complete proof of our results to Cen et al. (2021). ",
+        "bbox": [
+            174,
+            463,
+            825,
+            560
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1.3 Related works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            582,
+            313,
+            595
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our work lies at the intersection of saddle-point optimization, game theory, and reinforcement learning. In what follows, we discuss a few topics that are closely related to ours. ",
+        "bbox": [
+            174,
+            606,
+            823,
+            633
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unregularized matrix game. Freund and Schapire (1999) showed that many standard methods such as GDA and MWU have a converging average duality gap at the rate of $O ( 1 / \\sqrt { T } )$ , which is improved to $O ( 1 / T )$ by considering optimistic variants of these methods, such as OGDA and OMWU (Rakhlin and Sridharan, 2013; Daskalakis et al., 2011; Syrgkanis et al., 2015). However, the last-iterate convergence of these methods are less understood until recently (Daskalakis and Panageas, 2018a; Wei et al., 2021b). In particular, under the assumption that the NE is unique for the unregularized matrix game, Daskalakis and Panageas (2018a) showed the asymptotic convergence of the last iterate of OMWU to the unique equilibrium, and Wei et al. (2021b) showed the last iterate of OMWU achieves a linear rate of convergence after an initial phase of sublinear convergence, however the rates therein can be highly pessimistic in terms of the problem dimension, while our rate for entropy-regularized OMWU is dimension-free up to logarithmic factors. ",
+        "bbox": [
+            174,
+            640,
+            825,
+            794
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Saddle-point optimization. Considerable progress has been made towards understanding OGDA and extragradient (EG) methods in the unconstrained convex-concave saddle-point optimization with general objective functions (Mokhtari et al., 2020a,b; Nemirovski, 2004; Liang and Stokes, 2019). However, the last-iterate convergence of constrained convex-concave saddle-point optimization still lacks theoretical understanding in general and most works fall short of characterizing a finite-time convergence result. In particular, Mertikopoulos et al. (2018a) demonstrated the asymptotic lastiterate convergence of EG, and Hsieh et al. (2019) investigated similar questions for single-call EG algorithms. Lei et al. (2021) showed that OMWU converges to the equilibrium locally without an explicit rate. Wei et al. (2021b) showed that the last-iterate of OGDA converges linearly for strongly-convex strongly-concave constrained saddle-point optimization with an explicit rate. ",
+        "bbox": [
+            174,
+            800,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Entropy regularization in RL and games. In single-agent RL, the role of entropy regularization as an algorithmic mechanism to encourage exploration and accelerate convergence has been investigated extensively (Neu et al., 2017; Geist et al., 2019; Mei et al., 2020; Cen et al., 2020; Lan, 2021; Zhan et al., 2021). Turning to the game setting, entropy regularization is used to account for imperfect information in the seminal work of McKelvey and Palfrey (1995) that introduced the QRE, and a few representative works on entropy and more general regularizations in games include Savas et al. (2019); Hofbauer and Sandholm (2002); Mertikopoulos and Sandholm (2016). ",
+        "bbox": [
+            173,
+            126,
+            825,
+            223
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Zero-sum Markov games. There have been a significant recent interest in developing provably efficient self-play algorithms for Markov games, including model-based algorithms (Perolat et al., 2015; Zhang et al., 2020), value-based algorithms (Bai and Jin, 2020; Xie et al., 2020), and policybased algorithms (Daskalakis et al., 2020; Wei et al., 2021a; Zhao et al., 2021). The iteration complexities in prior works (Perolat et al., 2015; Daskalakis et al., 2020; Wei et al., 2021a; Zhao et al., 2021) depend on various notions of concentrability coefficient and therefore can scale quite pessimistically with the problem dimension. Our approach can be regarded as a policy-based algorithm to approximate value iteration, which can be implemented in a decentralized manner with symmetric and multiplicative updates from both players, and the iteration complexity is almost independent of the size of the state-action space. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Notation. We denote by $\\Delta ( \\mathcal { A } )$ the probability simplex over the set $\\mathcal { A }$ . We overload the functions such as $\\log ( \\cdot )$ and $\\exp ( \\cdot )$ to take vector inputs with the understanding that the function is applied in an entrywise manner. For instance, given any vector $z = [ z _ { i } ] _ { 1 \\leq i \\leq n } \\in \\mathbb { R } ^ { n }$ , the notation $\\exp ( z )$ denotes $\\exp ( z ) : = [ \\exp ( z _ { i } ) ] _ { 1 \\leq i \\leq n }$ ; other functions are defined analogously. Given two probability distributions $\\mu$ and $\\mu ^ { \\prime }$ over $\\mathcal { A }$ , the KL divergence from $\\mu ^ { \\prime }$ to $\\mu$ is defined by $\\begin{array} { r } { \\mathsf { K L } ( \\mu \\parallel \\mu ^ { \\prime } ) : = \\sum _ { a \\in \\mathcal { A } } \\mu ( a ) \\log \\frac { \\mu ( a ) } { \\mu ^ { \\prime } ( a ) } } \\end{array}$ . Given a matrix $A$ , $\\| A \\| _ { \\infty }$ is used to denote entrywise maximum norm, namely, $\\left\\| A \\right\\| _ { \\infty } = \\operatorname* { m a x } _ { i , j } \\left| A _ { i , j } \\right|$ . The all-one vector is denoted as 1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2 Zero-sum matrix games with entropy regularization ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We first consider a two-player zero-sum game with bilinear objective and probability simplex constraints, and demonstrate the positive role of entropy regularization in solving this problem. Throughout this paper, let $\\mathcal { A } = \\{ 1 , \\dotsc , m \\}$ and $\\boldsymbol { B } = \\{ 1 , \\ldots , \\bar { n } \\}$ be the action spaces of each player. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 Background and problem formulation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Zero-sum two-player matrix game. The focal point of this subsection is a constrained two-player zero-sum matrix game, which can be formulated as the following min-max problem (or saddle point optimization problem): ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/a84cc1579eae155c735ed137faef7ebb0f81235573bb5194136ab35bb171c683.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\mu \\in \\Delta ( \\mathcal { A } ) } \\operatorname* { m i n } _ { \\nu \\in \\Delta ( \\mathcal { B } ) } f ( \\mu , \\nu ) : = \\mu ^ { \\top } A \\nu ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            656,
+            609,
+            684
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\textbf { \\textit { A } } \\in \\mathbb { R } ^ { m \\times n }$ denotes the payoff matrix, $\\mu ~ \\in ~ \\Delta ( { \\mathcal { A } } )$ and $\\nu ~ \\in ~ \\Delta ( B )$ stand for the mixed/randomized policies of each player, defined respectively as distributions over the probability simplex $\\Delta ( \\mathcal { A } )$ and $\\Delta ( B )$ . A pair of policies $( \\mu ^ { \\star } , \\nu ^ { \\star } )$ is said to be a Nash equilibrium (NE) of (1) if $f ( \\mu ^ { \\star } , \\nu ) \\geq f ( \\mu ^ { \\star } , \\nu ^ { \\star } ) \\geq f ( \\mu , \\nu ^ { \\star } )$ for all $( \\mu , \\nu ) \\in \\Delta ( \\mathcal { A } ) \\times \\Delta ( \\mathcal { B } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Entropy-regularized zero-sum two-player matrix game. There is no shortage of scenarios where the payoff matrix $A$ might not be known perfectly. In an attempt to accommodate imperfect knowledge of $A$ , McKelvey and Palfrey (1995) proposed a seminal extension to the Nash equilibrium called the quantal response equilibrium $( Q R E )$ when the payoffs are perturbed by Gumbel-distributed noise. Formally, this amounts to solving the following matrix game with entropy regularization (Mertikopoulos and Sandholm, 2016): ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b4d50912a83264c73c0021a71bd120be521f76f0af08da4df50bf194bda49736.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\mu \\in \\Delta ( \\mathcal { A } ) } \\operatorname* { m i n } _ { \\nu \\in \\Delta ( \\mathcal { B } ) } f _ { \\tau } ( \\mu , \\nu ) : = \\mu ^ { \\top } A \\nu + \\tau \\mathcal { H } ( \\mu ) - \\tau \\mathcal { H } ( \\nu ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            847,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\mathcal { H } ( \\pi ) = - \\sum _ { i } \\pi _ { i } \\log ( \\pi _ { i } ) } \\end{array}$ denotes the Shannon entropy of a distribution $\\pi$ , and $\\tau \\geq 0$ is the regularization parameter. As is well known, the optimal solution $( \\mu _ { \\tau } ^ { \\star } , \\nu _ { \\tau } ^ { \\star } )$ to (2), dubbed as the QRE, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "is unique whenever $\\tau > 0$ (due to the presence of strong concavity/convexity), which satisfies the following fixed point equations: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/71555ba46498e3920949431f5517f0cfdf53213e02f54c9ada168880b5c7fd49.jpg",
+        "text": "$$\n\\begin{array} { r } { \\biggr \\{ \\mu _ { \\tau } ^ { \\star } ( a ) = \\frac { \\exp ( [ A \\nu _ { \\tau } ^ { \\star } ] a / \\tau ) } { \\sum _ { a = 1 } ^ { m } \\exp ( [ A \\nu _ { \\tau } ^ { \\star } ] a / \\tau ) } \\propto \\exp ( [ A \\nu _ { \\tau } ^ { \\star } ] a / \\tau ) , \\qquad \\mathrm { f o r ~ a l l } \\ a \\in \\mathcal { A } , } \\\\ { \\nu _ { \\tau } ^ { \\star } ( b ) = \\frac { \\exp ( - [ A ^ { \\top } \\mu _ { \\tau } ^ { \\star } ] b / \\tau ) } { \\sum _ { b = 1 } ^ { n } \\exp ( - [ A ^ { \\top } \\mu _ { \\tau } ^ { \\star } ] b / \\tau ) } \\propto \\exp ( - [ A ^ { \\top } \\mu _ { \\tau } ^ { \\star } ] b / \\tau ) , \\quad \\mathrm { f o r ~ a l l } \\ b \\in \\mathcal { B } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Goal. We aim to efficiently compute the QRE of the entropy-regularized matrix game in a decentralized manner, and investigate how an efficient solver of QRE can be leveraged to find a NE of the unregularized matrix game (1). Namely, we only assume access to “first-order information” as opposed to full knowledge of the payoff matrix $A$ or the actions of the opponent. The information received by each player is formally described in the following sampling oracle. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Sampling oracle for matrix games). For any policy pair $( \\mu , \\nu )$ and payoff matrix $A$ the sampling oracle returns the exact values of $\\mu ^ { \\top } A$ and $A \\nu$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Additional notation. For notational convenience, we let $\\zeta$ represent the concatenation of $\\mu \\in \\mathbb { R } ^ { | \\mathcal { A } | }$ and $\\nu \\in \\mathbb { R } ^ { | B | }$ , namely, $\\zeta = ( \\mu , \\nu )$ . The solution to (2), which is specified in (3), is denoted by $\\zeta _ { \\tau } ^ { \\star } = ( \\mu _ { \\tau } ^ { \\star } , \\nu _ { \\tau } ^ { \\star } )$ . For any $\\zeta = ( \\mu , \\nu )$ and $\\zeta ^ { \\prime } = ( \\mu ^ { \\prime } , \\nu ^ { \\prime } )$ , we shall often abuse the notation and let $\\mathsf { K L } \\big ( \\zeta \\| \\zeta ^ { \\prime } \\big ) = \\mathsf { K L } \\big ( \\mu \\| \\mu ^ { \\prime } \\big ) + \\mathsf { K L } \\big ( \\nu \\| \\nu ^ { \\prime } \\big )$ . The duality gap of the entropy-regularized matrix game (2) at $\\zeta = ( \\mu , \\nu )$ is defined as $\\begin{array} { r } { \\mathsf { D u a l G a p } _ { \\tau } ( \\zeta ) = \\operatorname* { m a x } _ { \\mu ^ { \\prime } \\in \\Delta ( A ) } f _ { \\tau } ( \\mu ^ { \\prime } , \\nu ) - \\operatorname* { m i n } _ { \\nu ^ { \\prime } \\in \\Delta ( B ) } f _ { \\tau } ( \\mu , \\nu ^ { \\prime } ) } \\end{array}$ which is clearly nonnegative and $\\mathsf { D u a l G a p } _ { \\tau } ( \\boldsymbol { \\zeta } _ { \\tau } ^ { \\star } ) = 0$ . Similarly, let the optimality gap of the entropyregularized matrix game (2) at $\\zeta = ( \\mu , \\nu )$ be $\\mathsf { O p t G a p } ( \\zeta ) = \\left| f _ { \\tau } ( \\mu , \\nu ) - f _ { \\tau } ( \\mu _ { \\tau } ^ { \\star } , \\nu _ { \\tau } ^ { \\star } ) \\right|$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.2 Proposed extragradient methods: PU and OMWU ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To begin, assume we are given a pair of policies $z _ { 1 } \\in \\Delta ( \\mathcal { A } )$ , $z _ { 2 } \\in \\Delta ( B )$ employed by each player respectively. If we proceed with fictitious play, i.e. player 1 (resp. player 2) aims to optimize its own policy by assuming the opponent’s policy is fixed as $z _ { 2 }$ (resp. $z _ { 1 }$ ), the saddle-point optimization problem (2) is then decoupled into two independent min/max optimization problems: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3ae3b2a7eb130747659835faa8d01046b129d51408a7d9c08d0dd4aa00a0a848.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\mu \\in \\Delta ( A ) } \\mu ^ { \\top } A z _ { 2 } + \\tau \\mathcal { H } ( \\mu ) - \\tau \\mathcal { H } ( z _ { 2 } ) \\qquad \\mathrm { a n d } \\qquad \\operatorname* { m i n } _ { \\nu \\in \\Delta ( B ) } z _ { 1 } ^ { \\top } A \\nu + \\tau \\mathcal { H } ( z _ { 1 } ) - \\tau \\mathcal { H } ( \\nu ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which are naturally solved via mirror descent / ascent with KL divergence. Specifically, one step of mirror descent / ascent takes the form ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a1e0157530e8a4c8bc9d0803138f0eba224ba8f93f9a46dce23f9f9876f0d1d5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\mu ^ { ( t + 1 ) } ( a ) \\propto \\mu ^ { ( t ) } ( a ) ^ { 1 - \\eta \\tau } \\exp ( \\eta [ A z _ { 2 } ] _ { a } ) , } & { \\mathrm { f o r ~ a l l ~ } a \\in \\mathcal { A } , } \\\\ { \\nu ^ { ( t + 1 ) } ( b ) \\propto \\nu ^ { ( t ) } ( b ) ^ { 1 - \\eta \\tau } \\exp ( - \\eta [ A ^ { \\top } z _ { 1 } ] _ { b } ) , } & { \\mathrm { f o r ~ a l l ~ } b \\in \\mathcal { B } , } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is the learning rate. The above update rule forms the basis of our algorithm design. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Motivation: a form of implicit updates with linear convergence. It turns out, if we could select the policy pair $( z _ { 1 } , z _ { 2 } ) = \\hat { \\zeta } ^ { ( t + 1 ) } : \\stackrel { \\bullet } { = } ( \\mu ^ { ( t + 1 ) } , \\nu ^ { ( t + 1 ) } )$ as the ones to be taken in the future, and call the resulting update rule as the Implicit Update (IU) method: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57dd0f2231a644892d0d759e737298493ccd260916dd0c48818739b8fbeb7aef.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left\\{ \\mu ^ { ( t + 1 ) } ( a ) \\propto \\mu ^ { ( t ) } ( a ) ^ { 1 - \\eta \\tau } \\exp ( \\eta [ A \\nu ^ { ( t + 1 ) } ] _ { a } ) , \\right. \\left. \\mathrm { f o r ~ a l l } \\ : a \\in \\mathcal { A } , \\right. } \\\\ { \\nu ^ { ( t + 1 ) } ( b ) \\propto \\nu ^ { ( t ) } ( b ) ^ { 1 - \\eta \\tau } \\exp ( - \\eta [ A ^ { \\top } \\mu ^ { ( t + 1 ) } ] _ { b } ) , \\left. \\mathrm { f o r ~ a l l } \\ : b \\in \\mathcal { B } . \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Though unrealistic — since it uses the future updates — it leads to a one-step convergence to the QRE when $\\eta = 1 / \\tau$ (see the optimality condition in (3)). Encouragingly, we have the following linear convergence guarantee of IU when adopting a general learning rate. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 (Linear convergence of IU). Assume $0 < \\eta \\leq 1 / \\tau$ , then for all $t \\geq 0$ , the iterates $\\zeta ^ { ( t ) } : = ( \\mu ^ { ( t ) } , \\nu ^ { ( t ) } )$ of the $I U$ method in (5) satisfy $\\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\parallel \\zeta ^ { ( t ) } \\big ) \\leq ( 1 - \\eta \\tau ) ^ { t } \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\parallel \\zeta ^ { ( 0 ) } \\big )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In words, the IU method achieves an appealing linear rate of convergence that is independent of the problem dimension. Motivated by this observation, we seek to design algorithms where the policies $( z _ { 1 } , z _ { 2 } )$ employed in (4) serve as good predictions of $( \\mu ^ { ( t + 1 ) } , \\nu ^ { ( t + \\bar { 1 ) } } )$ , such that the resulting algorithms are both practical and retain the appealing convergence rate of IU. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposed algorithms. We propose two extragradient algorithms for solving the entropy-regularized matrix game, namely the Predictive Update $( P U )$ method and the Optimistic Multiplicative Weights ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: The PU method ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2: The OMWU method ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1 initialization: $\\mu ^ { ( 0 ) }$ , $\\nu ^ { ( 0 ) }$ ",
+        "bbox": [
+            163,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2 for $t = 0 , 1 , 2 , \\cdots$ do ",
+        "bbox": [
+            160,
+            128,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2 for $t = 0 , 1 , 2 , \\cdots$ do ",
+        "bbox": [
+            485,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 Update $\\bar { \\mu }$ and $\\bar { \\nu }$ according to ",
+        "bbox": [
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+            143,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 ",
+        "bbox": [
+            485,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Update $\\bar { \\mu }$ and $\\bar { \\nu }$ according to ",
+        "bbox": [
+            517,
+            143,
+            714,
+            156
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2682d9ed9998363f783340595232da07def6ae1b62593530707c1b063a745524.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left\\{ \\bar { \\mu } ^ { ( t + 1 ) } ( a ) \\propto { \\mu } ^ { ( t ) } ( a ) ^ { 1 - \\eta \\tau } \\exp ( \\eta [ A \\nu ^ { ( t ) } ] _ { a } ) , \\right. \\qquad } \\\\ { \\left. \\bar { \\nu } ^ { ( t + 1 ) } ( b ) \\propto \\nu ^ { ( t ) } ( b ) ^ { 1 - \\eta \\tau } \\exp ( - \\eta [ A ^ { \\top } { \\mu } ^ { ( t ) } ] _ { b } ) . \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07b2e5847f06f569d996bfff2cef68ba4e7e99ebccd9cf274448fbddd63318e4.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left\\{ \\bar { \\mu } ^ { ( t + 1 ) } ( a ) \\propto { \\mu } ^ { ( t ) } ( a ) ^ { 1 - \\eta \\tau } \\exp ( \\eta [ A \\bar { \\nu } ^ { ( t ) } ] _ { a } ) , \\right. \\mathrm { ~ } } \\\\ { \\left. \\bar { \\nu } ^ { ( t + 1 ) } ( b ) \\propto \\nu ^ { ( t ) } ( b ) ^ { 1 - \\eta \\tau } \\exp ( - \\eta [ A ^ { \\top } \\bar { \\mu } ^ { ( t ) } ] _ { b } ) . \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Update $\\mu$ and $\\nu$ according to ",
+        "bbox": [
+            160,
+            210,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Update $\\mu$ and $\\nu$ according to ",
+        "bbox": [
+            524,
+            210,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/213e7eea9f5e11eb13db31a851e4ce7321f6ff73c1f9234656441c96d93ddf5f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\mu ^ { ( t + 1 ) } ( a ) \\propto \\mu ^ { ( t ) } ( a ) ^ { 1 - \\eta \\tau } \\exp ( \\eta [ A \\bar { \\nu } ^ { ( t + 1 ) } ] _ { a } ) , } \\\\ { \\nu ^ { ( t + 1 ) } ( b ) \\propto \\nu ^ { ( t ) } ( b ) ^ { 1 - \\eta \\tau } \\exp ( - \\eta [ A ^ { \\top } \\bar { \\mu } ^ { ( t + 1 ) } ] _ { b } ) . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/46d9ad77b55d1cc670745026c47b3d350c5530aabb855c6c7b775b9a5c3422f4.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\mu ^ { ( t + 1 ) } ( a ) \\propto \\mu ^ { ( t ) } ( a ) ^ { 1 - \\eta \\tau } \\exp ( \\eta [ A \\bar { \\nu } ^ { ( t + 1 ) } ] _ { a } ) , } \\\\ { \\nu ^ { ( t + 1 ) } ( b ) \\propto \\nu ^ { ( t ) } ( b ) ^ { 1 - \\eta \\tau } \\exp ( - \\eta [ A ^ { \\top } \\bar { \\mu } ^ { ( t + 1 ) } ] _ { b } ) . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Update (OMWU) method, the latter adapted from Rakhlin and Sridharan (2013); Daskalakis et al. (2011). Detailed procedures can be found in Algorithm 1 and Algorithm 2, respectively. On a high level, both algorithms maintain two intertwined sequences $\\{ ( \\mu ^ { ( \\bar { t } ) } , \\nu ^ { ( t ) } ) \\} _ { t \\geq 0 }$ and $\\{ ( \\bar { \\mu } ^ { ( \\dot { t } ) } , \\bar { \\nu } ^ { ( t ) } ) \\} _ { t \\geq 0 }$ , and in each iteration $t = 0 , 1 , \\ldots$ , proceed in two steps: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• The midpoint $( \\bar { \\mu } ^ { ( t + 1 ) } , \\bar { \\nu } ^ { ( t + 1 ) } )$ serves as a prediction of $( \\mu ^ { ( t + 1 ) } , \\nu ^ { ( t + 1 ) } )$ by running one step of mirror descent / ascent (cf. (4)) from either $( z _ { 1 } , z _ { 2 } ) = ( \\mu ^ { ( t ) } , \\nu ^ { ( t ) } )$ (for PU) or $( z _ { 1 } , z _ { 2 } ) = ( \\bar { \\mu } ^ { ( t ) } , \\bar { \\nu } ^ { ( t ) } )$ (for OMWU). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• The update of $( \\boldsymbol { \\mu } ^ { ( t + 1 ) } , \\boldsymbol { \\nu } ^ { ( t + 1 ) } )$ then mimics the implicit update (5) using the prediction $( \\bar { \\mu } ^ { ( t + 1 ) } , \\bar { \\nu } ^ { ( t + 1 ) } )$ obtained above. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When the proposed algorithms converge, both $( \\mu ^ { ( t ) } , \\nu ^ { ( t ) } )$ and $( \\bar { \\mu } ^ { ( t ) } , \\bar { \\nu } ^ { ( t ) } )$ converge to the same point. The two players are completely symmetric and adopt the same learning rate, and require only first-order information provided by the sampling oracle. While the two algorithms resemble each other in many aspects, a key difference lies in the query and use of the sampling oracle: in each iteration, OMWU makes a single call to the sampling oracle for gradient evaluation, while PU calls the sampling oracle twice. It is worth noting that, when $\\tau = 0$ (i.e., no entropy regularization is enforced), the OMWU method in Algorithm 2 reduces to the method analyzed in Rakhlin and Sridharan (2013); Daskalakis and Panageas (2018a); Wei et al. (2021b) without entropy regularization. ",
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+    {
+        "type": "text",
+        "text": "Remark 1. It is worth highlighting that the proposed algorithms are different from Mertikopoulos et al. (2018a), as the extragradient is only applied to the bilinear term but not the entropy regularization term. This seemingly small, but important, difference leads to a more concise closed-form update rule and a cleaner analysis, as shall be seen momentarily. ",
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+    {
+        "type": "text",
+        "text": "2.3 Performance guarantees ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We are now positioned to present our main theorem concerning the last-iterate convergence of PU and OMWU for solving (2). ",
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+        "type": "text",
+        "text": "Theorem 1 (Last-iterate convergence of PU and OMWU). Suppose that the learning rates $\\eta = \\eta _ { \\mathsf { P U } }$ of $P U$ in Algorithm $I$ and $\\eta = \\eta$ OMWU of OMWU in Algorithm 2 satisfy ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1da1f00199904a4b6c97f5e9c76f9a3df834798c6291155ebe93e90d726b2886.jpg",
+        "text": "$$\n0 < \\eta _ { \\mathsf { P U } } \\leq \\frac { 1 } { \\tau + 2 \\left\\| A \\right\\| _ { \\infty } } , a n d 0 < \\eta _ { \\mathsf { O M W U } } \\leq \\operatorname* { m i n } \\left\\{ \\frac { 1 } { 2 \\tau + 2 \\left\\| A \\right\\| _ { \\infty } } , \\frac { 1 } { 4 \\left\\| A \\right\\| _ { \\infty } } \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Then for any $t \\geq 0$ , the iterates $\\zeta ^ { ( t ) } = ( \\mu ^ { ( t ) } , \\nu ^ { ( t ) } )$ and $\\bar { \\zeta } ^ { ( t ) } = ( \\bar { \\mu } ^ { ( t ) } , \\bar { \\nu } ^ { ( t ) } )$ of $P U$ and OMWU achieve ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Linear convergence of policies in KL divergence and entrywise log-ratios: ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/9c9ca3a937c7498ac455080e8128e2e20c04105f9efdeb52eea00b6054ed9608.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\operatorname* { m a x } \\left\\{ \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\| \\zeta ^ { ( t ) } \\big ) , \\frac { 1 } { 2 } \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\| \\bar { \\zeta } ^ { ( t + 1 ) } \\big ) \\right\\} \\leq ( 1 - \\eta \\tau ) ^ { t } \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\| \\zeta ^ { ( 0 ) } \\big ) , } \\\\ & { \\left\\| \\log \\frac { \\zeta ^ { ( t ) } } { \\zeta _ { \\tau } ^ { \\star } } \\right\\| _ { \\infty } \\leq 2 ( 1 - \\eta \\tau ) ^ { t } \\left\\| \\log \\frac { \\zeta ^ { ( 0 ) } } { \\zeta _ { \\tau } ^ { \\star } } \\right\\| _ { \\infty } + \\frac { 8 \\| A \\| _ { \\infty } } { \\tau } ( 1 - \\eta \\tau ) ^ { t / 2 } \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\| \\zeta ^ { ( 0 ) } \\big ) ^ { 1 / 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Linear convergence of values in optimality and duality gaps: ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3702b093fedebbe58ca32ee1f3510f7779f94f8298da5387a3f661e7ff847f92.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathsf { O p t G a p } _ { \\tau } ( \\bar { \\zeta } ^ { ( t ) } ) \\leq \\eta ^ { - 1 } \\cdot \\frac { 1 } { 1 - ( \\tau + \\| A \\| _ { \\infty } ) \\eta } \\cdot \\frac { ( 1 - \\eta \\tau ) ^ { t } } { 1 - ( 1 - \\eta \\tau ) ^ { t } } \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\| \\zeta ^ { ( 0 ) } \\big ) , } \\\\ & { \\mathsf { D u a l G a p } _ { \\tau } ( \\bar { \\zeta } ^ { ( t ) } ) \\leq \\left( \\eta ^ { - 1 } + 2 \\tau ^ { - 1 } \\| A \\| _ { \\infty } ^ { 2 } \\right) ( 1 - \\eta \\tau ) ^ { t - 1 } \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\| \\zeta ^ { ( 0 ) } \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Remark 2. Setting $\\mu ^ { ( 0 ) }$ and $\\nu ^ { ( 0 ) }$ to be uniform policies leads to a universal bound ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/9dc3554546cd0d7e8714d2a8742b8e7b54e42d716313e7ff2b236bd90a374b64.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathsf { K L } \\big ( \\zeta _ { \\tau } ^ { \\star } \\| \\zeta ^ { ( 0 ) } \\big ) = \\log | \\mathcal { A } | + \\log | \\mathcal { B } | - \\mathcal { H } ( \\mu _ { \\tau } ^ { \\star } ) - \\mathcal { H } ( \\nu _ { \\tau } ^ { \\star } ) \\leq \\log | \\mathcal { A } | + \\log | \\mathcal { B } | . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Remark 3. Similar results continue to hold even when the two players use different regularization parameters $\\tau _ { \\mu } , \\tau _ { \\nu } > 0$ in (2), as long as the regularization parameter $\\tau$ is replaced by max $\\{ \\tau _ { \\mu } , \\tau _ { \\nu } \\}$ in the upper bounds of the learning rate, and the contraction parameter is replaced by $1 - \\operatorname* { m i n } \\{ \\tau _ { \\mu } , \\tau _ { \\nu } \\} \\eta$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 1 characterizes the convergence of the last-iterates $\\zeta ^ { ( t ) }$ and $\\bar { \\zeta } ^ { ( t ) }$ of PU and OMWU as long as the learning rate lies within the specified ranges. While PU doubles the number of calls to the sampling oracle, it also allows roughly as large as twice the learning rate compared with OMWU (cf. (6)). Compared with the vast literature analyzing the average-iterate performance of variants of extragradient methods, our results contribute towards characterizing the last-iterate convergence of multiplicative update methods in the presence of entropy regularization and simplex constraints, which to the best of our knowledge, are the first of its kind. Several remarks are in order. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Linear convergence to QRE. To achieve an $\\epsilon$ -accurate estimate of the QRE in terms of the KL divergence, the bound (7a) tells that it is sufficient to take ",
+        "bbox": [
+            173,
+            377,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be090bb26a96971777947c1ec6bf114494d258ce86d67ce09fbd7459d4b4d822.jpg",
+        "text": "$$\n\\frac { 1 } { \\eta \\tau } \\log \\left( \\frac { \\log | \\cal { A } | + \\log | \\cal { B } | } { \\epsilon } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "iterations using either PU or OMWU. Notably, this iteration complexity does not depend on any hidden constants and only depends double logarithmically on the cardinality of action spaces, which is almost dimension-free. Maximizing the learning rate, the iteration complexity is bounded by $( 1 + \\| A \\| _ { \\infty } / \\tau ) \\log ( 1 / \\epsilon )$ (modulo log factors), which only depends on the ratio $\\| A \\| _ { \\infty } / \\tau$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Entrywise error of the policy log-ratios. Both PU and OMWU enjoy strong entrywise guarantees in the sense we can guarantee the convergence of the $\\ell _ { \\infty }$ norm of the log-ratios between the learned policy pair and the QRE at the same dimension-free linear rate (cf. (7b)), which suggests the policy pair converges in a somewhat uniform manner across the entire action space. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Linear convergence of optimality and duality gaps. Our theorem also establishes the last-iterate convergence of the game values in terms of the optimality gap (cf. (7c)) and the duality gap (cf. (7d)) for both PU and OMWU. In particular, as will be seen, bounding the optimality gap of matrix games turns out to be the key enabler for generalizing our algorithms to Markov games, and bounding the duality gap allows to directly translate our results to finding a NE of unregularized matrix games. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Last-iterate convergence to approximate NE. The entropy-regularized matrix game can be thought as a smooth surrogate of the unregularized matrix game (1); in particular, it is possible to find an $\\epsilon$ -NE by setting $\\tau$ sufficiently small in (2). According to (Zhang et al., 2020, Definition 2.1), a policy pair $\\bar { \\zeta } = ( \\mu , \\bar { \\nu } )$ is an $\\epsilon$ -NE if it satisfies $\\begin{array} { r } { \\mathsf { D u a l G a p } ( \\zeta ) : = \\operatorname* { m a x } _ { \\mu ^ { \\prime } \\in \\Delta ( A ) } f ( \\mu ^ { \\prime } , \\nu ) - \\operatorname* { m i n } _ { \\nu ^ { \\prime } \\in \\Delta ( B ) } f ( \\mu , \\nu ^ { \\prime } ) \\leq \\epsilon } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Observe that setting $\\begin{array} { r } { \\tau = \\frac { \\epsilon / 4 } { \\log | \\mathcal { A } | + \\log | \\mathcal { B } | } } \\end{array}$ guarantees that $| f _ { \\tau } ( \\mu , \\nu ) - f ( \\mu , \\nu ) | < \\epsilon / 4$ uniformly over $( \\mu , \\nu ) \\in \\Delta ( \\mathcal { A } ) \\times \\Delta ( \\mathcal { B } )$ in view of the boundedness of the Shannon entropy $\\mathcal { H } ( \\cdot )$ . Theorem 7 (cf. (7d)) also ensures that our proposed algorithms find an approximate QRE $\\bar { \\zeta } ^ { ( T ) }$ such that $\\mathsf { D u a l G a p } _ { \\tau } \\bigl ( \\bar { \\zeta } ^ { ( T ) } \\bigr ) \\leq$ $\\epsilon / 2$ after taking $\\begin{array} { r } { T = \\widetilde { O } \\left( \\frac { 1 } { \\eta \\epsilon } \\right) } \\end{array}$ iterations, which is no more than $\\begin{array} { r } { \\widetilde { O } \\left( 1 + \\frac { \\| A \\| _ { \\infty } } { \\epsilon } \\right) } \\end{array}$ iterations with optimized learning rates. It follows immediately that ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "[ $\\mathtt { \\mathtt { M a l G a p } } ( \\bar { \\zeta } ^ { ( T ) } ) \\le \\mathtt { D u a l G a p } _ { \\tau } ( \\bar { \\zeta } ^ { ( T ) } ) + \\operatorname* { m a x } _ { \\mu ^ { \\prime } , \\nu ^ { \\prime } } \\Big | f _ { \\tau } ( \\mu ^ { \\prime } , \\bar { \\nu } ^ { ( T ) } ) - f _ { \\tau } \\big ( \\bar { \\mu } ^ { ( T ) } , \\nu ^ { \\prime } \\big ) - \\big ( f ( \\mu ^ { \\prime } , \\bar { \\nu } ^ { ( T ) } ) - f ( \\bar { \\mu } ^ { ( T ) } , \\nu ^ { \\prime } ) \\big ) \\Big | \\le \\epsilon ,$ and therefore $\\bar { \\zeta } ^ { ( T ) }$ is an $\\mathrm { \\epsilon - N E }$ . Intriguingly, unlike prior work (Daskalakis and Panageas, 2018a; Wei et al., 2021b) that analyzed the last-iterate convergence of OMWU in the unregularized setting $\\mathit { \\Pi } _ { \\mathcal { T } } = 0 \\mathit { \\Pi } _ { \\mathcal { c } }$ ), our last-iterate convergence does not require the NE of (1) to be unique. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Rationality. Another attractive feature of the algorithms developed above is being rational (as introduced in Bowling and Veloso (2001)) in the sense that the algorithm returns the best-response policy of one player when the opponent takes any fixed stationary policy. More specially, in terms of matrix games, when player 2 sticks to a stationary policy $\\nu$ , the update of player 1 reduces to $\\mu ^ { ( t + 1 ) } ( a ) \\bar { \\propto } \\mu ^ { ( t ) } ( a ) ^ { 1 - \\bar { \\eta tau } } \\exp ( \\eta [ A \\nu ] _ { a } )$ . In this case, Theorem 1 can be established in exactly the same fashion by restricting attention only to the updates of $\\mu ^ { ( t ) }$ . ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/4e4a6167f1bf0473e8d7a714c8b874f38c7ea875b79a1295462ea5a711766939.jpg",
+        "image_caption": [
+            "Figure 1: Performance illustration of the PU and OMWU methods for solving entropy-regularized matrix games with $| \\mathcal { A } | = | \\mathcal { B } | = 1 0 0$ , where the entries of the payoff matrix $A$ is generated independently from the uniform distribution on $[ - 1 , 1 ]$ . The learning rates are fixed as $\\eta = 0 . 1$ . The left panel plots various error metrics of convergence w.r.t. the iteration count with $\\tau = 0 . 0 1$ , while the right panel plots these error metrics at 1000-th iteration with different choices of $\\tau$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "No-regret learning of OMWU. Besides convergence to equilibria, in game-theoretical settings, it is often desirable to design and implement no-regret algorithms, which are capable of providing black-box guarantees over arbitrary sequences played by the opponent (Cesa-Bianchi and Lugosi, 2006; Rakhlin and Sridharan, 2013). Fortunately, it turns out that entropy regularization not only accelerates the convergence, but also enables no-regret learning somewhat “for free”: it encourages exploration by putting a positive mass on every action, therefore guards against adversaries. By using a properly chosen learning rate schedule, the proposed OMWU (Algorithm 2) can be further established as a no-regret algorithm; the details can be found in (Cen et al., 2021). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3 Zero-sum Markov games with entropy regularization ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            569,
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+            585
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Leveraging the success of PU and OMWU in solving the entropy-regularized matrix games, this section extends our current analysis to solve the zero-sum two-player Markov game with entropy regularization, which is again formulated as finding the equilibrium of a saddle-point optimization problem. We start by introducing its basic setup, which will be followed by the proposed policy extragradient method with its theoretical guarantees. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.1 Background and problem formulation ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            676,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We consider a discounted Markov Game (MG) which is defined as $\\mathcal { M } = \\{ { S , A , B , P , r , \\gamma } \\}$ , with discrete state space $s$ , action spaces of two players $\\mathcal { A }$ and $\\boldsymbol { B }$ , transition probability $P$ , reward function $r : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B }  [ 0 , 1 ]$ and discount factor $\\gamma \\in [ 0 , 1 )$ . A policy $\\mu : { \\mathcal { S } }  \\Delta ( { \\mathcal { A } } )$ (resp. $\\nu : S \\to \\Delta ( B ) )$ defines how player 1 (resp. player 2) reacts to a given state $s$ , where the probability of taking action $a \\in { \\mathcal { A } }$ (resp. $b \\in B ,$ ) is $\\mu ( a | s )$ (resp. $\\nu ( b | s ) )$ . The transition probability kernel $P : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B }  \\Delta ( \\mathcal { S } )$ defines the dynamics of the Markov game, where $P ( s ^ { \\prime } | s , a , b )$ specifies the probability of transiting to state $s ^ { \\prime }$ from state $s$ when the players take actions $a$ and $b$ respectively. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Motivated by entropy regularization in Markov decision processes (MDP) (Geist et al., 2019), we consider an entropy-regularized variant of MG, where the value function is defined as ",
+        "bbox": [
+            173,
+            800,
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd4673555690bc9e7c2c134b940cbf6ce609f112d808752990bda7448b1a8d47.jpg",
+        "text": "$$\nV _ { \\tau } ^ { \\mu , \\nu } ( s ) : = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\left( r ( s _ { t } , a _ { t } , b _ { t } ) - \\tau \\log \\mu ( a _ { t } | s _ { t } ) + \\tau \\log \\nu ( b _ { t } | s _ { t } ) \\right) ~ \\middle | s _ { 0 } = s \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where the quantity $\\tau \\geq 0$ denotes the regularization parameter, and the expectation is evaluated over the randomness of the transition kernel as well as the policies. The regularized Q-function $Q _ { \\tau } ^ { \\mu , \\nu }$ of a ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "policy pair $( \\mu , \\nu )$ is related to $V _ { \\tau } ^ { \\mu , \\nu }$ as ",
+        "bbox": [
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+            90,
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3de3e826aa012c6e58901a8e3b22025d2112634453e74fd01df633f9fe9af64.jpg",
+        "text": "$$\n\\begin{array} { r } { Q _ { \\tau } ^ { \\mu , \\nu } ( s , a , b ) = r ( s , a , b ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim P ( \\cdot \\mid s , a , b ) } \\bigl [ V _ { \\tau } ^ { \\mu , \\nu } \\bigl ( s ^ { \\prime } \\bigr ) \\bigr ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We will call $V _ { \\tau } ^ { \\mu , \\nu }$ and $Q _ { \\tau } ^ { \\mu , \\nu }$ the soft value function and soft $Q$ -function, respectively. A policy pair $( \\mu _ { \\tau } ^ { \\star } , \\nu _ { \\tau } ^ { \\star } )$ is said to be the quantal response equilibrium (QRE) of the entropy-regularized MG, if its value attains the minimax value of the entropy-regularized MG over all states $s \\in S$ , i.e. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2ba603658bc41dc17088723550a675658f813d3ce0cc45bf0c47f8124f2dac5.jpg",
+        "text": "$$\nV _ { \\tau } ^ { \\star } ( s ) = \\operatorname* { m a x } _ { \\mu } \\operatorname* { m i n } _ { \\nu } V _ { \\tau } ^ { \\mu , \\nu } ( s ) = \\operatorname* { m i n } _ { \\nu } \\operatorname* { m a x } _ { \\mu } V _ { \\tau } ^ { \\mu , \\nu } ( s ) : = V _ { \\tau } ^ { \\mu _ { \\tau } ^ { \\star } , \\nu _ { \\tau } ^ { \\star } } ( s ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\cal V } _ { \\tau } ^ { \\star }$ is called the optimal minimax soft value function, and similarly $Q _ { \\tau } ^ { \\star } : = Q _ { \\tau } ^ { \\mu _ { \\tau } ^ { \\star } , \\nu _ { \\tau } ^ { \\star } }$ is called the optimal minimax soft Q-function. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Goal. Our goal is to find the QRE of the entropy-regularized MG in a decentralized manner where the players only observe its own reward without accessing the opponent’s actions. By setting the regularization parameter sufficiently small $\\tau$ , this also allows us to find an approximate NE of the unregularized MG. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "3.2 From value iteration to policy extragradient methods ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Entropy-regularized value iteration. It is known that classical dynamic programming approaches such as value iteration can be extended to solve MG (Perolat et al., 2015), where each iteration amounts to solving a series of matrix games for each state. Similar to the single-agent case (Cen et al., 2020), we can extend these approaches to solve the entropy-regularized MG. Setting the stage, let us introduce the per-state $\\mathbf { Q }$ -value matrix $Q ( s ) : = Q ( s , \\cdot , \\cdot ) \\bar { \\in \\mathbb { R } ^ { | \\mathcal { A } | \\times | \\mathcal { B } | } }$ for every $s \\in S$ , where the element indexed by the action pair $( a , b )$ is $Q ( s , a , b )$ . Similarly, we define the per-state policies $\\mu ( s ) : = \\mu ( \\cdot | s ) \\in \\Delta ( { \\dot { A } } )$ and $\\nu ( s ) : = \\nu ( \\cdot | s ) \\in \\Delta ( B )$ for both players. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In parallel to the original Bellman operator, we denote the soft Bellman operator $\\mathcal { T } _ { \\tau }$ as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/2059d579c93d1602d1e72439a83460122b2d3c51c9a3214e4c20249b2456a9c9.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\tau } ( Q ) ( s , a , b ) : = r ( s , a , b ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim P ( \\cdot \\vert s , a , b ) } \\left[ \\operatorname* { m a x } _ { \\mu ( s ^ { \\prime } ) \\in \\Delta ( A ) } \\operatorname* { m i n } _ { \\nu ( s ^ { \\prime } ) \\in \\Delta ( B ) } f _ { \\tau } \\left( Q ( s ^ { \\prime } ) ; \\mu ( s ^ { \\prime } ) , \\nu ( s ^ { \\prime } ) \\right) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where for each per-state Q-value matrix $Q ( s )$ , we introduce an entropy-regularized matrix game in the form of ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "equation",
+        "img_path": "images/be159e0ed6417306b53ac4f96ca81f6ad60044ec1cf69fdcd8c0e3698c23ba8a.jpg",
+        "text": "$$\n\\displaystyle \\operatorname* { m a x } _ { \\mu \\in \\Delta ( A ) } \\operatorname* { m i n } _ { \\nu \\in \\Delta ( B ) } f _ { \\tau } \\big ( Q ( s ) ; \\mu ( s ) , \\nu ( s ) \\big ) : = \\mu ( s ) ^ { \\top } Q ( s ) \\nu ( s ) - \\tau \\mathcal { H } ( \\mu ( s ) ) + \\tau \\mathcal { H } ( \\nu ( s ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The entropy-regularized value iteration then proceeds as ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0dfa08fcfd9718653016b28f0cfebe97143b66b36a3bdfdbe1308aa1e3e265a4.jpg",
+        "text": "$$\nQ ^ { ( t + 1 ) } = T _ { \\tau } ( Q ^ { ( t ) } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $Q ^ { ( 0 ) }$ is an initialization. By definition, the optimal minimax soft Q-function obeys ${ \\cal T } _ { \\tau } ( Q _ { \\tau } ^ { \\star } ) =$ $Q _ { \\tau } ^ { \\star }$ and therefore corresponds to the fix point of the soft Bellman operator. Given the above entropyregularized value iteration, the following lemma states its iterates contract linearly to the optimal minimax soft Q-function at a rate of the discount factor $\\gamma$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. The entropy-regularized value iteration (10) converges at a linear rate, i.e. $\\parallel Q ^ { ( t ) } -$   \n$Q _ { \\tau } ^ { \\star } \\| _ { \\infty } \\leq \\gamma ^ { t } \\| Q ^ { ( 0 ) } - Q _ { \\tau } ^ { \\star } \\| _ { \\infty }$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Approximate value iteration via policy extragradient methods. Proposition 2 suggests that the optimal minimax soft Q-function of the entropy-regularized MG can be found by solving a series of entropy-regularized matrix games induced by $\\bar { \\{ Q ^ { ( t ) } \\} } _ { t \\geq 0 }$ in (10), a task that can be accomplished by adopting the fast extragradient methods developed earlier. To proceed, we first define the following sampling oracle, which makes it rigorous that the proposed algorithm does not require access to the Q-function of the entire MG, but only its own single-agent Q-function when playing against the opponent’s policy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 2 (Sampling oracle for Markov games). Given any policy pair $\\mu ( s ) , \\nu ( s )$ and $Q$ -value matrix $Q ( s )$ for any $s \\in S$ , the sampling oracle returns ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82747ffa22810c00af6604f8c9488a585bb5fc2f6fc901ae3a034f3572b482f0.jpg",
+        "text": "$$\n[ Q ( s ) \\nu ( s ) ] _ { a } = \\mathbb { E } _ { b \\sim \\nu ( s ) } \\left[ Q ( s , a , b ) \\right] , \\qquad a n d \\qquad [ Q ( s ) ^ { \\top } \\mu ( s ) ] _ { b } = \\mathbb { E } _ { a \\sim \\mu ( s ) } \\left[ Q ( s , a , b ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "for any $a \\in { \\mathcal { A } }$ and $b \\in B$ . ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Algorithm 3: Policy Extragradient Method for Entropy-regularized Markov Game ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "1 initialization: $Q ^ { ( 0 ) } = 0$ .   \n2 for $t = 0 , 1 , 2 , \\cdots , T _ { \\mathrm { m a i n } } \\mathrm { { \\bf d o } }$   \n3 Let $Q ^ { ( t ) }$ denote $\\begin{array} { r } { Q ^ { ( t ) } ( s , a , b ) = r ( s , a , b ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim P ( \\cdot | s , a , b ) } V ^ { ( t ) } ( s ^ { \\prime } ) . } \\end{array}$ (11)   \n4 Invoke PU (Algorithm 1) or OMWU (Algorithm 2) for $T _ { \\mathrm { s u b } }$ iterations to solve the following entropy-regularized matrix game for every state $s$ , where the initialization is set as uniform distributions: $\\operatorname* { m a x } _ { \\mu ( s ) \\in \\Delta ( A ) } \\operatorname* { m i n } _ { \\nu ( s ) \\in \\Delta ( B ) } f _ { \\tau } \\big ( Q ^ { ( t ) } ( s ) ; \\mu ( s ) , \\nu ( s ) \\big ) .$ Return the last iterate $\\bar { \\mu } ^ { ( t , T _ { \\mathrm { s u b } } ) } ( s ) , \\bar { \\nu } ^ { ( t , T _ { \\mathrm { s u b } } ) } ( s )$ .   \n5 Set $V ^ { ( t + 1 ) } ( s ) = f _ { \\tau } \\left( Q ^ { ( t ) } ( s ) ; \\bar { \\mu } ^ { ( t , \\bar { T } _ { \\mathrm { s u b } } ) } ( s ) , \\bar { \\nu } ^ { ( t , \\bar { T } _ { \\mathrm { s u b } } ) } ( s ) \\right)$ . ",
+        "bbox": [
+            158,
+            113,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Encouragingly, by judiciously setting the number of iterations in both the outer loop (for updating the Q-value matrices) and the inner loop (for updating the QRE of the corresponding Q-value matrix), we are guaranteed to find the QRE of the entropy-regularized MG in a small number of iterations without solving the iteration-varying matrix games exactly, as dictated by the following theorem. ",
+        "bbox": [
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+            337,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Assume $| { \\mathcal { A } } | \\geq | { \\mathcal { B } } |$ and $\\tau \\leq 1$ . Setting $\\begin{array} { r } { \\eta = \\frac { 1 - \\gamma } { 2 ( 1 + \\tau ( \\log | \\mathcal { A } | + 1 - \\gamma ) ) } } \\end{array}$ , the total iterations (namely, the product $T _ { \\mathrm { m a i n } } \\cdot T _ { \\mathrm { s u b , } }$ ) required for Algorithm 3 to achieve $\\left\\| Q ^ { ( { \\vec { T } _ { \\operatorname* { m i n } } } ) } - Q _ { \\tau } ^ { \\star } \\right\\| _ { \\infty } \\leq \\epsilon$ is at most $\\begin{array} { r } { O \\left( \\frac { ( \\log | \\cal { A } | + 1 / \\tau ) } { ( 1 - \\gamma ) ^ { 2 } } \\left( \\log \\frac { \\log | \\cal { A } | } { ( 1 - \\gamma ) \\epsilon } \\right) ^ { 2 } \\right) } \\end{array}$ . ",
+        "bbox": [
+            173,
+            398,
+            825,
+            468
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 ensures that within $\\begin{array} { r } { \\widetilde O \\left( \\frac { 1 } { \\tau ( 1 - \\gamma ) ^ { 2 } } \\log ^ { 2 } \\left( \\frac { 1 } { \\epsilon } \\right) \\right) } \\end{array}$ iterations, Algorithm 3 finds a pair of policies whose value is close to the optimal minimax soft Q-function $Q _ { \\tau } ^ { \\star }$ in an entrywise manner to a prescribed accuracy $\\epsilon$ . Remarkably, the iteration complexity is independent of the dimensions of the state space and the action space (up to log factors). ",
+        "bbox": [
+            174,
+            481,
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+            546
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Remark 4 (Duality gap and solving the unregularized MG). Solving the entropy-regularized MG provides a viable strategy to find an $\\epsilon$ -approximate NE of the unregularized MG, where the optimality of a policy pair is typically gauged by the duality gap. Fortunately, this can be achieved similar to the case of matrix games, and we refer interested readers to Cen et al. (2021) for details. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            632,
+            305,
+            648
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper develops provably efficient policy extragradient methods (PU and OMWU) for entropyregularized matrix games and Markov games, whose last iterates are guaranteed to converge linearly to the quantal response equilibrium at a linear rate. Encouragingly, the rate of convergence is independent of the dimension of the problem, i.e. the sizes of the space space and the action space. In addition, the last iterates of the proposed algorithms can also be used to locate Nash equilibria for the unregularized competitive games without assuming the uniqueness of the Nash equilibria by judiciously tuning the amount of regularization. This work opens up interesting opportunities for further investigations of policy extragradient methods for solving competitive games. For example, can we develop a two-time-scale policy extragradient algorithms for Markov games where the Qfunction is updated simultaneously with the policy but potentially at a different time scale, using samples, such as in an actor-critic algorithm (Konda and Tsitsiklis, 2000)? ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "S. Cen and Y. Chi are supported in part by the grants ONR N00014-18-1-2142 and N00014-19-1- 2404, ARO W911NF-18-1-0303, NSF CCF-1901199, CCF-2007911 and CCF-2106778. Y. Wei is supported in part by the NSF grants CCF-2007911, DMS-2147546/2015447 and CCF-2106778. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \nArora, S., Hazan, E., and Kale, S. (2012). The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(1):121–164.   \nBai, Y. and Jin, C. (2020). Provable self-play algorithms for competitive reinforcement learning. In International Conference on Machine Learning, pages 551–560. PMLR.   \nBowling, M. and Veloso, M. (2001). Rational and convergent learning in stochastic games. In Proceedings of the 17th international joint conference on Artificial intelligence-Volume 2, pages 1021–1026.   \nCen, S., Cheng, C., Chen, Y., Wei, Y., and Chi, Y. (2020). Fast global convergence of natural policy gradient methods with entropy regularization. arXiv preprint arXiv:2007.06558.   \nCen, S., Wei, Y., and Chi, Y. (2021). Fast policy extragradient methods for competitive games with entropy regularization. arXiv preprint arXiv:2105.15186.   \nCesa-Bianchi, N. and Lugosi, G. (2006). Prediction, learning, and games. 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Proceedings of the National Academy of Sciences, 39(10):1095–1100.   \nSyrgkanis, V., Agarwal, A., Luo, H., and Schapire, R. E. (2015). Fast convergence of regularized learning in games. In Proceedings of the 28th International Conference on Neural Information Processing Systems-Volume 2, pages 2989–2997.   \nTseng, P. (1995). On linear convergence of iterative methods for the variational inequality problem. Journal of Computational and Applied Mathematics, 60(1-2):237–252.   \nWei, C.-Y., Lee, C.-W., Zhang, M., and Luo, H. (2021a). Last-iterate convergence of decentralized optimistic gradient descent/ascent in infinite-horizon competitive Markov games. arXiv preprint arXiv:2102.04540.   \nWei, C.-Y., Lee, C.-W., Zhang, M., and Luo, H. (2021b). Linear last-iterate convergence in constrained saddle-point optimization. In International Conference on Learning Representations (ICLR).   \nXie, Q., Chen, Y., Wang, Z., and Yang, Z. (2020). Learning zero-sum simultaneous-move Markov games using function approximation and correlated equilibrium. In Conference on Learning Theory, pages 3674–3682. PMLR.   \nYadav, A., Shah, S., Xu, Z., Jacobs, D., and Goldstein, T. (2017). Stabilizing adversarial nets with prediction methods. arXiv preprint arXiv:1705.07364.   \nZhan, W., Cen, S., Huang, B., Chen, Y., Lee, J. D., and Chi, Y. (2021). Policy mirror descent for regularized reinforcement learning: A generalized framework with linear convergence. arXiv preprint arXiv:2105.11066.   \nZhang, K., Kakade, S., Basar, T., and Yang, L. (2020). Model-based multi-agent RL in zero-sum Markov games with near-optimal sample complexity. Advances in Neural Information Processing Systems, 33.   \nZhao, Y., Tian, Y., Lee, J. D., and Du, S. S. (2021). Provably efficient policy gradient methods for two-player zero-sum Markov games. arXiv preprint arXiv:2102.08903. ",
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parse/train/Hke-WTVtwr/Hke-WTVtwr.md

@@ -0,0 +1,387 @@
+# ENCODING WORD ORDER IN COMPLEX EMBEDDINGS
+
+Benyou Wang ∗ University of Padua wang@dei.unipd.it
+
+Donghao Zhao ∗ Tianjin University zhaodh@tju.edu.cn
+
+Christina Lioma University of Copenhagen chrh@di.ku.dk
+
+Qiuchi Li University of Padua qiuchili@dei.unipd.it
+
+Peng Zhang † Tianjin University pzhang@tju.edu.cn
+
+Jakob Grue Simonsen University of Copenhagen simonsen@di.ku.dk
+
+# ABSTRACT
+
+Sequential word order is important when processing text. Currently, neural networks (NNs) address this by modeling word position using position embeddings. The problem is that position embeddings capture the position of individual words, but not the ordered relationship (e.g., adjacency or precedence) between individual word positions. We present a novel and principled solution for modeling both the global absolute positions of words and their order relationships. Our solution generalizes word embeddings, previously defined as independent vectors, to continuous word functions over a variable (position). The benefit of continuous functions over variable positions is that word representations shift smoothly with increasing positions. Hence, word representations in different positions can correlate with each other in a continuous function. The general solution of these functions is extended to complex-valued domain due to richer representations. We extend CNN, RNN and Transformer NNs to complex-valued versions to incorporate our complex embedding (we make all code available). Experiments 1 on text classification, machine translation and language modeling show gains over both classical word embeddings and position-enriched word embeddings. To our knowledge, this is the first work in NLP to link imaginary numbers in complexvalued representations to concrete meanings (i.e., word order).
+
+# 1 INTRODUCTION
+
+When processing text, the sequential structure of language is important, but can be computationally costly to model with neural networks (NNs) (Socher et al., 2011) due to the difficulty in parallelization. This has been alleviated by modeling word sequence not on the NN architecture level, but by adding position embeddings on the feature level. This has been done by the convolutional sequence model (ConvSeq) (Gehring et al., 2017) and the Transformer model (Vaswani et al., 2017) that replaces recurrent and convolution operations with purely attention mechanisms. More generally, vanilla position embeddings (Gehring et al., 2017) assume that individual word positions are independent and do not consider relations between neighbouring word positions. We posit that both the global absolute positions of words and their inner sequential and adjacent relationships are crucial in language. This is supported by recent empirical findings by Shaw et al. (2018) and Dai et al. (2019) who show the importance of modeling distance between sequential elements, and explicitly use extra relative position encodings to capture the relative-distance relationship of words.
+
+We present a novel and principled approach to model both the global absolute positions of words and their inner sequential and adjacent relationships as follows: we extend each word embedding, previously defined as an independent vector, as a continuous function over an independent variable i.e., position. The benefit of continuous functions over variable positions is that word representations shift smoothly with increasing positions. Hence, word representations in different positions can correlate with each other in a continuous function. Fig. 1 illustrates this type of word representation with a three-dimensional complex-valued embedding, where the amplitudes $\{ r _ { 1 } , r _ { 2 } , r _ { 3 } \}$ denote semantic aspects corresponding to classical word vectors, and periods $\{ p _ { 1 } , p _ { 2 } , p _ { 3 } \}$ denote how sensitive the word is to positional information. We further discuss the necessary properties of these functions to model sequential information and obtain a general solution in the form of a complexvalued embedding. Interestingly, there is a direct connection between a specific case of our general embedding and the well-known positional encoding in Vaswani et al. (2017) (see App. A).
+
+![](images/ee1df64c33d8fcbca82c0506d4ba4959600580db8fbc24d9ca169bde95813e90.jpg)  
+Figure 1: 3-dimensional complex embedding for a single word in different positions. The three wave functions (setting the initial phases as zero) show the real part of the embedding; the imaginary part has a $\frac { \pi } { 2 }$ phase difference and shows the same curves with its real-valued counterpart. The $\mathbf { X }$ -axis denotes the absolute position of a word and the y-axis denotes the value of each element in its word vector. Colours mark different dimensions of the embedding. The three cross points between the functions and each vertical line (corresponding to a specific position pos) represent the embedding for this word in the pos-th position.
+
+We contribute (i) a novel paradigm that extends word vectors as continuous functions over changing variables like word position, and (ii) a general word embedding that models word order in a mathematically-sound manner. We integrate our complex word embeddings in state-of-the-art (SOTA) NN architectures (CNN, RNN, Transformer and experimentally find that it yields gains over both classical word embeddings and position-enriched word embeddings in text classification, machine translation and language modeling. Note that this is the first work in NLP to link imaginary numbers in complex-valued representation to concrete meanings (i.e., word order).
+
+# 2 MODELLING WORD ORDER IN EMBEDDING SPACE
+
+A Word Embedding (WE) generally defines a map $f _ { w e } : \mathbb { N }  \mathbb { R } ^ { D }$ from a discrete word index to a $D$ -dimensional real-valued vector and $\mathbb { N } = \{ 0 , 1 , 2 , \ldots \}$ . Similarly, a Position Embedding (PE) (Gehring et al., 2017; Vaswani et al., 2017) defines another map $f _ { p e } : \mathbb { N }  \mathbb { R } ^ { D }$ from a discrete position index to a vector. The final embedding for word $w _ { j }$ $\mathbf { \Phi } _ { w _ { j } } \in \mathbb { W }$ with index $j$ in a given vocabulary W) in the pos-th position in a sentence is usually constructed by the sum
+
+$$
+f ( j , p o s ) = f _ { w e } ( j ) + f _ { p e } ( p o s ) ,
+$$
+
+and $f ( j , p o s ) \in \mathbb { R } ^ { D }$ . Since both the word embedding map $f _ { w }$ and the position embedding map $f _ { p }$ only take integer values as word indexes or position indexes, embedding vectors for individual words or positions are trained independently. The independent training for each word vector is reasonable, since a word index is based on the order of a given arbitrary vocabulary and does not capture any specific sequential relationship with its neighboring words. However, the position index captures an ordered relationship, for instance adjacency or precedence, leading to the problem that position embeddings in individual positions (Gehring et al., 2017) are independent of each other; the ordered relationship between positions is not modelled. We refer to this as the position independence problem. This problem becomes more crucial when position embeddings are used in position-insensitive NNs, e.g., FastText (Mikolov et al., 2013b), ConvSeq (Gehring et al., 2017) and Transformer (Vaswani et al., 2017), because it is hard for such position-insensitive NNs with vanilla position embeddings (Gehring et al., 2017) to infer that $w _ { j _ { 1 } }$ in the pos-th position is close to $w _ { j _ { 2 } }$ in the $p o s + 1$ -th position, or that $w _ { j _ { 1 } }$ precedes $w _ { j _ { 2 } }$ ; instead, it is only inferred that $w _ { j _ { 1 } }$ and $w _ { j _ { 2 } }$ are in different positions, while the relative distance between them is almost unknown. Thus vanilla position embeddings (Gehring et al., 2017) cannot fully capture the sequential aspect of language.
+
+Next, we first introduce the necessary properties to model word order in embeddings, and then give a unique solution to meet such properties.
+
+# 2.1 EXTENDING VECTORS TO FUNCTIONS
+
+In the general definition in Eq. 1, each dimension of the position embedding is obtained based on the discrete position indexes $\{ 0 , 1 , 2 , . . . , \mathrm { p o s } , . . . \}$ . This makes it difficult to model the ordered relationship between the positions. One solution to this problem is to build continuous functions over a variable (i.e., position index) to represent a specific word in an individual dimension. Formally, we define a general embedding as
+
+$$
+f ( j , \mathrm { p o s } ) = \pmb { g } _ { j } ( \mathrm { p o s } ) \in \mathbb { R } ^ { D } ,
+$$
+
+where ${ \pmb g } _ { \mathcal { I } }$ is short for $\pmb { g } _ { w e } ( j ) \in ( \mathcal { F } ) ^ { D }$ , indicating $D$ functions over position index pos, and $g _ { w e } ( \cdot ) :$ : $\mathbb { N } \to ( \mathcal { F } ) ^ { D }$ is a mapping from a word index to $D$ functions. By expanding the $D$ dimension of ${ \pmb g } _ { \mathcal { I } }$ , a word $w _ { j }$ in the pos-th position can be represented as a $D$ -dimensional vector as shown in
+
+$$
+[ g _ { j , 1 } ( \mathbf { p o s } ) , g _ { j , 2 } ( \mathbf { p o s } ) , . . . , g _ { j , D } ( \mathbf { p o s } ) ] \in \mathbb { R } ^ { D } ,
+$$
+
+in which $\forall g _ { j , d } ( \cdot ) \in \mathcal { F } : \mathbb { N } \to \mathbb { R } , d \in \{ 1 , 2 , . . . , D \}$ is a function over the position index pos. To move the word $w _ { j }$ from the current position $p o s$ to another one $p o s ^ { \prime }$ , it needs only replace the variable pos to $p o s ^ { \prime }$ without changing ${ \pmb g } _ { \mathcal { I } }$ .
+
+Functions for words, especially continuous functions, are expected to capture smooth transformation from a position to its adjacent position therefore modeling word order. The position-independent position embedding (Gehring et al., 2017) can be considered as a special case of our definition when it only takes independent values for individual positions in the embedding function.
+
+# 2.2 PROPERTIES FOR THE FUNCTIONS TO CAPTURE WORD ORDER
+
+Relative distance is hard to compute because position indices are not visible in NNs after vector embedding (discrete position indices are necessarily embedded as vectors like words to be backpropagated with the gradient). Hence, we claim that the modeling of relative distance in NNs should be position-free: absolute position indices cannot be directly accessed in intermediate layers. Instead of processing position-free operations in NNs to capture relative distance between words, prior work (Shaw et al., 2018; Dai et al., 2019) first calculates the relative distance between words, and then feeds the relative distance as an additional feature or as embeddings/weights to NNs, instead of directly feeding with the raw position indices.
+
+Assume that words are embedded into $\mathbb { R } ^ { D }$ , and let, for $1 \leq d \leq D$ , the function $g _ { j , d } : \mathbb { N }  \mathbb { R }$ be the embedding function giving the $d$ -th coordinate of the representation of word $w _ { j }$ (i.e., $g _ { j , d } ( \mathfrak { p o s } )$ is the $d$ -th coordinate of the embedding of $w _ { j }$ if it occurs at position pos. In the following, we simply write $g$ instead of $g _ { j , d }$ when there is no risk of confusion. Ideally, one would like there to exist a function Tran $\operatorname { s f o r m } _ { n } : \mathbb { R }  \mathbb { R }$ that transforms the embedding of any word at some position pos to the embedding of a word at position $\boldsymbol { \mathrm { p o s } } + n$ such that Transform $^ { 1 } n$ is only dependent on the embedded value itself, but independent of the position pos, that is $\forall \mathbf { p o s } : g ( \mathbf { p o s } + n ) { \overline { { = \mathrm { T r a n s f o r m } _ { n } ( g ( \mathbf { p o s } ) ) } } }$ .
+
+Prior work in NLP (Li et al., 2019), Information Retrieval (Van Rijsbergen, 2004) and Machine Learning (Trabelsi et al., 2017) has shown the usefulness of complex numbers as richer representations. Complex word embeddings (Wang et al., 2019; Li et al., 2019; Li et al., 2018) have been used to model language. To investigate the potential of complex-valued representation, we extend the target domains of $g ( \cdot )$ from $\mathbb { R } ^ { D }$ to $\mathbb { C } ^ { D }$ without losing generality, since real-valued numbers are specific complex numbers with their imaginary part being zero. This property regarding “position-free offset transformation” in complex-valued domains is formally defined in Property 1 below.
+
+Property 1. Position-free offset transformation: An embedding function $g : \mathbb { N } \to \mathbb { C }$ is said to be a position-free offset transformation if there exists a function Transform : $\mathbb { N } \times \mathbb { C } \to \mathbb { C }$ (called the witness) such that for all $n \geq 1$ , the function Transfor $\mathfrak { m } _ { n } ( \cdot ) = \mathrm { T r a n s f o r m } ( n , \cdot )$ satisfies $\forall \mathsf { p o s } \in$ $\mathbb { N } : g ( \mathrm { p o s } + n ) = \mathrm { T r a n s f o r m } _ { n } ( g ( \mathrm { p o s } ) )$ . A position-free offset transformation $g$ is said to be linearly witnessed if there is a function $w : \mathbb { N }  \mathbb { C }$ such that $g$ has a witness Transform satisfying, for all $n$ , Transform $\operatorname { \mathrm { 1 } } ( n , \operatorname { p o s } ) = \operatorname { T r a n s f o r m } _ { n } ( \operatorname { p o s } ) = w ( n )$ (i.e., each Transform $_ n$ is a linear function).
+
+Additionally, a boundedness property is necessary to ensure that the position embedding can deal with text of any length (pos could be large in a long document).
+
+Property 2. Boundedness: The function over the variable position should be bounded, i.e. $\exists \delta \in$   
+$\overline { { \mathbb { R } ^ { + } , \forall \mathrm { p o s } \in \mathbb { N } } } , | g ( \mathrm { p o s } ) | \leq \delta$ .
+
+Formally, we prove the following claim that there is a unique solution that meets Properties 1 and 2 under the condition that the embedding function is linearly witnessed. We use linear functions because they are well-understood and simple with a single floating-point operation in NNs.
+
+Claim 1. A function $g : \mathbb { N }  \mathbb { C }$ is a bounded and linearly witnessed position-free offset transformation iff it is on the form $g ( p o s ) = z _ { 2 } z _ { 1 } ^ { p o s }$ for $z _ { 1 } , z _ { 2 } \in \mathbb { C }$ with $| z _ { 1 } | \le 1$ .
+
+Proof. Assume that $g$ is a bounded and linearly witnessed position-free offset transformation. Then, by linear witnessing, we have for all pos, $n _ { 1 } , n _ { 2 } \in \mathbb { N }$ :
+
+$$
+\begin{array} { r } { w ( n _ { 1 } ) w ( n _ { 2 } ) g ( \mathrm { p o s } ) = w ( n _ { 2 } ) g ( \mathrm { p o s } + n _ { 1 } ) = g ( \mathrm { p o s } + n _ { 1 } + n _ { 2 } ) \qquad } \\ { = \mathrm { T r a n s f o r m } _ { n _ { 1 } + n _ { 2 } } ( g ( \mathrm { p o s } ) ) = w ( n _ { 1 } + n _ { 2 } ) g ( \mathrm { p o s } ) } \end{array}
+$$
+
+whence $w ( n _ { 1 } + n _ { 2 } ) = w ( n _ { 1 } ) w ( n _ { 2 } )$ . Write $w ( 1 ) = z _ { 1 }$ and $g ( 0 ) = z _ { 2 }$ . As $n _ { 1 } , n _ { 2 } \in \mathbb { N }$ were arbitrary, Furthermore, observe that for we have $w ( n ) = ( w ( 1 ) ) ^ { n } = z _ { 1 } ^ { n }$ $\mathsf { p o s } \geq 1$ for all , we have $n \in \mathbb N$ . But then $g ( \mathrm { p o s } ) = g ( 1 + \mathrm { p o s } - 1 ) = w ( \mathrm { p o s } ) g ( 0 ) = z _ { 1 } ^ { \mathrm { p o s } } z _ { 2 } =$ $g ( \mathrm { p o s } + n ) = w ( n ) g ( \mathrm { p o s } ) = z _ { 1 } ^ { n } g ( \mathrm { p o s } )$ . $z _ { 2 } z _ { 1 } ^ { \mathrm { p o s } }$ . For $\mathrm { p o s } = 0$ , $g ( 0 ) = z _ { 2 } = z _ { 2 } z _ { 1 } ^ { 0 }$ , whence $g ( \mathrm { p o s } ) = z _ { 2 } z _ { 1 } ^ { \mathrm { p o s } }$ 1 2 , as desired. Observe that if $| z _ { 1 } | > 1$ $g ( { \mathfrak { p o s } } )$ unbouwith hence w. Then, $| z _ { 1 } | \le 1$ hat , w $g$ is once $g ( \mathsf { p o s } ) \stackrel { - } { = } z _ { 2 } z _ { 1 } ^ { \mathsf { p o s } }$ $| z _ { 1 } | \le 1$ $| g ( \mathrm { p o s } ) | \stackrel { \cdot } { \leq } | z _ { 2 } z _ { 1 } ^ { \mathrm { p o s } } | \leq | z _ { 2 } | | \dot { z } _ { 1 } ^ { \mathrm { p o s } } | \leq | z _ { 2 } |$ $g$ is bounded. Define, for each $n \in \mathbb N$ , $w ( n ) = z _ { 1 } ^ { n }$ and Transf ${ \mathrm { y r m } } _ { n } ( { \mathrm { p o s } } ) = w ( n ) { \mathrm { p o s } }$ . Then, for all pos, $n \in \mathbb { N }$ ,
+
+$$
+g ( \mathsf { p o s } + n ) = z _ { 2 } z _ { 1 } ^ { \mathsf { p o s } + n } = z _ { 2 } z _ { 1 } ^ { \mathsf { p o s } } z _ { 1 } ^ { n } = g ( \mathsf { p o s } ) z _ { 1 } ^ { n } = \mathrm { T r a n s f o r m } _ { n } ( g ( \mathsf { p o s } ) )
+$$
+
+showing that $g$ is a linearly witnessed position-free offset transformation.
+
+For any $z \in \mathbb { C }$ , we may write $z = r e ^ { i \theta } = r ( \cos \theta + i \sin \theta )$ . Thus, for the general form of the embedding $g$ from Theorem 1, we have:
+
+$$
+g ( \mathsf { p o s } ) = z _ { 2 } z _ { 1 } ^ { \mathsf { p o s } } = r _ { 2 } e ^ { i \theta _ { 2 } } ( r _ { 1 } e ^ { i \theta _ { 1 } } ) ^ { \mathsf { p o s } } = r _ { 2 } r _ { 1 } ^ { \mathsf { p o s } } e ^ { i ( \theta _ { 2 } + \theta _ { 1 } \mathsf { p o s } ) } ~ \mathrm { s u b j e c t } ~ \mathsf { t o } ~ | r _ { 1 } | \leq 1
+$$
+
+In implementations, the above definition of $g$ will lead to an optimization problem due to the constraint $| r _ { 1 } | \le 1$ . A natural and simple way to avoid this is to fix $r _ { 1 } = 1$ ; note that $| e ^ { i x } | \equiv 1$ , thus automatically satisfying the constraint, in contrast to a real-valued embedding where one would need to explicitly devise functions satisfying the constraint. Finally, Eq. 4 can be written in the simplified form: $g ( \mathrm { p o s } ) = r e ^ { i ( \omega \mathrm { p o s } + \theta ) }$ . Thus, one can think of $g$ as embedding positions counterclockwise on a complex circle of radius $r$ with a fixed period $\dot { \boldsymbol { r } }$ is the amplitude term, $\theta$ is the initial phase term, $\frac { \omega } { 2 \pi }$ is the frequency, and $\frac { 2 \pi } { \omega }$ is the period term).
+
+# 2.3 COMPLEX-VALUED WORD EMBEDDING
+
+We now define our complex-valued word embedding $g$ as a map taking a word index $j$ and position word index pos to $\mathbb { C } ^ { D }$ . For a word $w _ { j }$ in position pos, our general complex-valued embedding is defined as $f ( j , \mathrm { p o s } ) = { { g } _ { j } } ( \mathrm { p o s } ) = { { r } _ { j } } { { e } ^ { i ( \omega _ { j } \mathrm { p o s } + { { \theta } _ { j } } ) } }$ . Therefore, $f ( j , { \mathfrak { p o s } } )$ is defined as:
+
+$$
+[ r _ { j , 1 } e ^ { i ( \omega _ { j , 1 } \mathrm { p o s } + \theta _ { j , 1 } ) } , . . . , r _ { j , 2 } e ^ { i ( \omega _ { j , 2 } \mathrm { p o s } + \theta _ { j , 2 } ) } , \cdot \cdot \cdot , r _ { j , D } e ^ { i ( \omega _ { j , D } \mathrm { p o s } + \theta _ { j , D } ) } ]
+$$
+
+Note that each coordinate $d$ $1 \leq d \leq D )$ has a separate amplitude $\boldsymbol { r } _ { j , d }$ , period $\begin{array} { r } { p _ { j , d } = \frac { 2 \pi } { \omega _ { j , d } } } \end{array}$ , and initial phase $\theta _ { j , d }$ . In Fig. 1 each dimension is represented as a wave which is parameterized by an amplitude, a period/frequency, and an initial phase. The trainable parameters of the embedding are the amplitudes vector $\boldsymbol { r } _ { j } = [ r _ { j , 1 } , . . . , r _ { j , D } ]$ , the period/frequency related weights $\omega _ { j } = [ \omega _ { j , 1 } , . . . , \omega _ { j , D } ] ,$ , and the initial phase vector $\pmb { \theta } _ { j } = [ \theta _ { j , 1 } , . . . , \theta _ { j , D } ]$ . Note that the mean values of $f ( j , \cdot )$ over all positions are linearly dependent on the amplitude. Observe that the period/frequency determines to what degree the word is sensitive to the position. With an extremely long period (i.e., $\omega _ { j }$ very small), the complex-valued embedding is approximately constant for all possible values of pos, and hence approximates a standard word embedding. Conversely, if the period is short, the embedding will be highly sensitive to the position argument.
+
+In our embedding, the mean vectors of $f ( j , \cdot )$ taken over all positions are linearly correlated to the amplitude embedding $\pmb { r } _ { j } = [ r _ { j , 1 } , . . . , r _ { j , K } ]$ with a coefficient $\frac { 2 } { \pi }$ . The amplitude $\boldsymbol { r } _ { j , d }$ of our embedding depends only on the word $w _ { j }$ (and coordinate $d$ ), not on the position of the word, whence one can think of the vector $g _ { p e } ( j , \mathrm { p o s } ) = [ e ^ { i ( \omega _ { j , 1 } \mathrm { p o s } + \theta _ { j , 1 } ) } , \cdot \cdot \cdot , e ^ { i ( \omega _ { j , D } \mathrm { p o s } + \theta _ { j , D } ) } ]$ as a “purely” positional embedding. Consequently, our complex embedding can be considered an element-wise multiplication between the word embedding $\bar { g _ { w e } ( j ) } = [ r _ { j , 1 } , . . . , \bar { r _ { j , K } } ]$ and position embedding $g _ { p e }$ .
+
+Table 1: Dataset Statistics. CV means 10-fold cross validation. The last 2 datasets come with train/dev/test splits.   
+
+<table><tr><td>Dataset</td><td>train</td><td>test</td><td>vocab.</td><td>task</td><td>Classes</td></tr><tr><td>CR (Hu &amp; Liu,2014)</td><td>4K</td><td>CV</td><td>6K</td><td>product reviews</td><td>2</td></tr><tr><td>MPQA (Wiebe et al., 2005)</td><td>11k</td><td>CV</td><td>6K</td><td>opinion polarity</td><td>2</td></tr><tr><td>SUBJ (Pang &amp; Lee,2005)</td><td>10k</td><td>CV</td><td>21k</td><td>subjectivity</td><td>2</td></tr><tr><td>MR (Pang &amp; Lee,2005)</td><td>11.9k</td><td>CV</td><td>20k</td><td>moviereviews</td><td>2</td></tr><tr><td>SST (Socher et al.,2013)</td><td>67k</td><td>2.2k</td><td>18k</td><td>movie reviews</td><td>2</td></tr><tr><td>TREC (Li&amp; Roth,2002)</td><td>5.4k</td><td>0.5k</td><td>10k</td><td>Question</td><td>6</td></tr></table>
+
+$$
+f ( j , \mathrm { p o s } ) = g _ { w e } ( j ) \odot g _ { p e } ( j , \mathrm { p o s } )
+$$
+
+Prior work (Gehring et al., 2017; Vaswani et al., 2017) uses mean-weight addition between word embeddings $f _ { w e }$ and position embeddings $f _ { p e }$ (all words share the weights). In our work, word embeddings and position embeddings are decoupled to some extent by element-wise multiplication and therefore the frequency/period terms (related to $\omega _ { j , d } )$ ) can adaptively adjust the importance between semantic and position information for each word and each dimension. In particular, with higher frequency (i.e., large $\omega _ { j , d } )$ , the final embedding will change dramatically with the changing positions, while it can be fixed for any positions with an extremely-small frequency (i.e., small $\omega _ { j , d } )$ . Interestingly, the well-known position embedding in Transformer (Vaswani et al., 2017) can be seen as a degraded version of one of our specific complex word embeddings (see the Appendix A).
+
+# 3 EXPERIMENTAL EVALUATION
+
+We evaluate our embeddings in text classification, machine translation and language modeling.
+
+# 3.1 TEXT CLASSIFICATION
+
+Experimental Setup. We use six popular text classification datasets: CR, MPQA, SUBJ, MR, SST, and TREC (see Tab. 1). We use accuracy as evaluation measure based on fixed train/dev/test splits or cross validation, as per prior work. We use Fasttext (Joulin et al., 2016), CNN (Kim, 2014), LSTM and Transformer (Vaswani et al., 2017) as NN baselines2. We use each of them: (1) without positional information; (2) with Vanilla Position Embeddings (PE) (randomly initialized and updated during training using the sum between word and position vectors (Gehring et al., 2017); (3) with Trigonometric Position Embeddings (TPE) (defining position embeddings as trigonometric functions as per Eq. 7); (4) with Complex-vanilla word embeddings (where the amplitude embedding is initialized by the pre-trained word vectors, and the phrase embedding is randomly initialized in a range from $- \pi$ to $\pi$ without considering word order (Wang et al., 2019)); and (5) with our order-aware complex-valued word embeddings, Complex-order (which encode position in the phase parts, train the periods, and where the amplitude embedding is also initialized by pretrained word vectors). For more details on the complex-valued extensions of NNs, see App. B and App. C.
+
+Our embedding generally has $3 \times D \times | \mathbb { W } |$ parameters with D-dimensional word vectors and $| \mathbb { W } |$ words, while previous work (Mikolov et al., 2013b; Pennington et al., 2014) usually employs only $D \times | \mathbb { W } |$ parameters for embedding lookup tables. To increase efficiency and facilitate fair comparison with previous work we set initial phases $\pmb { \theta } _ { j } = [ \theta _ { j , 1 } , . . . , \theta _ { j , D } ]$ to a shared constant value (such as zero). Furthermore, the period vectors $\omega _ { j , d }$ depend on word index $j$ with length $\lvert \mathbb { W } \rvert$ and the coordinate index $d$ with length $D$ . To decrease the number of parameters, one can either use a word-sharing scheme (i.e., $\omega _ { j , d } = \omega _ { \cdot , d } )$ ), or a dimension-sharing scheme $( \omega _ { j , d } = \omega _ { j , \cdot } )$ , leading to $| \mathbb { W } | * D + | \bar { \mathbb { W } } |$ and $| \mathbb { W } | * D + D$ parameters in total for the embedding layer.
+
+Table 2: Text classification accuracy without position embeddings, with random position embeddings (PE), with trigonometric position embeddings (TPE), with complex-valued NNs without position embeddings (complex-vanilla), and with our complex-order embeddings. Superscripts $\ S$ , †, ‡ and ∗ mean a significant improvement over a baseline without position embeddings §, $\mathrm { P } \bar { \mathrm { E } } ^ { \dagger }$ , $\mathrm { \Delta T P E ^ { \ddagger } }$ and Complex-vanilla ∗ using Wilcoxon’s signed-rank test $\mathrm { p } { < } 0 . 0 5$ .   
+
+<table><tr><td>Method</td><td>MR</td><td>SUBJ</td><td>CR</td><td>MPQA</td><td>SST</td><td>TREC</td></tr><tr><td>Fasttext</td><td>0.765</td><td>0.916</td><td>0.789</td><td>0.874</td><td>0.788</td><td>0.874</td></tr><tr><td>Fasttext-PE</td><td>0.774</td><td>0.922</td><td>0.789</td><td>0.882</td><td>0.791</td><td>0.874</td></tr><tr><td>Fasttext-TPE</td><td>0.776</td><td>0.921</td><td>0.796</td><td>0.884</td><td>0.792</td><td>0.88</td></tr><tr><td>Fasttext-Complex-vanilla</td><td>0.773</td><td>0.918</td><td>0.79</td><td>0.867</td><td>0.803</td><td>0.872</td></tr><tr><td>Fasttext-Complex-order</td><td>0.7878+t*</td><td>0.9298†t*</td><td>0.800$+*</td><td>0.8898†t*</td><td>0.809S十**</td><td>0.8928+**</td></tr><tr><td>LSTM</td><td>0.775</td><td>0.896</td><td>0.813</td><td>0.887</td><td>0.807</td><td>0.858</td></tr><tr><td>LSTM-PE</td><td>0.778</td><td>0.915</td><td>0.822</td><td>0.889</td><td>0.811</td><td>0.858</td></tr><tr><td>LSTM-TPE</td><td>0.776</td><td>0.912</td><td>0.814</td><td>0.888</td><td>0.813</td><td>0.865</td></tr><tr><td>LSTM-Complex-vanilla</td><td>0.765</td><td>0.907</td><td>0.810</td><td>0.823</td><td>0.784</td><td>0.784</td></tr><tr><td>LSTM-Complex-order</td><td>0.7908+t*</td><td>0.9268+*</td><td>0.828S+*</td><td>0.8978†**</td><td>0.8198†**</td><td>0.869$+**</td></tr><tr><td>CNN</td><td>0.809</td><td>0.928</td><td>0.830</td><td>0.894</td><td>0.856</td><td>0.898</td></tr><tr><td>CNN-PE</td><td>0.816</td><td>0.938</td><td>0.831</td><td>0.897</td><td>0.856</td><td>0.890</td></tr><tr><td>CNN-TPE</td><td>0.815</td><td>0.938</td><td>0.836</td><td>0.896</td><td>0.838</td><td>0.918</td></tr><tr><td>CNN-Complex-vanilla</td><td>0.811</td><td>0.937</td><td>0.825</td><td>0.878</td><td>0.823</td><td>0.900</td></tr><tr><td>CNN-Complex-order</td><td>0.8258+t*</td><td>0.9518十**</td><td>0.852$十**</td><td>0.9068†**</td><td>0.8648+*</td><td>0.9398+**</td></tr><tr><td>Transformer w/o position embedding</td><td>0.669</td><td>0.847</td><td>0.735</td><td>0.716</td><td>0.736</td><td>0.802</td></tr><tr><td>Transformer-PE</td><td>0.737</td><td>0.859</td><td>0.751</td><td>0.722</td><td>0.753</td><td>0.820</td></tr><tr><td>Transformer-TPE (Vaswani et al., 2017)</td><td>0.731</td><td>0.863</td><td>0.762</td><td>0.723</td><td>0.761</td><td>0.834</td></tr><tr><td>Transformer-Complex-vanilla</td><td>0.715</td><td>0.848</td><td>0.753</td><td>0.786</td><td>0.742</td><td>0.856</td></tr><tr><td>Transformer-Complex-order</td><td>0.7468+*</td><td>0.895$+*</td><td>0.806$+**</td><td>0.863$+**</td><td>0.813$†*</td><td>0.8968†**</td></tr></table>
+
+Table 3: Text classification accuracy. $\star$ means that scores are reported from other papers.   
+
+<table><tr><td>Method</td><td>MR</td><td>SUBJ</td><td>CR</td><td>MPQA</td><td>SST</td><td>TREC</td></tr><tr><td>Word2vec Bow (Conneau et al.,2017) *</td><td>0.777</td><td>0.909</td><td>0.798</td><td>0.883</td><td>0.797</td><td>0.836</td></tr><tr><td>Sent2Vec (Pagliardini et al., 2017) *</td><td>0.763</td><td>0.912</td><td>0.791</td><td>0.872</td><td>0.802</td><td>0.858</td></tr><tr><td>QuickThoughts (Logeswaran &amp; Lee,2018)*</td><td>0.824</td><td>0.948</td><td>0.860</td><td>0.902</td><td>1</td><td>0.928</td></tr><tr><td>InferSent (Conneau et al., 2017) *</td><td>0.811</td><td>0.924</td><td>0.863</td><td>0.902</td><td>0.846</td><td>0.882</td></tr><tr><td>QPDN (Wang et al., 2019) *</td><td>0.801</td><td>0.927</td><td>0.810</td><td>0.870</td><td>0.839</td><td>0.882</td></tr></table>
+
+We search the hyper parameters from a parameter pool, with batch size in $\{ 3 2 , 6 4 , 1 2 8 \}$ , learning rate in $\{ 0 . 0 0 1 , 0 . 0 0 0 1 , 0 . 0 0 0 0 1 \}$ , L2-regularization rate in $\{ 0 , 0 . 0 0 1 , 0 . 0 0 0 1 \}$ , and number of hidden layer units in $\{ 1 2 0 , 1 2 8 \}$ . We use pre-trained 300-dimensional vectors from word2vec (Mikolov et al., 2013a) in all models except for Transformers. The models with trainable trigonometric position embedding produce nearly identical results compared to the non-trainable version, therefore we report the result of fixed position embeddings as per Vaswani et al. (2017). We adopt narrow convolution and max pooling in CNN, with number of filters in $\{ 6 4 , 1 2 8 \}$ , and size of filters in $\{ 3 , 4 , 5 \}$ . In all Transformer models, we only use the encoder layer to extract feature information, where the layer is 1, dimension of word and inner hidden are 256 and 512 respectively, and head number is 8.
+
+Results. The results are shown in Tab. 2. Our complex-order embeddings outperform all other variations at all times. This gain in effectiveness comes at a negligible (or non-existent) cost in efficiency (it varies per NN architecture – see Fig. 2). CNNs are the best performing NN as expected following Bai et al. (2018). Tranformer NNs benefit the most from our complex-order embeddings, most likely because they are our weakest baseline. To contextualise these results, Tab. 3 shows classification accuracy of five typical approaches on the same datasets (as reported in the original papers). Our complex-order embeddings outperform all methods, except for the CR dataset, where InferSent is marginally better. Overall, our approach is on a par with the SOTA in embeddings.
+
+We perform an ablation test (Tab. 4) on Transformer because it is the most common NN to be used with position embeddings. The two period-sharing schemas (dimension-sharing and word-sharing)
+
+![](images/3e281f61b64e81bd85bc32ab1d241a5dd6a5878639505a4ffd1356dda400c6a3.jpg)  
+Figure 2: Computation time (seconds) per epoch in Tensorflow on TITAN X GPU.
+
+Table 4: Ablation test for Transformer, showing the effect of (i) the definition of embedding layer $( f _ { d } ( j , \mathsf { p o s } ) )$ , and (ii) whether the real-part and imaginary transition share the weights, i.e., $\Re ( W ^ { Q / K / V } ) = \Im ( W ^ { Q / K / V } )$ .   
+
+<table><tr><td>Method</td><td colspan="2">Setting</td><td rowspan="2">Params</td><td rowspan="2">Accuracy</td><td rowspan="2">△</td></tr><tr><td></td><td>fa(j,pos)</td><td> share in WQ/K/V</td></tr><tr><td>Transformer-complex-order</td><td>rjde（wjdpos）</td><td>×</td><td>8.33M</td><td>0.813</td><td>=</td></tr><tr><td>adding initial phases</td><td>rj,dei(wj,dpos+0j,d）</td><td>×</td><td>11.89M</td><td>0.785</td><td>-0.028</td></tr><tr><td>dimension-sharing period schema</td><td>Tj,dewj,.pos</td><td>×</td><td>5.82M</td><td>0.797</td><td>-0.016</td></tr><tr><td>word-sharing period schema</td><td>Tj,deiw.,pos</td><td>×</td><td>5.81M</td><td>0.805</td><td>-0.008</td></tr><tr><td>dimension-sharing amplitude schema</td><td>rj,eiwj,.pos</td><td>×</td><td>5.82M</td><td>0.798</td><td>-0.015</td></tr><tr><td>word-sharing amplitude schema</td><td>r.deiw..apos</td><td>×</td><td>5.81M</td><td>0.804</td><td>-0.009</td></tr><tr><td>w/t encoding positions (complex-vanilla)</td><td>Tjdewjd</td><td>×</td><td>9.38M</td><td>0.764</td><td>-0.049</td></tr><tr><td>dimension-sharing period schema</td><td>Tj,dewj,.pos</td><td>√</td><td>4.77M</td><td>0.794</td><td>-0.019</td></tr><tr><td>word-sharing period schema</td><td>rj,dew.,apos</td><td>√</td><td>4.76M</td><td>0.797</td><td>-0.016</td></tr><tr><td>dimension-sharing amplitude schema</td><td>rj,eiwj,.pos</td><td>√</td><td>4.77M</td><td>0.792</td><td>-0.021</td></tr><tr><td>word-sharing amplitude schema</td><td>r.,deiw.,apos</td><td>√</td><td>4.76M</td><td>0.801</td><td>-0.012</td></tr><tr><td>w/t encoding positions (complex-vanilla)</td><td>rjdeiwjd</td><td>√</td><td>8.33M</td><td>0.743</td><td>-0.07</td></tr><tr><td>vanilla Transformer (Vaswani et al., 2017)</td><td>WEj,d+PEd</td><td>-</td><td>4.1M</td><td>0.761</td><td>-0.052</td></tr></table>
+
+slightly drop performance, because fewer parameters limit the representative power. Adding initial phases also hurts performance, although we observed that the loss could decrease faster in early epochs compared to the setting without offset. The negative effect of initial phases may be due to periodicity, and $\omega$ cannot be directly regularized with L2-norm penalties. The sharing schemes slightly decrease the performance with less parameters. More details of the learned periods/frequencies (e.g. the distributions of periods/frequencies and case studies) are shown in App. D.
+
+Note that the word-sharing schema outperform the Vanilla Transformer, (both have a comparable number of parameters). If we choose $\Re ( \mathrm { \dot { W } } ^ { Q / K / V } ) = \Im ( W ^ { Q / K / V } )$ , the additional parameters in the embedding layers will affect much less the whole parameter scale in the multiple-layer Transformer, since a embedding layer is only used in the first layer instead of the following Transformer layers.
+
+# 3.2 MACHINE TRANSLATION
+
+Experimental Setup. We use the standard WMT 2016 English-German dataset (Sennrich et al., 2016), whose training set consists of 29,000 sentence pairs. We use four baselines: basic Attentional encoder-decoder (AED) (Bahdanau et al., 2014); AED with Byte-pair encoding (BPE) subword segmentation for open-vocabulary translation (Sennrich et al., 2016); AED with extra linguistic features (morphological, part-of-speech, and syntactic dependency labels) (Sennrich & Haddow, 2016); and a 6-layer Transformer. Our approach (Transformer Complex-order) uses a batch size of 64, a head of 8, 6 layers, the rate of dropout is 0.1, and the dimension of the word embedding is 512. The embedding layer does not use initial phases, i.e., following $f ( j , p o s ) = r _ { j } e ^ { i ( \omega _ { j } p o s ) }$ . We evaluate MT performance with the Bilingual Evaluation Understudy (BLEU) measure.
+
+Table 6: Language modeling results. $\star$ marks scores reported from other papers.   
+
+<table><tr><td>Method</td><td>BLEU</td></tr><tr><td>AED (Bahdanau et al., 2014) *</td><td>26.8</td></tr><tr><td rowspan="4">AED+Linguistic (Sennrich &amp; Haddow,2016) ★ AED+BPE(Sennrich et al.,2016) * Transformer (Ma et al.,2019) *</td><td>28.4</td></tr><tr><td>34.2</td></tr><tr><td></td></tr><tr><td>34.5</td></tr><tr><td>Transformer complex vanilla Transformer Complex-order</td><td>34.7 35.8</td></tr></table>
+
+Table 5: Machine translation results. $\star$ marks scores reported from other papers.   
+
+<table><tr><td>Method BPC</td></tr><tr><td>BN-LSTM(Cooijmans et al., 2016) * 1.36 LN HM-LSTM(Chung et al., 2016) * 1.29</td></tr><tr><td>RHN (Zilly et al., 2017) * 1.27 Large mLSTM(Krause et al.,2016) * 1.27</td></tr><tr><td>Transformer XL 6L (Dai et al., 2019) 1.29 Transformercomplex vanilla 1.30 Transformer XL Complex-order 6L 1.26</td></tr></table>
+
+Results Tab. 5 shows the MT results. Our approach outperforms all baselines. Two things are worth noting: (1) Both the vanilla Transformer and our Transformer Complex-order outperform the three Attentional encoder-decoder baselines which are based on an LSTM encoder and decoder, even when AED uses additional features. (2) Our Transformer Complex-Order outperforms the Vanilla Transformer and complex-vanilla Transformer by 1.3 and 1.1 in absolute BLEU score respectively.
+
+# 3.3 LANGUAGE MODELING
+
+Experimental Setup. We use the text8 (Mahoney, 2011) dataset, consisting of English Wikipedia articles. The text is lowercased from a to z, and space. The dataset contains 100M characters (90M for training, 5M for dev, and 5M for testing, as per Mikolov et al. (2012)). We use as baselines BN-LSTM, LN HM-LSTM RHN and Large mLSTM, which are typical recurrent NNs for language modeling in this dataset. We evaluate performance with the Bits Per Character (BPC) measure, (the lower, the better). We run the coder in Dai et al. (2019) with 6 layers for Transformer XL 6L; our model, named Transformer XL complex-order, directly replaces the word embedding with our proposed embedding under the same setting. We choose 6 layers due to limitations in computing resources. For Transformer XL Complex-order, all other parameter settings are as for Transformer XL (Dai et al., 2019). Our complex-order model does not use initial phases.
+
+Results. We see in Tab. 6 that our method outperforms all baselines. The first four baselines (BNLSTM, LN HM-LSTM, RHN and Large mLSTM) are based on recurrent NNs and rely on different regulation methods to become applicable in multiple-layer recurrent architectures. Transformerbased architectures can easily be stacked with multiple layers due to their advantages in parallelization, however the vanilla Transformer does not outperform the multiple-layer recurrent baselines, most likely due to its limitation of 6 layers. Our Transformer XL Complex-order outperforms its vanilla counterpart under the 6-layer setting (and also strong recurrent network baselines), demonstrating that our embedding also generalizes well in tasks with long-term dependency. With limited resources, slightly increasing the parameters in the feature layer like our proposed embedding could be more beneficial than stacking more layers with linearly increasing parameters.
+
+# 4 RELATED WORK
+
+Complex-valued NNs are not new (Georgiou & Koutsougeras, 1992; Kim & Adalı, 2003; Hirose, 2003). Complex-valued weights have been used in NNs, motivated by biology (Reichert & Serre, 2013), and also as signal processing in speech recognition (Shi et al., 2006). More recently, Arjovsky et al. (2016) shifted RNNs into the complex domain and Wolter & Yao (2018) proposed a novel complex gated recurrent cell. Trabelsi et al. (2017) also developped a complex-valued NN for computer vision and audio processing.
+
+Complex numbers have also been applied to text processing like (Van Rijsbergen, 2004; Melucci, 2015; Blacoe et al., 2013). Trouillon et al. (2016) adopt complex embedding for entities in Knowledge Graph Completion to represent antisymmetric relations with Hermitian dot product. Li et al. (2019); Wang et al. (2019) extend word embeddings to complex-valued fashion in quantum probability driven NNs, seeing the overview in Wang et al. (2019). However, the physical meaning of both the complex-valued entity and word embeddings is unknown, since a complex number was considered as two real numbers in a black-box learning paradigm. Our work first links the phase in complex numbers to word position to define concrete physical meaning in document representations.
+
+# 5 CONCLUSIONS
+
+We extended word vectors to word functions with a variable i.e. position, to model the smooth shift among sequential word positions and therefore implicitly capture relative distances between words. These functions are well-defined to model the relative distances, therefore we derive a general solution in complex-valued fashion. Interestingly, the position embedding in Vaswani et al. (2017) can be considered a simplified version of our approach. We extend CNN, RNN and Transformer NNs to complex-valued versions to incorporate our complex embedding. Experiments on text classification, machine translation and language modeling show that our embeddings are more effective than vanilla position embeddings (Vaswani et al., 2017).
+
+# ACKNOWLEDGMENTS
+
+We thank Massimo Melucci and Emanuele Di Buccio for their helpful comments, Xindian Ma for his detailed experimental suggestions.
+
+This work is supported by the Quantum Access and Retrieval Theory (QUARTZ) project, which has received funding from the European Union‘s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 721321. Peng Zhang and Donghao Zhang are supported in part by Natural Science Foundation of China (grant No. 61772363, U1636203)
+
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+Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D. Manning, Andrew $\mathrm { N g }$ , and Christopher Potts. Recursive Deep Models for Semantic Compositionality Over a Sentiment Treebank. EMNLP, pp. 1631–1642, 2013.
+
+Chiheb Trabelsi, Olexa Bilaniuk, Ying Zhang, Dmitriy Serdyuk, Sandeep Subramanian, Joao Fe- ˜ lipe Santos, Soroush Mehri, Negar Rostamzadeh, Yoshua Bengio, and Christopher J Pal. Deep complex networks. arXiv preprint arXiv:1705.09792, 2017.
+
+Theo Trouillon, Johannes Welbl, Sebastian Riedel, ´ Eric Gaussier, and Guillaume Bouchard. Com- ´ plex embeddings for simple link prediction. In International Conference on Machine Learning, pp. 2071–2080, 2016.
+
+Cornelis Joost Van Rijsbergen. The geometry of information retrieval. Cambridge University Press, 2004.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017.
+
+Benyou Wang, Emanuele Di Buccio, and Massimo Melucci. Representing words in vector space and beyond. In Quantum-Like Models for Information Retrieval and Decision-Making, pp. 83–113. Springer, 2019.
+
+Benyou Wang, Qiuchi Li, Massimo Melucci, and Dawei Song. Semantic hilbert space for text representation learning. In The World Wide Web Conference on, pp. 3293–3299, 2019.
+
+Janyce Wiebe, Theresa Wilson, and Claire Cardie. Annotating Expressions of Opinions and Emotions in Language. Language Resources and Evaluation, 39(2-3):165–210, May 2005. ISSN 1574-020X, 1572-8412. doi: 10.1007/s10579-005-7880-9. URL http://link.springer. com/10.1007/s10579-005-7880-9.
+
+Moritz Wolter and Angela Yao. Gated complex recurrent neural networks. arXiv preprint arXiv:1806.08267, 2018.
+
+Julian Georg Zilly, Rupesh Kumar Srivastava, Jan Koutn´ık, and Jurgen Schmidhuber. Recurrent ¨ highway networks. In Proceedings of the 34th International Conference on Machine LearningVolume 70, pp. 4189–4198. JMLR. org, 2017.
+
+# A LINKING TO THE POSITION EMBEDDINGS IN (VASWANI ET AL., 2017)
+
+Vaswani et al. (2017) proposed a new initialization for position embedding, resulting in comparable performance with previous one (Gehring et al., 2017) even without fine-tuning. The position embedding is empirically selected as
+
+$$
+\begin{array} { r } { P E _ { 2 k } ( \cdot , p o s ) = \sin ( p o s / 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } ) } \\ { P E _ { 2 k + 1 } ( \cdot , p o s ) = \cos ( p o s / 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } ) } \end{array}
+$$
+
+Where pos is the position index, $2 k$ and $2 k + 1$ is the dimension index and $d _ { m o d e l }$ is the dimension size of embedding. The reason for choosing this position embedding was not well-explained and its general extension is unknown, leading to some difficulties to improve it.
+
+We claim that the proposed position embedding in (Vaswani et al., 2017) is a degraded version of one of our specific complex word embedding in word-sharing schema (i.e., $\omega _ { j , d } = \omega _ { \cdot , d } )$ ), in which $p _ { j , k } =$ $2 \pi \times 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } }$ and the initial phases are set as zero. In our complex-valued position embedding, let $f _ { p e , k } ( \cdot , p o s ) = e ^ { i \times 1 0 0 0 ^ { 2 k / d _ { m o d e l } } } = \cos ( 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } p o s ) + i \sin ( 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } p o s )$ . Note that there exists a bi-jection between $P E ( \cdot , p o s )$ and $f _ { p e , k } ( \cdot , p o s )$ :
+
+$$
+\begin{array} { r } { P E _ { 2 k } ( \cdot , p o s ) = \Im ( f _ { p e , k } ( \cdot , p o s ) ) , } \\ { P E _ { 2 k + 1 } ( \cdot , p o s ) = \Re ( f _ { p e , k } ( \cdot , p o s ) ) } \end{array}
+$$
+
+where $\mathfrak { R }$ and $\mathfrak { F }$ are the operations to take the real and imaginary part of a complex-valued number. Its inverse transformation is
+
+$$
+f _ { p e , k } ( \cdot , p o s ) = P E _ { 2 k + 1 } ( \cdot , p o s ) + i P E _ { 2 k } ( \cdot , p o s )
+$$
+
+In our overall embedding, each dimension $f _ { k } ( j , p o s ) = f _ { w e , k } ( j ) \odot f _ { p e , k } ( \cdot , p o s )$ in our approaches, while it is $E _ { k } ( j , p o s ) \stackrel { \_ } { = } W E _ { k } ( j ) + P E _ { k } ( \cdot , \bar { p o s } )$ in (Gehring et al., 2017; Vaswani et al., 2017). Hence the position embedding in (Vaswani et al., 2017) is equivalent, albeit not identical, to our complex-valued position embedding with $p _ { j , k } = 2 \pi \times 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } }$ . It is therefore a particular case of our complex-valued position embedding, the word-sharing schema in which all words share the same period at a certain dimension, i.e, $p _ { j , k } = p _ { \cdot , k }$ is irrelevant to the choice of $j$ .
+
+# B INTEGRATING COMPLEX-VALUED EMBEDDING TO GENERAL NEURAL NETWORKS
+
+Neural networks are typically given real numbers as inputs and return real numbers as outputs. To accommodate complex numbers as in- and output, we devise a complex-valued version of various neural network layers i.e. complex-valued FastText with dense layer, CNN, and RNN. Unlike existing complex-valued neural networks (Trabelsi et al., 2017; Wolter & Yao, 2018), our feature layers are also converted into complex-valued layers.
+
+Complex-valued FastText FastTest (Joulin et al., 2016) is a simple and efficient neural network architecture using a dense layer over the sum of all word embeddings for general text classification. For a linear dense layer, i.e., $\boldsymbol z = \mathrm { d e n s e } ( \boldsymbol x + i \boldsymbol y )$ , where $\pmb { x } + i \pmb { y }$ and $_ z$ denote the complex-valued in- and output, respectively. Let $\pmb { W } = \pmb { A } + i \pmb { B }$ and $b = c + i d$ be complex-valued linear weights and bias, respectively. Then, the complex-valued dense layer is given by:
+
+$$
+z = \sigma \left( A x - B y + c \right) + i \sigma ( B x + A y + d )
+$$
+
+where $\sigma$ is a real-valued activation function such as the sigmoid function. By rewriting (10) in matrix form (Trabelsi et al., 2017), we obtain:
+
+$$
+{ \biggl [ } \Re ( z ) { \biggr ] } = { \biggl [ } { \sigma } ( A x - B y + c ) { \biggr ] }
+$$
+
+where, for $z = x + i y , \Re ( z ) = x$ and $\Im ( z ) = y$ . To save parameters and fairly compare with our real-valued baselines, the weights for real-part and imaginary-part input can be shared, i.e., $\pmb { A } = \pmb { B } , c = d$ .
+
+Complex-valued CNN For the complex-valued version of the convolution operation Trabelsi et al. (2017), we similarly define a complex-valued convolution with separate real and imaginary kernels $\pmb { A }$ and $\textbf {  { B } }$ , to compute convolutions on the real and imaginary parts of the input in Eq. 10. A complex-valued CNN network is constructed by stacking the operations based on a complex-valued convolution kernel, adding a complex dense layer in (10), and taking their norm as the final prediction in the last layer.
+
+Complex-valued RNN The basic complex RNN formulation is:
+
+$$
+h _ { t } ^ { C } = f \left( W ^ { h } h _ { t - 1 } + W ^ { z } z _ { t } + b \right)
+$$
+
+where ${ \boldsymbol { z } } _ { t }$ and $\boldsymbol { h } _ { t }$ represent the complex-valued input and complex-value hidden state vectors at time $t , b$ is a complex-valued bias, $W ^ { h }$ and $W ^ { z }$ are complex-valued weight transitions for hidden state and input state, and $f \left( z \right) = \sigma \left( \Re \left( z \right) \right) + i \sigma \left( \Im \left( z \right) \right)$ is the activation function. The multiplication $W ^ { h } h _ { t - 1 }$ and $W ^ { z } z _ { t }$ is computed as defined in (10) above. Similarly, the complex-valued gates are used in LSTM via operations as in (12). In the final layer, a $l 2$ -norm operation is adopted to obtain a real-valued loss for backpropagation.
+
+Complex-valued Transformer The main components in the Transformer are self-attention sublayers and position-wise feed-forward (FFN) sublayers. A self-attention sublayer employs $h$ attention heads and the concatenation of all heads is used as the output followed by a parameterized linear transformation. For a sequence embedded as complex-valued vector $i n p u t = \{ \bar { \pmb { w } } _ { 1 } , \pmb { w } _ { 2 } , . . . , \pmb { w } _ { n } \}$ , the output of each head is computed as a weighted sum of a linear transformation of the input sequence itself, namely
+
+$$
+o u t p u t _ { i } = \sum _ { j } a _ { i , j } { \pmb w } _ { j } { \pmb W } ^ { V } ,
+$$
+
+where ${ \pmb w } _ { j }$ is a complex-valued vector and $W ^ { V }$ is a complex linear transformation; therefore outputi is also complex. Hence, output is a sequence of complex-valued vectors with the same shape as input. The weight coefficient, $a _ { i , j }$ , which is defined as in real-valued domain, is calculated as the softmax of the product between complex-valued query vectors and key vectors: $a _ { i , j } = \mathrm { s o f t m a x } \frac { e _ { i , j } } { \sum _ { i = 1 } ^ { n } e _ { i , j } }$ and
+
+$$
+{ e } _ { i , j } = \sqrt { \frac { \Re ( z ) ^ { 2 } + \Im ( z ) ^ { 2 } } { n } } , z = \left( { w } _ { i } { W } ^ { Q } \right) \left( { w } _ { j } { W } ^ { K } \right) ^ { \dag } ,
+$$
+
+$\mathbf { Z }$ is a complex number since ${ \pmb w } _ { i }$ and ${ \pmb w } _ { j }$ are complex-valued vectors and $W ^ { Q } , W ^ { K }$ are complexvalued transformation. To extend another variant of Transformer called Transformer XL, we keep its original relative position embedding and additionally replace its word embedding with our proposed embedding.
+
+Correspondingly, the FFN sublayer can easily be extended to a complex-valued version by replacing the real-valued layers with complex-valued ones. We use batch-normalization separately for the real and imaginary parts.
+
+# C IMPLEMENTATION OF THE PROPOSED EMBEDDING
+
+Words functions are implemented in neural networks by storing the function parameters $\{ r , \omega , \theta \}$ and then construct the values based on the arguments. Based on the definition, the implementation of the proposed embedding can easily be implemented with only modifying the embedding layer. We list the basic code to construct our general embedding as below:
+
+i m p o r t t o r c h   
+impor t math   
+c l a s s ComplexNN ( t o r c h . nn . M o d ul e ) : d e f i n i t ( s e l f , o p t ) : s u p e r ( ComplexNN , s e l f ) . i n i t ( ) s e l f . word emb $=$ t o r c h . nn . E m b e d d i n g ( o p t . n t o k e n , o p t . d m o d e l ) s e l f . f r e q u e n c y e m b $=$ t o r c h . nn . E m b e d d i n g ( o p t . n t o k e n , o p t . d m o d e l ) s e l f . i n i t i a l p h a s e e m b $=$ t o r c h . nn . E m b e d d i n g ( o p t . n t o k e n , o p t . d m o d e l )
+
+![](images/e18d67190d27ad89d72343f0f7e580d4499ce23a054a7022eb0448e518ff1704.jpg)  
+Figure 3: The distribution of the $\delta _ { j }$ . Higher values mean that the word representations are more sensitive to the word positions.
+
+d e f g e t e m b e d d i n g ( s e l f , x ) :
+
+a m p l i t u d e $=$ s e l f . w o r d e m b ( x ) f r e q u e n c y $=$ s e l f . f r e q u e n c y e m b ( x ) s e l f . i n i t i a l p h a s e e m b . w e i g h t $=$ t o r c h . n n . P a r a m e t e r ( s e l f . i n i t i a l p h a s e e m $\%$ ( $^ 2 \ast$ m a t h . p i ) ) s e n t l e $\mathbf { \epsilon } _ { 1 } = \mathbf { X }$ . s i z e $( - 1 )$ p o s s e q $=$ t o r c h . a r a n g e ( 1 , s e n t l e n + 1 , 1 . 0 , d e v i c e $=$ a m p l i t u d e . d e v i c e ) p o s s e q $=$ p o s s e q . u n s q u e e z e ( 0 ) . u n s q u e e z e $( - 1 )$ p o s s e q $=$ p o s s e q . r e p e a t ( [ x . s i z e ( 0 ) , 1 , a m p l i t u d e . s i z e ( − 1 ) ] ) d i m e n s i o n b a i s $=$ s e l f . i n i t i a l p h a s e e m b ( x ) e n c o u t p u t p h a s e $=$ t o r c h . mul ( p o s s e q , f r e q u e n c y ) $^ +$ d i m e n s i o n b a i s e n c o u t p u t r e a l $=$ a m p l i t u d e $^ *$ t o r c h . c o s ( e n c o u t p u t p h a s e ) e n c o u t p u t i m a g e $=$ a m p l i t u d e $^ *$ t o r c h . s i n ( e n c o u t p u t p h a s e ) # r e t u r n t o r c h . c a t ( [ e n c o u t p u t r e a l , e n c o u t p u t i m a g e ] , − 1 ) r e t u r n e n c o u t p u t r e a l , e n c o u t p u t i m a g e d e f f o r w a r d ( s e l f , $\mathbf { X }$ ) : r e t u r n s e l f . g e t e m b e d d i n g ( x )
+
+Note that both the frequency vectors $\omega$ and initial-phase vectors $\pmb { \theta }$ can be shared between words or dimensions, to save parameters. The proposed embedding can be also used in real-valued neural networks if one directly concatenates the real-part numbers and imaginary-part numbers as a doublesize real-valued vector; therefore it could easily be extended in any existing networks without any complex-valued components. For instance, it could be a good extension for Transformer based pretrained models like (Devlin et al., 2018) by enriching the feature layer.
+
+# D VISUALIZATION OF FREQUENCIES/PERIODS
+
+After training, we obtain the frequency vector $\omega$ for each word. For each word, the mean value of the absolute frequency values, i.e., $\begin{array} { r } { \delta _ { j } = \frac { 1 } { | D | } \sum _ { d = 1 } ^ { D } | \omega _ { j , d } | } \end{array}$ is considered as a metric to test the positional sensitivities of the word, since a period value could be negative during training. The density of the $\delta _ { j }$ is shown in Fig. 3.
+
+Words with the 50 greatest, and the 50 smallest, frequencies in the SST dataset are shown in Tab. 7. For the words with greatest frequencies, most of them are strong sentiment words like “worst”
+
+,“stupid” and “powerful”; a reason for this may be that such words appear in many positions in many documents during training, and thus they are more sensitive to the positions. Conversely, there are fewer words expressing strong sentiment among words with smaller frequencies, as shown in the second row.
+
+Table 7: Words with greatest frequencies and frequencies periods (based on $\delta _ { j }$ ) in SST (a sentiment classification task), all words are converted to lower-case. The strong sentiment words are bold based on manual labeling.   
+
+<table><tr><td>words</td><td></td></tr><tr><td>greatest frequencies in descending order</td><td>worst solid stupid powerful mess wonderful remarkable suffers intoxicating thoughtful rare captures portrait gem frontal terrific unique wannabe witty lousy pointless contrived none worse refreshingly charming inventive amazing junk incoherent refreshing mediocre unfunny thinks enjoyed heartbreaking delightfully crisp brilliant heart</td></tr><tr><td>smallest frequencies in ascending order</td><td>spirit perfectly nowhere mistake engrossing fashioned excellent unexpected wonderfully means slowly proposal schemes roiling juliette titles fabric superstar ah wow choreographed tastelessness beg fabulous muccino jacobi legendary jae rate example code sensation counter deaths hall eun drug mctiernan storylines cellophane wild motion ups trick comedy entertained mission frightening witnesses snoots liners african groan satisfaction calm saturday estranged holm refuses inquisitive</td></tr></table>

parse/train/Hke-WTVtwr/Hke-WTVtwr_content_list.json

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+        "text": "ENCODING WORD ORDER IN COMPLEX EMBEDDINGS ",
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+        "type": "text",
+        "text": "Benyou Wang ∗ University of Padua wang@dei.unipd.it ",
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+        "text": "Donghao Zhao ∗ Tianjin University zhaodh@tju.edu.cn ",
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+        "text": "Christina Lioma University of Copenhagen chrh@di.ku.dk ",
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+    {
+        "type": "text",
+        "text": "Qiuchi Li University of Padua qiuchili@dei.unipd.it ",
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+    {
+        "type": "text",
+        "text": "Peng Zhang † Tianjin University pzhang@tju.edu.cn ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jakob Grue Simonsen University of Copenhagen simonsen@di.ku.dk ",
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+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Sequential word order is important when processing text. Currently, neural networks (NNs) address this by modeling word position using position embeddings. The problem is that position embeddings capture the position of individual words, but not the ordered relationship (e.g., adjacency or precedence) between individual word positions. We present a novel and principled solution for modeling both the global absolute positions of words and their order relationships. Our solution generalizes word embeddings, previously defined as independent vectors, to continuous word functions over a variable (position). The benefit of continuous functions over variable positions is that word representations shift smoothly with increasing positions. Hence, word representations in different positions can correlate with each other in a continuous function. The general solution of these functions is extended to complex-valued domain due to richer representations. We extend CNN, RNN and Transformer NNs to complex-valued versions to incorporate our complex embedding (we make all code available). Experiments 1 on text classification, machine translation and language modeling show gains over both classical word embeddings and position-enriched word embeddings. To our knowledge, this is the first work in NLP to link imaginary numbers in complexvalued representations to concrete meanings (i.e., word order). ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "When processing text, the sequential structure of language is important, but can be computationally costly to model with neural networks (NNs) (Socher et al., 2011) due to the difficulty in parallelization. This has been alleviated by modeling word sequence not on the NN architecture level, but by adding position embeddings on the feature level. This has been done by the convolutional sequence model (ConvSeq) (Gehring et al., 2017) and the Transformer model (Vaswani et al., 2017) that replaces recurrent and convolution operations with purely attention mechanisms. More generally, vanilla position embeddings (Gehring et al., 2017) assume that individual word positions are independent and do not consider relations between neighbouring word positions. We posit that both the global absolute positions of words and their inner sequential and adjacent relationships are crucial in language. This is supported by recent empirical findings by Shaw et al. (2018) and Dai et al. (2019) who show the importance of modeling distance between sequential elements, and explicitly use extra relative position encodings to capture the relative-distance relationship of words. ",
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+        "text": "We present a novel and principled approach to model both the global absolute positions of words and their inner sequential and adjacent relationships as follows: we extend each word embedding, previously defined as an independent vector, as a continuous function over an independent variable i.e., position. The benefit of continuous functions over variable positions is that word representations shift smoothly with increasing positions. Hence, word representations in different positions can correlate with each other in a continuous function. Fig. 1 illustrates this type of word representation with a three-dimensional complex-valued embedding, where the amplitudes $\\{ r _ { 1 } , r _ { 2 } , r _ { 3 } \\}$ denote semantic aspects corresponding to classical word vectors, and periods $\\{ p _ { 1 } , p _ { 2 } , p _ { 3 } \\}$ denote how sensitive the word is to positional information. We further discuss the necessary properties of these functions to model sequential information and obtain a general solution in the form of a complexvalued embedding. Interestingly, there is a direct connection between a specific case of our general embedding and the well-known positional encoding in Vaswani et al. (2017) (see App. A). ",
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+        "image_caption": [
+            "Figure 1: 3-dimensional complex embedding for a single word in different positions. The three wave functions (setting the initial phases as zero) show the real part of the embedding; the imaginary part has a $\\frac { \\pi } { 2 }$ phase difference and shows the same curves with its real-valued counterpart. The $\\mathbf { X }$ -axis denotes the absolute position of a word and the y-axis denotes the value of each element in its word vector. Colours mark different dimensions of the embedding. The three cross points between the functions and each vertical line (corresponding to a specific position pos) represent the embedding for this word in the pos-th position. "
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+    {
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+        "text": "We contribute (i) a novel paradigm that extends word vectors as continuous functions over changing variables like word position, and (ii) a general word embedding that models word order in a mathematically-sound manner. We integrate our complex word embeddings in state-of-the-art (SOTA) NN architectures (CNN, RNN, Transformer and experimentally find that it yields gains over both classical word embeddings and position-enriched word embeddings in text classification, machine translation and language modeling. Note that this is the first work in NLP to link imaginary numbers in complex-valued representation to concrete meanings (i.e., word order). ",
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+        "type": "text",
+        "text": "2 MODELLING WORD ORDER IN EMBEDDING SPACE ",
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+        "text": "A Word Embedding (WE) generally defines a map $f _ { w e } : \\mathbb { N }  \\mathbb { R } ^ { D }$ from a discrete word index to a $D$ -dimensional real-valued vector and $\\mathbb { N } = \\{ 0 , 1 , 2 , \\ldots \\}$ . Similarly, a Position Embedding (PE) (Gehring et al., 2017; Vaswani et al., 2017) defines another map $f _ { p e } : \\mathbb { N }  \\mathbb { R } ^ { D }$ from a discrete position index to a vector. The final embedding for word $w _ { j }$ $\\mathbf { \\Phi } _ { w _ { j } } \\in \\mathbb { W }$ with index $j$ in a given vocabulary W) in the pos-th position in a sentence is usually constructed by the sum ",
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+        "text": "$$\nf ( j , p o s ) = f _ { w e } ( j ) + f _ { p e } ( p o s ) ,\n$$",
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+        "type": "text",
+        "text": "and $f ( j , p o s ) \\in \\mathbb { R } ^ { D }$ . Since both the word embedding map $f _ { w }$ and the position embedding map $f _ { p }$ only take integer values as word indexes or position indexes, embedding vectors for individual words or positions are trained independently. The independent training for each word vector is reasonable, since a word index is based on the order of a given arbitrary vocabulary and does not capture any specific sequential relationship with its neighboring words. However, the position index captures an ordered relationship, for instance adjacency or precedence, leading to the problem that position embeddings in individual positions (Gehring et al., 2017) are independent of each other; the ordered relationship between positions is not modelled. We refer to this as the position independence problem. This problem becomes more crucial when position embeddings are used in position-insensitive NNs, e.g., FastText (Mikolov et al., 2013b), ConvSeq (Gehring et al., 2017) and Transformer (Vaswani et al., 2017), because it is hard for such position-insensitive NNs with vanilla position embeddings (Gehring et al., 2017) to infer that $w _ { j _ { 1 } }$ in the pos-th position is close to $w _ { j _ { 2 } }$ in the $p o s + 1$ -th position, or that $w _ { j _ { 1 } }$ precedes $w _ { j _ { 2 } }$ ; instead, it is only inferred that $w _ { j _ { 1 } }$ and $w _ { j _ { 2 } }$ are in different positions, while the relative distance between them is almost unknown. Thus vanilla position embeddings (Gehring et al., 2017) cannot fully capture the sequential aspect of language. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we first introduce the necessary properties to model word order in embeddings, and then give a unique solution to meet such properties. ",
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+    {
+        "type": "text",
+        "text": "2.1 EXTENDING VECTORS TO FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the general definition in Eq. 1, each dimension of the position embedding is obtained based on the discrete position indexes $\\{ 0 , 1 , 2 , . . . , \\mathrm { p o s } , . . . \\}$ . This makes it difficult to model the ordered relationship between the positions. One solution to this problem is to build continuous functions over a variable (i.e., position index) to represent a specific word in an individual dimension. Formally, we define a general embedding as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9af78adb2dadc1a549d01b8486ae40b138cd2f0c29508ab1ba281b078d55c6d.jpg",
+        "text": "$$\nf ( j , \\mathrm { p o s } ) = \\pmb { g } _ { j } ( \\mathrm { p o s } ) \\in \\mathbb { R } ^ { D } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where ${ \\pmb g } _ { \\mathcal { I } }$ is short for $\\pmb { g } _ { w e } ( j ) \\in ( \\mathcal { F } ) ^ { D }$ , indicating $D$ functions over position index pos, and $g _ { w e } ( \\cdot ) :$ : $\\mathbb { N } \\to ( \\mathcal { F } ) ^ { D }$ is a mapping from a word index to $D$ functions. By expanding the $D$ dimension of ${ \\pmb g } _ { \\mathcal { I } }$ , a word $w _ { j }$ in the pos-th position can be represented as a $D$ -dimensional vector as shown in ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11c83652acf7a3b0ac5f911a9040fc2e1ef56561033821f7ca3bc19bbd96067f.jpg",
+        "text": "$$\n[ g _ { j , 1 } ( \\mathbf { p o s } ) , g _ { j , 2 } ( \\mathbf { p o s } ) , . . . , g _ { j , D } ( \\mathbf { p o s } ) ] \\in \\mathbb { R } ^ { D } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "in which $\\forall g _ { j , d } ( \\cdot ) \\in \\mathcal { F } : \\mathbb { N } \\to \\mathbb { R } , d \\in \\{ 1 , 2 , . . . , D \\}$ is a function over the position index pos. To move the word $w _ { j }$ from the current position $p o s$ to another one $p o s ^ { \\prime }$ , it needs only replace the variable pos to $p o s ^ { \\prime }$ without changing ${ \\pmb g } _ { \\mathcal { I } }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Functions for words, especially continuous functions, are expected to capture smooth transformation from a position to its adjacent position therefore modeling word order. The position-independent position embedding (Gehring et al., 2017) can be considered as a special case of our definition when it only takes independent values for individual positions in the embedding function. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.2 PROPERTIES FOR THE FUNCTIONS TO CAPTURE WORD ORDER",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Relative distance is hard to compute because position indices are not visible in NNs after vector embedding (discrete position indices are necessarily embedded as vectors like words to be backpropagated with the gradient). Hence, we claim that the modeling of relative distance in NNs should be position-free: absolute position indices cannot be directly accessed in intermediate layers. Instead of processing position-free operations in NNs to capture relative distance between words, prior work (Shaw et al., 2018; Dai et al., 2019) first calculates the relative distance between words, and then feeds the relative distance as an additional feature or as embeddings/weights to NNs, instead of directly feeding with the raw position indices. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Assume that words are embedded into $\\mathbb { R } ^ { D }$ , and let, for $1 \\leq d \\leq D$ , the function $g _ { j , d } : \\mathbb { N }  \\mathbb { R }$ be the embedding function giving the $d$ -th coordinate of the representation of word $w _ { j }$ (i.e., $g _ { j , d } ( \\mathfrak { p o s } )$ is the $d$ -th coordinate of the embedding of $w _ { j }$ if it occurs at position pos. In the following, we simply write $g$ instead of $g _ { j , d }$ when there is no risk of confusion. Ideally, one would like there to exist a function Tran $\\operatorname { s f o r m } _ { n } : \\mathbb { R }  \\mathbb { R }$ that transforms the embedding of any word at some position pos to the embedding of a word at position $\\boldsymbol { \\mathrm { p o s } } + n$ such that Transform $^ { 1 } n$ is only dependent on the embedded value itself, but independent of the position pos, that is $\\forall \\mathbf { p o s } : g ( \\mathbf { p o s } + n ) { \\overline { { = \\mathrm { T r a n s f o r m } _ { n } ( g ( \\mathbf { p o s } ) ) } } }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Prior work in NLP (Li et al., 2019), Information Retrieval (Van Rijsbergen, 2004) and Machine Learning (Trabelsi et al., 2017) has shown the usefulness of complex numbers as richer representations. Complex word embeddings (Wang et al., 2019; Li et al., 2019; Li et al., 2018) have been used to model language. To investigate the potential of complex-valued representation, we extend the target domains of $g ( \\cdot )$ from $\\mathbb { R } ^ { D }$ to $\\mathbb { C } ^ { D }$ without losing generality, since real-valued numbers are specific complex numbers with their imaginary part being zero. This property regarding “position-free offset transformation” in complex-valued domains is formally defined in Property 1 below. ",
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+    },
+    {
+        "type": "text",
+        "text": "Property 1. Position-free offset transformation: An embedding function $g : \\mathbb { N } \\to \\mathbb { C }$ is said to be a position-free offset transformation if there exists a function Transform : $\\mathbb { N } \\times \\mathbb { C } \\to \\mathbb { C }$ (called the witness) such that for all $n \\geq 1$ , the function Transfor $\\mathfrak { m } _ { n } ( \\cdot ) = \\mathrm { T r a n s f o r m } ( n , \\cdot )$ satisfies $\\forall \\mathsf { p o s } \\in$ $\\mathbb { N } : g ( \\mathrm { p o s } + n ) = \\mathrm { T r a n s f o r m } _ { n } ( g ( \\mathrm { p o s } ) )$ . A position-free offset transformation $g$ is said to be linearly witnessed if there is a function $w : \\mathbb { N }  \\mathbb { C }$ such that $g$ has a witness Transform satisfying, for all $n$ , Transform $\\operatorname { \\mathrm { 1 } } ( n , \\operatorname { p o s } ) = \\operatorname { T r a n s f o r m } _ { n } ( \\operatorname { p o s } ) = w ( n )$ (i.e., each Transform $_ n$ is a linear function). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Additionally, a boundedness property is necessary to ensure that the position embedding can deal with text of any length (pos could be large in a long document). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Property 2. Boundedness: The function over the variable position should be bounded, i.e. $\\exists \\delta \\in$   \n$\\overline { { \\mathbb { R } ^ { + } , \\forall \\mathrm { p o s } \\in \\mathbb { N } } } , | g ( \\mathrm { p o s } ) | \\leq \\delta$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, we prove the following claim that there is a unique solution that meets Properties 1 and 2 under the condition that the embedding function is linearly witnessed. We use linear functions because they are well-understood and simple with a single floating-point operation in NNs. ",
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+    {
+        "type": "text",
+        "text": "Claim 1. A function $g : \\mathbb { N }  \\mathbb { C }$ is a bounded and linearly witnessed position-free offset transformation iff it is on the form $g ( p o s ) = z _ { 2 } z _ { 1 } ^ { p o s }$ for $z _ { 1 } , z _ { 2 } \\in \\mathbb { C }$ with $| z _ { 1 } | \\le 1$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "Proof. Assume that $g$ is a bounded and linearly witnessed position-free offset transformation. Then, by linear witnessing, we have for all pos, $n _ { 1 } , n _ { 2 } \\in \\mathbb { N }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/02fe029bb38f319e348753ab7e72854b6e995d5a6b9ab63dac2933107e0b98c4.jpg",
+        "text": "$$\n\\begin{array} { r } { w ( n _ { 1 } ) w ( n _ { 2 } ) g ( \\mathrm { p o s } ) = w ( n _ { 2 } ) g ( \\mathrm { p o s } + n _ { 1 } ) = g ( \\mathrm { p o s } + n _ { 1 } + n _ { 2 } ) \\qquad } \\\\ { = \\mathrm { T r a n s f o r m } _ { n _ { 1 } + n _ { 2 } } ( g ( \\mathrm { p o s } ) ) = w ( n _ { 1 } + n _ { 2 } ) g ( \\mathrm { p o s } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "whence $w ( n _ { 1 } + n _ { 2 } ) = w ( n _ { 1 } ) w ( n _ { 2 } )$ . Write $w ( 1 ) = z _ { 1 }$ and $g ( 0 ) = z _ { 2 }$ . As $n _ { 1 } , n _ { 2 } \\in \\mathbb { N }$ were arbitrary, Furthermore, observe that for we have $w ( n ) = ( w ( 1 ) ) ^ { n } = z _ { 1 } ^ { n }$ $\\mathsf { p o s } \\geq 1$ for all , we have $n \\in \\mathbb N$ . But then $g ( \\mathrm { p o s } ) = g ( 1 + \\mathrm { p o s } - 1 ) = w ( \\mathrm { p o s } ) g ( 0 ) = z _ { 1 } ^ { \\mathrm { p o s } } z _ { 2 } =$ $g ( \\mathrm { p o s } + n ) = w ( n ) g ( \\mathrm { p o s } ) = z _ { 1 } ^ { n } g ( \\mathrm { p o s } )$ . $z _ { 2 } z _ { 1 } ^ { \\mathrm { p o s } }$ . For $\\mathrm { p o s } = 0$ , $g ( 0 ) = z _ { 2 } = z _ { 2 } z _ { 1 } ^ { 0 }$ , whence $g ( \\mathrm { p o s } ) = z _ { 2 } z _ { 1 } ^ { \\mathrm { p o s } }$ 1 2 , as desired. Observe that if $| z _ { 1 } | > 1$ $g ( { \\mathfrak { p o s } } )$ unbouwith hence w. Then, $| z _ { 1 } | \\le 1$ hat , w $g$ is once $g ( \\mathsf { p o s } ) \\stackrel { - } { = } z _ { 2 } z _ { 1 } ^ { \\mathsf { p o s } }$ $| z _ { 1 } | \\le 1$ $| g ( \\mathrm { p o s } ) | \\stackrel { \\cdot } { \\leq } | z _ { 2 } z _ { 1 } ^ { \\mathrm { p o s } } | \\leq | z _ { 2 } | | \\dot { z } _ { 1 } ^ { \\mathrm { p o s } } | \\leq | z _ { 2 } |$ $g$ is bounded. Define, for each $n \\in \\mathbb N$ , $w ( n ) = z _ { 1 } ^ { n }$ and Transf ${ \\mathrm { y r m } } _ { n } ( { \\mathrm { p o s } } ) = w ( n ) { \\mathrm { p o s } }$ . Then, for all pos, $n \\in \\mathbb { N }$ , ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/6d0795a5607bb360d12a9ff9b0f12a4fbbe5da63bed7601073fa004070938f20.jpg",
+        "text": "$$\ng ( \\mathsf { p o s } + n ) = z _ { 2 } z _ { 1 } ^ { \\mathsf { p o s } + n } = z _ { 2 } z _ { 1 } ^ { \\mathsf { p o s } } z _ { 1 } ^ { n } = g ( \\mathsf { p o s } ) z _ { 1 } ^ { n } = \\mathrm { T r a n s f o r m } _ { n } ( g ( \\mathsf { p o s } ) )\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "showing that $g$ is a linearly witnessed position-free offset transformation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For any $z \\in \\mathbb { C }$ , we may write $z = r e ^ { i \\theta } = r ( \\cos \\theta + i \\sin \\theta )$ . Thus, for the general form of the embedding $g$ from Theorem 1, we have: ",
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+    {
+        "type": "equation",
+        "img_path": "images/446f31a81a4ea34d6d45ae9ae88f3f4b1bfa136ba94f9eedde5d432bc81f94df.jpg",
+        "text": "$$\ng ( \\mathsf { p o s } ) = z _ { 2 } z _ { 1 } ^ { \\mathsf { p o s } } = r _ { 2 } e ^ { i \\theta _ { 2 } } ( r _ { 1 } e ^ { i \\theta _ { 1 } } ) ^ { \\mathsf { p o s } } = r _ { 2 } r _ { 1 } ^ { \\mathsf { p o s } } e ^ { i ( \\theta _ { 2 } + \\theta _ { 1 } \\mathsf { p o s } ) } ~ \\mathrm { s u b j e c t } ~ \\mathsf { t o } ~ | r _ { 1 } | \\leq 1\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In implementations, the above definition of $g$ will lead to an optimization problem due to the constraint $| r _ { 1 } | \\le 1$ . A natural and simple way to avoid this is to fix $r _ { 1 } = 1$ ; note that $| e ^ { i x } | \\equiv 1$ , thus automatically satisfying the constraint, in contrast to a real-valued embedding where one would need to explicitly devise functions satisfying the constraint. Finally, Eq. 4 can be written in the simplified form: $g ( \\mathrm { p o s } ) = r e ^ { i ( \\omega \\mathrm { p o s } + \\theta ) }$ . Thus, one can think of $g$ as embedding positions counterclockwise on a complex circle of radius $r$ with a fixed period $\\dot { \\boldsymbol { r } }$ is the amplitude term, $\\theta$ is the initial phase term, $\\frac { \\omega } { 2 \\pi }$ is the frequency, and $\\frac { 2 \\pi } { \\omega }$ is the period term). ",
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+    {
+        "type": "text",
+        "text": "2.3 COMPLEX-VALUED WORD EMBEDDING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now define our complex-valued word embedding $g$ as a map taking a word index $j$ and position word index pos to $\\mathbb { C } ^ { D }$ . For a word $w _ { j }$ in position pos, our general complex-valued embedding is defined as $f ( j , \\mathrm { p o s } ) = { { g } _ { j } } ( \\mathrm { p o s } ) = { { r } _ { j } } { { e } ^ { i ( \\omega _ { j } \\mathrm { p o s } + { { \\theta } _ { j } } ) } }$ . Therefore, $f ( j , { \\mathfrak { p o s } } )$ is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4200795c7256b106cf6b11d3f6f812f89fa1ca3dcf9833290c41d544ffbced76.jpg",
+        "text": "$$\n[ r _ { j , 1 } e ^ { i ( \\omega _ { j , 1 } \\mathrm { p o s } + \\theta _ { j , 1 } ) } , . . . , r _ { j , 2 } e ^ { i ( \\omega _ { j , 2 } \\mathrm { p o s } + \\theta _ { j , 2 } ) } , \\cdot \\cdot \\cdot , r _ { j , D } e ^ { i ( \\omega _ { j , D } \\mathrm { p o s } + \\theta _ { j , D } ) } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that each coordinate $d$ $1 \\leq d \\leq D )$ has a separate amplitude $\\boldsymbol { r } _ { j , d }$ , period $\\begin{array} { r } { p _ { j , d } = \\frac { 2 \\pi } { \\omega _ { j , d } } } \\end{array}$ , and initial phase $\\theta _ { j , d }$ . In Fig. 1 each dimension is represented as a wave which is parameterized by an amplitude, a period/frequency, and an initial phase. The trainable parameters of the embedding are the amplitudes vector $\\boldsymbol { r } _ { j } = [ r _ { j , 1 } , . . . , r _ { j , D } ]$ , the period/frequency related weights $\\omega _ { j } = [ \\omega _ { j , 1 } , . . . , \\omega _ { j , D } ] ,$ , and the initial phase vector $\\pmb { \\theta } _ { j } = [ \\theta _ { j , 1 } , . . . , \\theta _ { j , D } ]$ . Note that the mean values of $f ( j , \\cdot )$ over all positions are linearly dependent on the amplitude. Observe that the period/frequency determines to what degree the word is sensitive to the position. With an extremely long period (i.e., $\\omega _ { j }$ very small), the complex-valued embedding is approximately constant for all possible values of pos, and hence approximates a standard word embedding. Conversely, if the period is short, the embedding will be highly sensitive to the position argument. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our embedding, the mean vectors of $f ( j , \\cdot )$ taken over all positions are linearly correlated to the amplitude embedding $\\pmb { r } _ { j } = [ r _ { j , 1 } , . . . , r _ { j , K } ]$ with a coefficient $\\frac { 2 } { \\pi }$ . The amplitude $\\boldsymbol { r } _ { j , d }$ of our embedding depends only on the word $w _ { j }$ (and coordinate $d$ ), not on the position of the word, whence one can think of the vector $g _ { p e } ( j , \\mathrm { p o s } ) = [ e ^ { i ( \\omega _ { j , 1 } \\mathrm { p o s } + \\theta _ { j , 1 } ) } , \\cdot \\cdot \\cdot , e ^ { i ( \\omega _ { j , D } \\mathrm { p o s } + \\theta _ { j , D } ) } ]$ as a “purely” positional embedding. Consequently, our complex embedding can be considered an element-wise multiplication between the word embedding $\\bar { g _ { w e } ( j ) } = [ r _ { j , 1 } , . . . , \\bar { r _ { j , K } } ]$ and position embedding $g _ { p e }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/71d6fddbcbb19c028e644b6683e1214f76462e01010b5cc7a01f43777ff7d1e8.jpg",
+        "table_caption": [
+            "Table 1: Dataset Statistics. CV means 10-fold cross validation. The last 2 datasets come with train/dev/test splits. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>train</td><td>test</td><td>vocab.</td><td>task</td><td>Classes</td></tr><tr><td>CR (Hu &amp; Liu,2014)</td><td>4K</td><td>CV</td><td>6K</td><td>product reviews</td><td>2</td></tr><tr><td>MPQA (Wiebe et al., 2005)</td><td>11k</td><td>CV</td><td>6K</td><td>opinion polarity</td><td>2</td></tr><tr><td>SUBJ (Pang &amp; Lee,2005)</td><td>10k</td><td>CV</td><td>21k</td><td>subjectivity</td><td>2</td></tr><tr><td>MR (Pang &amp; Lee,2005)</td><td>11.9k</td><td>CV</td><td>20k</td><td>moviereviews</td><td>2</td></tr><tr><td>SST (Socher et al.,2013)</td><td>67k</td><td>2.2k</td><td>18k</td><td>movie reviews</td><td>2</td></tr><tr><td>TREC (Li&amp; Roth,2002)</td><td>5.4k</td><td>0.5k</td><td>10k</td><td>Question</td><td>6</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "img_path": "images/4313ba26dee6f520c296fda6cc12724c2fafd66f5247b95630e90d1ad1825a8f.jpg",
+        "text": "$$\nf ( j , \\mathrm { p o s } ) = g _ { w e } ( j ) \\odot g _ { p e } ( j , \\mathrm { p o s } )\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Prior work (Gehring et al., 2017; Vaswani et al., 2017) uses mean-weight addition between word embeddings $f _ { w e }$ and position embeddings $f _ { p e }$ (all words share the weights). In our work, word embeddings and position embeddings are decoupled to some extent by element-wise multiplication and therefore the frequency/period terms (related to $\\omega _ { j , d } )$ ) can adaptively adjust the importance between semantic and position information for each word and each dimension. In particular, with higher frequency (i.e., large $\\omega _ { j , d } )$ , the final embedding will change dramatically with the changing positions, while it can be fixed for any positions with an extremely-small frequency (i.e., small $\\omega _ { j , d } )$ . Interestingly, the well-known position embedding in Transformer (Vaswani et al., 2017) can be seen as a degraded version of one of our specific complex word embeddings (see the Appendix A). ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTAL EVALUATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate our embeddings in text classification, machine translation and language modeling. ",
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+    {
+        "type": "text",
+        "text": "3.1 TEXT CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experimental Setup. We use six popular text classification datasets: CR, MPQA, SUBJ, MR, SST, and TREC (see Tab. 1). We use accuracy as evaluation measure based on fixed train/dev/test splits or cross validation, as per prior work. We use Fasttext (Joulin et al., 2016), CNN (Kim, 2014), LSTM and Transformer (Vaswani et al., 2017) as NN baselines2. We use each of them: (1) without positional information; (2) with Vanilla Position Embeddings (PE) (randomly initialized and updated during training using the sum between word and position vectors (Gehring et al., 2017); (3) with Trigonometric Position Embeddings (TPE) (defining position embeddings as trigonometric functions as per Eq. 7); (4) with Complex-vanilla word embeddings (where the amplitude embedding is initialized by the pre-trained word vectors, and the phrase embedding is randomly initialized in a range from $- \\pi$ to $\\pi$ without considering word order (Wang et al., 2019)); and (5) with our order-aware complex-valued word embeddings, Complex-order (which encode position in the phase parts, train the periods, and where the amplitude embedding is also initialized by pretrained word vectors). For more details on the complex-valued extensions of NNs, see App. B and App. C. ",
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+        "type": "text",
+        "text": "Our embedding generally has $3 \\times D \\times | \\mathbb { W } |$ parameters with D-dimensional word vectors and $| \\mathbb { W } |$ words, while previous work (Mikolov et al., 2013b; Pennington et al., 2014) usually employs only $D \\times | \\mathbb { W } |$ parameters for embedding lookup tables. To increase efficiency and facilitate fair comparison with previous work we set initial phases $\\pmb { \\theta } _ { j } = [ \\theta _ { j , 1 } , . . . , \\theta _ { j , D } ]$ to a shared constant value (such as zero). Furthermore, the period vectors $\\omega _ { j , d }$ depend on word index $j$ with length $\\lvert \\mathbb { W } \\rvert$ and the coordinate index $d$ with length $D$ . To decrease the number of parameters, one can either use a word-sharing scheme (i.e., $\\omega _ { j , d } = \\omega _ { \\cdot , d } )$ ), or a dimension-sharing scheme $( \\omega _ { j , d } = \\omega _ { j , \\cdot } )$ , leading to $| \\mathbb { W } | * D + | \\bar { \\mathbb { W } } |$ and $| \\mathbb { W } | * D + D$ parameters in total for the embedding layer. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/777c66237d8f2c17c53e01b5aeff65834a3b57f5fcc7a6323d0ff3d017987acc.jpg",
+        "table_caption": [
+            "Table 2: Text classification accuracy without position embeddings, with random position embeddings (PE), with trigonometric position embeddings (TPE), with complex-valued NNs without position embeddings (complex-vanilla), and with our complex-order embeddings. Superscripts $\\ S$ , †, ‡ and ∗ mean a significant improvement over a baseline without position embeddings §, $\\mathrm { P } \\bar { \\mathrm { E } } ^ { \\dagger }$ , $\\mathrm { \\Delta T P E ^ { \\ddagger } }$ and Complex-vanilla ∗ using Wilcoxon’s signed-rank test $\\mathrm { p } { < } 0 . 0 5$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>MR</td><td>SUBJ</td><td>CR</td><td>MPQA</td><td>SST</td><td>TREC</td></tr><tr><td>Fasttext</td><td>0.765</td><td>0.916</td><td>0.789</td><td>0.874</td><td>0.788</td><td>0.874</td></tr><tr><td>Fasttext-PE</td><td>0.774</td><td>0.922</td><td>0.789</td><td>0.882</td><td>0.791</td><td>0.874</td></tr><tr><td>Fasttext-TPE</td><td>0.776</td><td>0.921</td><td>0.796</td><td>0.884</td><td>0.792</td><td>0.88</td></tr><tr><td>Fasttext-Complex-vanilla</td><td>0.773</td><td>0.918</td><td>0.79</td><td>0.867</td><td>0.803</td><td>0.872</td></tr><tr><td>Fasttext-Complex-order</td><td>0.7878+t*</td><td>0.9298†t*</td><td>0.800$+*</td><td>0.8898†t*</td><td>0.809S十**</td><td>0.8928+**</td></tr><tr><td>LSTM</td><td>0.775</td><td>0.896</td><td>0.813</td><td>0.887</td><td>0.807</td><td>0.858</td></tr><tr><td>LSTM-PE</td><td>0.778</td><td>0.915</td><td>0.822</td><td>0.889</td><td>0.811</td><td>0.858</td></tr><tr><td>LSTM-TPE</td><td>0.776</td><td>0.912</td><td>0.814</td><td>0.888</td><td>0.813</td><td>0.865</td></tr><tr><td>LSTM-Complex-vanilla</td><td>0.765</td><td>0.907</td><td>0.810</td><td>0.823</td><td>0.784</td><td>0.784</td></tr><tr><td>LSTM-Complex-order</td><td>0.7908+t*</td><td>0.9268+*</td><td>0.828S+*</td><td>0.8978†**</td><td>0.8198†**</td><td>0.869$+**</td></tr><tr><td>CNN</td><td>0.809</td><td>0.928</td><td>0.830</td><td>0.894</td><td>0.856</td><td>0.898</td></tr><tr><td>CNN-PE</td><td>0.816</td><td>0.938</td><td>0.831</td><td>0.897</td><td>0.856</td><td>0.890</td></tr><tr><td>CNN-TPE</td><td>0.815</td><td>0.938</td><td>0.836</td><td>0.896</td><td>0.838</td><td>0.918</td></tr><tr><td>CNN-Complex-vanilla</td><td>0.811</td><td>0.937</td><td>0.825</td><td>0.878</td><td>0.823</td><td>0.900</td></tr><tr><td>CNN-Complex-order</td><td>0.8258+t*</td><td>0.9518十**</td><td>0.852$十**</td><td>0.9068†**</td><td>0.8648+*</td><td>0.9398+**</td></tr><tr><td>Transformer w/o position embedding</td><td>0.669</td><td>0.847</td><td>0.735</td><td>0.716</td><td>0.736</td><td>0.802</td></tr><tr><td>Transformer-PE</td><td>0.737</td><td>0.859</td><td>0.751</td><td>0.722</td><td>0.753</td><td>0.820</td></tr><tr><td>Transformer-TPE (Vaswani et al., 2017)</td><td>0.731</td><td>0.863</td><td>0.762</td><td>0.723</td><td>0.761</td><td>0.834</td></tr><tr><td>Transformer-Complex-vanilla</td><td>0.715</td><td>0.848</td><td>0.753</td><td>0.786</td><td>0.742</td><td>0.856</td></tr><tr><td>Transformer-Complex-order</td><td>0.7468+*</td><td>0.895$+*</td><td>0.806$+**</td><td>0.863$+**</td><td>0.813$†*</td><td>0.8968†**</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/de7f52f5ee6760b22f8a39852d68d9c1d6d44ac42d72ca7c92dbece30230b7bb.jpg",
+        "table_caption": [
+            "Table 3: Text classification accuracy. $\\star$ means that scores are reported from other papers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>MR</td><td>SUBJ</td><td>CR</td><td>MPQA</td><td>SST</td><td>TREC</td></tr><tr><td>Word2vec Bow (Conneau et al.,2017) *</td><td>0.777</td><td>0.909</td><td>0.798</td><td>0.883</td><td>0.797</td><td>0.836</td></tr><tr><td>Sent2Vec (Pagliardini et al., 2017) *</td><td>0.763</td><td>0.912</td><td>0.791</td><td>0.872</td><td>0.802</td><td>0.858</td></tr><tr><td>QuickThoughts (Logeswaran &amp; Lee,2018)*</td><td>0.824</td><td>0.948</td><td>0.860</td><td>0.902</td><td>1</td><td>0.928</td></tr><tr><td>InferSent (Conneau et al., 2017) *</td><td>0.811</td><td>0.924</td><td>0.863</td><td>0.902</td><td>0.846</td><td>0.882</td></tr><tr><td>QPDN (Wang et al., 2019) *</td><td>0.801</td><td>0.927</td><td>0.810</td><td>0.870</td><td>0.839</td><td>0.882</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We search the hyper parameters from a parameter pool, with batch size in $\\{ 3 2 , 6 4 , 1 2 8 \\}$ , learning rate in $\\{ 0 . 0 0 1 , 0 . 0 0 0 1 , 0 . 0 0 0 0 1 \\}$ , L2-regularization rate in $\\{ 0 , 0 . 0 0 1 , 0 . 0 0 0 1 \\}$ , and number of hidden layer units in $\\{ 1 2 0 , 1 2 8 \\}$ . We use pre-trained 300-dimensional vectors from word2vec (Mikolov et al., 2013a) in all models except for Transformers. The models with trainable trigonometric position embedding produce nearly identical results compared to the non-trainable version, therefore we report the result of fixed position embeddings as per Vaswani et al. (2017). We adopt narrow convolution and max pooling in CNN, with number of filters in $\\{ 6 4 , 1 2 8 \\}$ , and size of filters in $\\{ 3 , 4 , 5 \\}$ . In all Transformer models, we only use the encoder layer to extract feature information, where the layer is 1, dimension of word and inner hidden are 256 and 512 respectively, and head number is 8. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results. The results are shown in Tab. 2. Our complex-order embeddings outperform all other variations at all times. This gain in effectiveness comes at a negligible (or non-existent) cost in efficiency (it varies per NN architecture – see Fig. 2). CNNs are the best performing NN as expected following Bai et al. (2018). Tranformer NNs benefit the most from our complex-order embeddings, most likely because they are our weakest baseline. To contextualise these results, Tab. 3 shows classification accuracy of five typical approaches on the same datasets (as reported in the original papers). Our complex-order embeddings outperform all methods, except for the CR dataset, where InferSent is marginally better. Overall, our approach is on a par with the SOTA in embeddings. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We perform an ablation test (Tab. 4) on Transformer because it is the most common NN to be used with position embeddings. The two period-sharing schemas (dimension-sharing and word-sharing) ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/3e281f61b64e81bd85bc32ab1d241a5dd6a5878639505a4ffd1356dda400c6a3.jpg",
+        "image_caption": [
+            "Figure 2: Computation time (seconds) per epoch in Tensorflow on TITAN X GPU. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            323,
+            101,
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b161fc29c882b041c0e94712ee052872c4b3dde58d2170a08f1847105500aa0e.jpg",
+        "table_caption": [
+            "Table 4: Ablation test for Transformer, showing the effect of (i) the definition of embedding layer $( f _ { d } ( j , \\mathsf { p o s } ) )$ , and (ii) whether the real-part and imaginary transition share the weights, i.e., $\\Re ( W ^ { Q / K / V } ) = \\Im ( W ^ { Q / K / V } )$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"2\">Setting</td><td rowspan=\"2\">Params</td><td rowspan=\"2\">Accuracy</td><td rowspan=\"2\">△</td></tr><tr><td></td><td>fa(j,pos)</td><td> share in WQ/K/V</td></tr><tr><td>Transformer-complex-order</td><td>rjde（wjdpos）</td><td>×</td><td>8.33M</td><td>0.813</td><td>=</td></tr><tr><td>adding initial phases</td><td>rj,dei(wj,dpos+0j,d）</td><td>×</td><td>11.89M</td><td>0.785</td><td>-0.028</td></tr><tr><td>dimension-sharing period schema</td><td>Tj,dewj,.pos</td><td>×</td><td>5.82M</td><td>0.797</td><td>-0.016</td></tr><tr><td>word-sharing period schema</td><td>Tj,deiw.,pos</td><td>×</td><td>5.81M</td><td>0.805</td><td>-0.008</td></tr><tr><td>dimension-sharing amplitude schema</td><td>rj,eiwj,.pos</td><td>×</td><td>5.82M</td><td>0.798</td><td>-0.015</td></tr><tr><td>word-sharing amplitude schema</td><td>r.deiw..apos</td><td>×</td><td>5.81M</td><td>0.804</td><td>-0.009</td></tr><tr><td>w/t encoding positions (complex-vanilla)</td><td>Tjdewjd</td><td>×</td><td>9.38M</td><td>0.764</td><td>-0.049</td></tr><tr><td>dimension-sharing period schema</td><td>Tj,dewj,.pos</td><td>√</td><td>4.77M</td><td>0.794</td><td>-0.019</td></tr><tr><td>word-sharing period schema</td><td>rj,dew.,apos</td><td>√</td><td>4.76M</td><td>0.797</td><td>-0.016</td></tr><tr><td>dimension-sharing amplitude schema</td><td>rj,eiwj,.pos</td><td>√</td><td>4.77M</td><td>0.792</td><td>-0.021</td></tr><tr><td>word-sharing amplitude schema</td><td>r.,deiw.,apos</td><td>√</td><td>4.76M</td><td>0.801</td><td>-0.012</td></tr><tr><td>w/t encoding positions (complex-vanilla)</td><td>rjdeiwjd</td><td>√</td><td>8.33M</td><td>0.743</td><td>-0.07</td></tr><tr><td>vanilla Transformer (Vaswani et al., 2017)</td><td>WEj,d+PEd</td><td>-</td><td>4.1M</td><td>0.761</td><td>-0.052</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "slightly drop performance, because fewer parameters limit the representative power. Adding initial phases also hurts performance, although we observed that the loss could decrease faster in early epochs compared to the setting without offset. The negative effect of initial phases may be due to periodicity, and $\\omega$ cannot be directly regularized with L2-norm penalties. The sharing schemes slightly decrease the performance with less parameters. More details of the learned periods/frequencies (e.g. the distributions of periods/frequencies and case studies) are shown in App. D. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that the word-sharing schema outperform the Vanilla Transformer, (both have a comparable number of parameters). If we choose $\\Re ( \\mathrm { \\dot { W } } ^ { Q / K / V } ) = \\Im ( W ^ { Q / K / V } )$ , the additional parameters in the embedding layers will affect much less the whole parameter scale in the multiple-layer Transformer, since a embedding layer is only used in the first layer instead of the following Transformer layers. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.2 MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            767,
+            388,
+            781
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Experimental Setup. We use the standard WMT 2016 English-German dataset (Sennrich et al., 2016), whose training set consists of 29,000 sentence pairs. We use four baselines: basic Attentional encoder-decoder (AED) (Bahdanau et al., 2014); AED with Byte-pair encoding (BPE) subword segmentation for open-vocabulary translation (Sennrich et al., 2016); AED with extra linguistic features (morphological, part-of-speech, and syntactic dependency labels) (Sennrich & Haddow, 2016); and a 6-layer Transformer. Our approach (Transformer Complex-order) uses a batch size of 64, a head of 8, 6 layers, the rate of dropout is 0.1, and the dimension of the word embedding is 512. The embedding layer does not use initial phases, i.e., following $f ( j , p o s ) = r _ { j } e ^ { i ( \\omega _ { j } p o s ) }$ . We evaluate MT performance with the Bilingual Evaluation Understudy (BLEU) measure. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0eb4f7bc17fac9dafe8f29328509ea3a49988b0cc118ca248f9f536791ad15c7.jpg",
+        "table_caption": [
+            "Table 6: Language modeling results. $\\star$ marks scores reported from other papers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>BLEU</td></tr><tr><td>AED (Bahdanau et al., 2014) *</td><td>26.8</td></tr><tr><td rowspan=\"4\">AED+Linguistic (Sennrich &amp; Haddow,2016) ★ AED+BPE(Sennrich et al.,2016) * Transformer (Ma et al.,2019) *</td><td>28.4</td></tr><tr><td>34.2</td></tr><tr><td></td></tr><tr><td>34.5</td></tr><tr><td>Transformer complex vanilla Transformer Complex-order</td><td>34.7 35.8</td></tr></table>",
+        "bbox": [
+            178,
+            147,
+            526,
+            252
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f123726c10f6426226fd420db516f924007181387d1d51653b5c3c6bedc2ef2a.jpg",
+        "table_caption": [
+            "Table 5: Machine translation results. $\\star$ marks scores reported from other papers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method BPC</td></tr><tr><td>BN-LSTM(Cooijmans et al., 2016) * 1.36 LN HM-LSTM(Chung et al., 2016) * 1.29</td></tr><tr><td>RHN (Zilly et al., 2017) * 1.27 Large mLSTM(Krause et al.,2016) * 1.27</td></tr><tr><td>Transformer XL 6L (Dai et al., 2019) 1.29 Transformercomplex vanilla 1.30 Transformer XL Complex-order 6L 1.26</td></tr></table>",
+        "bbox": [
+            534,
+            141,
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+            257
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results Tab. 5 shows the MT results. Our approach outperforms all baselines. Two things are worth noting: (1) Both the vanilla Transformer and our Transformer Complex-order outperform the three Attentional encoder-decoder baselines which are based on an LSTM encoder and decoder, even when AED uses additional features. (2) Our Transformer Complex-Order outperforms the Vanilla Transformer and complex-vanilla Transformer by 1.3 and 1.1 in absolute BLEU score respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.3 LANGUAGE MODELING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            358,
+            379,
+            372
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Experimental Setup. We use the text8 (Mahoney, 2011) dataset, consisting of English Wikipedia articles. The text is lowercased from a to z, and space. The dataset contains 100M characters (90M for training, 5M for dev, and 5M for testing, as per Mikolov et al. (2012)). We use as baselines BN-LSTM, LN HM-LSTM RHN and Large mLSTM, which are typical recurrent NNs for language modeling in this dataset. We evaluate performance with the Bits Per Character (BPC) measure, (the lower, the better). We run the coder in Dai et al. (2019) with 6 layers for Transformer XL 6L; our model, named Transformer XL complex-order, directly replaces the word embedding with our proposed embedding under the same setting. We choose 6 layers due to limitations in computing resources. For Transformer XL Complex-order, all other parameter settings are as for Transformer XL (Dai et al., 2019). Our complex-order model does not use initial phases. ",
+        "bbox": [
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+            383,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results. We see in Tab. 6 that our method outperforms all baselines. The first four baselines (BNLSTM, LN HM-LSTM, RHN and Large mLSTM) are based on recurrent NNs and rely on different regulation methods to become applicable in multiple-layer recurrent architectures. Transformerbased architectures can easily be stacked with multiple layers due to their advantages in parallelization, however the vanilla Transformer does not outperform the multiple-layer recurrent baselines, most likely due to its limitation of 6 layers. Our Transformer XL Complex-order outperforms its vanilla counterpart under the 6-layer setting (and also strong recurrent network baselines), demonstrating that our embedding also generalizes well in tasks with long-term dependency. With limited resources, slightly increasing the parameters in the feature layer like our proposed embedding could be more beneficial than stacking more layers with linearly increasing parameters. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            690,
+            344,
+            707
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Complex-valued NNs are not new (Georgiou & Koutsougeras, 1992; Kim & Adalı, 2003; Hirose, 2003). Complex-valued weights have been used in NNs, motivated by biology (Reichert & Serre, 2013), and also as signal processing in speech recognition (Shi et al., 2006). More recently, Arjovsky et al. (2016) shifted RNNs into the complex domain and Wolter & Yao (2018) proposed a novel complex gated recurrent cell. Trabelsi et al. (2017) also developped a complex-valued NN for computer vision and audio processing. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Complex numbers have also been applied to text processing like (Van Rijsbergen, 2004; Melucci, 2015; Blacoe et al., 2013). Trouillon et al. (2016) adopt complex embedding for entities in Knowledge Graph Completion to represent antisymmetric relations with Hermitian dot product. Li et al. (2019); Wang et al. (2019) extend word embeddings to complex-valued fashion in quantum probability driven NNs, seeing the overview in Wang et al. (2019). However, the physical meaning of both the complex-valued entity and word embeddings is unknown, since a complex number was considered as two real numbers in a black-box learning paradigm. Our work first links the phase in complex numbers to word position to define concrete physical meaning in document representations. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We extended word vectors to word functions with a variable i.e. position, to model the smooth shift among sequential word positions and therefore implicitly capture relative distances between words. These functions are well-defined to model the relative distances, therefore we derive a general solution in complex-valued fashion. Interestingly, the position embedding in Vaswani et al. (2017) can be considered a simplified version of our approach. We extend CNN, RNN and Transformer NNs to complex-valued versions to incorporate our complex embedding. Experiments on text classification, machine translation and language modeling show that our embeddings are more effective than vanilla position embeddings (Vaswani et al., 2017). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Massimo Melucci and Emanuele Di Buccio for their helpful comments, Xindian Ma for his detailed experimental suggestions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work is supported by the Quantum Access and Retrieval Theory (QUARTZ) project, which has received funding from the European Union‘s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 721321. Peng Zhang and Donghao Zhang are supported in part by Natural Science Foundation of China (grant No. 61772363, U1636203) ",
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+        "text": "A LINKING TO THE POSITION EMBEDDINGS IN (VASWANI ET AL., 2017) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Vaswani et al. (2017) proposed a new initialization for position embedding, resulting in comparable performance with previous one (Gehring et al., 2017) even without fine-tuning. The position embedding is empirically selected as ",
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+    {
+        "type": "equation",
+        "img_path": "images/18234793b3ffc63153ac63f3b07e3f300c4f122f0e2692fe53c4e74e604a276b.jpg",
+        "text": "$$\n\\begin{array} { r } { P E _ { 2 k } ( \\cdot , p o s ) = \\sin ( p o s / 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } ) } \\\\ { P E _ { 2 k + 1 } ( \\cdot , p o s ) = \\cos ( p o s / 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } ) } \\end{array}\n$$",
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+        "type": "text",
+        "text": "Where pos is the position index, $2 k$ and $2 k + 1$ is the dimension index and $d _ { m o d e l }$ is the dimension size of embedding. The reason for choosing this position embedding was not well-explained and its general extension is unknown, leading to some difficulties to improve it. ",
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+        "page_idx": 11
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+    {
+        "type": "text",
+        "text": "We claim that the proposed position embedding in (Vaswani et al., 2017) is a degraded version of one of our specific complex word embedding in word-sharing schema (i.e., $\\omega _ { j , d } = \\omega _ { \\cdot , d } )$ ), in which $p _ { j , k } =$ $2 \\pi \\times 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } }$ and the initial phases are set as zero. In our complex-valued position embedding, let $f _ { p e , k } ( \\cdot , p o s ) = e ^ { i \\times 1 0 0 0 ^ { 2 k / d _ { m o d e l } } } = \\cos ( 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } p o s ) + i \\sin ( 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } } p o s )$ . Note that there exists a bi-jection between $P E ( \\cdot , p o s )$ and $f _ { p e , k } ( \\cdot , p o s )$ : ",
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+            825,
+            353
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce1e7d7d5a8e8bd0be6504b3fdce0e2b05007d59a4113e47e09cdc41334ddb9b.jpg",
+        "text": "$$\n\\begin{array} { r } { P E _ { 2 k } ( \\cdot , p o s ) = \\Im ( f _ { p e , k } ( \\cdot , p o s ) ) , } \\\\ { P E _ { 2 k + 1 } ( \\cdot , p o s ) = \\Re ( f _ { p e , k } ( \\cdot , p o s ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            367,
+            617,
+            405
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathfrak { R }$ and $\\mathfrak { F }$ are the operations to take the real and imaginary part of a complex-valued number. Its inverse transformation is ",
+        "bbox": [
+            174,
+            406,
+            821,
+            434
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/73c83963e37e356ccac9f1beece911eb94ee770c88669cc2d4f3ecc5d7df34db.jpg",
+        "text": "$$\nf _ { p e , k } ( \\cdot , p o s ) = P E _ { 2 k + 1 } ( \\cdot , p o s ) + i P E _ { 2 k } ( \\cdot , p o s )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            440,
+            661,
+            458
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In our overall embedding, each dimension $f _ { k } ( j , p o s ) = f _ { w e , k } ( j ) \\odot f _ { p e , k } ( \\cdot , p o s )$ in our approaches, while it is $E _ { k } ( j , p o s ) \\stackrel { \\_ } { = } W E _ { k } ( j ) + P E _ { k } ( \\cdot , \\bar { p o s } )$ in (Gehring et al., 2017; Vaswani et al., 2017). Hence the position embedding in (Vaswani et al., 2017) is equivalent, albeit not identical, to our complex-valued position embedding with $p _ { j , k } = 2 \\pi \\times 1 0 0 0 0 ^ { 2 k / d _ { m o d e l } }$ . It is therefore a particular case of our complex-valued position embedding, the word-sharing schema in which all words share the same period at a certain dimension, i.e, $p _ { j , k } = p _ { \\cdot , k }$ is irrelevant to the choice of $j$ . ",
+        "bbox": [
+            173,
+            469,
+            826,
+            556
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B INTEGRATING COMPLEX-VALUED EMBEDDING TO GENERAL NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            577,
+            779,
+            608
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Neural networks are typically given real numbers as inputs and return real numbers as outputs. To accommodate complex numbers as in- and output, we devise a complex-valued version of various neural network layers i.e. complex-valued FastText with dense layer, CNN, and RNN. Unlike existing complex-valued neural networks (Trabelsi et al., 2017; Wolter & Yao, 2018), our feature layers are also converted into complex-valued layers. ",
+        "bbox": [
+            173,
+            623,
+            825,
+            694
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Complex-valued FastText FastTest (Joulin et al., 2016) is a simple and efficient neural network architecture using a dense layer over the sum of all word embeddings for general text classification. For a linear dense layer, i.e., $\\boldsymbol z = \\mathrm { d e n s e } ( \\boldsymbol x + i \\boldsymbol y )$ , where $\\pmb { x } + i \\pmb { y }$ and $_ z$ denote the complex-valued in- and output, respectively. Let $\\pmb { W } = \\pmb { A } + i \\pmb { B }$ and $b = c + i d$ be complex-valued linear weights and bias, respectively. Then, the complex-valued dense layer is given by: ",
+        "bbox": [
+            173,
+            708,
+            825,
+            780
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/53de9ce9bf8de49eeca107551940bd7611309416116e1ee648f96a3557cb0a0e.jpg",
+        "text": "$$\nz = \\sigma \\left( A x - B y + c \\right) + i \\sigma ( B x + A y + d )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            785,
+            653,
+            803
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is a real-valued activation function such as the sigmoid function. By rewriting (10) in matrix form (Trabelsi et al., 2017), we obtain: ",
+        "bbox": [
+            174,
+            808,
+            821,
+            837
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/01faabd0975590a2a37566da11434014bfbc29f94dde966000f6cc151a5e403b.jpg",
+        "text": "$$\n{ \\biggl [ } \\Re ( z ) { \\biggr ] } = { \\biggl [ } { \\sigma } ( A x - B y + c ) { \\biggr ] }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            840,
+            606,
+            876
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where, for $z = x + i y , \\Re ( z ) = x$ and $\\Im ( z ) = y$ . To save parameters and fairly compare with our real-valued baselines, the weights for real-part and imaginary-part input can be shared, i.e., $\\pmb { A } = \\pmb { B } , c = d$ . ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Complex-valued CNN For the complex-valued version of the convolution operation Trabelsi et al. (2017), we similarly define a complex-valued convolution with separate real and imaginary kernels $\\pmb { A }$ and $\\textbf {  { B } }$ , to compute convolutions on the real and imaginary parts of the input in Eq. 10. A complex-valued CNN network is constructed by stacking the operations based on a complex-valued convolution kernel, adding a complex dense layer in (10), and taking their norm as the final prediction in the last layer. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Complex-valued RNN The basic complex RNN formulation is: ",
+        "bbox": [
+            173,
+            202,
+            607,
+            217
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56b40debdb8bcfe755f8890fa267df347982b6635b593811567d82720a1931b1.jpg",
+        "text": "$$\nh _ { t } ^ { C } = f \\left( W ^ { h } h _ { t - 1 } + W ^ { z } z _ { t } + b \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            222,
+            612,
+            242
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\boldsymbol { z } } _ { t }$ and $\\boldsymbol { h } _ { t }$ represent the complex-valued input and complex-value hidden state vectors at time $t , b$ is a complex-valued bias, $W ^ { h }$ and $W ^ { z }$ are complex-valued weight transitions for hidden state and input state, and $f \\left( z \\right) = \\sigma \\left( \\Re \\left( z \\right) \\right) + i \\sigma \\left( \\Im \\left( z \\right) \\right)$ is the activation function. The multiplication $W ^ { h } h _ { t - 1 }$ and $W ^ { z } z _ { t }$ is computed as defined in (10) above. Similarly, the complex-valued gates are used in LSTM via operations as in (12). In the final layer, a $l 2$ -norm operation is adopted to obtain a real-valued loss for backpropagation. ",
+        "bbox": [
+            173,
+            247,
+            825,
+            332
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Complex-valued Transformer The main components in the Transformer are self-attention sublayers and position-wise feed-forward (FFN) sublayers. A self-attention sublayer employs $h$ attention heads and the concatenation of all heads is used as the output followed by a parameterized linear transformation. For a sequence embedded as complex-valued vector $i n p u t = \\{ \\bar { \\pmb { w } } _ { 1 } , \\pmb { w } _ { 2 } , . . . , \\pmb { w } _ { n } \\}$ , the output of each head is computed as a weighted sum of a linear transformation of the input sequence itself, namely ",
+        "bbox": [
+            173,
+            345,
+            825,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46a7e39cd995f7ff8468017cfdf5c3ec1990a3339a362390a8b9beb91692c702.jpg",
+        "text": "$$\no u t p u t _ { i } = \\sum _ { j } a _ { i , j } { \\pmb w } _ { j } { \\pmb W } ^ { V } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            428,
+            589,
+            462
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\pmb w } _ { j }$ is a complex-valued vector and $W ^ { V }$ is a complex linear transformation; therefore outputi is also complex. Hence, output is a sequence of complex-valued vectors with the same shape as input. The weight coefficient, $a _ { i , j }$ , which is defined as in real-valued domain, is calculated as the softmax of the product between complex-valued query vectors and key vectors: $a _ { i , j } = \\mathrm { s o f t m a x } \\frac { e _ { i , j } } { \\sum _ { i = 1 } ^ { n } e _ { i , j } }$ and ",
+        "bbox": [
+            173,
+            465,
+            825,
+            542
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ddd7d4e72917a127a89ebd1ccac157624d13881d396572841bf2b40b7f9bc09e.jpg",
+        "text": "$$\n{ e } _ { i , j } = \\sqrt { \\frac { \\Re ( z ) ^ { 2 } + \\Im ( z ) ^ { 2 } } { n } } , z = \\left( { w } _ { i } { W } ^ { Q } \\right) \\left( { w } _ { j } { W } ^ { K } \\right) ^ { \\dag } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            547,
+            676,
+            583
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { Z }$ is a complex number since ${ \\pmb w } _ { i }$ and ${ \\pmb w } _ { j }$ are complex-valued vectors and $W ^ { Q } , W ^ { K }$ are complexvalued transformation. To extend another variant of Transformer called Transformer XL, we keep its original relative position embedding and additionally replace its word embedding with our proposed embedding. ",
+        "bbox": [
+            173,
+            589,
+            825,
+            647
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Correspondingly, the FFN sublayer can easily be extended to a complex-valued version by replacing the real-valued layers with complex-valued ones. We use batch-normalization separately for the real and imaginary parts. ",
+        "bbox": [
+            174,
+            652,
+            823,
+            695
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C IMPLEMENTATION OF THE PROPOSED EMBEDDING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            715,
+            625,
+            732
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Words functions are implemented in neural networks by storing the function parameters $\\{ r , \\omega , \\theta \\}$ and then construct the values based on the arguments. Based on the definition, the implementation of the proposed embedding can easily be implemented with only modifying the embedding layer. We list the basic code to construct our general embedding as below: ",
+        "bbox": [
+            174,
+            747,
+            825,
+            803
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "i m p o r t t o r c h   \nimpor t math   \nc l a s s ComplexNN ( t o r c h . nn . M o d ul e ) : d e f i n i t ( s e l f , o p t ) : s u p e r ( ComplexNN , s e l f ) . i n i t ( ) s e l f . word emb $=$ t o r c h . nn . E m b e d d i n g ( o p t . n t o k e n , o p t . d m o d e l ) s e l f . f r e q u e n c y e m b $=$ t o r c h . nn . E m b e d d i n g ( o p t . n t o k e n , o p t . d m o d e l ) s e l f . i n i t i a l p h a s e e m b $=$ t o r c h . nn . E m b e d d i n g ( o p t . n t o k e n , o p t . d m o d e l ) ",
+        "bbox": [
+            174,
+            810,
+            852,
+            912
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/e18d67190d27ad89d72343f0f7e580d4499ce23a054a7022eb0448e518ff1704.jpg",
+        "image_caption": [
+            "Figure 3: The distribution of the $\\delta _ { j }$ . Higher values mean that the word representations are more sensitive to the word positions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            320,
+            123,
+            655,
+            310
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "d e f g e t e m b e d d i n g ( s e l f , x ) : ",
+        "bbox": [
+            210,
+            376,
+            446,
+            390
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "a m p l i t u d e $=$ s e l f . w o r d e m b ( x ) f r e q u e n c y $=$ s e l f . f r e q u e n c y e m b ( x ) s e l f . i n i t i a l p h a s e e m b . w e i g h t $=$ t o r c h . n n . P a r a m e t e r ( s e l f . i n i t i a l p h a s e e m $\\%$ ( $^ 2 \\ast$ m a t h . p i ) ) s e n t l e $\\mathbf { \\epsilon } _ { 1 } = \\mathbf { X }$ . s i z e $( - 1 )$ p o s s e q $=$ t o r c h . a r a n g e ( 1 , s e n t l e n + 1 , 1 . 0 , d e v i c e $=$ a m p l i t u d e . d e v i c e ) p o s s e q $=$ p o s s e q . u n s q u e e z e ( 0 ) . u n s q u e e z e $( - 1 )$ p o s s e q $=$ p o s s e q . r e p e a t ( [ x . s i z e ( 0 ) , 1 , a m p l i t u d e . s i z e ( − 1 ) ] ) d i m e n s i o n b a i s $=$ s e l f . i n i t i a l p h a s e e m b ( x ) e n c o u t p u t p h a s e $=$ t o r c h . mul ( p o s s e q , f r e q u e n c y ) $^ +$ d i m e n s i o n b a i s e n c o u t p u t r e a l $=$ a m p l i t u d e $^ *$ t o r c h . c o s ( e n c o u t p u t p h a s e ) e n c o u t p u t i m a g e $=$ a m p l i t u d e $^ *$ t o r c h . s i n ( e n c o u t p u t p h a s e ) # r e t u r n t o r c h . c a t ( [ e n c o u t p u t r e a l , e n c o u t p u t i m a g e ] , − 1 ) r e t u r n e n c o u t p u t r e a l , e n c o u t p u t i m a g e d e f f o r w a r d ( s e l f , $\\mathbf { X }$ ) : r e t u r n s e l f . g e t e m b e d d i n g ( x ) ",
+        "bbox": [
+            210,
+            400,
+            874,
+            683
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Note that both the frequency vectors $\\omega$ and initial-phase vectors $\\pmb { \\theta }$ can be shared between words or dimensions, to save parameters. The proposed embedding can be also used in real-valued neural networks if one directly concatenates the real-part numbers and imaginary-part numbers as a doublesize real-valued vector; therefore it could easily be extended in any existing networks without any complex-valued components. For instance, it could be a good extension for Transformer based pretrained models like (Devlin et al., 2018) by enriching the feature layer. ",
+        "bbox": [
+            173,
+            693,
+            825,
+            776
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D VISUALIZATION OF FREQUENCIES/PERIODS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            796,
+            571,
+            813
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "After training, we obtain the frequency vector $\\omega$ for each word. For each word, the mean value of the absolute frequency values, i.e., $\\begin{array} { r } { \\delta _ { j } = \\frac { 1 } { | D | } \\sum _ { d = 1 } ^ { D } | \\omega _ { j , d } | } \\end{array}$ is considered as a metric to test the positional sensitivities of the word, since a period value could be negative during training. The density of the $\\delta _ { j }$ is shown in Fig. 3. ",
+        "bbox": [
+            174,
+            827,
+            825,
+            888
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Words with the 50 greatest, and the 50 smallest, frequencies in the SST dataset are shown in Tab. 7. For the words with greatest frequencies, most of them are strong sentiment words like “worst” ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": ",“stupid” and “powerful”; a reason for this may be that such words appear in many positions in many documents during training, and thus they are more sensitive to the positions. Conversely, there are fewer words expressing strong sentiment among words with smaller frequencies, as shown in the second row. ",
+        "bbox": [
+            174,
+            103,
+            826,
+            159
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/2d3112ebe37ae3445f9917a47031f8a67ac78b06bafa42d5c4d1684d7c0f1d0d.jpg",
+        "table_caption": [
+            "Table 7: Words with greatest frequencies and frequencies periods (based on $\\delta _ { j }$ ) in SST (a sentiment classification task), all words are converted to lower-case. The strong sentiment words are bold based on manual labeling. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>words</td><td></td></tr><tr><td>greatest frequencies in descending order</td><td>worst solid stupid powerful mess wonderful remarkable suffers intoxicating thoughtful rare captures portrait gem frontal terrific unique wannabe witty lousy pointless contrived none worse refreshingly charming inventive amazing junk incoherent refreshing mediocre unfunny thinks enjoyed heartbreaking delightfully crisp brilliant heart</td></tr><tr><td>smallest frequencies in ascending order</td><td>spirit perfectly nowhere mistake engrossing fashioned excellent unexpected wonderfully means slowly proposal schemes roiling juliette titles fabric superstar ah wow choreographed tastelessness beg fabulous muccino jacobi legendary jae rate example code sensation counter deaths hall eun drug mctiernan storylines cellophane wild motion ups trick comedy entertained mission frightening witnesses snoots liners african groan satisfaction calm saturday estranged holm refuses inquisitive</td></tr></table>",
+        "bbox": [
+            173,
+            171,
+            913,
+            314
+        ],
+        "page_idx": 14
+    }
+]

parse/train/WigDnV-_Gq/WigDnV-_Gq.md

@@ -0,0 +1,303 @@
+# BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation
+
+Mingguo He Renmin University of China mingguo@ruc.edu.cn
+
+Zhewei Wei∗ Renmin University of China zhewei@ruc.edu.cn
+
+Zengfeng Huang Fudan University huangzf@fudan.edu.cn
+
+Hongteng $\mathbf { X } \mathbf { u } ^ { * }$ Renmin University of China hongtengxu@ruc.edu.cn
+
+# Abstract
+
+Many representative graph neural networks, e.g., GPR-GNN and ChebNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose BernNet, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order- $K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks. Code is available at https://github.com/ivam-he/BernNet.
+
+# 1 Introduction
+
+Graph neural networks (GNNs) have received extensive attention from researchers due to their   
+excellent performance on various graph learning tasks such as social analysis [24, 17, 29], drug   
+discovery [12, 25], traffic forecasting [18, 3, 6], recommendation system [38, 32] and computer   
+vision [39, 4]. Recent studies suggest that many popular GNNs operate as polynomial graph spectral   
+filters [7, 13, 5, 16, 2, 35]. Specifically, we denote an undirected graph with node set $V$ and edge   
+set $E$ as $G = ( V , E )$ , whose adjacency matrix is A. Given a signal $\mathbf { x } = [ x ] \in R ^ { n }$ on the graph, $n = | V |$ weightshe diag, where is the symmetric normalizednother equivalent polynomial normalized adjacency matrix   
+$\scriptstyle \sum _ { k = 0 } ^ { K } w _ { k } \mathbf { L } ^ { k } \mathbf { x }$ $w _ { k }$ $\mathbf { L } = \mathbf { I } - \mathbf { D } ^ { - 1 / 2 } \mathbf { A } \mathbf { D } ^ { - 1 / 2 }$ $G$ $\mathbf { D }$ $\scriptstyle \sum _ { k = 0 } ^ { K } c _ { k } \mathbf { P } ^ { k } \mathbf { x }$ $\mathbf { P } = \mathbf { D } ^ { - 1 / 2 } \mathbf { A } \mathbf { D } ^ { - 1 / 2 }$ $c _ { k }$
+
+![](images/190ec5391113ac944665522e1adf4c8cf8fc31bf8e9a0938f191d278b12a49dd.jpg)  
+Figure 1: An illustration of the proposed BernNet.
+
+We can broadly categorize the GNNs applying the above filtering operation into two classes, depending on whether they design the filter weights or learn them based on observed graphs. Some representative models in these two classes are shown below.
+
+• The GNNs driven by designing filters: GCN [13] uses a simplified first-order Chebyshev polynomial, which is proven to be a low-pass filter [1, 31, 34, 41]. APPNP [14] utilizes Personalized PageRank (PPR) to set the filter weights and achieves a low-pass filter as well [15, 41]. GNN-LF/HF [41] designs filter weights from the perspective of graph optimization functions, which can simulate high- and low-pass filters.
+
+• The GNNs driven by learning filters: ChebNet [7] approximates the filtering operation with Chebyshev polynomials, and learns a filter via trainable weights of the Chebyshev basis. GPR-GNN [5] learns a polynomial filter by directly performing gradient descent on the filter weights, which can derive high- or low-pass filters. ARMA [2] learns a rational filter via the family of Auto-Regressive Moving Average filters [21].
+
+Although the above GNNs achieve some encouraging results in various graph modeling tasks, they still suffer from two major drawbacks. Firstly, most existing methods focus on designing or learning simple filters (e.g., low- and/or high-pass filters), while real-world applications often require much more complex filters such as band-rejection and comb filters. To the best of our knowledge, none of the existing work supports designing arbitrary interpretable spectral filters. The GNNs driven by learning filters can learn arbitrary filters in theory, but they cannot intuitively show what filters they have learned. In other words, their interpretability is poor. For example, GPR-GNN [5] learns the filter weights $w _ { k }$ ’s but only proves a small subset of the learnt weight sequences corresponds to low- or high-pass filters. Secondly, the GNNs often design their filters empirically or learn the filter weights without any necessary constraints. As a result, their filter weights often have poor controllability. For example, GNN-LF/HF [41] designs its filters with a complex and non-intuitive polynomial with difficult-to-tune hyperparameters. The multi-layer GCN/SGC [13, 31] leads to “ill-posed” filters (i.e., those deriving negative spectral responses). Additionally, the filters learned by GPR-GNN [5] or ChebNet [7] have a chance to be ill-posed as well.
+
+To overcome the above issues, we propose a novel graph neural network called BernNet, which provides an effective algorithmic framework for designing and learning arbitrary graph spectral filters. As illustrated in Figure 1, for an arbitrary spectral filter $h : [ 0 , 2 ] \mapsto [ 0 , 1 ]$ over the spectrum of the symmetric normalized Laplacian $\mathbf { L }$ , our BernNet approximates $h$ by a $K$ -order Bernstein polynomial approximation, i.e., $\begin{array} { r } { \bar { h } ( \lambda ) = \sum _ { k = 0 } ^ { K } \theta _ { k } b _ { k } ^ { K } ( \lambda ) } \end{array}$ . The non-negative coefficients $\{ \theta _ { k } \} _ { k = 0 } ^ { K }$ of the Bernstein basis $\{ b _ { k } ^ { K } ( \lambda ) \} _ { k = 0 } ^ { K }$ work as the model parameter, which can be interpreted as $h ( 2 k / K )$ , $k = 0 , \ldots , K$ (i.e., the filter values uniformly sampled from [0, 2]). By designing or learning the $\theta _ { k }$ ’s, we can obtain various spectral filters, whose filtering operation can be formulated as $\begin{array} { r l } { ~ } & { { } \sum _ { k = 0 } ^ { K } \theta _ { k } \frac { 1 } { 2 ^ { K } } \binom { K } { k } ( 2 \mathbf { I } - \mathbf { L } ) ^ { K - k } \mathbf { L } ^ { k } \mathbf { x } } \end{array}$ , where $\mathbf { x }$ is the graph signal. We further demonstrate the rationality of our BernNet from the perspective of graph optimization — any valid polynomial filers, i.e., those polynomial functions mapping $[ 0 , 2 ]$ to $[ 0 , 1 ]$ , can always be expressed by our BernNet, and accordingly, the filters learned by our BernNet are always valid. Finally, we conduct experiments to demonstrate that 1) BernNet can learn arbitrary graph spectral filters (e.g., band-rejection, comb, low-band-pass, etc.), and 2) BernNet achieves superior performance on real-world datasets.
+
+# 2 BernNet
+
+# 2.1 Bernstein approximation of spectral filters
+
+Given an arbitrary filter function $h : [ 0 , 2 ] \mapsto [ 0 , 1 ]$ , let $\mathbf { L } = \mathbf { U } \mathbf { A } \mathbf { U } ^ { T }$ denote the eigendecomposition of the symmetric normalized Laplacian matrix $\mathbf { L }$ , where $\mathbf { U }$ is the matrix of eigenvectors and $\Lambda =$ $d i a g [ \lambda _ { 1 } , . . . , \lambda _ { n } ]$ is the diagonal matrix of eigenvalues. We use
+
+$$
+h ( { \bf L } ) { \bf x } = { \bf U } h ( { \boldsymbol \Lambda } ) { \bf U } ^ { T } { \bf x } = { \bf U } d i a g [ h ( \lambda _ { 1 } ) , . . . , h ( \lambda _ { n } ) ] { \bf U } ^ { T } { \bf x }
+$$
+
+to denote a spectral filter on graph signal $\mathbf { x }$ . The key of our work is approximate $h ( \mathbf { L } )$ (or, equivalently, $h ( \lambda ) )$ . For this purpose, we leverage the Bernstein basis and Bernstein polynomial approximation defined below.
+
+Definition 2.1 ( [10]). (Bernstein polynomial approximation) Given an arbitrary continuous function $f ( t )$ on $t \in [ 0 , 1 ]$ , the Bernstein polynomial approximation (of order $K$ ) for $f$ is defined as
+
+$$
+p _ { K } ( t ) : = \sum _ { k = 0 } ^ { K } \theta _ { k } \cdot b _ { k } ^ { K } ( t ) = \sum _ { k = 0 } ^ { K } f \left( \frac { k } { K } \right) \cdot \binom { K } { k } ( 1 - t ) ^ { K - k } t ^ { k } .
+$$
+
+Here, for $k = 0 , . . . , K$ $\begin{array} { r } { K , b _ { k } ^ { K } ( t ) = \binom { K } { k } ( 1 - t ) ^ { K - k } t ^ { k } } \end{array}$ is the $k$ -th Bernstein base, and $\textstyle \theta _ { k } = f ( { \frac { k } { K } } )$ is the function value at $k / K$ , which works as the coefficient of $b _ { k } ^ { K } ( t )$ .
+
+Lemma 2.1 ( [10]). Given an arbitrary continuous function $f ( t )$ on $t \in [ 0 , 1 ] ,$ , let $p _ { K } ( t )$ denote the Bernstein approximation of $f ( t )$ as defined in Equation (2). We have $p _ { K } ( t )  f ( t )$ as $K  \infty$ .
+
+For the filter function $h : [ 0 , 2 ] \mapsto [ 0 , 1 ]$ , we let $\begin{array} { r } { t = \frac { \lambda } { 2 } } \end{array}$ and $f ( t ) = h ( 2 t )$ , so that the Bernstein polynomial approximation becomes applicable, where $\theta _ { k } = f ( k / K ) = h ( 2 k / K )$ and $b _ { k } ^ { K } ( t ) =$ $\begin{array} { r } { b _ { k } ^ { K } ( \frac { \lambda } { 2 } ) \ = \ \binom { K } { k } ( 1 - \frac { \lambda } { 2 } ) ^ { K - k } ( \frac { \lambda } { 2 } ) ^ { k } } \end{array}$ for $k = 1 , . . . , K$ . Consequently, we can approximate $h ( \lambda )$ by $\begin{array} { r } { p _ { K } ( \lambda / 2 ) = \sum _ { k = 0 } ^ { K } \theta _ { k } { \binom { K } { k } } ( 1 - \frac { \lambda } { 2 } ) ^ { K - k } \left( \frac { \lambda } { 2 } \right) ^ { k } = \sum _ { k = 0 } ^ { K } \theta _ { k } \frac { 1 } { 2 ^ { K } } { \binom { K } { k } } ( 2 - \lambda ) ^ { K - k } \lambda ^ { k } . } \end{array}$ , and Lemma 2.1 ensures that $p _ { K } ( \lambda / 2 ) \to h ( \lambda )$ as $K  \infty$ .
+
+Replacing $\{ h ( \lambda _ { i } ) \} _ { i = 1 } ^ { n }$ with $\{ p _ { K } ( \lambda _ { i } / 2 ) \} _ { i = 1 } ^ { n }$ , we approximate the spectral filter $h ( \mathbf { L } )$ in Equation (1) as Udiag $[ p _ { K } ( \lambda _ { 1 } / 2 ) , . . . , p _ { K } ( \lambda _ { n } / 2 ) ] \mathbf { U } ^ { T }$ and derive the proposed BernNet. In particular, given a graph signal $\mathbf { x }$ , the convolutional operator of our BernNet is defined as follows:
+
+$$
+\mathbf { z } = \underbrace { \mathbf { U } d i a g [ p _ { K } ( \lambda _ { 1 } / 2 ) , . . . , p _ { K } ( \lambda _ { n } / 2 ) ] \mathbf { U } ^ { T } } _ { \mathrm { B e r n N e t } } \mathbf { U } ^ { T } \mathbf { x } = \sum _ { k = 0 } ^ { K } \theta _ { k } { \frac { 1 } { 2 ^ { K } } } { \binom { K } { k } } ( 2 \mathbf { I } - \mathbf { L } ) ^ { K - k } \mathbf { L } ^ { k } \mathbf { x }
+$$
+
+where each coefficient $\theta _ { k }$ can be either set to $h ( 2 k / K )$ to approximate a predetermined filter $h$ , or learnt from the graph structure and signal in an end-to-end fashion. As a natural extension of Lemma 2.1, our BernNet owns the following proposition.
+
+Proposition 2.1. For an arbitrary continuous filter function $h : [ 0 , 2 ]  [ 0 , 1 ] ,$ , by setting $\theta _ { k } =$ $h ( 2 \bar { k } / K ) , k = 0 , \dots , K$ , the $\mathbf { z }$ in Equation (3) satisfies ${ \bf z }  h ( { \bf L } ) { \bf x } $ as $K  \infty$ .
+
+Proof. According to the above derivation, we have $\begin{array} { r } { p _ { K } ( \lambda / 2 ) = \sum _ { k = 0 } ^ { K } \theta _ { k } \binom { K } { k } ( 1 - \frac { \lambda } { 2 } ) ^ { K - k } \left( \frac { \lambda } { 2 } \right) ^ { k } = } \end{array}$ $\begin{array} { r } { \sum _ { k = 0 } ^ { K } \theta _ { k } \frac { 1 } { 2 ^ { K } } \binom { K } { k } ( 2 - \lambda ) ^ { K - k } \lambda ^ { k } } \end{array}$ , and Lemma 2.1 ensures that $p _ { K } ( \lambda / 2 ) \to h ( \lambda )$ as $\theta _ { k } = h ( 2 k / K )$ and $K  \infty$ .
+
+Consequently, we have
+
+$\mathbf { z } = \mathbf { U } d i a g [ p _ { K } ( \lambda _ { 1 } / 2 ) , . . . , p _ { K } ( \lambda _ { n } / 2 ) ] \mathbf { U } ^ { T } \mathbf { x } \to \mathbf { U } d i a g [ h ( \lambda _ { 1 } ) , . . . , h ( \lambda _ { n } ) ] \mathbf { U } ^ { T } \mathbf { x } = h ( \mathbf { L } )$ as $\theta _ { k } = h ( 2 k / K )$ and $K  \infty$ .
+
+# 2.2 Realizing existing filters with BernNet.
+
+As shown in Proposition 2.1, our BernNet can approximate arbitrary continuous spectral filters with sufficient precision. Below we give some representative examples of how our BernNet exactly realizes existing filters that are commonly used in GNNs.
+
+Table 1: Realizing commonly used filters with BernNet.   
+
+<table><tr><td>Filter types</td><td>Filter h(λ)</td><td>0k for k=0,...,K</td><td>Bernstein approximation pk(）</td><td>BernNet</td></tr><tr><td>All-pass</td><td>1</td><td>0=1</td><td>1</td><td>I</td></tr><tr><td>Linear low-pass</td><td>1-&gt;/2</td><td>0=1-k/K</td><td>1-λ/2</td><td>1</td></tr><tr><td>Linear high-pass</td><td>入/2</td><td>0=k/K</td><td>入/2</td><td></td></tr><tr><td>Impulse low-pass</td><td>(入）</td><td>0=1and other 0k =0</td><td>(1-X/2)K</td><td></td></tr><tr><td>Impulse high-pass</td><td>8(入）</td><td>0K=1and other 0k =0</td><td>（入/2）K</td><td>美K</td></tr><tr><td>Impulse band-pass</td><td>8(入）</td><td>0K/2 =1and other 0k =0</td><td>（K2）(1-&gt;/2)K/2(/2）K/2</td><td>（K/2)（21-L）K/2LK/2</td></tr></table>
+
+• All-pass filter $h ( \lambda ) = 1$ . We set $\theta _ { k } \ = \ 1$ for $k = 0 , \ldots , K$ , and the approximation $p _ { K } ( \frac { \lambda } { 2 } ) = 1$ is exactly the same with $h ( \lambda )$ . Accordingly, our BernNet becomes an identity matrix, which realizes the all-pass filter perfectly.
+
+• Linear low-pass filter $h ( \lambda ) = 1 - \lambda / 2$ . We set $\theta _ { k } = 1 - k / K$ for $k = 0 , \ldots , K$ and obtain $p _ { K } ( \frac { \lambda } { 2 } ) = 1 - \lambda / 2$ . The BernNet becomes $\begin{array} { r } { \sum _ { k = 0 } ^ { K } \frac { ( K - k ) } { K } \frac { 1 } { 2 ^ { K } } \binom { K } { k } ( 2 \mathbf { I } - \mathbf { L } ) ^ { K - k } \mathbf { L } ^ { k } = \mathbf { I } - \frac { 1 } { 2 } \mathbf { L } } \end{array}$ which achieves the linear low-pass filter exactly. Note that $\begin{array} { r } { \mathbf { I } - \frac { 1 } { 2 } \mathbf { L } = \frac { 1 } { 2 } ( \mathbf { I } + \mathbf { P } ) } \end{array}$ is also the same as the graph convolutional network (GCN) before renormalization [13].
+
+• Linear high-pass filter $h ( \lambda ) = \lambda / 2$ . Similarly, we can set $\theta _ { k } = k / K$ for $k = 0 , \ldots , K$ to get a perfect approximation $\begin{array} { r } { p _ { K } ( \frac { \lambda } { 2 } ) = \frac { \lambda } { 2 } } \end{array}$ , and the BernNet becomes $\scriptstyle { \frac { 1 } { 2 } } \mathbf { L }$ .
+
+Note that even for those non-continuous spectral filters, e.g., the impulse low/high/band-pass filters, our BernNet can also provide good approximations (with sufficient large $K$ ).
+
+• Impulse low-pass filter $h ( \lambda ) = \delta _ { 0 } ( \lambda )$ .† We set $\theta _ { 0 } = 1$ and $\theta _ { k } \ = \ 0$ for $k \neq 0$ , and $\begin{array} { r } { p _ { K } \bar { ( \frac { \lambda } { 2 } ) } = ( 1 - \frac { \bar { \lambda } } { 2 } ) ^ { K } } \end{array}$ . Accordingly, the BernNet becomes $\begin{array} { r } { \frac { 1 } { 2 ^ { K } } ( 2 \mathbf { I } - \mathbf { L } ) ^ { K } } \end{array}$ , deriving an $K$ -layer linear low-pass filter.
+
+• Impulse high-pass filter $h ( \lambda ) = \delta _ { 2 } ( \lambda )$ . We set $\theta _ { K } = 1$ and $\theta _ { k } = 0$ for $k \neq K$ , and $\begin{array} { r } { p _ { K } ( \frac { \lambda } { 2 } ) = ( \frac { \lambda } { 2 } ) ^ { K } } \end{array}$ . The BernNet becomes $\scriptstyle { \frac { 1 } { 2 ^ { K } } } \mathbf { L } ^ { K }$ , i.e., an $K$ -layer linear high-pass filter.
+
+• Impulse band-pass filter $h ( \lambda ) = \delta _ { 1 } ( \lambda )$ . Similarly, we set $\theta _ { K / 2 } = 1$ and $\theta _ { k } = 0$ for $k \neq$ $K / 2$ , and $\begin{array} { r } { p _ { K } ( \frac { \lambda } { 2 } ) = \binom { K } { K / 2 } ( 1 - \lambda / 2 ) ^ { K / 2 } ( \lambda / 2 ) ^ { K / 2 } } \end{array}$ . The BernNet becomes $\frac { 1 } { 2 ^ { K } } \binom { K } { K / 2 } ( 2 \mathbf { I } -$ $\mathbf { L } ) ^ { K / 2 } \mathbf { L } ^ { K / 2 }$ , which can be explained as stacking a $K / 2$ -layer linear low-pass filter and a $K / 2$ -layer linear high-pass filter. Obviously, $K$ should be an even number in this case.
+
+Table 1 summarizes the design of the BernNet for the filters above. We can find that an appealing advantage of our BernNet is that its coefficients are highly correlated with the spectral property of the target filter. In particular, we can determine to pass or reject the spectral signal with $\textstyle \lambda \approx { \frac { { \bar { 2 } } k } { K } }$ by using a large or small property provid $\theta _ { k }$ because each Bernstein base useful guidance when designi $b _ { k } ^ { K } ( \lambda )$ corresponds to a “bump” located at rs, which enhances the interpretabili $\textstyle { \frac { 2 k } { K } }$ . This of our BernNet.
+
+# 2.3 Learning complex filters with BernNet
+
+Besides designing the above typical filters, our BernNet can express more complex filters, such as band-pass, band-rejection, comb, low-band-pass filters, etc. Moreover, given the graph signals before and after applying such filters (i.e., the x’s and the corresponding $\mathbf { z }$ ’s), our BernNet can learn their approximations in an end-to-end manner. Specifically, given the pairs $\{ \mathbf { x } , \mathbf { z } \}$ , we learn the coefficients $\{ \theta _ { k } \} _ { k = 0 } ^ { K }$ of the BernNet by gradient descent. More implementation details can be found at the experimental section below. Figure 2 illustrates the four complex filters and the approximations we learned (The low-band pass filter is $h ( \lambda ) = I _ { [ 0 , 0 . 5 ] } ( \lambda ) + \hat { \exp { ( - 1 0 0 ( \lambda - 0 . 5 ) ^ { 2 } ) } } \hat { I } _ { ( 0 . 5 , 1 ) } ( \lambda ) +$ $\exp { ( - 5 0 ( \lambda - 1 . 5 ) ^ { 2 } ) } I _ { [ 1 , 2 ] } ( \lambda )$ , where $I _ { \Omega } ( \lambda ) = 1$ when $\lambda \in \Omega$ , otherwise $I _ { \Omega } ( \lambda ) = 0 )$ . In general, our BernNet can learn a smoothed approximation of these complex filters, and the approximation precision improves with the increase of the order $K$ . Note that although the BernNet cannot pinpoint the exact peaks of the comb filter or drop to 0 for the valleys of comb or low-band-pass filters due to the limitation of $K$ , it still significantly outperforms other GNNs for learning such complex filters.
+
+![](images/79be059be4cf245f9f112beb0e01c5cd45d48d8fea28b73516bb37b15d002002.jpg)  
+Figure 2: Illustrations of four complex filters and their approximations learnt by BernNet.
+
+# 3 BernNet in the Lens of Graph Optimization
+
+In this section, we motivate BernNet from the perspective of graph optimization. In particular, we show that any polynomial filter that attempts to approximate a valid filter has to take the form of BernNet.
+
+# 3.1 A generalized graph optimization problem
+
+Given a $n$ -dimensional graph signal $\mathbf { x }$ , we consider a generalized graph optimization problem
+
+$$
+\operatorname* { m i n } _ { \mathbf { z } } f ( \mathbf { z } ) = ( 1 - \alpha ) \mathbf { z } ^ { T } \boldsymbol { \gamma } ( \mathbf { L } ) \mathbf { z } + \alpha \| \mathbf { z } - \mathbf { x } \| _ { 2 } ^ { 2 }
+$$
+
+where $\alpha \in [ 0 , 1 )$ is a trade-off parameter, $\mathbf { z } \in R ^ { n }$ denotes the propagated representation of the input graph signal $\mathbf { x }$ , and $\gamma ( \mathbf { L } )$ denotes an energy function of $\mathbf { L }$ , determining the rate of propagation [28]. Generally, $\gamma ( \cdot )$ operates on the spectral of $\mathbf { L }$ , and we have $\gamma ( { \bf L } ) = { \bf U } { \bar { d } } i a g [ \gamma ( \lambda _ { 1 } ) , \dot { \bf \omega } . \dot { \bf \omega } , \dot { \gamma ( \lambda _ { n } ) } ] { \bf U } ^ { T }$ .
+
+We can model the polynomial filtering operation of existing GNNs with the optimal solution of Equation (4). For example, if we set $\begin{array} { r } { \gamma ( \mathbf { L } ) = \mathbf { L } } \end{array}$ , then the optimization function (4) becomes $f ( \mathbf { \bar { z } } ) = ( 1 - \alpha ) \mathbf { z } ^ { T } \mathbf { L } \mathbf { z } + \acute { \alpha } \| \mathbf { z } - \mathbf { x } \| _ { 2 } ^ { 2 }$ , a well-known convex graph optimization function proposed by Zhou et al. [40]. $f ( \mathbf { z } )$ takes the minimum when the derivativ e ∂f(z)z = 2(1 − α)Lz + 2α (z − x) = 0, which solves to
+
+$$
+\mathbf { z } ^ { * } = \alpha \left( \mathbf { I } - ( 1 - \alpha ) ( \mathbf { I } - \mathbf { L } ) \right) ^ { - 1 } \mathbf { x } = \sum _ { k = 0 } ^ { \infty } \alpha ( 1 - \alpha ) ^ { k } \left( \mathbf { I } - \mathbf { L } \right) ^ { k } \mathbf { x } = \sum _ { k = 0 } ^ { \infty } \alpha ( 1 - \alpha ) ^ { k } \mathbf { P } ^ { k } \mathbf { x } .
+$$
+
+By taking a suffix sum $\begin{array} { r } { \sum _ { k = 0 } ^ { K } \alpha ( 1 - \alpha ) ^ { k } \mathbf { P } ^ { k } \mathbf { x } } \end{array}$ , we obtain the polynomial filtering operation for APPNP [14]. Zhu et al. [41] further show that GCN [13], DAGNN [19], and JKNet [36] can be interpreted by the optimization function (4) with $\gamma ( \mathbf { L } ) = \mathbf { L }$ .
+
+The generalized form of Equation (4) allows us to simulate more complex polynomial filtering operation. For example, let $\alpha = 0 . 5$ and $\gamma ( \mathbf { L } ) = e ^ { t \mathbf { L } } - \mathbf { I }$ , a heat kernel with $t$ as the temperature parameter. Then $f ( \mathbf { z } )$ takes the minimum when the derivative $\begin{array} { r } { \frac { \partial f ( \mathbf { z } ) } { \partial \mathbf { z } } = \left( e ^ { t \mathbf { L } } - \mathbf { I } \right) \mathbf { z } + \mathbf { z } - \mathbf { x } = \mathbf { 0 } } \end{array}$ , which solves to
+
+$$
+\mathbf { z } ^ { * } = e ^ { - t \mathbf { L } } \mathbf { x } = e ^ { - t ( \mathbf { I } - \mathbf { P } ) } \mathbf { x } = \sum _ { k = 0 } ^ { \infty } e ^ { - t } \frac { t ^ { k } } { k ! } \mathbf { P } ^ { k } \mathbf { x } .
+$$
+
+By taking a suffix sum $\textstyle \sum _ { k = 0 } ^ { K } e ^ { - t } { \frac { t ^ { k } } { k ! } } \mathbf { P } ^ { k } \mathbf { x }$ , we obtain the polynomial filtering operation for the heat kernal based GNN such as GDC [15] and GraphHeat [34].
+
+# 3.2 Non-negative constraint on polynomial filters
+
+A natural question is that, does an arbitrary energy function $\gamma ( \mathbf { L } )$ correspond to a valid or ill-posed spectral filter? Conversely, does any polynomial filtering operation $\scriptstyle \sum _ { k = 0 } ^ { K } w _ { k } \mathbf { L } ^ { k } \mathbf { x }$ correspond to the optimal solution of the optimization function (4) for some energy function $\gamma ( \mathbf { L } )$ ?
+
+As it turns out, there is a “minimum requirement” for the energy function $\gamma ( \mathbf { L } )$ ; $\gamma ( \mathbf { L } )$ has to be positive semidefinite. In particular, if $\gamma ( \mathbf { L } )$ is not positive semidefinite, then the optimization function $f ( \mathbf { z } )$ is not convex, and the solution to $\begin{array} { r } { \frac { \partial f ( \mathbf { z } ) } { \partial \mathbf { z } } = 0 } \end{array}$ may corresponds to a saddle point. Furthermore, without the positive semidefinite constraint on $\gamma ( \mathbf { L } )$ , $f ( \mathbf { z } )$ may goes to $- \infty$ as we set $\mathbf { z }$ to be a multiple of the eigenvector corresponding to the negative eigenvalue.
+
+Non-negative polynomial filters. Given a positive semidefinite energy function $\gamma ( \mathbf { L } )$ , we now consider how the corresponding polynomial filtering operatio n PKk=0 wkLkx should look like. Recall that we assume $\boldsymbol \gamma ( \mathbf { L } ) = \mathbf { U } d i a g [ \gamma ( \lambda _ { 1 } ) , . . . , \gamma ( \lambda _ { n } ) ] \mathbf { U } ^ { T }$ . By the positive semidefinite constraint, we have $\gamma ( \lambda ) \geq 0$ for $\lambda \in [ 0 , 2 ]$ . Since the objective function $f ( \mathbf { z } )$ is convex, it takes the minimum when $\begin{array} { r } { \frac { \partial f ( \mathbf { z } ) } { \partial \mathbf { z } } = 2 ( 1 - \alpha ) \gamma ( \mathbf { L } ) \mathbf { z } + 2 \alpha \left( \mathbf { z } - \mathbf { x } \right) = \mathbf { 0 } } \end{array}$ . Accordingly, the optimum $z ^ { * }$ can be derived as
+
+$$
+\alpha \left( \alpha \mathbf { I } + ( 1 - \alpha ) \gamma ( \mathbf { L } ) \right) ^ { - 1 } \mathbf { x } = \mathbf { U } d i a g \left[ \frac { \alpha } { \alpha + ( 1 - \alpha ) \gamma ( \lambda _ { 1 } ) } , . . . , \frac { \alpha } { \alpha + ( 1 - \alpha ) \gamma ( \lambda _ { n } ) } \right] \mathbf { U } ^ { T } \mathbf { x } .
+$$
+
+Let h(λ) = αα+(1−α)γ(λ) denote the exact spectral filter, and $\begin{array} { r } { g ( \lambda ) = \sum _ { k = 0 } ^ { K } w _ { k } \lambda ^ { k } } \end{array}$ denote a polynomial approximation of $h ( \lambda )$ (e.g. the suffix sum of $h ( \lambda )$ ’s taylor expansion). Since $\gamma ( \lambda ) \geq 0$ when the polynomial filter $\lambda \in [ 0 , 2 ]$ , we have $\begin{array} { r } { 0 \leq \dot { h } ( \lambda ) \leq \frac { \alpha } { \alpha + ( 1 - \alpha ) \cdot 0 } = 1 } \end{array}$ $\begin{array} { r } { g ( \lambda ) = \sum _ { k = 0 } ^ { K } w _ { k } \lambda ^ { k } } \end{array}$ 0  also satisfies for $\lambda \in [ 0 , 2 ]$ $0 \leq g ( \lambda ) \leq 1$ . Consequently, it is natural to assume .
+
+Constraint 3.1. Assuming the energy function $\gamma ( \mathbf { L } )$ is positive semidefinite, a polynomial filter $\begin{array} { r } { g ( \lambda ) = \sum _ { k = 0 } ^ { K } w _ { k } \lambda ^ { k } } \end{array}$ approximating the optimal solution to Equation (4) has to satisfy
+
+$$
+0 \leq g ( \lambda ) = \sum _ { k = 0 } ^ { K } w _ { k } \lambda ^ { k } \leq 1 , \forall \lambda \in [ 0 , 2 ] .
+$$
+
+While Constraint 3.1 seems to be simple and intuitive, some of the existing GNN may not satisfies this constraint. For example, GCN uses $\mathbf { \bar { z } } = \mathbf { P x } = \left( \mathbf { I } - \mathbf { L } \right) \mathbf { x }$ , which corresponds to a polynomial filter $g ( \lambda ) = 1 - \lambda$ that takes negative value when $\lambda > 1$ , violating Constraint 3.1. As shown in [31], the renormalization trick $\tilde { \mathbf { P } } = \left( \mathbf { I } + \mathbf { D } \right) ^ { - 1 / 2 } \left( \mathbf { I } + \mathbf { A } \right) \left( \mathbf { I } + \mathbf { D } \right) ^ { - 1 / 2 }$ shrinks the spectral and thus reliefs the problem. However, $g ( \lambda )$ may still take negative value as the maximum eigenvalue of $\tilde { \mathbf { L } } = \mathbf { I } - \tilde { \mathbf { P } }$ is still larger than 1.
+
+# 3.3 Non-negative polynomials and Bernstein basis
+
+Constraint 3.1 motivates us to design polynomial filters g(λ) = PKk=0 wkλk such that 0 ≤ g(λ) ≤ 1 when . The part is trivial, as we can always rescale each by a factor of $\scriptstyle \sum _ { k = 0 } ^ { K } | w _ { k } | 2 ^ { k }$ . The $g ( \lambda ) \geq 0$ part, however, requires more elaboration. Note that we can not simply set $w _ { k } \geq 0$ for each $k = 0 \ldots , K$ , since it is shown in [5] that such polynomials only correspond to low-pass filters.
+
+As it turns out, the Bernstein basis has the following nice property: a polynomial that is non-negative on a certain interval can always be expressed as a non-negative linear combination of Bernstein basis. Specifically, we have the following lemma.
+
+Lemma 3.1 ([23]). Assume a polynomial $\textstyle p ( x ) = \sum _ { k = 0 } ^ { K } \theta _ { k } x ^ { k }$ satisfies $p ( x ) \geq 0$ for $x \in [ 0 , 1 ]$ . Then there exists a sequence of non-negative coefficients $\theta _ { k }$ , ${ \bf \ddot { \boldsymbol { k } } } = 0 , \dots , K$ , such that
+
+$$
+p ( x ) = \sum _ { k = 0 } ^ { K } \theta _ { k } b _ { k } ^ { K } ( x ) = \sum _ { k = 0 } ^ { K } \theta _ { k } \binom { K } { k } ( 1 - x ) ^ { K - k } x ^ { k }
+$$
+
+Lemma 3.1 suggests that to approximate a valid filter, the polynomial filter $g ( \lambda )$ has to be a nonnegative linear combination of Bernstein basis. Specifically, by setting $x = \lambda / 2$ , the filter $g ( \lambda )$ that satisfies $g ( \lambda ) \geq 0$ for $\lambda \in [ 0 , 2 ]$ can be expressed as
+
+$$
+g ( \lambda ) : = p \left( \frac { \lambda } { 2 } \right) = \sum _ { k = 0 } ^ { K } \theta _ { k } \frac { 1 } { 2 ^ { K } } \binom { K } { k } ( 2 - \lambda ) ^ { K - k } \lambda ^ { k } .
+$$
+
+Consequently, any valid polynomial filter that approximate the optimal solution of (4) with positive semidefinite energy function $\gamma ( \mathbf { L } )$ has to take the following form: $\begin{array} { r } { \mathbf { z } = \sum _ { k = 0 } ^ { K } \theta _ { k } \frac { 1 } { 2 ^ { K } } \binom { \bar { K } } { k } ( 2 \mathbf { I } - } \end{array}$ $\mathbf { L } ) ^ { K - k } \mathbf { L } ^ { k } \mathbf { x }$ . This observation motivates our BernNet from the perspective of graph optimization — any valid polynomial filers, i.e., the $g : [ 0 , 2 ] \mapsto [ 0 , 1 ]$ , can always be expressed by BernNet, and accordingly, the filters learned by our BernNet are always valid.
+
+![](images/a04bbe15dede6bfda46a24f4347c73fa94999d261b7cfd0556e6ed2a7fcce68c.jpg)  
+Figure 3: A input image and the filtering results.
+
+Table 2: Average sum of squared error and $R ^ { 2 }$ score in parentheses.   
+
+<table><tr><td></td><td>Low-pass</td><td>High-pass</td><td>Band-pass</td><td>Band-rejection</td><td>Comb</td></tr><tr><td></td><td>exp(-10x2)</td><td>1-exp(-10x²)</td><td>exp(-10(λ -1)2)</td><td>1 -exp(-10(λ-1)²)</td><td>|sin(πλ)l</td></tr><tr><td>GCN</td><td>3.4799(.9872)</td><td>67.6635(.2364)</td><td>25.8755(.1148)</td><td>21.0747(.9438)</td><td>50.5120(.2977)</td></tr><tr><td>GAT</td><td>2.3574(.9905)</td><td>21.9618(.7529)</td><td>14.4326(.4823)</td><td>12.6384(.9652)</td><td>23.1813(.6957)</td></tr><tr><td>GPR-GNN</td><td>0.4169(.9984)</td><td>0.0943(.9986)</td><td>3.5121(.8551)</td><td>3.7917(.9905)</td><td>4.6549(.9311)</td></tr><tr><td>ARMA</td><td>1.8478(.9932)</td><td>1.8632(.9793)</td><td>7.6922(.7098)</td><td>8.2732(.9782)</td><td>15.1214(.7975)</td></tr><tr><td>ChebNet</td><td>0.8220(.9973)</td><td>0.7867(.9903)</td><td>2.2722(.9104)</td><td>2.5296(.9934)</td><td>4.0735(.9447)</td></tr><tr><td>BernNet</td><td>0.0314(.9999)</td><td>0.0113(.9999)</td><td>0.0411(.9984)</td><td>0.9313(.9973)</td><td>0.9982(.9868)</td></tr></table>
+
+# 4 Related Work
+
+Graph neural networks (GNNs) can be broadly divided into spectral-based GNNs and spatial-based GNNs [33].
+
+Spectral-based GNNs design spectral graph filters in the spectral domain. ChebNet [7] uses Chebyshev polynomial to approximate a filter. GCN [13] simplifies the Chebyshev filter with the first-order approximation. GraphHeat [34] uses heat kernel to design a graph filter. APPNP [14] utilizes Personalized PageRank (PPR) to set the filter weights. GPR-GNN [5] learns the polynomial filters via gradient descent on the polynomial coefficients. ARMA [2] learns a rational filter via the family of Auto-Regressive Moving Average filters [21]. AdaGNN [9] learns simple filters across multiple layers with a single parameter for each feature channel at each layer. As aforementioned, these methods mainly focus on designing low- or high-pass filters or learning filters without any constraints, which may lead to misspecified even ill-posed filters.
+
+On the other hand, spatial-based GNNs directly propagate and aggregate graph information in the spatial domain. From this perspective, GCN [13] can be explained as the aggregation of the one-hop neighbor information on the graph. GAT [30] uses the attention mechanism to learn aggregation weights. Recently, Balcilar et al. [1] bridge the gap between spectral-based and spatial-based GNNs and unify GNNs in the same framework. Their work shows that the GNNs can be interpreted as sophisticated data-driven filters. This motivates the design of the proposed BernNet, which can learn arbitrary non-negative spectral filters from real-world graph signals.
+
+# 5 Experiments
+
+In this section, we conduct experiments to evaluate BernNet’s capability to learn arbitrary filters as well as the performance of BernNet on real datasets. All the experiments are conducted on a machine with an NVIDIA TITAN V GPU (12GB memory), Intel Xeon CPU $( 2 . 2 0 \mathrm { G H z } )$ , and 512GB of RAM.
+
+# 5.1 Learning filters from the signal
+
+We conduct an empirical analysis on 50 real images with the resolution of $1 0 0 \times 1 0 0$ from the Image Processing Toolbox in Matlab. We conduct independent experiments on these 50 images and report the average of the evaluation index. Following the experimental setting in [1], we regard each image as a 2D regular 4-neighborhood grid graph. The graph structure translates to an 1 $) , 0 0 0 \times 1 0 , 0 0 0$ adjacency matrix while the pixel intensity translates to a 10, 000-dimensional signal vector.
+
+Table 3: Dataset statistics.   
+
+<table><tr><td></td><td>Cora</td><td>CiteSeer</td><td>PubMed</td><td>Computers</td><td>Photo</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Texas</td><td>Cornell</td></tr><tr><td>Nodes</td><td>2708</td><td>3327</td><td>19717</td><td>13752</td><td>7650</td><td>2277</td><td>5201</td><td>7600</td><td>183</td><td>183</td></tr><tr><td>Edges</td><td>5278</td><td>4552</td><td>44324</td><td>245861</td><td>119081</td><td>31371</td><td>198353</td><td>26659</td><td>279</td><td>277</td></tr><tr><td>Features</td><td>1433</td><td>3703</td><td>500</td><td>767</td><td>745</td><td>2325</td><td>2089</td><td>932</td><td>1703</td><td>1703</td></tr><tr><td>Classes</td><td>7</td><td>6</td><td>5</td><td>10</td><td>8</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>
+
+For each of the 50 images, we apply 5 different filters (low-pass, high-pass, band-pass, band-rejection and comb) to the spectral domain of its signal. The formula of each filter is shown in Table 2. Recall that applying a low-pass filter $\exp ( - 1 0 \lambda ^ { 2 } )$ to the spectral domain $\mathbf { L } = \mathbf { U } d i a g \left[ \lambda _ { 1 } , \ldots , \lambda _ { n } \right] \mathbf { U } ^ { \top }$ means applying Udiag $\left[ \exp ( - 1 0 \lambda _ { 1 } ^ { 2 } ) , \dots , \exp ( - 1 0 \lambda _ { n } ^ { 2 } ) \right] \mathbf { U } ^ { \top }$ to the graph signal. Figure 3 shows the one of the input image and the corresponding filtering results.
+
+In this task, we use the original graph signal as the input and the filtering signal to supervise the training process. The goal is to minimize the square error between output and the filtering signal by learning the correct filter. We evaluate BernNet against five popular GNN models: GCN [13], GAT [30], GPR-GNN [5], ARMA [2] and ChebNet [7]. To ensure fairness, we use two convolutional units and a linear output layer for all models. We train all models with approximately $2 \mathrm { k }$ trainable parameters and tune the hidden units to ensure they have similar parameters. Following [1], we discard any regularization or dropout and simply force the GNN to learn the input-output relation. For all models, we set the maximum number of epochs to 2000 and stop the training if the loss does not drop for 100 consecutive times and use Adam optimization with a 0.01 learning rate without decay. Models do not use the position information of the picture pixels. We use a mask to cover the edge nodes of the picture, so the problem can be regarded as a simple regression problem. For BernNet, we use a two-layer model, with each layer sharing the same set of $\theta _ { k }$ for $k = 0 , \ldots , K$ and set $K = 1 0$ . For GPR-GNN, we use the officially released code (see the supplementary materials for URL and commit numbers) and set the order of polynomial filter $K = 1 0$ . Other baseline models are based on Pytorch Geometric implementation [11]. The more detailed experiments setting can be found in the Appendix.
+
+Table 2 shows the average of the sum of squared error (lower the better) and the $R ^ { 2 }$ scores (higher the better). We first observe that GCN and GAT can only handle low-pass filters, which concurs with the theoretical analysis in [1]. GPR-GNN, ARMA and ChebNet can learn different filters from the signals. However, BernNet consistently outperformed these models by a large margin on all tasks in terms of both metrics. We attribute this quality to BernNet’s ability to tune the coefficients $\theta _ { k }$ ’s, which directly correspond to the uniformly sampled filter values.
+
+# 5.2 Node classification on real-world datasets
+
+We now evaluate the performance of BernNet against the competitors on real-world datasets. Following [5], we include three citation graph Cora, CiteSeer and PubMed [27, 37], and the Amazon co-purchase graph Computers and Photo [20]. As shown in [5] these 5 datasets are homophilic graphs on which the connected nodes tend to share the same label. We also include the Wikipedia graph Chameleon and Squirrel [26], the Actor co-occurrence graph, and webpage graphs Texas and Cornell from WebKB‡ [22]. These 5 datasets are heterophilic datasets on which connected nodes tend to have different labels. We summarize the statistics of these datasets in Table 3.
+
+Following [5], we perform full-supervised node classification task with each model, where we randomly split the node set into train/validation/test set with ratio $6 0 \% / 2 0 \% / 2 0 \%$ . For fairness, we generate 10 random splits by random seeds and evaluate all models on the same splits, and report the average metric for each model.
+
+We compare BernNet with 6 baseline models: MLP, GCN [13], GAT [30], APPNP [14], ChebNet [7], and GPR-GNN [5]. For GPR-GNN, we use the officially released code (see the supplementary materials for URL and commit numbers) and set the order of polynomial filter $K = 1 0$ . For other models, we use the corresponding Pytorch Geometric library implementations [11]. For BernNet, we
+
+![](images/765778b1d4ff270459c69b13aef9961eb2a095d359a571028f9ab9fa76f0416a.jpg)  
+Figure 4: Filters learnt from real-world datasets by BernNet.
+
+Table 4: Results on real world benchmark datasets: Mean accuracy $( \% ) \pm 9 5 \%$ confidence interval.   
+
+<table><tr><td></td><td>GCN</td><td>GAT</td><td>APPNP</td><td>MLP</td><td>ChebNet</td><td>GPR-GNN</td><td>BernNet</td></tr><tr><td>Cora</td><td>87.14±1.01</td><td>88.03±0.79</td><td>88.14±0.73</td><td>76.96±0.95</td><td>86.67±0.82</td><td>88.57±0.69</td><td>88.52±0.95</td></tr><tr><td>CiteSeer</td><td>79.86±0.67</td><td>80.52±0.71</td><td>80.47±0.74</td><td>76.58±0.88</td><td>79.11±0.75</td><td>80.12±0.83</td><td>80.09±0.79</td></tr><tr><td>PubMed</td><td>86.74±0.27</td><td>87.04±0.24</td><td>88.12±0.31</td><td>85.94±0.22</td><td>87.95±0.28</td><td>88.46±0.33</td><td>88.48±0.41</td></tr><tr><td>Computers</td><td>83.32±0.33</td><td>83.32±0.39</td><td>85.32±0.37</td><td>82.85±0.38</td><td>87.54±0.43</td><td>86.85±0.25</td><td>87.64±0.44</td></tr><tr><td>Photo</td><td>88.26±0.73</td><td>90.94±0.68</td><td>88.51±0.31</td><td>84.72±0.34</td><td>93.77±0.32</td><td>93.85±0.28</td><td>93.63±0.35</td></tr><tr><td>Chameleon</td><td>59.61±2.21</td><td>63.13±1.93</td><td>51.84±1.82</td><td>46.85±1.51</td><td>59.28±1.25</td><td>67.28±1.09</td><td>68.29±1.58</td></tr><tr><td>Actor</td><td>33.23±1.16</td><td>33.93±2.47</td><td>39.66±0.55</td><td>40.19±0.56</td><td>37.61±0.89</td><td>39.92±0.67</td><td>41.79±101</td></tr><tr><td>Squirrel</td><td>46.78±0.87</td><td>44.49±0.88</td><td>34.71±0.57</td><td>31.03±1.18</td><td>40.55±0.42</td><td>50.15±1.92</td><td>51.35±0.73</td></tr><tr><td>Texas</td><td>77.38±3.28</td><td>80.82±2.13</td><td>90.98±1.64</td><td>91.45±1.14</td><td>86.22±2.45</td><td>92.95±1.31</td><td>93.12±0.65</td></tr><tr><td>Cornell</td><td>65.90±4.43</td><td>78.21±2.95</td><td>91.81±1.96</td><td>90.82±1.63</td><td>83.93±2.13</td><td>91.37±1.81</td><td>92.13±1.64</td></tr></table>
+
+use the following propagation process:
+
+$$
+\mathbf { Z } = \sum _ { k = 0 } ^ { K } \theta _ { k } \frac { 1 } { 2 ^ { K } } \binom { K } { k } ( 2 \mathbf { I } - \mathbf { L } ) ^ { K - k } \mathbf { L } ^ { k } f \left( \mathbf { X } \right) ,
+$$
+
+where $f ( \mathbf { X } )$ is a 2-layer MLP with 64 hidden units on the feature matrix $\mathbf { X }$ . Note that this propagation process is almost identical to that of APPNP or GPR-GNN. The only difference is that we substitute the Generalized PageRank polynomial with Bernstein polynomial. We set the $K = 1 0$ and use different learning rate and dropout for the linear layer and the propagation layer. For all models, we optimal leaning rate over $\{ 0 . 0 \bar { 0 } 1 , 0 . 0 0 2 , 0 . 0 1 , 0 . 0 5 \}$ and weight decay $\{ 0 . 0 , \dot { 0 } . 0 0 0 5 \}$ . More detailed experimental settings are discussed in Appendix.
+
+We use the micro-F1 score with a $9 5 \%$ confidence interval as the evaluation metric. The relevant results are summarized in Table 4. Boldface letters indicate the best result for the given confidence interval. We observe that BernNet provides the best results on seven out of the ten datasets. On the other three datasets, BernNet also achieves competitive results against SOTA methods.
+
+More interestingly, this experiment also shows BernNet can learn complex filters from real-world datasets with only the supervision of node labels. Figure 4 plots some of the filters BernNet learnt in the training process. On Actor, BernNet learns an all-pass-alike filter, which concurs with the fact that MLP outperforms all other baselines on this dataset. On Chameleon and Squirrel, BernNet learns two comb-alike filters. Given that BernNet outperforms all competitors by at least $1 \%$ on these two datasets, it may suggest that comb-alike filters are necessary for Chameleon and Squirrel. Figure 5 shows the Coefficients $\theta _ { k }$ learnt from real-world datasets by BernNet. When comparing Figures 4 and 5, we observe that the curves of filters and curves of coefficients are almost the same. This is because BernNet’s coefficients are highly correlated with the spectral property of the target filter, which indicates BernNet Bernnet has strong interpretability.
+
+Finally, we present the train time for each method in Table 5. BernNet is slower than other methods due to its quadratic dependence on the degree $K$ . However, compared to the SOTA method GPRGNN, the margin is generally less than 2, which is often acceptable in practice. In theory, both ChebNet [7] and GPR-GNN [5] are linear time complexity related to propagation step $K$ , but BernNet is quadratic time complexity related to $K$ . Delgado et al. [8] show that Bernstein approximation can be evaluated in linear time related to $K$ using the corner cutting algorithm. However, BernNet can not use this algorithm directly, because we need to multiply signal $\mathbf { x }$ . How to convert BernNet to linear complexity will be a problem worth studying in the future.
+
+![](images/b0dcb357edbc74b9b1067c37e462472b96686afdfe6fe7fdb0d4e52b97d5bd83.jpg)  
+Figure 5: Coefficients $\theta _ { k }$ learnt from real-world datasets by BernNet.
+
+Table 5: Average running time per epoch (ms)/average total running time (s).   
+
+<table><tr><td></td><td>GCN</td><td>GAT</td><td>APPNP</td><td>MLP</td><td>ChebNet</td><td>GPR-GNN</td><td>BernNet</td></tr><tr><td>Cora</td><td>4.59/1.62</td><td>9.56/2.03</td><td>7.16/2.32</td><td>3.06/0.93</td><td>6.25/1.76</td><td>9.94/2.21</td><td>19.71/5.47</td></tr><tr><td>CiteSeer</td><td>4.63/1.95</td><td>9.93/2.21</td><td>7.79/2.77</td><td>2.95/1.09</td><td>8.28/2.56</td><td>11.16/2.37</td><td>22.36/6.32</td></tr><tr><td>PubMed</td><td>5.12/1.87</td><td>16.16/3.41</td><td>8.21/2.63</td><td>2.91/1.61</td><td>18.04/3.03</td><td>10.45/2.81</td><td>22.02/8.19</td></tr><tr><td>Computers</td><td>5.72/2.52</td><td>30.91/7.85</td><td>9.19/3.48</td><td>3.47/1.31</td><td>20.64/9.64</td><td>16.05/4.38</td><td>28.83/8.69</td></tr><tr><td>Photo</td><td>5.08/2.63</td><td>19.97/5.41</td><td>8.69/4.18</td><td>3.67/1.66</td><td>13.25/7.02</td><td>13.96/3.94</td><td>24.69/7.37</td></tr><tr><td>Chameleon</td><td>4.93/0.99</td><td>13.11/2.66</td><td>7.93/1.62</td><td>3.14/0.63</td><td>10.92/2.25</td><td>10.93/2.41</td><td>22.54/4.75</td></tr><tr><td>Actor</td><td>5.43/1.09</td><td>11.94/2.45</td><td>8.46/1.71</td><td>3.82/0.77</td><td>7.99/1.62</td><td>11.57/2.35</td><td>23.34/5.81</td></tr><tr><td>Squirrel</td><td>5.61/1.13</td><td>22.76/4.91</td><td>8.01/1.61</td><td>3.41/0.69</td><td>38.12/7.78</td><td>9.87/5.56</td><td>25.58/9.23</td></tr><tr><td>Texas</td><td>4.58/0.92</td><td>9.65/1.96</td><td>7.83/1.63</td><td>3.19/0.65</td><td>6.51/1.34</td><td>10.45/2.16</td><td>23.35/4.81</td></tr><tr><td>Cornell</td><td>4.83/0.97</td><td>9.79/1.99</td><td>8.23/1.68</td><td>3.25/0.66</td><td>5.85/1.22</td><td>9.86/2.05</td><td>22.23/5.26</td></tr></table>
+
+# 6 Conclusion
+
+This paper proposes BernNet, a graph neural network that provides a simple and intuitive mechanism for designing and learning an arbitrary spectral filter via Bernstein polynomial approximation. Compared to previous methods, BernNet can approximate complex filters such as band-rejection and comb filters, and can provide better interpretability. Furthermore, the polynomial filters designed and learned by BernNet are always valid. Experiments show that BernNet outperforms SOTA methods in terms of effectiveness on both synthetic and real-world datasets. For future work, an interesting direction is to improve the efficiency of BernNet.
+
+# Broader Impact
+
+The proposed BernNet algorithm addresses the challenge of designing and learning arbitrary spectral filters on graphs. We consider this algorithm a general technical and theoretical contribution, without any foreseeable specific impacts. For applications in bioinformatics, computer vision, and natural language processing, applying the BernNet algorithm may improve the performance of existing GNN models. We leave the exploration of other potential impacts to future work.
+
+# Acknowledgments and Disclosure of Funding
+
+Zhewei Wei was supported in part by National Natural Science Foundation of China (No. 61972401, No. 61932001 and No. 61832017), by Beijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098, by Alibaba Group through Alibaba Innovative Research Program, and by CCF-Baidu Open the Fund (NO.2021PP15002000). Zengfeng Huang was supported by National Natural Science Foundation of China Grant No. 61802069, and by Shanghai Science and Technology Commission Grant No. 17JC1420200. Hongteng Xu was supported by Tencent AI Lab Rhino-Bird Joint Research Program. This work is supported by China Unicom Innovation Ecological Cooperation Plan and by Intelligent Social Governance Platform, Major Innovation $\&$ Planning Interdisciplinary Platform for the “Double-First Class” Initiative, Renmin University of China. We also wish to acknowledge the support provided and contribution made by Public Policy and Decision-making Research Lab of Renmin University of China.
+
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parse/train/WigDnV-_Gq/WigDnV-_Gq_content_list.json

@@ -0,0 +1,1330 @@
+[
+    {
+        "type": "text",
+        "text": "BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation ",
+        "text_level": 1,
+        "bbox": [
+            179,
+            122,
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+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mingguo He Renmin University of China mingguo@ruc.edu.cn ",
+        "bbox": [
+            264,
+            226,
+            450,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zhewei Wei∗ Renmin University of China zhewei@ruc.edu.cn ",
+        "bbox": [
+            547,
+            226,
+            735,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zengfeng Huang Fudan University huangzf@fudan.edu.cn ",
+        "bbox": [
+            267,
+            289,
+            441,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hongteng $\\mathbf { X } \\mathbf { u } ^ { * }$ Renmin University of China hongtengxu@ruc.edu.cn ",
+        "bbox": [
+            542,
+            289,
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+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            367,
+            535,
+            382
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many representative graph neural networks, e.g., GPR-GNN and ChebNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose BernNet, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order- $K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks. Code is available at https://github.com/ivam-he/BernNet. ",
+        "bbox": [
+            233,
+            401,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            638,
+            310,
+            656
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks (GNNs) have received extensive attention from researchers due to their   \nexcellent performance on various graph learning tasks such as social analysis [24, 17, 29], drug   \ndiscovery [12, 25], traffic forecasting [18, 3, 6], recommendation system [38, 32] and computer   \nvision [39, 4]. Recent studies suggest that many popular GNNs operate as polynomial graph spectral   \nfilters [7, 13, 5, 16, 2, 35]. Specifically, we denote an undirected graph with node set $V$ and edge   \nset $E$ as $G = ( V , E )$ , whose adjacency matrix is A. Given a signal $\\mathbf { x } = [ x ] \\in R ^ { n }$ on the graph, $n = | V |$ weightshe diag, where is the symmetric normalizednother equivalent polynomial normalized adjacency matrix   \n$\\scriptstyle \\sum _ { k = 0 } ^ { K } w _ { k } \\mathbf { L } ^ { k } \\mathbf { x }$ $w _ { k }$ $\\mathbf { L } = \\mathbf { I } - \\mathbf { D } ^ { - 1 / 2 } \\mathbf { A } \\mathbf { D } ^ { - 1 / 2 }$ $G$ $\\mathbf { D }$ $\\scriptstyle \\sum _ { k = 0 } ^ { K } c _ { k } \\mathbf { P } ^ { k } \\mathbf { x }$ $\\mathbf { P } = \\mathbf { D } ^ { - 1 / 2 } \\mathbf { A } \\mathbf { D } ^ { - 1 / 2 }$ $c _ { k }$ ",
+        "bbox": [
+            173,
+            672,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/190ec5391113ac944665522e1adf4c8cf8fc31bf8e9a0938f191d278b12a49dd.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of the proposed BernNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            88,
+            790,
+            231
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We can broadly categorize the GNNs applying the above filtering operation into two classes, depending on whether they design the filter weights or learn them based on observed graphs. Some representative models in these two classes are shown below. ",
+        "bbox": [
+            176,
+            281,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• The GNNs driven by designing filters: GCN [13] uses a simplified first-order Chebyshev polynomial, which is proven to be a low-pass filter [1, 31, 34, 41]. APPNP [14] utilizes Personalized PageRank (PPR) to set the filter weights and achieves a low-pass filter as well [15, 41]. GNN-LF/HF [41] designs filter weights from the perspective of graph optimization functions, which can simulate high- and low-pass filters. ",
+        "bbox": [
+            217,
+            342,
+            823,
+            411
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• The GNNs driven by learning filters: ChebNet [7] approximates the filtering operation with Chebyshev polynomials, and learns a filter via trainable weights of the Chebyshev basis. GPR-GNN [5] learns a polynomial filter by directly performing gradient descent on the filter weights, which can derive high- or low-pass filters. ARMA [2] learns a rational filter via the family of Auto-Regressive Moving Average filters [21]. ",
+        "bbox": [
+            218,
+            422,
+            823,
+            493
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although the above GNNs achieve some encouraging results in various graph modeling tasks, they still suffer from two major drawbacks. Firstly, most existing methods focus on designing or learning simple filters (e.g., low- and/or high-pass filters), while real-world applications often require much more complex filters such as band-rejection and comb filters. To the best of our knowledge, none of the existing work supports designing arbitrary interpretable spectral filters. The GNNs driven by learning filters can learn arbitrary filters in theory, but they cannot intuitively show what filters they have learned. In other words, their interpretability is poor. For example, GPR-GNN [5] learns the filter weights $w _ { k }$ ’s but only proves a small subset of the learnt weight sequences corresponds to low- or high-pass filters. Secondly, the GNNs often design their filters empirically or learn the filter weights without any necessary constraints. As a result, their filter weights often have poor controllability. For example, GNN-LF/HF [41] designs its filters with a complex and non-intuitive polynomial with difficult-to-tune hyperparameters. The multi-layer GCN/SGC [13, 31] leads to “ill-posed” filters (i.e., those deriving negative spectral responses). Additionally, the filters learned by GPR-GNN [5] or ChebNet [7] have a chance to be ill-posed as well. ",
+        "bbox": [
+            173,
+            511,
+            825,
+            704
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To overcome the above issues, we propose a novel graph neural network called BernNet, which provides an effective algorithmic framework for designing and learning arbitrary graph spectral filters. As illustrated in Figure 1, for an arbitrary spectral filter $h : [ 0 , 2 ] \\mapsto [ 0 , 1 ]$ over the spectrum of the symmetric normalized Laplacian $\\mathbf { L }$ , our BernNet approximates $h$ by a $K$ -order Bernstein polynomial approximation, i.e., $\\begin{array} { r } { \\bar { h } ( \\lambda ) = \\sum _ { k = 0 } ^ { K } \\theta _ { k } b _ { k } ^ { K } ( \\lambda ) } \\end{array}$ . The non-negative coefficients $\\{ \\theta _ { k } \\} _ { k = 0 } ^ { K }$ of the Bernstein basis $\\{ b _ { k } ^ { K } ( \\lambda ) \\} _ { k = 0 } ^ { K }$ work as the model parameter, which can be interpreted as $h ( 2 k / K )$ , $k = 0 , \\ldots , K$ (i.e., the filter values uniformly sampled from [0, 2]). By designing or learning the $\\theta _ { k }$ ’s, we can obtain various spectral filters, whose filtering operation can be formulated as $\\begin{array} { r l } { ~ } & { { } \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\frac { 1 } { 2 ^ { K } } \\binom { K } { k } ( 2 \\mathbf { I } - \\mathbf { L } ) ^ { K - k } \\mathbf { L } ^ { k } \\mathbf { x } } \\end{array}$ , where $\\mathbf { x }$ is the graph signal. We further demonstrate the rationality of our BernNet from the perspective of graph optimization — any valid polynomial filers, i.e., those polynomial functions mapping $[ 0 , 2 ]$ to $[ 0 , 1 ]$ , can always be expressed by our BernNet, and accordingly, the filters learned by our BernNet are always valid. Finally, we conduct experiments to demonstrate that 1) BernNet can learn arbitrary graph spectral filters (e.g., band-rejection, comb, low-band-pass, etc.), and 2) BernNet achieves superior performance on real-world datasets. ",
+        "bbox": [
+            173,
+            712,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BernNet ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            89,
+            276,
+            106
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Bernstein approximation of spectral filters ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            119,
+            509,
+            136
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given an arbitrary filter function $h : [ 0 , 2 ] \\mapsto [ 0 , 1 ]$ , let $\\mathbf { L } = \\mathbf { U } \\mathbf { A } \\mathbf { U } ^ { T }$ denote the eigendecomposition of the symmetric normalized Laplacian matrix $\\mathbf { L }$ , where $\\mathbf { U }$ is the matrix of eigenvectors and $\\Lambda =$ $d i a g [ \\lambda _ { 1 } , . . . , \\lambda _ { n } ]$ is the diagonal matrix of eigenvalues. We use ",
+        "bbox": [
+            173,
+            143,
+            825,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/307cc9a3d1fe81593f26a9044a69a609c09b6a1c5b9c38117f8c94f9bdb2fda0.jpg",
+        "text": "$$\nh ( { \\bf L } ) { \\bf x } = { \\bf U } h ( { \\boldsymbol \\Lambda } ) { \\bf U } ^ { T } { \\bf x } = { \\bf U } d i a g [ h ( \\lambda _ { 1 } ) , . . . , h ( \\lambda _ { n } ) ] { \\bf U } ^ { T } { \\bf x }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            191,
+            684,
+            210
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "to denote a spectral filter on graph signal $\\mathbf { x }$ . The key of our work is approximate $h ( \\mathbf { L } )$ (or, equivalently, $h ( \\lambda ) )$ . For this purpose, we leverage the Bernstein basis and Bernstein polynomial approximation defined below. ",
+        "bbox": [
+            176,
+            214,
+            825,
+            257
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 ( [10]). (Bernstein polynomial approximation) Given an arbitrary continuous function $f ( t )$ on $t \\in [ 0 , 1 ]$ , the Bernstein polynomial approximation (of order $K$ ) for $f$ is defined as ",
+        "bbox": [
+            171,
+            260,
+            826,
+            289
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6426e4cd56a6efb4df1ff686832f2bc5d91d9f6b6d1fcb6d4343298dae118c9f.jpg",
+        "text": "$$\np _ { K } ( t ) : = \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\cdot b _ { k } ^ { K } ( t ) = \\sum _ { k = 0 } ^ { K } f \\left( \\frac { k } { K } \\right) \\cdot \\binom { K } { k } ( 1 - t ) ^ { K - k } t ^ { k } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            292,
+            704,
+            337
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here, for $k = 0 , . . . , K$ $\\begin{array} { r } { K , b _ { k } ^ { K } ( t ) = \\binom { K } { k } ( 1 - t ) ^ { K - k } t ^ { k } } \\end{array}$ is the $k$ -th Bernstein base, and $\\textstyle \\theta _ { k } = f ( { \\frac { k } { K } } )$ is the function value at $k / K$ , which works as the coefficient of $b _ { k } ^ { K } ( t )$ . ",
+        "bbox": [
+            176,
+            340,
+            820,
+            375
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2.1 ( [10]). Given an arbitrary continuous function $f ( t )$ on $t \\in [ 0 , 1 ] ,$ , let $p _ { K } ( t )$ denote the Bernstein approximation of $f ( t )$ as defined in Equation (2). We have $p _ { K } ( t )  f ( t )$ as $K  \\infty$ . ",
+        "bbox": [
+            171,
+            376,
+            823,
+            406
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For the filter function $h : [ 0 , 2 ] \\mapsto [ 0 , 1 ]$ , we let $\\begin{array} { r } { t = \\frac { \\lambda } { 2 } } \\end{array}$ and $f ( t ) = h ( 2 t )$ , so that the Bernstein polynomial approximation becomes applicable, where $\\theta _ { k } = f ( k / K ) = h ( 2 k / K )$ and $b _ { k } ^ { K } ( t ) =$ $\\begin{array} { r } { b _ { k } ^ { K } ( \\frac { \\lambda } { 2 } ) \\ = \\ \\binom { K } { k } ( 1 - \\frac { \\lambda } { 2 } ) ^ { K - k } ( \\frac { \\lambda } { 2 } ) ^ { k } } \\end{array}$ for $k = 1 , . . . , K$ . Consequently, we can approximate $h ( \\lambda )$ by $\\begin{array} { r } { p _ { K } ( \\lambda / 2 ) = \\sum _ { k = 0 } ^ { K } \\theta _ { k } { \\binom { K } { k } } ( 1 - \\frac { \\lambda } { 2 } ) ^ { K - k } \\left( \\frac { \\lambda } { 2 } \\right) ^ { k } = \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\frac { 1 } { 2 ^ { K } } { \\binom { K } { k } } ( 2 - \\lambda ) ^ { K - k } \\lambda ^ { k } . } \\end{array}$ , and Lemma 2.1 ensures that $p _ { K } ( \\lambda / 2 ) \\to h ( \\lambda )$ as $K  \\infty$ . ",
+        "bbox": [
+            173,
+            415,
+            825,
+            500
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Replacing $\\{ h ( \\lambda _ { i } ) \\} _ { i = 1 } ^ { n }$ with $\\{ p _ { K } ( \\lambda _ { i } / 2 ) \\} _ { i = 1 } ^ { n }$ , we approximate the spectral filter $h ( \\mathbf { L } )$ in Equation (1) as Udiag $[ p _ { K } ( \\lambda _ { 1 } / 2 ) , . . . , p _ { K } ( \\lambda _ { n } / 2 ) ] \\mathbf { U } ^ { T }$ and derive the proposed BernNet. In particular, given a graph signal $\\mathbf { x }$ , the convolutional operator of our BernNet is defined as follows: ",
+        "bbox": [
+            174,
+            505,
+            825,
+            549
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d8c2a5ca52d36a788530f215dc6aeff5dce30ab2fd1299b32b0c16440bfe80b.jpg",
+        "text": "$$\n\\mathbf { z } = \\underbrace { \\mathbf { U } d i a g [ p _ { K } ( \\lambda _ { 1 } / 2 ) , . . . , p _ { K } ( \\lambda _ { n } / 2 ) ] \\mathbf { U } ^ { T } } _ { \\mathrm { B e r n N e t } } \\mathbf { U } ^ { T } \\mathbf { x } = \\sum _ { k = 0 } ^ { K } \\theta _ { k } { \\frac { 1 } { 2 ^ { K } } } { \\binom { K } { k } } ( 2 \\mathbf { I } - \\mathbf { L } ) ^ { K - k } \\mathbf { L } ^ { k } \\mathbf { x }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            553,
+            766,
+            602
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where each coefficient $\\theta _ { k }$ can be either set to $h ( 2 k / K )$ to approximate a predetermined filter $h$ , or learnt from the graph structure and signal in an end-to-end fashion. As a natural extension of Lemma 2.1, our BernNet owns the following proposition. ",
+        "bbox": [
+            176,
+            606,
+            825,
+            648
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2.1. For an arbitrary continuous filter function $h : [ 0 , 2 ]  [ 0 , 1 ] ,$ , by setting $\\theta _ { k } =$ $h ( 2 \\bar { k } / K ) , k = 0 , \\dots , K$ , the $\\mathbf { z }$ in Equation (3) satisfies ${ \\bf z }  h ( { \\bf L } ) { \\bf x } $ as $K  \\infty$ . ",
+        "bbox": [
+            169,
+            650,
+            823,
+            680
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof. According to the above derivation, we have $\\begin{array} { r } { p _ { K } ( \\lambda / 2 ) = \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\binom { K } { k } ( 1 - \\frac { \\lambda } { 2 } ) ^ { K - k } \\left( \\frac { \\lambda } { 2 } \\right) ^ { k } = } \\end{array}$ $\\begin{array} { r } { \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\frac { 1 } { 2 ^ { K } } \\binom { K } { k } ( 2 - \\lambda ) ^ { K - k } \\lambda ^ { k } } \\end{array}$ , and Lemma 2.1 ensures that $p _ { K } ( \\lambda / 2 ) \\to h ( \\lambda )$ as $\\theta _ { k } = h ( 2 k / K )$ and $K  \\infty$ . ",
+        "bbox": [
+            173,
+            695,
+            825,
+            746
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consequently, we have ",
+        "bbox": [
+            174,
+            751,
+            326,
+            766
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { z } = \\mathbf { U } d i a g [ p _ { K } ( \\lambda _ { 1 } / 2 ) , . . . , p _ { K } ( \\lambda _ { n } / 2 ) ] \\mathbf { U } ^ { T } \\mathbf { x } \\to \\mathbf { U } d i a g [ h ( \\lambda _ { 1 } ) , . . . , h ( \\lambda _ { n } ) ] \\mathbf { U } ^ { T } \\mathbf { x } = h ( \\mathbf { L } )$ as $\\theta _ { k } = h ( 2 k / K )$ and $K  \\infty$ . ",
+        "bbox": [
+            173,
+            770,
+            779,
+            808
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Realizing existing filters with BernNet. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            843,
+            480,
+            859
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As shown in Proposition 2.1, our BernNet can approximate arbitrary continuous spectral filters with sufficient precision. Below we give some representative examples of how our BernNet exactly realizes existing filters that are commonly used in GNNs. ",
+        "bbox": [
+            174,
+            868,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/9dc5140848e3d3617d3a5af27f72597b412a99a0910ba6857a332cf25b5d8f8f.jpg",
+        "table_caption": [
+            "Table 1: Realizing commonly used filters with BernNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Filter types</td><td>Filter h(λ)</td><td>0k for k=0,...,K</td><td>Bernstein approximation pk(）</td><td>BernNet</td></tr><tr><td>All-pass</td><td>1</td><td>0=1</td><td>1</td><td>I</td></tr><tr><td>Linear low-pass</td><td>1-&gt;/2</td><td>0=1-k/K</td><td>1-λ/2</td><td>1</td></tr><tr><td>Linear high-pass</td><td>入/2</td><td>0=k/K</td><td>入/2</td><td></td></tr><tr><td>Impulse low-pass</td><td>(入）</td><td>0=1and other 0k =0</td><td>(1-X/2)K</td><td></td></tr><tr><td>Impulse high-pass</td><td>8(入）</td><td>0K=1and other 0k =0</td><td>（入/2）K</td><td>美K</td></tr><tr><td>Impulse band-pass</td><td>8(入）</td><td>0K/2 =1and other 0k =0</td><td>（K2）(1-&gt;/2)K/2(/2）K/2</td><td>（K/2)（21-L）K/2LK/2</td></tr></table>",
+        "bbox": [
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+            109,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• All-pass filter $h ( \\lambda ) = 1$ . We set $\\theta _ { k } \\ = \\ 1$ for $k = 0 , \\ldots , K$ , and the approximation $p _ { K } ( \\frac { \\lambda } { 2 } ) = 1$ is exactly the same with $h ( \\lambda )$ . Accordingly, our BernNet becomes an identity matrix, which realizes the all-pass filter perfectly. ",
+        "bbox": [
+            217,
+            231,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Linear low-pass filter $h ( \\lambda ) = 1 - \\lambda / 2$ . We set $\\theta _ { k } = 1 - k / K$ for $k = 0 , \\ldots , K$ and obtain $p _ { K } ( \\frac { \\lambda } { 2 } ) = 1 - \\lambda / 2$ . The BernNet becomes $\\begin{array} { r } { \\sum _ { k = 0 } ^ { K } \\frac { ( K - k ) } { K } \\frac { 1 } { 2 ^ { K } } \\binom { K } { k } ( 2 \\mathbf { I } - \\mathbf { L } ) ^ { K - k } \\mathbf { L } ^ { k } = \\mathbf { I } - \\frac { 1 } { 2 } \\mathbf { L } } \\end{array}$ which achieves the linear low-pass filter exactly. Note that $\\begin{array} { r } { \\mathbf { I } - \\frac { 1 } { 2 } \\mathbf { L } = \\frac { 1 } { 2 } ( \\mathbf { I } + \\mathbf { P } ) } \\end{array}$ is also the same as the graph convolutional network (GCN) before renormalization [13]. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Linear high-pass filter $h ( \\lambda ) = \\lambda / 2$ . Similarly, we can set $\\theta _ { k } = k / K$ for $k = 0 , \\ldots , K$ to get a perfect approximation $\\begin{array} { r } { p _ { K } ( \\frac { \\lambda } { 2 } ) = \\frac { \\lambda } { 2 } } \\end{array}$ , and the BernNet becomes $\\scriptstyle { \\frac { 1 } { 2 } } \\mathbf { L }$ . ",
+        "bbox": [
+            218,
+            345,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that even for those non-continuous spectral filters, e.g., the impulse low/high/band-pass filters, our BernNet can also provide good approximations (with sufficient large $K$ ). ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Impulse low-pass filter $h ( \\lambda ) = \\delta _ { 0 } ( \\lambda )$ .† We set $\\theta _ { 0 } = 1$ and $\\theta _ { k } \\ = \\ 0$ for $k \\neq 0$ , and $\\begin{array} { r } { p _ { K } \\bar { ( \\frac { \\lambda } { 2 } ) } = ( 1 - \\frac { \\bar { \\lambda } } { 2 } ) ^ { K } } \\end{array}$ . Accordingly, the BernNet becomes $\\begin{array} { r } { \\frac { 1 } { 2 ^ { K } } ( 2 \\mathbf { I } - \\mathbf { L } ) ^ { K } } \\end{array}$ , deriving an $K$ -layer linear low-pass filter. ",
+        "bbox": [
+            217,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Impulse high-pass filter $h ( \\lambda ) = \\delta _ { 2 } ( \\lambda )$ . We set $\\theta _ { K } = 1$ and $\\theta _ { k } = 0$ for $k \\neq K$ , and $\\begin{array} { r } { p _ { K } ( \\frac { \\lambda } { 2 } ) = ( \\frac { \\lambda } { 2 } ) ^ { K } } \\end{array}$ . The BernNet becomes $\\scriptstyle { \\frac { 1 } { 2 ^ { K } } } \\mathbf { L } ^ { K }$ , i.e., an $K$ -layer linear high-pass filter. ",
+        "bbox": [
+            218,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Impulse band-pass filter $h ( \\lambda ) = \\delta _ { 1 } ( \\lambda )$ . Similarly, we set $\\theta _ { K / 2 } = 1$ and $\\theta _ { k } = 0$ for $k \\neq$ $K / 2$ , and $\\begin{array} { r } { p _ { K } ( \\frac { \\lambda } { 2 } ) = \\binom { K } { K / 2 } ( 1 - \\lambda / 2 ) ^ { K / 2 } ( \\lambda / 2 ) ^ { K / 2 } } \\end{array}$ . The BernNet becomes $\\frac { 1 } { 2 ^ { K } } \\binom { K } { K / 2 } ( 2 \\mathbf { I } -$ $\\mathbf { L } ) ^ { K / 2 } \\mathbf { L } ^ { K / 2 }$ , which can be explained as stacking a $K / 2$ -layer linear low-pass filter and a $K / 2$ -layer linear high-pass filter. Obviously, $K$ should be an even number in this case. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes the design of the BernNet for the filters above. We can find that an appealing advantage of our BernNet is that its coefficients are highly correlated with the spectral property of the target filter. In particular, we can determine to pass or reject the spectral signal with $\\textstyle \\lambda \\approx { \\frac { { \\bar { 2 } } k } { K } }$ by using a large or small property provid $\\theta _ { k }$ because each Bernstein base useful guidance when designi $b _ { k } ^ { K } ( \\lambda )$ corresponds to a “bump” located at rs, which enhances the interpretabili $\\textstyle { \\frac { 2 k } { K } }$ . This of our BernNet. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 Learning complex filters with BernNet ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            693,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Besides designing the above typical filters, our BernNet can express more complex filters, such as band-pass, band-rejection, comb, low-band-pass filters, etc. Moreover, given the graph signals before and after applying such filters (i.e., the x’s and the corresponding $\\mathbf { z }$ ’s), our BernNet can learn their approximations in an end-to-end manner. Specifically, given the pairs $\\{ \\mathbf { x } , \\mathbf { z } \\}$ , we learn the coefficients $\\{ \\theta _ { k } \\} _ { k = 0 } ^ { K }$ of the BernNet by gradient descent. More implementation details can be found at the experimental section below. Figure 2 illustrates the four complex filters and the approximations we learned (The low-band pass filter is $h ( \\lambda ) = I _ { [ 0 , 0 . 5 ] } ( \\lambda ) + \\hat { \\exp { ( - 1 0 0 ( \\lambda - 0 . 5 ) ^ { 2 } ) } } \\hat { I } _ { ( 0 . 5 , 1 ) } ( \\lambda ) +$ $\\exp { ( - 5 0 ( \\lambda - 1 . 5 ) ^ { 2 } ) } I _ { [ 1 , 2 ] } ( \\lambda )$ , where $I _ { \\Omega } ( \\lambda ) = 1$ when $\\lambda \\in \\Omega$ , otherwise $I _ { \\Omega } ( \\lambda ) = 0 )$ . In general, our BernNet can learn a smoothed approximation of these complex filters, and the approximation precision improves with the increase of the order $K$ . Note that although the BernNet cannot pinpoint the exact peaks of the comb filter or drop to 0 for the valleys of comb or low-band-pass filters due to the limitation of $K$ , it still significantly outperforms other GNNs for learning such complex filters. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/79be059be4cf245f9f112beb0e01c5cd45d48d8fea28b73516bb37b15d002002.jpg",
+        "image_caption": [
+            "Figure 2: Illustrations of four complex filters and their approximations learnt by BernNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 BernNet in the Lens of Graph Optimization ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we motivate BernNet from the perspective of graph optimization. In particular, we show that any polynomial filter that attempts to approximate a valid filter has to take the form of BernNet. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 A generalized graph optimization problem ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given a $n$ -dimensional graph signal $\\mathbf { x }$ , we consider a generalized graph optimization problem ",
+        "bbox": [
+            173,
+            385,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd35f489ee490033710e3c8b2ca4ce8719ed7c233f214b3ccdc7624314b379f4.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { z } } f ( \\mathbf { z } ) = ( 1 - \\alpha ) \\mathbf { z } ^ { T } \\boldsymbol { \\gamma } ( \\mathbf { L } ) \\mathbf { z } + \\alpha \\| \\mathbf { z } - \\mathbf { x } \\| _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            406,
+            645,
+            431
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha \\in [ 0 , 1 )$ is a trade-off parameter, $\\mathbf { z } \\in R ^ { n }$ denotes the propagated representation of the input graph signal $\\mathbf { x }$ , and $\\gamma ( \\mathbf { L } )$ denotes an energy function of $\\mathbf { L }$ , determining the rate of propagation [28]. Generally, $\\gamma ( \\cdot )$ operates on the spectral of $\\mathbf { L }$ , and we have $\\gamma ( { \\bf L } ) = { \\bf U } { \\bar { d } } i a g [ \\gamma ( \\lambda _ { 1 } ) , \\dot { \\bf \\omega } . \\dot { \\bf \\omega } , \\dot { \\gamma ( \\lambda _ { n } ) } ] { \\bf U } ^ { T }$ . ",
+        "bbox": [
+            173,
+            436,
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+            479
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can model the polynomial filtering operation of existing GNNs with the optimal solution of Equation (4). For example, if we set $\\begin{array} { r } { \\gamma ( \\mathbf { L } ) = \\mathbf { L } } \\end{array}$ , then the optimization function (4) becomes $f ( \\mathbf { \\bar { z } } ) = ( 1 - \\alpha ) \\mathbf { z } ^ { T } \\mathbf { L } \\mathbf { z } + \\acute { \\alpha } \\| \\mathbf { z } - \\mathbf { x } \\| _ { 2 } ^ { 2 }$ , a well-known convex graph optimization function proposed by Zhou et al. [40]. $f ( \\mathbf { z } )$ takes the minimum when the derivativ e ∂f(z)z = 2(1 − α)Lz + 2α (z − x) = 0, which solves to ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f4e8724c4364b397e5b247559a6190ef11ed795d9809d061ba24aa35d2ce95a8.jpg",
+        "text": "$$\n\\mathbf { z } ^ { * } = \\alpha \\left( \\mathbf { I } - ( 1 - \\alpha ) ( \\mathbf { I } - \\mathbf { L } ) \\right) ^ { - 1 } \\mathbf { x } = \\sum _ { k = 0 } ^ { \\infty } \\alpha ( 1 - \\alpha ) ^ { k } \\left( \\mathbf { I } - \\mathbf { L } \\right) ^ { k } \\mathbf { x } = \\sum _ { k = 0 } ^ { \\infty } \\alpha ( 1 - \\alpha ) ^ { k } \\mathbf { P } ^ { k } \\mathbf { x } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            560,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By taking a suffix sum $\\begin{array} { r } { \\sum _ { k = 0 } ^ { K } \\alpha ( 1 - \\alpha ) ^ { k } \\mathbf { P } ^ { k } \\mathbf { x } } \\end{array}$ , we obtain the polynomial filtering operation for APPNP [14]. Zhu et al. [41] further show that GCN [13], DAGNN [19], and JKNet [36] can be interpreted by the optimization function (4) with $\\gamma ( \\mathbf { L } ) = \\mathbf { L }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The generalized form of Equation (4) allows us to simulate more complex polynomial filtering operation. For example, let $\\alpha = 0 . 5$ and $\\gamma ( \\mathbf { L } ) = e ^ { t \\mathbf { L } } - \\mathbf { I }$ , a heat kernel with $t$ as the temperature parameter. Then $f ( \\mathbf { z } )$ takes the minimum when the derivative $\\begin{array} { r } { \\frac { \\partial f ( \\mathbf { z } ) } { \\partial \\mathbf { z } } = \\left( e ^ { t \\mathbf { L } } - \\mathbf { I } \\right) \\mathbf { z } + \\mathbf { z } - \\mathbf { x } = \\mathbf { 0 } } \\end{array}$ , which solves to ",
+        "bbox": [
+            174,
+            660,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/999514e44500a83bd5be65c2574fdbebae72ab79b440bf043769d6fa9efaef97.jpg",
+        "text": "$$\n\\mathbf { z } ^ { * } = e ^ { - t \\mathbf { L } } \\mathbf { x } = e ^ { - t ( \\mathbf { I } - \\mathbf { P } ) } \\mathbf { x } = \\sum _ { k = 0 } ^ { \\infty } e ^ { - t } \\frac { t ^ { k } } { k ! } \\mathbf { P } ^ { k } \\mathbf { x } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            714,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By taking a suffix sum $\\textstyle \\sum _ { k = 0 } ^ { K } e ^ { - t } { \\frac { t ^ { k } } { k ! } } \\mathbf { P } ^ { k } \\mathbf { x }$ , we obtain the polynomial filtering operation for the heat kernal based GNN such as GDC [15] and GraphHeat [34]. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Non-negative constraint on polynomial filters ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A natural question is that, does an arbitrary energy function $\\gamma ( \\mathbf { L } )$ correspond to a valid or ill-posed spectral filter? Conversely, does any polynomial filtering operation $\\scriptstyle \\sum _ { k = 0 } ^ { K } w _ { k } \\mathbf { L } ^ { k } \\mathbf { x }$ correspond to the optimal solution of the optimization function (4) for some energy function $\\gamma ( \\mathbf { L } )$ ? ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As it turns out, there is a “minimum requirement” for the energy function $\\gamma ( \\mathbf { L } )$ ; $\\gamma ( \\mathbf { L } )$ has to be positive semidefinite. In particular, if $\\gamma ( \\mathbf { L } )$ is not positive semidefinite, then the optimization function $f ( \\mathbf { z } )$ is not convex, and the solution to $\\begin{array} { r } { \\frac { \\partial f ( \\mathbf { z } ) } { \\partial \\mathbf { z } } = 0 } \\end{array}$ may corresponds to a saddle point. Furthermore, without the positive semidefinite constraint on $\\gamma ( \\mathbf { L } )$ , $f ( \\mathbf { z } )$ may goes to $- \\infty$ as we set $\\mathbf { z }$ to be a multiple of the eigenvector corresponding to the negative eigenvalue. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Non-negative polynomial filters. Given a positive semidefinite energy function $\\gamma ( \\mathbf { L } )$ , we now consider how the corresponding polynomial filtering operatio n PKk=0 wkLkx should look like. Recall that we assume $\\boldsymbol \\gamma ( \\mathbf { L } ) = \\mathbf { U } d i a g [ \\gamma ( \\lambda _ { 1 } ) , . . . , \\gamma ( \\lambda _ { n } ) ] \\mathbf { U } ^ { T }$ . By the positive semidefinite constraint, we have $\\gamma ( \\lambda ) \\geq 0$ for $\\lambda \\in [ 0 , 2 ]$ . Since the objective function $f ( \\mathbf { z } )$ is convex, it takes the minimum when $\\begin{array} { r } { \\frac { \\partial f ( \\mathbf { z } ) } { \\partial \\mathbf { z } } = 2 ( 1 - \\alpha ) \\gamma ( \\mathbf { L } ) \\mathbf { z } + 2 \\alpha \\left( \\mathbf { z } - \\mathbf { x } \\right) = \\mathbf { 0 } } \\end{array}$ . Accordingly, the optimum $z ^ { * }$ can be derived as ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24f28c6a27df4ff8cc35201b1407d7e14288356e7100dfbc628bca44193ac547.jpg",
+        "text": "$$\n\\alpha \\left( \\alpha \\mathbf { I } + ( 1 - \\alpha ) \\gamma ( \\mathbf { L } ) \\right) ^ { - 1 } \\mathbf { x } = \\mathbf { U } d i a g \\left[ \\frac { \\alpha } { \\alpha + ( 1 - \\alpha ) \\gamma ( \\lambda _ { 1 } ) } , . . . , \\frac { \\alpha } { \\alpha + ( 1 - \\alpha ) \\gamma ( \\lambda _ { n } ) } \\right] \\mathbf { U } ^ { T } \\mathbf { x } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            222,
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+            256
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Let h(λ) = αα+(1−α)γ(λ) denote the exact spectral filter, and $\\begin{array} { r } { g ( \\lambda ) = \\sum _ { k = 0 } ^ { K } w _ { k } \\lambda ^ { k } } \\end{array}$ denote a polynomial approximation of $h ( \\lambda )$ (e.g. the suffix sum of $h ( \\lambda )$ ’s taylor expansion). Since $\\gamma ( \\lambda ) \\geq 0$ when the polynomial filter $\\lambda \\in [ 0 , 2 ]$ , we have $\\begin{array} { r } { 0 \\leq \\dot { h } ( \\lambda ) \\leq \\frac { \\alpha } { \\alpha + ( 1 - \\alpha ) \\cdot 0 } = 1 } \\end{array}$ $\\begin{array} { r } { g ( \\lambda ) = \\sum _ { k = 0 } ^ { K } w _ { k } \\lambda ^ { k } } \\end{array}$ 0  also satisfies for $\\lambda \\in [ 0 , 2 ]$ $0 \\leq g ( \\lambda ) \\leq 1$ . Consequently, it is natural to assume . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Constraint 3.1. Assuming the energy function $\\gamma ( \\mathbf { L } )$ is positive semidefinite, a polynomial filter $\\begin{array} { r } { g ( \\lambda ) = \\sum _ { k = 0 } ^ { K } w _ { k } \\lambda ^ { k } } \\end{array}$ approximating the optimal solution to Equation (4) has to satisfy ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d51bb84d20535b7727d17b8d0c550f61bfe18a153b2b35d45b9e26d47caad0ef.jpg",
+        "text": "$$\n0 \\leq g ( \\lambda ) = \\sum _ { k = 0 } ^ { K } w _ { k } \\lambda ^ { k } \\leq 1 , \\forall \\lambda \\in [ 0 , 2 ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While Constraint 3.1 seems to be simple and intuitive, some of the existing GNN may not satisfies this constraint. For example, GCN uses $\\mathbf { \\bar { z } } = \\mathbf { P x } = \\left( \\mathbf { I } - \\mathbf { L } \\right) \\mathbf { x }$ , which corresponds to a polynomial filter $g ( \\lambda ) = 1 - \\lambda$ that takes negative value when $\\lambda > 1$ , violating Constraint 3.1. As shown in [31], the renormalization trick $\\tilde { \\mathbf { P } } = \\left( \\mathbf { I } + \\mathbf { D } \\right) ^ { - 1 / 2 } \\left( \\mathbf { I } + \\mathbf { A } \\right) \\left( \\mathbf { I } + \\mathbf { D } \\right) ^ { - 1 / 2 }$ shrinks the spectral and thus reliefs the problem. However, $g ( \\lambda )$ may still take negative value as the maximum eigenvalue of $\\tilde { \\mathbf { L } } = \\mathbf { I } - \\tilde { \\mathbf { P } }$ is still larger than 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 Non-negative polynomials and Bernstein basis ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Constraint 3.1 motivates us to design polynomial filters g(λ) = PKk=0 wkλk such that 0 ≤ g(λ) ≤ 1 when . The part is trivial, as we can always rescale each by a factor of $\\scriptstyle \\sum _ { k = 0 } ^ { K } | w _ { k } | 2 ^ { k }$ . The $g ( \\lambda ) \\geq 0$ part, however, requires more elaboration. Note that we can not simply set $w _ { k } \\geq 0$ for each $k = 0 \\ldots , K$ , since it is shown in [5] that such polynomials only correspond to low-pass filters. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As it turns out, the Bernstein basis has the following nice property: a polynomial that is non-negative on a certain interval can always be expressed as a non-negative linear combination of Bernstein basis. Specifically, we have the following lemma. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.1 ([23]). Assume a polynomial $\\textstyle p ( x ) = \\sum _ { k = 0 } ^ { K } \\theta _ { k } x ^ { k }$ satisfies $p ( x ) \\geq 0$ for $x \\in [ 0 , 1 ]$ . Then there exists a sequence of non-negative coefficients $\\theta _ { k }$ , ${ \\bf \\ddot { \\boldsymbol { k } } } = 0 , \\dots , K$ , such that ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/4d70723a86e3d339b5e2fdb626e31bd8a97cd63885ffc673851aedeaeeeb5383.jpg",
+        "text": "$$\np ( x ) = \\sum _ { k = 0 } ^ { K } \\theta _ { k } b _ { k } ^ { K } ( x ) = \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\binom { K } { k } ( 1 - x ) ^ { K - k } x ^ { k }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lemma 3.1 suggests that to approximate a valid filter, the polynomial filter $g ( \\lambda )$ has to be a nonnegative linear combination of Bernstein basis. Specifically, by setting $x = \\lambda / 2$ , the filter $g ( \\lambda )$ that satisfies $g ( \\lambda ) \\geq 0$ for $\\lambda \\in [ 0 , 2 ]$ can be expressed as ",
+        "bbox": [
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+            744,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58c0408e57fa0739cc2955cd29474512c1d5bf68fe4e57a05facb7cf5597abee.jpg",
+        "text": "$$\ng ( \\lambda ) : = p \\left( \\frac { \\lambda } { 2 } \\right) = \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\frac { 1 } { 2 ^ { K } } \\binom { K } { k } ( 2 - \\lambda ) ^ { K - k } \\lambda ^ { k } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            790,
+            668,
+            834
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Consequently, any valid polynomial filter that approximate the optimal solution of (4) with positive semidefinite energy function $\\gamma ( \\mathbf { L } )$ has to take the following form: $\\begin{array} { r } { \\mathbf { z } = \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\frac { 1 } { 2 ^ { K } } \\binom { \\bar { K } } { k } ( 2 \\mathbf { I } - } \\end{array}$ $\\mathbf { L } ) ^ { K - k } \\mathbf { L } ^ { k } \\mathbf { x }$ . This observation motivates our BernNet from the perspective of graph optimization — any valid polynomial filers, i.e., the $g : [ 0 , 2 ] \\mapsto [ 0 , 1 ]$ , can always be expressed by BernNet, and accordingly, the filters learned by our BernNet are always valid. ",
+        "bbox": [
+            173,
+            835,
+            826,
+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/a04bbe15dede6bfda46a24f4347c73fa94999d261b7cfd0556e6ed2a7fcce68c.jpg",
+        "image_caption": [
+            "Figure 3: A input image and the filtering results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            94,
+            810,
+            193
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/e6b3f5dec4812af694d4e40684edb2c340965c0c8c071bcfc52e4b4c749872e4.jpg",
+        "table_caption": [
+            "Table 2: Average sum of squared error and $R ^ { 2 }$ score in parentheses. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Low-pass</td><td>High-pass</td><td>Band-pass</td><td>Band-rejection</td><td>Comb</td></tr><tr><td></td><td>exp(-10x2)</td><td>1-exp(-10x²)</td><td>exp(-10(λ -1)2)</td><td>1 -exp(-10(λ-1)²)</td><td>|sin(πλ)l</td></tr><tr><td>GCN</td><td>3.4799(.9872)</td><td>67.6635(.2364)</td><td>25.8755(.1148)</td><td>21.0747(.9438)</td><td>50.5120(.2977)</td></tr><tr><td>GAT</td><td>2.3574(.9905)</td><td>21.9618(.7529)</td><td>14.4326(.4823)</td><td>12.6384(.9652)</td><td>23.1813(.6957)</td></tr><tr><td>GPR-GNN</td><td>0.4169(.9984)</td><td>0.0943(.9986)</td><td>3.5121(.8551)</td><td>3.7917(.9905)</td><td>4.6549(.9311)</td></tr><tr><td>ARMA</td><td>1.8478(.9932)</td><td>1.8632(.9793)</td><td>7.6922(.7098)</td><td>8.2732(.9782)</td><td>15.1214(.7975)</td></tr><tr><td>ChebNet</td><td>0.8220(.9973)</td><td>0.7867(.9903)</td><td>2.2722(.9104)</td><td>2.5296(.9934)</td><td>4.0735(.9447)</td></tr><tr><td>BernNet</td><td>0.0314(.9999)</td><td>0.0113(.9999)</td><td>0.0411(.9984)</td><td>0.9313(.9973)</td><td>0.9982(.9868)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks (GNNs) can be broadly divided into spectral-based GNNs and spatial-based GNNs [33]. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Spectral-based GNNs design spectral graph filters in the spectral domain. ChebNet [7] uses Chebyshev polynomial to approximate a filter. GCN [13] simplifies the Chebyshev filter with the first-order approximation. GraphHeat [34] uses heat kernel to design a graph filter. APPNP [14] utilizes Personalized PageRank (PPR) to set the filter weights. GPR-GNN [5] learns the polynomial filters via gradient descent on the polynomial coefficients. ARMA [2] learns a rational filter via the family of Auto-Regressive Moving Average filters [21]. AdaGNN [9] learns simple filters across multiple layers with a single parameter for each feature channel at each layer. As aforementioned, these methods mainly focus on designing low- or high-pass filters or learning filters without any constraints, which may lead to misspecified even ill-posed filters. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, spatial-based GNNs directly propagate and aggregate graph information in the spatial domain. From this perspective, GCN [13] can be explained as the aggregation of the one-hop neighbor information on the graph. GAT [30] uses the attention mechanism to learn aggregation weights. Recently, Balcilar et al. [1] bridge the gap between spectral-based and spatial-based GNNs and unify GNNs in the same framework. Their work shows that the GNNs can be interpreted as sophisticated data-driven filters. This motivates the design of the proposed BernNet, which can learn arbitrary non-negative spectral filters from real-world graph signals. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we conduct experiments to evaluate BernNet’s capability to learn arbitrary filters as well as the performance of BernNet on real datasets. All the experiments are conducted on a machine with an NVIDIA TITAN V GPU (12GB memory), Intel Xeon CPU $( 2 . 2 0 \\mathrm { G H z } )$ , and 512GB of RAM. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.1 Learning filters from the signal ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            814,
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+            829
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conduct an empirical analysis on 50 real images with the resolution of $1 0 0 \\times 1 0 0$ from the Image Processing Toolbox in Matlab. We conduct independent experiments on these 50 images and report the average of the evaluation index. Following the experimental setting in [1], we regard each image as a 2D regular 4-neighborhood grid graph. The graph structure translates to an 1 $) , 0 0 0 \\times 1 0 , 0 0 0$ adjacency matrix while the pixel intensity translates to a 10, 000-dimensional signal vector. ",
+        "bbox": [
+            174,
+            842,
+            825,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/ce85e616d16afd0ab08a3ea37e2f184d604adbfffa7c10ba9865ece27cc01890.jpg",
+        "table_caption": [
+            "Table 3: Dataset statistics. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Cora</td><td>CiteSeer</td><td>PubMed</td><td>Computers</td><td>Photo</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Texas</td><td>Cornell</td></tr><tr><td>Nodes</td><td>2708</td><td>3327</td><td>19717</td><td>13752</td><td>7650</td><td>2277</td><td>5201</td><td>7600</td><td>183</td><td>183</td></tr><tr><td>Edges</td><td>5278</td><td>4552</td><td>44324</td><td>245861</td><td>119081</td><td>31371</td><td>198353</td><td>26659</td><td>279</td><td>277</td></tr><tr><td>Features</td><td>1433</td><td>3703</td><td>500</td><td>767</td><td>745</td><td>2325</td><td>2089</td><td>932</td><td>1703</td><td>1703</td></tr><tr><td>Classes</td><td>7</td><td>6</td><td>5</td><td>10</td><td>8</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For each of the 50 images, we apply 5 different filters (low-pass, high-pass, band-pass, band-rejection and comb) to the spectral domain of its signal. The formula of each filter is shown in Table 2. Recall that applying a low-pass filter $\\exp ( - 1 0 \\lambda ^ { 2 } )$ to the spectral domain $\\mathbf { L } = \\mathbf { U } d i a g \\left[ \\lambda _ { 1 } , \\ldots , \\lambda _ { n } \\right] \\mathbf { U } ^ { \\top }$ means applying Udiag $\\left[ \\exp ( - 1 0 \\lambda _ { 1 } ^ { 2 } ) , \\dots , \\exp ( - 1 0 \\lambda _ { n } ^ { 2 } ) \\right] \\mathbf { U } ^ { \\top }$ to the graph signal. Figure 3 shows the one of the input image and the corresponding filtering results. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this task, we use the original graph signal as the input and the filtering signal to supervise the training process. The goal is to minimize the square error between output and the filtering signal by learning the correct filter. We evaluate BernNet against five popular GNN models: GCN [13], GAT [30], GPR-GNN [5], ARMA [2] and ChebNet [7]. To ensure fairness, we use two convolutional units and a linear output layer for all models. We train all models with approximately $2 \\mathrm { k }$ trainable parameters and tune the hidden units to ensure they have similar parameters. Following [1], we discard any regularization or dropout and simply force the GNN to learn the input-output relation. For all models, we set the maximum number of epochs to 2000 and stop the training if the loss does not drop for 100 consecutive times and use Adam optimization with a 0.01 learning rate without decay. Models do not use the position information of the picture pixels. We use a mask to cover the edge nodes of the picture, so the problem can be regarded as a simple regression problem. For BernNet, we use a two-layer model, with each layer sharing the same set of $\\theta _ { k }$ for $k = 0 , \\ldots , K$ and set $K = 1 0$ . For GPR-GNN, we use the officially released code (see the supplementary materials for URL and commit numbers) and set the order of polynomial filter $K = 1 0$ . Other baseline models are based on Pytorch Geometric implementation [11]. The more detailed experiments setting can be found in the Appendix. ",
+        "bbox": [
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+            289,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows the average of the sum of squared error (lower the better) and the $R ^ { 2 }$ scores (higher the better). We first observe that GCN and GAT can only handle low-pass filters, which concurs with the theoretical analysis in [1]. GPR-GNN, ARMA and ChebNet can learn different filters from the signals. However, BernNet consistently outperformed these models by a large margin on all tasks in terms of both metrics. We attribute this quality to BernNet’s ability to tune the coefficients $\\theta _ { k }$ ’s, which directly correspond to the uniformly sampled filter values. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Node classification on real-world datasets ",
+        "text_level": 1,
+        "bbox": [
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+            623,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We now evaluate the performance of BernNet against the competitors on real-world datasets. Following [5], we include three citation graph Cora, CiteSeer and PubMed [27, 37], and the Amazon co-purchase graph Computers and Photo [20]. As shown in [5] these 5 datasets are homophilic graphs on which the connected nodes tend to share the same label. We also include the Wikipedia graph Chameleon and Squirrel [26], the Actor co-occurrence graph, and webpage graphs Texas and Cornell from WebKB‡ [22]. These 5 datasets are heterophilic datasets on which connected nodes tend to have different labels. We summarize the statistics of these datasets in Table 3. ",
+        "bbox": [
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+            651,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Following [5], we perform full-supervised node classification task with each model, where we randomly split the node set into train/validation/test set with ratio $6 0 \\% / 2 0 \\% / 2 0 \\%$ . For fairness, we generate 10 random splits by random seeds and evaluate all models on the same splits, and report the average metric for each model. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compare BernNet with 6 baseline models: MLP, GCN [13], GAT [30], APPNP [14], ChebNet [7], and GPR-GNN [5]. For GPR-GNN, we use the officially released code (see the supplementary materials for URL and commit numbers) and set the order of polynomial filter $K = 1 0$ . For other models, we use the corresponding Pytorch Geometric library implementations [11]. For BernNet, we ",
+        "bbox": [
+            174,
+            818,
+            825,
+            873
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/765778b1d4ff270459c69b13aef9961eb2a095d359a571028f9ab9fa76f0416a.jpg",
+        "image_caption": [
+            "Figure 4: Filters learnt from real-world datasets by BernNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            94,
+            813,
+            203
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/63e83abd549a8e0b4e6706eb749430e54629fffaf1ae38ab3e1947f15f70b7fe.jpg",
+        "table_caption": [
+            "Table 4: Results on real world benchmark datasets: Mean accuracy $( \\% ) \\pm 9 5 \\%$ confidence interval. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GCN</td><td>GAT</td><td>APPNP</td><td>MLP</td><td>ChebNet</td><td>GPR-GNN</td><td>BernNet</td></tr><tr><td>Cora</td><td>87.14±1.01</td><td>88.03±0.79</td><td>88.14±0.73</td><td>76.96±0.95</td><td>86.67±0.82</td><td>88.57±0.69</td><td>88.52±0.95</td></tr><tr><td>CiteSeer</td><td>79.86±0.67</td><td>80.52±0.71</td><td>80.47±0.74</td><td>76.58±0.88</td><td>79.11±0.75</td><td>80.12±0.83</td><td>80.09±0.79</td></tr><tr><td>PubMed</td><td>86.74±0.27</td><td>87.04±0.24</td><td>88.12±0.31</td><td>85.94±0.22</td><td>87.95±0.28</td><td>88.46±0.33</td><td>88.48±0.41</td></tr><tr><td>Computers</td><td>83.32±0.33</td><td>83.32±0.39</td><td>85.32±0.37</td><td>82.85±0.38</td><td>87.54±0.43</td><td>86.85±0.25</td><td>87.64±0.44</td></tr><tr><td>Photo</td><td>88.26±0.73</td><td>90.94±0.68</td><td>88.51±0.31</td><td>84.72±0.34</td><td>93.77±0.32</td><td>93.85±0.28</td><td>93.63±0.35</td></tr><tr><td>Chameleon</td><td>59.61±2.21</td><td>63.13±1.93</td><td>51.84±1.82</td><td>46.85±1.51</td><td>59.28±1.25</td><td>67.28±1.09</td><td>68.29±1.58</td></tr><tr><td>Actor</td><td>33.23±1.16</td><td>33.93±2.47</td><td>39.66±0.55</td><td>40.19±0.56</td><td>37.61±0.89</td><td>39.92±0.67</td><td>41.79±101</td></tr><tr><td>Squirrel</td><td>46.78±0.87</td><td>44.49±0.88</td><td>34.71±0.57</td><td>31.03±1.18</td><td>40.55±0.42</td><td>50.15±1.92</td><td>51.35±0.73</td></tr><tr><td>Texas</td><td>77.38±3.28</td><td>80.82±2.13</td><td>90.98±1.64</td><td>91.45±1.14</td><td>86.22±2.45</td><td>92.95±1.31</td><td>93.12±0.65</td></tr><tr><td>Cornell</td><td>65.90±4.43</td><td>78.21±2.95</td><td>91.81±1.96</td><td>90.82±1.63</td><td>83.93±2.13</td><td>91.37±1.81</td><td>92.13±1.64</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "use the following propagation process: ",
+        "bbox": [
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+            435,
+            428,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bc3e86ae34d642196fbdc687dfd60857e436eb1f46d12dd1e07dd14d0ddfafb9.jpg",
+        "text": "$$\n\\mathbf { Z } = \\sum _ { k = 0 } ^ { K } \\theta _ { k } \\frac { 1 } { 2 ^ { K } } \\binom { K } { k } ( 2 \\mathbf { I } - \\mathbf { L } ) ^ { K - k } \\mathbf { L } ^ { k } f \\left( \\mathbf { X } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            454,
+            647,
+            498
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\mathbf { X } )$ is a 2-layer MLP with 64 hidden units on the feature matrix $\\mathbf { X }$ . Note that this propagation process is almost identical to that of APPNP or GPR-GNN. The only difference is that we substitute the Generalized PageRank polynomial with Bernstein polynomial. We set the $K = 1 0$ and use different learning rate and dropout for the linear layer and the propagation layer. For all models, we optimal leaning rate over $\\{ 0 . 0 \\bar { 0 } 1 , 0 . 0 0 2 , 0 . 0 1 , 0 . 0 5 \\}$ and weight decay $\\{ 0 . 0 , \\dot { 0 } . 0 0 0 5 \\}$ . More detailed experimental settings are discussed in Appendix. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We use the micro-F1 score with a $9 5 \\%$ confidence interval as the evaluation metric. The relevant results are summarized in Table 4. Boldface letters indicate the best result for the given confidence interval. We observe that BernNet provides the best results on seven out of the ten datasets. On the other three datasets, BernNet also achieves competitive results against SOTA methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "More interestingly, this experiment also shows BernNet can learn complex filters from real-world datasets with only the supervision of node labels. Figure 4 plots some of the filters BernNet learnt in the training process. On Actor, BernNet learns an all-pass-alike filter, which concurs with the fact that MLP outperforms all other baselines on this dataset. On Chameleon and Squirrel, BernNet learns two comb-alike filters. Given that BernNet outperforms all competitors by at least $1 \\%$ on these two datasets, it may suggest that comb-alike filters are necessary for Chameleon and Squirrel. Figure 5 shows the Coefficients $\\theta _ { k }$ learnt from real-world datasets by BernNet. When comparing Figures 4 and 5, we observe that the curves of filters and curves of coefficients are almost the same. This is because BernNet’s coefficients are highly correlated with the spectral property of the target filter, which indicates BernNet Bernnet has strong interpretability. ",
+        "bbox": [
+            173,
+            655,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, we present the train time for each method in Table 5. BernNet is slower than other methods due to its quadratic dependence on the degree $K$ . However, compared to the SOTA method GPRGNN, the margin is generally less than 2, which is often acceptable in practice. In theory, both ChebNet [7] and GPR-GNN [5] are linear time complexity related to propagation step $K$ , but BernNet is quadratic time complexity related to $K$ . Delgado et al. [8] show that Bernstein approximation can be evaluated in linear time related to $K$ using the corner cutting algorithm. However, BernNet can not use this algorithm directly, because we need to multiply signal $\\mathbf { x }$ . How to convert BernNet to linear complexity will be a problem worth studying in the future. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/b0dcb357edbc74b9b1067c37e462472b96686afdfe6fe7fdb0d4e52b97d5bd83.jpg",
+        "image_caption": [
+            "Figure 5: Coefficients $\\theta _ { k }$ learnt from real-world datasets by BernNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            94,
+            813,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/a0ba4c43ac3275e9c568953872efbc59a96ece653711eb4b73c5eefb4ab518f1.jpg",
+        "table_caption": [
+            "Table 5: Average running time per epoch (ms)/average total running time (s). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GCN</td><td>GAT</td><td>APPNP</td><td>MLP</td><td>ChebNet</td><td>GPR-GNN</td><td>BernNet</td></tr><tr><td>Cora</td><td>4.59/1.62</td><td>9.56/2.03</td><td>7.16/2.32</td><td>3.06/0.93</td><td>6.25/1.76</td><td>9.94/2.21</td><td>19.71/5.47</td></tr><tr><td>CiteSeer</td><td>4.63/1.95</td><td>9.93/2.21</td><td>7.79/2.77</td><td>2.95/1.09</td><td>8.28/2.56</td><td>11.16/2.37</td><td>22.36/6.32</td></tr><tr><td>PubMed</td><td>5.12/1.87</td><td>16.16/3.41</td><td>8.21/2.63</td><td>2.91/1.61</td><td>18.04/3.03</td><td>10.45/2.81</td><td>22.02/8.19</td></tr><tr><td>Computers</td><td>5.72/2.52</td><td>30.91/7.85</td><td>9.19/3.48</td><td>3.47/1.31</td><td>20.64/9.64</td><td>16.05/4.38</td><td>28.83/8.69</td></tr><tr><td>Photo</td><td>5.08/2.63</td><td>19.97/5.41</td><td>8.69/4.18</td><td>3.67/1.66</td><td>13.25/7.02</td><td>13.96/3.94</td><td>24.69/7.37</td></tr><tr><td>Chameleon</td><td>4.93/0.99</td><td>13.11/2.66</td><td>7.93/1.62</td><td>3.14/0.63</td><td>10.92/2.25</td><td>10.93/2.41</td><td>22.54/4.75</td></tr><tr><td>Actor</td><td>5.43/1.09</td><td>11.94/2.45</td><td>8.46/1.71</td><td>3.82/0.77</td><td>7.99/1.62</td><td>11.57/2.35</td><td>23.34/5.81</td></tr><tr><td>Squirrel</td><td>5.61/1.13</td><td>22.76/4.91</td><td>8.01/1.61</td><td>3.41/0.69</td><td>38.12/7.78</td><td>9.87/5.56</td><td>25.58/9.23</td></tr><tr><td>Texas</td><td>4.58/0.92</td><td>9.65/1.96</td><td>7.83/1.63</td><td>3.19/0.65</td><td>6.51/1.34</td><td>10.45/2.16</td><td>23.35/4.81</td></tr><tr><td>Cornell</td><td>4.83/0.97</td><td>9.79/1.99</td><td>8.23/1.68</td><td>3.25/0.66</td><td>5.85/1.22</td><td>9.86/2.05</td><td>22.23/5.26</td></tr></table>",
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+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+        "text": "This paper proposes BernNet, a graph neural network that provides a simple and intuitive mechanism for designing and learning an arbitrary spectral filter via Bernstein polynomial approximation. Compared to previous methods, BernNet can approximate complex filters such as band-rejection and comb filters, and can provide better interpretability. Furthermore, the polynomial filters designed and learned by BernNet are always valid. Experiments show that BernNet outperforms SOTA methods in terms of effectiveness on both synthetic and real-world datasets. For future work, an interesting direction is to improve the efficiency of BernNet. ",
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+    {
+        "type": "text",
+        "text": "Broader Impact ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The proposed BernNet algorithm addresses the challenge of designing and learning arbitrary spectral filters on graphs. We consider this algorithm a general technical and theoretical contribution, without any foreseeable specific impacts. For applications in bioinformatics, computer vision, and natural language processing, applying the BernNet algorithm may improve the performance of existing GNN models. We leave the exploration of other potential impacts to future work. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "text": "Zhewei Wei was supported in part by National Natural Science Foundation of China (No. 61972401, No. 61932001 and No. 61832017), by Beijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098, by Alibaba Group through Alibaba Innovative Research Program, and by CCF-Baidu Open the Fund (NO.2021PP15002000). Zengfeng Huang was supported by National Natural Science Foundation of China Grant No. 61802069, and by Shanghai Science and Technology Commission Grant No. 17JC1420200. Hongteng Xu was supported by Tencent AI Lab Rhino-Bird Joint Research Program. This work is supported by China Unicom Innovation Ecological Cooperation Plan and by Intelligent Social Governance Platform, Major Innovation $\\&$ Planning Interdisciplinary Platform for the “Double-First Class” Initiative, Renmin University of China. We also wish to acknowledge the support provided and contribution made by Public Policy and Decision-making Research Lab of Renmin University of China. ",
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+        "text": "References   \n[1] Muhammet Balcilar, Pierre Héroux, Benoit Gaüzère, Sébastien Adam, and Paul Honeine. Analyzing the expressive power of graph neural networks in a spectral perspective. In ICLR, 2021.   \n[2] Filippo Maria Bianchi, Daniele Grattarola, Lorenzo Livi, and Cesare Alippi. Graph neural networks with convolutional arma filters. TPAMI, 2021.   \n[3] Toon Bogaerts, Antonio D Masegosa, Juan S Angarita-Zapata, Enrique Onieva, and Peter Hellinckx. A graph cnn-lstm neural network for short and long-term traffic forecasting based on trajectory data. Transportation Research Part C: Emerging Technologies, 2020.   \n[4] Zhao-Min Chen, Xiu-Shen Wei, Peng Wang, and Yanwen Guo. Multi-label image recognition with graph convolutional networks. In CVPR, 2019.   \n[5] Eli Chien, Jianhao Peng, Pan Li, and Olgica Milenkovic. Adaptive universal generalized pagerank graph neural network. In ICLR, 2021.   \n[6] Zhiyong Cui, Kristian Henrickson, Ruimin Ke, and Yinhai Wang. 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parse/train/rklklCVYvB/rklklCVYvB.md

@@ -0,0 +1,350 @@
+# TIME2VEC: LEARNING A VECTOR REPRESENTATION OF TIME
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Time is an important feature in many applications involving events that occur synchronously and/or asynchronously. To effectively consume time information, recent studies have focused on designing new architectures. In this paper, we take an orthogonal but complementary approach by providing a model-agnostic vector representation for time, called $T i m e 2 V e c$ , that can be easily imported into many existing and future architectures and improve their performances. We show on a range of models and problems that replacing the notion of time with its Time2Vec representation improves the performance of the final model.
+
+# 1 INTRODUCTION
+
+In building machine learning models, “time” is often an important feature. Examples include predicting daily sales for a company based on the date (and other available features), predicting the time for a patient’s next health event based on their medical history, and predicting the song a person is interested in listening to based on their listening history. The input for problems involving time can be considered as a sequence where, rather than being identically and independently distributed (iid), there exists a dependence across time (and/or space) among the data points. The sequence can be either synchronous, i.e. sampled at regular intervals, or asynchronous, i.e. sampled at different points in time. In both cases, time may be an important feature. For predicting daily sales, for instance, it may be useful to know if it is a holiday or not. For predicting the time for a patient’s next encounter, it is important to know the (asynchronous) times of their previous visits.
+
+Recurrent neural networks (RNNs) do not typically treat time itself as a feature, typically assuming that inputs are synchronous. When time is known to be a relevant feature, it is often fed in as yet another input dimension (Choi et al., 2016; Du et al., 2016; Li et al., 2018b). In practice, RNNs often fail at effectively making use of time as a feature. To help the RNN make better use of time, several researchers design hand-crafted features of time that suit their specific problem and feed those features into the RNN (Choi et al., 2016; Baytas et al., 2017; Kwon et al., 2019). Hand-crafting features, however, can be expensive and requires domain expertise about the problem.
+
+Many recent studies aim at obviating the need for hand-crafting features by proposing generalpurpose—as opposed to problem specific—architectures that better handle time (Neil et al., 2016; Zhu et al., 2017; Mei & Eisner, 2017; Hu & Qi, 2017; Upadhyay et al., 2018; Li et al., 2018a). We follow an orthogonal but complementary approach to these recent studies by developing a generalpurpose model-agnostic representation for time that can be potentially used in any architecture. In particular, we develop a learnable vector representation (or embedding) for time as a vector representation can be easily combined with many models or architectures. We call this vector representation Time2Vec. To validate the effectiveness of Time2Vec, we conduct experiments on several (synthesized and real-world) datasets and integrate it with several architectures. Our main result is to show that on a range of problems and architectures that consume time, using Time2Vec instead of the time itself offers a boost in performance.
+
+# 2 RELATED WORK
+
+There is a long history of algorithms for predictive modeling in time series analysis. They include auto-regressive techniques (Akaike, 1969) that predict future measurements in a sequence based on a window of past measurements. Since it is not always clear how long the window of past measurements should be, hidden Markov models (Rabiner & Juang, 1986), dynamic Bayesian networks (Murphy & Russell, 2002), and dynamic conditional random fields (Sutton et al., 2007) use hidden states as a finite memory that can remember information arbitrarily far in the past. These models can be seen as special cases of recurrent neural networks (Hochreiter & Schmidhuber, 1997). They typically assume that inputs are synchronous, i.e. arrive at regular time intervals, and that the underlying process is stationary with respect to time. It is possible to aggregate asynchronous events into time-bins and to use synchronous models over the bins (Lipton et al., 2016; Anumula et al., 2018). Asynchronous events can also be directly modeled with point processes (e.g., Poisson, Cox, and Hawkes point processes) (Daley & Vere-Jones, 2007; Laub et al., 2015; Xiao et al., 2017; Li et al., 2018a; Xiao et al., 2018) and continuous time normalizing flows (Chen et al., 2018). Alternatively, one can also interpolate or make predictions at arbitrary time stamps with Gaussian processes (Rasmussen, 2004) or support vector regression (Drucker et al., 1997).
+
+Our goal is not to propose a new model for time series analysis, but instead to propose a representation of time in the form of a vector embedding that can be used by many models. Vector embedding has been previously successfully used for other domains such as text (Mikolov et al., 2013; Pennington et al., 2014), (knowledge) graphs (Grover & Leskovec, 2016; Nickel et al., 2016; Kazemi & Poole, 2018), and positions (Vaswani et al., 2017; Gehring et al., 2017). Our approach is related to time decomposition techniques that encode a temporal signal into a set of frequencies (Cohen, 1995). However, instead of using a fixed set of frequencies as in Fourier transforms (Bracewell & Bracewell, 1986), we allow the frequencies to be learned. We take inspiration from the neural decomposition of Godfrey & Gashler (2018) (and similarly (Gashler & Ashmore, 2016)). For time-series analysis, Godfrey & Gashler (2018) decompose a 1D signal of time into several sine functions and a linear function to extrapolate (or interpolate) the given signal. We follow a similar intuition but instead of decomposing a 1D signal of time into its components, we transform the time itself and feed its transformation into the model that is to consume the time information. Our approach corresponds to the technique of Godfrey & Gashler (2018) when applied to regression tasks in 1D signals, but it is more general since we learn a representation that can be shared across many signals and can be fed to many models for tasks beyond regression.
+
+While there is a body of literature on designing neural networks with sine activations (Lapedes & Farber, 1987; Sopena et al., 1999; Wong et al., 2002; Mingo et al., 2004; Liu et al., 2016), our work uses sine only for transforming time; the rest of the network uses other activations. There is also a set of techniques that consider time as yet another feature and concatenate time (or some hand designed features of time such as log and/or inverse of delta time) with the input (Choi et al., 2016; Li et al., 2017; Du et al., 2016; Baytas et al., 2017; Kwon et al., 2019; Trivedi et al., 2017; Kumar et al., 2018; Ma et al., 2018). Kazemi et al. (2019) survey several such approaches for dynamic (knowledge) graphs. These models can directly benefit from our proposed vector embedding, Time2Vec, by concatenating Time2Vec with the input instead of their time features. Other works (Neil et al., 2016; Zhu et al., 2017; Mei & Eisner, 2017; Hu & Qi, 2017; Upadhyay et al., 2018; Li et al., 2018a) propose new neural architectures that take into account time (or some features of time). We show how Time2Vec can be used in one of these architectures to better exploit temporal information; it can be potentially used in other architectures as well.
+
+# 3 BACKGROUND & NOTATION
+
+We use lower-case letters to denote scalars, bold lower-case letters to denote vectors, and bold upper-case letters to denote matrices. We represent the $i ^ { t h }$ element of the vector $\pmb { r }$ as $\pmb { r } [ i ]$ . For two vectors $\pmb { r }$ and $\pmb { s }$ , we use $[ \pmb { r } ; \pmb { s } ]$ to represent their concatenation and $\pmb { r } \odot \pmb { s }$ to represent element-wise (Hadamard) multiplication of the two vectors. Throughout the paper, we use $\tau$ to represent a scalar notion of time (e.g., absolute time or time from the last event) and $\tau$ for a vector of time features.
+
+Long Short Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) is considered one of the most successful RNN architectures for sequence modeling. A formulation of the original LSTM model and a variant of it based on peepholes (Gers & Schmidhuber, 2000) is presented in Appendix C. When time is a relevant feature, the easiest way to handle time is to consider it as just another feature (or extract some engineered features from it), concatenate the time features with the input, and use the standard LSTM model (or some other sequence model) (Choi et al., 2016; Du et al., 2016; Li et al.,
+
+2018b). In this paper, we call this model $L S T M + T .$ Another way of handling time is by changing the formulation of the standard LSTM. Zhu et al. (2017) developed one such formulation, named TimeLSTM, by adding time gates to the architecture of the LSTM with peepholes. They proposed three architectures namely TLSTM1, TLSTM2, TLSTM3. A description of TLSTM1 and TLSTM3 can be found in Appendix C (we skipped TLSTM2 as it is quite similar to TLSTM3).
+
+# 4 TIME2VEC
+
+A common approach to deal with time in different applications is to apply some hand-crafted function(s) $f _ { 1 } , \ldots , f _ { m }$ to $\tau$ ， $\mathit { \Pi } _ { \tau }$ can be absolute time, time from last event, etc.), concatenate the outputs $f _ { 1 } ( \tau ) , \dots , f _ { m } ( \tau )$ with the rest of the input features $\pmb { x }$ , and feed the resulting vector $[ { \pmb x } ; f _ { 1 } ( \tau ) ; \dots ; f _ { m } ( \tau ) ]$ to a sequence model (see Section 2 for references). This approach requires hand-crafting useful functions of time which may be difficult (or impossible) in several applications, and the hand-crafted functions may not be optimal for the task at hand. Instead of hand-crafting functions of time, we devise a representation of time which can be used to approximate any function through learnable parameters. Such a representation offers two advantages: 1- it obviates the need for hand-crafting functions of time and 2- it provides the grounds for learning suitable functions of time based on the data. As vector representations can be efficiently integrated with the current deep learning architectures, we employ a vector representation for time.
+
+Our proposed representation leverages the Fourier sine series (Arfken & Weber, 1999) according to which any 1D function can be approximated in a given interval using a weighted sum of sinusoids with appropriate frequencies (and phase-shifts). We include $k$ sinusoids of the form $s i n ( \omega _ { i } \tau + \varphi _ { i } )$ in our vector representation where $\omega _ { i }$ and $\varphi _ { i }$ are learnable parameters1. That is, we concatenate the input features $\pmb { x }$ with $k$ sinusoids and feed the concatenation $[ \pmb { x } ; s i n ( \omega _ { 1 } \tau + \varphi _ { 1 } ) ; \dots ; s i n ( \omega _ { k } \tau + \varphi _ { k } ) ]$ into a sequence model. Different functions of time can be created using these sinusoids by taking a weighted sum of them with different weights. We allow the weights of the sequence model to combine the sinusoids and create functions of time suitable for the task. If we expand the output $\begin{array} { r } { \pmb { a } ( \tau , k ) [ j ] = \gamma _ { j } + \sum _ { i = 1 } ^ { k } \theta _ { j , i } \sin \left( \omega _ { i } \tau + \varphi _ { i } \right) } \end{array}$ fore, wheures p $\theta _ { j , i ^ { \mathrm { S } } }$ g an activation function), it  are the first layer weights and  on the temporal features). Ea $\gamma _ { j }$ $\pmb { x }$ $\mathbf { \pmb { a } } ( \tau , k ) [ j ]$ operates on the input features $\pmb { x }$ as well as a learned function $\begin{array} { r } { f _ { j } ( \tau ) = \sum _ { i = 1 } ^ { k } \theta _ { j , i } \sin { \left( \omega _ { i } \tau + \varphi _ { i } \right) } } \end{array}$ of time, as opposed to a hand-crafted function2. Following Godfrey & Gashler (2018), to facilitate approximating functions with non-periodic patterns and help with generalization, we also include a linear projection of time in our vector representation. We name our vector representation of time Time2Vec. Time2Vec of $\tau$ , denoted as $\mathbf { t } 2 \mathbf { v } ( \tau )$ , is a vector of size $k + 1$ defined as follows:
+
+$$
+\mathbf t 2 \mathbf v ( \tau ) [ i ] = { \left\{ \begin{array} { l l } { \omega _ { i } \tau + \varphi _ { i } , } & { { \mathrm { i f } } i = 0 . } \\ { \sin { ( \omega _ { i } \tau + \varphi _ { i } ) } , } & { { \mathrm { i f } } 1 \leq i \leq k . } \end{array} \right. }
+$$
+
+where $\mathbf { t } 2 \mathbf { v } ( \tau ) [ i ]$ is the $i ^ { t h }$ element of $\mathbf { t } 2 \mathbf { v } ( \tau )$ and $\omega _ { i } \mathbf { s }$ and $\varphi _ { i } \mathbf { s }$ are learnable parameters.
+
+The use of sine functions is inspired in part by Vaswani et al. (2017)’s positional encoding. Consider a sequence of items (e.g., a sequence of words) $\{ I _ { 1 } , I _ { 2 } , \ldots , I _ { N } \}$ and a vector representation ${ \pmb v } _ { I _ { j } } \in \mathbb { R } ^ { d }$ for the $j ^ { t h }$ item $I _ { j }$ in the sequence. Vaswani et al. (2017) added $\sin { ( j / 1 0 0 0 0 ^ { k / d } ) }$ to ${ \pmb v } _ { I _ { j } } [ k ]$ if $k$ is even and $\sin { ( j / 1 0 0 0 0 ^ { k / d } + \pi / 2 ) }$ if $k$ is odd so that the resulting vector includes information about the position of the item in the sequence. These sine functions are called the positional encoding. Intuitively, positions can be considered as the times and the items can be considered as the events happening at that time. Thus, Time2Vec can be considered as representing continuous time, instead of discrete positions, using sine functions. The sine functions in Time2Vec also enable capturing periodic behaviors which is not a goal in positional encoding. We feed Time2Vec as an input to the model (or to some gate in the model) instead of adding it to other vector representations. Unlike positional encoding, we show in our experiments that learning the frequencies and phase-shifts of sine functions in Time2Vec result in better performance compared to fixing them.
+
+# 4.1 PROPERTIES OF TIME2VEC
+
+We review some of the interesting and desired properties of Time2Vec.
+
+Periodicity: In many scenarios, some events occur periodically. The amount of sales of a store, for instance, may be higher on weekends or holidays. Weather condition usually follows a periodic pattern over different seasons (Gashler & Ashmore, 2016). Some other events may be non-periodic but only happen after a point in time and/or become more probable as time proceeds. For instance, some diseases are more likely for older ages.
+
+The period of $\sin \left( \omega _ { i } \tau + \varphi _ { i } \right)$ is $\frac { 2 \pi } { \omega _ { i } }$ , i.e. it has the same value for $\tau$ and $\textstyle { \tau + { \frac { 2 \pi } { \omega _ { i } } } }$ . Therefore, the sine functions in Time2Vec help capture periodic behaviors without the need for feature engineering. For instance, a sine function $\sin { \left( \omega \tau + \varphi \right) }$ with $\begin{array} { r } { \omega = \frac { 2 \pi } { 7 } } \end{array}$ repeats every 7 days (assuming $\tau$ indicates days) and can be potentially used to model weekly patterns. Furthermore, unlike other basis functions which may show strange behaviors for extrapolation (see, e.g., (Poole et al., 2014)), sine functions are expected to work well for extrapolating to future and out of sample data (Vaswani et al., 2017). The linear term represents the progression of time and can be used for capturing non-periodic patterns in the input that depend on time.
+
+Invariance to Time Rescaling: Since time can be measured in different scales (e.g., days, hours, seconds, etc.), another important property of a representation for time is invariance to time rescaling (see, e.g., (Tallec & Ollivier, 2018)). A class $\mathcal { C }$ of models is invariant to time rescaling if for any model $\mathcal { M } _ { 1 } \in \mathcal { C }$ and any scalar $\alpha > 0$ , there exists a model $\mathcal { M } _ { 2 } \in \mathcal { C }$ that behaves on $\alpha \tau$ ( $\mathit { \Pi } _ { \tau }$ scaled by $\alpha$ ) in the same way $\mathcal { M } _ { 1 }$ behaves on original $\tau \mathrm { s }$ . Proposition 1 establishes the invariance of Time2Vec to time rescaling. The proof is in Appendix D.
+
+Proposition 1. Time2Vec is invariant to time rescaling.
+
+Simplicity: A representation for time should be easily consumable by different models and architectures. A matrix representation, for instance, may be difficult to consume as it cannot be easily appended with the other inputs. By selecting a vector representation for time, we ensure easy integration with deep learning architectures.
+
+# 5 EXPERIMENTS & RESULTS
+
+We use the following datasets:
+
+1) Synthesized data: We create a toy dataset to use for explanatory experiments. The inputs in this dataset are the integers between 1 and 365. Input integers that are multiples of 7 belong to class one and the other integers belong to class two. The first $7 5 \%$ is used for training and the last $2 5 \%$ for testing. This dataset is inspired by the periodic patterns (e.g., weekly or monthly) that often exist in daily-collected data; the input integers can be considered as the days.
+
+2) Event-MNIST: Sequential (event-based) MNIST is a common benchmark in sequence modeling literature (see, e.g., (Bellec et al., 2018; Campos et al., 2018; Fatahi et al., 2016)). We create a sequential event-based version of MNIST by flattening the images and recording the position of the pixels whose intensities are larger than a threshold (0.9 in our experiment). Following this transformation, each image will be represented as an array of increasing numbers such as $[ t _ { 1 } , t _ { 2 } , t _ { 3 } , \ldots , t _ { m } ]$ . We consider these values as the event times and use them to classify the images. As in other sequence modeling works, our aim in building this dataset is not to beat the state-of-the-art on the MNIST dataset; our aim is to provide a dataset where the only input is time and different representations for time can be compared when extraneous variables (confounders) are eliminated as much as possible.
+
+3) N TIDIGITS18 (Anumula et al., 2018): The dataset includes audio spikes of the TIDIGITS spoken digit dataset (Leonard & Doddington, 1993) recorded by the binaural 64-channel silicon cochlea sensor. Each sample is a sequence of $( t , c )$ tuples where $t$ represents time and $c$ denotes the index of active frequency channel at time $t$ . The labels are sequences of 1 to 7 connected digits with a vocabulary consisting of 11 digits (i.e. “zero” to “nine” plus “oh”) and the goal is to classify the spoken digit based on the given sequence of active channels. We use the reduced version of the dataset where only the single digit samples are used for training and testing. The reduced dataset has a total of 2,464 training and 2,486 test samples.
+
+![](images/d284eb34fa7cec683a41bc11dcf77dc40c6bbb62c4cfea0ed98d33c7730f82d8.jpg)  
+Figure 1: Comparing $\mathrm { L S T M + T }$ and LSTM+Time2Vec on several datasets.
+
+4) Stack Overflow (SOF): This dataset contains sequences of badges obtained by stack overflow users and the timestamps at which the badges were obtained3. We used the subset released by Du et al. (2016) containing $\sim 6 K$ users, 22 event types (badges), and $\sim 4 8 0 K$ events. Given a sequence $[ ( b _ { 1 } ^ { u } , t _ { 1 } ^ { u } ) , ( b _ { 2 } ^ { u } , t _ { 2 } ^ { u } ) , . . . , ( b _ { n } ^ { u } , t _ { n } ^ { u } ) ]$ for each user $u$ where $b _ { i } ^ { u }$ is the badge id and $t _ { i } ^ { u }$ is the timestamp when $u$ received this badge id, the task is to predict the badge the user will obtain at time $t _ { k + 1 } ^ { u }$ .
+
+5) Last.FM: This dataset contains a history of listening habits for Last.FM users (Celma, 2010). We used the code released by Zhu et al. (2017) to pre-process the data. The dataset contains $\sim 1 K$ users, 5000 event types (songs), and $\sim 8 1 9 K$ events. The prediction problem is similar to the SOF dataset but with dynamic updating (see, (Zhu et al., 2017) for details).
+
+6) CiteULike: This dataset contains data about what and when a user posted on citeulike website4. The original dataset has about 8000 samples. Similar to Last.FM, we used the pre-processing used by Zhu et al. (2017) to select $\sim 1 . 6 K$ sequences with 5000 event types (papers) and $\sim 3 6 K$ events. The task for this dataset is similar to that for Last.FM.
+
+Measures: For classification tasks, we report accuracy corresponding to the percentage of correctly classified examples. For recommendation tasks, we report Recall $@ q$ and $M R R @ q$ . Following Zhu et al. (2017), to generate a recommendation list, we sample $k - 1$ random items and add the correct item to the sampled list resulting in a list of $k$ items. Then our model ranks these $k$ items. Looking only at the top ten recommendations, Recall $@ \mathbf { q }$ corresponds to the percentage of recommendation lists where the correct item is in the top $q$ ; $\mathbf { M R R } @ \mathbf { q }$ (reported in Appendix B) corresponds to the mean of the inverses of the rankings of the correct items where the inverse rank is considered 0 if the item does not appear in top $q$ recommendations. For Last.FM and CiteULike, following Zhu et al. (2017) we report Recall $@ 1 0$ and $\mathbf { M R R } @ 1 0$ . For SOF, we report Recall $\textcircled { \alpha } 3$ and MRR as there are only 22 event types and Recall $@ 1 0$ and MRR $@ 1 0$ are not informative enough. The detail of the implementations is presented in Appendix A.
+
+# 5.1 ON THE EFFECTIVENESS OF TIME2VEC
+
+Fig. 1 represents the obtained results of comparing $L S T M + T i m e 2 V e c$ with $L S T M + T$ on several datasets with different properties and statistics. On all datasets, replacing time with Time2Vec improves the performance in most cases and never deteriorates it; in many cases, LSTM $^ +$ Time2Vec performs consistently better than $\mathrm { L S T M + T } .$ . Anumula et al. (2018) mention that $\mathrm { L S T M + T }$ fails on N TIDIGITS18 as the dataset contains very long sequences. By feeding better features to the LSTM rather than relying on the LSTM to extract them, Time2Vec helps better optimize the LSTM and offers higher accuracy (and lower variance) compared to $\mathrm { L S T M + T } .$ Besides N TIDIGITS18, SOF also contains somewhat long sequences and long time horizons. The results on these two datasets indicate that Time2Vec can be effective for datasets with long sequences and time horizons.
+
+![](images/a5e4bb8f547dcf576f3780aed266cd6e8c6666d13534634fe9224fe3dd4d675d.jpg)  
+Figure 2: Comparing TLSTM1 and TLSTM3 on Last.FM and CiteULike in terms of Recall $@$ 10 with and without Time2Vec.
+
+To verify if Time2Vec can be integrated with other architectures and improve their performance, we integrate it with TLSTM1 and TLSTM3, two recent and powerful models for handling asynchronous events. We replaced their notion $\tau$ of time with $\mathbf { t } 2 \mathbf { v } ( \tau )$ and replaced the vectors getting multiplied to $\tau$ with matrices accordingly. The updated formulations are presented in Appendix C. The obtained results in Fig. 2 for TLSTM1 and TLSTM3 on Last.FM and CiteULike demonstrates that replacing time with Time2Vec for both TLSTM1 and TLSTM3 improves the performance.
+
+# 5.2 MODEL VARIANTS & ABLATION STUDY
+
+Other activation functions: Inspired by Fourier sine series and by positional encoding, we used sine activations in Eq. 1. To evaluate how sine activations compare to other activation functions for our setting, we repeated the experiment on Event-MNIST in Section 5.1 when using non-periodic activations such as Sigmoid, Tanh, and rectified linear units (ReLU) (Nair & Hinton, 2010), and periodic activations such as mod and triangle. We fixed the length of the Time2Vec to $6 4 + 1$ , i.e. 64 units with a non-linear transformation and 1 unit with a linear transformation. From the results shown in Fig. 5(a), it can be observed that the periodic activation functions (sine, mod, and triangle) outperform the non-periodic ones. Other than not being able to capture periodic behaviors, we believe one of the main reasons why these non-periodic activation functions do not perform well is because as time goes forward and becomes larger, Sigmoid and Tanh saturate and ReLU either goes to zero or explodes. Among periodic activation functions, sine outperforms the other two.
+
+Fixed frequencies and phase-shifts: Vaswani et al. (2017) mention that learning sine frequencies and phase-shifts for their positional encoding gives the same performance as fixing frequencies to exponentially-decaying values and phase-shifts to 0 and $\frac { \pi } { 2 }$ . This raises the question of whether learning the sine frequencies and phase-shifts of Time2Vec from data offer any advantage compared to fixing them. To answer this question, we compare three models on Event-MNIST when using Time2Vec of length $1 6 + 1$ : 1- fixing $\mathbf { t } 2 \mathbf { v } ( \tau ) [ n ]$ to sin $\left( { \frac { 2 \pi n } { 1 6 } } \right)$ for $n \leq 1 6$ , 2- fixing the frequencies and phase shifts according to Vaswani et al. (2017)’s positional encoding, and 3- learning the frequencies and phase-shifts from the data. Fig. 5(b) represents our obtained results. The obtained results in Fig. 5(b) show that learning the frequencies and phase-shifts rather than fixing them helps improve the performance of the model.
+
+Modeling Periodic Behaviours: To measure how well Time2Vec performs in capturing periodic behaviours, we trained a model on our synthesized dataset where the input integer (day) is used as
+
+(a) A weighted sum of the sinusoids in Time2Vec oscillating every 7 days.
+
+![](images/57cf77dc0ea113bc62ab8a494db033a617854e6fab9b8bd56bf5d83b74ac3f8d.jpg)  
+Figure 3: The models learned for our synthesized dataset before the final activation. The red dots represent the points to be classified as 1.
+
+![](images/46452851a214ffea2d551fa40a2dfa6df34da8dca372ab0536941f637a1d1e1a.jpg)  
+(b) A weighted sum of the sinusoids in Time2Vec oscillating every 14 days.
+
+the time for Time2Vec and a fully connected layer is used on top of the Time2Vec to predict the class. That is, the probability of one of the classes is a sigmoid of a weighted sum of the Time2Vec elements. Fig. 3 (a) shows a the learned function for the days in the test set where the weights, frequencies and phase-shifts are learned from the data. The red dots on the figure represent multiples of 7. It can be observed that Time2Vec successfully learns the correct period and oscillates every 7 days. The phase-shifts have been learned in a way that all multiples of 7 are placed on the positive peaks of the signal to facilitate separating them from the other days. Looking at the learned frequency and phase-shift for the sine functions across several runs, we observed that in many runs one of the main sine functions has a frequency around $\begin{array} { r } { 0 . 8 9 8 \approx \frac { 2 \pi } { 7 } } \end{array}$ and a phase-shift around $1 . 5 6 \approx \frac { \pi } { 2 }$ , thus learning to oscillate every 7 days and shifting by $\frac { \pi } { 2 }$ to make sure multiples of 7 end up at the peaks of the signal. Fig. 4 shows the initial and learned sine frequencies for one run. It can be viewed that at the beginning, the weights and frequencies are random numbers. But after training, only the desired frequency $\textstyle { \left( { \frac { 2 \pi } { 7 } } \right) }$ has a high weight (and the 0 frequency which gets subsumed into the bias). The model perfectly classifies the examples in the test set which represents the sine functions in Time2Vec can be used effectively for extrapolation and out of sample times assuming that the test set follows similar periodic patterns as the train set5. We added some noise to our labels by flipping $5 \%$ of the labels selected at random and observed a similar performance in most runs.
+
+To test invariance to time rescaling, we multiplied the inputs by 2 and observed that in many runs, the frequency of one of the main sine functions was around $\begin{array} { r } { 0 . 4 4 \dot { 8 } \approx \frac { 2 \pi } { 2 * 7 } } \end{array}$ thus oscillating every 14 days. An example of a combination of signals learned to oscillate every 14 days is in Fig. 3 (b).
+
+The use of periodicity in sine functions: It has been argued that when sine activations are used, only a monotonically increasing (or decreasing) part of it is used and the periodic part is ignored (Giambattista Parascandolo, 2017). When we use Time2Vec, however, the periodicity of the sine functions are also being used and seem to be key to the effectiveness of the Time2Vec representation. Fig. 5(c) shows some statistics on the frequencies learned for Event-MNIST where we count the number of learned frequencies that fall within intervals of lengths 0.1 centered at $[ 0 . 0 5 , 0 . 1 5 , \ldots , 0 . 9 5 ]$ ] (all learned frequencies are between 0 and 1). The figure contains two peaks at 0.35 and 0.85. Since the input to the sine functions for this problem can have a maximum value of 784 (number of pixels in an image), sine functions with frequencies around 0.35 and 0.85 finish (almost) 44 and 106 full periods. The smallest learned frequency is 0.029 which finishes (almost) 3.6 full periods. These values indicate that the model is indeed using the periodicity of the sine functions, not just a monotonically increasing (or decreasing) part of them.
+
+![](images/087e83ffaf5f5917afade18bb06245b15156d18257d36717c25fb05ca79d2905.jpg)  
+Figure 4: (a) Initial vs. (b) learned weights and frequencies for our synthesized dataset.
+
+![](images/8da9c3d24f891893c3a0d28a36f5cd502358906524cea96ca5fb9d75114ada8c.jpg)  
+Figure 5: An ablation study of several components in Time2Vec. (a) Comparing different activation functions for Time2Vec on Event-MNIST. Sigmoid and Tanh almost overlap. (b) Comparing frequencies fixed to equally-spaced values, frequencies fixed according to positional encoding (Vaswani et al., 2017), and learned frequencies on Event-MNIST. (c) A histogram of the frequencies learned in Time2Vec for Event-MNIST. The $\mathbf { X }$ -axis represents frequency intervals and the y-axis represents the number of frequencies in that interval. (d) The performance of TLSTM3 $+$ Time2Vec on CiteULike in terms of Recall $@ 1 0$ with and without the linear term.
+
+The Linear Term: To see the effect of the linear term in Time2Vec, we repeated the experiment for Event-MNIST when the linear term is removed from Time2VecWe observed that the results were not ˙ affected substantially, thus showing that the linear term may not be helpful for Event-MNIST. This might be due to the simplicity of the Event-MNIST dataset. Then we conducted a similar experiment for TLSTM3 on CiteULike (which is a more challenging dataset) and obtained the results in Fig. 5(d). From these results, we can see that the linear term helps facilitate learning functions of time that can be effectively consumed by the model.
+
+# 6 CONCLUSION & FUTURE WORK
+
+In many tasks for synchronous and asynchronous event predictions, time is an important feature. Previous work has mainly resorted to applying hand-crafted functions to time and concatenating these functions with the rest of the input features. In this work, we presented an approach that automatically learns these functions from data. In particular, we developed Time2Vec, a vector representation for time, using sine and linear activations and showed the effectiveness of this representation across several datasets and several tasks. In the majority of our experiments, Time2Vec improved our results, while the remaining results were not hindered by its application. While sine functions have been argued to complicate the optimization (Lapedes & Farber, 1987; Giambattista Parascandolo, 2017), we did not experience such a complication except for the experiment in Subsection 5.2 on our synthesized dataset when using only a few sine functions. We hypothesize that the main reasons include combining sine functions with a powerful model (e.g., LSTM) and using many sine functions which reduces the distance to the goal (see, e.g., (Neyshabur et al., 2019)). We leave a deeper theoretical analysis of this hypothesis, development of better optimizers, and experimenting with other representations for time as future work.
+
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+
+![](images/00dbf51aef6dc414379150092b4e2229c8a9e041d859b21a9f3ec08e5d3c547f.jpg)  
+Figure 6: Comparing $\mathrm { L S T M + T }$ and LSTM $\cdot +$ Time2Vec on Event-MNIST.
+
+# A IMPLEMENTATION DETAIL
+
+For the experiments on Event-MNIST, N TIDIGITS18 and SOF, we implemented6 our model in PyTorch Paszke et al. (2017). We used Adam optimizer Kingma & Ba (2014) with a learning rate of 0.001. For Event-MNIST and SOF, we fixed the hidden size of the LSTM to 128. For N TIDIGITS18, due to its smaller train set, we fixed the hidden size to 64. We allowed each model 200 epochs. We used a batch size of 512 for Event-MNIST and 128 for N TIDIGITS18 and SOF. For the experiments on Last.FM and CiteULike, we used the code released by Zhu et al. $( 2 0 1 7 ) ^ { 7 }$ without any modifications, except replacing $\tau$ with $\mathbf { t } 2 \mathbf { v } ( \tau )$ . The only other thing we changed in their code was to change the SAMPLE TIME variable from 3 to 20. SAMPLE TIME controls the number of times we do sampling to compute Recall $@ 1 0$ and $\mathbf { M R R } @ 1 0$ . We experienced a high variation when sampling only 3 times so we increased the number of times we sample to 20 to make the results more robust. For both Last.FM and CiteULike, Adagrad optimizer is used with a learning rate of 0.01, vocabulary size is 5000, and the maximum length of the sequence is 200. For Last.FM, the hidden size of the LSTM is 128 and for CiteULike, it is 256. For all except the synthesized dataset, we shifted the event times such that the first event of each sequence starts at time 0.
+
+For the fairness of the experiments, we made sure the competing models for all our experiments have an (almost) equal number of parameters. For instance, since adding Time2Vec as an input to the LSTM increases the number of model parameters compared to just adding time as a feature, we reduced the hidden size of the LSTM for this model to ensure the number of model parameters stays (almost) the same. For the experiments involving Time2Vec, unless stated otherwise, we tried vectors with 16, 32 and 64 sine functions (and one linear term). We reported the vector length offering the best performance in the main text. The results for other vector lengths can be found in Appendix B. For the synthetic dataset, we use Adam optimizer with a learning rate of 0.001 without any regularization. The length of the Time2Vec vector is 32.
+
+![](images/76aa07786e64a3837fd9f14d434cd6ac184d1ae6940f879a76d6c3a580a792b6.jpg)  
+Figure 7: Comparing $\mathrm { L S T M + T }$ and LSTM $+$ Time2Vec on Event-MNIST and raw N TIDIGITS18.
+
+![](images/d3a24b60f8c8f0493e1757dd3dcc0c4e89eab8a986fb46521120a9364e6b6fa6.jpg)  
+Figure 8: Comparing LSTM+T and LSTM+Time2Vec on SOF.
+
+![](images/b6da3e307b8b779ebf7c7096fdb4a9b4227a850e1255c49ed128d8cbe5db8464.jpg)  
+Figure 9: Comparing LSTM $+ \mathrm { T }$ and LSTM $^ +$ Time2Vec on Last.FM.
+
+# B MORE RESULTS
+
+We ran experiments on other versions of the N TIDIGITS18 dataset as well. Following Anumula et al. (2018), we converted the raw event data to event-binned features by virtue of aggregating active channels through a period of time in which a pre-defined number of events occur. The outcome of binning is thus consecutive frames each with multiple but a fixed number of active channels. In our experiments, we used event-binning with 100 events per frame. For this variant of the dataset, we compared $\mathrm { L S T M + T }$ and LSTM+Time2Vec similar to the experiments in Section 5.1. The obtained results were on-par. Then, similar to Event-MNIST, we only fed as input the times at which events occurred (i.e. we removed the channels from the input). We allowed the models 1000 epochs to make sure they converge. The obtained results are presented in Fig. 6. It can be viewed that Time2Vec provides an effective representation for time and LSTM $+$ Time2Vec outperforms LSTM $+ \mathrm { T }$ on this dataset.
+
+In the main text, for the experiments involving Time2Vec, we tested Time2Vec vectors with 16, 32 and 64 sinusoids and reported the best one for the clarity of the diagrams. Here, we show the results for all frequencies. Figures 7, 8, 9, and 10 compare LSTM $+ \mathrm { T }$ and LSTM $^ +$ Time2Vec for our datasets. Figures 11, and 12 compare TLSTM1 with TLSTM1 $^ +$ Time2Vec on Last.FM and CiteULike. Figures 13, and 14 compare TLSTM3 with TLSTM1 $^ +$ Time2Vec on Last.FM and CiteULike. In most cases, Time2Vec with 64 sinusoids outperforms (or gives on-par results with) the cases with 32 or 16 sinusoids. An exception is TLSTM3 where 16 sinusoids works best. We believe that is because TLSTM3 has two time gates and adding, e.g., 64 temporal components (corresponding to the sinusoids) to each gate makes it overfit to the temporal signals.
+
+# C LSTM ARCHITECTURES
+
+The original LSTM model can be neatly defined with the following equations:
+
+$$
+\begin{array} { r l } & { { i _ { j } } = \sigma \left( W _ { i } { { \bf { x } } _ { j } } + U _ { i } { { h _ { j - 1 } } } + { { \bf { b } } _ { i } } \right) } \\ & { { f _ { j } } = \sigma \left( { W _ { f } { { \bf { x } } _ { j } } + U _ { f } { { h _ { j - 1 } } } + { { b _ { f } } } } \right) } \\ & { { { { \overline { { c } } } _ { j } } } = T a n h \left( { W _ { c } { { \bf { x } } _ { j } } + U _ { c } { { h _ { j - 1 } } } + { { b _ { c } } } } \right) } \\ & { { { { \overline { { c _ { j } } } } } } = { f _ { t } } \odot { { \bf { c } } _ { j - 1 } } + { i _ { j } } \odot { { { \overline { { { c _ { j } } } } } } } } \\ &  { { { \sigma _ { j } } } = \sigma \left( { W _ { o } { { \bf { x } } _ { j } } + U _ { o } { { h _ { j - 1 } } } + { { b _ { o } } } } \right) } \\ & { { { { h _ { j } } } = { { o _ { j } } } \odot T a n h \left( { { c _ { j } } } \right) } } \end{array}
+$$
+
+![](images/40d16ef249ff694d5fa296be84bbd3e8a1eabbff493ebb5ca30448726d171f10.jpg)  
+Figure 10: Comparing LSTM+T and LSTM $^ +$ Time2Vec on CiteULike.
+
+![](images/1ee8a037142439db5e80aadffce7ae031a58fd252a1ea45770669fbfec34052f.jpg)  
+Figure 11: TLSTM1’s performance on Last.FM with and without Time2Vec.
+
+Here $\mathbf { \delta } _ { i , \mathrm { ~ \tiny ~ \left. ~ \right. ~ } }$ , $\pmb { f } _ { t }$ , and $\mathbf { \sigma } _ { \pmb { o } _ { t } }$ represent the input, forget and output gates respectively, while $\mathbf { c } _ { t }$ is the memory cell and $\pmb { h } _ { t }$ is the hidden state. $\sigma$ and T anh represent the Sigmoid and hyperbolic tangent activation functions respectively. We refer to $\pmb { x } _ { j }$ as the $j ^ { \bar { t } h }$ event.
+
+Peepholes: Gers & Schmidhuber (2000) introduced a variant of the LSTM architecture where the input, forget, and output gates peek into the memory cell. In this variant, ${ \pmb w } _ { p i } \odot { \pmb c } _ { j - 1 } , { \pmb w } _ { p f } \odot { \pmb c } _ { j - 1 }$ and ${ \pmb w } _ { p o } \odot { \pmb c } _ { j }$ are added to the linear parts of Eq. (2), (3), and (6) respectively, where ${ \pmb w } _ { p i } , { \pmb w } _ { p f }$ , and ${ \pmb w } _ { p o }$ are learnable parameters.
+
+$\mathbf { L S T M + T } \mathbf { : }$ Let $\tau _ { j }$ represent the time features for the $j ^ { t h }$ event in the input and let $\pmb { x } _ { j } ^ { \prime } = [ \pmb { x } _ { j } ; \pmb { \tau } _ { j } ]$ Then $\mathrm { L S T M + T }$ uses the exact same equations as the standard LSTM (denoted above) except that $\pmb { x } _ { j }$ is replaced with $\pmb { x } _ { j } ^ { \prime }$ .
+
+TimeLSTM: We explain TLSTM1 and TLSTM3 which have been used in our experiments. For clarity of writing, we do not include the peephole terms in the equations but they are used in the experiments. In TLSTM1, a new time gate is introduced as in Eq. equation 8 and Eq. equation 5 and equation 6 are updated to Eq. equation 9 and equation 10 respectively:
+
+$$
+\begin{array} { r l } & { \pmb { t } _ { j } = \sigma \left( \pmb { W } _ { t } \pmb { x } _ { j } + \sigma \left( \pmb { u } _ { t } \tau _ { j } \right) + \pmb { b } _ { t } \right) } \\ & { \pmb { c } _ { j } = \pmb { f } _ { j } \odot \pmb { c } _ { j - 1 } + \pmb { i } _ { j } \odot \pmb { t } _ { j } \odot \overline { { \pmb { c } _ { j } } } } \\ & { \pmb { o } _ { j } = \sigma \left( \pmb { W } _ { o } \pmb { x } _ { j } + \pmb { v } _ { t } \tau _ { j } + \pmb { U } _ { o } \pmb { h } _ { j - 1 } + \pmb { b } _ { o } \right) } \end{array}
+$$
+
+$t _ { j }$ controls the influence of the current input on the prediction and makes the required information from timing history get stored on the cell state. TLSTM3 uses two time gates:
+
+$$
+\begin{array} { r } { \pmb { t 1 } _ { j } = \sigma \left( \pmb { W } _ { t 1 } \pmb { x } _ { j } + \sigma \left( \pmb { u } _ { t 1 } \tau _ { j } \right) + \pmb { b } _ { t 1 } \right) } \\ { \pmb { t 2 } _ { j } = \sigma \left( \pmb { W } _ { t 2 } \pmb { x } _ { j } + \sigma \left( \pmb { u } _ { t 2 } \tau _ { j } \right) + \pmb { b } _ { t 2 } \right) } \end{array}
+$$
+
+where the elements of $W _ { t 1 }$ are constrained to be non-positive. $\pmb { t 1 }$ is used for controlling the influence of the last consumed item and $\pmb { t 2 }$ stores the $\tau \mathbf { S }$ thus enabling modeling long range dependencies. TLSTM3 couples the input and forget gates following Greff et al. (2017) along with the $\pmb { t 1 }$ and $\pmb { t 2 }$ gates and replaces Eq. (5) to (7) with the following:
+
+$$
+\begin{array} { r l } & { \tilde { \mathbf { c } } _ { j } = \left( 1 - i _ { j } \odot t { \bf 1 } _ { j } \right) \odot \mathbf { c } _ { j - 1 } + i _ { j } \odot t { \bf 1 } _ { j } \odot \bar { \mathbf { c } } _ { j } } \\ & { \mathbf { c } _ { j } = \left( 1 - i _ { j } \right) \odot \mathbf { c } _ { j - 1 } + i _ { j } \odot t { \bf 2 } _ { j } \odot \bar { \mathbf { c } } _ { j } } \\ & { \mathbf { o } _ { j } = \sigma \left( W _ { o } \mathbf { x } _ { j } + \mathbf { v } _ { t } \tau _ { j } + U _ { o } \mathbf { h } _ { j - 1 } + \mathbf { b } _ { o } \right) } \\ & { \mathbf { h } _ { j } = o _ { j } \odot T a n h \left( \tilde { \mathbf { c } } _ { j } \right) } \end{array}
+$$
+
+![](images/2792f646e18eaf88109b2d7268bb9be9b2f1ad4ca537fb0c4e885ec977ca9f2c.jpg)  
+Figure 12: TLSTM1’s performance on CiteULike with and without Time2Vec.
+
+![](images/63be7796c2b8a7c7b18e4cff05ef2e9180793681d99a0b276524b82af87f436c.jpg)  
+Figure 13: TLSTM3’s performance on Last.FM with and without Time2Vec.
+
+![](images/da71988ac8cf0d1afe94a485ec7ac9e71ec909c5b026ed379f6867a32deef577.jpg)  
+Figure 14: TLSTM3’s performance on CiteULike with and without Time2Vec.
+
+Zhu et al. (2017) use $\tau _ { j } = \Delta t _ { j }$ in their experiments, where $\Delta t _ { j }$ is the duration between the current and the last event.
+
+TimeLSTM $^ +$ Time2Vec: To replace time in TLSTM1 with Time2Vec, we modify Eq. (8) and (10) as follows:
+
+$$
+\begin{array} { r l } & { \pmb { t _ { j } } = \sigma \left( \pmb { W _ { t } } \pmb { x _ { j } } + \sigma \left( \pmb { U _ { t } } \pmb { \mathrm { t } } 2 \mathbf { v } ( \tau ) \right) + \pmb { b _ { t } } \right) } \\ & { \pmb { o _ { j } } = \sigma ( \pmb { W _ { o } } \pmb { x _ { j } } + \pmb { V _ { t } } \pmb { \mathrm { t } } 2 \mathbf { v } ( \tau ) + \pmb { U _ { o } } \pmb { h _ { j - 1 } } + \pmb { b _ { o } } ) } \end{array}
+$$
+
+i.e., $\tau$ is replaced with $\mathbf { t } 2 \mathbf { v } ( \tau )$ , $\mathbf { \pmb { u } } _ { t }$ is replaced with $\boldsymbol { U } _ { t }$ , and ${ \pmb v } _ { t }$ is replaced with $V _ { t }$ . Similarly, for TLSTM3 we modify Eq. (11), (12) and (15) as follows:
+
+$$
+\begin{array} { r } { t \pmb { 1 } _ { j } = \sigma \left( \pmb { W } _ { t 1 } \pmb { x } _ { j } + \sigma \left( \pmb { U } _ { t 1 } \pmb { \mathrm { t } } 2 \pmb { \mathrm { v } } ( \tau ) \right) + \pmb { b } _ { t 1 } \right) } \\ { t \pmb { 2 } _ { j } = \sigma \left( \pmb { W } _ { t 2 } \pmb { x } _ { j } + \sigma \left( \pmb { U } _ { t 2 } \pmb { \mathrm { t } } 2 \pmb { \mathrm { v } } ( \tau ) \right) + \pmb { b } _ { t 2 } \right) } \\ { \pmb { o } _ { j } = \sigma ( \pmb { W } _ { o } \pmb { x } _ { j } + \pmb { V } _ { t } \pmb { \mathrm { t } } 2 \pmb { \mathrm { v } } ( \tau ) + \pmb { U } _ { o } \pmb { h } _ { j - 1 } + \pmb { b } _ { o } ) } \end{array}
+$$
+
+# D PROOFS
+
+Proposition 1. Time2Vec is invariant to time rescaling.
+
+Proof. Consider the following Time2Vec representation $\mathcal { M } _ { 1 }$ :
+
+$$
+\mathbf t 2 \mathbf v ( \tau ) [ i ] = { \left\{ \begin{array} { l l } { \omega _ { i } \tau + \varphi _ { i } , } & { { \mathrm { i f } } i = 0 . } \\ { \sin { ( \omega _ { i } \tau + \varphi _ { i } ) } , } & { { \mathrm { i f } } 1 \leq i \leq k . } \end{array} \right. }
+$$
+
+Replacing $\tau$ with $\alpha \cdot \tau$ (for $\alpha > 0$ ), the Time2Vec representation updates as follows:
+
+$$
+\mathbf { t } 2 \mathbf { v } ( \alpha \cdot \tau ) [ i ] = { \left\{ \begin{array} { l l } { \omega _ { i } ( \alpha \cdot \tau ) + \varphi _ { i } , } & { { \mathrm { i f } } i = 0 . } \\ { \sin { ( \omega _ { i } ( \alpha \cdot \tau ) + \varphi _ { i } ) } , } & { { \mathrm { i f } } 1 \leq i \leq k . } \end{array} \right. }
+$$
+
+Consider anosame way as Time2Vec representation . This proves that Time2 $\mathcal { M } _ { 2 }$ with frequencies s invariant to time $\begin{array} { r } { \omega _ { i } ^ { \prime } = \frac { \omega _ { i } } { \alpha } } \end{array}$ . Then g. $\mathcal { M } _ { 2 }$ behaves in the $\mathcal { M } _ { 1 }$

parse/train/rklklCVYvB/rklklCVYvB_content_list.json

@@ -0,0 +1,1878 @@
+[
+    {
+        "type": "text",
+        "text": "TIME2VEC: LEARNING A VECTOR REPRESENTATION OF TIME ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            823,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Time is an important feature in many applications involving events that occur synchronously and/or asynchronously. To effectively consume time information, recent studies have focused on designing new architectures. In this paper, we take an orthogonal but complementary approach by providing a model-agnostic vector representation for time, called $T i m e 2 V e c$ , that can be easily imported into many existing and future architectures and improve their performances. We show on a range of models and problems that replacing the notion of time with its Time2Vec representation improves the performance of the final model. ",
+        "bbox": [
+            233,
+            267,
+            766,
+            378
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            406,
+            336,
+            422
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In building machine learning models, “time” is often an important feature. Examples include predicting daily sales for a company based on the date (and other available features), predicting the time for a patient’s next health event based on their medical history, and predicting the song a person is interested in listening to based on their listening history. The input for problems involving time can be considered as a sequence where, rather than being identically and independently distributed (iid), there exists a dependence across time (and/or space) among the data points. The sequence can be either synchronous, i.e. sampled at regular intervals, or asynchronous, i.e. sampled at different points in time. In both cases, time may be an important feature. For predicting daily sales, for instance, it may be useful to know if it is a holiday or not. For predicting the time for a patient’s next encounter, it is important to know the (asynchronous) times of their previous visits. ",
+        "bbox": [
+            174,
+            439,
+            825,
+            578
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks (RNNs) do not typically treat time itself as a feature, typically assuming that inputs are synchronous. When time is known to be a relevant feature, it is often fed in as yet another input dimension (Choi et al., 2016; Du et al., 2016; Li et al., 2018b). In practice, RNNs often fail at effectively making use of time as a feature. To help the RNN make better use of time, several researchers design hand-crafted features of time that suit their specific problem and feed those features into the RNN (Choi et al., 2016; Baytas et al., 2017; Kwon et al., 2019). Hand-crafting features, however, can be expensive and requires domain expertise about the problem. ",
+        "bbox": [
+            174,
+            585,
+            825,
+            683
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many recent studies aim at obviating the need for hand-crafting features by proposing generalpurpose—as opposed to problem specific—architectures that better handle time (Neil et al., 2016; Zhu et al., 2017; Mei & Eisner, 2017; Hu & Qi, 2017; Upadhyay et al., 2018; Li et al., 2018a). We follow an orthogonal but complementary approach to these recent studies by developing a generalpurpose model-agnostic representation for time that can be potentially used in any architecture. In particular, we develop a learnable vector representation (or embedding) for time as a vector representation can be easily combined with many models or architectures. We call this vector representation Time2Vec. To validate the effectiveness of Time2Vec, we conduct experiments on several (synthesized and real-world) datasets and integrate it with several architectures. Our main result is to show that on a range of problems and architectures that consume time, using Time2Vec instead of the time itself offers a boost in performance. ",
+        "bbox": [
+            174,
+            689,
+            825,
+            842
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            863,
+            344,
+            878
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There is a long history of algorithms for predictive modeling in time series analysis. They include auto-regressive techniques (Akaike, 1969) that predict future measurements in a sequence based on a window of past measurements. Since it is not always clear how long the window of past measurements should be, hidden Markov models (Rabiner & Juang, 1986), dynamic Bayesian networks (Murphy & Russell, 2002), and dynamic conditional random fields (Sutton et al., 2007) use hidden states as a finite memory that can remember information arbitrarily far in the past. These models can be seen as special cases of recurrent neural networks (Hochreiter & Schmidhuber, 1997). They typically assume that inputs are synchronous, i.e. arrive at regular time intervals, and that the underlying process is stationary with respect to time. It is possible to aggregate asynchronous events into time-bins and to use synchronous models over the bins (Lipton et al., 2016; Anumula et al., 2018). Asynchronous events can also be directly modeled with point processes (e.g., Poisson, Cox, and Hawkes point processes) (Daley & Vere-Jones, 2007; Laub et al., 2015; Xiao et al., 2017; Li et al., 2018a; Xiao et al., 2018) and continuous time normalizing flows (Chen et al., 2018). Alternatively, one can also interpolate or make predictions at arbitrary time stamps with Gaussian processes (Rasmussen, 2004) or support vector regression (Drucker et al., 1997). ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            284
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our goal is not to propose a new model for time series analysis, but instead to propose a representation of time in the form of a vector embedding that can be used by many models. Vector embedding has been previously successfully used for other domains such as text (Mikolov et al., 2013; Pennington et al., 2014), (knowledge) graphs (Grover & Leskovec, 2016; Nickel et al., 2016; Kazemi & Poole, 2018), and positions (Vaswani et al., 2017; Gehring et al., 2017). Our approach is related to time decomposition techniques that encode a temporal signal into a set of frequencies (Cohen, 1995). However, instead of using a fixed set of frequencies as in Fourier transforms (Bracewell & Bracewell, 1986), we allow the frequencies to be learned. We take inspiration from the neural decomposition of Godfrey & Gashler (2018) (and similarly (Gashler & Ashmore, 2016)). For time-series analysis, Godfrey & Gashler (2018) decompose a 1D signal of time into several sine functions and a linear function to extrapolate (or interpolate) the given signal. We follow a similar intuition but instead of decomposing a 1D signal of time into its components, we transform the time itself and feed its transformation into the model that is to consume the time information. Our approach corresponds to the technique of Godfrey & Gashler (2018) when applied to regression tasks in 1D signals, but it is more general since we learn a representation that can be shared across many signals and can be fed to many models for tasks beyond regression. ",
+        "bbox": [
+            174,
+            291,
+            826,
+            513
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While there is a body of literature on designing neural networks with sine activations (Lapedes & Farber, 1987; Sopena et al., 1999; Wong et al., 2002; Mingo et al., 2004; Liu et al., 2016), our work uses sine only for transforming time; the rest of the network uses other activations. There is also a set of techniques that consider time as yet another feature and concatenate time (or some hand designed features of time such as log and/or inverse of delta time) with the input (Choi et al., 2016; Li et al., 2017; Du et al., 2016; Baytas et al., 2017; Kwon et al., 2019; Trivedi et al., 2017; Kumar et al., 2018; Ma et al., 2018). Kazemi et al. (2019) survey several such approaches for dynamic (knowledge) graphs. These models can directly benefit from our proposed vector embedding, Time2Vec, by concatenating Time2Vec with the input instead of their time features. Other works (Neil et al., 2016; Zhu et al., 2017; Mei & Eisner, 2017; Hu & Qi, 2017; Upadhyay et al., 2018; Li et al., 2018a) propose new neural architectures that take into account time (or some features of time). We show how Time2Vec can be used in one of these architectures to better exploit temporal information; it can be potentially used in other architectures as well. ",
+        "bbox": [
+            174,
+            520,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND & NOTATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            728,
+            439,
+            744
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We use lower-case letters to denote scalars, bold lower-case letters to denote vectors, and bold upper-case letters to denote matrices. We represent the $i ^ { t h }$ element of the vector $\\pmb { r }$ as $\\pmb { r } [ i ]$ . For two vectors $\\pmb { r }$ and $\\pmb { s }$ , we use $[ \\pmb { r } ; \\pmb { s } ]$ to represent their concatenation and $\\pmb { r } \\odot \\pmb { s }$ to represent element-wise (Hadamard) multiplication of the two vectors. Throughout the paper, we use $\\tau$ to represent a scalar notion of time (e.g., absolute time or time from the last event) and $\\tau$ for a vector of time features. ",
+        "bbox": [
+            174,
+            762,
+            823,
+            833
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Long Short Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) is considered one of the most successful RNN architectures for sequence modeling. A formulation of the original LSTM model and a variant of it based on peepholes (Gers & Schmidhuber, 2000) is presented in Appendix C. When time is a relevant feature, the easiest way to handle time is to consider it as just another feature (or extract some engineered features from it), concatenate the time features with the input, and use the standard LSTM model (or some other sequence model) (Choi et al., 2016; Du et al., 2016; Li et al., ",
+        "bbox": [
+            174,
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+            825,
+            922
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2018b). In this paper, we call this model $L S T M + T .$ Another way of handling time is by changing the formulation of the standard LSTM. Zhu et al. (2017) developed one such formulation, named TimeLSTM, by adding time gates to the architecture of the LSTM with peepholes. They proposed three architectures namely TLSTM1, TLSTM2, TLSTM3. A description of TLSTM1 and TLSTM3 can be found in Appendix C (we skipped TLSTM2 as it is quite similar to TLSTM3). ",
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+        "type": "text",
+        "text": "4 TIME2VEC ",
+        "text_level": 1,
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+        "text": "A common approach to deal with time in different applications is to apply some hand-crafted function(s) $f _ { 1 } , \\ldots , f _ { m }$ to $\\tau$ ， $\\mathit { \\Pi } _ { \\tau }$ can be absolute time, time from last event, etc.), concatenate the outputs $f _ { 1 } ( \\tau ) , \\dots , f _ { m } ( \\tau )$ with the rest of the input features $\\pmb { x }$ , and feed the resulting vector $[ { \\pmb x } ; f _ { 1 } ( \\tau ) ; \\dots ; f _ { m } ( \\tau ) ]$ to a sequence model (see Section 2 for references). This approach requires hand-crafting useful functions of time which may be difficult (or impossible) in several applications, and the hand-crafted functions may not be optimal for the task at hand. Instead of hand-crafting functions of time, we devise a representation of time which can be used to approximate any function through learnable parameters. Such a representation offers two advantages: 1- it obviates the need for hand-crafting functions of time and 2- it provides the grounds for learning suitable functions of time based on the data. As vector representations can be efficiently integrated with the current deep learning architectures, we employ a vector representation for time. ",
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+        "text": "Our proposed representation leverages the Fourier sine series (Arfken & Weber, 1999) according to which any 1D function can be approximated in a given interval using a weighted sum of sinusoids with appropriate frequencies (and phase-shifts). We include $k$ sinusoids of the form $s i n ( \\omega _ { i } \\tau + \\varphi _ { i } )$ in our vector representation where $\\omega _ { i }$ and $\\varphi _ { i }$ are learnable parameters1. That is, we concatenate the input features $\\pmb { x }$ with $k$ sinusoids and feed the concatenation $[ \\pmb { x } ; s i n ( \\omega _ { 1 } \\tau + \\varphi _ { 1 } ) ; \\dots ; s i n ( \\omega _ { k } \\tau + \\varphi _ { k } ) ]$ into a sequence model. Different functions of time can be created using these sinusoids by taking a weighted sum of them with different weights. We allow the weights of the sequence model to combine the sinusoids and create functions of time suitable for the task. If we expand the output $\\begin{array} { r } { \\pmb { a } ( \\tau , k ) [ j ] = \\gamma _ { j } + \\sum _ { i = 1 } ^ { k } \\theta _ { j , i } \\sin \\left( \\omega _ { i } \\tau + \\varphi _ { i } \\right) } \\end{array}$ fore, wheures p $\\theta _ { j , i ^ { \\mathrm { S } } }$ g an activation function), it  are the first layer weights and  on the temporal features). Ea $\\gamma _ { j }$ $\\pmb { x }$ $\\mathbf { \\pmb { a } } ( \\tau , k ) [ j ]$ operates on the input features $\\pmb { x }$ as well as a learned function $\\begin{array} { r } { f _ { j } ( \\tau ) = \\sum _ { i = 1 } ^ { k } \\theta _ { j , i } \\sin { \\left( \\omega _ { i } \\tau + \\varphi _ { i } \\right) } } \\end{array}$ of time, as opposed to a hand-crafted function2. Following Godfrey & Gashler (2018), to facilitate approximating functions with non-periodic patterns and help with generalization, we also include a linear projection of time in our vector representation. We name our vector representation of time Time2Vec. Time2Vec of $\\tau$ , denoted as $\\mathbf { t } 2 \\mathbf { v } ( \\tau )$ , is a vector of size $k + 1$ defined as follows: ",
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+        "type": "equation",
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+        "text": "$$\n\\mathbf t 2 \\mathbf v ( \\tau ) [ i ] = { \\left\\{ \\begin{array} { l l } { \\omega _ { i } \\tau + \\varphi _ { i } , } & { { \\mathrm { i f } } i = 0 . } \\\\ { \\sin { ( \\omega _ { i } \\tau + \\varphi _ { i } ) } , } & { { \\mathrm { i f } } 1 \\leq i \\leq k . } \\end{array} \\right. }\n$$",
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+        "type": "text",
+        "text": "where $\\mathbf { t } 2 \\mathbf { v } ( \\tau ) [ i ]$ is the $i ^ { t h }$ element of $\\mathbf { t } 2 \\mathbf { v } ( \\tau )$ and $\\omega _ { i } \\mathbf { s }$ and $\\varphi _ { i } \\mathbf { s }$ are learnable parameters. ",
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+        "text": "The use of sine functions is inspired in part by Vaswani et al. (2017)’s positional encoding. Consider a sequence of items (e.g., a sequence of words) $\\{ I _ { 1 } , I _ { 2 } , \\ldots , I _ { N } \\}$ and a vector representation ${ \\pmb v } _ { I _ { j } } \\in \\mathbb { R } ^ { d }$ for the $j ^ { t h }$ item $I _ { j }$ in the sequence. Vaswani et al. (2017) added $\\sin { ( j / 1 0 0 0 0 ^ { k / d } ) }$ to ${ \\pmb v } _ { I _ { j } } [ k ]$ if $k$ is even and $\\sin { ( j / 1 0 0 0 0 ^ { k / d } + \\pi / 2 ) }$ if $k$ is odd so that the resulting vector includes information about the position of the item in the sequence. These sine functions are called the positional encoding. Intuitively, positions can be considered as the times and the items can be considered as the events happening at that time. Thus, Time2Vec can be considered as representing continuous time, instead of discrete positions, using sine functions. The sine functions in Time2Vec also enable capturing periodic behaviors which is not a goal in positional encoding. We feed Time2Vec as an input to the model (or to some gate in the model) instead of adding it to other vector representations. Unlike positional encoding, we show in our experiments that learning the frequencies and phase-shifts of sine functions in Time2Vec result in better performance compared to fixing them. ",
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+        "type": "text",
+        "text": "4.1 PROPERTIES OF TIME2VEC",
+        "text_level": 1,
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+        "text": "We review some of the interesting and desired properties of Time2Vec. ",
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+        "text": "Periodicity: In many scenarios, some events occur periodically. The amount of sales of a store, for instance, may be higher on weekends or holidays. Weather condition usually follows a periodic pattern over different seasons (Gashler & Ashmore, 2016). Some other events may be non-periodic but only happen after a point in time and/or become more probable as time proceeds. For instance, some diseases are more likely for older ages. ",
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+        "text": "The period of $\\sin \\left( \\omega _ { i } \\tau + \\varphi _ { i } \\right)$ is $\\frac { 2 \\pi } { \\omega _ { i } }$ , i.e. it has the same value for $\\tau$ and $\\textstyle { \\tau + { \\frac { 2 \\pi } { \\omega _ { i } } } }$ . Therefore, the sine functions in Time2Vec help capture periodic behaviors without the need for feature engineering. For instance, a sine function $\\sin { \\left( \\omega \\tau + \\varphi \\right) }$ with $\\begin{array} { r } { \\omega = \\frac { 2 \\pi } { 7 } } \\end{array}$ repeats every 7 days (assuming $\\tau$ indicates days) and can be potentially used to model weekly patterns. Furthermore, unlike other basis functions which may show strange behaviors for extrapolation (see, e.g., (Poole et al., 2014)), sine functions are expected to work well for extrapolating to future and out of sample data (Vaswani et al., 2017). The linear term represents the progression of time and can be used for capturing non-periodic patterns in the input that depend on time. ",
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+        "type": "text",
+        "text": "Invariance to Time Rescaling: Since time can be measured in different scales (e.g., days, hours, seconds, etc.), another important property of a representation for time is invariance to time rescaling (see, e.g., (Tallec & Ollivier, 2018)). A class $\\mathcal { C }$ of models is invariant to time rescaling if for any model $\\mathcal { M } _ { 1 } \\in \\mathcal { C }$ and any scalar $\\alpha > 0$ , there exists a model $\\mathcal { M } _ { 2 } \\in \\mathcal { C }$ that behaves on $\\alpha \\tau$ ( $\\mathit { \\Pi } _ { \\tau }$ scaled by $\\alpha$ ) in the same way $\\mathcal { M } _ { 1 }$ behaves on original $\\tau \\mathrm { s }$ . Proposition 1 establishes the invariance of Time2Vec to time rescaling. The proof is in Appendix D. ",
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+        "type": "text",
+        "text": "Proposition 1. Time2Vec is invariant to time rescaling. ",
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+        "type": "text",
+        "text": "Simplicity: A representation for time should be easily consumable by different models and architectures. A matrix representation, for instance, may be difficult to consume as it cannot be easily appended with the other inputs. By selecting a vector representation for time, we ensure easy integration with deep learning architectures. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS & RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use the following datasets: ",
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+    {
+        "type": "text",
+        "text": "1) Synthesized data: We create a toy dataset to use for explanatory experiments. The inputs in this dataset are the integers between 1 and 365. Input integers that are multiples of 7 belong to class one and the other integers belong to class two. The first $7 5 \\%$ is used for training and the last $2 5 \\%$ for testing. This dataset is inspired by the periodic patterns (e.g., weekly or monthly) that often exist in daily-collected data; the input integers can be considered as the days. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2) Event-MNIST: Sequential (event-based) MNIST is a common benchmark in sequence modeling literature (see, e.g., (Bellec et al., 2018; Campos et al., 2018; Fatahi et al., 2016)). We create a sequential event-based version of MNIST by flattening the images and recording the position of the pixels whose intensities are larger than a threshold (0.9 in our experiment). Following this transformation, each image will be represented as an array of increasing numbers such as $[ t _ { 1 } , t _ { 2 } , t _ { 3 } , \\ldots , t _ { m } ]$ . We consider these values as the event times and use them to classify the images. As in other sequence modeling works, our aim in building this dataset is not to beat the state-of-the-art on the MNIST dataset; our aim is to provide a dataset where the only input is time and different representations for time can be compared when extraneous variables (confounders) are eliminated as much as possible. ",
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+    {
+        "type": "text",
+        "text": "3) N TIDIGITS18 (Anumula et al., 2018): The dataset includes audio spikes of the TIDIGITS spoken digit dataset (Leonard & Doddington, 1993) recorded by the binaural 64-channel silicon cochlea sensor. Each sample is a sequence of $( t , c )$ tuples where $t$ represents time and $c$ denotes the index of active frequency channel at time $t$ . The labels are sequences of 1 to 7 connected digits with a vocabulary consisting of 11 digits (i.e. “zero” to “nine” plus “oh”) and the goal is to classify the spoken digit based on the given sequence of active channels. We use the reduced version of the dataset where only the single digit samples are used for training and testing. The reduced dataset has a total of 2,464 training and 2,486 test samples. ",
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+    {
+        "type": "image",
+        "img_path": "images/d284eb34fa7cec683a41bc11dcf77dc40c6bbb62c4cfea0ed98d33c7730f82d8.jpg",
+        "image_caption": [
+            "Figure 1: Comparing $\\mathrm { L S T M + T }$ and LSTM+Time2Vec on several datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4) Stack Overflow (SOF): This dataset contains sequences of badges obtained by stack overflow users and the timestamps at which the badges were obtained3. We used the subset released by Du et al. (2016) containing $\\sim 6 K$ users, 22 event types (badges), and $\\sim 4 8 0 K$ events. Given a sequence $[ ( b _ { 1 } ^ { u } , t _ { 1 } ^ { u } ) , ( b _ { 2 } ^ { u } , t _ { 2 } ^ { u } ) , . . . , ( b _ { n } ^ { u } , t _ { n } ^ { u } ) ]$ for each user $u$ where $b _ { i } ^ { u }$ is the badge id and $t _ { i } ^ { u }$ is the timestamp when $u$ received this badge id, the task is to predict the badge the user will obtain at time $t _ { k + 1 } ^ { u }$ . ",
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+    {
+        "type": "text",
+        "text": "5) Last.FM: This dataset contains a history of listening habits for Last.FM users (Celma, 2010). We used the code released by Zhu et al. (2017) to pre-process the data. The dataset contains $\\sim 1 K$ users, 5000 event types (songs), and $\\sim 8 1 9 K$ events. The prediction problem is similar to the SOF dataset but with dynamic updating (see, (Zhu et al., 2017) for details). ",
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+    {
+        "type": "text",
+        "text": "6) CiteULike: This dataset contains data about what and when a user posted on citeulike website4. The original dataset has about 8000 samples. Similar to Last.FM, we used the pre-processing used by Zhu et al. (2017) to select $\\sim 1 . 6 K$ sequences with 5000 event types (papers) and $\\sim 3 6 K$ events. The task for this dataset is similar to that for Last.FM. ",
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+    {
+        "type": "text",
+        "text": "Measures: For classification tasks, we report accuracy corresponding to the percentage of correctly classified examples. For recommendation tasks, we report Recall $@ q$ and $M R R @ q$ . Following Zhu et al. (2017), to generate a recommendation list, we sample $k - 1$ random items and add the correct item to the sampled list resulting in a list of $k$ items. Then our model ranks these $k$ items. Looking only at the top ten recommendations, Recall $@ \\mathbf { q }$ corresponds to the percentage of recommendation lists where the correct item is in the top $q$ ; $\\mathbf { M R R } @ \\mathbf { q }$ (reported in Appendix B) corresponds to the mean of the inverses of the rankings of the correct items where the inverse rank is considered 0 if the item does not appear in top $q$ recommendations. For Last.FM and CiteULike, following Zhu et al. (2017) we report Recall $@ 1 0$ and $\\mathbf { M R R } @ 1 0$ . For SOF, we report Recall $\\textcircled { \\alpha } 3$ and MRR as there are only 22 event types and Recall $@ 1 0$ and MRR $@ 1 0$ are not informative enough. The detail of the implementations is presented in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "5.1 ON THE EFFECTIVENESS OF TIME2VEC ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Fig. 1 represents the obtained results of comparing $L S T M + T i m e 2 V e c$ with $L S T M + T$ on several datasets with different properties and statistics. On all datasets, replacing time with Time2Vec improves the performance in most cases and never deteriorates it; in many cases, LSTM $^ +$ Time2Vec performs consistently better than $\\mathrm { L S T M + T } .$ . Anumula et al. (2018) mention that $\\mathrm { L S T M + T }$ fails on N TIDIGITS18 as the dataset contains very long sequences. By feeding better features to the LSTM rather than relying on the LSTM to extract them, Time2Vec helps better optimize the LSTM and offers higher accuracy (and lower variance) compared to $\\mathrm { L S T M + T } .$ Besides N TIDIGITS18, SOF also contains somewhat long sequences and long time horizons. The results on these two datasets indicate that Time2Vec can be effective for datasets with long sequences and time horizons. ",
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+    {
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+        "img_path": "images/a5e4bb8f547dcf576f3780aed266cd6e8c6666d13534634fe9224fe3dd4d675d.jpg",
+        "image_caption": [
+            "Figure 2: Comparing TLSTM1 and TLSTM3 on Last.FM and CiteULike in terms of Recall $@$ 10 with and without Time2Vec. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "To verify if Time2Vec can be integrated with other architectures and improve their performance, we integrate it with TLSTM1 and TLSTM3, two recent and powerful models for handling asynchronous events. We replaced their notion $\\tau$ of time with $\\mathbf { t } 2 \\mathbf { v } ( \\tau )$ and replaced the vectors getting multiplied to $\\tau$ with matrices accordingly. The updated formulations are presented in Appendix C. The obtained results in Fig. 2 for TLSTM1 and TLSTM3 on Last.FM and CiteULike demonstrates that replacing time with Time2Vec for both TLSTM1 and TLSTM3 improves the performance. ",
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+    {
+        "type": "text",
+        "text": "5.2 MODEL VARIANTS & ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Other activation functions: Inspired by Fourier sine series and by positional encoding, we used sine activations in Eq. 1. To evaluate how sine activations compare to other activation functions for our setting, we repeated the experiment on Event-MNIST in Section 5.1 when using non-periodic activations such as Sigmoid, Tanh, and rectified linear units (ReLU) (Nair & Hinton, 2010), and periodic activations such as mod and triangle. We fixed the length of the Time2Vec to $6 4 + 1$ , i.e. 64 units with a non-linear transformation and 1 unit with a linear transformation. From the results shown in Fig. 5(a), it can be observed that the periodic activation functions (sine, mod, and triangle) outperform the non-periodic ones. Other than not being able to capture periodic behaviors, we believe one of the main reasons why these non-periodic activation functions do not perform well is because as time goes forward and becomes larger, Sigmoid and Tanh saturate and ReLU either goes to zero or explodes. Among periodic activation functions, sine outperforms the other two. ",
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+        "type": "text",
+        "text": "Fixed frequencies and phase-shifts: Vaswani et al. (2017) mention that learning sine frequencies and phase-shifts for their positional encoding gives the same performance as fixing frequencies to exponentially-decaying values and phase-shifts to 0 and $\\frac { \\pi } { 2 }$ . This raises the question of whether learning the sine frequencies and phase-shifts of Time2Vec from data offer any advantage compared to fixing them. To answer this question, we compare three models on Event-MNIST when using Time2Vec of length $1 6 + 1$ : 1- fixing $\\mathbf { t } 2 \\mathbf { v } ( \\tau ) [ n ]$ to sin $\\left( { \\frac { 2 \\pi n } { 1 6 } } \\right)$ for $n \\leq 1 6$ , 2- fixing the frequencies and phase shifts according to Vaswani et al. (2017)’s positional encoding, and 3- learning the frequencies and phase-shifts from the data. Fig. 5(b) represents our obtained results. The obtained results in Fig. 5(b) show that learning the frequencies and phase-shifts rather than fixing them helps improve the performance of the model. ",
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+        "type": "text",
+        "text": "Modeling Periodic Behaviours: To measure how well Time2Vec performs in capturing periodic behaviours, we trained a model on our synthesized dataset where the input integer (day) is used as ",
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+        "text": "(a) A weighted sum of the sinusoids in Time2Vec oscillating every 7 days. ",
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+    {
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+        "img_path": "images/57cf77dc0ea113bc62ab8a494db033a617854e6fab9b8bd56bf5d83b74ac3f8d.jpg",
+        "image_caption": [
+            "Figure 3: The models learned for our synthesized dataset before the final activation. The red dots represent the points to be classified as 1. "
+        ],
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+        "img_path": "images/46452851a214ffea2d551fa40a2dfa6df34da8dca372ab0536941f637a1d1e1a.jpg",
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+            "(b) A weighted sum of the sinusoids in Time2Vec oscillating every 14 days. "
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+        "text": "the time for Time2Vec and a fully connected layer is used on top of the Time2Vec to predict the class. That is, the probability of one of the classes is a sigmoid of a weighted sum of the Time2Vec elements. Fig. 3 (a) shows a the learned function for the days in the test set where the weights, frequencies and phase-shifts are learned from the data. The red dots on the figure represent multiples of 7. It can be observed that Time2Vec successfully learns the correct period and oscillates every 7 days. The phase-shifts have been learned in a way that all multiples of 7 are placed on the positive peaks of the signal to facilitate separating them from the other days. Looking at the learned frequency and phase-shift for the sine functions across several runs, we observed that in many runs one of the main sine functions has a frequency around $\\begin{array} { r } { 0 . 8 9 8 \\approx \\frac { 2 \\pi } { 7 } } \\end{array}$ and a phase-shift around $1 . 5 6 \\approx \\frac { \\pi } { 2 }$ , thus learning to oscillate every 7 days and shifting by $\\frac { \\pi } { 2 }$ to make sure multiples of 7 end up at the peaks of the signal. Fig. 4 shows the initial and learned sine frequencies for one run. It can be viewed that at the beginning, the weights and frequencies are random numbers. But after training, only the desired frequency $\\textstyle { \\left( { \\frac { 2 \\pi } { 7 } } \\right) }$ has a high weight (and the 0 frequency which gets subsumed into the bias). The model perfectly classifies the examples in the test set which represents the sine functions in Time2Vec can be used effectively for extrapolation and out of sample times assuming that the test set follows similar periodic patterns as the train set5. We added some noise to our labels by flipping $5 \\%$ of the labels selected at random and observed a similar performance in most runs. ",
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+    {
+        "type": "text",
+        "text": "To test invariance to time rescaling, we multiplied the inputs by 2 and observed that in many runs, the frequency of one of the main sine functions was around $\\begin{array} { r } { 0 . 4 4 \\dot { 8 } \\approx \\frac { 2 \\pi } { 2 * 7 } } \\end{array}$ thus oscillating every 14 days. An example of a combination of signals learned to oscillate every 14 days is in Fig. 3 (b). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The use of periodicity in sine functions: It has been argued that when sine activations are used, only a monotonically increasing (or decreasing) part of it is used and the periodic part is ignored (Giambattista Parascandolo, 2017). When we use Time2Vec, however, the periodicity of the sine functions are also being used and seem to be key to the effectiveness of the Time2Vec representation. Fig. 5(c) shows some statistics on the frequencies learned for Event-MNIST where we count the number of learned frequencies that fall within intervals of lengths 0.1 centered at $[ 0 . 0 5 , 0 . 1 5 , \\ldots , 0 . 9 5 ]$ ] (all learned frequencies are between 0 and 1). The figure contains two peaks at 0.35 and 0.85. Since the input to the sine functions for this problem can have a maximum value of 784 (number of pixels in an image), sine functions with frequencies around 0.35 and 0.85 finish (almost) 44 and 106 full periods. The smallest learned frequency is 0.029 which finishes (almost) 3.6 full periods. These values indicate that the model is indeed using the periodicity of the sine functions, not just a monotonically increasing (or decreasing) part of them. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/087e83ffaf5f5917afade18bb06245b15156d18257d36717c25fb05ca79d2905.jpg",
+        "image_caption": [
+            "Figure 4: (a) Initial vs. (b) learned weights and frequencies for our synthesized dataset. "
+        ],
+        "image_footnote": [],
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/8da9c3d24f891893c3a0d28a36f5cd502358906524cea96ca5fb9d75114ada8c.jpg",
+        "image_caption": [
+            "Figure 5: An ablation study of several components in Time2Vec. (a) Comparing different activation functions for Time2Vec on Event-MNIST. Sigmoid and Tanh almost overlap. (b) Comparing frequencies fixed to equally-spaced values, frequencies fixed according to positional encoding (Vaswani et al., 2017), and learned frequencies on Event-MNIST. (c) A histogram of the frequencies learned in Time2Vec for Event-MNIST. The $\\mathbf { X }$ -axis represents frequency intervals and the y-axis represents the number of frequencies in that interval. (d) The performance of TLSTM3 $+$ Time2Vec on CiteULike in terms of Recall $@ 1 0$ with and without the linear term. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The Linear Term: To see the effect of the linear term in Time2Vec, we repeated the experiment for Event-MNIST when the linear term is removed from Time2VecWe observed that the results were not ˙ affected substantially, thus showing that the linear term may not be helpful for Event-MNIST. This might be due to the simplicity of the Event-MNIST dataset. Then we conducted a similar experiment for TLSTM3 on CiteULike (which is a more challenging dataset) and obtained the results in Fig. 5(d). From these results, we can see that the linear term helps facilitate learning functions of time that can be effectively consumed by the model. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION & FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In many tasks for synchronous and asynchronous event predictions, time is an important feature. Previous work has mainly resorted to applying hand-crafted functions to time and concatenating these functions with the rest of the input features. In this work, we presented an approach that automatically learns these functions from data. In particular, we developed Time2Vec, a vector representation for time, using sine and linear activations and showed the effectiveness of this representation across several datasets and several tasks. In the majority of our experiments, Time2Vec improved our results, while the remaining results were not hindered by its application. While sine functions have been argued to complicate the optimization (Lapedes & Farber, 1987; Giambattista Parascandolo, 2017), we did not experience such a complication except for the experiment in Subsection 5.2 on our synthesized dataset when using only a few sine functions. We hypothesize that the main reasons include combining sine functions with a powerful model (e.g., LSTM) and using many sine functions which reduces the distance to the goal (see, e.g., (Neyshabur et al., 2019)). We leave a deeper theoretical analysis of this hypothesis, development of better optimizers, and experimenting with other representations for time as future work. ",
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Corentin Tallec and Yann Ollivier. Can recurrent neural networks warp time? In International Conference on Learning Representation (ICLR), 2018. ",
+        "bbox": [
+            173,
+            603,
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+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "Rakshit Trivedi, Hanjun Dai, Yichen Wang, and Le Song. Know-evolve: Deep temporal reasoning for dynamic knowledge graphs. In ICML, pp. 3462–3471, 2017. ",
+        "bbox": [
+            171,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Utkarsh Upadhyay, Abir De, and Manuel Gomez-Rodriguez. Deep reinforcement learning of marked temporal point processes. In NeurIPS, 2018. ",
+        "bbox": [
+            174,
+            681,
+            823,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NeurIPS, 2017. ",
+        "bbox": [
+            173,
+            720,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kwok-wo Wong, Chi-sing Leung, and Sheng-jiang Chang. Handwritten digit recognition using multilayer feedforward neural networks with periodic and monotonic activation functions. In Pattern Recognition, volume 3, pp. 106–109. IEEE, 2002. ",
+        "bbox": [
+            174,
+            760,
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+            804
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Shuai Xiao, Mehrdad Farajtabar, Xiaojing Ye, Junchi Yan, Le Song, and Hongyuan Zha. Wasserstein learning of deep generative point process models. In NeurIPS, 2017. ",
+        "bbox": [
+            169,
+            813,
+            825,
+            843
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Shuai Xiao, Hongteng Xu, Junchi Yan, Mehrdad Farajtabar, Xiaokang Yang, Le Song, and Hongyuan Zha. Learning conditional generative models for temporal point processes. In AAAI, 2018. ",
+        "bbox": [
+            171,
+            852,
+            823,
+            883
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yu Zhu, Hao Li, Yikang Liao, Beidou Wang, Ziyu Guan, Haifeng Liu, and Deng Cai. What to do next: Modeling user behaviors by time-lstm. In IJCAI, pp. 3602–3608, 2017. ",
+        "bbox": [
+            173,
+            892,
+            823,
+            921
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/00dbf51aef6dc414379150092b4e2229c8a9e041d859b21a9f3ec08e5d3c547f.jpg",
+        "image_caption": [
+            "Figure 6: Comparing $\\mathrm { L S T M + T }$ and LSTM $\\cdot +$ Time2Vec on Event-MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            380,
+            113,
+            604,
+            228
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAIL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            287,
+            429,
+            304
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the experiments on Event-MNIST, N TIDIGITS18 and SOF, we implemented6 our model in PyTorch Paszke et al. (2017). We used Adam optimizer Kingma & Ba (2014) with a learning rate of 0.001. For Event-MNIST and SOF, we fixed the hidden size of the LSTM to 128. For N TIDIGITS18, due to its smaller train set, we fixed the hidden size to 64. We allowed each model 200 epochs. We used a batch size of 512 for Event-MNIST and 128 for N TIDIGITS18 and SOF. For the experiments on Last.FM and CiteULike, we used the code released by Zhu et al. $( 2 0 1 7 ) ^ { 7 }$ without any modifications, except replacing $\\tau$ with $\\mathbf { t } 2 \\mathbf { v } ( \\tau )$ . The only other thing we changed in their code was to change the SAMPLE TIME variable from 3 to 20. SAMPLE TIME controls the number of times we do sampling to compute Recall $@ 1 0$ and $\\mathbf { M R R } @ 1 0$ . We experienced a high variation when sampling only 3 times so we increased the number of times we sample to 20 to make the results more robust. For both Last.FM and CiteULike, Adagrad optimizer is used with a learning rate of 0.01, vocabulary size is 5000, and the maximum length of the sequence is 200. For Last.FM, the hidden size of the LSTM is 128 and for CiteULike, it is 256. For all except the synthesized dataset, we shifted the event times such that the first event of each sequence starts at time 0. ",
+        "bbox": [
+            173,
+            324,
+            825,
+            518
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the fairness of the experiments, we made sure the competing models for all our experiments have an (almost) equal number of parameters. For instance, since adding Time2Vec as an input to the LSTM increases the number of model parameters compared to just adding time as a feature, we reduced the hidden size of the LSTM for this model to ensure the number of model parameters stays (almost) the same. For the experiments involving Time2Vec, unless stated otherwise, we tried vectors with 16, 32 and 64 sine functions (and one linear term). We reported the vector length offering the best performance in the main text. The results for other vector lengths can be found in Appendix B. For the synthetic dataset, we use Adam optimizer with a learning rate of 0.001 without any regularization. The length of the Time2Vec vector is 32. ",
+        "bbox": [
+            173,
+            526,
+            825,
+            650
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/76aa07786e64a3837fd9f14d434cd6ac184d1ae6940f879a76d6c3a580a792b6.jpg",
+        "image_caption": [
+            "Figure 7: Comparing $\\mathrm { L S T M + T }$ and LSTM $+$ Time2Vec on Event-MNIST and raw N TIDIGITS18. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            250,
+            761,
+            738,
+            893
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/d3a24b60f8c8f0493e1757dd3dcc0c4e89eab8a986fb46521120a9364e6b6fa6.jpg",
+        "image_caption": [
+            "Figure 8: Comparing LSTM+T and LSTM+Time2Vec on SOF. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            103,
+            753,
+            218
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b6da3e307b8b779ebf7c7096fdb4a9b4227a850e1255c49ed128d8cbe5db8464.jpg",
+        "image_caption": [
+            "Figure 9: Comparing LSTM $+ \\mathrm { T }$ and LSTM $^ +$ Time2Vec on Last.FM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            263,
+            753,
+            376
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B MORE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            428,
+            343,
+            444
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We ran experiments on other versions of the N TIDIGITS18 dataset as well. Following Anumula et al. (2018), we converted the raw event data to event-binned features by virtue of aggregating active channels through a period of time in which a pre-defined number of events occur. The outcome of binning is thus consecutive frames each with multiple but a fixed number of active channels. In our experiments, we used event-binning with 100 events per frame. For this variant of the dataset, we compared $\\mathrm { L S T M + T }$ and LSTM+Time2Vec similar to the experiments in Section 5.1. The obtained results were on-par. Then, similar to Event-MNIST, we only fed as input the times at which events occurred (i.e. we removed the channels from the input). We allowed the models 1000 epochs to make sure they converge. The obtained results are presented in Fig. 6. It can be viewed that Time2Vec provides an effective representation for time and LSTM $+$ Time2Vec outperforms LSTM $+ \\mathrm { T }$ on this dataset. ",
+        "bbox": [
+            173,
+            459,
+            825,
+            612
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the main text, for the experiments involving Time2Vec, we tested Time2Vec vectors with 16, 32 and 64 sinusoids and reported the best one for the clarity of the diagrams. Here, we show the results for all frequencies. Figures 7, 8, 9, and 10 compare LSTM $+ \\mathrm { T }$ and LSTM $^ +$ Time2Vec for our datasets. Figures 11, and 12 compare TLSTM1 with TLSTM1 $^ +$ Time2Vec on Last.FM and CiteULike. Figures 13, and 14 compare TLSTM3 with TLSTM1 $^ +$ Time2Vec on Last.FM and CiteULike. In most cases, Time2Vec with 64 sinusoids outperforms (or gives on-par results with) the cases with 32 or 16 sinusoids. An exception is TLSTM3 where 16 sinusoids works best. We believe that is because TLSTM3 has two time gates and adding, e.g., 64 temporal components (corresponding to the sinusoids) to each gate makes it overfit to the temporal signals. ",
+        "bbox": [
+            173,
+            619,
+            826,
+            744
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C LSTM ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            763,
+            413,
+            781
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The original LSTM model can be neatly defined with the following equations: ",
+        "bbox": [
+            173,
+            795,
+            684,
+            810
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fdba7f72699780e83d0754365b7c4430972689522e7f4240089608f877b56d5e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { { i _ { j } } = \\sigma \\left( W _ { i } { { \\bf { x } } _ { j } } + U _ { i } { { h _ { j - 1 } } } + { { \\bf { b } } _ { i } } \\right) } \\\\ & { { f _ { j } } = \\sigma \\left( { W _ { f } { { \\bf { x } } _ { j } } + U _ { f } { { h _ { j - 1 } } } + { { b _ { f } } } } \\right) } \\\\ & { { { { \\overline { { c } } } _ { j } } } = T a n h \\left( { W _ { c } { { \\bf { x } } _ { j } } + U _ { c } { { h _ { j - 1 } } } + { { b _ { c } } } } \\right) } \\\\ & { { { { \\overline { { c _ { j } } } } } } = { f _ { t } } \\odot { { \\bf { c } } _ { j - 1 } } + { i _ { j } } \\odot { { { \\overline { { { c _ { j } } } } } } } } \\\\ &  { { { \\sigma _ { j } } } = \\sigma \\left( { W _ { o } { { \\bf { x } } _ { j } } + U _ { o } { { h _ { j - 1 } } } + { { b _ { o } } } } \\right) } \\\\ & { { { { h _ { j } } } = { { o _ { j } } } \\odot T a n h \\left( { { c _ { j } } } \\right) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            815,
+            619,
+            922
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/40d16ef249ff694d5fa296be84bbd3e8a1eabbff493ebb5ca30448726d171f10.jpg",
+        "image_caption": [
+            "Figure 10: Comparing LSTM+T and LSTM $^ +$ Time2Vec on CiteULike. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            104,
+            754,
+            218
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1ee8a037142439db5e80aadffce7ae031a58fd252a1ea45770669fbfec34052f.jpg",
+        "image_caption": [
+            "Figure 11: TLSTM1’s performance on Last.FM with and without Time2Vec. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            263,
+            753,
+            376
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Here $\\mathbf { \\delta } _ { i , \\mathrm { ~ \\tiny ~ \\left. ~ \\right. ~ } }$ , $\\pmb { f } _ { t }$ , and $\\mathbf { \\sigma } _ { \\pmb { o } _ { t } }$ represent the input, forget and output gates respectively, while $\\mathbf { c } _ { t }$ is the memory cell and $\\pmb { h } _ { t }$ is the hidden state. $\\sigma$ and T anh represent the Sigmoid and hyperbolic tangent activation functions respectively. We refer to $\\pmb { x } _ { j }$ as the $j ^ { \\bar { t } h }$ event. ",
+        "bbox": [
+            174,
+            431,
+            825,
+            473
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Peepholes: Gers & Schmidhuber (2000) introduced a variant of the LSTM architecture where the input, forget, and output gates peek into the memory cell. In this variant, ${ \\pmb w } _ { p i } \\odot { \\pmb c } _ { j - 1 } , { \\pmb w } _ { p f } \\odot { \\pmb c } _ { j - 1 }$ and ${ \\pmb w } _ { p o } \\odot { \\pmb c } _ { j }$ are added to the linear parts of Eq. (2), (3), and (6) respectively, where ${ \\pmb w } _ { p i } , { \\pmb w } _ { p f }$ , and ${ \\pmb w } _ { p o }$ are learnable parameters. ",
+        "bbox": [
+            174,
+            479,
+            825,
+            536
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { L S T M + T } \\mathbf { : }$ Let $\\tau _ { j }$ represent the time features for the $j ^ { t h }$ event in the input and let $\\pmb { x } _ { j } ^ { \\prime } = [ \\pmb { x } _ { j } ; \\pmb { \\tau } _ { j } ]$ Then $\\mathrm { L S T M + T }$ uses the exact same equations as the standard LSTM (denoted above) except that $\\pmb { x } _ { j }$ is replaced with $\\pmb { x } _ { j } ^ { \\prime }$ . ",
+        "bbox": [
+            173,
+            544,
+            825,
+            588
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "TimeLSTM: We explain TLSTM1 and TLSTM3 which have been used in our experiments. For clarity of writing, we do not include the peephole terms in the equations but they are used in the experiments. In TLSTM1, a new time gate is introduced as in Eq. equation 8 and Eq. equation 5 and equation 6 are updated to Eq. equation 9 and equation 10 respectively: ",
+        "bbox": [
+            173,
+            593,
+            825,
+            650
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d141487b621fc6f4f1a8f24bf1d5a72bcdc466b0ef14263944d0f292fa94efc.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\pmb { t } _ { j } = \\sigma \\left( \\pmb { W } _ { t } \\pmb { x } _ { j } + \\sigma \\left( \\pmb { u } _ { t } \\tau _ { j } \\right) + \\pmb { b } _ { t } \\right) } \\\\ & { \\pmb { c } _ { j } = \\pmb { f } _ { j } \\odot \\pmb { c } _ { j - 1 } + \\pmb { i } _ { j } \\odot \\pmb { t } _ { j } \\odot \\overline { { \\pmb { c } _ { j } } } } \\\\ & { \\pmb { o } _ { j } = \\sigma \\left( \\pmb { W } _ { o } \\pmb { x } _ { j } + \\pmb { v } _ { t } \\tau _ { j } + \\pmb { U } _ { o } \\pmb { h } _ { j - 1 } + \\pmb { b } _ { o } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            655,
+            630,
+            712
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$t _ { j }$ controls the influence of the current input on the prediction and makes the required information from timing history get stored on the cell state. TLSTM3 uses two time gates: ",
+        "bbox": [
+            169,
+            713,
+            821,
+            742
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f535d695e47b83109ae5489cf96fecb4e7ca91a05d5cd24ad2150d121996e929.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { t 1 } _ { j } = \\sigma \\left( \\pmb { W } _ { t 1 } \\pmb { x } _ { j } + \\sigma \\left( \\pmb { u } _ { t 1 } \\tau _ { j } \\right) + \\pmb { b } _ { t 1 } \\right) } \\\\ { \\pmb { t 2 } _ { j } = \\sigma \\left( \\pmb { W } _ { t 2 } \\pmb { x } _ { j } + \\sigma \\left( \\pmb { u } _ { t 2 } \\tau _ { j } \\right) + \\pmb { b } _ { t 2 } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            747,
+            619,
+            786
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where the elements of $W _ { t 1 }$ are constrained to be non-positive. $\\pmb { t 1 }$ is used for controlling the influence of the last consumed item and $\\pmb { t 2 }$ stores the $\\tau \\mathbf { S }$ thus enabling modeling long range dependencies. TLSTM3 couples the input and forget gates following Greff et al. (2017) along with the $\\pmb { t 1 }$ and $\\pmb { t 2 }$ gates and replaces Eq. (5) to (7) with the following: ",
+        "bbox": [
+            173,
+            789,
+            825,
+            844
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/90fc15217213a6d5c9172dc6649fc072fc9a6ccea7fc32e5973a0019da2c51f2.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\tilde { \\mathbf { c } } _ { j } = \\left( 1 - i _ { j } \\odot t { \\bf 1 } _ { j } \\right) \\odot \\mathbf { c } _ { j - 1 } + i _ { j } \\odot t { \\bf 1 } _ { j } \\odot \\bar { \\mathbf { c } } _ { j } } \\\\ & { \\mathbf { c } _ { j } = \\left( 1 - i _ { j } \\right) \\odot \\mathbf { c } _ { j - 1 } + i _ { j } \\odot t { \\bf 2 } _ { j } \\odot \\bar { \\mathbf { c } } _ { j } } \\\\ & { \\mathbf { o } _ { j } = \\sigma \\left( W _ { o } \\mathbf { x } _ { j } + \\mathbf { v } _ { t } \\tau _ { j } + U _ { o } \\mathbf { h } _ { j - 1 } + \\mathbf { b } _ { o } \\right) } \\\\ & { \\mathbf { h } _ { j } = o _ { j } \\odot T a n h \\left( \\tilde { \\mathbf { c } } _ { j } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            849,
+            643,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/2792f646e18eaf88109b2d7268bb9be9b2f1ad4ca537fb0c4e885ec977ca9f2c.jpg",
+        "image_caption": [
+            "Figure 12: TLSTM1’s performance on CiteULike with and without Time2Vec. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            106,
+            754,
+            218
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/63be7796c2b8a7c7b18e4cff05ef2e9180793681d99a0b276524b82af87f436c.jpg",
+        "image_caption": [
+            "Figure 13: TLSTM3’s performance on Last.FM with and without Time2Vec. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            263,
+            753,
+            377
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/da71988ac8cf0d1afe94a485ec7ac9e71ec909c5b026ed379f6867a32deef577.jpg",
+        "image_caption": [
+            "Figure 14: TLSTM3’s performance on CiteULike with and without Time2Vec. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            421,
+            754,
+            536
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Zhu et al. (2017) use $\\tau _ { j } = \\Delta t _ { j }$ in their experiments, where $\\Delta t _ { j }$ is the duration between the current and the last event. ",
+        "bbox": [
+            173,
+            589,
+            823,
+            619
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "TimeLSTM $^ +$ Time2Vec: To replace time in TLSTM1 with Time2Vec, we modify Eq. (8) and (10) as follows: ",
+        "bbox": [
+            173,
+            625,
+            820,
+            654
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d02e576b19854ff438c8fa35503eb75838e3e240401a86843011ef2bd05618f6.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\pmb { t _ { j } } = \\sigma \\left( \\pmb { W _ { t } } \\pmb { x _ { j } } + \\sigma \\left( \\pmb { U _ { t } } \\pmb { \\mathrm { t } } 2 \\mathbf { v } ( \\tau ) \\right) + \\pmb { b _ { t } } \\right) } \\\\ & { \\pmb { o _ { j } } = \\sigma ( \\pmb { W _ { o } } \\pmb { x _ { j } } + \\pmb { V _ { t } } \\pmb { \\mathrm { t } } 2 \\mathbf { v } ( \\tau ) + \\pmb { U _ { o } } \\pmb { h _ { j - 1 } } + \\pmb { b _ { o } } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            659,
+            647,
+            698
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "i.e., $\\tau$ is replaced with $\\mathbf { t } 2 \\mathbf { v } ( \\tau )$ , $\\mathbf { \\pmb { u } } _ { t }$ is replaced with $\\boldsymbol { U } _ { t }$ , and ${ \\pmb v } _ { t }$ is replaced with $V _ { t }$ . Similarly, for TLSTM3 we modify Eq. (11), (12) and (15) as follows: ",
+        "bbox": [
+            171,
+            702,
+            823,
+            731
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84d83c3a74908778584095bde89293133a25fef3282065a2c781738c6a468257.jpg",
+        "text": "$$\n\\begin{array} { r } { t \\pmb { 1 } _ { j } = \\sigma \\left( \\pmb { W } _ { t 1 } \\pmb { x } _ { j } + \\sigma \\left( \\pmb { U } _ { t 1 } \\pmb { \\mathrm { t } } 2 \\pmb { \\mathrm { v } } ( \\tau ) \\right) + \\pmb { b } _ { t 1 } \\right) } \\\\ { t \\pmb { 2 } _ { j } = \\sigma \\left( \\pmb { W } _ { t 2 } \\pmb { x } _ { j } + \\sigma \\left( \\pmb { U } _ { t 2 } \\pmb { \\mathrm { t } } 2 \\pmb { \\mathrm { v } } ( \\tau ) \\right) + \\pmb { b } _ { t 2 } \\right) } \\\\ { \\pmb { o } _ { j } = \\sigma ( \\pmb { W } _ { o } \\pmb { x } _ { j } + \\pmb { V } _ { t } \\pmb { \\mathrm { t } } 2 \\pmb { \\mathrm { v } } ( \\tau ) + \\pmb { U } _ { o } \\pmb { h } _ { j - 1 } + \\pmb { b } _ { o } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            734,
+            645,
+            792
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            808,
+            277,
+            824
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Time2Vec is invariant to time rescaling. ",
+        "bbox": [
+            173,
+            839,
+            537,
+            854
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof. Consider the following Time2Vec representation $\\mathcal { M } _ { 1 }$ : ",
+        "bbox": [
+            173,
+            869,
+            576,
+            886
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e0e4237ab1de27c0c3c25f046c569a3e1fd5c070b949deaa94a76a300aa589f.jpg",
+        "text": "$$\n\\mathbf t 2 \\mathbf v ( \\tau ) [ i ] = { \\left\\{ \\begin{array} { l l } { \\omega _ { i } \\tau + \\varphi _ { i } , } & { { \\mathrm { i f } } i = 0 . } \\\\ { \\sin { ( \\omega _ { i } \\tau + \\varphi _ { i } ) } , } & { { \\mathrm { i f } } 1 \\leq i \\leq k . } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            892,
+            650,
+            928
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Replacing $\\tau$ with $\\alpha \\cdot \\tau$ (for $\\alpha > 0$ ), the Time2Vec representation updates as follows: ",
+        "bbox": [
+            173,
+            102,
+            725,
+            119
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e11041b52eceeeece7ea224fd9bd116dd3ea688cea78cffb8cf1c01fdb7b36c.jpg",
+        "text": "$$\n\\mathbf { t } 2 \\mathbf { v } ( \\alpha \\cdot \\tau ) [ i ] = { \\left\\{ \\begin{array} { l l } { \\omega _ { i } ( \\alpha \\cdot \\tau ) + \\varphi _ { i } , } & { { \\mathrm { i f } } i = 0 . } \\\\ { \\sin { ( \\omega _ { i } ( \\alpha \\cdot \\tau ) + \\varphi _ { i } ) } , } & { { \\mathrm { i f } } 1 \\leq i \\leq k . } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            126,
+            678,
+            161
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Consider anosame way as Time2Vec representation . This proves that Time2 $\\mathcal { M } _ { 2 }$ with frequencies s invariant to time $\\begin{array} { r } { \\omega _ { i } ^ { \\prime } = \\frac { \\omega _ { i } } { \\alpha } } \\end{array}$ . Then g. $\\mathcal { M } _ { 2 }$ behaves in the $\\mathcal { M } _ { 1 }$ ",
+        "bbox": [
+            174,
+            169,
+            826,
+            199
+        ],
+        "page_idx": 15
+    }
+]

parse/train/rytstxWAW/rytstxWAW.md

@@ -0,0 +1,522 @@
+# FASTGCN: FAST LEARNING WITH GRAPH CONVOLUTIONAL NETWORKS VIA IMPORTANCE SAMPLING
+
+Jie Chen∗, Tengfei Ma∗, Cao Xiao  
+IBM Research  
+chenjie@us.ibm.com, Tengfei.Ma1@ibm.com, cxiao@us.ibm.com
+
+# ABSTRACT
+
+The graph convolutional networks (GCN) recently proposed by Kipf and Welling are an effective graph model for semi-supervised learning. This model, however, was originally designed to be learned with the presence of both training and test data. Moreover, the recursive neighborhood expansion across layers poses time and memory challenges for training with large, dense graphs. To relax the requirement of simultaneous availability of test data, we interpret graph convolutions as integral transforms of embedding functions under probability measures. Such an interpretation allows for the use of Monte Carlo approaches to consistently estimate the integrals, which in turn leads to a batched training scheme as we propose in this work—FastGCN. Enhanced with importance sampling, FastGCN not only is efficient for training but also generalizes well for inference. We show a comprehensive set of experiments to demonstrate its effectiveness compared with GCN and related models. In particular, training is orders of magnitude more efficient while predictions remain comparably accurate.
+
+# 1 INTRODUCTION
+
+Graphs are universal representations of pairwise relationship. Many real world data come naturally in the form of graphs; e.g., social networks, gene expression networks, and knowledge graphs. To improve the performance of graph-based learning tasks, such as node classification and link prediction, recently much effort is made to extend well-established network architectures, including recurrent neural networks (RNN) and convolutional neural networks (CNN), to graph data; see, e.g., Bruna et al. (2013); Duvenaud et al. (2015); Li et al. (2015); Jain et al. (2015); Henaff et al. (2015); Niepert et al. (2016); Kipf & Welling (2016a;b).
+
+Whereas learning feature representations for graphs is an important subject among this effort, here, we focus on the feature representations for graph vertices. In this vein, the closest work that applies a convolution architecture is the graph convolutional network (GCN) (Kipf & Welling, 2016a;b). Borrowing the concept of a convolution filter for image pixels or a linear array of signals, GCN uses the connectivity structure of the graph as the filter to perform neighborhood mixing. The architecture may be elegantly summarized by the following expression:
+
+$$
+H ^ { ( l + 1 ) } = \sigma ( \hat { A } H ^ { ( l ) } W ^ { ( l ) } ) ,
+$$
+
+where $\hat { A }$ is some normalization of the graph adjacency matrix, $H ^ { ( l ) }$ contains the embedding (rowwise) of the graph vertices in the $l$ th layer, $\mathbf { \overline { { W } } } ^ { ( l ) }$ is a parameter matrix, and $\sigma$ is nonlinearity.
+
+As with many graph algorithms, the adjacency matrix encodes the pairwise relationship for both training and test data. The learning of the model as well as the embedding is performed for both data simultaneously, at least as the authors proposed. For many applications, however, test data may not be readily available, because the graph may be constantly expanding with new vertices (e.g. new members of a social network, new products to a recommender system, and new drugs for functionality tests). Such scenarios require an inductive scheme that learns a model from only a training set of vertices and that generalizes well to any augmentation of the graph.
+
+A more severe challenge for GCN is that the recursive expansion of neighborhoods across layers incurs expensive computations in batched training. Particularly for dense graphs and powerlaw graphs, the expansion of the neighborhood for a single vertex quickly fills up a large portion of the graph. Then, a usual mini-batch training will involve a large amount of data for every batch, even with a small batch size. Hence, scalability is a pressing issue to resolve for GCN to be applicable to large, dense graphs.
+
+To address both challenges, we propose to view graph convolutions from a different angle and interpret them as integral transforms of embedding functions under probability measures. Such a view provides a principled mechanism for inductive learning, starting from the formulation of the loss to the stochastic version of the gradient. Specifically, we interpret that graph vertices are iid samples of some probability distribution and write the loss and each convolution layer as integrals with respect to vertex embedding functions. Then, the integrals are evaluated through Monte Carlo approximation that defines the sample loss and the sample gradient. One may further alter the sampling distribution (as in importance sampling) to reduce the approximation variance.
+
+The proposed approach, coined FastGCN, not only rids the reliance on the test data but also yields a controllable cost for per-batch computation. At the time of writing, we notice a newly published work GraphSAGE (Hamilton et al., 2017) that proposes also the use of sampling to reduce the computational footprint of GCN. Our sampling scheme is more economic, resulting in a substantial saving in the gradient computation, as will be analyzed in more detail in Section 3.3. Experimental results in Section 4 indicate that the per-batch computation of FastGCN is more than an order of magnitude faster than that of GraphSAGE, while classification accuracies are highly comparable.
+
+# 2 RELATED WORK
+
+Over the past few years, several graph-based convolution network models emerged for addressing applications of graph-structured data, such as the representation of molecules (Duvenaud et al., 2015). An important stream of work is built on spectral graph theory (Bruna et al., 2013; Henaff et al., 2015; Defferrard et al., 2016). They define parameterized filters in the spectral domain, inspired by graph Fourier transform. These approaches learn a feature representation for the whole graph and may be used for graph classification.
+
+Another line of work learns embeddings for graph vertices, for which Goyal & Ferrara (2017) is a recent survey that covers comprehensively several categories of methods. A major category consists of factorization based algorithms that yield the embedding through matrix factorizations; see, e.g., Roweis & Saul (2000); Belkin & Niyogi (2001); Ahmed et al. (2013); Cao et al. (2015); Ou et al. (2016). These methods learn the representations of training and test data jointly. Another category is random walk based methods (Perozzi et al., 2014; Grover & Leskovec, 2016) that compute node representations through exploration of neighborhoods. LINE (Tang et al., 2015) is also such a technique that is motivated by the preservation of the first and second-order proximities. Meanwhile, there appear a few deep neural network architectures, which better capture the nonlinearity within graphs, such as SDNE (Wang et al., 2016). As motivated earlier, GCN (Kipf & Welling, 2016a) is the model on which our work is based.
+
+The most relevant work to our approach is GraphSAGE (Hamilton et al., 2017), which learns node representations through aggregation of neighborhood information. One of the proposed aggregators employs the GCN architecture. The authors also acknowledge the memory bottleneck of GCN and hence propose an ad hoc sampling scheme to restrict the neighborhood size. Our sampling approach is based on a different and more principled formulation. The major distinction is that we sample vertices rather than neighbors. The resulting computational savings are analyzed in Section 3.3.
+
+# 3 TRAINING AND INFERENCE THROUGH SAMPLING
+
+One striking difference between GCN and many standard neural network architectures is the lack of independence in the sample loss. Training algorithms such as SGD and its batch generalization are designed based on the additive nature of the loss function with respect to independent data samples. For graphs, on the other hand, each vertex is convolved with all its neighbors and hence defining a sample gradient that is efficient to compute is beyond straightforward.
+
+Concretely, consider the standard SGD scenario where the loss is the expectation of some function $g$ with respect to a data distribution $D$ :
+
+$$
+L = \operatorname { E } _ { x \sim D } [ g ( W ; x ) ] .
+$$
+
+Here, $W$ denotes the model parameter to be optimized. Of course, the data distribution is generally unknown and one instead minimizes the empirical loss through accessing $n$ iid samples $x _ { 1 } , \ldots , x _ { n }$ :
+
+$$
+L _ { \mathrm { e m p } } = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } g ( W ; x _ { i } ) , \qquad x _ { i } \sim D , \ \forall i .
+$$
+
+In each step of SGD, the gradient is approximated by $\nabla g ( W ; x _ { i } )$ , an (assumed) unbiased sample of $\nabla L$ . One may interpret that each gradient step makes progress toward the sample loss $g ( W ; x _ { i } )$ . The sample loss and the sample gradient involve only one single sample $x _ { i }$ .
+
+For graphs, one may no longer leverage the independence and compute the sample gradient $\nabla g ( W ; x _ { i } )$ by discarding the information of $i$ ’s neighboring vertices and their neighbors, recursively. We therefore seek an alternative formulation. In order to cast the learning problem under the same sampling framework, let us assume that there is a (possibly infinite) graph $G ^ { \prime }$ with the vertex set $V ^ { \prime }$ associated with a probability space $( V ^ { \prime } , F , P )$ , such that for the given graph $G$ , it is an induced subgraph of $G ^ { \prime }$ and its vertices are iid samples of $V ^ { \prime }$ according to the probability measure $P$ . For the probability space, $V ^ { \prime }$ serves as the sample space and $F$ may be any event space (e.g., the power set $F = 2 ^ { V ^ { \prime } }$ ). The probability measure $P$ defines a sampling distribution.
+
+To resolve the problem of lack of independence caused by convolution, we interpret that each layer of the network defines an embedding function of the vertices (random variable) that are tied to the same probability measure but are independent. See Figure 1. Specifically, recall the architecture of GCN
+
+$$
+\tilde { H } ^ { ( l + 1 ) } = \hat { A } H ^ { ( l ) } W ^ { ( l ) } , \quad H ^ { ( l + 1 ) } = \sigma ( \tilde { H } ^ { ( l + 1 ) } ) , \quad l = 0 , \dots , M - 1 , \quad L = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } g ( H ^ { ( M ) } ( i , : ) ) .
+$$
+
+For the functional generalization, we write
+
+$$
+\tilde { h } ^ { ( l + 1 ) } ( v ) = \int \hat { A } ( v , u ) h ^ { ( l ) } ( u ) W ^ { ( l ) } d P ( u ) , \quad h ^ { ( l + 1 ) } ( v ) = \sigma ( \tilde { h } ^ { ( l + 1 ) } ( v ) ) , \quad l = 0 , \ldots , M - 1 ,
+$$
+
+$$
+L = \mathrm { E } _ { v \sim P } [ g ( h ^ { ( M ) } ( v ) ) ] = \int g ( h ^ { ( M ) } ( v ) ) d P ( v ) .
+$$
+
+Here, $u$ and $v$ are independent random variables, both of which have the same probability measure $P$ . The function $\it { h ^ { ( l ) } }$ is interpreted as the embedding function from the lth layer. The embedding functions from two consecutive layers are related through convolution, expressed as an integral transform, where the kernel $\hat { A } ( v , u )$ corresponds to the $( v , u )$ element of the matrix $\hat { A }$ . The loss is the expectation of $g ( h ^ { ( M ) } )$ for the final embedding $h ^ { ( M ) }$ . Note that the integrals are not the usual Riemann–Stieltjes integrals, because the variables $u$ and $v$ are graph vertices but not real numbers; however, this distinction is only a matter of formalism.
+
+Writing GCN in the functional form allows for evaluating the integrals in the Monte Carlo manner, which leads to a batched training algorithm and also to a natural separation of training and test data, as in inductive learning. For each layer $l$ , we use $t _ { l }$ iid samples $u _ { 1 } ^ { ( l ) } , \ldots , u _ { t _ { l } } ^ { ( l ) } \sim P$ to approximately evaluate the integral transform (2); that is,
+
+$$
+\tilde { h } _ { t _ { l + 1 } } ^ { ( l + 1 ) } ( v ) : = \frac { 1 } { t _ { l } } \sum _ { j = 1 } ^ { t _ { l } } \hat { A } ( v , u _ { j } ^ { ( l ) } ) h _ { t _ { l } } ^ { ( l ) } ( u _ { j } ^ { ( l ) } ) W ^ { ( l ) } , \quad h _ { t _ { l + 1 } } ^ { ( l + 1 ) } ( v ) : = \sigma ( \tilde { h } _ { t _ { l + 1 } } ^ { ( l + 1 ) } ( v ) ) , \quad l = 0 , \ldots , M - 1 ,
+$$
+
+with the convention $h _ { t _ { 0 } } ^ { ( 0 ) } \equiv h ^ { ( 0 ) }$ . Then, the loss $L$ in (3) admits an estimator
+
+$$
+L _ { t _ { 0 } , t _ { 1 } , \dots , t _ { M } } : = \frac { 1 } { t _ { M } } \sum _ { i = 1 } ^ { t _ { M } } g ( h _ { t _ { M } } ^ { ( M ) } ( u _ { i } ^ { ( M ) } ) ) .
+$$
+
+The follow result establishes that the estimator is consistent. The proof is a recursive application of the law of large numbers and the continuous mapping theorem; it is given in the appendix.
+
+![](images/cf270237a65cf7e32f03eb558345d72b4d11db5bcbb3bcbf03aebfc5bedf4549.jpg)  
+Figure 1: Two views of GCN. On the left (graph convolution view), each circle represents a graph vertex. On two consecutive rows, a circle $i$ is connected (in gray line) with circle $j$ if the two corresponding vertices in the graph are connected. A convolution layer uses the graph connectivity structure to mix the vertex features/embeddings. On the right (integral transform view), the embedding function in the next layer is an integral transform (illustrated by the orange fanout shape) of the one in the previous layer. For the proposed method, all integrals (including the loss function) are evaluated by using Monte Carlo sampling. Correspondingly in the graph view, vertices are subsampled in a bootstrapping manner in each layer to approximate the convolution. The sampled portions are collectively denoted by the solid blue circles and the orange lines.
+
+Theorem 1. If $g$ and $\sigma$ are continuous, then
+
+$$
+\operatorname * { l i m } _ { t _ { 0 } , t _ { 1 } , . . . , t _ { M }  \infty } { \cal L } _ { t _ { 0 } , t _ { 1 } , . . . , t _ { M } } = { \cal L } w i t h p r o b a b i l i t y o n e .
+$$
+
+In practical use, we are given a graph whose vertices are already assumed to be samples. Hence, we will need bootstrapping to obtain a consistent estimate. In particular, for the network architecture (1), the output $H ^ { ( M ) }$ is split into batches as usual. We will still use $u _ { 1 } ^ { ( M ) } , \dots , u _ { t _ { M } } ^ { ( M ) }$ to denote a batch of vertices, which come from the given graph. For each batch, we sample (with replacement) uniformly each layer and obtain samples $u _ { i } ^ { ( l ) }$ , $i = 1 , \ldots , t _ { l }$ , $l = 0 , \ldots , M - 1$ . Such a procedure is equivalent to uniformly sampling the rows of $H ^ { ( l ) }$ for each $l$ . Then, we obtain the batch loss
+
+$$
+L _ { \mathrm { b a t c h } } = \frac { 1 } { t _ { M } } \sum _ { i = 1 } ^ { t _ { M } } g ( H ^ { ( M ) } ( u _ { i } ^ { ( M ) } , : ) ) ,
+$$
+
+where, recursively,
+
+$$
+H ^ { ( l + 1 ) } ( v , : ) = \sigma \left( \frac { n } { t _ { l } } \sum _ { j = 1 } ^ { t _ { l } } \hat { A } ( v , u _ { j } ^ { ( l ) } ) H ^ { ( l ) } ( u _ { j } ^ { ( l ) } , : ) W ^ { ( l ) } \right) , \quad l = 0 , \ldots , M - 1 .
+$$
+
+Here, the $n$ inside the activation function $\sigma$ is the number of vertices in the given graph and is used to account for the normalization difference between the matrix form (1) and the integral form (2). The corresponding batch gradient may be straightforwardly obtained through applying the chain rule on each $H ^ { ( l ) }$ . See Algorithm 1.
+
+# 3.1 VARIANCE REDUCTION
+
+As for any estimator, one is interested in improving its variance. Whereas computing the full variance is highly challenging because of nonlinearity in all the layers, it is possible to consider each single layer and aim at improving the varianceSpecifically, consider for the lth layer, the function $\tilde { h } _ { t _ { l + 1 } } ^ { ( l + 1 ) } ( v )$ bedding function before nonlinearity. as an approximation to the convolution $\begin{array} { r l } { \int \hat { A } ( v , u ) h _ { t _ { l } } ^ { ( l ) } ( u ) W ^ { ( l ) } d P ( u ) } \end{array}$ . When taking $t _ { l + 1 }$ samples $v = u _ { 1 } ^ { ( l + 1 ) } , \dots , u _ { t _ { l + 1 } } ^ { ( l + 1 ) }$ , the sample average of by this lay $\tilde { h } _ { t _ { l + 1 } } ^ { ( l + 1 ) } ( v )$ admits a variance that captures the deviation from the eventual loss contributed, we seek an improvement of this variance. Now that we consider each layer separately, we will do the following change of notation to keep the expressions less cumbersome:
+
+# Algorithm 1 FastGCN batched training (one epoch)
+
+1: for each batch do   
+2: For each layer $l$ , sample uniformly $t _ { l }$ vertices $u _ { 1 } ^ { ( l ) } , \ldots , u _ { t _ { l } } ^ { ( l ) }$   
+3: for each layer $l$ do $\triangleright$ Compute batch gradient $\nabla L _ { \mathrm { b a t c h } }$   
+4: If $v$ is sampled in the next layer,
+
+$$
+\nabla \tilde { H } ^ { ( l + 1 ) } ( v , : ) \gets \frac { n } { t _ { l } } \sum _ { j = 1 } ^ { t _ { l } } \hat { A } ( v , u _ { j } ^ { ( l ) } ) \nabla \Big \{ H ^ { ( l ) } ( u _ { j } ^ { ( l ) } , : ) W ^ { ( l ) } \Big \}
+$$
+
+5: end for   
+6: $W  W - \eta \nabla L _ { \mathrm { b a t c h } }$   
+7: end for
+
+. SGD step
+
+<table><tr><td></td><td>Function</td><td>Samples</td><td>Num. samples</td></tr><tr><td>Layer l +1; random variable u</td><td></td><td>（+1）→Ui 山i</td><td>t+1→s</td></tr><tr><td>Layer l; random variable u</td><td>m (u)W(l) → x(u)</td><td>uj (1) →uj</td><td>t→t</td></tr></table>
+
+Under the joint distribution of $v$ and $u$ , the aforementioned sample average is
+
+$$
+G : = \frac { 1 } { s } \sum _ { i = 1 } ^ { s } y ( v _ { i } ) = \frac { 1 } { s } \sum _ { i = 1 } ^ { s } \left( \frac { 1 } { t } \sum _ { j = 1 } ^ { t } \hat { A } ( v _ { i } , u _ { j } ) x ( u _ { j } ) \right) .
+$$
+
+First, we have the following result.
+
+Proposition 2. The variance of $G$ admits
+
+$$
+\mathrm { V a r } \{ G \} = R + \frac { 1 } { s t } \int \int \hat { A } ( v , u ) ^ { 2 } x ( u ) ^ { 2 } d P ( u ) d P ( v ) ,
+$$
+
+where
+
+$$
+R = \frac { 1 } { s } \left( 1 - \frac { 1 } { t } \right) \int e ( v ) ^ { 2 } d P ( v ) - \frac { 1 } { s } \left( \int e ( v ) d P ( v ) \right) ^ { 2 } \quad a n d \quad e ( v ) = \int \hat { A } ( v , u ) x ( u ) d P ( u ) .
+$$
+
+The variance (6) consists of two parts. The first part $R$ leaves little room for improvement, because the sampling in the $v$ space is not done in this layer. The second part (the double integral), on the other hand, depends on how the $u _ { j }$ ’s in this layer are sampled. The current result (6) is the consequence of sampling $u _ { j }$ ’s by using the probability measure $P$ . One may perform importance sampling, altering the sampling distribution to reduce variance. Specifically, let $Q ( u )$ be the new probability measure, where the $u _ { j }$ ’s are drawn from. We hence define the new sample average approximation
+
+$$
+y _ { Q } ( v ) : = \frac { 1 } { t } \sum _ { j = 1 } ^ { t } \hat { A } ( v , u _ { j } ) x ( u _ { j } ) \left( \frac { d P ( u ) } { d Q ( u ) } \bigg | _ { u _ { j } } \right) , \qquad u _ { 1 } , \ldots , u _ { t } \sim Q ,
+$$
+
+and the quantity of interest
+
+$$
+G _ { Q } : = \frac { 1 } { s } \sum _ { i = 1 } ^ { s } y _ { Q } ( v _ { i } ) = \frac { 1 } { s } \sum _ { i = 1 } ^ { s } \left( \frac { 1 } { t } \sum _ { j = 1 } ^ { t } \hat { A } ( v _ { i } , u _ { j } ) x ( u _ { j } ) \left( \left. \frac { d P ( u ) } { d Q ( u ) } \right| _ { u _ { j } } \right) \right) .
+$$
+
+Clearly, the expectation of $G _ { Q }$ is the same as that of $G$ , regardless of the new measure $Q$ . The following result gives the optimal $Q$ .
+
+# Theorem 3. If
+
+$$
+d Q ( u ) = \frac { b ( u ) | x ( u ) | d P ( u ) } { \int b ( u ) | x ( u ) | d P ( u ) } \quad w h e r e \quad b ( u ) = \left[ \int \hat { A } ( v , u ) ^ { 2 } d P ( v ) \right] ^ { \frac { 1 } { 2 } } ,
+$$
+
+then the variance of $G _ { Q }$ admits
+
+$$
+\mathrm { V a r } \{ G _ { Q } \} = R + { \frac { 1 } { s t } } \left[ \int b ( u ) | x ( u ) | d P ( u ) \right] ^ { 2 } ,
+$$
+
+where $R$ is defined in Proposition 2. The variance is minimum among all choices of $Q$ .
+
+A drawback of defining the sampling distribution $Q$ in this manner is that it involves $| x ( u ) |$ , which constantly changes during training. It corresponds to the product of the embedding matrix $H ^ { ( l ) }$ and the parameter matrix $\mathsf { \bar { W } } ^ { ( l ) }$ . The parameter matrix is updated in every iteration; and the matrix product is expensive to compute. Hence, the cost of computing the optimal measure $Q$ is quite high.
+
+As a compromise, we consider a different choice of $Q$ , which involves only $b ( u )$ . The following proposition gives the precise definition. The resulting variance may or may not be smaller than (6). In practice, however, we find that it is almost always helpful.
+
+# Proposition 4. If
+
+$$
+d Q ( u ) = \frac { b ( u ) ^ { 2 } d P ( u ) } { \int b ( u ) ^ { 2 } d P ( u ) }
+$$
+
+where $b ( u )$ is defined in (7), then the variance of $G _ { Q }$ admits
+
+$$
+\mathrm { V a r } \{ G _ { Q } \} = R + \frac { 1 } { s t } \int b ( u ) ^ { 2 } d P ( u ) \int x ( u ) ^ { 2 } d P ( u ) ,
+$$
+
+where $R$ is defined in Proposition 2.
+
+With this choice of the probability measure $Q$ , the ratio $d Q ( u ) / d P ( u )$ is proportional to $b ( u ) ^ { 2 }$ , which is simply the integral of $\hat { A } ( v , u ) ^ { 2 }$ with respect to $v$ . In practical use, for the network architecture (1), we define a probability mass function for all the vertices in the given graph:
+
+$$
+q ( u ) = \| \hat { A } ( : , u ) \| ^ { 2 } / \sum _ { u ^ { \prime } \in V } \| \hat { A } ( : , u ^ { \prime } ) \| ^ { 2 } , \quad u \in V
+$$
+
+and sample $t$ vertices $u _ { 1 } , \ldots , u _ { t }$ according to this distribution. From the expression of $q$ , we see that it has no dependency on $l$ ; that is, the sampling distribution is the same for all layers. To summarize, the batch loss $L _ { \mathrm { b a t c h } }$ in (4) now is recursively expanded as
+
+$$
+H ^ { ( l + 1 ) } ( v , : ) = \sigma \left( \frac { 1 } { t _ { l } } \sum _ { j = 1 } ^ { t _ { l } } \frac { \hat { A } ( v , u _ { j } ^ { ( l ) } ) H ^ { ( l ) } ( u _ { j } ^ { ( l ) } , : ) W ^ { ( l ) } } { q ( u _ { j } ^ { ( l ) } ) } \right) , \quad u _ { j } ^ { ( l ) } \sim q , \quad l = 0 , \ldots , M - 1 .
+$$
+
+The major difference between (5) and (10) is that the former obtains samples uniformly whereas the latter according to $q$ . Accordingly, the scaling inside the summation changes. The corresponding batch gradient may be straightforwardly obtained through applying the chain rule on each $\bar { H } ^ { ( l ) }$ . See Algorithm 2.
+
+# Algorithm 2 FastGCN batched training (one epoch), improved version
+
+1: For each vertex $u$ , compute sampling probability $q ( u ) \propto \lVert \hat { A } ( : , u ) \rVert ^ { 2 }$   
+2: for each batch do For each layer l, sample tl vertices u(l)1 , . $u _ { 1 } ^ { ( l ) } , \ldots , u _ { t _ { l } } ^ { ( l ) }$ according to distribution $q$   
+4: . Compute batch gradient $\nabla L _ { \mathrm { b a t c h } }$   
+5: If $v$ is sampled in the next layer,
+
+$$
+\nabla \tilde { H } ^ { ( l + 1 ) } ( v , : ) \gets \frac { 1 } { t _ { l } } \sum _ { j = 1 } ^ { t _ { l } } \frac { \hat { A } ( v , u _ { j } ^ { ( l ) } ) } { q ( u _ { j } ^ { ( l ) } ) } \nabla \Big \{ H ^ { ( l ) } ( u _ { j } ^ { ( l ) } , : ) W ^ { ( l ) } \Big \}
+$$
+
+. SGD step
+
+# 3.2 INFERENCE
+
+The sampling approach described in the preceding subsection clearly separates out test data from training. Such an approach is inductive, as opposed to transductive that is common for many graph algorithms. The essence is to cast the set of graph vertices as iid samples of a probability distribution, so that the learning algorithm may use the gradient of a consistent estimator of the loss to perform parameter update. Then, for inference, the embedding of a new vertex may be either computed by using the full GCN architecture (1), or approximated through sampling as is done in parameter learning. Generally, using the full architecture is more straightforward and easier to implement.
+
+# 3.3 COMPARISON WITH GRAPHSAGE
+
+GraphSAGE (Hamilton et al., 2017) is a newly proposed architecture for generating vertex embeddings through aggregating neighborhood information. It shares the same memory bottleneck with GCN, caused by recursive neighborhood expansion. To reduce the computational footprint, the authors propose restricting the immediate neighborhood size for each layer. Using our notation for the sample size, if one samples $t _ { l }$ neighbors for each vertex in the lth layer, then the size of the expanded neighborhood is, in the worst case, the product of the $t _ { l }$ ’s. On the other hand, FastGCN samples vertices rather than neighbors in each layer. Then, the total number of involved vertices is at most the sum of the $t _ { l }$ ’s, rather than the product. See experimental results in Section 4 for the order-of-magnitude saving in actual computation time.
+
+# 4 EXPERIMENTS
+
+We follow the experiment setup in Kipf & Welling (2016a) and Hamilton et al. (2017) to demonstrate the effective use of FastGCN, comparing with the original GCN model as well as GraphSAGE, on the following benchmark tasks: (1) classifying research topics using the Cora citation data set (McCallum et al., 2000); (2) categorizing academic papers with the Pubmed database; and (3) predicting the community structure of a social network modeled with Reddit posts. These data sets are downloaded from the accompany websites of the aforementioned references. The graphs have increasingly more nodes and higher node degrees, representative of the large and dense setting under which our method is motivated. Statistics are summarized in Table 1. We adjusted the training/validation/test split of Cora and Pubmed to align with the supervised learning scenario. Specifically, all labels of the training examples are used for training, as opposed to only a small portion in the semi-supervised setting (Kipf & Welling, 2016a). Such a split is coherent with that of the other data set, Reddit, used in the work of GraphSAGE. Additional experiments using the original split of Cora and Pubmed are reported in the appendix.
+
+Table 1: Dataset Statistics   
+
+<table><tr><td>Dataset</td><td>Nodes</td><td>Edges</td><td>Classes</td><td>Features</td><td>Training/Validation/Test</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>7</td><td>1,433</td><td>1,208/500/1,000</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,338</td><td>3</td><td>500</td><td>18,217/500/1,000</td></tr><tr><td>Reddit</td><td>232,965</td><td>11,606,919</td><td>41</td><td>602</td><td>152,410/23,699/55,334</td></tr></table>
+
+Implementation details are as following. All networks (including those under comparison) contain two layers as usual. The codes of GraphSAGE and GCN are downloaded from the accompany websites and the latter is adapted for FastGCN. Inference with FastGCN is done with the full GCN network, as mentioned in Section 3.2. Further details are contained in the appendix.
+
+We first consider the use of sampling in FastGCN. The left part of Table 2 (columns under “Sampling”) lists the time and classification accuracy as the number of samples increases. For illustration purpose, we equalize the sample size on both layers. Clearly, with more samples, the per-epoch training time increases, but the accuracy (as measured by using micro F1 scores) also improves generally.
+
+An interesting observation is that given input features $H ^ { ( 0 ) }$ , the product $\hat { A } H ^ { ( 0 ) }$ in the bottom layer does not change, which means that the chained expansion of the gradient with respect to $W ^ { ( 0 ) }$ in the last step is a constant throughout training. Hence, one may precompute the product rather than sampling this layer to gain efficiency. The compared results are listed on the right part of Table 2 (columns under “Precompute”). One sees that the training time substantially decreases while the accuracy is comparable. Hence, all the experiments that follow use precomputation.
+
+Table 2: Benefit of precomputing $\hat { A } H ^ { ( 0 ) }$ for the input layer. Data set: Pubmed. Training time is in seconds, per-epoch (batch size 1024). Accuracy is measured by using micro F1 score.   
+
+<table><tr><td colspan="2">Sampling</td><td rowspan="2">F1</td><td rowspan="2">Precompute Time F1</td></tr><tr><td>t1</td><td>Time</td></tr><tr><td>5</td><td>0.737</td><td>0.859</td><td>0.139 0.849</td></tr><tr><td>10</td><td>0.755</td><td>0.863</td><td>0.141 0.870</td></tr><tr><td>25</td><td>0.760</td><td>0.873</td><td>0.144 0.879</td></tr><tr><td>50</td><td>0.774</td><td>0.864</td><td>0.142 0.880</td></tr></table>
+
+![](images/c89d2f6c01abefbd0d3dad500867153e7990fb5dff5c5b4811cd9fb2ea82f634.jpg)  
+Figure 2: Prediction accuracy: uniform versus importance sampling. The three data sets from top to bottom are ordered the same as Table 1.
+
+Next, we compare the sampling approaches for FastGCN: uniform and importance sampling. Figure 2 summarizes the prediction accuracy under both approaches. It shows that importance sampling consistently yields higher accuracy than does uniform sampling. Since the altered sampling distribution (see Proposition 4 and Algorithm 2) is a compromise alternative of the optimal distribution that is impractical to use, this result suggests that the variance of the used sampling indeed is smaller than that of uniform sampling; i.e., the term (9) stays closer to (8) than does (6). A possible reason is that $b ( u )$ correlates with $| x ( u ) |$ . Hence, later experiments will apply importance sampling.
+
+We now demonstrate that the proposed method is significantly faster than the original GCN as well as GraphSAGE, while maintaining comparable prediction performance. See Figure 3. The bar heights indicate the per-batch training time, in the log scale. One sees that GraphSAGE is a substantial improvement of GCN for large and dense graphs (e.g., Reddit), although for smaller ones (Cora and Pubmed), GCN trains faster. FastGCN is the fastest, with at least an order of magnitude improvement compared with the runner up (except for Cora), and approximately two orders of magnitude speed up compared with the slowest. Here, the training time of FastGCN is with respect to the sample size that achieves the best prediction accuracy. As seen from the table on the right, this accuracy is highly comparable with the best of the other two methods.
+
+<table><tr><td colspan="3">Micro F1 Score</td></tr><tr><td>Cora</td><td>Pubmed</td><td>Reddit</td></tr><tr><td>FastGCN</td><td>0.850 0.880</td><td>0.937</td></tr><tr><td>GraphSAGE-GCN</td><td>0.829 0.849</td><td>0.923</td></tr><tr><td>GraphSAGE-mean</td><td>0.822 0.888</td><td>0.946</td></tr><tr><td>GCN (batched)</td><td>0.851 0.867</td><td>0.930</td></tr><tr><td>GCN (original)</td><td>0.865 0.875</td><td>NA</td></tr></table>
+
+![](images/aee78eb0f79ba7f4845f3f9257458cc251c039184f7aaea9e2ecb0576c444894.jpg)  
+Figure 3: Per-batch training time in seconds (left) and prediction accuracy (right). For timing, GraphSAGE refers to GraphSAGE-GCN in Hamilton et al. (2017). The timings of using other aggregators, such as GraphSAGE-mean, are similar. GCN refers to using batched learning, as opposed to the original version that is nonbatched; for more details of the implementation, see the appendix. The nonbatched version of GCN runs out of memory on the large graph Reddit. The sample sizes for FastGCN are 400, 100, and 400, respectively for the three data sets.
+
+In the discussion period, the authors of GraphSAGE offered an improved implementation of their codes and alerted that GraphSAGE was better suited for massive graphs. The reason is that for small graphs, the sample size (recalling that it is the product across layers) is comparable to the graph size and hence improvement is marginal; moreover, sampling overhead might then adversely affect the timing. For fair comparison, the authors of GraphSAGE kept the sampling strategy but improved the implementation of their original codes by eliminating redundant calculations of the sampled nodes. Now the per-batch training time of GraphSAGE compares more favorably on the smallest graph Cora; see Table 3. Note that this implementation does not affect large graphs (e.g., Reddit) and our observation of orders of magnitude faster training remains valid.
+
+Table 3: Further comparison of per-batch training time (in seconds) with new implementation of GraphSAGE for small graphs. The new implementation is in PyTorch whereas the rest are in TensorFlow.   
+
+<table><tr><td></td><td>Cora</td><td>Pubmed</td><td>Reddit</td></tr><tr><td>FastGCN</td><td>0.0084</td><td>0.0047</td><td>0.0129</td></tr><tr><td>GraphSAGE-GCN (old impl)</td><td>1.1630</td><td>0.3579</td><td>0.4260</td></tr><tr><td>GraphSAGE-GCN (new impl)</td><td>0.0380</td><td>0.3989</td><td>NA</td></tr><tr><td>GCN (batched)</td><td>0.0166</td><td>0.0815</td><td>2.1731</td></tr></table>
+
+# 5 CONCLUSIONS
+
+We have presented FastGCN, a fast improvement of the GCN model recently proposed by Kipf & Welling (2016a) for learning graph embeddings. It generalizes transductive training to an inductive manner and also addresses the memory bottleneck issue of GCN caused by recursive expansion of neighborhoods. The crucial ingredient is a sampling scheme in the reformulation of the loss and the gradient, well justified through an alternative view of graph convoluntions in the form of integral transforms of embedding functions. We have compared the proposed method with additionally GraphSAGE (Hamilton et al., 2017), a newly published work that also proposes using sampling to restrict the neighborhood size, although the two sampling schemes substantially differ in both algorithm and cost. Experimental results indicate that our approach is orders of magnitude faster than GCN and GraphSAGE, while maintaining highly comparable prediction performance with the two.
+
+The simplicity of the GCN architecture allows for a natural interpretation of graph convolutions in terms of integral transforms. Such a view, yet, generalizes to many graph models whose formulations are based on first-order neighborhoods, examples of which include MoNet that applies to (meshed) manifolds (Monti et al., 2017), as well as many message-passing neural networks (see e.g., Scarselli et al. (2009); Gilmer et al. (2017)). The proposed work elucidates the basic Monte Carlo ingredients for consistently estimating the integrals. When generalizing to other networks aforementioned, an additional effort is to investigate whether and how variance reduction may improve the estimator, a possibly rewarding avenue of future research.
+
+# REFERENCES
+
+Amr Ahmed, Nino Shervashidze, Shravan Narayanamurthy, Vanja Josifovski, and Alexander J. Smola. Distributed large-scale natural graph factorization. In Proceedings of the 22Nd International Conference on World Wide Web, WWW ’13, pp. 37–48, 2013. ISBN 978-1-4503-2035-1.   
+Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic, NIPS’01, pp. 585–591, 2001.   
+Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. CoRR, abs/1312.6203, 2013.   
+Shaosheng Cao, Wei Lu, and Qiongkai Xu. Grarep: Learning graph representations with global structural information. In Proceedings of the 24th ACM International on Conference on Information and Knowledge Management, CIKM ’15, pp. 891–900, 2015. ISBN 978-1-4503-3794-6.
+
+Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on ¨ graphs with fast localized spectral filtering. CoRR, abs/1606.09375, 2016.
+
+David K Duvenaud, Dougal Maclaurin, Jorge Iparraguirre, Rafael Bombarell, Timothy Hirzel, Alan Aspuru-Guzik, and Ryan P Adams. Convolutional networks on graphs for learning molecular fingerprints. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett (eds.), Advances in Neural Information Processing Systems 28, pp. 2224–2232. Curran Associates, Inc., 2015.
+
+J. Gilmer, S.S. Schoenholz, P.F. Riley, O. Vinyals, and G.E. Dahl. Neural message passing for quantum chemistry. In ICML, 2017.
+
+Palash Goyal and Emilio Ferrara. Graph embedding techniques, applications, and performance: A survey. CoRR, abs/1705.02801, 2017.
+
+Aditya Grover and Jure Leskovec. Node2vec: Scalable feature learning for networks. In Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, pp. 855–864, 2016. ISBN 978-1-4503-4232-2.
+
+William L. Hamilton, Rex Ying, and Jure Leskovec. Inductive representation learning on large graphs. CoRR, abs/1706.02216, 2017.
+
+Mikael Henaff, Joan Bruna, and Yann LeCun. Deep convolutional networks on graph-structured data. CoRR, abs/1506.05163, 2015.
+
+Ashesh Jain, Amir Roshan Zamir, Silvio Savarese, and Ashutosh Saxena. Structural-rnn: Deep learning on spatio-temporal graphs. CoRR, abs/1511.05298, 2015.
+
+Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. CoRR, abs/1609.02907, 2016a.
+
+TN. Kipf and M. Welling. Variational graph auto-encoders. In NIPS Workshop on Bayesian Deep Learning. 2016b.
+
+Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard S. Zemel. Gated graph sequence neural networks. CoRR, abs/1511.05493, 2015.
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+Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore. Automating the construction of internet portals with machine learning. Inf. Retr., 3(2):127–163, July 2000. ISSN 1386-4564.
+
+F. Monti, D. Boscaini, J. Masci, E. Rodala, J. Svoboda, and M.M. Bronstein. Geometric deep learning on graphs and manifolds using mixture model CNNs. In CVPR, 2017.
+
+Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning convolutional neural networks for graphs. CoRR, abs/1605.05273, 2016.
+
+Mingdong Ou, Peng Cui, Jian Pei, Ziwei Zhang, and Wenwu Zhu. Asymmetric transitivity preserving graph embedding. In Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, pp. 1105–1114, 2016. ISBN 978-1-4503- 4232-2.
+
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’14, pp. 701–710, 2014. ISBN 978-1-4503-2956-9.
+
+Sam T. Roweis and Lawrence K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323–2326, 2000. ISSN 0036-8075. doi: 10.1126/science.290.5500. 2323.
+
+F. Scarselli, M. Gori, A.C. Tsoi, M. Hagenbuchner, and G. Monfardini. The graph neural network model. IEEE Transactions on Neural Networks, 20, 2009.
+
+Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large-scale information network embedding. In Proceedings of the 24th International Conference on World Wide Web, WWW ’15, pp. 1067–1077, 2015. ISBN 978-1-4503-3469-3.
+
+Daixin Wang, Peng Cui, and Wenwu Zhu. Structural deep network embedding. In Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, pp. 1225–1234, 2016. ISBN 978-1-4503-4232-2.
+
+# A PROOFS
+
+Proof of Theorem 1. Because the samples $u _ { j } ^ { ( 0 ) }$ are iid, by the strong law of large numbers,
+
+$$
+\tilde { h } _ { t _ { 1 } } ^ { ( 1 ) } ( v ) = \frac { 1 } { t _ { 0 } } \sum _ { j = 1 } ^ { t _ { 0 } } \hat { A } ( v , u _ { j } ^ { ( 0 ) } ) h ^ { ( 0 ) } ( u _ { j } ^ { ( 0 ) } ) W ^ { ( 0 ) }
+$$
+
+converges almost surely to $\tilde { h } ^ { ( 1 ) } ( v )$ . Then, because the activation function $\sigma$ is continuous, the continuous mapping theorem implies that $h _ { t _ { 1 } } ^ { ( 1 ) } ( v ) \ = \ \sigma ( \tilde { h } _ { t _ { 1 } } ^ { ( 1 ) } ( v ) )$ converges almost surely to $h ^ { ( 1 ) } ( v ) = \sigma ( \tilde { h } ^ { ( 1 ) } ( v ) )$ . Thus, $\begin{array} { r } { \int \hat { A } ( v , u ) h _ { t _ { 1 } } ^ { ( 1 ) } ( u ) W ^ { ( 1 ) } d P ( u ) } \end{array}$ converges almost surely to $\tilde { h } ^ { ( 2 ) } ( v ) =$ $\begin{array} { r } { \int \hat { A } ( v , u ) h ^ { ( 1 ) } ( u ) W ^ { ( 1 ) } d P ( u ) } \end{array}$ , where note that the probability space is with respect to the 0th layer and hence has nothing to do with that of the variable $u$ or $v$ in this statement. Similarly,
+
+$$
+\tilde { h } _ { t _ { 2 } } ^ { ( 2 ) } ( v ) = \frac { 1 } { t _ { 1 } } \sum _ { j = 1 } ^ { t _ { 1 } } \hat { A } ( v , u _ { j } ^ { ( 1 ) } ) h _ { t _ { 1 } } ^ { ( 1 ) } ( u _ { j } ^ { ( 1 ) } ) W ^ { ( 1 ) }
+$$
+
+converges almost surely to $\begin{array} { r } { \int \hat { A } ( v , u ) h _ { t _ { 1 } } ^ { ( 1 ) } ( u ) W ^ { ( 1 ) } d P ( u ) } \end{array}$ and thus to $\tilde { h } ^ { ( 2 ) } ( v )$ . A simple induction completes the rest of the proof.
+
+Proof of Proposition 2. Conditioned on $v$ , the expectation of $y ( v )$ is
+
+$$
+\operatorname { E } [ y ( v ) | v ] = \int { \hat { A } } ( v , u ) x ( u ) d P ( u ) = e ( v ) ,
+$$
+
+and the variance is $1 / t$ times that of $\hat { A } ( v , u ) x ( u )$ , i.e.,
+
+$$
+\mathrm { V a r } \{ y ( v ) | v \} = \frac { 1 } { t } \left( \int \hat { A } ( v , u ) ^ { 2 } x ( u ) ^ { 2 } d P ( u ) - e ( v ) ^ { 2 } \right) .
+$$
+
+Instantiating (11) and (12) with iid samples $v _ { 1 } , \ldots , v _ { s } \sim P$ and taking variance and expectation in the front, respectively, we obtain
+
+$$
+\left\{ \mathop { \mathrm { A r } } \left\{ \left[ \frac { 1 } { s } \sum _ { i = 1 } ^ { s } y ( v _ { i } ) \right] v _ { 1 } , \ldots , v _ { s } \right\} \right\} = \mathop { \mathrm { V a r } } \left\{ \frac { 1 } { s } \sum _ { i = 1 } ^ { s } e ( v _ { i } ) \right\} = \frac { 1 } { s } \int e ( v ) ^ { 2 } d P ( v ) - \frac { 1 } { s } \left( \int e ( v ) d P ( v ) \right) =
+$$
+
+and
+
+$$
+\mathbb { E } \left[ \mathrm { V a r } \left\{ \frac { 1 } { s } \sum _ { i = 1 } ^ { s } y ( v _ { i } ) \Bigg | v _ { 1 } , \dots , v _ { s } \right\} \right] = \frac { 1 } { s t } \iint \hat { A } ( v , u ) ^ { 2 } x ( u ) ^ { 2 } d P ( u ) d P ( v ) - \frac { 1 } { s t } \int e ( v ) ^ { 2 } d P ( v ) .
+$$
+
+Then, applying the law of total variance
+
+$$
+ \operatorname { a r } \{ { \frac { 1 } { s } } \sum _ { i = 1 } ^ { s } y ( v _ { i } ) \} = \operatorname { V a r } \{ \operatorname { E } [  { \frac { 1 } { s } } \sum _ { i = 1 } ^ { s } y ( v _ { i } ) | v _ { 1 } , \ldots , v _ { s } ] \} + \operatorname { E } [ \operatorname { V a r } \{ { \frac { 1 } { s } } \sum _ { i = 1 } ^ { s } y ( v _ { i } ) { \Bigg | } v _ { 1 } , \ldots , v _ { s } \} ] ,
+$$
+
+we conclude the proof.
+
+Proof of Theorem 3. Conditioned on $v$ , the variance of $y _ { Q } ( v )$ is $1 / t$ times that of
+
+$$
+{ \hat { A } } ( v , u ) x ( u ) { \frac { d P ( u ) } { d Q ( u ) } } \quad ( { \mathrm { w h e r e ~ } } u \sim Q ) ,
+$$
+
+i.e.,
+
+$$
+\mathrm { V a r } \{ y _ { Q } ( v ) | v \} = \frac { 1 } { t } \left( \int \frac { \hat { A } ( v , u ) ^ { 2 } { x } ( u ) ^ { 2 } d P ( u ) ^ { 2 } } { d Q ( u ) } - e ( v ) ^ { 2 } \right) .
+$$
+
+Then, following the proof of Proposition 2, the overall variance is
+
+$$
+\mathrm { V a r } \{ G _ { Q } \} = R + \frac { 1 } { s t } \iint \frac { \hat { A } ( v , u ) ^ { 2 } x ( u ) ^ { 2 } d P ( u ) ^ { 2 } d P ( v ) } { d Q ( u ) } = R + \frac { 1 } { s t } \int \frac { b ( u ) ^ { 2 } x ( u ) ^ { 2 } d P ( u ) ^ { 2 } } { d Q ( u ) } .
+$$
+
+Hence, the optimal $d Q ( u )$ must be proportional to $b ( u ) | x ( u ) | d P ( u )$ . Because it also must integrate to unity, we have
+
+$$
+d Q ( u ) = \frac { b ( u ) | x ( u ) | d P ( u ) } { \int b ( u ) | x ( u ) | d P ( u ) } ,
+$$
+
+in which case
+
+$$
+\mathrm { V a r } \{ G _ { Q } \} = R + { \frac { 1 } { s t } } \left[ \int b ( u ) | x ( u ) | d P ( u ) \right] ^ { 2 } .
+$$
+
+Proof of Proposition 4. Conditioned on $v$ , the variance of $y _ { Q } ( v )$ is $1 / t$ times that of
+
+$$
+\hat { A } ( v , u ) x ( u ) \frac { d P ( u ) } { d Q ( u ) } = \frac { \hat { A } ( v , u ) \mathrm { s g n } ( x ( u ) ) } { b ( u ) } \int b ( u ) | x ( u ) | d P ( u ) ,
+$$
+
+i.e.,
+
+$$
+\mathrm { V a r } \{ y _ { Q } ( v ) | v \} = \frac { 1 } { t } \left( \left[ \int b ( u ) | x ( u ) | d P ( u ) \right] ^ { 2 } \int \frac { \hat { A } ( v , u ) ^ { 2 } } { b ( u ) ^ { 2 } } d Q ( u ) - e ( v ) ^ { 2 } \right) .
+$$
+
+The rest of the proof follows that of Proposition 2.
+
+# B ADDITIONAL EXPERIMENT DETAILS
+
+# B.1 BASELINES
+
+GCN: The original GCN cannot work on very large graphs (e.g., Reddit). So we modified it into a batched version by simply removing the sampling in our FastGCN (i.e., using all the nodes instead of sampling a few in each batch). For relatively small graphs (Cora and Pubmed), we also compared the results with the original GCN.
+
+GraphSAGE: For training time comparison, we use GraphSAGE-GCN that employs GCN as the aggregator. It is also the fastest version among all choices of the aggregators. For accuracy comparison, we also compared with GraphSAGE-mean. We used the codes from https: //github.com/williamleif/GraphSAGE. Following the setting of Hamilton et al. (2017), we use two layers with neighborhood sample sizes $S _ { 1 } = 2 5$ and $S _ { 2 } = 1 0$ . For fair comparison with our method, the batch size is set to be the same as FastGCN, and the hidden dimension is 128.
+
+# B.2 EXPERIMENT SETUP
+
+Datasets: The Cora and Pubmed data sets are from https://github.com/tkipf/gcn. As we explained in the paper, we kept the validation index and test index unchanged but changed the training index to use all the remaining nodes in the graph. The Reddit data is from http: //snap.stanford.edu/graphsage/.
+
+Experiment Setting: We preformed hyperparameter selection for the learning rate and model dimension. We swept learning rate in the set $\left. 0 . 0 1 , 0 . 0 0 1 , 0 . 0 0 0 1 \right.$ . The hidden dimension of FastGCN for Reddit is set as 128, and for the other two data sets, it is 16. The batch size is 256 for Cora and Reddit, and 1024 for Pubmed. Dropout rate is set as 0. We use Adam as the optimization method for training. In the test phase, we use the trained parameters and all the graph nodes instead of sampling. For more details please check our codes in a temporary git repository https://github.com/matenure/FastGCN.
+
+Hardware: Running time is compared on a single machine with 4-core 2.5 GHz Intel Core i7, and 16G RAM.
+
+# C ADDITIONAL EXPERIMENTS
+
+C.1 TRAINING TIME COMPARISON
+
+Figure 3 in the main text compares the per-batch training time for different methods. Here, we list the total training time for reference. It is impacted by the convergence of SGD, whose contributing factors include learning rate, batch size, and sample size. See Table 4. Although the orders-ofmagnitude speedup of per-batch time is slightly weakened by the convergence speed, one still sees a substantial advantage of the proposed method in the overall training time. Note that even though the original GCN trains faster than the batched version, it does not scale because of memory limitation. Hence, a fair comparison should be gauged with the batched version. We additionally show in Figure 4 the evolution of prediction accuracy as training progresses.
+
+Table 4: Total training time (in seconds).   
+
+<table><tr><td></td><td>Cora</td><td>Pubmed</td><td>Reddit</td></tr><tr><td>FastGCN</td><td>2.7</td><td>15.5</td><td>638.6</td></tr><tr><td>GraphSAGE-GCN</td><td>72.4</td><td>259.6</td><td>3318.5</td></tr><tr><td>GCN (batched)</td><td>6.9</td><td>210.8</td><td>58346.6</td></tr><tr><td>GCN (original)</td><td>1.7</td><td>21.4</td><td>NA</td></tr></table>
+
+![](images/1008ca8b5638fd3d161b1aa4b3d754c0fea24b24f49755b163298dbf769a672d.jpg)  
+Figure 4: Training/test accuracy versus training time. From left to right, the data sets are Cora, Pubmed, and Reddit, respectively.
+
+# C.2 ORIGINAL DATA SPLIT FOR CORA AND PUBMED
+
+As explained in Section 4, we increased the number of labels used for training in Cora and Pubmed, to align with the supervised learning setting of Reddit. For reference, here we present results by using the original data split with substantially fewer training labels. We also fork a separate version of FastGCN, called FastGCN-transductive, that uses both training and test data for learning. See Table 5.
+
+The results for GCN are consistent with those reported by Kipf & Welling (2016a). Because labeled data are scarce, the training of GCN is quite fast. FastGCN beats it only on Pubmed. The accuracy results of FastGCN are inferior to GCN, also because of the limited number of training labels. The transductive version FastGCN-transductive matches the accuracy of that of GCN. The results for GraphSAGE are curious. We suspect that the model significantly overfits the data, because perfect training accuracy (i.e., 1) is attained.
+
+One may note a subtlety that the training of GCN (original) is slower than what is reported in Table 4, even though fewer labels are used here. The reason is that we adopt the same hyperparameters as in Kipf & Welling (2016a) to reproduce the F1 scores of their work, whereas for Table 4, a better learning rate is found that boosts the performance on the new split of the data, in which case GCN (original) converges faster.
+
+Table 5: Total training time and test accuracy for Cora and Pubmed, original data split. Time is in seconds.   
+
+<table><tr><td></td><td colspan="2">Cora</td><td colspan="2">Pubmed</td></tr><tr><td></td><td>Time</td><td>F1</td><td>Time</td><td>F1</td></tr><tr><td>FastGCN</td><td>2.52</td><td>0.723</td><td>0.97</td><td>0.721</td></tr><tr><td>FastGCN-transductive</td><td>5.88</td><td>0.818</td><td>8.97</td><td>0.776</td></tr><tr><td>GraphSAGE-GCN</td><td>107.95</td><td>0.334</td><td>39.34</td><td>0.386</td></tr><tr><td>GCN (original)</td><td>2.18</td><td>0.814</td><td>32.65</td><td>0.795</td></tr></table>
+
+# D CONVERGENCE
+
+Strictly speaking, the training algorithms proposed in Section 3 do not precisely follow the existing theory of SGD, because the gradient estimator, though consistent, is biased. In this section, we fill the gap by deriving a convergence result. Similar to the case of standard SGD where the convergence rate depends on the properties of the objective function, here we analyze only a simple case; a comprehensive treatment is out of the scope of the present work. For convenience, we will need a separate system of notations and the same notations appearing in the main text may bear a different meaning here. We abbreviate “with probability one” to “w.p.1” for short.
+
+We use $f ( x )$ to denote the objective function and assume that it is differentiable. Differentiability is not a restriction because for the nondifferentiable case, the analysis that follows needs simply change the gradient to the subgradient. The key assumption made on $f$ is that it is $l$ -strictly convex; that is, there exists a positive real number $l$ such that
+
+$$
+f ( x ) - f ( y ) \geq \langle \nabla f ( y ) , x - y \rangle + { \frac { l } { 2 } } \| x - y \| ^ { 2 } ,
+$$
+
+for all $x$ and $y$ . We use $g$ to denote the gradient estimator. Specifically, denote by $g ( x ; \xi _ { N } )$ , with $\xi _ { N }$ being a random variable, a strongly consistent estimator of $\nabla f ( x )$ ; that is,
+
+$$
+\operatorname* { l i m } _ { N \to \infty } g ( x ; \xi _ { N } ) = \nabla f ( x ) \quad { \mathrm { w . p . 1 } } .
+$$
+
+Moreover, we consider the SGD update rule
+
+$$
+x _ { k + 1 } = x _ { k } - \gamma _ { k } \ g ( x _ { k } ; \xi _ { N } ^ { ( k ) } ) ,
+$$
+
+where (k) is an indepedent sample of $\xi _ { N }$ for the $k$ th update. The following result states that the update converges on the order of $O ( 1 / k )$ .
+
+Theorem 5. Let $x ^ { * }$ be the (global) minimum of $f$ and assume that $\| \nabla f ( x ) \|$ is uniformly bounded by some constant $G > 0$ . $I f \gamma _ { k } = ( l k ) ^ { - 1 }$ , then there exists a sequence $B _ { k }$ with
+
+$$
+B _ { k } \leq \frac { \operatorname* { m a x } \{ \| x _ { 1 } - x ^ { * } \| ^ { 2 } , ~ G ^ { 2 } / l ^ { 2 } \} } { k }
+$$
+
+such that $\| x _ { k } - x ^ { * } \| ^ { 2 } \to B _ { k }$ w.p.1.
+
+Proof. Expanding $\| x _ { k + 1 } - x ^ { * } \| ^ { 2 }$ by using the update rule (14), we obtain
+
+$$
+\| x _ { k + 1 } - x ^ { * } \| ^ { 2 } = \| x _ { k } - x ^ { * } \| ^ { 2 } - 2 \gamma _ { k } \langle g _ { k } , x _ { k } - x ^ { * } \rangle + \gamma _ { k } ^ { 2 } \| g _ { k } \| ^ { 2 } ,
+$$
+
+where $g _ { k } \equiv g ( x _ { k } ; \xi _ { N } ^ { ( k ) } )$ . Because for a given $x _ { k } , \textrm { } g _ { k }$ converges to $\nabla f ( x _ { k } )$ w.p.1, we have that conditioned on $x _ { k }$ ,
+
+$$
+\| x _ { k + 1 } - x ^ { * } \| ^ { 2 } \to \| x _ { k } - x ^ { * } \| ^ { 2 } - 2 \gamma _ { k } \langle \nabla f ( x _ { k } ) , x _ { k } - x ^ { * } \rangle + \gamma _ { k } ^ { 2 } \| \nabla f ( x _ { k } ) \| ^ { 2 } \quad \mathrm { w . p . 1 } .
+$$
+
+On the other hand, applying the strict convexity (13), by first taking $x = x _ { k } , y = x ^ { * }$ and then taking $x = x ^ { * } , y = x _ { k }$ , we obtain
+
+$$
+\langle \nabla f ( x _ { k } ) , x _ { k } - x ^ { * } \rangle \geq l \| x _ { k } - x ^ { * } \| ^ { 2 } .
+$$
+
+Substituting (16) to (15), we have that conditioned on $x _ { k }$ ,
+
+$$
+\| x _ { k + 1 } - x ^ { * } \| ^ { 2 } \to C _ { k } \quad { \mathrm { w . p . 1 } }
+$$
+
+for some
+
+$$
+\begin{array} { r } { C _ { k } \leq ( 1 - 2 l \gamma _ { k } ) \| x _ { k } - x ^ { * } \| ^ { 2 } + \gamma _ { k } ^ { 2 } G ^ { 2 } = ( 1 - 2 / k ) \| x _ { k } - x ^ { * } \| ^ { 2 } + G ^ { 2 } / ( l ^ { 2 } k ^ { 2 } ) . } \end{array}
+$$
+
+Now consider the randomness of $x _ { k }$ and apply induction. For the base case $k = 2$ , the theorem clearly holds with $B _ { 2 } = C _ { 1 }$ . If the theorem holds for $k = T$ , let $L = \operatorname* { m a x } \{ \| x _ { 1 } - x ^ { * } \| ^ { 2 } , \ G ^ { 2 } / l ^ { 2 } \}$ . Then, taking the probabilistic limit of $x _ { T }$ on both sides of (17), we have that $C _ { T }$ converges w.p.1 to some limit that is less than or equal to $( 1 - 2 / T ) ( L / T ) + G ^ { 2 } / ( l ^ { 2 } T ^ { 2 } ) \leq L / ( T + 1 )$ . Letting this limit be $B _ { T + 1 }$ , we complete the induction proof. □