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+# LEARNING COMPOSITIONAL KOOPMAN OPERATORSFOR MODEL-BASED CONTROL
+
+Yunzhu Li∗ Hao He∗ MIT CSAIL MIT CSAIL
+
+Jiajun Wu MIT CSAIL
+
+Dina Katabi MIT CSAIL
+
+Antonio Torralba MIT CSAIL
+
+# ABSTRACT
+
+Finding an embedding space for a linear approximation of a nonlinear dynamical system enables efficient system identification and control synthesis. The Koopman operator theory lays the foundation for identifying the nonlinear-to-linear coordinate transformations with data-driven methods. Recently, researchers have proposed to use deep neural networks as a more expressive class of basis functions for calculating the Koopman operators. These approaches, however, assume a fixed dimensional state space; they are therefore not applicable to scenarios with a variable number of objects. In this paper, we propose to learn compositional Koopman operators, using graph neural networks to encode the state into objectcentric embeddings and using a block-wise linear transition matrix to regularize the shared structure across objects. The learned dynamics can quickly adapt to new environments of unknown physical parameters and produce control signals to achieve a specified goal. Our experiments on manipulating ropes and controlling soft robots show that the proposed method has better efficiency and generalization ability than existing baselines.
+
+# 1 INTRODUCTION
+
+Simulating and controlling complex dynamical systems, such as ropes or soft robots, relies on two key features of the dynamics model: first, it needs to be efficient for system identification and motor control; second, it needs to be generalizable to a complex, constantly evolving environments.
+
+In practice, computational models for complex, nonlinear dynamical systems are often not efficient enough for real-time control (Mayne, 2000). The Koopman operator theory suggests that identifying nonlinear-to-linear coordinate transformations allows efficient linear approximation of nonlinear systems (Williams et al., 2015; Mauroy & Goncalves, 2016). Fast as they are, however, existing papers on Koopman operators focus on a single dynamical system, making it hard to generalize to cases where there are a variable number of components.
+
+In contrast, recent advances in approximating dynamics models with deep nets have demonstrated its power in characterizing complex, generic environments. In particular, a few recent papers have explored the use of graph nets in dynamics modeling, taking into account the state of each object as well as their interactions. This allows their models to generalize to scenarios with a variable number of objects (Battaglia et al., 2016; Chang et al., 2017). Despite their strong generalization power, they are not as efficient in system identification and control, because deep nets are heavily over-parameterized, making optimization time-consuming and sample-inefficient.
+
+In this paper, we propose compositional Koopman operators, integrating Koopman operators with graph networks for generalizable and efficient dynamics modeling. We build on the idea of encoding states into object-centric embeddings with graph neural networks, which ensures generalization power. But instead of using over-parameterized neural nets to model state transition, we identify the Koopman matrix and control matrix from data as a linear approximation of the nonlinear dynamical system. The linear approximation allows efficient system identification and control synthesis.
+
+The main challenge of extending Koopman theory to multi-object systems is scalability. The number of parameters in the Koopman matrix scales quadratically with the number of objects, which harms the learning efficiency and leads to overfitting. To tackle this issue, we exploit the structure of the underlying system and use the same block-wise Koopman sub-matrix for object pairs of the same relation. This significantly reduces the number of parameters that need to be identified by making it independent of the size of the system.
+
+![](images/324bececc168c0745244997b7ffa1ced5381091c5ea6bfc968579a1b4ad86135.jpg)  
+Figure 1: Overview of our model. A graph neural network $\phi$ takes in the current state of the physical system $\mathbf { \boldsymbol { x } } ^ { t }$ , and generates object-centric representations in the Koopman space $g ^ { t }$ . We then use the block-wise Koopman matrix $K$ and control matrix $L$ identified from equation 6 or equation 8 to predict the Koopman embeddings in the next time step $\mathbf g ^ { t + 1 }$ . Note that in $K$ and $L$ , object pairs of the same relation share the same sub-matrix. Another graph neural network $\psi$ maps $\mathbf g ^ { t + 1 }$ back to the original state space, i.e., $\boldsymbol { x } ^ { t + 1 }$ . The mapping between $g ^ { \hat { t } }$ and $\mathbf g ^ { t + 1 }$ is linear and is shared across all time steps, where we can iteratively apply $K$ and $L$ to the Koopman embeddings and roll multiple steps into the future. The formulation enables efficient system identification and control synthesis.
+
+Our experiments include simulating and controlling ropes of variable lengths and soft robots of different shapes. The compositional Koopman operators are significantly more accurate than the state-of-the-art learned physics engines (Battaglia et al., 2016; Li et al., 2019b), and faster when adapting to new environments of unknown physical parameters. Our method also outperforms vanilla deep Koopman methods (Lusch et al., 2018; Morton et al., 2018) and Koopman models with manually-designed basis functions, which shows the advantages of using a structured Koopman matrix and graph neural networks. Please see our project page for demonstrating videos.
+
+# 2 RELATED WORK
+
+Koopman operators. The Koopman operator formalism of dynamical systems is rooted in the seminal works of Koopman and Von Neumann in the early 1930s (Koopman, 1931; Koopman & Neumann, 1932). The core idea is to map the state of a nonlinear dynamical system to an embedding space, over which we can linearly propagate into the future. Researchers have proposed various algorithms to explore the Koopman spectral properties from data. A large portion of them are in the class of dynamic mode decomposition (DMD) (Rowley et al., 2009; Schmid, 2010; Tu et al., 2014; Williams et al., 2015; Arbabi & Mezic, 2017). The linear representation will enable efficient prediction, estimation, and control using tools from linear dynamical systems (Williams et al., 2016; Proctor et al., 2018; Mauroy & Goncalves, 2019; Korda & Mezic´, 2018). People have been using hand-designed Koopman observables for various modeling and control tasks (Brunton et al., 2016; Kaiser et al., 2017; Abraham et al., 2017; Bruder et al., 2019b; Arbabi et al., 2018). Some recent works have applied the method to the real world and successfully control soft robots with great precision (Bruder et al., 2019a; Mamakoukas et al., 2019).
+
+However, hand-crafted basis functions sometimes fail to generalize to more complex environments. Learning these functions from data using neural nets turns out to generate a more expressive invariant subspace (Lusch et al., 2018; Takeishi et al., 2017) and has achieved successes in fluid control (Morton et al., 2018). Morton et al. (2019) has also extended the framework to account for uncertainty in the system by inferring a distribution over observations. Our model differs by explicitly modeling the compositionality of the underlying system with graph networks. It generalizes better to environments of a variable number of objects or soft robots of different shapes.
+
+Learning-based physical simulators. Battaglia et al. (2016) and Chang et al. (2017) first explored learning a simulator from data by approximating object interactions with neural networks. These models are no longer bounded to hard-coded physical rules, and can adapt to scenarios where the underlying physics is unknown. Please refer to Battaglia et al. (2018) for a full review. Recently, Mrowca et al. (2018) and Li et al. (2019a) extended these models to approximate particle dynamics of deformable shapes and fluids. Flexible as they are, these models become less efficient during model adaptation in complex scenarios, because the optimization of neural networks usually needs a lot of samples and compute, which limits its use in an online setting. Nagabandi et al. (2019a;b) proposed to use meta-learning for online adaptation, and have shown to be effective in simulated robots and a real legged millirobot. However, it is not clear whether their methods can generalize to systems with variable numbers of instances. The use of graph nets and Koopman operators in our model allows better generalization ability and enables efficient system identification as we only need to identify the transition matrices, which is essentially a least-square problem and can be solved very efficiently.
+
+People have also used the learned physics engines for planning and control. Many previous papers in this direction learn a latent dynamics model together with a policy in a model-based reinforcement learning setup (Racaniere et al. \` , 2017; Hamrick et al., 2017; Pascanu et al., 2017; Hafner et al., 2019); a few alternatives use the learned model in model-predictive control (MPC) (Sanchez-Gonzalez et al., 2018; Li et al., 2019b; Janner et al., 2019). In this paper, we leverage the fact that the embeddings in the Koopman space are propagating linearly through time, which allows us to formulate the control problem as quadratic programming and optimize the control signals much more efficiently.
+
+# 3 APPROACH
+
+We first present the basics of Koopman operators: for a nonlinear dynamical system, the Koopman observation functions can map the state space to an embedding space where the dynamics become linear. We then discuss the compositional nature of physical systems and show how graph networks can be used to capture the compositionality.
+
+# 3.1 THE KOOPMAN OPERATORS
+
+Let $\pmb { x } ^ { t } \in \mathcal { X } \subset \mathbb { R } ^ { n }$ be the state vector for the system at time step $t$ . We consider a non-linear discrete-time dynamical system described by ${ \pmb x } ^ { t + 1 } = F ( { \pmb x } ^ { t } )$ . The Koopman operator (Koopman, 1931), denoted as $\mathcal { K } : \mathcal { F }  \mathcal { F }$ , is a linear transformation defined by $\kappa g \triangleq g \circ F$ , where $\mathcal { F }$ is the collection of all functions (also referred to as observables) that form an infinite-dimensional Hilbert space. For every function $g : \mathcal { X }  \mathbb { R }$ belonging to $\mathcal { F }$ , we have
+
+$$
+( \mathcal { K } g ) ( \pmb { x } ^ { t } ) = g ( F ( \pmb { x } ^ { t } ) ) = g ( \pmb { x } ^ { t + 1 } ) ,
+$$
+
+making the function space $\mathcal { F }$ invariant under the action of the Koopman operator.
+
+Although the theory guarantees the existence of the Koopman operator, its use in practice is limited by its infinite dimensionality. Most often, we assume there is an invariant subspace $\mathcal { G }$ of the Koopman operator. It spans by a set of base observation functions $\{ g _ { 1 } , \cdots , g _ { m } \}$ and satisfies that $\kappa g \in \mathcal G$ for any $g \in { \mathcal { G } }$ . With a slightly abuse of the notation, we now use $\bar { g } ( \pmb { x } ^ { \hat { t } } ) : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ to represent $[ g _ { 1 } ( { \pmb x } ^ { t } ) , \cdot \cdot \cdot , g _ { m } ( { \pmb x } ^ { t } ) ] ^ { T }$ . By constraining the Koopman operator on this invariant subspace, we get a finite-dimensional linear operator $K \in \mathbb { R } ^ { m \times m }$ that we refer as the Koopman matrix.
+
+Traditionally, people hand-craft base observation functions from the knowledge of underlying physics. The system identification problem is then reduced to finding the Koopman matrix $K$ , which can be solved by linear regression given historical data of the system. Recently, researchers have also explored data-driven methods that automatically find the Koopman invariant subspace via representing the base observation functions $g ( { \pmb x } )$ via deep neural networks.
+
+Although the original Koopman theory does not consider the system with external control inputs, researchers have found that linearly injecting control signals to the Koopman observation space can give us good numerical performance (Brunton et al., 2016; Bruder et al., 2019a). Mathematically, considering a dynamical system, $\pmb { x } ^ { t + 1 } = F ( \pmb { x } ^ { t } , \pmb { u } ^ { t } )$ , with an external control input ${ \mathbf { } } _ { { \mathbf { } } { \mathbf { } } } { \mathbf { } } _ { { \mathbf { } } } t$ , we aim to find the Koopman observation functions and the linear dynamics model in the form of
+
+$$
+g ( \pmb { x } ^ { t + 1 } ) = K g ( \pmb { x } ^ { t } ) + L \pmb { u } ^ { t } ,
+$$
+
+where the coefficient matrix $L$ is referred to as the control matrix.
+
+# 3.2 COMPOSITIONAL KOOPMAN OPERATORS
+
+The dynamics of a physical system are governed by physical rules, which are usually shared across different subcomponents in the system. Explicitly modeling such compositionality enables more efficient system identification and control synthesis and provides better generalization ability.
+
+Motivating example. Consider a system with $N$ balls moving in a 2D plane, each pair connected by a linear spring. Assume all balls have mass 1 and all springs share the same stiffness coefficient $k$ . We denote the $i$ ’s ball’s position as $( x _ { i } , y _ { i } )$ and its velocity as $( { \dot { x } } _ { i } , { \dot { y } } _ { i } )$ . For ball $i$ , equation 3 describes its dynamics, where $\mathbf { \Phi } _ { \pmb { x } _ { i } } \triangleq [ x _ { i } , y _ { i } , \dot { x } _ { i } , \dot { y } _ { i } ] ^ { T }$ denotes ball $i$ ’s state:
+
+$$
+\begin{array}{c} \dot { v } _ { i } = \left[ \begin{array} { c } { \dot { x } _ { i } } \\ { \dot { y } _ { i } } \\ { \dot { x } _ { i } } \\ { \dot { y } _ { i } } \end{array} \right] = \left[ \begin{array} { c c c c c } { \dot { x } _ { i } } & { 0 } & { 1 } & { 0 } \\ { \dot { y } _ { i } } & { 0 } & { 0 } & { 1 } \\ { \sum _ { j = 1 } ^ { N } k ( x _ { j } - x _ { i } ) } \\ { \sum _ { j = 1 } ^ { N } k ( y _ { j } - y _ { i } ) } \end{array} \right] = \underbrace { \left[ \begin{array} { c c c c c } { 0 } & { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 1 } \\ { k - N k } & { 0 } & { 0 } & { 0 } \\ { 0 } & { k - N k } & { 0 } & { 0 } \end{array} \right] } _ { \triangleq \mathcal { A } } \left[ \begin{array} { c } { x _ { i } } \\ { y _ { i } } \\ { \dot { x } _ { i } } \end{array} \right] + \sum _ { j \neq i } \left[ \underbrace { 0 } _ { k } _ { 0 } \begin{ c c c c } { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } \\ { k } & { 0 } & { 0 } & { 0 } \\ { 0 } & { k } & { 0 } & { 0 } \end{array} \right] \left[ \begin{array} { c } { x _ { j } } \\ { y _ { j } } \\ { \dot { x } _ { j } } \end{array} \right]
+$$
+
+We can represent the state of the whole system using the union of every ball’s state, where ${ \textbf { \em x } } =$ $[ \pmb { x } _ { 1 } , \cdots , \pmb { \dot { x } } _ { N } ] ^ { T }$ . Then the transition matrix is essentially a block matrix, where the matrix parameters are shared among the diagonal or off-diagonal blocks as shown in equation 4:
+
+$$
+{ \dot { \pmb x } } = \left[ \begin{array} { c } { \dot { \pmb x } _ { 1 } } \\ { \dot { \pmb x } _ { 2 } } \\ { \vdots } \\ { \dot { \pmb x } _ { N } } \end{array} \right] = \left[ \begin{array} { c c c c } { { \cal A } } & { { \cal B } } & { \cdots } & { { \cal B } } \\ { { \cal B } } & { { \cal A } } & { \cdots } & { { \cal B } } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { { \cal B } } & { { \cal B } } & { \cdots } & { { \cal A } } \end{array} \right] \left[ \begin{array} { c } { { \pmb x } _ { 1 } } \\ { { \pmb x } _ { 2 } } \\ { \vdots } \\ { { \pmb x } _ { N } } \end{array} \right] .
+$$
+
+Based on the linear spring system, we make three observations for multi-object systems.
+
+The system state is composed of the state of each individual object. The dimension of the whole system scales linearly with the number of objects. We formulate the system state by concatenating the state of every object, corresponding to an object-centric state representation. • The transition matrix has a block-wise substructure. After assuming an object-centric state representation, the transition matrix naturally has a block-wise structure as shown in equation 4. • The same physical interactions share the same transition block. The blocks in the transition matrix encode actual interactions and generalize across systems. $A$ and $B$ govern the dynamics of the linear spring system, and are shared by systems with a different number of objects.
+
+These observations inspire us to exploit the structure of multi-object systems, instead of learning separate models for systems that contains different numbers of balls.
+
+Compositional Koopman operators. Motivated by the linear spring system, we want to inject a good inductive bias to incorporate compositionality when applying the Koopman theory. This allows better generalization ability and more efficient system identification and better controller design. Figure 1 shows an overview of our model.
+
+Considering a system with $N$ objects, we denote $\mathbf { \boldsymbol { x } } ^ { t }$ as the system state at time $t$ and $\boldsymbol { x } _ { i } ^ { t }$ is the state of the $\romannumeral 1$ ’th object. We further denote $\mathbf { \boldsymbol { g } } ^ { t } \triangleq \mathbf { \boldsymbol { g } } ( \mathbf { \boldsymbol { x } } ^ { t } )$ as the embedding of the state in the Koopman invariant space. In the rest of the paper, we call $g ^ { t }$ the Koopman embedding. Based on the observation we made in the case of linear spring system, we propose the following assumptions on the compositional structure of the Koopman embedding and the Koopman matrix.
+
+• The Koopman embedding of the system is composed of the Koopman embedding of every objects. Similar to the decomposition in the state space, we assume the Koopman embedding can be divided into object-centric sub-embeddings, i.e. $\pmb { g } ^ { t } \in \mathbb { R } ^ { N m }$ denoting the concatenation of $g _ { 1 } ^ { t } , \cdots , g _ { N } ^ { t }$ , where we use $\pmb { g } _ { i } ^ { t } = g _ { i } ( \pmb { x } ^ { t } ) \in \mathbb { R } ^ { m }$ as the Koopman embedding for the $i ^ { \because }$ ’th object.
+
+The Koopman matrix has a block-wise structure. It is natural to think the Koopman matrix is composed of block matrices after assuming an object-centric Koopman embeddings. In equation 5, $K _ { i j } \ \in \mathbb { R } ^ { m \times m }$ and $L _ { i j } \in \mathbb { R } ^ { m \times l }$ are blocks of the Koopman matrix and the control matrix, where $l$ is the dimension of the action for each object and $\pmb { u } ^ { t } \in \mathbb { R } ^ { N l }$ is the concatenation of $\boldsymbol { \mathbf { \mathit { u } } } _ { 1 } ^ { t } , \cdots , \boldsymbol { \mathbf { \mathit { u } } } _ { N } ^ { t }$ denoting the total control signal at time $t$ :
+
+$$
+\left[ \begin{array} { c } { { { \bf g } _ { 1 } ^ { t + 1 } } } \\ { { \vdots } } \\ { { \vdots } } \\ { { { \bf g } _ { N } ^ { t + 1 } } } \end{array} \right] = \left[ \begin{array} { c c c c } { { { \cal K } _ { 1 1 } } } & { { \cdots } } & { { { \cal K } _ { 1 N } } } \\ { { \vdots } } & { { \ddots } } & { { \vdots } } \\ { { { \cal K } _ { N 1 } } } & { { \cdots } } & { { { \cal K } _ { N N } } } \end{array} \right] \left[ \begin{array} { c } { { { \bf g } _ { 1 } ^ { t } } } \\ { { \vdots } } \\ { { { \bf g } _ { N } ^ { t } } } \end{array} \right] + \left[ \begin{array} { c c c c } { { { \cal L } _ { 1 1 } } } & { { \cdots } } & { { { \cal L } _ { 1 N } } } \\ { { \vdots } } & { { \ddots } } & { { \vdots } } \\ { { { \cal L } _ { N 1 } } } & { { \cdots } } & { { { \cal L } _ { N N } } } \end{array} \right] \left[ \begin{array} { c } { { { \bf u } _ { 1 } ^ { t } } } \\ { { \vdots } } \\ { { \vdots } } \\ { { { \bf u } _ { N } ^ { t } } } \end{array} \right] .
+$$
+
+As we have seen in the case of linear spring system, those matrix blocks are not independent, but some of them share the same set of values.
+
+• The same physical interactions shall share the same transition block. The equivalence between the blocks should reflect the equivalence of the interactions, where we use the same transition sub-matrix for object pairs of the same relation. For example, if the system is composed of $N$ identical objects interacting with the same relation, then, by symmetry, all the diagonal blocks should be the same, while all the off-diagonal blocks should also be the same. The repetitive structure allows us to efficiently identify the values using least squares regression.
+
+# 3.3 LEARNING THE KOOPMAN EMBEDDINGS USING GRAPH NEURAL NETWORKS
+
+For a graph that contains , where vertic $N$ nt the syesent obj m at time s and ed $t$ us $G ^ { t } = ( O ^ { t } , R )$ $O ^ { t } = \{ o _ { i } ^ { t } \} _ { i = 1 } ^ { N }$ $\bar { R \mathbf { \Psi } } = \{ r _ { k } \} _ { k = 1 } ^ { N ^ { 2 } }$ represent pair-wise relations. Specifically, ${ \pmb O } _ { i } ^ { t } = ( \bar { \pmb x } _ { i } ^ { t } , { \pmb a } _ { i } ^ { o } )$ $\boldsymbol { x } _ { i } ^ { t }$ is the state of object $i$ and $\pmb { a } _ { i } ^ { o }$ is a one-hot vector indicating the object type, e.g., fixed or movable. For relation, we have $\pmb { r } _ { k } = ( u _ { k } , v _ { k } , \pmb { a } _ { k } ^ { r } ) , 1 \leq u _ { k } , v _ { k } \leq N$ , where $u _ { k }$ and $v _ { k }$ are integers denoting the end points of this directed edge, and $\pmb { a } _ { k } ^ { r }$ is a one-hot vector denoting the type of the relation $k$ .
+
+We use a graph neural network similar to Interaction Networks (IN) (Battaglia et al., 2016) to generate object-centric Koopman embeddings. IN defines an object function $f _ { O }$ and a relation function $f _ { R }$ to model objects and their relations in a compositional way. Similar to a message passing procedure, we calculate the edge effect $\pmb { e } _ { k } ^ { t } = f _ { R } ( \bar { \pmb { o } } _ { u _ { k } } ^ { t } , \pmb { o } _ { v _ { k } } ^ { t } , \pmb { a } _ { k } ^ { r } ) _ { k = 1 \dots N ^ { 2 } }$ , and node effect $\begin{array} { r } { \pmb { g } _ { i } ^ { t } = f _ { O } ( \pmb { o } _ { i } ^ { t } , \sum _ { k \in \mathcal { N } _ { i } } \pmb { e } _ { k } ^ { t } ) _ { i = 1 \dots N } } \end{array}$ , where ${ \mathcal { N } } _ { i }$ denotes the relations that point to the object $i$ and $\{ g _ { i } ^ { t } \}$ are the derived Koopman embeddings. We use this graph neural network, denoted as $\phi$ , to represent our Koopman observation function.
+
+System identification. For a sequence of observations $\widetilde { \pmb { x } } = \{ \pmb { x } ^ { 1 } , \cdots , \pmb { x } ^ { T } \}$ from time 1 to time $T$ , we first map them to the Koopman space as $\widetilde { \pmb { g } } = [ \pmb { g } ^ { 1 } , \dotsb { \mathrm { ~ , ~ } } \pmb { g } ^ { T } ]$ using the graph encoder $\phi$ , where $g ^ { t } = \phi ( \pmb { x } ^ { t } )$ . We use $\pmb { g } ^ { i : j }$ eto denote the sub-sequence $[ \pmb { g } ^ { i } , \cdots , \pmb { g } ^ { j } ]$ . To identify the Koopman matrix, we solve the linear regression $\operatorname* { m i n } _ { K } { \| K g ^ { 1 : T - 1 } - g ^ { \overleftarrow { 2 } : T } \| _ { 2 } }$ . As a result, $K { \dot { = } } g ^ { 2 : T } ( { \dot { g } } ^ { 1 : T - 1 } ) ^ { \dagger }$ will asymptotically approach the Koopman operator $\kappa$ with an increasing $T$ . For cases where there are control inputs $\widetilde { \pmb { u } } = [ \pmb { u } ^ { 1 } , \dotsb , \pmb { u } ^ { T - 1 } ]$ , the calculation of the Koopman matrix and the control matrix is eessentially solving a least squares problem w.r.t. the objective
+
+$$
+\operatorname* { m i n } _ { K , L } | | K g ^ { 1 : T - 1 } + L \widetilde { \pmb { u } } - \pmb { g } ^ { 2 : T } | | _ { 2 } .
+$$
+
+As we mentioned in the Section 3.2, the dimension of the Koopman space is linear to the number of objects in the system, i.e., $\widetilde { \pmb g } \in \mathbb R ^ { N m \times T }$ and $K \in \mathbb { R } ^ { N m \times N m }$ . If we do not enforce any structure on the Koopman matrix $K$ e, we will have to identify $N ^ { 2 } m ^ { 2 }$ parameters. Instead, we can significantly reduce the number by leveraging the assumption on the structure of $K$ . Assume we know some blocks $( \{ K _ { i j } \} )$ of the matrix $K$ are shared and in total there are $h$ different kinds of blocks, which we denote as $\hat { K } \in \mathbb { R } ^ { h \times m \times m }$ . Then, the number of parameter to be identified reduce to $h m ^ { 2 }$ . Usually, $h$ does not depend on $N$ , and is much smaller than $N ^ { 2 }$ . Now, for each block $K _ { i j }$ , we have a one-hot vector $\sigma _ { i j } \in \{ 0 , 1 \} ^ { h }$ indicating its type, i.e., $K _ { i j } = \sigma _ { i j } \hat { K } \in \mathbb { R } ^ { m \times m }$ . Finally, as shown in equation 7, we represent the Koopman matrix as the product of the index tensor $\sigma$ and the parameter tensor $\hat { K }$ :
+
+$$
+K = \sigma \otimes { \hat { K } } = \left[ \begin{array} { c c c } { \sigma _ { 1 1 } { \hat { K } } } & { \cdots } & { \sigma _ { 1 N } { \hat { K } } } \\ { \vdots } & { \ddots } & { \vdots } \\ { \sigma _ { N 1 } { \hat { K } } } & { \cdots } & { \sigma _ { N N } { \hat { K } } } \end{array} \right] , \mathrm { w h e r e } \sigma = \left[ \begin{array} { c c c } { \sigma _ { 1 1 } } & { \cdots } & { \sigma _ { 1 N } } \\ { \vdots } & { \ddots } & { \vdots } \\ { \sigma _ { N 1 } } & { \cdots } & { \sigma _ { N N } } \end{array} \right] \in \mathbb { R } ^ { N \times N \times h } .
+$$
+
+Similar to the Koopman matrix, we assume the same block structure in the control matrix $L$ and denote its parameter as $\hat { L } \in \mathbb { R } ^ { h \times m \times l }$ . The least squares problem of identifying $\hat { K }$ and $\hat { L }$ becomes
+
+$$
+\operatorname* { m i n } _ { \hat { K } , \hat { L } } \| ( \sigma \otimes \hat { K } ) \pmb { g } ^ { 1 : T - 1 } + ( \sigma \otimes \hat { L } ) \widetilde { \pmb { u } } - \pmb { g } ^ { 2 : T } \| _ { 2 } ,
+$$
+
+where $\sigma \otimes \hat { K } \in \mathbb { R } ^ { N m \times N m }$ , $\sigma \otimes \hat { L } \in \mathbb { R } ^ { N m \times N l }$ , $\pmb { g } ^ { 1 : T - 1 } \in \mathbb { R } ^ { N m \times ( T - 1 ) }$ and $\widetilde { \pmb { u } } \in \mathbb { R } ^ { N l \times ( T - 1 ) }$ . Since ethe linear least squares problems described in equation 6 and equation 8 have analytical solutions, performing system identification using our method is very efficient.
+
+Training GNN models. To make predictions on the states, we use a graph decoder $\psi$ to map the Koopman embeddings back to the original state space. In total, we have three losses to train the graph encoder and decoder. The first term is the auto-encoding loss
+
+$$
+\mathcal { L } _ { \mathrm { a e } } = \frac { 1 } { T } \sum _ { i } ^ { T } \| \psi ( \phi ( \pmb { x } ^ { i } ) ) - \pmb { x } ^ { i } \| .
+$$
+
+The second term is the prediction loss. To calculate it, we rollout in the Koopman space and denote the embeddings as $\hat { \pmb g } ^ { 1 } \overset { ^ { } } { = } \pmb g ^ { 1 }$ , and $\hat { \pmb { g } } ^ { t + 1 } = K \hat { \pmb { g } } ^ { t } + L \pmb { u } ^ { t }$ , for $t = 1 , \cdots , T - 1$ . The prediction loss is defined as the difference between the decoded states and the actual states, i.e.,
+
+$$
+\mathcal { L } _ { \mathrm { p r e d } } = \frac { 1 } { T } \sum _ { i = 1 } ^ { T } \Vert \psi ( \hat { \pmb g } ^ { i } ) - \pmb x ^ { i } \Vert .
+$$
+
+Third, we employ a metric loss to encourage the Koopman embeddings preserving the distance in the original state space. The loss is defined as the absolute error between the distances measured in the Koopman space and that in the original space, i.e.,
+
+$$
+{ \mathcal { L } } _ { \mathrm { m e t r i c } } = \sum _ { i j } \left| \left\| { \pmb { g } } ^ { i } - { \pmb { g } } ^ { j } \right\| - \left\| { \pmb { x } } ^ { i } - { \pmb { x } } ^ { j } \right\| \right| .
+$$
+
+Having Koopman embeddings that perserves the distance in the state space is important as we are using the distance in the Koopman space to define the cost function for downstream control tasks.
+
+The final training loss is simply the combination of all the terms above: $\mathcal { L } = \mathcal { L } _ { \mathrm { a e } } + \lambda _ { 1 } \mathcal { L } _ { \mathrm { p r e d } } + \lambda _ { 2 } \mathcal { L } _ { \mathrm { m e t r i c } }$ . We then minimize the loss $\mathcal { L }$ by optimizing the parameters in the graph encoder $\phi$ and graph decoder $\psi$ using stochastic gradient descent. Once the model is trained, it can be used for system identification, future prediction, and control synthesis.
+
+# 3.4 CONTROL
+
+For a control task, the goal is to synthesize a sequence of control inputs $\mathbf { \pmb { u } } ^ { 1 : T }$ that minimize $C =$ $\textstyle \sum _ { t = 1 } ^ { T } c _ { t } ( \pmb { x } ^ { t } , \pmb { u } ^ { t } )$ , the total incurred cost, where $c _ { t } ( \pmb { x } ^ { t } , \pmb { u } ^ { t } )$ is the instantaneous cost. For example, considering the control task of reaching a desired state $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } } ^ { * }$ at time $T$ , we can design the following instantaneous cost, $c _ { t } ( \pmb { x } ^ { t } , \pmb { u } ^ { t } ) = \mathbb { 1 } _ { [ t = T ] } \bar { \lVert \pmb { x } ^ { t } - \pmb { x } ^ { * } \rVert _ { 2 } ^ { 2 } + \lambda \lVert \pmb { u } ^ { t } \rVert _ { 2 } ^ { 2 } }$ . The first term promotes the control sequence that matches the state to the goal, while the second term regularizes the control signals.
+
+Open-loop control via quadratic programming (QP). Our model maps the original nonlinear dynamics to a linear dynamical system. We can then solve the control task by solving a linear control problem. With the assumption that the Koopman embeddings preserve the distance measure, we define the control cost as $c _ { t } ( { \pmb g } ^ { t } , { \pmb u } ^ { t } ) = \mathbb { 1 } _ { [ t = T ] } \| { \pmb g } ^ { t } - { \pmb g } ^ { * } \| _ { 2 } ^ { 2 } + \lambda \| { \pmb u } ^ { t } \| _ { 2 } ^ { 2 }$ . As a result, we reduce the problem to minimizing a quadratic cost function $\begin{array} { r } { C = \sum _ { t = 1 } ^ { T } c _ { t } ( g ^ { t } , \pmb { u } ^ { t } ) } \end{array}$ over variables $\{ g ^ { t } , \pmb { u } ^ { t } \} _ { t = 1 } ^ { T }$ , where $\pmb { g } ^ { 1 } = \phi ( \pmb { x } ^ { 1 } )$ and ${ \pmb g } ^ { * } = \phi ( { \pmb x } ^ { * } )$
+
+Model predictive control (MPC). Solving the QP gives us control signals, which might not be good enough for long-term control as the prediction error accumulates. We can combine it with Model Predictive Control, assuming feedback from the environment every $\tau$ steps.
+
+# 4 EXPERIMENTS
+
+Environments. We evaluate our method by assessing how well it can simulate and control ropes and soft robots. Specifically, we consider three environments. (1) Rope (Figure 2a): the top mass of a rope is fixed to a specific height. We apply force to the top mass to move it in a horizontal line. The rest of the masses are free to move according to internal force and gravity. (2) Soft (Figure 2b): we aim to control a soft robot that is consist of soft blocks. Blocks in dark grey are rigid and those in light blue are soft blocks. Each one of the dark blue blocks is soft but have an actuator inside that can contract or expand the block. One of the blocks is pinned to the ground, as shown using the red dots. (3) Swim (Figure 2c): instead of pinning the soft robot to the ground, we let the robot swim in fluids. The colors shown in this environment have the same meaning as in Soft.
+
+![](images/71263117fabd577db5465647abbac468c4da70a4fa5de4c94cdec174d7baecd9.jpg)  
+Figure 2: Qualitative results. Top: our model prediction matches the ground truth over a long period. Bottom: for control, we use red dots or frames to indicate the goal. We apply the control signals generated from our identified model to the original simulator, which allows the agent to achieve the goal accurately. Please refer to our supplementary video for more results.
+
+Observation space. In the Rope environment, each mass on the rope is considered as an object. The observation of each mass is its position and velocity in the 2D plane, which has a dimension of 4. In total, a rope with $N$ masses has an observation space of dimension $4 N$ . In both the Soft and the Swim environments, each quadrilateral is considered as an object. For each quadrilateral, we have access to the positions and velocities of the four corners. Thus for a soft robot containing $N$ quadrilaterals, we have a $4 \times 4 \times N = 1 6 N$ dimensional observation.
+
+Baselines. We compare our model to the following baselines: Interaction Networks (Battaglia et al., 2016) (IN), Propagation Networks (Li et al., 2019b) (PN) and Koopman method with handcrafted Koopman base functions (KPM). IN and PN are the state-of-the-art learning-based physical simulators, and we evaluate their adaptation ability by finetuning their parameters on a small sequence of observations from the testing environment. Similar to our method, KPM fits a linear dynamics in the Koopman space. Instead of learning Koopman observations from data, KPM uses polynomials of the original states as the basis functions. In our setting, we set the maximum order of the polynomials to be three to make the dimension of the hand-crafted Koopman embeddings match our model’s.
+
+Data generation. We generate 10,000 episodes for Rope and 50,000 episodes for Soft and Swim. Among them, $9 0 \%$ are used for training, and the rest for testing. Each episode has 100 time steps. In the dataset, the physical systems have a various number of objects from 5 to 9, i.e. the ropes have 5 to 9 masses while the soft robots in Soft and Swim environments have 5 to 9 quadrilaterals. To evaluate the model’s extrapolating generalization ability, for each environment, we generate an extra dataset with the same size as the test set while containing systems consist of 10 to 14 objects.
+
+![](images/f0fdc1a405d982574e72ebf82281193b8558e35f505a30bae1a8d27cd079858f.jpg)  
+Figure 3: Quantitative results on simulation. The $x$ axis shows time steps. The solid lines indicate medians and the transparent regions are the interquartile ranges of simulation errors. Our method significantly outperforms the baselines in all testing environments.
+
+![](images/64d97fdca400c7934ba7f32ddaca8ae4060677260c2492205a69120a934edfd3.jpg)  
+Figure 4: Quantitative results on control and ablation studies on model hyperparameters. Left: box-plots show the distributions of control errors. The yellow line in the box indicates the median. Our model consistently achieves smaller errors in all environments against KPM. Right: our model’s simulation errors with different amount of data for system identification (d) and different dimensions of the Koopman space (e).
+
+Training and evaluation protocols. All models are trained using Adam optimizer (Kingma & Ba, 2015) with a learning rate of $1 0 ^ { - 4 }$ and a batch size of 8. $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are 1.0 and 0.3, respectively, for our model. For both our model and the baselines, we apply 400K iterations of gradient steps in the Rope environment and 580K iterations in the Soft and Swim environment. Our model is trained on the sub-sequence of length 64 from the training set, and IN/PN aims at minimizing the L1 distance between their prediction and the ground truth. During test time, the models have to adapt to a new environment of unknown physical parameters, where they have access to a short sequence of observations and the opportunity to adjust their models’ parameters. Our model uses 8 episodes to identify the transition matrix via least-square regression. IN/PN update the model’s parameters by minimizing the distance between the model’s prediction and the actual observation using a gradient step of length $1 0 ^ { - 4 }$ for 5 iterations. For evaluation, we use two metrics: simulation error and control error. For a given episode, the simulation error at time step $t$ is defined as the mean squared error between the model prediction $\hat { \mathbf { x } } ^ { t }$ and the ground truth $\mathbf { \boldsymbol { x } } ^ { t }$ . For control, we pick the initial frame $\mathbf { \boldsymbol { x } } ^ { 0 }$ and the $t ^ { \star }$ th frame $\mathbf { \boldsymbol { x } } ^ { t }$ from a episode. Then we ask the model to generate a control sequence of length $t$ to transfer the system from the initial state $\mathbf { \boldsymbol { x } } ^ { 0 }$ to the target state $\mathbf { \boldsymbol { x } } ^ { t }$ . The control error is defined as the mean squared distance between the target state and the state of the system at time $t$ .
+
+# 4.1 SIMULATION
+
+Figure 2 shows qualitative results on simulation. Our model accurately predicts system dynamics for more than 100 steps. For Rope, the small prediction error comes from the slight delay of the force propagation inside the rope; hence, the tail of the rope usually has a larger error. For Soft, our model captures the interaction between the body parts and generates accurate prediction over the global movements of the robot. The error mainly comes from the misalignment of some local components.
+
+We evaluate the models by predicting 100 steps into the future on 500 trajectories and Figure 3 shows quantitative results. IN and PN do not work well in the Rope and Swim environments due to insufficient system identification ability. The KPM baseline performs poorly in the Rope and
+
+Soft environments indicating the limited power of polynomial Koopman base functions. Our model significantly outperforms all the baselines.
+
+# 4.2 CONTROL
+
+We compare our model with KPM, the Koopman baseline using polynomial basis. In Rope, we ask the models to perform open-loop control where it only solves the QP once at the beginning. The length of the control sequence is 40. When it comes to Soft/Swim, each model is asked to generate control signals of 64 steps, and we allow the model to receive feedback after 32 steps. Thus every model has a second chance to correct its control sequence by solving the QP again at the time step 32.
+
+As shown in Figure 2, our model leverages the inertia of the rope and matches the target state accurately. As for controlling a soft body swinging on the ground or swimming in the water, our model can move each part (the boxes) of the body to the exact target position. The small control error comes from the slight misalignment of the orientation and the size of the body parts. Figure 4 shows that quantitatively our model outperforms KPM, too.
+
+# 4.3 ABLATION STUDY
+
+Structure of the Koopman matrix. We explore three different structures of the Koopman matrix, Block, Diag and None, to understand its effect on the learned dynamics. None assumes no structure in the Koopman matrix. Diag assumes a diagonal block structure of $K$ : all off-diagonal blocks $( K _ { i j }$ where $i \neq j$ ) are zeros and all diagonal blocks share the same values. Block predefines a block-wise structure, decided by the relation between the objects as introduced in Section 3.3.
+
+Table 1 includes our model’s simulation error and control error with different Koopman matrix structures in Rope. All models are trained in the Rope environment with 5 to 9 masses. Besides the result on the test set, we also report models’ extrapolation performance in parentheses, where the model is evaluated on systems with more masses than training, i.e., 10 to 14 masses.
+
+Table 1: Ablation study results on the Koopman matrix structure (Rope environment). For simulation, we show the Mean Squared Error between the prediction and the ground truth at $T = 1 0 0$ , whereas for control, we show the performance with a horizon of length 40. The numbers in parentheses show the performance on extrapolation.
+
+Our model with Block structure consistently achieves a smaller error in all settings. Diag assumes an overly simplified structure, leading to larger errors and failing to make reasonable controls. None has comparable simulation errors but larger control errors. Without the structure in the Koopman matrix, it overfits the data and makes the resulting linear dynamics less amiable to the control.
+
+<table><tr><td></td><td>Simulation</td><td>Control</td></tr><tr><td>Diag</td><td>0.133 (0.174)</td><td>2.337 (2.809)</td></tr><tr><td>None</td><td>0.117 (0.083)</td><td>1.522 (1.288)</td></tr><tr><td>Block</td><td>0.105 (0.075)</td><td>0.854 (1.101)</td></tr></table>
+
+Hyperparameters. In our main experiments, we set the dimension of the Koopman embedding to $m = 3 2$ per object. Online system identification requires 800 data samples for each training/test case. To understand our model’s performance under different hyperparameters, we vary the dimension of the Koopman embedding from 8 to 64 and the number of data samples used for system identification from 200 to 1,600. Figure 4d shows that more data for system identification leads to better simulation results. Figure 4e shows that dimension 16 gives the best results on simulation. It may suggest that the intrinsic dimension of the Koopman invariant space of the Rope system is around 16 per object.
+
+# 5 CONCLUSION
+
+Compositionality is common in our daily life. Many ordinary objects contain repetitive subcomponents: ropes and soft robots, as shown in this paper, granular materials such as coffee beans and lego blocks, and deformable objects such as cloth and modeling clay. These objects are known to be very challenging for manipulation using traditional methods, while our formulation opens up a new direction by combining deep Koopman operators with graph neural networks. By leveraging the compositional structure in the Koopman operator via graph neural nets, our model can efficiently manipulate deformable objects such as ropes and soft robots, and generalize to systems with variable numbers of components. We hope this work could encourage more endeavors in modeling larger and more complex systems by integrating the power of the Koopman theory and the expressiveness of neural networks.
+
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+
+# A ENVIRONMENT AND MODEL DETAILS
+
+Interaction types. In our experiments, interactions are considered different if the types are different or the objects involved have different physical properties.
+
+In the Rope environment, the top mass has a fixed height and is considered differently from the other masses. Thus, we have 2 types of self-interactions for the top mass and the non-top masses. In addition, we have 8 types of interactions between different objects. The objects on a relation could be either top mass or non-top mass. It is a combination of 4. And the interaction may happen between two nearby masses or masses that are two-hop away. In total, the number of interactions between different objects is $4 \times 2 = 8$ .
+
+In the Soft environments, there are four types of quadrilaterals: rigid, soft, actuated, and fixed. We have four types of self-interactions correspondingly. For the interactions between objects, we add edges between two quadrilaterals only if they are connected by a point or edge. Connection from different directions are considered as different relations. There are 8 different directions, up, down, left, right, up-left, down-left, up-right, down-right. The relation types also encode the type of receiver object. Thus, in total, there are $( 8 + 1 ) \times 4 = 3 6$ types of relations between different objects.
+
+In the Swim environment, there are three types of quadrilaterals: rigid, soft, and actuated. Similar to the Soft environment, we use different edge types for different connecting directions; hence, the number of edge type is $( 8 + 1 ) \times 3 = 2 7$ .
+
+# B ADDITIONAL EXPERIMENTS
+
+Comparison with a classical physical simulator optimized using back-box optimization. We have performed comparisons with a classical physical simulator optimized using black-box optimization (Delingette, 1998) by assuming different levels of knowledge over the ground truth model.
+
+If we assume that we know the ground truth model, where we only need to identify relevant physical parameters during the system identification stage, Bayesian Optimization (Snoek et al., 2012) (BO) can give us a reasonable estimate of the physical parameters. However, BO requires much more time to achieve a comparable performance with our method in the Rope environment: 0.43 vs. 180 seconds averaged over 100 trails (Ours vs. BO).
+
+If we are unsure about the ground truth model and we approximate the system using a set of points linked by springs and dampers, BO does not work as well. In our additional experiments, we approximate the Rope environment using a chained spring-mass system, say $n$ masses and $n - 1$ springs. While taking much more time, BO still cannot give us a satisfying result: simulation error 0.046 vs. 0.084 and control error 0.854 vs. 2.547 (Ours vs. BO).
+
+Ablation study on the effectiveness of the metric loss. The internal linear structure allows us to solve the control problem using quadratic programming, where the objective function for control is defined in the embedding space (Section 3.4); hence, it is desirable to have Koopman embeddings that preserve the distance in the original state space. In Section 3.3, we introduce a metric loss to promote learning a Koopman embedding that keeps the distance measurement.
+
+To demonstrate the effect of the metric loss, we compare the models trained with and without the metric loss. We establish the comparison using two measurements, the distance preservation and the prediction accuracy. To evaluate how well the Koopman embeddings preserve the distance, we compute the distribution of the log-ratio of the distance in the Koopman space and in the original state space, i.e., $\log \left( { \frac { \| { \pmb { g } } ^ { i } - { \pmb { g } } ^ { j } \| _ { 2 } } { \| { \pmb { x } } ^ { i } - { \pmb { x } } ^ { j } \| _ { 2 } } } \right)$ . For the model prediction accuracy, we show the simulation errors.
+
+We perform the experiments in the Rope environment and show the result in Figure 5. On the left, we show the ratio of distance in the learned Koopman space and the distance in the original state space. The model trained with metric loss has a log distance ratio that significantly more concentrates on 0. It means the metric loss effectively regularizes the model to preserve the distance. On the right, we show the simulation errors of the two models, which indicate that two models have comparable prediction performance. Metric loss effectively enhances the property of distance-preserving while not making a big sacrifice on the accuracy of the dynamics modeling.
+
+![](images/f00d555af41a4f7c7e11cf5d9b806fe2c47cfd9aed456f875f1dcb1ec4ae1f89.jpg)  
+Figure 5: Ablation study on the metric loss in the Rope environment. (a) shows the distributions of the logarithm distance ratio, i.e, $\log \left( { \frac { \| { \pmb { g } } ^ { i } - { \pmb { g } } ^ { j } \| _ { 2 } } { \| { \pmb { x } } ^ { i } - { \pmb { x } } ^ { j } \| _ { 2 } } } \right)$ The model trained with metric loss has a distance ratio much more concentrated to 1, which indicates it preserves the distance much better than the counterpart. (b) illustrates the simulation error of two models, where their performance is on par. (c) shows that the model trained using the metric loss performs better control.
+
+![](images/574fbfd0e376cdc34bb544d0ad7177a1053f7d961bda58f59dcd9e9e43a0b42a.jpg)  
+Figure 6: Modeling rope with known physical parameters. We show the comparison between our model and IN/PN in scenarios where we have access to the ground truth physical parameters. In this case, IN and PN slightly outperform our method due to the internal linear structure in our model. However, in the real world, we do not always know the physical parameters and their values, which makes our method preferable when adapting to new environments.
+
+Experiments in a known physical parameter setting. Our setting is different from the settings in the original IN and PN papers that we do not assume we know the physical parameters and their values, such as stiffness, mass, and gravity. Instead, the parameters are embedded in the transition matrices during the system identification stage (Section 3.3). If the model has access to the underlying physical parameters, as expected, IN and PN slightly outperform our method as the internal linear structure limits our model’s expressiveness, as shown in Figure 6. In the real world, however, the underlying physical parameters are not always known, which makes our model a better choice when adapting to unseen environments.

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+        "type": "text",
+        "text": "LEARNING COMPOSITIONAL KOOPMAN OPERATORSFOR MODEL-BASED CONTROL",
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+        "text": "Yunzhu Li∗ Hao He∗ MIT CSAIL MIT CSAIL ",
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+        "text": "Finding an embedding space for a linear approximation of a nonlinear dynamical system enables efficient system identification and control synthesis. The Koopman operator theory lays the foundation for identifying the nonlinear-to-linear coordinate transformations with data-driven methods. Recently, researchers have proposed to use deep neural networks as a more expressive class of basis functions for calculating the Koopman operators. These approaches, however, assume a fixed dimensional state space; they are therefore not applicable to scenarios with a variable number of objects. In this paper, we propose to learn compositional Koopman operators, using graph neural networks to encode the state into objectcentric embeddings and using a block-wise linear transition matrix to regularize the shared structure across objects. The learned dynamics can quickly adapt to new environments of unknown physical parameters and produce control signals to achieve a specified goal. Our experiments on manipulating ropes and controlling soft robots show that the proposed method has better efficiency and generalization ability than existing baselines. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Simulating and controlling complex dynamical systems, such as ropes or soft robots, relies on two key features of the dynamics model: first, it needs to be efficient for system identification and motor control; second, it needs to be generalizable to a complex, constantly evolving environments. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In practice, computational models for complex, nonlinear dynamical systems are often not efficient enough for real-time control (Mayne, 2000). The Koopman operator theory suggests that identifying nonlinear-to-linear coordinate transformations allows efficient linear approximation of nonlinear systems (Williams et al., 2015; Mauroy & Goncalves, 2016). Fast as they are, however, existing papers on Koopman operators focus on a single dynamical system, making it hard to generalize to cases where there are a variable number of components. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast, recent advances in approximating dynamics models with deep nets have demonstrated its power in characterizing complex, generic environments. In particular, a few recent papers have explored the use of graph nets in dynamics modeling, taking into account the state of each object as well as their interactions. This allows their models to generalize to scenarios with a variable number of objects (Battaglia et al., 2016; Chang et al., 2017). Despite their strong generalization power, they are not as efficient in system identification and control, because deep nets are heavily over-parameterized, making optimization time-consuming and sample-inefficient. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In this paper, we propose compositional Koopman operators, integrating Koopman operators with graph networks for generalizable and efficient dynamics modeling. We build on the idea of encoding states into object-centric embeddings with graph neural networks, which ensures generalization power. But instead of using over-parameterized neural nets to model state transition, we identify the Koopman matrix and control matrix from data as a linear approximation of the nonlinear dynamical system. The linear approximation allows efficient system identification and control synthesis. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The main challenge of extending Koopman theory to multi-object systems is scalability. The number of parameters in the Koopman matrix scales quadratically with the number of objects, which harms the learning efficiency and leads to overfitting. To tackle this issue, we exploit the structure of the underlying system and use the same block-wise Koopman sub-matrix for object pairs of the same relation. This significantly reduces the number of parameters that need to be identified by making it independent of the size of the system. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/324bececc168c0745244997b7ffa1ced5381091c5ea6bfc968579a1b4ad86135.jpg",
+        "image_caption": [
+            "Figure 1: Overview of our model. A graph neural network $\\phi$ takes in the current state of the physical system $\\mathbf { \\boldsymbol { x } } ^ { t }$ , and generates object-centric representations in the Koopman space $g ^ { t }$ . We then use the block-wise Koopman matrix $K$ and control matrix $L$ identified from equation 6 or equation 8 to predict the Koopman embeddings in the next time step $\\mathbf g ^ { t + 1 }$ . Note that in $K$ and $L$ , object pairs of the same relation share the same sub-matrix. Another graph neural network $\\psi$ maps $\\mathbf g ^ { t + 1 }$ back to the original state space, i.e., $\\boldsymbol { x } ^ { t + 1 }$ . The mapping between $g ^ { \\hat { t } }$ and $\\mathbf g ^ { t + 1 }$ is linear and is shared across all time steps, where we can iteratively apply $K$ and $L$ to the Koopman embeddings and roll multiple steps into the future. The formulation enables efficient system identification and control synthesis. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Our experiments include simulating and controlling ropes of variable lengths and soft robots of different shapes. The compositional Koopman operators are significantly more accurate than the state-of-the-art learned physics engines (Battaglia et al., 2016; Li et al., 2019b), and faster when adapting to new environments of unknown physical parameters. Our method also outperforms vanilla deep Koopman methods (Lusch et al., 2018; Morton et al., 2018) and Koopman models with manually-designed basis functions, which shows the advantages of using a structured Koopman matrix and graph neural networks. Please see our project page for demonstrating videos. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Koopman operators. The Koopman operator formalism of dynamical systems is rooted in the seminal works of Koopman and Von Neumann in the early 1930s (Koopman, 1931; Koopman & Neumann, 1932). The core idea is to map the state of a nonlinear dynamical system to an embedding space, over which we can linearly propagate into the future. Researchers have proposed various algorithms to explore the Koopman spectral properties from data. A large portion of them are in the class of dynamic mode decomposition (DMD) (Rowley et al., 2009; Schmid, 2010; Tu et al., 2014; Williams et al., 2015; Arbabi & Mezic, 2017). The linear representation will enable efficient prediction, estimation, and control using tools from linear dynamical systems (Williams et al., 2016; Proctor et al., 2018; Mauroy & Goncalves, 2019; Korda & Mezic´, 2018). People have been using hand-designed Koopman observables for various modeling and control tasks (Brunton et al., 2016; Kaiser et al., 2017; Abraham et al., 2017; Bruder et al., 2019b; Arbabi et al., 2018). Some recent works have applied the method to the real world and successfully control soft robots with great precision (Bruder et al., 2019a; Mamakoukas et al., 2019). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, hand-crafted basis functions sometimes fail to generalize to more complex environments. Learning these functions from data using neural nets turns out to generate a more expressive invariant subspace (Lusch et al., 2018; Takeishi et al., 2017) and has achieved successes in fluid control (Morton et al., 2018). Morton et al. (2019) has also extended the framework to account for uncertainty in the system by inferring a distribution over observations. Our model differs by explicitly modeling the compositionality of the underlying system with graph networks. It generalizes better to environments of a variable number of objects or soft robots of different shapes. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning-based physical simulators. Battaglia et al. (2016) and Chang et al. (2017) first explored learning a simulator from data by approximating object interactions with neural networks. These models are no longer bounded to hard-coded physical rules, and can adapt to scenarios where the underlying physics is unknown. Please refer to Battaglia et al. (2018) for a full review. Recently, Mrowca et al. (2018) and Li et al. (2019a) extended these models to approximate particle dynamics of deformable shapes and fluids. Flexible as they are, these models become less efficient during model adaptation in complex scenarios, because the optimization of neural networks usually needs a lot of samples and compute, which limits its use in an online setting. Nagabandi et al. (2019a;b) proposed to use meta-learning for online adaptation, and have shown to be effective in simulated robots and a real legged millirobot. However, it is not clear whether their methods can generalize to systems with variable numbers of instances. The use of graph nets and Koopman operators in our model allows better generalization ability and enables efficient system identification as we only need to identify the transition matrices, which is essentially a least-square problem and can be solved very efficiently. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "People have also used the learned physics engines for planning and control. Many previous papers in this direction learn a latent dynamics model together with a policy in a model-based reinforcement learning setup (Racaniere et al. \\` , 2017; Hamrick et al., 2017; Pascanu et al., 2017; Hafner et al., 2019); a few alternatives use the learned model in model-predictive control (MPC) (Sanchez-Gonzalez et al., 2018; Li et al., 2019b; Janner et al., 2019). In this paper, we leverage the fact that the embeddings in the Koopman space are propagating linearly through time, which allows us to formulate the control problem as quadratic programming and optimize the control signals much more efficiently. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3 APPROACH",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We first present the basics of Koopman operators: for a nonlinear dynamical system, the Koopman observation functions can map the state space to an embedding space where the dynamics become linear. We then discuss the compositional nature of physical systems and show how graph networks can be used to capture the compositionality. ",
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+    {
+        "type": "text",
+        "text": "3.1 THE KOOPMAN OPERATORS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Let $\\pmb { x } ^ { t } \\in \\mathcal { X } \\subset \\mathbb { R } ^ { n }$ be the state vector for the system at time step $t$ . We consider a non-linear discrete-time dynamical system described by ${ \\pmb x } ^ { t + 1 } = F ( { \\pmb x } ^ { t } )$ . The Koopman operator (Koopman, 1931), denoted as $\\mathcal { K } : \\mathcal { F }  \\mathcal { F }$ , is a linear transformation defined by $\\kappa g \\triangleq g \\circ F$ , where $\\mathcal { F }$ is the collection of all functions (also referred to as observables) that form an infinite-dimensional Hilbert space. For every function $g : \\mathcal { X }  \\mathbb { R }$ belonging to $\\mathcal { F }$ , we have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/3a8e76fc026fbc8a9628fba449ad693c778802b04c87f1240707bc33f04c3fa4.jpg",
+        "text": "$$\n( \\mathcal { K } g ) ( \\pmb { x } ^ { t } ) = g ( F ( \\pmb { x } ^ { t } ) ) = g ( \\pmb { x } ^ { t + 1 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "making the function space $\\mathcal { F }$ invariant under the action of the Koopman operator. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Although the theory guarantees the existence of the Koopman operator, its use in practice is limited by its infinite dimensionality. Most often, we assume there is an invariant subspace $\\mathcal { G }$ of the Koopman operator. It spans by a set of base observation functions $\\{ g _ { 1 } , \\cdots , g _ { m } \\}$ and satisfies that $\\kappa g \\in \\mathcal G$ for any $g \\in { \\mathcal { G } }$ . With a slightly abuse of the notation, we now use $\\bar { g } ( \\pmb { x } ^ { \\hat { t } } ) : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { m }$ to represent $[ g _ { 1 } ( { \\pmb x } ^ { t } ) , \\cdot \\cdot \\cdot , g _ { m } ( { \\pmb x } ^ { t } ) ] ^ { T }$ . By constraining the Koopman operator on this invariant subspace, we get a finite-dimensional linear operator $K \\in \\mathbb { R } ^ { m \\times m }$ that we refer as the Koopman matrix. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Traditionally, people hand-craft base observation functions from the knowledge of underlying physics. The system identification problem is then reduced to finding the Koopman matrix $K$ , which can be solved by linear regression given historical data of the system. Recently, researchers have also explored data-driven methods that automatically find the Koopman invariant subspace via representing the base observation functions $g ( { \\pmb x } )$ via deep neural networks. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Although the original Koopman theory does not consider the system with external control inputs, researchers have found that linearly injecting control signals to the Koopman observation space can give us good numerical performance (Brunton et al., 2016; Bruder et al., 2019a). Mathematically, considering a dynamical system, $\\pmb { x } ^ { t + 1 } = F ( \\pmb { x } ^ { t } , \\pmb { u } ^ { t } )$ , with an external control input ${ \\mathbf { } } _ { { \\mathbf { } } { \\mathbf { } } } { \\mathbf { } } _ { { \\mathbf { } } } t$ , we aim to find the Koopman observation functions and the linear dynamics model in the form of ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aeca0dca83c29a28d6c02256bc3be82f02c63e9cef0928ecc2b69454c458b877.jpg",
+        "text": "$$\ng ( \\pmb { x } ^ { t + 1 } ) = K g ( \\pmb { x } ^ { t } ) + L \\pmb { u } ^ { t } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the coefficient matrix $L$ is referred to as the control matrix. ",
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+        "type": "text",
+        "text": "3.2 COMPOSITIONAL KOOPMAN OPERATORS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The dynamics of a physical system are governed by physical rules, which are usually shared across different subcomponents in the system. Explicitly modeling such compositionality enables more efficient system identification and control synthesis and provides better generalization ability. ",
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+    {
+        "type": "text",
+        "text": "Motivating example. Consider a system with $N$ balls moving in a 2D plane, each pair connected by a linear spring. Assume all balls have mass 1 and all springs share the same stiffness coefficient $k$ . We denote the $i$ ’s ball’s position as $( x _ { i } , y _ { i } )$ and its velocity as $( { \\dot { x } } _ { i } , { \\dot { y } } _ { i } )$ . For ball $i$ , equation 3 describes its dynamics, where $\\mathbf { \\Phi } _ { \\pmb { x } _ { i } } \\triangleq [ x _ { i } , y _ { i } , \\dot { x } _ { i } , \\dot { y } _ { i } ] ^ { T }$ denotes ball $i$ ’s state: ",
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+    {
+        "type": "equation",
+        "img_path": "images/ba598beba883d6e68bbcc5d18dae1e2c55afbff48b50f3cbf2ee7257a942ce60.jpg",
+        "text": "$$\n\\begin{array}{c} \\dot { v } _ { i } = \\left[ \\begin{array} { c } { \\dot { x } _ { i } } \\\\ { \\dot { y } _ { i } } \\\\ { \\dot { x } _ { i } } \\\\ { \\dot { y } _ { i } } \\end{array} \\right] = \\left[ \\begin{array} { c c c c c } { \\dot { x } _ { i } } & { 0 } & { 1 } & { 0 } \\\\ { \\dot { y } _ { i } } & { 0 } & { 0 } & { 1 } \\\\ { \\sum _ { j = 1 } ^ { N } k ( x _ { j } - x _ { i } ) } \\\\ { \\sum _ { j = 1 } ^ { N } k ( y _ { j } - y _ { i } ) } \\end{array} \\right] = \\underbrace { \\left[ \\begin{array} { c c c c c } { 0 } & { 0 } & { 1 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { 1 } \\\\ { k - N k } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { k - N k } & { 0 } & { 0 } \\end{array} \\right] } _ { \\triangleq \\mathcal { A } } \\left[ \\begin{array} { c } { x _ { i } } \\\\ { y _ { i } } \\\\ { \\dot { x } _ { i } } \\end{array} \\right] + \\sum _ { j \\neq i } \\left[ \\underbrace { 0 } _ { k } _ { 0 } \\begin{ c c c c } { 0 } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { 0 } \\\\ { k } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { k } & { 0 } & { 0 } \\end{array} \\right] \\left[ \\begin{array} { c } { x _ { j } } \\\\ { y _ { j } } \\\\ { \\dot { x } _ { j } } \\end{array} \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We can represent the state of the whole system using the union of every ball’s state, where ${ \\textbf { \\em x } } =$ $[ \\pmb { x } _ { 1 } , \\cdots , \\pmb { \\dot { x } } _ { N } ] ^ { T }$ . Then the transition matrix is essentially a block matrix, where the matrix parameters are shared among the diagonal or off-diagonal blocks as shown in equation 4: ",
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+        "type": "equation",
+        "img_path": "images/d1f5758e2a724846dac9354c50b68b1a0fcc46cf2a705d726c3ea659fcd0c517.jpg",
+        "text": "$$\n{ \\dot { \\pmb x } } = \\left[ \\begin{array} { c } { \\dot { \\pmb x } _ { 1 } } \\\\ { \\dot { \\pmb x } _ { 2 } } \\\\ { \\vdots } \\\\ { \\dot { \\pmb x } _ { N } } \\end{array} \\right] = \\left[ \\begin{array} { c c c c } { { \\cal A } } & { { \\cal B } } & { \\cdots } & { { \\cal B } } \\\\ { { \\cal B } } & { { \\cal A } } & { \\cdots } & { { \\cal B } } \\\\ { \\vdots } & { \\vdots } & { \\ddots } & { \\vdots } \\\\ { { \\cal B } } & { { \\cal B } } & { \\cdots } & { { \\cal A } } \\end{array} \\right] \\left[ \\begin{array} { c } { { \\pmb x } _ { 1 } } \\\\ { { \\pmb x } _ { 2 } } \\\\ { \\vdots } \\\\ { { \\pmb x } _ { N } } \\end{array} \\right] .\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Based on the linear spring system, we make three observations for multi-object systems. ",
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+    {
+        "type": "text",
+        "text": "The system state is composed of the state of each individual object. The dimension of the whole system scales linearly with the number of objects. We formulate the system state by concatenating the state of every object, corresponding to an object-centric state representation. • The transition matrix has a block-wise substructure. After assuming an object-centric state representation, the transition matrix naturally has a block-wise structure as shown in equation 4. • The same physical interactions share the same transition block. The blocks in the transition matrix encode actual interactions and generalize across systems. $A$ and $B$ govern the dynamics of the linear spring system, and are shared by systems with a different number of objects. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "These observations inspire us to exploit the structure of multi-object systems, instead of learning separate models for systems that contains different numbers of balls. ",
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+        "type": "text",
+        "text": "Compositional Koopman operators. Motivated by the linear spring system, we want to inject a good inductive bias to incorporate compositionality when applying the Koopman theory. This allows better generalization ability and more efficient system identification and better controller design. Figure 1 shows an overview of our model. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Considering a system with $N$ objects, we denote $\\mathbf { \\boldsymbol { x } } ^ { t }$ as the system state at time $t$ and $\\boldsymbol { x } _ { i } ^ { t }$ is the state of the $\\romannumeral 1$ ’th object. We further denote $\\mathbf { \\boldsymbol { g } } ^ { t } \\triangleq \\mathbf { \\boldsymbol { g } } ( \\mathbf { \\boldsymbol { x } } ^ { t } )$ as the embedding of the state in the Koopman invariant space. In the rest of the paper, we call $g ^ { t }$ the Koopman embedding. Based on the observation we made in the case of linear spring system, we propose the following assumptions on the compositional structure of the Koopman embedding and the Koopman matrix. ",
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+        "type": "text",
+        "text": "• The Koopman embedding of the system is composed of the Koopman embedding of every objects. Similar to the decomposition in the state space, we assume the Koopman embedding can be divided into object-centric sub-embeddings, i.e. $\\pmb { g } ^ { t } \\in \\mathbb { R } ^ { N m }$ denoting the concatenation of $g _ { 1 } ^ { t } , \\cdots , g _ { N } ^ { t }$ , where we use $\\pmb { g } _ { i } ^ { t } = g _ { i } ( \\pmb { x } ^ { t } ) \\in \\mathbb { R } ^ { m }$ as the Koopman embedding for the $i ^ { \\because }$ ’th object. ",
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+    {
+        "type": "text",
+        "text": "The Koopman matrix has a block-wise structure. It is natural to think the Koopman matrix is composed of block matrices after assuming an object-centric Koopman embeddings. In equation 5, $K _ { i j } \\ \\in \\mathbb { R } ^ { m \\times m }$ and $L _ { i j } \\in \\mathbb { R } ^ { m \\times l }$ are blocks of the Koopman matrix and the control matrix, where $l$ is the dimension of the action for each object and $\\pmb { u } ^ { t } \\in \\mathbb { R } ^ { N l }$ is the concatenation of $\\boldsymbol { \\mathbf { \\mathit { u } } } _ { 1 } ^ { t } , \\cdots , \\boldsymbol { \\mathbf { \\mathit { u } } } _ { N } ^ { t }$ denoting the total control signal at time $t$ : ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27279256958bdd2e14f0803d000bf979c3482f16a1446c046a59417c0b7031b6.jpg",
+        "text": "$$\n\\left[ \\begin{array} { c } { { { \\bf g } _ { 1 } ^ { t + 1 } } } \\\\ { { \\vdots } } \\\\ { { \\vdots } } \\\\ { { { \\bf g } _ { N } ^ { t + 1 } } } \\end{array} \\right] = \\left[ \\begin{array} { c c c c } { { { \\cal K } _ { 1 1 } } } & { { \\cdots } } & { { { \\cal K } _ { 1 N } } } \\\\ { { \\vdots } } & { { \\ddots } } & { { \\vdots } } \\\\ { { { \\cal K } _ { N 1 } } } & { { \\cdots } } & { { { \\cal K } _ { N N } } } \\end{array} \\right] \\left[ \\begin{array} { c } { { { \\bf g } _ { 1 } ^ { t } } } \\\\ { { \\vdots } } \\\\ { { { \\bf g } _ { N } ^ { t } } } \\end{array} \\right] + \\left[ \\begin{array} { c c c c } { { { \\cal L } _ { 1 1 } } } & { { \\cdots } } & { { { \\cal L } _ { 1 N } } } \\\\ { { \\vdots } } & { { \\ddots } } & { { \\vdots } } \\\\ { { { \\cal L } _ { N 1 } } } & { { \\cdots } } & { { { \\cal L } _ { N N } } } \\end{array} \\right] \\left[ \\begin{array} { c } { { { \\bf u } _ { 1 } ^ { t } } } \\\\ { { \\vdots } } \\\\ { { \\vdots } } \\\\ { { { \\bf u } _ { N } ^ { t } } } \\end{array} \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            869,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As we have seen in the case of linear spring system, those matrix blocks are not independent, but some of them share the same set of values. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• The same physical interactions shall share the same transition block. The equivalence between the blocks should reflect the equivalence of the interactions, where we use the same transition sub-matrix for object pairs of the same relation. For example, if the system is composed of $N$ identical objects interacting with the same relation, then, by symmetry, all the diagonal blocks should be the same, while all the off-diagonal blocks should also be the same. The repetitive structure allows us to efficiently identify the values using least squares regression. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 LEARNING THE KOOPMAN EMBEDDINGS USING GRAPH NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For a graph that contains , where vertic $N$ nt the syesent obj m at time s and ed $t$ us $G ^ { t } = ( O ^ { t } , R )$ $O ^ { t } = \\{ o _ { i } ^ { t } \\} _ { i = 1 } ^ { N }$ $\\bar { R \\mathbf { \\Psi } } = \\{ r _ { k } \\} _ { k = 1 } ^ { N ^ { 2 } }$ represent pair-wise relations. Specifically, ${ \\pmb O } _ { i } ^ { t } = ( \\bar { \\pmb x } _ { i } ^ { t } , { \\pmb a } _ { i } ^ { o } )$ $\\boldsymbol { x } _ { i } ^ { t }$ is the state of object $i$ and $\\pmb { a } _ { i } ^ { o }$ is a one-hot vector indicating the object type, e.g., fixed or movable. For relation, we have $\\pmb { r } _ { k } = ( u _ { k } , v _ { k } , \\pmb { a } _ { k } ^ { r } ) , 1 \\leq u _ { k } , v _ { k } \\leq N$ , where $u _ { k }$ and $v _ { k }$ are integers denoting the end points of this directed edge, and $\\pmb { a } _ { k } ^ { r }$ is a one-hot vector denoting the type of the relation $k$ . ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We use a graph neural network similar to Interaction Networks (IN) (Battaglia et al., 2016) to generate object-centric Koopman embeddings. IN defines an object function $f _ { O }$ and a relation function $f _ { R }$ to model objects and their relations in a compositional way. Similar to a message passing procedure, we calculate the edge effect $\\pmb { e } _ { k } ^ { t } = f _ { R } ( \\bar { \\pmb { o } } _ { u _ { k } } ^ { t } , \\pmb { o } _ { v _ { k } } ^ { t } , \\pmb { a } _ { k } ^ { r } ) _ { k = 1 \\dots N ^ { 2 } }$ , and node effect $\\begin{array} { r } { \\pmb { g } _ { i } ^ { t } = f _ { O } ( \\pmb { o } _ { i } ^ { t } , \\sum _ { k \\in \\mathcal { N } _ { i } } \\pmb { e } _ { k } ^ { t } ) _ { i = 1 \\dots N } } \\end{array}$ , where ${ \\mathcal { N } } _ { i }$ denotes the relations that point to the object $i$ and $\\{ g _ { i } ^ { t } \\}$ are the derived Koopman embeddings. We use this graph neural network, denoted as $\\phi$ , to represent our Koopman observation function. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "System identification. For a sequence of observations $\\widetilde { \\pmb { x } } = \\{ \\pmb { x } ^ { 1 } , \\cdots , \\pmb { x } ^ { T } \\}$ from time 1 to time $T$ , we first map them to the Koopman space as $\\widetilde { \\pmb { g } } = [ \\pmb { g } ^ { 1 } , \\dotsb { \\mathrm { ~ , ~ } } \\pmb { g } ^ { T } ]$ using the graph encoder $\\phi$ , where $g ^ { t } = \\phi ( \\pmb { x } ^ { t } )$ . We use $\\pmb { g } ^ { i : j }$ eto denote the sub-sequence $[ \\pmb { g } ^ { i } , \\cdots , \\pmb { g } ^ { j } ]$ . To identify the Koopman matrix, we solve the linear regression $\\operatorname* { m i n } _ { K } { \\| K g ^ { 1 : T - 1 } - g ^ { \\overleftarrow { 2 } : T } \\| _ { 2 } }$ . As a result, $K { \\dot { = } } g ^ { 2 : T } ( { \\dot { g } } ^ { 1 : T - 1 } ) ^ { \\dagger }$ will asymptotically approach the Koopman operator $\\kappa$ with an increasing $T$ . For cases where there are control inputs $\\widetilde { \\pmb { u } } = [ \\pmb { u } ^ { 1 } , \\dotsb , \\pmb { u } ^ { T - 1 } ]$ , the calculation of the Koopman matrix and the control matrix is eessentially solving a least squares problem w.r.t. the objective ",
+        "bbox": [
+            173,
+            458,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43fd9b32206052f36735904835c30a5f13d0f36885e6576abfc24ccadf1193c3.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { K , L } | | K g ^ { 1 : T - 1 } + L \\widetilde { \\pmb { u } } - \\pmb { g } ^ { 2 : T } | | _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            564,
+            607,
+            590
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As we mentioned in the Section 3.2, the dimension of the Koopman space is linear to the number of objects in the system, i.e., $\\widetilde { \\pmb g } \\in \\mathbb R ^ { N m \\times T }$ and $K \\in \\mathbb { R } ^ { N m \\times N m }$ . If we do not enforce any structure on the Koopman matrix $K$ e, we will have to identify $N ^ { 2 } m ^ { 2 }$ parameters. Instead, we can significantly reduce the number by leveraging the assumption on the structure of $K$ . Assume we know some blocks $( \\{ K _ { i j } \\} )$ of the matrix $K$ are shared and in total there are $h$ different kinds of blocks, which we denote as $\\hat { K } \\in \\mathbb { R } ^ { h \\times m \\times m }$ . Then, the number of parameter to be identified reduce to $h m ^ { 2 }$ . Usually, $h$ does not depend on $N$ , and is much smaller than $N ^ { 2 }$ . Now, for each block $K _ { i j }$ , we have a one-hot vector $\\sigma _ { i j } \\in \\{ 0 , 1 \\} ^ { h }$ indicating its type, i.e., $K _ { i j } = \\sigma _ { i j } \\hat { K } \\in \\mathbb { R } ^ { m \\times m }$ . Finally, as shown in equation 7, we represent the Koopman matrix as the product of the index tensor $\\sigma$ and the parameter tensor $\\hat { K }$ : ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4df5d2c2ed611751b6b43f92b79d081d0688e4b5fb33bc2c35ce086a8f9f4075.jpg",
+        "text": "$$\nK = \\sigma \\otimes { \\hat { K } } = \\left[ \\begin{array} { c c c } { \\sigma _ { 1 1 } { \\hat { K } } } & { \\cdots } & { \\sigma _ { 1 N } { \\hat { K } } } \\\\ { \\vdots } & { \\ddots } & { \\vdots } \\\\ { \\sigma _ { N 1 } { \\hat { K } } } & { \\cdots } & { \\sigma _ { N N } { \\hat { K } } } \\end{array} \\right] , \\mathrm { w h e r e } \\sigma = \\left[ \\begin{array} { c c c } { \\sigma _ { 1 1 } } & { \\cdots } & { \\sigma _ { 1 N } } \\\\ { \\vdots } & { \\ddots } & { \\vdots } \\\\ { \\sigma _ { N 1 } } & { \\cdots } & { \\sigma _ { N N } } \\end{array} \\right] \\in \\mathbb { R } ^ { N \\times N \\times h } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            738,
+            784,
+            799
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Similar to the Koopman matrix, we assume the same block structure in the control matrix $L$ and denote its parameter as $\\hat { L } \\in \\mathbb { R } ^ { h \\times m \\times l }$ . The least squares problem of identifying $\\hat { K }$ and $\\hat { L }$ becomes ",
+        "bbox": [
+            173,
+            805,
+            825,
+            837
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f2db5b0b11e0fdc28528d0a52362578d7af5ebc60560d5fe605f404ef68c9f9.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\hat { K } , \\hat { L } } \\| ( \\sigma \\otimes \\hat { K } ) \\pmb { g } ^ { 1 : T - 1 } + ( \\sigma \\otimes \\hat { L } ) \\widetilde { \\pmb { u } } - \\pmb { g } ^ { 2 : T } \\| _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            842,
+            648,
+            872
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma \\otimes \\hat { K } \\in \\mathbb { R } ^ { N m \\times N m }$ , $\\sigma \\otimes \\hat { L } \\in \\mathbb { R } ^ { N m \\times N l }$ , $\\pmb { g } ^ { 1 : T - 1 } \\in \\mathbb { R } ^ { N m \\times ( T - 1 ) }$ and $\\widetilde { \\pmb { u } } \\in \\mathbb { R } ^ { N l \\times ( T - 1 ) }$ . Since ethe linear least squares problems described in equation 6 and equation 8 have analytical solutions, performing system identification using our method is very efficient. ",
+        "bbox": [
+            174,
+            880,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Training GNN models. To make predictions on the states, we use a graph decoder $\\psi$ to map the Koopman embeddings back to the original state space. In total, we have three losses to train the graph encoder and decoder. The first term is the auto-encoding loss ",
+        "bbox": [
+            173,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee22ff9c1861be5fc72b30121791deb93da221d00ae9e1e7acee15e987772ce4.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { a e } } = \\frac { 1 } { T } \\sum _ { i } ^ { T } \\| \\psi ( \\phi ( \\pmb { x } ^ { i } ) ) - \\pmb { x } ^ { i } \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            154,
+            604,
+            198
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The second term is the prediction loss. To calculate it, we rollout in the Koopman space and denote the embeddings as $\\hat { \\pmb g } ^ { 1 } \\overset { ^ { } } { = } \\pmb g ^ { 1 }$ , and $\\hat { \\pmb { g } } ^ { t + 1 } = K \\hat { \\pmb { g } } ^ { t } + L \\pmb { u } ^ { t }$ , for $t = 1 , \\cdots , T - 1$ . The prediction loss is defined as the difference between the decoded states and the actual states, i.e., ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/914ee7763ca915bf878eb4864780cf0449126e8ccc6313f4f0e1a4ce74a4bbc2.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { p r e d } } = \\frac { 1 } { T } \\sum _ { i = 1 } ^ { T } \\Vert \\psi ( \\hat { \\pmb g } ^ { i } ) - \\pmb x ^ { i } \\Vert .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            256,
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+            299
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Third, we employ a metric loss to encourage the Koopman embeddings preserving the distance in the original state space. The loss is defined as the absolute error between the distances measured in the Koopman space and that in the original space, i.e., ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee726c65e21dae565f014959c4c5051daa1f537b0a49ee58644a4188d9cee8b0.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { m e t r i c } } = \\sum _ { i j } \\left| \\left\\| { \\pmb { g } } ^ { i } - { \\pmb { g } } ^ { j } \\right\\| - \\left\\| { \\pmb { x } } ^ { i } - { \\pmb { x } } ^ { j } \\right\\| \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Having Koopman embeddings that perserves the distance in the state space is important as we are using the distance in the Koopman space to define the cost function for downstream control tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The final training loss is simply the combination of all the terms above: $\\mathcal { L } = \\mathcal { L } _ { \\mathrm { a e } } + \\lambda _ { 1 } \\mathcal { L } _ { \\mathrm { p r e d } } + \\lambda _ { 2 } \\mathcal { L } _ { \\mathrm { m e t r i c } }$ . We then minimize the loss $\\mathcal { L }$ by optimizing the parameters in the graph encoder $\\phi$ and graph decoder $\\psi$ using stochastic gradient descent. Once the model is trained, it can be used for system identification, future prediction, and control synthesis. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 CONTROL ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For a control task, the goal is to synthesize a sequence of control inputs $\\mathbf { \\pmb { u } } ^ { 1 : T }$ that minimize $C =$ $\\textstyle \\sum _ { t = 1 } ^ { T } c _ { t } ( \\pmb { x } ^ { t } , \\pmb { u } ^ { t } )$ , the total incurred cost, where $c _ { t } ( \\pmb { x } ^ { t } , \\pmb { u } ^ { t } )$ is the instantaneous cost. For example, considering the control task of reaching a desired state $\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } } ^ { * }$ at time $T$ , we can design the following instantaneous cost, $c _ { t } ( \\pmb { x } ^ { t } , \\pmb { u } ^ { t } ) = \\mathbb { 1 } _ { [ t = T ] } \\bar { \\lVert \\pmb { x } ^ { t } - \\pmb { x } ^ { * } \\rVert _ { 2 } ^ { 2 } + \\lambda \\lVert \\pmb { u } ^ { t } \\rVert _ { 2 } ^ { 2 } }$ . The first term promotes the control sequence that matches the state to the goal, while the second term regularizes the control signals. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Open-loop control via quadratic programming (QP). Our model maps the original nonlinear dynamics to a linear dynamical system. We can then solve the control task by solving a linear control problem. With the assumption that the Koopman embeddings preserve the distance measure, we define the control cost as $c _ { t } ( { \\pmb g } ^ { t } , { \\pmb u } ^ { t } ) = \\mathbb { 1 } _ { [ t = T ] } \\| { \\pmb g } ^ { t } - { \\pmb g } ^ { * } \\| _ { 2 } ^ { 2 } + \\lambda \\| { \\pmb u } ^ { t } \\| _ { 2 } ^ { 2 }$ . As a result, we reduce the problem to minimizing a quadratic cost function $\\begin{array} { r } { C = \\sum _ { t = 1 } ^ { T } c _ { t } ( g ^ { t } , \\pmb { u } ^ { t } ) } \\end{array}$ over variables $\\{ g ^ { t } , \\pmb { u } ^ { t } \\} _ { t = 1 } ^ { T }$ , where $\\pmb { g } ^ { 1 } = \\phi ( \\pmb { x } ^ { 1 } )$ and ${ \\pmb g } ^ { * } = \\phi ( { \\pmb x } ^ { * } )$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Model predictive control (MPC). Solving the QP gives us control signals, which might not be good enough for long-term control as the prediction error accumulates. We can combine it with Model Predictive Control, assuming feedback from the environment every $\\tau$ steps. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Environments. We evaluate our method by assessing how well it can simulate and control ropes and soft robots. Specifically, we consider three environments. (1) Rope (Figure 2a): the top mass of a rope is fixed to a specific height. We apply force to the top mass to move it in a horizontal line. The rest of the masses are free to move according to internal force and gravity. (2) Soft (Figure 2b): we aim to control a soft robot that is consist of soft blocks. Blocks in dark grey are rigid and those in light blue are soft blocks. Each one of the dark blue blocks is soft but have an actuator inside that can contract or expand the block. One of the blocks is pinned to the ground, as shown using the red dots. (3) Swim (Figure 2c): instead of pinning the soft robot to the ground, we let the robot swim in fluids. The colors shown in this environment have the same meaning as in Soft. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/71263117fabd577db5465647abbac468c4da70a4fa5de4c94cdec174d7baecd9.jpg",
+        "image_caption": [
+            "Figure 2: Qualitative results. Top: our model prediction matches the ground truth over a long period. Bottom: for control, we use red dots or frames to indicate the goal. We apply the control signals generated from our identified model to the original simulator, which allows the agent to achieve the goal accurately. Please refer to our supplementary video for more results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Observation space. In the Rope environment, each mass on the rope is considered as an object. The observation of each mass is its position and velocity in the 2D plane, which has a dimension of 4. In total, a rope with $N$ masses has an observation space of dimension $4 N$ . In both the Soft and the Swim environments, each quadrilateral is considered as an object. For each quadrilateral, we have access to the positions and velocities of the four corners. Thus for a soft robot containing $N$ quadrilaterals, we have a $4 \\times 4 \\times N = 1 6 N$ dimensional observation. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines. We compare our model to the following baselines: Interaction Networks (Battaglia et al., 2016) (IN), Propagation Networks (Li et al., 2019b) (PN) and Koopman method with handcrafted Koopman base functions (KPM). IN and PN are the state-of-the-art learning-based physical simulators, and we evaluate their adaptation ability by finetuning their parameters on a small sequence of observations from the testing environment. Similar to our method, KPM fits a linear dynamics in the Koopman space. Instead of learning Koopman observations from data, KPM uses polynomials of the original states as the basis functions. In our setting, we set the maximum order of the polynomials to be three to make the dimension of the hand-crafted Koopman embeddings match our model’s. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Data generation. We generate 10,000 episodes for Rope and 50,000 episodes for Soft and Swim. Among them, $9 0 \\%$ are used for training, and the rest for testing. Each episode has 100 time steps. In the dataset, the physical systems have a various number of objects from 5 to 9, i.e. the ropes have 5 to 9 masses while the soft robots in Soft and Swim environments have 5 to 9 quadrilaterals. To evaluate the model’s extrapolating generalization ability, for each environment, we generate an extra dataset with the same size as the test set while containing systems consist of 10 to 14 objects. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/f0fdc1a405d982574e72ebf82281193b8558e35f505a30bae1a8d27cd079858f.jpg",
+        "image_caption": [
+            "Figure 3: Quantitative results on simulation. The $x$ axis shows time steps. The solid lines indicate medians and the transparent regions are the interquartile ranges of simulation errors. Our method significantly outperforms the baselines in all testing environments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/64d97fdca400c7934ba7f32ddaca8ae4060677260c2492205a69120a934edfd3.jpg",
+        "image_caption": [
+            "Figure 4: Quantitative results on control and ablation studies on model hyperparameters. Left: box-plots show the distributions of control errors. The yellow line in the box indicates the median. Our model consistently achieves smaller errors in all environments against KPM. Right: our model’s simulation errors with different amount of data for system identification (d) and different dimensions of the Koopman space (e). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Training and evaluation protocols. All models are trained using Adam optimizer (Kingma & Ba, 2015) with a learning rate of $1 0 ^ { - 4 }$ and a batch size of 8. $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ are 1.0 and 0.3, respectively, for our model. For both our model and the baselines, we apply 400K iterations of gradient steps in the Rope environment and 580K iterations in the Soft and Swim environment. Our model is trained on the sub-sequence of length 64 from the training set, and IN/PN aims at minimizing the L1 distance between their prediction and the ground truth. During test time, the models have to adapt to a new environment of unknown physical parameters, where they have access to a short sequence of observations and the opportunity to adjust their models’ parameters. Our model uses 8 episodes to identify the transition matrix via least-square regression. IN/PN update the model’s parameters by minimizing the distance between the model’s prediction and the actual observation using a gradient step of length $1 0 ^ { - 4 }$ for 5 iterations. For evaluation, we use two metrics: simulation error and control error. For a given episode, the simulation error at time step $t$ is defined as the mean squared error between the model prediction $\\hat { \\mathbf { x } } ^ { t }$ and the ground truth $\\mathbf { \\boldsymbol { x } } ^ { t }$ . For control, we pick the initial frame $\\mathbf { \\boldsymbol { x } } ^ { 0 }$ and the $t ^ { \\star }$ th frame $\\mathbf { \\boldsymbol { x } } ^ { t }$ from a episode. Then we ask the model to generate a control sequence of length $t$ to transfer the system from the initial state $\\mathbf { \\boldsymbol { x } } ^ { 0 }$ to the target state $\\mathbf { \\boldsymbol { x } } ^ { t }$ . The control error is defined as the mean squared distance between the target state and the state of the system at time $t$ . ",
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+    {
+        "type": "text",
+        "text": "4.1 SIMULATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Figure 2 shows qualitative results on simulation. Our model accurately predicts system dynamics for more than 100 steps. For Rope, the small prediction error comes from the slight delay of the force propagation inside the rope; hence, the tail of the rope usually has a larger error. For Soft, our model captures the interaction between the body parts and generates accurate prediction over the global movements of the robot. The error mainly comes from the misalignment of some local components. ",
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+    {
+        "type": "text",
+        "text": "We evaluate the models by predicting 100 steps into the future on 500 trajectories and Figure 3 shows quantitative results. IN and PN do not work well in the Rope and Swim environments due to insufficient system identification ability. The KPM baseline performs poorly in the Rope and ",
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+    {
+        "type": "text",
+        "text": "Soft environments indicating the limited power of polynomial Koopman base functions. Our model significantly outperforms all the baselines. ",
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+    {
+        "type": "text",
+        "text": "4.2 CONTROL ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We compare our model with KPM, the Koopman baseline using polynomial basis. In Rope, we ask the models to perform open-loop control where it only solves the QP once at the beginning. The length of the control sequence is 40. When it comes to Soft/Swim, each model is asked to generate control signals of 64 steps, and we allow the model to receive feedback after 32 steps. Thus every model has a second chance to correct its control sequence by solving the QP again at the time step 32. ",
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+    {
+        "type": "text",
+        "text": "As shown in Figure 2, our model leverages the inertia of the rope and matches the target state accurately. As for controlling a soft body swinging on the ground or swimming in the water, our model can move each part (the boxes) of the body to the exact target position. The small control error comes from the slight misalignment of the orientation and the size of the body parts. Figure 4 shows that quantitatively our model outperforms KPM, too. ",
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+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Structure of the Koopman matrix. We explore three different structures of the Koopman matrix, Block, Diag and None, to understand its effect on the learned dynamics. None assumes no structure in the Koopman matrix. Diag assumes a diagonal block structure of $K$ : all off-diagonal blocks $( K _ { i j }$ where $i \\neq j$ ) are zeros and all diagonal blocks share the same values. Block predefines a block-wise structure, decided by the relation between the objects as introduced in Section 3.3. ",
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+    },
+    {
+        "type": "text",
+        "text": "Table 1 includes our model’s simulation error and control error with different Koopman matrix structures in Rope. All models are trained in the Rope environment with 5 to 9 masses. Besides the result on the test set, we also report models’ extrapolation performance in parentheses, where the model is evaluated on systems with more masses than training, i.e., 10 to 14 masses. ",
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+    },
+    {
+        "type": "text",
+        "text": "Table 1: Ablation study results on the Koopman matrix structure (Rope environment). For simulation, we show the Mean Squared Error between the prediction and the ground truth at $T = 1 0 0$ , whereas for control, we show the performance with a horizon of length 40. The numbers in parentheses show the performance on extrapolation. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our model with Block structure consistently achieves a smaller error in all settings. Diag assumes an overly simplified structure, leading to larger errors and failing to make reasonable controls. None has comparable simulation errors but larger control errors. Without the structure in the Koopman matrix, it overfits the data and makes the resulting linear dynamics less amiable to the control. ",
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+    {
+        "type": "table",
+        "img_path": "images/f9b768d6a872aa4b8c11ac0e774b2929d75d198248c6819b004ae778153ee3f2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Simulation</td><td>Control</td></tr><tr><td>Diag</td><td>0.133 (0.174)</td><td>2.337 (2.809)</td></tr><tr><td>None</td><td>0.117 (0.083)</td><td>1.522 (1.288)</td></tr><tr><td>Block</td><td>0.105 (0.075)</td><td>0.854 (1.101)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters. In our main experiments, we set the dimension of the Koopman embedding to $m = 3 2$ per object. Online system identification requires 800 data samples for each training/test case. To understand our model’s performance under different hyperparameters, we vary the dimension of the Koopman embedding from 8 to 64 and the number of data samples used for system identification from 200 to 1,600. Figure 4d shows that more data for system identification leads to better simulation results. Figure 4e shows that dimension 16 gives the best results on simulation. It may suggest that the intrinsic dimension of the Koopman invariant space of the Rope system is around 16 per object. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Compositionality is common in our daily life. Many ordinary objects contain repetitive subcomponents: ropes and soft robots, as shown in this paper, granular materials such as coffee beans and lego blocks, and deformable objects such as cloth and modeling clay. These objects are known to be very challenging for manipulation using traditional methods, while our formulation opens up a new direction by combining deep Koopman operators with graph neural networks. By leveraging the compositional structure in the Koopman operator via graph neural nets, our model can efficiently manipulate deformable objects such as ropes and soft robots, and generalize to systems with variable numbers of components. We hope this work could encourage more endeavors in modeling larger and more complex systems by integrating the power of the Koopman theory and the expressiveness of neural networks. ",
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+    {
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+            196
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A ENVIRONMENT AND MODEL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            526,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Interaction types. In our experiments, interactions are considered different if the types are different or the objects involved have different physical properties. ",
+        "bbox": [
+            173,
+            135,
+            823,
+            164
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the Rope environment, the top mass has a fixed height and is considered differently from the other masses. Thus, we have 2 types of self-interactions for the top mass and the non-top masses. In addition, we have 8 types of interactions between different objects. The objects on a relation could be either top mass or non-top mass. It is a combination of 4. And the interaction may happen between two nearby masses or masses that are two-hop away. In total, the number of interactions between different objects is $4 \\times 2 = 8$ . ",
+        "bbox": [
+            174,
+            170,
+            825,
+            253
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the Soft environments, there are four types of quadrilaterals: rigid, soft, actuated, and fixed. We have four types of self-interactions correspondingly. For the interactions between objects, we add edges between two quadrilaterals only if they are connected by a point or edge. Connection from different directions are considered as different relations. There are 8 different directions, up, down, left, right, up-left, down-left, up-right, down-right. The relation types also encode the type of receiver object. Thus, in total, there are $( 8 + 1 ) \\times 4 = 3 6$ types of relations between different objects. ",
+        "bbox": [
+            174,
+            261,
+            825,
+            344
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the Swim environment, there are three types of quadrilaterals: rigid, soft, and actuated. Similar to the Soft environment, we use different edge types for different connecting directions; hence, the number of edge type is $( 8 + 1 ) \\times 3 = 2 7$ . ",
+        "bbox": [
+            176,
+            351,
+            823,
+            393
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            414,
+            442,
+            430
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Comparison with a classical physical simulator optimized using back-box optimization. We have performed comparisons with a classical physical simulator optimized using black-box optimization (Delingette, 1998) by assuming different levels of knowledge over the ground truth model. ",
+        "bbox": [
+            174,
+            446,
+            825,
+            488
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "If we assume that we know the ground truth model, where we only need to identify relevant physical parameters during the system identification stage, Bayesian Optimization (Snoek et al., 2012) (BO) can give us a reasonable estimate of the physical parameters. However, BO requires much more time to achieve a comparable performance with our method in the Rope environment: 0.43 vs. 180 seconds averaged over 100 trails (Ours vs. BO). ",
+        "bbox": [
+            174,
+            496,
+            825,
+            565
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "If we are unsure about the ground truth model and we approximate the system using a set of points linked by springs and dampers, BO does not work as well. In our additional experiments, we approximate the Rope environment using a chained spring-mass system, say $n$ masses and $n - 1$ springs. While taking much more time, BO still cannot give us a satisfying result: simulation error 0.046 vs. 0.084 and control error 0.854 vs. 2.547 (Ours vs. BO). ",
+        "bbox": [
+            174,
+            573,
+            825,
+            642
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ablation study on the effectiveness of the metric loss. The internal linear structure allows us to solve the control problem using quadratic programming, where the objective function for control is defined in the embedding space (Section 3.4); hence, it is desirable to have Koopman embeddings that preserve the distance in the original state space. In Section 3.3, we introduce a metric loss to promote learning a Koopman embedding that keeps the distance measurement. ",
+        "bbox": [
+            174,
+            664,
+            825,
+            734
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the effect of the metric loss, we compare the models trained with and without the metric loss. We establish the comparison using two measurements, the distance preservation and the prediction accuracy. To evaluate how well the Koopman embeddings preserve the distance, we compute the distribution of the log-ratio of the distance in the Koopman space and in the original state space, i.e., $\\log \\left( { \\frac { \\| { \\pmb { g } } ^ { i } - { \\pmb { g } } ^ { j } \\| _ { 2 } } { \\| { \\pmb { x } } ^ { i } - { \\pmb { x } } ^ { j } \\| _ { 2 } } } \\right)$ . For the model prediction accuracy, we show the simulation errors. ",
+        "bbox": [
+            174,
+            741,
+            825,
+            820
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We perform the experiments in the Rope environment and show the result in Figure 5. On the left, we show the ratio of distance in the learned Koopman space and the distance in the original state space. The model trained with metric loss has a log distance ratio that significantly more concentrates on 0. It means the metric loss effectively regularizes the model to preserve the distance. On the right, we show the simulation errors of the two models, which indicate that two models have comparable prediction performance. Metric loss effectively enhances the property of distance-preserving while not making a big sacrifice on the accuracy of the dynamics modeling. ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f00d555af41a4f7c7e11cf5d9b806fe2c47cfd9aed456f875f1dcb1ec4ae1f89.jpg",
+        "image_caption": [
+            "Figure 5: Ablation study on the metric loss in the Rope environment. (a) shows the distributions of the logarithm distance ratio, i.e, $\\log \\left( { \\frac { \\| { \\pmb { g } } ^ { i } - { \\pmb { g } } ^ { j } \\| _ { 2 } } { \\| { \\pmb { x } } ^ { i } - { \\pmb { x } } ^ { j } \\| _ { 2 } } } \\right)$ The model trained with metric loss has a distance ratio much more concentrated to 1, which indicates it preserves the distance much better than the counterpart. (b) illustrates the simulation error of two models, where their performance is on par. (c) shows that the model trained using the metric loss performs better control. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            104,
+            818,
+            241
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/574fbfd0e376cdc34bb544d0ad7177a1053f7d961bda58f59dcd9e9e43a0b42a.jpg",
+        "image_caption": [
+            "Figure 6: Modeling rope with known physical parameters. We show the comparison between our model and IN/PN in scenarios where we have access to the ground truth physical parameters. In this case, IN and PN slightly outperform our method due to the internal linear structure in our model. However, in the real world, we do not always know the physical parameters and their values, which makes our method preferable when adapting to new environments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            362,
+            354,
+            632,
+            477
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Experiments in a known physical parameter setting. Our setting is different from the settings in the original IN and PN papers that we do not assume we know the physical parameters and their values, such as stiffness, mass, and gravity. Instead, the parameters are embedded in the transition matrices during the system identification stage (Section 3.3). If the model has access to the underlying physical parameters, as expected, IN and PN slightly outperform our method as the internal linear structure limits our model’s expressiveness, as shown in Figure 6. In the real world, however, the underlying physical parameters are not always known, which makes our model a better choice when adapting to unseen environments. ",
+        "bbox": [
+            173,
+            577,
+            825,
+            689
+        ],
+        "page_idx": 13
+    }
+]

parse/train/H1xQSjCqFQ/H1xQSjCqFQ.md

@@ -0,0 +1,264 @@
+# EXCITATION DROPOUT: ENCOURAGING PLASTICITYIN DEEP NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a guided dropout regularizer for deep networks based on the evidence of a network prediction: the firing of neurons in specific paths. In this work, we utilize the evidence at each neuron to determine the probability of dropout, rather than dropping out neurons uniformly at random as in standard dropout. In essence, we dropout with higher probability those neurons which contribute more to decision making at training time. This approach penalizes high saliency neurons that are most relevant for model prediction, i.e. those having stronger evidence. By dropping such high-saliency neurons, the network is forced to learn alternative paths in order to maintain loss minimization, resulting in a plasticity-like behavior, a characteristic of human brains too. We demonstrate better generalization ability, an increased utilization of network neurons, and a higher resilience to network compression using several metrics over four image/video recognition benchmarks.
+
+# 1 INTRODUCTION
+
+Dropout [Hinton et al. (2012); Srivastava et al. (2014)] is a classical regularization technique that is used in many state-of-the-art deep neural networks, typically applied to fully-connected layers. Standard Dropout selects a fraction of neurons to randomly drop out by zeroing their forward signal. In this work, we propose a scheme for biasing this selection. Our scheme utilizes the contribution of neurons to the prediction made by the network at a certain training iteration stage.
+
+Dropout can be interpreted as model averaging technique that avoids overfitting on training data, allowing for better generalization on unseen test data. A recent variant of dropout that targets improved generalization ability is Curriculum Dropout [Morerio et al. (2017)]. It targets adjusting the dropout rate by exponentially increasing the unit suppression rate during training, answering the question How many neurons to drop out over time? Like Standard Dropout [Hinton et al. (2012); Srivastava et al. (2014)], Curriculum Dropout selects the neurons to be dropped randomly. In this work, however, we target at determining how the dropped neurons are selected, answering the question Which neurons to drop out?
+
+Our approach is inspired by brain plasticity [Hebb (2005); Song et al. (2000); Mittal et al. (2018); Miconi et al. (2018)]. We deliberately, and temporarily, paralyze/injure neurons to enforce learning alternative paths in a deep network. At training time, neurons that are more relevant to the current prediction are given a higher dropout probability. The relevance of a neuron for making a certain prediction is quantified using Excitation Backprop, a top-down saliency approach proposed by Zhang et al. (2016). Excitation Backprop conveniently yields a probability distribution at each layer that reflects neuron saliency, or neuron contribution to the prediction being made. This is utilized in the pipeline of our approach, named Excitation Dropout, which is summarized in Fig. 1.
+
+In particular, we study how this approach improves generalization through utilizing more network’s neurons for image classification. We report an increased recognition rate for both CNN models that are fine-tuned and trained from scratch. This improvement is validated on four image/video recognition datasets, and ranges from $1 . 1 \% - 6 . 3 \%$ over state-of-the-art Curriculum Dropout.
+
+Next, we examine the effect of our approach on network utilization. Mittal et al. (2018) and Ma et al. (2017) introduce metrics that measure network utilization. We show a consistent increased network utilization using Excitation Dropout on four image/video recognition datasets. For example, averaged over all four benchmarks, we get $7 6 . 5 5 \%$ reduction in conservative filters, filters whose parameters do not change significantly during training, as compared to Standard Dropout.
+
+![](images/a6cd4c0b90572493c61bafb0d0a027deae64fbc8a02b1ea8e9ccd622c7fe66b8.jpg)  
+Figure 1: Training pipeline of Excitation Dropout. Step 1: A minibatch goes through the standard forward pass. Step 2: Backward EB is performed until the specified dropout layer; this gives a neuron saliency map at the dropout layer in the form of a probability distribution. Step 3: The probability distribution is used to generate a binary mask for each image of the batch based on a Bernoulli distribution determining whether each neuron will be dropped out or not. Step 4: A forward pass is performed from the specified dropout layer to the end of the network, zeroing the activations of the dropped out neurons. Step 5: The standard backward pass is performed to update model weights.
+
+Finally, we study network resilience to neuron dropping at test time. We observe that training with Excitation Dropout leads to models that are a lot more robust when layers are shrunk/compressed by removing units. We demonstrate this when dropping the most relevant neurons, the least relevant neurons, and with a random dropping selection. This can be quite desirable for compressing/distilling [Hinton et al. (2015)] a model, e.g. for deployment on mobile devices.
+
+In summary, by encouraging plasticity-like behavior, our contributions are threefold:
+
+1. Better generalization on test data.   
+2. Higher utilization of network neurons.   
+3. Resilience to network compression.
+
+# 2 RELATED WORK
+
+Dropout was first introduced by Hinton et al. (2012) and Srivastava et al. (2014) as a way to prevent neural units from co-adapting too much on the training data by randomly omitting subsets of neurons at each iteration of the training phase.
+
+Some follow-up works have explored different schemes for determining how much dropout is applied to neurons/weights. Wager et al. (2013) described the dropout mechanism in terms of an adaptive regularization, establishing a connection to the AdaGrad algorithm. Inspired by information theoretic principles, Achille & Soatto (2018) propose Information Dropout, a generalization dropout which can be automatically adapted to the data. Kingma et al. (2015) showed that a relationship between dropout and Bayesian inference can be extended when the dropout rates are directly learned from the data. Kang et al. (2017) introduces Shakeout which instead of randomly discarding units as dropout does, it randomly enhances or reverses each units contribution to the next layer. Wan et al. (2013) introduced the DropConnect framework, adding dynamic sparsity on the weights of a deep model. DropConnect generalized Standard Dropout by randomly dropping the weights rather than the neuron activations in the network. Rennie et al. (2014) proposed a time scheduling for the retaining probability for the neurons in the network. The presented adaptive regularization scheme smoothly decreased in time the number of neurons turned off during training.
+
+Recently, Morerio et al. (2017) proposed Curriculum Dropout to adjust the dropout rate in the opposite direction, exponentially increasing unit suppression rate during training, leading to a better generalization on unseen data.
+
+Other works focus on which neurons to drop out. Dropout is usually applied to fully-connected layers of a deep network. Conversely, Wu & Gu (2015) studied the effect of dropout in convolutional and pooling layers. The selection of neurons to drop depends on the layer where they reside. In contrast, we select neurons within a layer based on their contribution. Wang & Manning (2013) demonstrate that sampling neurons from a Gaussian approximation gave an order of magnitude speedup and more stability during training. Li et al. (2016) proposed to use multinomial sampling for dropout, i.e. keeping neurons according to a multinomial distribution with specific probabilities for different neurons. Ba & Frey (2013) jointly trained a binary belief network with a neural network to regularize its hidden units by selectively setting activations to zero accordingly to their magnitude. While this takes into consideration the magnitude of the forward activations, it does not take into consideration the relationship of these activations to the ground-truth. In contrast, we drop neurons based on how they contribute to a network’s decision.
+
+We compare our results against Morerio et al. (2017). To the best of our knowledge, we are the first to probabilistically select neurons to dropout based on their task-relevance.
+
+# 3 METHOD
+
+# 3.1 BACKGROUND
+
+Saliency maps that quantize the importance of class-specific neurons for an input image are instrumental to our proposed scheme. Popular approaches include Class Activation Maps (CAM) [Zhou et al. (2016)], Gradient-weighted Class Activation Mapping (Grad-CAM) [Selvaraju et al. (2017)], and Excitation Backprop (EB) [Zhang et al. (2017)]. A thorough analysis of all saliency methods is out of the scope of this work, and the saliency problem in general is far from solved. We choose to use EB since it produces a valid probability distribution for each network layer. The saliency maps obtained using this approach are evaluated for spatial localization of objects and demonstrate the ability of pointing to the right region of an image [Zhang et al. (2017)].
+
+In a standard CNN, the forward activation of neuron $a _ { j }$ is computed by $\begin{array} { r } { \widehat { a } _ { j } = \phi ( \sum _ { i } w _ { i j } \widehat { a } _ { i } + b _ { i } ) } \end{array}$ , where $\widehat { a } _ { i }$ is the activation coming from the previous layer, $\phi$ b bis a nonlinear activation function, $w _ { i j }$ and $b _ { i }$ bare the weight from neuron $i$ to neuron $j$ and the added bias at layer $i$ , respectively. EB devises a backpropagation formulation able to reconstruct the evidence used by a deep model to make decisions. It computes the probability of each neuron recursively using conditional probabilities $P ( a _ { i } | a _ { j } )$ in a top-down order starting from a probability distribution over the output units, as follows:
+
+$$
+P ( a _ { i } ) = \sum _ { a _ { j } \in \mathcal { P } _ { i } } P ( a _ { i } | a _ { j } ) P ( a _ { j } )
+$$
+
+where $\mathcal { P } _ { i }$ is the parent node set of $a _ { i }$ . EB passes top-down signals through excitatory connections having non-negative activations, excluding from the competition inhibitory ones. EB is designed with an assumption of non-negative activations. Most modern CNNs use ReLU activation functions, which satisfy this assumption. Therefore, negative weights can be assumed to not positively contribute to the final prediction. Assuming $C _ { j }$ the child node set of $a _ { j }$ , for each $a _ { i } \in C _ { j }$ , the conditional winning probability $P ( a _ { i } | a _ { j } )$ is defined as
+
+$$
+P ( a _ { i } | a _ { j } ) = \left\{ \begin{array} { l l } { Z _ { j } \widehat { a } _ { i } w _ { i j } , } & { \mathrm { i f } \ w _ { i j } \geq 0 , } \\ { 0 , } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+where $Z _ { j }$ is a normalization factor such that $\begin{array} { r } { \sum _ { a _ { i } \in \mathcal { C } _ { j } } P ( a _ { i } | a _ { j } ) = 1 } \end{array}$ . Recursively propagating the top-down signal and preserving the sum of backpropagated probabilities, it is possible to highlight the salient neurons in each layer using Eqn. 1, i.e. neurons that mostly contribute to a specific task. We will refer to the distribution of $P ( a _ { i } )$ as $p _ { E B } ( a _ { i } )$ .
+
+![](images/13b48eaa2c04c820150b59d7e2cb1113153194ce36691f10b5f7572a5b0e1d83.jpg)  
+Figure 2: The retaining probability, $p$ , as a function of the Excitation Backprop probability $p _ { E B }$ . This plot was created using $N = 1 0$ and a base retaining probability $P = 0 . 5$ . In this case, when the saliency of neurons is uniform, i.e. $p _ { E B } = 0 . 1$ , then $p = P$ as marked in the figure.
+
+# 3.2 EXCITATION DROPOUT
+
+In the standard formulation of dropout [Hinton et al. (2012); Srivastava et al. (2014)], the suppression of a neuron in a given layer is modeled by a Bernoulli random variable $0 < p \leq 1$ where $p$ is defined as the probability of retaining a neuron. Given a specific layer where dropout is applied, during the training phase, each neuron is turned off with a probability $1 - p$ .
+
+We argue for a different approach that is guided in the way it selects neurons to be dropped. In a training iteration, certain paths have high excitation contributing to the resulting classification, while other regions of the network have low response. We encourage learning alternative paths (plasticity) through the temporary damaging of the currently highly excited path. We re-define the probability of retaining a neuron as a function of its contribution in the currently highly excited path
+
+$$
+p = 1 - \frac { ( 1 - P ) * ( N - 1 ) * p _ { E B } } { ( ( 1 - P ) * N - 1 ) * p _ { E B } + P }
+$$
+
+where $p _ { E B }$ is the probability backpropagated through the EB formulation (Eqn. 1) in layer $l$ , $P$ is the base probability of retaining a neuron when all neurons are equally contributing to the prediction and $N$ is the number of neurons in a fully-connected layer $l$ or the number of filters in a convolutional layer $l$ . The retaining probability defined in Eqn. 3 drops neurons which contribute the most to the recognition of a specific class, with higher probability. Dropping out highly relevant neurons, we retain less relevant ones and thus encourage them to awaken. We also study how this compares to dropping the least relevant neurons (Adaptive Dropout by Ba & Frey (2013)) in the Appendix.
+
+Fig. 2 shows $p$ as a function of $p _ { E B }$ . To gain some intuition for Eqn. 3, we can look more closely at the graph: 1) If neuron $a _ { i }$ has $p _ { E B } ( a _ { i } ) = 1$ : This results in a retaining probability of $p = 0$ . We do not want to keep a neuron which has a high contribution to the correct label. 2) If neuron $a _ { i }$ has $p _ { E B } ( a _ { i } ) = 0$ : This results in a retaining probability of $p = 1$ . We want to keep a neuron which has not contributed to the correct classification of an image. 3) If neuron $a _ { i }$ has $\dot { p } _ { E B } ( a _ { i } ) = 1 / N$ , i.e. $p _ { E B }$ is a uniform probability distribution: This results in a retaining probability $p = P$ . We want to keep a neuron with base probability $P$ since all neurons contribute equally.
+
+Eqn. 3 provides a dropout probability for each neuron, which is then used as the parameter of a Bernoulli distribution giving a binary dropout mask. During training, each image in a batch leads to different excitatory connections in the network and therefore has a different $p _ { E B }$ distribution, consequently leading to a different dropout mask. Fig. 1 presents the pipeline of Excitation Dropout at training time, and a run-time analysis is presented in the Appendix.
+
+# 4 EXPERIMENTS
+
+In this section, we present how Excitation Dropout improves the generalization ability on four image/video recognition datasets in fully connected layers of different architectures. We then present an analysis of how Excitation Dropout affects the utilization of network neurons on the same datasets. Finally, we examine the resilience of a model trained using Excitation Dropout to network compression.
+
+![](images/7f31a87e4102999f1f968caeeb79910d91cac21faf742b96537e81b511da59aa.jpg)  
+Figure 3: We compare the test accuracy of different dropout training strategies on four image/video recognition datasets: Cifar10, Cifar100, Caltech256, UCF101. Results presented here are averaged over five trained models and the standard deviation is depicted around the mean curve using a lighter shade. Excitation Dropout performs best after convergence compared to the other strategies.
+
+# 4.1 DATASETS AND ARCHITECTURES
+
+We present results on four image/video recognition datasets. Cifar10 and Cifar100 [Krizhevsky (2009)] are image recognition datasets, each consisting of $6 0 0 0 0 3 2 \times 3 2$ tiny RGB natural images. Cifar10 images are distributed over 10 classes with 6000 images per class, and Cifar100 images are distributed over 100 classes with 600 images per class. Training and test splits contain $5 0 K$ and $1 0 K$ images, respectively. We feed the network with the original image dimensions. Caltech256 [Griffin et al. (2007)] is an image recognition dataset consisting 31000 RGB images divided in 256 classes. We consider five different random splits of 50 train images and 20 testing images for each class. Images were reshaped to $1 2 8 \times 1 2 8$ pixel to feed the network. UCF101 [Soomro et al. (2012)] is a video action recognition dataset based on 13320 actions belonging to 101 action classes. For this dataset we consider a frame-based action recognition task. The images are resized to $2 2 4 \times 2 2 4$ and $2 2 7 \times 2 2 7$ to fit the input layers of the VGG and AlexNet architectures, respectively.
+
+We present results on four architectures. Relatively shallow architectures are trained from scratch, and deeper popular architectures are fine-tuned after being pre-trained on ImageNet [Deng et al. (2009)]. Models trained from scratch: We train the CNN-2 architecture used in Morerio et al. (2017), the state-of-the-art dropout variant, for comparison purposes. This architecture consists of three convolutional and two fully-connected layers (see Appendix). We train this network from scratch for $1 0 0 K$ iterations on the datasets: Cifar-10, Cifar-100 and Caltech-256. We use minibatches of 100 images and fix the learning rate to be $1 0 ^ { - 3 }$ , decreasing to $1 0 ^ { - 4 }$ after $2 5 K$ iterations.
+
+Fine-tuned models: We fine-tune the commonly used architectures: AlexNet [Krizhevsky et al. (2012)], VGG16 and VGG19 [Simonyan & Zisserman (2014)] pre-trained on ImageNet. We finetune the models for a frame by frame action recognition task on UCF101. The learning rate is fixed to $1 0 ^ { - 3 }$ for all the processes. We fine-tune AlexNet for $5 K$ while VGG16 and VGG19 for $3 0 K$ iterations. We use a batch size of 128 and 50 images for AlexNet and VGG16/19, respectively.
+
+<table><tr><td>Architecture</td><td>No Dropout (%)</td><td>Standard Dropout (%)</td><td>Curriculum Dropout (%)</td><td>Excitation Dropout (%)</td></tr><tr><td>VGG16</td><td>69.37</td><td>71.93 (+2.56%)</td><td>72.14 (+2.77%)</td><td>73.23 (+3.86%)</td></tr><tr><td>VGG19</td><td>71.32</td><td>72.52 (+1.29%)</td><td>73.18 (+1.86%)</td><td>74.34 (+3.02%)</td></tr><tr><td>AlexNet</td><td>62.89</td><td>64.50 (+1.61%)</td><td>64.55 (+1.66%)</td><td>67.56 (+4.67%)</td></tr></table>
+
+Table 1: Test accuracy comparison between No, Standard, Curriculum and Excitation Dropout in the $f c 6$ layer of three architectures: AlexNet, VGG16 and VGG19, fine-tuned for the action recognition task on UCF101. The numbers reported are the final test accuracies together with the improvements (in parenthesis) with respect to No Dropout, averaged over five trained models.
+
+Table 2: Different metrics to reflect the usage of network capacity in the first fully-connected layer of the CNN-2 architecture consisting of 2048 neurons and the VGG16 consisting of 4096 neurons. Results presented here are averaged over five trained models for each of the datasets: Cifar10, Cifar100, Caltech256 and UCF101 $\sigma$ in brackets). Excitation Dropout consistently produces more neurons with non-zero activations, has a more spread saliency map leading to a lower saliency peak, has a higher entropy of both activations and saliency, and has a lower number of conservative filters; all reflecting an improved utilization of the network neurons using Excitation Dropout.   
+
+<table><tr><td>Dataset</td><td>Metric</td><td>Standard Dropout</td><td>Curriculum Dropout</td><td>Excitation Dropout</td></tr><tr><td rowspan="4">Cripfi</td><td># Neurons ON Peak pEB</td><td>1194 (±153) 0.011 (±0.004)</td><td>1169 (±61) 0.009 (±0.001)</td><td>1325 (±61) 0.003 (±0.0002)</td></tr><tr><td>Entropy of Activations</td><td>3.55 (±0.72)</td><td>3.50 (±0.12)</td><td></td></tr><tr><td>Entropy of pEB</td><td></td><td></td><td>4.29 (±0.28)</td></tr><tr><td>Conservative Filters△=0.25</td><td>3.28 (±0.56) 1204 (±37)</td><td>3.32 (±0.13) 959 (±34)</td><td>4.26 (±0.26) 124(±22)</td></tr><tr><td rowspan="4">cormfi</td><td># Neurons ON Peak pEB</td><td>453 (±183) 0.011 (±0.0004)</td><td>460 (±75) 0.012 (±0.0004)</td><td>943 (±131) 0.005 (±0.0005)</td></tr><tr><td>Entropy of Activations</td><td>1.67 (±0.31)</td><td>1.70 (±0.29)</td><td>3.21 (±0.44)</td></tr><tr><td>Entropy of pEB</td><td>1.64 (±0.27)</td><td>1.67 (±0.26)</td><td>3.17 (±0.41)</td></tr><tr><td>Conservative Filters△=0.30</td><td>2048 (±51)</td><td>2038 (±44)</td><td>14 (±13)</td></tr><tr><td rowspan="5">Ceresst5</td><td># Neurons ON Peak pEB</td><td>412 (±126) 0.014 (±0.0007)</td><td>471 (±146)</td><td>702 (±171)</td></tr><tr><td></td><td></td><td>0.013 (±0.0006)</td><td>0.007 (±0.0003)</td></tr><tr><td>Entropy of Activations</td><td>1.63 (±0.32)</td><td>1.84 (±0.35)</td><td>2.63 (±0.23)</td></tr><tr><td>Entropy of pEB</td><td>1.58 (±0.29)</td><td>1.77 (±0.31)</td><td>2.59 (±0.22)</td></tr><tr><td>Conservative Filters△=1.25</td><td>2048 (±46)</td><td>2048 (±49)</td><td>1671 (±31)</td></tr><tr><td rowspan="5">IUIIIN</td><td># Neurons ON Peak pEB</td><td>1120 (±25)</td><td>1143 (±22)</td><td>1404 (±37)</td></tr><tr><td></td><td>0.007 (±0.0002)</td><td>0.007 (±0.0002)</td><td>0.004 (±0.0002)</td></tr><tr><td>Entropy of Activations</td><td>2.04 (±0.23)</td><td>2.08 (±0.21)</td><td>2.51 (±0.18)</td></tr><tr><td>Entropy of pEB</td><td>1.92 (±0.22)</td><td>1.95 (±0.20)</td><td>2.42 (±0.18)</td></tr><tr><td>Conservative Filters△=0.15</td><td>3599 (±66)</td><td>3859 (±53)</td><td>44(±36)</td></tr></table>
+
+# 4.2 SETUP AND RESULTS: GENERALIZATION
+
+In this section we compare the performance of Excitation Dropout to that of No Dropout, Standard Dropout, and Curriculum Dropout [Morerio et al. (2017)]. We train a CNN-2 model from scratch on the datasets: Cifar10, Cifar100, Caltech256. Fig. 3 depicts the test accuracies over training iterations for the three datasets averaged over five trained models. After convergence, Excitation Dropout demonstrates a significant improvement in performance compared to other methods. We hypothesize that Excitation Dropout takes longer to converge due to the additional loop (Steps 2-4 in Fig. 1) introduced in the learning process, and due to the learning of the alternative paths. We note that Excitation Dropout, during training, uses a different binary mask for each image in a minibatch, while in Standard Dropout, one random mask is employed per minibatch. To prove that it is precisely the fact that masks reflective of the particular input give rise to a boost in accuracy, and not the fact that different masks are used for different images, we add a comparison with Standard Dropout having a different random mask for each image. We refer to this accuracy as ‘Standard Dropout $^ +$ Mask/Img’ in the plots. As expected, the latter approach is comparable to Standard Dropout in performance.
+
+![](images/2fa319d5f33636ca112a15cb60fa1f1b00877fc4929fb69c77caab7893d92bb1.jpg)  
+Figure 4: Visualizations for a VGG16 network fine-tuned on UCF101. The middle columns display the saliency map over the same video frame of the action HorseRiding while incrementally switching off the most $k$ relevant/salient neurons $( k = 0 , 1 0 0 , 2 0 0 , \ldots , 5 0 0 )$ in the $f c 6$ layer at test time. Excitation Dropout shows more robustness when more neurons are switched off. This is demonstrated through its ability to recover more of the saliency map even when a high percentage of the most salient neurons is dropped-out. This ability reflects the alternative learnt paths. Histograms of the leftmost and rightmost saliency maps are presented to demonstrate that Excitation Dropout has a wider range of saliency values.
+
+Next, we evaluate the effectiveness of Excitation Dropout on popular network architectures that employ dropout layers: AlexNet, VGG16, VGG19. This is done by fine-tuning on the video recognition dataset UCF101. Fig. 3 shows superior Excitation Dropout performance on AlexNet fine-tuned on UCF101. Table 1 presents more comparative results on other deep architectures by reporting the accuracy after convergence. Again, Excitation Dropout demonstrates higher generalizability on the test data for all architectures.
+
+For fair comparison, we set $p = 0 . 5$ for Standard Dropout and $P = 0 . 5$ for the base retaining probability of Excitation Dropout in all experiments1. We perform dropout in the first fully-connected layer of the networks (fc1 for CNN-2 and $f c 6$ for AlexNet and VGGs) for Standard, Curriculum, and Excitation Dropout. For Curriculum Dropout we fix the parameter $\gamma$ to $5 * 1 0 ^ { - 4 }$ as in Morerio et al. (2017).
+
+# 4.3 SETUP AND RESULTS: UTILIZATION OF NETWORK NEURONS
+
+In this section we examine how Excitation Dropout expands the network’s utilization of neurons through the learnt alternative paths for a certain task.
+
+Mittal et al. (2018) introduced scoring functions to rank the filters in specific network layers including the average percentage of zero activations, a metric to count how many neurons have zero activations, and the entropy of activations, a metric to measure how much information is contained in the neurons of a layer. We analogously compute the Neurons ON which is the average number of non-zero activations, the entropy of $p _ { E B }$ which is higher when the probability distribution is spread out over more neurons in a layer. We also compute the peak $p _ { E B }$ which is expected to be lower on a more spread distribution. Moreover, Ma et al. (2017) introduced conservative filters: filters whose parameters do not change significantly during training. Conservative filters reduce the effective number of parameters in a CNN and may limit the CNNs modeling capacity for the target task.
+
+![](images/20343932c634f72f2c6c178a0f022656d412a353549077273e95e78cdc80aa0c.jpg)  
+Figure 5: Robustness of predicted ground-truth class probabilities as more neurons are droppedout for UCF101 test images. We fine-tune VGG16 with Excitation, Curriculum, Standard, and No Dropout at the $f c 6$ layer, averaging results over five trained models. The standard deviation is depicted around the mean curve using a lighter shade. At test time, we switch off starting from the most relevant neurons with respect to $p _ { c }$ (left), from the least relevant neurons with respect to $p _ { c }$ (center), and $k$ random neurons (right). In all scenarios, Excitation Dropout shows more robustness to network compression (dropping $f c$ neurons $\equiv$ removing filters).
+
+A conservative filter is a filter $k$ in layer $n$ whose weights have changed by $\Delta _ { n } ^ { k } = \| \hat { w } _ { n } ^ { k } - w _ { n } ^ { k } \|$ , where $\Delta _ { n } ^ { k }$ is less than a threshold $\Delta$ (empirically set).
+
+We evaluate the presented metrics for Excitation Dropout and compare against Standard and Curriculum Dropout in Table 2. This is done on the same datasets and architectures considered in Sec. 4.2. All metrics are computed for the first fully-connected layer of the CNN-2 and VGG16 nets consisting of 2048 and 4096 neurons, respectively. We compute each metric over the test set of each dataset. Excitation Dropout consistently outperforms Standard and Curriculum Dropout in all the metrics over all datasets. Excitation Dropout shows a higher number of active neurons, a higher entropy over activations, a probability distribution $p _ { E B }$ that is more spread (higher entropy over $p _ { E B } )$ among the neurons of the layer, leading to a lower peak probability of $p _ { E B }$ and therefore less specialized neurons. Averaging models having less specialized neurons results in higher robustness to information loss. We also observe a significantly smaller number of conservative filters when using Excitation Dropout. Fewer filters remain unchanged, i.e. do not sufficiently learn anything far from the random initialization. These results show that the models trained with Excitation Dropout were trained to be more informative, i.e. the contribution for the final classification task is provided by a higher number of neurons in the network, reflecting the alternative learnt paths. An analysis of such metrics over the training iterations is presented in the Appendix.
+
+# 4.4 SETUP AND RESULTS: RESILIENCE TO COMPRESSION
+
+In this section, we simulate ‘Brain Damage’ by dropping out neurons at test time. Fig. 4 demonstrates how a network utilizes the learnt alternative paths to capture the evidence of the class HorseRiding in a video frame of the UCF101 dataset. Given a VGG16 model fine-tuned with Excitation, Curriculum, Standard, and No Dropout at the $f c 6$ layer, we show the excitation saliency map obtained at the conv5-1 layer as we drop out a fixed number of the most relevant neurons from the same layer dropout is performed upon during training. A neuron is considered to be more relevant if it has a higher $p _ { E B }$ . In the first column of frames of Fig. 4, the original saliency maps for the different models are shown. As already highlighted in Table 2, the original saliency map obtained from the model trained with Excitation Dropout is more spread as compared to that of the other schemes, which present more pronounced red peaks. In the following columns of Fig. 4, we present the saliency maps the model is able to restore when the 100, 200, 300, 400, 500 most relevant neurons are dropped-out. Despite the increasing number of relevant neurons being dropped-out, Excitation Dropout is capable of restoring more of the saliency map contributing to HorseRiding. This means that the network with Excitation Dropout was trained to find alternative paths which belong to the same HorseRiding-relevant cues of the image. Despite the fact that we are considering the worst-case scenario, where we are switching off the most relevant neurons at test time, Excitation Dropout shows most robustness. More examples are presented in the Appendix.
+
+While Fig. 4 visualizes one example qualitatively, Fig. 5 presents a complete quantitative analysis on the entire test set after training is complete. We study how the predicted ground-truth (GT) probability changes as more neurons are dropped-out at test time. On the left we present the worst case when the neurons dropped are the most relevant to the prediction. The horizontal axis in the graph represents $p _ { c }$ , where $0 \leq p _ { c } \leq 1$ is the cumulative sum of $p _ { E B }$ of neurons which will be switched off starting from the most ‘important’. The analysis is performed for $p _ { c } = \{ 0 , 0 . 0 5 , \hdots , 0 . 9 0 , 0 . 9 5 \}$ . In the center, we present an analogous analysis starting to drop from the ‘least’ relevant neurons. On the right, we present the random case (more realistic) when $k$ neurons $( k = 0 , 1 2 8 , 2 5 6 , \ldots , 4 0 9 6 )$ are randomly switched off. As we drop more neurons, Excitation Dropout (purple curves) is capable of maintaining a much less steep decline of GT probability, indicating more robustness against network compression. Cifar10, Cifar100, and Caltech256 show similar behavior (see analogous plots in the Appendix).
+
+# 5 CONCLUSION
+
+We propose a new regularization scheme that encourages the learning of alternative paths in a neural network by deliberately paralyzing high-saliency neurons that contribute more to a network’s prediction during training. In experiments on four image/video recognition datasets, and on different architectures, we demonstrate that our approach yields better generalization on unseen data, higher utilization of network neurons, and higher resilience to network compression.
+
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+
+# APPENDIX
+
+CNN-2 architecture. Table 3 shows the details of the CNN-2 architecture adopted for Cifar10, Cifar100, and Caltech256 experiments. The size of the softmax layer depends upon the number of classes for each dataset.
+
+Table 3: Details of the CNN-2 architecture used for experiments on the Cifar10, Cifar100, and Caltech-256 datasets.   
+
+<table><tr><td>Layer Type</td><td>Layer Size</td><td>Filter Size</td><td>Padding/Stride</td></tr><tr><td>conv</td><td>96 filters</td><td>5x5</td><td>2/1</td></tr><tr><td>max pool</td><td></td><td>3x3</td><td>0/2</td></tr><tr><td>conv</td><td>128 filters</td><td>5x5</td><td>2/1</td></tr><tr><td>max pool</td><td></td><td>3x3</td><td>0/2</td></tr><tr><td>conv</td><td>256 filters</td><td>5x5</td><td>2/1</td></tr><tr><td>max pool</td><td></td><td>3x3</td><td>0/2</td></tr><tr><td>fc</td><td>2048 units</td><td></td><td></td></tr><tr><td>fc</td><td>2048 units</td><td></td><td></td></tr><tr><td>softmax</td><td>#classes</td><td></td><td></td></tr></table>
+
+Least vs. most relevant neurons. Popular dropout methods (e.g. Adaptive Dropout Ba & Frey (2013)) drop useless neurons with low activations during training. In this work we motivate and demonstrate that dropping neurons based on their Excitation Backprop (EB) probability has added benefits. Please note that we are not simply considering neuron activation. We demonstrate the performance of Excitation Dropout compared to the variant Adaptive Dropout [Ba & Frey (2013)] in Table 4. In essence, Adaptive and Excitation Dropout are opposites by dropping the least and most important neurons, respectively.
+
+<table><tr><td>Dataset</td><td>Adaptive Dropout</td><td>Excitation Dropout</td></tr><tr><td>Cifar10</td><td>76.82%</td><td>81.94%</td></tr><tr><td>Cifar100</td><td>44.55%</td><td>52.04%</td></tr><tr><td>Caltech256</td><td>23.32%</td><td>35.77%</td></tr><tr><td>UCF101</td><td>71.76%</td><td>73.23%</td></tr></table>
+
+Table 4: Accuracy comparison between Adaptive Dropout and Excitation Dropout. The numbers reported in this table are the average test set accuracy over five trained models for each dataset.
+
+Model complexity. Excitation Dropout consistently outperforms Standard Dropout (SD), with zero increase in test-time computational complexity. In training, there is a moderate increase in computation: in the worst case, Excitation Dropout will take double (same O-notation complexity) the training time of SD. This will happen when the utilized Excitation Dropout maps are at the first layer of the network. If a middle layer map is used, Excitation Dropout requires an additional partial forward-backward pass. We use maps of fc layers close to the end of the network to reduce this overhead. Table 5 presents a run-time analysis for the two main architectures used in this work and compares it to that of Standard Dropout.
+
+Table 5: Run-time comparison: Average time of 100 iterations (in seconds, batch size $\scriptstyle = 5 0$ ) for a Caffe python layer on a GTX Titan X GPU and Intel(R) Xeon(R) CPU E5-2650 v3 ${ \textcircled { a } } ~ 2 . 3 0 \mathrm { G H z }$ . In parenthesis is the percentage increase with respect to SD.   
+
+<table><tr><td></td><td>Standard Dropout</td><td>Excitation Dropout</td></tr><tr><td rowspan="2">CNN-2 (1 iter) Run-time VGG16 (1 iter)</td><td>0.1532 ± 0.0064</td><td>0.1885 ± 0.0070 (+23%)</td></tr><tr><td>2.2928 士 0.0297</td><td>2.8202 ± 0.0312 (+23%)</td></tr></table>
+
+Sensitivity analysis. In this section we present a sensitivity analysis over the dropout rate hyperparameter. In our work, we implement Excitation Dropout with a base retaining probability $P$ , and we compare that to standard dropout with a retaining probability $p$ , where $P = p$ . If Excitation
+
+Dropout produced a uniform probability distribution over the desired layer, then every node would have a retain probability equal to the base probability $P$ . For completeness, we add a sensitivity analysis of the parameters $p$ and $P$ in Table 6.
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Dropout Scheme</td><td rowspan=1 colspan=1>0.25Dropout</td><td rowspan=1 colspan=1>0.5Dropout</td><td rowspan=1 colspan=1>0.75Dropout</td></tr><tr><td rowspan=1 colspan=1>Cifar10</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>79.16%81.38%</td><td rowspan=1 colspan=1>80.13%81.94%</td><td rowspan=1 colspan=1>81.19%81.55%</td></tr><tr><td rowspan=1 colspan=1>Cifar100</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>48.44%53.23%</td><td rowspan=1 colspan=1>50.36%52.04%</td><td rowspan=1 colspan=1>51.64%51.87%</td></tr><tr><td rowspan=1 colspan=1>Caltech256</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>26.23%33.60%</td><td rowspan=1 colspan=1>28.73%35.77%</td><td rowspan=1 colspan=1>32.51%36.81%</td></tr><tr><td rowspan=1 colspan=1>UCF101</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>71.01%73.56%</td><td rowspan=1 colspan=1>71.93%73.23%</td><td rowspan=1 colspan=1>72.92%73.06%</td></tr></table>
+
+Table 6: Hyper-parameter sensitivity analysis for the Standard Dropout probability, and the Excitation Dropout base dropout probability. The accuracy is reported on the test set of each dataset. The retaining probability $p$ or $P$ is one minus the dropout rate.
+
+Excitation Dropout in Convolutional Layers. Excitation Dropout is a generic formulation that can be applied to any neural network layer. For a convolutional layer, a generic convolutional activation map is in the form of $[ w , h , N ]$ , where $N$ is the number of feature maps while $w$ and $h$ are the spatial dimensions. To apply Excitation Dropout to a convolutional layer, first $p _ { E B }$ is computed for each feature map $N$ as the sum of $p _ { E B }$ across spatial locations $w$ and $h$ . Specific 2D feature maps are then dropped-out following Eqn. 3. We test Excitation Dropout at conv3 of the CNN-2 architecture and obtain the following accuracy results for Cifar10: No Dropout $7 6 . 9 1 \%$ ; Excitation Dropout at conv3 $7 8 . 0 1 \%$ ; Excitation Dropout at $f c 1 8 1 . 9 4 \%$ . Again, we observe an improvement respect to No Dropout, but consistent with the literature [Hinton et al. (2012); Srivastava et al. (2014)], the improvement is not as large as using dropout in fully connected layers.
+
+Validation on Multiple Architectures. In this section we report additional results for Caltech256 and Cifar10 datasets considering different architectures. Table 7 reports the results of fine-tuning the deep architectures AlexNet, VGG16 and VGG19 on Caltech256 for the task of image classification. Finally, we adopt the WideResNet (WRN-28-10) architecture, a ResNet style architecture which combines batch normalization and dropout regularization techniques. WideResNet is used to obtain state-of-the-art results on Cifar10. We replace the Standard Dropout layer in the network with Excitation Dropout obtaining $3 . 8 8 \%$ test error on Cifar10 (vs. $4 . 1 7 \%$ Zagoruyko et al.s published result, BMVC16). Therefore, Excitation Dropout gives state-of-the-art result on Cifar10.
+
+<table><tr><td>Architecture</td><td>No Dropout (%)</td><td>Standard Dropout (%)</td><td>Curriculum Dropout (%)</td><td>Excitation Dropout (%)</td></tr><tr><td>VGG16</td><td>77.97</td><td>79.11 (+1.14%)</td><td>79.31 (+1.34%)</td><td>79.92 (+1.95%)</td></tr><tr><td>VGG19</td><td>77.98</td><td>79.46 (+1.48%)</td><td>79.66 (+1.68%)</td><td>80.65 (+2.67%)</td></tr><tr><td>AlexNet</td><td>66.48</td><td>68.10 (+1.62%)</td><td>68.71 (+2.23%)</td><td>69.66 (+3.18%)</td></tr></table>
+
+Table 7: Test accuracy comparison between No, Standard, Curriculum, and Excitation Dropout in the $f c 6$ layer of three architectures: AlexNet, VGG16 and VGG19, fine-tuned for the image recognition task on Caltech256. The numbers reported are the final test accuracies together with the improvements (in parenthesis) with respect to No Dropout, averaged over five trained models.
+
+Additional Visualizations. In this section, we present more visualizations similar to that of Fig. 4 in the main manuscript. Visualizations for a VGG16 network fine-tuned on UCF101 for the actions: PlayingFlute, PlayingSitar, and GolfSwing are presented in Fig.s 6,7,8, respectively. Every column displays the saliency map over the same video frame of an action while incrementally switching off the most $k$ relevant neurons $( k = 0 , 1 0 0 , 2 0 0 , \ldots , 5 0 0 )$ . Excitation Dropout is more robust over higher number of switched off neurons. This is demonstrated through its ability to recover the saliency map even when a high percentage of the most salient neurons is dropped-out. This is done through the alternative learnt paths, which are reflected in the higher number of non-zero activations that remain after dropout compared to other dropout strategies.
+
+![](images/055b57f36761045374b24b4e4a4b2d1285b7f8d934ccb246737784c0755cd3a8.jpg)  
+Figure 6: Despite the increasing number of relevant neurons being dropped out at test time, Excitation Dropout is capable of restoring more of the saliency map contributing to the specific class: PlayingFlute
+
+Additional Plots. In this section we present the $f c$ compression results for three additional datasets: Cifar10, Cifar100, and Caltech256 (Fig. 9), using CNN-2. We study how the predicted ground-truth (GT) probability changes as more neurons are dropped-out at test time. On the left we present the worst case when the neurons dropped are the most relevant to the prediction. The horizontal axis in the graph represents $p _ { c }$ , where $0 ~ \leq ~ p _ { c } ~ \leq ~ 1$ is the cumulative sum of $p _ { E B }$ of neurons which will be switched off starting from the most ‘important’. The analysis is performed for $\mathit { p _ { c } } =$ $\{ 0 , 0 . 0 5 , \hdots , 0 . 9 0 , 0 . 9 5 \}$ . In the center, we present an analogous analysis starting to drop from the ‘least’ relevant neurons. On the right, we present the random case (more realistic) when $k$ neurons $( k \ : = \ : 0 , 1 2 8 , 2 5 6 , \ldots , 2 0 4 8 )$ are randomly switched off. As we drop more neurons, Excitation Dropout (purple curves) is capable of maintaining a much less steep decline of GT probability.
+
+Metric Analysis During Training. In this section we report an extended analysis of the metrics: # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ during training. Excitation Dropout shows a higher number of active neurons, a higher entropy over activations, a probability distribution $p _ { E B }$ that is more spread (higher entropy over $p _ { E B } .$ ) among the neurons of the layer, leading to a lower peak probability of $p _ { E B }$ and therefore less specialized neurons. These results are observed to have consistent trends over all training iterations for all datasets considered (see Fig.s 10, 11, 12, and 13)
+
+![](images/bb0accbe64bc2d329717bcbd3f546cffb09cd66b200f52d0cdbf8b1d5d8990dc.jpg)  
+Figure 7: Despite the increasing number of relevant neurons being dropped out at test time, Excitation Dropout is capable of restoring more of the saliency map contributing to the specific class: PlayingSitar
+
+![](images/b60b944e6ef04732af9d663f8e4f5ba75d38f389d8475003393d14663bf20fb2.jpg)  
+Figure 8: Despite the increasing number of relevant neurons being dropped out at test time, Excitation Dropout is capable of restoring more of the saliency map contributing to the specific class: GolfSwing
+
+![](images/d426f098174c920b4b0eae13cbe3bedc148dd693ddd5326114b164e5de773e3d.jpg)  
+Figure 9: Robustness of predicted ground-truth class probabilities as more neurons are dropped out for each dataset’s test images. We train CNN-2 from scratch with Excitation, Curriculum, Standard, and No Dropout at the $f c 1$ layer, averaging results over five trained models. The standard deviation is depicted around the mean curve using a lighter shade. Left: the most relevant neurons with respect to the $p _ { c }$ threshold are switched off. Center: the least relevant neurons with respect to the $p _ { c }$ threshold are switched off. Right: $k$ neurons are randomly switched off. In all scenarios, Excitation Dropout shows more robustness to network compression (dropping $f c$ neurons $\equiv$ removing filters).
+
+![](images/7d2b678f17f563faaba44ab1f82dc9d0810fca08df08728b18cc44fd35a8a1d5.jpg)  
+Figure 10: Cifar10. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ over time during training.
+
+![](images/63d436b008fc255652a6d92c375699f6e0682d163a681d2e8fefc422cef6adff.jpg)  
+Figure 11: Cifar100. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ over time during training.
+
+![](images/418febd7aef2668302078c9442b398ef17a961a4366e5c54f07890830b5f3ef3.jpg)  
+Figure 12: Caltech256. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of ${ p } _ { E B }$ over time during training.
+
+![](images/106f92fa791ab8ee0ea5ded0bcc3cf155c219d427b49c5549f99c12629246e83.jpg)  
+Figure 13: UCF101. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ over time during training.

parse/train/H1xQSjCqFQ/H1xQSjCqFQ_content_list.json

@@ -0,0 +1,1433 @@
+[
+    {
+        "type": "text",
+        "text": "EXCITATION DROPOUT: ENCOURAGING PLASTICITYIN DEEP NEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            112,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            184,
+            398,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            248,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a guided dropout regularizer for deep networks based on the evidence of a network prediction: the firing of neurons in specific paths. In this work, we utilize the evidence at each neuron to determine the probability of dropout, rather than dropping out neurons uniformly at random as in standard dropout. In essence, we dropout with higher probability those neurons which contribute more to decision making at training time. This approach penalizes high saliency neurons that are most relevant for model prediction, i.e. those having stronger evidence. By dropping such high-saliency neurons, the network is forced to learn alternative paths in order to maintain loss minimization, resulting in a plasticity-like behavior, a characteristic of human brains too. We demonstrate better generalization ability, an increased utilization of network neurons, and a higher resilience to network compression using several metrics over four image/video recognition benchmarks. ",
+        "bbox": [
+            233,
+            280,
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+            446
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            336,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dropout [Hinton et al. (2012); Srivastava et al. (2014)] is a classical regularization technique that is used in many state-of-the-art deep neural networks, typically applied to fully-connected layers. Standard Dropout selects a fraction of neurons to randomly drop out by zeroing their forward signal. In this work, we propose a scheme for biasing this selection. Our scheme utilizes the contribution of neurons to the prediction made by the network at a certain training iteration stage. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dropout can be interpreted as model averaging technique that avoids overfitting on training data, allowing for better generalization on unseen test data. A recent variant of dropout that targets improved generalization ability is Curriculum Dropout [Morerio et al. (2017)]. It targets adjusting the dropout rate by exponentially increasing the unit suppression rate during training, answering the question How many neurons to drop out over time? Like Standard Dropout [Hinton et al. (2012); Srivastava et al. (2014)], Curriculum Dropout selects the neurons to be dropped randomly. In this work, however, we target at determining how the dropped neurons are selected, answering the question Which neurons to drop out? ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our approach is inspired by brain plasticity [Hebb (2005); Song et al. (2000); Mittal et al. (2018); Miconi et al. (2018)]. We deliberately, and temporarily, paralyze/injure neurons to enforce learning alternative paths in a deep network. At training time, neurons that are more relevant to the current prediction are given a higher dropout probability. The relevance of a neuron for making a certain prediction is quantified using Excitation Backprop, a top-down saliency approach proposed by Zhang et al. (2016). Excitation Backprop conveniently yields a probability distribution at each layer that reflects neuron saliency, or neuron contribution to the prediction being made. This is utilized in the pipeline of our approach, named Excitation Dropout, which is summarized in Fig. 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In particular, we study how this approach improves generalization through utilizing more network’s neurons for image classification. We report an increased recognition rate for both CNN models that are fine-tuned and trained from scratch. This improvement is validated on four image/video recognition datasets, and ranges from $1 . 1 \\% - 6 . 3 \\%$ over state-of-the-art Curriculum Dropout. ",
+        "bbox": [
+            174,
+            819,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Next, we examine the effect of our approach on network utilization. Mittal et al. (2018) and Ma et al. (2017) introduce metrics that measure network utilization. We show a consistent increased network utilization using Excitation Dropout on four image/video recognition datasets. For example, averaged over all four benchmarks, we get $7 6 . 5 5 \\%$ reduction in conservative filters, filters whose parameters do not change significantly during training, as compared to Standard Dropout. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a6cd4c0b90572493c61bafb0d0a027deae64fbc8a02b1ea8e9ccd622c7fe66b8.jpg",
+        "image_caption": [
+            "Figure 1: Training pipeline of Excitation Dropout. Step 1: A minibatch goes through the standard forward pass. Step 2: Backward EB is performed until the specified dropout layer; this gives a neuron saliency map at the dropout layer in the form of a probability distribution. Step 3: The probability distribution is used to generate a binary mask for each image of the batch based on a Bernoulli distribution determining whether each neuron will be dropped out or not. Step 4: A forward pass is performed from the specified dropout layer to the end of the network, zeroing the activations of the dropped out neurons. Step 5: The standard backward pass is performed to update model weights. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finally, we study network resilience to neuron dropping at test time. We observe that training with Excitation Dropout leads to models that are a lot more robust when layers are shrunk/compressed by removing units. We demonstrate this when dropping the most relevant neurons, the least relevant neurons, and with a random dropping selection. This can be quite desirable for compressing/distilling [Hinton et al. (2015)] a model, e.g. for deployment on mobile devices. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, by encouraging plasticity-like behavior, our contributions are threefold: ",
+        "bbox": [
+            173,
+            561,
+            725,
+            577
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Better generalization on test data.   \n2. Higher utilization of network neurons.   \n3. Resilience to network compression. ",
+        "bbox": [
+            210,
+            588,
+            485,
+            640
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            662,
+            344,
+            679
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Dropout was first introduced by Hinton et al. (2012) and Srivastava et al. (2014) as a way to prevent neural units from co-adapting too much on the training data by randomly omitting subsets of neurons at each iteration of the training phase. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Some follow-up works have explored different schemes for determining how much dropout is applied to neurons/weights. Wager et al. (2013) described the dropout mechanism in terms of an adaptive regularization, establishing a connection to the AdaGrad algorithm. Inspired by information theoretic principles, Achille & Soatto (2018) propose Information Dropout, a generalization dropout which can be automatically adapted to the data. Kingma et al. (2015) showed that a relationship between dropout and Bayesian inference can be extended when the dropout rates are directly learned from the data. Kang et al. (2017) introduces Shakeout which instead of randomly discarding units as dropout does, it randomly enhances or reverses each units contribution to the next layer. Wan et al. (2013) introduced the DropConnect framework, adding dynamic sparsity on the weights of a deep model. DropConnect generalized Standard Dropout by randomly dropping the weights rather than the neuron activations in the network. Rennie et al. (2014) proposed a time scheduling for the retaining probability for the neurons in the network. The presented adaptive regularization scheme smoothly decreased in time the number of neurons turned off during training. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recently, Morerio et al. (2017) proposed Curriculum Dropout to adjust the dropout rate in the opposite direction, exponentially increasing unit suppression rate during training, leading to a better generalization on unseen data. ",
+        "bbox": [
+            176,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Other works focus on which neurons to drop out. Dropout is usually applied to fully-connected layers of a deep network. Conversely, Wu & Gu (2015) studied the effect of dropout in convolutional and pooling layers. The selection of neurons to drop depends on the layer where they reside. In contrast, we select neurons within a layer based on their contribution. Wang & Manning (2013) demonstrate that sampling neurons from a Gaussian approximation gave an order of magnitude speedup and more stability during training. Li et al. (2016) proposed to use multinomial sampling for dropout, i.e. keeping neurons according to a multinomial distribution with specific probabilities for different neurons. Ba & Frey (2013) jointly trained a binary belief network with a neural network to regularize its hidden units by selectively setting activations to zero accordingly to their magnitude. While this takes into consideration the magnitude of the forward activations, it does not take into consideration the relationship of these activations to the ground-truth. In contrast, we drop neurons based on how they contribute to a network’s decision. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We compare our results against Morerio et al. (2017). To the best of our knowledge, we are the first to probabilistically select neurons to dropout based on their task-relevance. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            417,
+            315,
+            433
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Saliency maps that quantize the importance of class-specific neurons for an input image are instrumental to our proposed scheme. Popular approaches include Class Activation Maps (CAM) [Zhou et al. (2016)], Gradient-weighted Class Activation Mapping (Grad-CAM) [Selvaraju et al. (2017)], and Excitation Backprop (EB) [Zhang et al. (2017)]. A thorough analysis of all saliency methods is out of the scope of this work, and the saliency problem in general is far from solved. We choose to use EB since it produces a valid probability distribution for each network layer. The saliency maps obtained using this approach are evaluated for spatial localization of objects and demonstrate the ability of pointing to the right region of an image [Zhang et al. (2017)]. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In a standard CNN, the forward activation of neuron $a _ { j }$ is computed by $\\begin{array} { r } { \\widehat { a } _ { j } = \\phi ( \\sum _ { i } w _ { i j } \\widehat { a } _ { i } + b _ { i } ) } \\end{array}$ , where $\\widehat { a } _ { i }$ is the activation coming from the previous layer, $\\phi$ b bis a nonlinear activation function, $w _ { i j }$ and $b _ { i }$ bare the weight from neuron $i$ to neuron $j$ and the added bias at layer $i$ , respectively. EB devises a backpropagation formulation able to reconstruct the evidence used by a deep model to make decisions. It computes the probability of each neuron recursively using conditional probabilities $P ( a _ { i } | a _ { j } )$ in a top-down order starting from a probability distribution over the output units, as follows: ",
+        "bbox": [
+            174,
+            565,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d8afb918dda2beef6d9efdb3fc549a0125da196d36bee6379877cecb3e9a1a1.jpg",
+        "text": "$$\nP ( a _ { i } ) = \\sum _ { a _ { j } \\in \\mathcal { P } _ { i } } P ( a _ { i } | a _ { j } ) P ( a _ { j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            667,
+            601,
+            704
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { P } _ { i }$ is the parent node set of $a _ { i }$ . EB passes top-down signals through excitatory connections having non-negative activations, excluding from the competition inhibitory ones. EB is designed with an assumption of non-negative activations. Most modern CNNs use ReLU activation functions, which satisfy this assumption. Therefore, negative weights can be assumed to not positively contribute to the final prediction. Assuming $C _ { j }$ the child node set of $a _ { j }$ , for each $a _ { i } \\in C _ { j }$ , the conditional winning probability $P ( a _ { i } | a _ { j } )$ is defined as ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7e9fe7d59ab150e407dd2faf0ffec8016d7b106324400cb15e0990886ff1e1d.jpg",
+        "text": "$$\nP ( a _ { i } | a _ { j } ) = \\left\\{ \\begin{array} { l l } { Z _ { j } \\widehat { a } _ { i } w _ { i j } , } & { \\mathrm { i f } \\ w _ { i j } \\geq 0 , } \\\\ { 0 , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            815,
+            620,
+            851
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $Z _ { j }$ is a normalization factor such that $\\begin{array} { r } { \\sum _ { a _ { i } \\in \\mathcal { C } _ { j } } P ( a _ { i } | a _ { j } ) = 1 } \\end{array}$ . Recursively propagating the top-down signal and preserving the sum of backpropagated probabilities, it is possible to highlight the salient neurons in each layer using Eqn. 1, i.e. neurons that mostly contribute to a specific task. We will refer to the distribution of $P ( a _ { i } )$ as $p _ { E B } ( a _ { i } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/13b48eaa2c04c820150b59d7e2cb1113153194ce36691f10b5f7572a5b0e1d83.jpg",
+        "image_caption": [
+            "Figure 2: The retaining probability, $p$ , as a function of the Excitation Backprop probability $p _ { E B }$ . This plot was created using $N = 1 0$ and a base retaining probability $P = 0 . 5$ . In this case, when the saliency of neurons is uniform, i.e. $p _ { E B } = 0 . 1$ , then $p = P$ as marked in the figure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 EXCITATION DROPOUT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            314,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the standard formulation of dropout [Hinton et al. (2012); Srivastava et al. (2014)], the suppression of a neuron in a given layer is modeled by a Bernoulli random variable $0 < p \\leq 1$ where $p$ is defined as the probability of retaining a neuron. Given a specific layer where dropout is applied, during the training phase, each neuron is turned off with a probability $1 - p$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We argue for a different approach that is guided in the way it selects neurons to be dropped. In a training iteration, certain paths have high excitation contributing to the resulting classification, while other regions of the network have low response. We encourage learning alternative paths (plasticity) through the temporary damaging of the currently highly excited path. We re-define the probability of retaining a neuron as a function of its contribution in the currently highly excited path ",
+        "bbox": [
+            173,
+            404,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd251c1dcfb46905f2b4b1e7649db12a474c0d72e53f74da2f82d06defa1ab4a.jpg",
+        "text": "$$\np = 1 - \\frac { ( 1 - P ) * ( N - 1 ) * p _ { E B } } { ( ( 1 - P ) * N - 1 ) * p _ { E B } + P }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            481,
+            633,
+            515
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { E B }$ is the probability backpropagated through the EB formulation (Eqn. 1) in layer $l$ , $P$ is the base probability of retaining a neuron when all neurons are equally contributing to the prediction and $N$ is the number of neurons in a fully-connected layer $l$ or the number of filters in a convolutional layer $l$ . The retaining probability defined in Eqn. 3 drops neurons which contribute the most to the recognition of a specific class, with higher probability. Dropping out highly relevant neurons, we retain less relevant ones and thus encourage them to awaken. We also study how this compares to dropping the least relevant neurons (Adaptive Dropout by Ba & Frey (2013)) in the Appendix. ",
+        "bbox": [
+            173,
+            521,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 shows $p$ as a function of $p _ { E B }$ . To gain some intuition for Eqn. 3, we can look more closely at the graph: 1) If neuron $a _ { i }$ has $p _ { E B } ( a _ { i } ) = 1$ : This results in a retaining probability of $p = 0$ . We do not want to keep a neuron which has a high contribution to the correct label. 2) If neuron $a _ { i }$ has $p _ { E B } ( a _ { i } ) = 0$ : This results in a retaining probability of $p = 1$ . We want to keep a neuron which has not contributed to the correct classification of an image. 3) If neuron $a _ { i }$ has $\\dot { p } _ { E B } ( a _ { i } ) = 1 / N$ , i.e. $p _ { E B }$ is a uniform probability distribution: This results in a retaining probability $p = P$ . We want to keep a neuron with base probability $P$ since all neurons contribute equally. ",
+        "bbox": [
+            173,
+            626,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Eqn. 3 provides a dropout probability for each neuron, which is then used as the parameter of a Bernoulli distribution giving a binary dropout mask. During training, each image in a batch leads to different excitatory connections in the network and therefore has a different $p _ { E B }$ distribution, consequently leading to a different dropout mask. Fig. 1 presents the pipeline of Excitation Dropout at training time, and a run-time analysis is presented in the Appendix. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            821,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present how Excitation Dropout improves the generalization ability on four image/video recognition datasets in fully connected layers of different architectures. We then present an analysis of how Excitation Dropout affects the utilization of network neurons on the same datasets. Finally, we examine the resilience of a model trained using Excitation Dropout to network compression. ",
+        "bbox": [
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+        "image_caption": [
+            "Figure 3: We compare the test accuracy of different dropout training strategies on four image/video recognition datasets: Cifar10, Cifar100, Caltech256, UCF101. Results presented here are averaged over five trained models and the standard deviation is depicted around the mean curve using a lighter shade. Excitation Dropout performs best after convergence compared to the other strategies. "
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+        "text": "4.1 DATASETS AND ARCHITECTURES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We present results on four image/video recognition datasets. Cifar10 and Cifar100 [Krizhevsky (2009)] are image recognition datasets, each consisting of $6 0 0 0 0 3 2 \\times 3 2$ tiny RGB natural images. Cifar10 images are distributed over 10 classes with 6000 images per class, and Cifar100 images are distributed over 100 classes with 600 images per class. Training and test splits contain $5 0 K$ and $1 0 K$ images, respectively. We feed the network with the original image dimensions. Caltech256 [Griffin et al. (2007)] is an image recognition dataset consisting 31000 RGB images divided in 256 classes. We consider five different random splits of 50 train images and 20 testing images for each class. Images were reshaped to $1 2 8 \\times 1 2 8$ pixel to feed the network. UCF101 [Soomro et al. (2012)] is a video action recognition dataset based on 13320 actions belonging to 101 action classes. For this dataset we consider a frame-based action recognition task. The images are resized to $2 2 4 \\times 2 2 4$ and $2 2 7 \\times 2 2 7$ to fit the input layers of the VGG and AlexNet architectures, respectively. ",
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+        "text": "We present results on four architectures. Relatively shallow architectures are trained from scratch, and deeper popular architectures are fine-tuned after being pre-trained on ImageNet [Deng et al. (2009)]. Models trained from scratch: We train the CNN-2 architecture used in Morerio et al. (2017), the state-of-the-art dropout variant, for comparison purposes. This architecture consists of three convolutional and two fully-connected layers (see Appendix). We train this network from scratch for $1 0 0 K$ iterations on the datasets: Cifar-10, Cifar-100 and Caltech-256. We use minibatches of 100 images and fix the learning rate to be $1 0 ^ { - 3 }$ , decreasing to $1 0 ^ { - 4 }$ after $2 5 K$ iterations. ",
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+        "text": "Fine-tuned models: We fine-tune the commonly used architectures: AlexNet [Krizhevsky et al. (2012)], VGG16 and VGG19 [Simonyan & Zisserman (2014)] pre-trained on ImageNet. We finetune the models for a frame by frame action recognition task on UCF101. The learning rate is fixed to $1 0 ^ { - 3 }$ for all the processes. We fine-tune AlexNet for $5 K$ while VGG16 and VGG19 for $3 0 K$ iterations. We use a batch size of 128 and 50 images for AlexNet and VGG16/19, respectively. ",
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+        "table_body": "<table><tr><td>Architecture</td><td>No Dropout (%)</td><td>Standard Dropout (%)</td><td>Curriculum Dropout (%)</td><td>Excitation Dropout (%)</td></tr><tr><td>VGG16</td><td>69.37</td><td>71.93 (+2.56%)</td><td>72.14 (+2.77%)</td><td>73.23 (+3.86%)</td></tr><tr><td>VGG19</td><td>71.32</td><td>72.52 (+1.29%)</td><td>73.18 (+1.86%)</td><td>74.34 (+3.02%)</td></tr><tr><td>AlexNet</td><td>62.89</td><td>64.50 (+1.61%)</td><td>64.55 (+1.66%)</td><td>67.56 (+4.67%)</td></tr></table>",
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+        "text": "Table 1: Test accuracy comparison between No, Standard, Curriculum and Excitation Dropout in the $f c 6$ layer of three architectures: AlexNet, VGG16 and VGG19, fine-tuned for the action recognition task on UCF101. The numbers reported are the final test accuracies together with the improvements (in parenthesis) with respect to No Dropout, averaged over five trained models. ",
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+    {
+        "type": "table",
+        "img_path": "images/0a64ecc03ed6c80889706e3b53d5c5aec57de1a1b5de1b88efc8aa8695f92aaa.jpg",
+        "table_caption": [
+            "Table 2: Different metrics to reflect the usage of network capacity in the first fully-connected layer of the CNN-2 architecture consisting of 2048 neurons and the VGG16 consisting of 4096 neurons. Results presented here are averaged over five trained models for each of the datasets: Cifar10, Cifar100, Caltech256 and UCF101 $\\sigma$ in brackets). Excitation Dropout consistently produces more neurons with non-zero activations, has a more spread saliency map leading to a lower saliency peak, has a higher entropy of both activations and saliency, and has a lower number of conservative filters; all reflecting an improved utilization of the network neurons using Excitation Dropout. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Metric</td><td>Standard Dropout</td><td>Curriculum Dropout</td><td>Excitation Dropout</td></tr><tr><td rowspan=\"4\">Cripfi</td><td># Neurons ON Peak pEB</td><td>1194 (±153) 0.011 (±0.004)</td><td>1169 (±61) 0.009 (±0.001)</td><td>1325 (±61) 0.003 (±0.0002)</td></tr><tr><td>Entropy of Activations</td><td>3.55 (±0.72)</td><td>3.50 (±0.12)</td><td></td></tr><tr><td>Entropy of pEB</td><td></td><td></td><td>4.29 (±0.28)</td></tr><tr><td>Conservative Filters△=0.25</td><td>3.28 (±0.56) 1204 (±37)</td><td>3.32 (±0.13) 959 (±34)</td><td>4.26 (±0.26) 124(±22)</td></tr><tr><td rowspan=\"4\">cormfi</td><td># Neurons ON Peak pEB</td><td>453 (±183) 0.011 (±0.0004)</td><td>460 (±75) 0.012 (±0.0004)</td><td>943 (±131) 0.005 (±0.0005)</td></tr><tr><td>Entropy of Activations</td><td>1.67 (±0.31)</td><td>1.70 (±0.29)</td><td>3.21 (±0.44)</td></tr><tr><td>Entropy of pEB</td><td>1.64 (±0.27)</td><td>1.67 (±0.26)</td><td>3.17 (±0.41)</td></tr><tr><td>Conservative Filters△=0.30</td><td>2048 (±51)</td><td>2038 (±44)</td><td>14 (±13)</td></tr><tr><td rowspan=\"5\">Ceresst5</td><td># Neurons ON Peak pEB</td><td>412 (±126) 0.014 (±0.0007)</td><td>471 (±146)</td><td>702 (±171)</td></tr><tr><td></td><td></td><td>0.013 (±0.0006)</td><td>0.007 (±0.0003)</td></tr><tr><td>Entropy of Activations</td><td>1.63 (±0.32)</td><td>1.84 (±0.35)</td><td>2.63 (±0.23)</td></tr><tr><td>Entropy of pEB</td><td>1.58 (±0.29)</td><td>1.77 (±0.31)</td><td>2.59 (±0.22)</td></tr><tr><td>Conservative Filters△=1.25</td><td>2048 (±46)</td><td>2048 (±49)</td><td>1671 (±31)</td></tr><tr><td rowspan=\"5\">IUIIIN</td><td># Neurons ON Peak pEB</td><td>1120 (±25)</td><td>1143 (±22)</td><td>1404 (±37)</td></tr><tr><td></td><td>0.007 (±0.0002)</td><td>0.007 (±0.0002)</td><td>0.004 (±0.0002)</td></tr><tr><td>Entropy of Activations</td><td>2.04 (±0.23)</td><td>2.08 (±0.21)</td><td>2.51 (±0.18)</td></tr><tr><td>Entropy of pEB</td><td>1.92 (±0.22)</td><td>1.95 (±0.20)</td><td>2.42 (±0.18)</td></tr><tr><td>Conservative Filters△=0.15</td><td>3599 (±66)</td><td>3859 (±53)</td><td>44(±36)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 SETUP AND RESULTS: GENERALIZATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we compare the performance of Excitation Dropout to that of No Dropout, Standard Dropout, and Curriculum Dropout [Morerio et al. (2017)]. We train a CNN-2 model from scratch on the datasets: Cifar10, Cifar100, Caltech256. Fig. 3 depicts the test accuracies over training iterations for the three datasets averaged over five trained models. After convergence, Excitation Dropout demonstrates a significant improvement in performance compared to other methods. We hypothesize that Excitation Dropout takes longer to converge due to the additional loop (Steps 2-4 in Fig. 1) introduced in the learning process, and due to the learning of the alternative paths. We note that Excitation Dropout, during training, uses a different binary mask for each image in a minibatch, while in Standard Dropout, one random mask is employed per minibatch. To prove that it is precisely the fact that masks reflective of the particular input give rise to a boost in accuracy, and not the fact that different masks are used for different images, we add a comparison with Standard Dropout having a different random mask for each image. We refer to this accuracy as ‘Standard Dropout $^ +$ Mask/Img’ in the plots. As expected, the latter approach is comparable to Standard Dropout in performance. ",
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+    {
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+        "img_path": "images/2fa319d5f33636ca112a15cb60fa1f1b00877fc4929fb69c77caab7893d92bb1.jpg",
+        "image_caption": [
+            "Figure 4: Visualizations for a VGG16 network fine-tuned on UCF101. The middle columns display the saliency map over the same video frame of the action HorseRiding while incrementally switching off the most $k$ relevant/salient neurons $( k = 0 , 1 0 0 , 2 0 0 , \\ldots , 5 0 0 )$ in the $f c 6$ layer at test time. Excitation Dropout shows more robustness when more neurons are switched off. This is demonstrated through its ability to recover more of the saliency map even when a high percentage of the most salient neurons is dropped-out. This ability reflects the alternative learnt paths. Histograms of the leftmost and rightmost saliency maps are presented to demonstrate that Excitation Dropout has a wider range of saliency values. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Next, we evaluate the effectiveness of Excitation Dropout on popular network architectures that employ dropout layers: AlexNet, VGG16, VGG19. This is done by fine-tuning on the video recognition dataset UCF101. Fig. 3 shows superior Excitation Dropout performance on AlexNet fine-tuned on UCF101. Table 1 presents more comparative results on other deep architectures by reporting the accuracy after convergence. Again, Excitation Dropout demonstrates higher generalizability on the test data for all architectures. ",
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+        "type": "text",
+        "text": "For fair comparison, we set $p = 0 . 5$ for Standard Dropout and $P = 0 . 5$ for the base retaining probability of Excitation Dropout in all experiments1. We perform dropout in the first fully-connected layer of the networks (fc1 for CNN-2 and $f c 6$ for AlexNet and VGGs) for Standard, Curriculum, and Excitation Dropout. For Curriculum Dropout we fix the parameter $\\gamma$ to $5 * 1 0 ^ { - 4 }$ as in Morerio et al. (2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 SETUP AND RESULTS: UTILIZATION OF NETWORK NEURONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we examine how Excitation Dropout expands the network’s utilization of neurons through the learnt alternative paths for a certain task. ",
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+    {
+        "type": "text",
+        "text": "Mittal et al. (2018) introduced scoring functions to rank the filters in specific network layers including the average percentage of zero activations, a metric to count how many neurons have zero activations, and the entropy of activations, a metric to measure how much information is contained in the neurons of a layer. We analogously compute the Neurons ON which is the average number of non-zero activations, the entropy of $p _ { E B }$ which is higher when the probability distribution is spread out over more neurons in a layer. We also compute the peak $p _ { E B }$ which is expected to be lower on a more spread distribution. Moreover, Ma et al. (2017) introduced conservative filters: filters whose parameters do not change significantly during training. Conservative filters reduce the effective number of parameters in a CNN and may limit the CNNs modeling capacity for the target task. ",
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+    {
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+        "img_path": "images/20343932c634f72f2c6c178a0f022656d412a353549077273e95e78cdc80aa0c.jpg",
+        "image_caption": [
+            "Figure 5: Robustness of predicted ground-truth class probabilities as more neurons are droppedout for UCF101 test images. We fine-tune VGG16 with Excitation, Curriculum, Standard, and No Dropout at the $f c 6$ layer, averaging results over five trained models. The standard deviation is depicted around the mean curve using a lighter shade. At test time, we switch off starting from the most relevant neurons with respect to $p _ { c }$ (left), from the least relevant neurons with respect to $p _ { c }$ (center), and $k$ random neurons (right). In all scenarios, Excitation Dropout shows more robustness to network compression (dropping $f c$ neurons $\\equiv$ removing filters). "
+        ],
+        "image_footnote": [],
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "A conservative filter is a filter $k$ in layer $n$ whose weights have changed by $\\Delta _ { n } ^ { k } = \\| \\hat { w } _ { n } ^ { k } - w _ { n } ^ { k } \\|$ , where $\\Delta _ { n } ^ { k }$ is less than a threshold $\\Delta$ (empirically set). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the presented metrics for Excitation Dropout and compare against Standard and Curriculum Dropout in Table 2. This is done on the same datasets and architectures considered in Sec. 4.2. All metrics are computed for the first fully-connected layer of the CNN-2 and VGG16 nets consisting of 2048 and 4096 neurons, respectively. We compute each metric over the test set of each dataset. Excitation Dropout consistently outperforms Standard and Curriculum Dropout in all the metrics over all datasets. Excitation Dropout shows a higher number of active neurons, a higher entropy over activations, a probability distribution $p _ { E B }$ that is more spread (higher entropy over $p _ { E B } )$ among the neurons of the layer, leading to a lower peak probability of $p _ { E B }$ and therefore less specialized neurons. Averaging models having less specialized neurons results in higher robustness to information loss. We also observe a significantly smaller number of conservative filters when using Excitation Dropout. Fewer filters remain unchanged, i.e. do not sufficiently learn anything far from the random initialization. These results show that the models trained with Excitation Dropout were trained to be more informative, i.e. the contribution for the final classification task is provided by a higher number of neurons in the network, reflecting the alternative learnt paths. An analysis of such metrics over the training iterations is presented in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "4.4 SETUP AND RESULTS: RESILIENCE TO COMPRESSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we simulate ‘Brain Damage’ by dropping out neurons at test time. Fig. 4 demonstrates how a network utilizes the learnt alternative paths to capture the evidence of the class HorseRiding in a video frame of the UCF101 dataset. Given a VGG16 model fine-tuned with Excitation, Curriculum, Standard, and No Dropout at the $f c 6$ layer, we show the excitation saliency map obtained at the conv5-1 layer as we drop out a fixed number of the most relevant neurons from the same layer dropout is performed upon during training. A neuron is considered to be more relevant if it has a higher $p _ { E B }$ . In the first column of frames of Fig. 4, the original saliency maps for the different models are shown. As already highlighted in Table 2, the original saliency map obtained from the model trained with Excitation Dropout is more spread as compared to that of the other schemes, which present more pronounced red peaks. In the following columns of Fig. 4, we present the saliency maps the model is able to restore when the 100, 200, 300, 400, 500 most relevant neurons are dropped-out. Despite the increasing number of relevant neurons being dropped-out, Excitation Dropout is capable of restoring more of the saliency map contributing to HorseRiding. This means that the network with Excitation Dropout was trained to find alternative paths which belong to the same HorseRiding-relevant cues of the image. Despite the fact that we are considering the worst-case scenario, where we are switching off the most relevant neurons at test time, Excitation Dropout shows most robustness. More examples are presented in the Appendix. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While Fig. 4 visualizes one example qualitatively, Fig. 5 presents a complete quantitative analysis on the entire test set after training is complete. We study how the predicted ground-truth (GT) probability changes as more neurons are dropped-out at test time. On the left we present the worst case when the neurons dropped are the most relevant to the prediction. The horizontal axis in the graph represents $p _ { c }$ , where $0 \\leq p _ { c } \\leq 1$ is the cumulative sum of $p _ { E B }$ of neurons which will be switched off starting from the most ‘important’. The analysis is performed for $p _ { c } = \\{ 0 , 0 . 0 5 , \\hdots , 0 . 9 0 , 0 . 9 5 \\}$ . In the center, we present an analogous analysis starting to drop from the ‘least’ relevant neurons. On the right, we present the random case (more realistic) when $k$ neurons $( k = 0 , 1 2 8 , 2 5 6 , \\ldots , 4 0 9 6 )$ are randomly switched off. As we drop more neurons, Excitation Dropout (purple curves) is capable of maintaining a much less steep decline of GT probability, indicating more robustness against network compression. Cifar10, Cifar100, and Caltech256 show similar behavior (see analogous plots in the Appendix). ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            250,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We propose a new regularization scheme that encourages the learning of alternative paths in a neural network by deliberately paralyzing high-saliency neurons that contribute more to a network’s prediction during training. In experiments on four image/video recognition datasets, and on different architectures, we demonstrate that our approach yields better generalization on unseen data, higher utilization of network neurons, and higher resilience to network compression. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            279,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "CNN-2 architecture. Table 3 shows the details of the CNN-2 architecture adopted for Cifar10, Cifar100, and Caltech256 experiments. The size of the softmax layer depends upon the number of classes for each dataset. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            176
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b13a212af68c98d79be551c5ff93385c2154e2f24b26e4c9ab29c583ac45fe4.jpg",
+        "table_caption": [
+            "Table 3: Details of the CNN-2 architecture used for experiments on the Cifar10, Cifar100, and Caltech-256 datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Layer Type</td><td>Layer Size</td><td>Filter Size</td><td>Padding/Stride</td></tr><tr><td>conv</td><td>96 filters</td><td>5x5</td><td>2/1</td></tr><tr><td>max pool</td><td></td><td>3x3</td><td>0/2</td></tr><tr><td>conv</td><td>128 filters</td><td>5x5</td><td>2/1</td></tr><tr><td>max pool</td><td></td><td>3x3</td><td>0/2</td></tr><tr><td>conv</td><td>256 filters</td><td>5x5</td><td>2/1</td></tr><tr><td>max pool</td><td></td><td>3x3</td><td>0/2</td></tr><tr><td>fc</td><td>2048 units</td><td></td><td></td></tr><tr><td>fc</td><td>2048 units</td><td></td><td></td></tr><tr><td>softmax</td><td>#classes</td><td></td><td></td></tr></table>",
+        "bbox": [
+            290,
+            188,
+            702,
+            330
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Least vs. most relevant neurons. Popular dropout methods (e.g. Adaptive Dropout Ba & Frey (2013)) drop useless neurons with low activations during training. In this work we motivate and demonstrate that dropping neurons based on their Excitation Backprop (EB) probability has added benefits. Please note that we are not simply considering neuron activation. We demonstrate the performance of Excitation Dropout compared to the variant Adaptive Dropout [Ba & Frey (2013)] in Table 4. In essence, Adaptive and Excitation Dropout are opposites by dropping the least and most important neurons, respectively. ",
+        "bbox": [
+            173,
+            401,
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+            500
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/9392d33ad8e22ee6bf81dc0d422ea8b0e6def21b412dd292bbe41f802e8f9185.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Adaptive Dropout</td><td>Excitation Dropout</td></tr><tr><td>Cifar10</td><td>76.82%</td><td>81.94%</td></tr><tr><td>Cifar100</td><td>44.55%</td><td>52.04%</td></tr><tr><td>Caltech256</td><td>23.32%</td><td>35.77%</td></tr><tr><td>UCF101</td><td>71.76%</td><td>73.23%</td></tr></table>",
+        "bbox": [
+            299,
+            512,
+            697,
+            587
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Accuracy comparison between Adaptive Dropout and Excitation Dropout. The numbers reported in this table are the average test set accuracy over five trained models for each dataset. ",
+        "bbox": [
+            173,
+            595,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Model complexity. Excitation Dropout consistently outperforms Standard Dropout (SD), with zero increase in test-time computational complexity. In training, there is a moderate increase in computation: in the worst case, Excitation Dropout will take double (same O-notation complexity) the training time of SD. This will happen when the utilized Excitation Dropout maps are at the first layer of the network. If a middle layer map is used, Excitation Dropout requires an additional partial forward-backward pass. We use maps of fc layers close to the end of the network to reduce this overhead. Table 5 presents a run-time analysis for the two main architectures used in this work and compares it to that of Standard Dropout. ",
+        "bbox": [
+            173,
+            640,
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+            753
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/21ea333c6079754168b24fb6de31479abeafe84126c18db8ffbf1e3f9ee00add.jpg",
+        "table_caption": [
+            "Table 5: Run-time comparison: Average time of 100 iterations (in seconds, batch size $\\scriptstyle = 5 0$ ) for a Caffe python layer on a GTX Titan X GPU and Intel(R) Xeon(R) CPU E5-2650 v3 ${ \\textcircled { a } } ~ 2 . 3 0 \\mathrm { G H z }$ . In parenthesis is the percentage increase with respect to SD. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Standard Dropout</td><td>Excitation Dropout</td></tr><tr><td rowspan=\"2\">CNN-2 (1 iter) Run-time VGG16 (1 iter)</td><td>0.1532 ± 0.0064</td><td>0.1885 ± 0.0070 (+23%)</td></tr><tr><td>2.2928 士 0.0297</td><td>2.8202 ± 0.0312 (+23%)</td></tr></table>",
+        "bbox": [
+            227,
+            767,
+            772,
+            813
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sensitivity analysis. In this section we present a sensitivity analysis over the dropout rate hyperparameter. In our work, we implement Excitation Dropout with a base retaining probability $P$ , and we compare that to standard dropout with a retaining probability $p$ , where $P = p$ . If Excitation ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dropout produced a uniform probability distribution over the desired layer, then every node would have a retain probability equal to the base probability $P$ . For completeness, we add a sensitivity analysis of the parameters $p$ and $P$ in Table 6. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/e18c17e9dc8dbfb87dcf2f6239c1b2539f22eb060f1b8bc0d613819da994e0cb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Dropout Scheme</td><td rowspan=1 colspan=1>0.25Dropout</td><td rowspan=1 colspan=1>0.5Dropout</td><td rowspan=1 colspan=1>0.75Dropout</td></tr><tr><td rowspan=1 colspan=1>Cifar10</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>79.16%81.38%</td><td rowspan=1 colspan=1>80.13%81.94%</td><td rowspan=1 colspan=1>81.19%81.55%</td></tr><tr><td rowspan=1 colspan=1>Cifar100</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>48.44%53.23%</td><td rowspan=1 colspan=1>50.36%52.04%</td><td rowspan=1 colspan=1>51.64%51.87%</td></tr><tr><td rowspan=1 colspan=1>Caltech256</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>26.23%33.60%</td><td rowspan=1 colspan=1>28.73%35.77%</td><td rowspan=1 colspan=1>32.51%36.81%</td></tr><tr><td rowspan=1 colspan=1>UCF101</td><td rowspan=1 colspan=1>StandardExcitation</td><td rowspan=1 colspan=1>71.01%73.56%</td><td rowspan=1 colspan=1>71.93%73.23%</td><td rowspan=1 colspan=1>72.92%73.06%</td></tr></table>",
+        "bbox": [
+            263,
+            157,
+            733,
+            301
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Hyper-parameter sensitivity analysis for the Standard Dropout probability, and the Excitation Dropout base dropout probability. The accuracy is reported on the test set of each dataset. The retaining probability $p$ or $P$ is one minus the dropout rate. ",
+        "bbox": [
+            176,
+            311,
+            823,
+            354
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Excitation Dropout in Convolutional Layers. Excitation Dropout is a generic formulation that can be applied to any neural network layer. For a convolutional layer, a generic convolutional activation map is in the form of $[ w , h , N ]$ , where $N$ is the number of feature maps while $w$ and $h$ are the spatial dimensions. To apply Excitation Dropout to a convolutional layer, first $p _ { E B }$ is computed for each feature map $N$ as the sum of $p _ { E B }$ across spatial locations $w$ and $h$ . Specific 2D feature maps are then dropped-out following Eqn. 3. We test Excitation Dropout at conv3 of the CNN-2 architecture and obtain the following accuracy results for Cifar10: No Dropout $7 6 . 9 1 \\%$ ; Excitation Dropout at conv3 $7 8 . 0 1 \\%$ ; Excitation Dropout at $f c 1 8 1 . 9 4 \\%$ . Again, we observe an improvement respect to No Dropout, but consistent with the literature [Hinton et al. (2012); Srivastava et al. (2014)], the improvement is not as large as using dropout in fully connected layers. ",
+        "bbox": [
+            174,
+            367,
+            825,
+            507
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Validation on Multiple Architectures. In this section we report additional results for Caltech256 and Cifar10 datasets considering different architectures. Table 7 reports the results of fine-tuning the deep architectures AlexNet, VGG16 and VGG19 on Caltech256 for the task of image classification. Finally, we adopt the WideResNet (WRN-28-10) architecture, a ResNet style architecture which combines batch normalization and dropout regularization techniques. WideResNet is used to obtain state-of-the-art results on Cifar10. We replace the Standard Dropout layer in the network with Excitation Dropout obtaining $3 . 8 8 \\%$ test error on Cifar10 (vs. $4 . 1 7 \\%$ Zagoruyko et al.s published result, BMVC16). Therefore, Excitation Dropout gives state-of-the-art result on Cifar10. ",
+        "bbox": [
+            173,
+            513,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/dcece945cce764c937fb8ed1e873d8f04bd56ec9e92806cf8987e426ea9672c0.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>No Dropout (%)</td><td>Standard Dropout (%)</td><td>Curriculum Dropout (%)</td><td>Excitation Dropout (%)</td></tr><tr><td>VGG16</td><td>77.97</td><td>79.11 (+1.14%)</td><td>79.31 (+1.34%)</td><td>79.92 (+1.95%)</td></tr><tr><td>VGG19</td><td>77.98</td><td>79.46 (+1.48%)</td><td>79.66 (+1.68%)</td><td>80.65 (+2.67%)</td></tr><tr><td>AlexNet</td><td>66.48</td><td>68.10 (+1.62%)</td><td>68.71 (+2.23%)</td><td>69.66 (+3.18%)</td></tr></table>",
+        "bbox": [
+            200,
+            637,
+            795,
+            710
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Test accuracy comparison between No, Standard, Curriculum, and Excitation Dropout in the $f c 6$ layer of three architectures: AlexNet, VGG16 and VGG19, fine-tuned for the image recognition task on Caltech256. The numbers reported are the final test accuracies together with the improvements (in parenthesis) with respect to No Dropout, averaged over five trained models. ",
+        "bbox": [
+            173,
+            720,
+            825,
+            776
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Additional Visualizations. In this section, we present more visualizations similar to that of Fig. 4 in the main manuscript. Visualizations for a VGG16 network fine-tuned on UCF101 for the actions: PlayingFlute, PlayingSitar, and GolfSwing are presented in Fig.s 6,7,8, respectively. Every column displays the saliency map over the same video frame of an action while incrementally switching off the most $k$ relevant neurons $( k = 0 , 1 0 0 , 2 0 0 , \\ldots , 5 0 0 )$ . Excitation Dropout is more robust over higher number of switched off neurons. This is demonstrated through its ability to recover the saliency map even when a high percentage of the most salient neurons is dropped-out. This is done through the alternative learnt paths, which are reflected in the higher number of non-zero activations that remain after dropout compared to other dropout strategies. ",
+        "bbox": [
+            173,
+            797,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/055b57f36761045374b24b4e4a4b2d1285b7f8d934ccb246737784c0755cd3a8.jpg",
+        "image_caption": [
+            "Figure 6: Despite the increasing number of relevant neurons being dropped out at test time, Excitation Dropout is capable of restoring more of the saliency map contributing to the specific class: PlayingFlute "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            99,
+            782,
+            449
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Additional Plots. In this section we present the $f c$ compression results for three additional datasets: Cifar10, Cifar100, and Caltech256 (Fig. 9), using CNN-2. We study how the predicted ground-truth (GT) probability changes as more neurons are dropped-out at test time. On the left we present the worst case when the neurons dropped are the most relevant to the prediction. The horizontal axis in the graph represents $p _ { c }$ , where $0 ~ \\leq ~ p _ { c } ~ \\leq ~ 1$ is the cumulative sum of $p _ { E B }$ of neurons which will be switched off starting from the most ‘important’. The analysis is performed for $\\mathit { p _ { c } } =$ $\\{ 0 , 0 . 0 5 , \\hdots , 0 . 9 0 , 0 . 9 5 \\}$ . In the center, we present an analogous analysis starting to drop from the ‘least’ relevant neurons. On the right, we present the random case (more realistic) when $k$ neurons $( k \\ : = \\ : 0 , 1 2 8 , 2 5 6 , \\ldots , 2 0 4 8 )$ are randomly switched off. As we drop more neurons, Excitation Dropout (purple curves) is capable of maintaining a much less steep decline of GT probability. ",
+        "bbox": [
+            173,
+            529,
+            825,
+            667
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Metric Analysis During Training. In this section we report an extended analysis of the metrics: # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ during training. Excitation Dropout shows a higher number of active neurons, a higher entropy over activations, a probability distribution $p _ { E B }$ that is more spread (higher entropy over $p _ { E B } .$ ) among the neurons of the layer, leading to a lower peak probability of $p _ { E B }$ and therefore less specialized neurons. These results are observed to have consistent trends over all training iterations for all datasets considered (see Fig.s 10, 11, 12, and 13) ",
+        "bbox": [
+            173,
+            674,
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+            772
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb0accbe64bc2d329717bcbd3f546cffb09cd66b200f52d0cdbf8b1d5d8990dc.jpg",
+        "image_caption": [
+            "Figure 7: Despite the increasing number of relevant neurons being dropped out at test time, Excitation Dropout is capable of restoring more of the saliency map contributing to the specific class: PlayingSitar "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            102,
+            782,
+            452
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/b60b944e6ef04732af9d663f8e4f5ba75d38f389d8475003393d14663bf20fb2.jpg",
+        "image_caption": [
+            "Figure 8: Despite the increasing number of relevant neurons being dropped out at test time, Excitation Dropout is capable of restoring more of the saliency map contributing to the specific class: GolfSwing "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            517,
+            782,
+            864
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/d426f098174c920b4b0eae13cbe3bedc148dd693ddd5326114b164e5de773e3d.jpg",
+        "image_caption": [
+            "Figure 9: Robustness of predicted ground-truth class probabilities as more neurons are dropped out for each dataset’s test images. We train CNN-2 from scratch with Excitation, Curriculum, Standard, and No Dropout at the $f c 1$ layer, averaging results over five trained models. The standard deviation is depicted around the mean curve using a lighter shade. Left: the most relevant neurons with respect to the $p _ { c }$ threshold are switched off. Center: the least relevant neurons with respect to the $p _ { c }$ threshold are switched off. Right: $k$ neurons are randomly switched off. In all scenarios, Excitation Dropout shows more robustness to network compression (dropping $f c$ neurons $\\equiv$ removing filters). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            234,
+            820,
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+        ],
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+        "img_path": "images/7d2b678f17f563faaba44ab1f82dc9d0810fca08df08728b18cc44fd35a8a1d5.jpg",
+        "image_caption": [
+            "Figure 10: Cifar10. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ over time during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/63d436b008fc255652a6d92c375699f6e0682d163a681d2e8fefc422cef6adff.jpg",
+        "image_caption": [
+            "Figure 11: Cifar100. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ over time during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
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+        "page_idx": 15
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+        "image_caption": [
+            "Figure 12: Caltech256. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of ${ p } _ { E B }$ over time during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "image_caption": [
+            "Figure 13: UCF101. # Neurons ON, Peak $p _ { E B }$ , Entropy of Activations, and Entropy of $p _ { E B }$ over time during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
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+        "page_idx": 16
+    }
+]

parse/train/dvSExzhjG9D/dvSExzhjG9D.md

@@ -0,0 +1,678 @@
+# MLR-SNET: TRANSFERABLE LR SCHEDULES FOR HETEROGENEOUS TASKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The learning rate (LR) is one of the most important hyper-parameters in stochastic gradient descent (SGD) for deep neural networks (DNN) training and generalization. However, current hand-designed LR schedules need to manually pre-specify a fixed form, which limits their ability to adapt to non-convex optimization problems due to the significant variation of training dynamics. Meanwhile, it always needs to search a proper LR schedule from scratch for new tasks. To address these issues, we propose to parameterize LR schedules with an explicit mapping formulation, called MLR-SNet. The learnable structure brings more flexibility for MLR-SNet to learn a proper LR schedule to comply with the training dynamics of DNN. Image and text classification benchmark experiments substantiate the capability of our method for achieving proper LR schedules. Moreover, the meta-learned MLR-SNet is plugand-play to generalize to new heterogeneous tasks. We transfer our meta-trained MLR-SNet to tasks like different training epochs, network architectures, datasets, especially large scale ImageNet dataset, and achieve comparable performance with hand-designed LR schedules. Finally, MLR-SNet can achieve better robustness when training data are biased with corrupted noise.
+
+# 1 INTRODUCTION
+
+Stochastic gradient descent (SGD) and its many variants (Robbins & Monro, 1951; Duchi et al., 2011; Zeiler, 2012; Tieleman & Hinton, 2012; Kingma & Ba, 2015), have been served as the cornerstone of modern machine learning with big data. It has been empirically shown that DNN achieves stateof-the-art generalization performance on a wide variety of tasks when trained with SGD (Zhang et al., 2017). Several recent researches observe that SGD tends to select the so-called flat minima (Hochreiter & Schmidhuber, 1997a; Keskar et al., 2017), which seems to generalize better in practice.
+
+Scheduling learning rate (LR) for SGD is one of the most widely studied aspects to help improve the SGD training for DNN. Specifically, it has been experimentally studied how the LR (Jastrzebski et al., 2017) influences mimima solutions found by SGD. Theoretically, Wu et al. (2018a) analyze that LR plays an important role in minima selection from a dynamical stability perspective. He et al. (2019) provide a PAC-Bayes generalization bound for DNN trained by SGD, which is correlated with LR. In a word, finding a proper LR schedule highly influences the generalization performance of DNN, which has been widely studied recently (Bengio, 2012; Schaul et al., 2013; Nar & Sastry, 2018).
+
+There mainly exist three kinds of hand-designed LR schedules: (1) Pre-defined LR policy is mostly used in current DNN training, like decaying or cyclic LR (Gower et al., 2019; Loshchilov & Hutter, 2017), and brings large improvements in training efficiency. Some theoretical works suggested that the decaying schedule can yield faster convergence (Ge et al., 2019; Davis et al., 2019) or avoid strict saddles (Lee et al., 2019; Panageas et al., 2019) under some mild conditions. (2) LR search methods in tranditional convex optimization (Nocedal & Wright, 2006) can be extended to DNN training by searching LR adaptively in each step, such as Polyak’s update rule (Rolinek & Martius, 2018), Frank-Wolfe algorithm (Berrada et al., 2019), and Armijo line-search (Vaswani et al., 2019), etc. (3) Adaptive gradient methods like Adam (Duchi et al., 2011; Tieleman & Hinton, 2012; Kingma & Ba, 2015), adapt LR for each parameters separately according to some gradient information.
+
+Although above LR schedules (as depicted in Fig. 1(a) and 1(b)) can achieve competitive results on their learning tasks, they still have evident deficiencies in practice. On the one hand, these policies need to manually pre-specify the form of LR schedules, suffering from the limited flexibility to adapt to non-convex optimization problems due to the significant variation of training dynamics. On the other hand, when solving new heterogeneous tasks, it always needs to search a proper LR schedule from scratch, as well as to tune their involving hyper-parameters. This process is time and computation expensive, which tends to further raise their application difficulty in real problems.
+
+![](images/31a8cc68343048639b13c613eb73d12f62289c16caf112acd97302eede998371.jpg)  
+Figure 1: Pre-set LR schedules for (a) image and (b) text classification. (c) Visualization of how we input current loss $\mathcal { L } _ { t }$ to MLR-SNet, which then outputs a proper LR $\alpha _ { t }$ to help SGD find a better minima. LR schedules learned by MLR-SNet on (d) image and (e) text classification. (f) We transfer LR schedules learned on CIFAR-10 to image (CIFAR-100) and text (Penn Treebank) classification, and the subfigure shows the predicted LR during training.
+
+To alleviate the aforementioned issues, this paper presents a model to learn a plug-and-play LR schedule. The main idea is to parameterize the LR schedule as a LSTM network (Hochreiter & Schmidhuber, 1997b), which is capable of dealing with such a long-term information dependent problem. As shown in Fig. 1(c), the proposed Meta-LR-Schedule-Net (MLR-SNet) learns an explicit loss-LR dependent relationship. In a nutshell, this paper makes the following three-fold contributions.
+
+(1) We propose a MLR-SNet to learn an adaptive LR schedule, which can adjust LR based on current training loss as well as the information delivered from past training histories stored in the MLR-SNet. Due to the parameterized form of the MLR-SNet, it can be more flexible than hand-designed policies to find a proper LR schedule for the specific learning task. Fig.1(d) and 1(e) show our learned LR schedules, which have similar tendency as pre-defined policies, but more variations at their locality. This validates the efficacy of our method for adaptively adjusting LR according to training dynamics.
+
+(2) With an explicit parameterized structure, the meta-trained MLR-SNet can be transferred to new heterogeneous tasks (meta-test stage), including different training epochs, network architectures and datasets. Experimental results verify that our plug-and-play LR schedules can achieve comparable performance, while do not have any hyper-parameters compared with tranditional LR schedules. This potentially saves large labor and computation cost in real world applications.
+
+(3) The MLR-SNet is meta-learned to improve generalization performance on unseen data. We validate that with the guidance of clean data, our MLR-SNet can achieve better robustness when training data are biased with corrupted noise than hand-designed LR schedules.
+
+# 2 RELATED WORK
+
+Meta learning for optimization. Meta learning has a long history in psychology (Ward, 1937; Lake et al., 2017). Meta learning for optimization can date back to 1980s-1990s (Schmidhuber, 1992; Bengio et al., 1991), aiming to meta-learn the optimization process of learning itself. Recently, Andrychowicz et al. (2016); Ravi & Larochelle (2017); Chen et al. (2017); Wichrowska et al. (2017); Li & Malik (2017); Lv et al. (2017) have attempted to scale this idea to larger DNN optimization problems. The main idea is to construct a meta-learner as the optimizer, which takes the gradients as input and outputs the whole updating rules. These approaches tend to make selecting appropriate training algorithms, scheduling LR and tuning other hyper-parameters in an automatic way. Except for solving continuous optimization problems, some works employ these ideas to other optimization problems, such as black-box functions (Chen et al., 2017), few-shot learning (Li et al., 2017), model’s curvature (Park & Oliva, 2019), evolution strategies (Houthooft et al., 2018), combinatorial functions (Rosenfeld et al., 2018), etc.
+
+Though faster in decreasing training loss than the traditional optimizers in some cases, the learned optimizers may not always generalize well to diverse problems, especially longer horizons (Lv et al., 2017) and large scale optimization problems (Wichrowska et al., 2017). Moreover, they can not be guaranteed to output a proper descent direction in each iteration for DNN training, since they assume all parameters share one small net and ignore the relationship between each parameters. Our proposed method attempts to learn an adaptive LR schedule rather than the whole update rules. This makes it easy to learn and the meta-learned LR schedule can be transferred to new heterogeneous tasks.
+
+HPO and LR schedule adaptation. Hyper-parameter optimization (HPO) was historically investigated by selecting proper values for algorithm hyper-parameters to obtain better performance on validation set (see (Hutter et al., 2019) for an overview). Typical methods include grid search, random search (Bergstra & Bengio, 2012), Bayesian optimization (Snoek et al., 2012), gradient-based methods (Franceschi et al., 2017; Shu et al., 2020a;b), etc. Recently, some works attempt to find a proper LR schedule under the framework of gradient-based HPO, which can be solved by bilevel optimization (Franceschi et al., 2017; Baydin et al., 2018). However, most HPO techniques tend to fall into short-horizon bias and easily find a bad minima (Wu et al., 2018b). Our MLR-SNet has an explicit function form, which makes the optimization of the LR schedules more robust and effective.
+
+Transfer to heterogeneous tasks. Transfer learning (Pan & Yang, 2009) aims to transfer knowledge obtained from source task to help the learning on the target task. Most transfer learning methods assume the source and target tasks consist of the same instance, feature or model spaces (Yang et al., 2020), which greatly limits their applications. Recently, meta learning (Finn et al., 2017) aims to learn common knowledge shared over a distribution of tasks, such that the learned knowledge can transfer to unseen heterogeneous tasks. Most meta learning approaches focus on few shot learning framework, while we attempt to extend it into a standard learning framwork. The hand-designed LR schedules and HPO methods just try to find a proper LR schedule for given tasks, and need to be learned from scratch for new tasks. However, our meta-learned MLR-SNet is plug-and-play, which can directly transfer how to schedule LR for SGD to heterogeneous tasks without additional learning.
+
+# 3 THE PROPOSED META-LR-SCHEDULE-NET (MLR-SNET) METHOD
+
+The problem of training DNN can be formulated as the following non-convex optimization problem,
+
+$$
+\operatorname* { m i n } _ { w \in \mathbb { R } ^ { n } } \mathcal { L } _ { T r } ( D _ { T r } ; w ) : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { L } _ { i } ^ { T r } ( w ) ,
+$$
+
+where $\mathcal { L } _ { i } ^ { T r }$ is the training loss function for data samples $i \in D _ { T r } = \{ 1 , 2 , \cdot \cdot \cdot , N \}$ , which characters the deviation of the model prediction from the data, and $w \in \mathbb { R } ^ { n }$ represents the parameters of the model (e.g., the weight matrices in DNN) to be optimized. SGD (Robbins & Monro, 1951; Polyak, 1964) and its variants, including Momentum (Tseng, 1998), Adagrad (Duchi et al., 2011), Adadelta (Zeiler, 2012), RMSprop (Tieleman & Hinton, 2012), Adam (Kingma & Ba, 2015), are often used for training DNN. In general, these algorithms can be summarized as the following formulation,
+
+$$
+\boldsymbol { w } _ { t + 1 } = \boldsymbol { w } _ { t } + \Delta \boldsymbol { w } _ { t } , \Delta \boldsymbol { w } _ { t } = \mathcal { O } _ { t } ( \nabla \mathcal { L } ^ { T r } ( \boldsymbol { w } _ { t } ) , \mathcal { H } _ { t } ; \Theta _ { t } ) ,
+$$
+
+where $w _ { t }$ is $t$ -th updating model parameters, $\nabla \mathcal { L } ^ { T r } ( w _ { t } )$ denotes the gradient of $\mathcal { L } ^ { T r }$ at $w _ { t }$ , $\mathcal { H } _ { t }$ represents the historical gradient information, and $\Theta _ { t }$ is the hyperparameter of the optimizer $\mathcal { O }$ , e.g., LR. To present our method’s efficiency, we focus on the following vanilla SGD formulation,
+
+$$
+w _ { t + 1 } = w _ { t } - \alpha _ { t } \left( \frac { 1 } { | B _ { t } | } \sum _ { i \in B _ { t } } \nabla \mathcal { L } _ { i } ^ { T r } ( w _ { t } ) \right) ,
+$$
+
+where $B _ { t } \subset D _ { T r }$ denotes the batch samples randomly sampled from the training dataset, $| B _ { t } |$ denotes the number of the sampled batch samples, and $\nabla \mathcal { L } _ { i } ^ { T \bar { r } } ( w _ { t } )$ denotes the gradient of sample $i$ computed at $w _ { t }$ and $\alpha _ { t }$ is the LR at $t$ -th iteration.
+
+# 3.1 EXISTING LR SCHEDULE STRATEGIES
+
+As Bengio (2012) demonstrated, the choice of LR remains central to effective DNN training with SGD. As mentioned in Section 1, a variety of hand-designed LR schedules have been proposed. Though they achieve competitive results on some learning tasks, they share several drawbacks: (1) The pre-defined LR schedules suffer from the limited flexibility to adapt to the significantly changed training dynamics for the non-convex optimization problems. (2) It needs to be learned from scratch to find a proper LR schedule for the new tasks, which raises their application difficulty in real problems.
+
+![](images/cc4db57ac378fbdd7859b31a87710ae964e5e5a3c7a93329ebf68329dd49205f.jpg)  
+Figure 2: The structure of our proposed MLR-SNet.
+
+![](images/7cbe75f0a3b859c4af0142f056a8a324d3ef343b5b9cfcf45d019d4dca05a919.jpg)  
+Figure 3: (Above) Train loss and (Below) test loss as a function of a point on a random ray starting at the solutions for different methods on CIFAR-100 with ResNet-18.
+
+Inspired by current meta-learning developments (Finn et al., 2017; Shu et al., 2018; 2019), some researches proposed to learn a generic optimizer from data (Andrychowicz et al., 2016; Ravi & Larochelle, 2017; Chen et al., 2017; Wichrowska et al., 2017; Li & Malik, 2017; Lv et al., 2017). The main idea is to learn a meta-learner as the optimizer to guide the learning of the whole updating rules. For example, Andrychowicz et al. (2016) try to replace Eq.(2) with the following formulation,
+
+$$
+\boldsymbol { w } _ { t + 1 } = \boldsymbol { w } _ { t } + g _ { t } , [ g _ { t } , h _ { t + 1 } ] ^ { T } = m ( \nabla _ { t } , h _ { t } ; \phi ) ,
+$$
+
+where $g _ { t }$ is the output of a LSTM net $m$ , parameterized by $\phi$ , whose state is $h _ { t }$ .This strategy can make selecting appropriate training algorithms, scheduling LR and tuning other hyper-parameters in a unified and automatic way. Though faster in decreasing training loss than the traditional optimizers in some cases, the learned optimizer may not always generalize well to more variant and diverse problems, like longer horizons (Lv et al., 2017) and large scale optimization problems (Wichrowska et al., 2017). Moreover, it can not guarantee to output a proper descent direction in each iteration for network training. This tends to further increase their application difficulty in real problems.
+
+Recently, some methods (Franceschi et al., 2017; Baydin et al., 2018) consider the following constrained optimization problem to search the optimal LR schedule $\alpha ^ { * }$ such that the produced models are associated with small validation error,
+
+$$
+\operatorname* { m i n } _ { \alpha = \{ \alpha _ { 0 } , \cdots , \alpha _ { T - 1 } \} } { \mathcal { L } } _ { V a l } ( D _ { V a l } , w _ { T } ) , s . t . w _ { t + 1 } = \phi _ { t } ( w _ { t } , \alpha _ { t } ) , t = 0 , 1 , \cdots , T - 1 ,
+$$
+
+where ${ \mathcal { L } } _ { V a l }$ denotes the validation loss function, $D _ { V a l } = \{ 1 , 2 , \cdots , M \}$ denotes hold-out validation set, $\alpha$ is to-be-solved hyper-parameter, $\phi _ { t } : \mathbb { R } ^ { n } \times \mathbb { R } _ { + } \to \mathbb { R } ^ { n }$ is a stochastic weight update dynamics, like the updating rule in Eq.(2) or the vanilla SGD in Eq.(3), and $T$ is the maximum iteration step. Though achieving comparable results on some tasks with hand-designed LR schedules, they can not directly transfer to new tasks, since they do not have an explict transferable structure form.
+
+# 3.2 PROPOSED META-LR-SCHEDULE-NET (MLR-SNET) METHOD
+
+To address aforementioned issues, the main idea is to design a meta-learner with an explicit mapping formulation to parameterize LR schedules as shown in Fig.1(c), called MLR-SNet. The parameterized structure can bring two benefits: 1) It gives more flexibility to learn a proper LR schedule to comply with the significantly changed training dynamics of DNN; 2) It makes the meta-learned LR schedules be transferable and plug-and-play, which can be applied to new heterogeneous tasks.
+
+Formulation of MLR-SNet. The computational graph of MLR-SNet is depicted in Fig.2(a). Let $\mathcal { A } ( \cdot ; \theta )$ denote the MLR-SNet, and then the updating equation of SGD in Eq.(3) can be rewritten as
+
+$$
+w _ { t + 1 } = w _ { t } - \mathcal { A } ( \mathcal { L } _ { t } ; \theta _ { t } ) \left( \frac { 1 } { | B _ { t } | } \sum _ { i \in B _ { t } } \nabla \mathcal { L } _ { i } ^ { T r } ( w _ { t } ) \right) , \mathcal { L } _ { t } = \frac { 1 } { | B _ { t } | } \sum _ { i \in B _ { t } } \mathcal { L } _ { i } ^ { T r } ( w _ { t } ) ,
+$$
+
+where $\theta _ { t }$ is the parameter of MLR-SNet at $t { \cdot }$ -th iteration $( t = 0 , \cdots , T - 1 )$ . At any iteration steps, $\mathcal { A } ( \cdot ; \theta )$ can learn an explicit loss-LR dependent relationship, such that the net can adaptively predict LR according to the current input loss $\scriptstyle { \mathcal { L } } _ { t }$ , as well as the historical information stored in the net. For every iteration step, the whole forward computation process is (as shown in Fig. 2(b))
+
+$$
+\left( \begin{array} { l } { i _ { t } } \\ { f _ { t } } \\ { o _ { t } } \\ { g _ { t } } \end{array} \right) = \left( \begin{array} { l } { \sigma } \\ { \sigma } \\ { \sigma } \\ { \operatorname { t a n h } } \end{array} \right) W _ { 2 } \left( \begin{array} { l } { \operatorname { R e L U } } \\ { \operatorname { R e L U } } \end{array} \right) W _ { 1 } \left( \begin{array} { l } { h _ { t - 1 } } \\ { \mathscr { L } _ { t } } \end{array} \right) , \begin{array} { l } { c _ { t } = f _ { t } \odot c _ { t - 1 } + i _ { t } \odot g _ { t } } \\ { h _ { t } = o _ { t } \odot \operatorname { t a n h } ( c _ { t } ) } \\ { p _ { t } = \sigma ( W _ { 3 } h _ { t } ) } \\ { \alpha _ { t } = \gamma \cdot p _ { t } } \end{array} ,
+$$
+
+where $i _ { t } , f _ { t } , o _ { t }$ denote the Input, Forget and Output gates, respectively. Different from vanilla LSTM, the input $h _ { t - 1 }$ and the training loss $\mathcal { L } _ { t }$ are preprocessed by a fully-connected layer $W _ { 1 }$ with ReLU activation function. Then it works as LSTM and obtains the output $h _ { t }$ . After that, the predicted value $p _ { t }$ is obtained by a linear transform $W _ { 3 }$ on the $h _ { t }$ with a Sigmoid activation function. Finally, we introduce a scale factor $\gamma ^ { 1 }$ to guarantee the final predicted LR located in the interval of $[ 0 , \gamma ]$ . Albeit simple, this net is known for dealing with such long-term information dependent problem, and thus capable of finding a proper LR schedule to comply with the complex variations of training dynamics.
+
+Meta-Train: adapting to the training dynamics of DNN. The MLR-SNet can be meta-trained to improve the generalization performance on unseen validation data for DNN training as follows:
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } _ { V a l } ( D _ { V a l } , w _ { T } ) , s . t . w _ { t + 1 } = \phi _ { t } ( w _ { t } , \mathcal { A } ( \mathcal { L } _ { t } ; \theta ) ) , t = 0 , 1 , \cdots , T - 1 .
+$$
+
+Now the important question is how to efficiently meta-learn the parameter $\theta$ for the MLR-SNet. We employ the online approximation technique in (Shu et al., 2019) to jointly update $\theta$ and model parameter $w$ to explore a proper LR schedule with better generalization for DNNs training. However, the step-wise optimization for $\theta$ is still expensive to handle large-scale datasets and DNN. Furthermore, we attempt to update $\theta$ after updating $w$ several steps $( T _ { v a l } )$ as summarized in Algorithm 1.
+
+Updating $\theta$ . When it does not satisfy the updating conditions, $\theta$ is fixed; otherwise, $\theta$ is updated using the model parameter $w _ { t }$ and MLR-SNet parameter $\theta _ { t }$ obtained in the last step by minimizing the validation loss defined in Eq.(8). Adam can be employed to optimize the validation loss, i.e.,
+
+$$
+\begin{array} { r } { \theta _ { t + 1 } = \theta _ { t } + A d a m ( \nabla _ { { \theta } } \mathcal { L } _ { V a l } ( D _ { m } , \hat { w } _ { t + 1 } ( { \theta } ) ) ; \eta _ { t } ) , } \end{array}
+$$
+
+where Adam denotes the Adam algorithm, whose input is the gradient of validation loss with respect to MLR-SNet parameter $\theta$ on $m$ mini-batch samples $D _ { m }$ from $D _ { V a l }$ . $\eta _ { t }$ denotes the LR of Adam. $\hat { w } _ { t + 1 } ( \theta ) ^ { 2 }$ is formulated on a mini-batch training samples $D _ { n }$ from $D _ { T r }$ as follows:
+
+$$
+\boldsymbol { \hat { w } } _ { t + 1 } ( \theta ) = \boldsymbol { w } _ { t } - \boldsymbol { \mathcal { A } } ( \boldsymbol { \mathcal { L } } _ { T r } ( D _ { n } , \boldsymbol { w } _ { t } ) ; \theta ) \cdot \nabla _ { \boldsymbol { w } } \boldsymbol { \mathcal { L } } _ { T r } ( D _ { n } , \boldsymbol { w } ) \big | _ { \boldsymbol { w } _ { t } } .
+$$
+
+Updating $w$ . Then, the updated $\theta _ { t + 1 }$ is employed to ameliorate the model parameter $w$ , i.e.,
+
+$$
+w _ { t + 1 } = w _ { t } - { \cal A } ( { \cal L } _ { T r } ( D _ { n } , w _ { t } ) ; \theta _ { t + 1 } ) \cdot \nabla _ { w } { \cal L } _ { T r } ( D _ { n } , w ) \big | _ { w _ { t } } .
+$$
+
+The whole meta-train learning algorithm can be summarized in Algorithm 1. All computations of gradients can be efficiently implemented by automatic differentiation libraries, like PyTorch (Paszke et al., 2019), and generalized to any DNN architectures. It can be seen that the MLR-SNet can be gradually optimized during the learning process and adjust the LR dynamically based on the training dynamics of DNNs.
+
+# Algorithm 1 The Meta-Train Algorithm of MLR-SNet
+
+Input: Training data $D _ { T r }$ , validation set $D _ { V a l }$ , batch size $n , m$ max iterations $T$ , updating period $T _ { v a l }$ .
+
+Output: Model parameter $w _ { T }$ and MLR-SNet parameter $\theta _ { T }$
+
+1: Initialize model parameter $_ { w _ { 0 } }$ and MLR-SNet parameter $\theta _ { 0 }$ .   
+2: for $t = 0$ to $T - 1$ do   
+3: $D _ { n } \gets .$ SampleMiniBatch $( D _ { T r } , n )$ .   
+4: if $t \% T _ { v a l } = 0$ , then   
+5: $D _ { m } \gets$ SampleMiniBatch $( D _ { V a l } , m )$ .   
+6: Update $\theta _ { t + 1 }$ by Eq. (9).   
+7: end if   
+8: Update $w _ { t + 1 }$ by Eq. (11).   
+9: end for
+
+Meta-Test: transferring to heterogeneous tasks. When we obtain the meta-learned MLR-SNet, it can be easily applied to new tasks. Now the new model parameter $u$ for the new task is updated by,
+
+$$
+\begin{array} { r } { u _ { t + 1 } = u _ { t } - \mathcal { A } ( \mathcal { L } _ { T r } ( D _ { n } , u _ { t } ) ; \theta ^ { * } ) \cdot \nabla _ { u } \mathcal { L } _ { T r } ( D _ { n } , u ) \big | _ { u _ { t } } , } \end{array}
+$$
+
+where $\theta ^ { * }$ is the parameter of the meta-learned MLR-SNet, which is fixed in the meta-test stage.
+
+# 4 EXPERIMENTAL RESULTS
+
+To evaluate the proposed MLR-SNet, we firstly conduct experiments to show our method is capable of finding proper LR schedules compared with baseline methods. Then we transfer the learned LR schedules to various tasks to show its superiority in generalization. Finally, we show our method behaves robust and stable when training data contain different data corruptions.
+
+1As we know that the performance of hand-designed LR schedules and HPO methods is very sensitive to the initial LR. To avoid carefully tuning the initial LR, we learn the LR schedules from an interval $[ 0 , \gamma ]$ , and now the initial LR is determined by the output of the MLR-SNet. We set $\gamma = 1$ for image tasks, and $\gamma = 4 0$ for text tasks in all our experiments to eliminate the influence of loss magnitude between two different tasks.
+
+2Notice that $\hat { w } _ { t + 1 } ( \boldsymbol { \theta } )$ here is a function of $\theta$ to guarantee the gradient in Eq.(9) to be able to compute.
+
+![](images/b56b3121979dc0d2d7d950c4c6d3be695f7608d2bcf3bc8f3534dc1c249c0a1f.jpg)  
+Figure 5: Test accuracy on CIFAR-100 of ResNet-18 with varying epochs.
+
+# 4.1 META-TRAIN: EVALUATION ON THE LR SCHEDULE LEARNED BY MLR-SNET
+
+Datasets and models. To verify general effectiveness of our method, we respectively train different models on four benchmark data, including ResNet-18 (He et al., 2016) on CIFAR-10, WideResNet28-10 (Zagoruyko & Komodakis, 2016) on CIFAR-100 (Krizhevsky, 2009), 2-layer LSTM and 3-layer LSTM on Penn Treebank (Marcus & Marcinkiewicz).
+
+Baselines. For image classification tasks, the compared methods include SGD with hand-designed LR schedules: 1) Fixed LR, 2) Exponential decay, 3) MultiStep decay, 4) SGD with restarts (SGDR) (Loshchilov & Hutter, 2017). Also, we compare with SGD with Momentum (SGDM) with above four LR schedules. The momentum is fixed as 0.9. Meanwhile, we compare with adaptive gradient method: 5)Adam, LR search method: 6) L4 (Rolinek & Martius, 2018), and current LR schedule adaptation methods: 7) hyper-gradient descent (HD) (Baydin et al., 2018), 8) real-time hyper-parameter optimization (RTHO) (Franceschi et al., 2017). For text classification tasks, we compare with 1) SGD and 2) Adam with LR tuned using a validation set. They drop the LR by a factor of 4 when the validation loss stops decreasing. Also, we compared with 3) L4, 4) HD, 5) RTHO. We run all experiments with 3 different seeds reporting accuracy. The detailed illustrations of experimental setting, and more experimental results are presented in Appendix B.
+
+Image tasks. Fig.4(a) and 4(b) show the classification accuracy on CIFAR-10 and CIFAR-100 test sets, respectively. It can be observed that: 1) our algorithm outperforms all other competing methods, and the learned LR schedules by MLR-SNet are presented in Fig.1(d), which have similar shape as the hand-designed policies, while with more elaborate variation details in locality for adapting training dynamics. 2) The Fixed LR has similar performance to other baselines at the early training, while falls into fluctuations at the later training. This implies that the Fixed LR can not finely adapt to such DNN training dynamics. 3) The MultiStep LR drops the LR at some epochs, and such elegant strategy overcomes the issue of Fixed LR and obtains higher and stabler performance at the later training. 3) The Exponential LR improves test performance faster at the early training than other baselines, while makes a slow progress due to smaller LR at the later training. 4) SGDR uses the cyclic LR, which needs more epochs to obtain a stable result. 5) Though Adam has an adaptive coordinate-specific LR, it behaves worse than MultiStep and Exponential LR as demonstrated in Wilson et al. (2017). An extra tuning is necessary for better performance. 6) L4 greedily searches LR locally to decrease loss, while the complex DNN training dynamics can not guarantee it to obtain a good minima. 7) HD and RTHO are able to achieve similar performance to hand-designed LR schedules. The LR schedules learned by L4, HD and RTHO can be found in supplementary material.
+
+![](images/ebad36202f0640a1ecc3101bc5245d6990d1def948fb5a5b16aef35f3f181db0.jpg)
+
+![](images/f98fbcd6f6975c7e95bded0f42d2a7b92e666a0e3c08317e77a4a113eeea219f.jpg)  
+Figure 6: Test accuracy of transferred LR schedules on different datasets.   
+Figure 7: Test accuracy on CIFAR-10 of different network architectures
+
+Since image tasks often use SGDM to train DNNs, Fig.4(d) and 4(e) show the results of baseline methods trained with SGDM, and they obtain a remarkable improvement than SGD. Though not using extra historical gradient information to help optimization, our method achieves comparable results with baselines by finding a proper LR schedule for SGD.
+
+Text tasks. Fig.4(c) and 4(f) show the test perplexity on the Penn Treebank with 2-layer and 3-layer LSTM, respectively. Adam and SGD heuristically drops LR when the validation loss stops decreasing. However, our MLR-SNet predicts LR according to training dynamics by minimizing the validation loss, which is a more intelligent way to employ the validation dataset. Thus our method achieves comparable or even better performance than Adam and SGD. The learned LR schedules of the MLR-SNet are presented in Fig.1(b), which have similar shape as the hand-designed policies. L4 often falls into a bad minima since it greedily searches LR locally. HD and RTHO directly optimize LR to improve the performance on validation dataset, obtaining the similar results as Adam and SGD. With an explicit strcuture, our method behaves more robust and efficient than HD and RTHO.
+
+Remark. Actually, the performance of the hand-design LR schedules can be regarded as the best/ upper performance bound. Since these strategies have been tested to work well for the specific tasks, and they are written into the standard deep learning library. For different image and text tasks, our MLR-SNet can achieve the similar or even a little better performance compared with the best baselines, demostrating the effectiveness and generality of our method.
+
+# 4.2 META-TEST: TRANSFERABILITY OF PLUG-AND-PLAY LR SCHEDULES
+
+The learned MLR-SNet is transferable and plug-and-play. Here we validate if MLR-SNet can transfer to new heterogeneous tasks. Since the methods L4,HD,RTHO in Section 4.1 are not able to generalize, we do not compare them here. Actually our results show superiority on image tasks beyond baseline methods when trained with SGD, here we present stronger baseline results in which compared methods are trained with SGDM. We use the MLR-SNet meta-learned on CIFAR-10 with ResNet-18 in Section 4.1 as the plug-and-play LR schedules for the following experiments.
+
+Transfer to different epochs. The plug-and-play MLR-SNet is meta-trained with epoch 200, and we transfer it to other different training epochs, e.g., 100, 400,1200. As shown in Fig.5, our MLR-SNet has the ability to train for longer horizons and achieves almost same performance as MultiStep LR. The slight shakes for epoch 1200 may due to that our MLR-SNet can learn the LR similar to SGDR locally. The Exponential LR has a little performance decreased for the longer epochs.
+
+Transfer to different datasets. We transfer the LR schedules meta-learned on CIFAR-10 to SVHN (Netzer et al., 2011), TinyImageNet 3, and Penn Treebank (Marcus & Marcinkiewicz). As shown in Fig.6, though datasets vary from image to text, our method can still obtain a relatively stable and comparable generalization performance for different tasks with baseline method.
+
+Transfer to different net architectures. We also transfer the LR schedules meta-learned on ResNet18 to light-weight nets ShuffleNetV2 (Ma et al., 2018), MobileNetV2 (Sandler et al., 2018) or NASNet (Zoph et al., 2018) 4. As shown in Fig.7, our method achieves almost similar results to SGDM with MultiStep or Exponential LR.
+
+![](images/4a809d823e724c41fa0ba6ec8995a200ac7b5432d3541f248fa710dffe352a3b.jpg)  
+Figure 8: Test accuracy on ImageNet with ResNet-50.
+
+![](images/9411d199054842d27b0cb63efc5b5322420f49a6aef638d498450bbc927c3b19.jpg)  
+Figure 9: Comparion of different meta-learners.
+
+![](images/316da9a019decee5ab2d9cfddec778d2dbebcf9fbc24252939fbd46f4ab25086.jpg)  
+Figure 10: Time consuming of different LR schedules methods.
+
+Table 1: Test accuracy $( \% )$ on CIFAR-10 and CIFAR-100 training set of different methods trained on CIFAR-10-C and CIFAR-100-C. Best and Last denote the results of the best and the last epoch.   
+
+<table><tr><td colspan="2">Datasets/Methods</td><td>Fixed</td><td>MultiStep</td><td>Exponential</td><td>SGDR</td><td>Adam</td><td>Ours(Train)</td></tr><tr><td rowspan="2">CIFAR-10-C</td><td>Best</td><td>79.78±3.95</td><td>85.52±1.72</td><td>83.48±1.45</td><td>85.94±1.52</td><td>81.45±1.42</td><td>86.04±1.51</td></tr><tr><td>Last</td><td>77.88±3.91</td><td>85.36±1.71</td><td>83.32±1.43</td><td>78.21±2.01</td><td>80.29±1.64</td><td>85.87±1.54</td></tr><tr><td rowspan="2">CIFAR-100-C</td><td>Best</td><td>46.74±3.03</td><td>52.26±2.58</td><td>49.72±1.97</td><td>52.54±2.49</td><td>45.45±1.94</td><td>52.56±2.26</td></tr><tr><td>Last</td><td>44.79±3.91</td><td>52.16±2.59</td><td>49.58±1.98</td><td>41.58±3.24</td><td>43.76±2.22</td><td>52.42±2.34</td></tr></table>
+
+Transfer to large scale optimization problem. To our best knowledge, only Wichrowska et al. (2017) attempted to use the learned optimizers to train DNN on ImageNet dataset (Deng et al., 2009) among existing learning-to-optimize literatures. However, it can only be executed for thousands of steps, and then its loss begins to increase dramatically, far from the optimization process in practice. We transfer the LR schedule meta-trained on CIFAR-10 with ResNet-18 to ImageNet dataset with ResNet-50 5. As shown in Fig.8, the validation accuracy of our method is competitive with those hand-designed LR schedules methods. This implies our method is capable of dealing with such large scale optimization problem, making learning-to-optimize ideas towards more practical applications.
+
+# 4.3 ROBUSTNESS ON DIFFERENT DATA CORRUPTIONS
+
+In this section, we validate that whether our MLR-SNet behaves robust against corrupted training data. To this aim, we design experiments as follows: we take CIFAR-10-C and CIFAR-100-C (Hendrycks & Dietterich, 2019) as our training set, consisting of 15 types of different generated corruptions on test images data of CIFAR-10/CIFAR-100, and the original training set of CIFAR-10/100 as test set. Though the original images of CIFAR-10/100-C are the same with the CIFAR-10/100 test set, different corruptions have changed the data distributions. To guarantee the calculated models finely generalize to test set, we choose the validation set as 10 clean images for each class. Each corruption can be roughly regarded as a task, and thus we obtain 15 models trained on CIFAR-10/100-C. Table 1 shows the mean test accuracy of 15 models ( $\pm$ std). As can be seen, our proposed MLR-SNet is capable of achieving better generalization performance on clean test data than baseline methods, which implies that our method behaves more robust and stable than the pre-set LR schedules when the learning tasks are changed. This is due to that our MLR-SNet has more flexibility to adapt the variation of the data distribution than the pre-set LR schedules, and it can find a proper LR schedule through minimizing the generalization error which is based on the knowledge specifically conveyed from the given validation data. The detailed illustrations of experimental setting, and the transferrablity experiment of meta-learned MLR-SNet are presented in supplementary material.
+
+# 5 SOME ANALYSIS OF MLR-SNET
+
+# 5.1 CONVERGENCE ANALYSIS OF MLR-SNET
+
+The preliminary experimental evaluations show that our method gives good convergence performance on various tasks. We find that the meta-learned LR schedules in our experiments follow a consistent trajectory as shown in Fig.1, sharing a similar tendency as the Exponential LR schedules. To provide a theoretical convergence analysis, we roughly assume that the LR predicted by MLR-SNet obey a Exponential LR form. The convergence analysis for DNN training can refer to (Li et al., 2020). Here, we provide a convergence analysis of the MLR-SNet training. The proof is listed in the Appendix A.
+
+Theorem 1 Suppose the loss function $\ell$ is Lipschitz smooth with respect to the model parameter $w$ with constant $L$ , and have $\rho$ -bounded gradients with respect to training/validation data. And the $\mathcal A ( \theta )$ is differential with a $\delta$ -bounded gradient and twice differential with its Hessian bounded by $\boldsymbol { B }$ . Let the learning rate $\alpha _ { t } = \mathcal { A } ( \theta _ { t } )$ predicted by MLR-SNet obey the exponential LR, i.e., $\alpha _ { t } = \alpha _ { 0 } \beta ^ { t } , \beta = ( \Gamma / T ) ^ { 1 / T } , \Gamma \geq 1 .$ . Let $\eta _ { t } = \eta$ for all $t \in [ T ]$ . If we use Adam algorithm to update MLR-SNet, we choose $\eta$ satisfied $\eta \leq \frac { \epsilon } { 2 L }$ and $\begin{array} { r } { 1 - \beta _ { 2 } \le \frac { \epsilon ^ { 2 } } { 1 6 \rho ^ { 2 } } } \end{array}$ , where $\beta _ { 2 }$ , $\epsilon$ are the hyperparameter of the Adam algorithm. Then for $\theta _ { t }$ generated using Adam, we have the following bound:
+
+$$
+\operatorname* { m i n } _ { 0 \leq t \leq T } \mathbb { E } [ \| \nabla \mathcal { L } _ { V a l } ( \hat { \mathbf { w } } _ { t } ( \theta _ { t } ) ) \| _ { 2 } ^ { 2 } ] \leq \mathcal { O } ( \frac { C \ln ( T ) } { T } + \sigma ^ { 2 } ) ,
+$$
+
+where $C$ is some constant independent of the convergence process, $\sigma$ is the variance of drawing uniformly mini-batch sample at random.
+
+# 5.2 THE STRUCTURE OF THE MLR-SNET
+
+We regard the LR scheduling as a long-term information dependent problem, and thus we parameterize the LR schedule as an LSTM network. As we known, MLP network can also learn an explicit mapping but ignore the temporal information. Here, we compare the performance of the two types of metalearners. As shown in Fig. 9, in the early training stage, both of them achieve the similar performance. While at the later training stage, the LSTM meta-learner brings a notable performance increase compared with MLP meta-learner. This may due to that the accumulated temporal information of the LSTM meta-learner can help find a more proper LR for such DNNs training.
+
+# 5.3 COMPUTATIONAL COMPLEXITY ANALYSIS
+
+In the meta-training stage, our MLR-SNet learning algorithm can be roughly regarded as requiring two extra full forward and backward passes of the network (step 6 in algorithm 1) in the presence of the normal network parameters update (step 8 in algorithm 1), together with the forward passes of MLR-SNet for every LR. Therefore compared to normal training, our method needs about $3 \times$ computation time for one iteration. Since we periodically update MLR-SNet after several iterations, this will not substantially increase the computational complexity compared with normal network training. In the meta-test stage, our transferred LR schedules predict LR for each iteration by a small MLR-SNet, whose computational cost should be significantly less than the cost of the normal network training. To empirically show the differences between hand-designed LR schedules and our method, we conduct experiments with ResNet-18 on CIFAR-10 and report the running time for all methods. All experiments are implemented on a computer with Intel Xeon(R) CPU E5-2686 v4 and a NVIDIA GeForce RTX 2080 8GB GPU. We follow the corresponding settings in Section 4.1, and results are shown in Figure 10. Except that RTHO costs significantly more time, other methods including MLR-SNet training and testing have similar time consuming. Our MLR-SNet takes barely longer time to complete the meta-training and meta-testing phase compared to hand-designed LR schedules. Therefore our method is completely capable of practical application.
+
+# 6 CONCLUSION AND DISCUSSION
+
+In this paper, we have proposed to learn an adaptive and transferrable LR schedule in a meta learning manner. To this aim, we design an LSTM-type meta-learner (MLR-SNet) to parameterize LR schedules, which gives more flexibility to adaptively learn a proper LR schedule to comply with the significantly complex training dynamics of DNN. Meanwhile, the meta-learned LR schedules are plug-and-play and transferrable, which can be transferred how to schedule LR for SGD to new heterogeneous tasks. Comprehensive experiments substantiate the superiority of our method on various image and text benchmarks in its adaptability, transferrability and robustness, as compared with current LR schedules policies. The MLR-SNet is highly practical as it requires negligible increase in the parameter size and computation time, and no transferrable cost for new tasks. We believe our proposed method has a potential to become a new tool to study how to design LR schedules to help improve current DNN training, as well as more practical applications.
+
+Recently, Keskar et al. (2017); Dinh et al. (2017) suggested that the width of a local optimum is related to generalization. Wider optima leads to better generalization. We use the visualization technique in (Izmailov et al., 2018) to visualize the "width" of the solutions for different LR schedules on CIFAR-100 with ResNet-18. As shown in Fig.3, our method lies a wide flat region of the train loss. This could explain the better generalization of our method compared with pre-set LR schedules. Deeper understandings on this point will be further investigated.
+
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+# A CONVERGENCE ANALYSIS OF THE MLR-SNET
+
+Lemma 1 Suppose the loss function $\ell$ is Lipschitz smooth with respect to the model parameter w with constant $L$ , and have $\rho$ -bounded gradients with respect to training/validation data. And the $\mathcal { A } ( \theta )$ is differential with a $\delta$ -bounded gradient and twice differential with its Hessian bounded by $\boldsymbol { B }$ . Then the gradient of MLR-SNet parameter $\theta$ with respect to loss is Lipschitz smooth.
+
+Proof The gradient of MLR-SNet parameter $\theta$ with respect to loss
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \theta } \ell _ { j } \big ( \hat { w } _ { t } ( \theta ) \big ) \vert _ { \theta _ { t } } = \frac { \partial \ell _ { j } \big ( \hat { w } _ { t } ( \theta ) \big ) } { \partial \hat { w } _ { t } ( \theta ) } \frac { \partial \hat { w } _ { t } ( \theta ) } { \partial A ( \theta ) } \frac { \partial A ( \theta ) } { \partial \theta } } \\ { \displaystyle = \frac { - \alpha _ { t } } { n } \sum _ { i = 1 } ^ { n } \left( \frac { \partial \ell _ { j } \big ( \hat { w } _ { t } ( \theta ) \big ) } { \partial \hat { w } _ { t } ( \theta ) } \frac { \partial \ell _ { i } \big ( w _ { t } \big ) } { \partial w _ { t } } \right) \frac { \partial A ( \theta ) } { \partial \theta } \vert _ { \theta _ { t } } , } \end{array}
+$$
+
+Let $\begin{array} { r } { G _ { i j } = \frac { \partial \ell _ { j } \left( \hat { w } _ { t } ( \theta ) \right) } { \partial \hat { w } _ { t } ( \theta ) } \frac { \partial \ell _ { i } ( w _ { t } ) } { \partial w _ { t } } } \end{array}$ , and then take gradient of $\theta$ in both sides of above equallity, we have
+
+$$
+\nabla _ { \theta ^ { 2 } } ^ { 2 } \ell _ { j } ( \hat { w } _ { t } ( \theta ) ) | _ { \theta _ { t } } = \frac { - \alpha _ { t } } { n } \sum _ { i = 1 } ^ { n } \left[ \frac { \partial G _ { i j } } { \partial \theta } \frac { \partial \mathcal { A } ( \theta ) } { \partial \theta } + G _ { i j } \frac { \partial \mathcal { A } ^ { 2 } ( \theta ) } { \partial \theta ^ { 2 } } \right] .
+$$
+
+For the first term in the right hand side, we have that
+
+$$
+\begin{array} { r l } { \displaystyle \left\| \frac { \partial G _ { i j } } { \partial \theta } \frac { \partial A ( \theta ) } { \partial \theta } \right\| \leq \delta \left\| \frac { \partial \ell _ { j } ( \hat { w } _ { t } ( \theta ) ) } { \partial \hat { w } _ { t } ( \theta ) \partial \theta } \frac { \partial \ell _ { i } ( w _ { t } ) } { \partial w _ { t } } \right\| } & { } \\ { \displaystyle } & { = \delta \left\| \frac { \partial } { \partial \hat { w } _ { t } ( \theta ) } \left( \frac { - \alpha _ { t } } { n } \sum _ { i = 1 } ^ { n } \left( \frac { \partial \ell _ { j } ( \hat { w } _ { t } ( \theta ) ) } { \partial \hat { w } _ { t } ( \theta ) } \frac { \partial \ell _ { i } ( w _ { t } ) } { \partial w _ { t } } \right) \frac { \partial A ( \theta ) } { \partial \theta } | _ { \theta _ { t } } \right) \frac { \partial \ell _ { i } ( w _ { t } ) } { \partial w _ { t } } \right\| } \\ { \displaystyle } & { = \delta \left\| \left( \frac { - \alpha _ { t } } { n } \sum _ { i = 1 } ^ { n } \left( \frac { \partial ^ { 2 } \ell _ { j } ( \hat { w } _ { t } ( \theta ) ) } { \partial \hat { w } _ { t } ^ { 2 } ( \theta ) } \frac { \partial \ell _ { i } ( w _ { t } ) } { \partial w _ { t } } \right) \frac { \partial A ( \theta ) } { \partial \theta } | _ { \theta _ { t } } \right) \frac { \partial \ell _ { i } ( w _ { t } ) } { \partial w _ { t } } \right\| } \\ { \displaystyle } & { \leq \alpha _ { t } L \rho ^ { 2 } \delta ^ { 2 } . } \end{array}
+$$
+
+For the second term in the right hand side, we have that
+
+$$
+\left. G _ { i j } \frac { \partial \mathcal { A } ^ { 2 } ( \theta ) } { \partial \theta ^ { 2 } } \right. \leq B \rho ^ { 2 }
+$$
+
+Combining the above two inequalities Eq.(15)(16), we have
+
+$$
+\| \nabla _ { \theta } \ell _ { j } ( \hat { w } _ { t } ( \theta ) ) | _ { \theta _ { t } } \| \leq \alpha \rho ^ { 2 } ( \alpha _ { t } L \delta ^ { 2 } + \mathcal { B } ) .
+$$
+
+Define $L _ { A } = \alpha \rho ^ { 2 } ( \alpha _ { t } L \delta ^ { 2 } + B )$ , and based on the Lagrange mean value theorem, we have:
+
+$$
+\| \nabla \mathcal { L } _ { V a l } ( \hat { \mathbf { w } } _ { t } ( \theta _ { 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { \mathbf { w } } _ { t } ( \theta _ { 2 } ) ) \| \le L _ { A } \left\| \theta _ { 1 } - \theta _ { 2 } \right\| .
+$$
+
+Thus the conclusion holds.
+
+Theorem 2 Suppose the loss function $\ell$ is Lipschitz smooth with respect to the model parameter $w$ with constant $L$ , and have $\rho$ -bounded gradients with respect to training/validation data. And the $\mathcal A ( \theta )$ is differential with a $\delta$ -bounded gradient and twice differential with its Hessian bounded by $\boldsymbol { B }$ . Let the learning rate $\alpha _ { t } = \mathcal { A } ( \theta _ { t } )$ predicted by MLR-SNet obey the exponential LR, i.e., $\alpha _ { t } = \alpha _ { 0 } \beta ^ { t } , \beta = ( \Gamma / T ) ^ { 1 / T } , \Gamma \geq 1 .$ . Let $\eta _ { t } = \eta$ for all $t \in [ T ]$ . If we use Adam algorithm to update MLR-SNet, we choose $\eta$ satisfied $\eta \leq \frac { \epsilon } { 2 L }$ and $\begin{array} { r } { 1 - \beta _ { 2 } \le \frac { \epsilon ^ { 2 } } { 1 6 \rho ^ { 2 } } } \end{array}$ , where $\beta _ { 2 }$ ,  are the hyperparameter of the Adam algorithm. Then for $\theta _ { t }$ generated using Adam, we have the following bound:
+
+$$
+\operatorname* { m i n } _ { 0 \leq t \leq T } \mathbb { E } [ \| \nabla \mathcal { L } _ { V a l } ( \hat { \mathbf { w } } _ { t } ( \theta _ { t } ) ) \| _ { 2 } ^ { 2 } ] \leq \mathcal { O } ( \frac { C \ln ( T ) } { T } + \sigma ^ { 2 } ) ,
+$$
+
+where $C$ is some constant independent of the convergence process, $\sigma$ is the variance of drawing uniformly mini-batch sample at random.
+
+Proof Suppose we have a small validation set with $M$ samples $\{ x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { M } \}$ , each associating with a validation loss function $\ell _ { i } ( w ( \theta ) )$ , where $w$ is the parameter of the model, and $\theta$ is the parameter of the MLR-SNet. The overall validation loss would $b e$ ,
+
+$$
+\mathcal { L } _ { V a l } ( w ) = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \ell _ { i } ( w ( \theta ) ) .
+$$
+
+According to the updating algorithm $^ { l }$ , we have:
+
+$$
+\begin{array} { r l } & { \quad \mathcal { L } _ { V a l } ( \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) } \\ & { = \underbrace { \{ \mathcal { L } _ { V a l } ( \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) \} } _ { ( a ) } + \underbrace { \{ \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) \} } _ { ( b ) } } \end{array}
+$$
+
+For term $( a )$
+
+$$
+\begin{array} { r l r } {  { \mathcal { L } _ { V a l } ( \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) } } \\ & { \leq \langle \nabla _ { w } \mathcal { L } _ { V a l } ( \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) ) , \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) - \hat { w } _ { t } ( \theta _ { t + 1 } ) \rangle + \displaystyle \frac { L } { 2 } \| \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) - \hat { w } _ { t } ( \theta _ { t + 1 } ) \| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+According to Eq $( 6 )$ , we have
+
+$$
+\hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) - \hat { w } _ { t } ( \theta _ { t + 1 } ) = - \alpha _ { t } \nabla _ { w } \mathcal { L } _ { T r } ^ { B } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) )
+$$
+
+where $\begin{array} { r } { \alpha _ { t } = \mathcal { A } ( \mathcal { L } _ { T r } ^ { B } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ; \theta _ { t } ) , \mathcal { L } _ { T r } ^ { B } ( w _ { t } ) = \frac { 1 } { | B _ { t } | } \sum _ { i \in B _ { t } } \nabla \mathcal { L } _ { i } ^ { T r } ( w _ { t } ) . } \end{array}$ . This can be written as
+
+$$
+\begin{array} { r } { \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) - \hat { w } _ { t } ( \theta _ { t + 1 } ) = - \alpha _ { t } \left[ \nabla _ { w } \mathcal { L } _ { T r } \big ( \hat { w } _ { t } ( \theta _ { t + 1 } ) \big ) + \xi ^ { ( t ) } \right] , } \end{array}
+$$
+
+where $\xi ^ { ( t ) } = \nabla _ { w } \mathcal { L } _ { T r } ^ { B } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) - \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) )$ . Since $B _ { t }$ is the mini-batch samples drawn uniformly from the entire data set, we have $\mathbb { E } [ \xi ^ { ( t ) } ] = 0$ . Furthermore, $\xi ^ { ( t ) }$ are i.i.d random variable with finite variance, since $B _ { t }$ are drawn i.i.d with a finite number of samples. Then Eq (22) can be written as
+
+$$
+\begin{array} { r l } & { \iota \leq \Big \langle \nabla _ { w } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) , - \alpha _ { t } \left[ \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) + \xi ^ { ( t ) } \right] \Big \rangle + \frac { L } { 2 } \left\| - \alpha _ { t } \left[ \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) + \xi ^ { ( t ) } \right] \right\| } \\ & { = \Big \langle \nabla _ { w } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) , - \alpha _ { t } \left[ \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) + \xi ^ { ( t ) } \right] \Big \rangle } \\ & { + \frac { L \alpha _ { t } ^ { 2 } } { 2 } \left[ \| \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) \| ^ { 2 } + \| \xi ^ { ( t ) } \| _ { 2 } ^ { 2 } - \langle \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) , \xi ^ { ( t ) } \rangle \right] } \\ & { \leq \Big \langle \nabla _ { w } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) , - \alpha _ { t } \left[ \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) + \xi ^ { ( t ) } \right] \Big \rangle + \frac { L } { 2 } \alpha _ { t } ^ { 2 } \left[ \rho ^ { 2 } + \| \xi ^ { ( t ) } \| _ { 2 } ^ { 2 } - \langle \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) \rangle \right] } \end{array}
+$$
+
+For term $( b )$ , according to Lemma $^ { l }$ , i.e., the validation loss is Lipschitz smooth with respect to the MLR-SNet parameter $\theta$ , for briefly denote $L$ .
+
+$$
+\begin{array} { r l } & { \quad \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) } \\ & { \quad \le \langle \nabla _ { \theta } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) , \theta _ { t + 1 } - \theta _ { t } \rangle + \displaystyle \frac { L } { 2 } \left\| \theta _ { t + 1 } - \theta _ { t } \right\| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+If we adopt Adam to update the parameter of MLR-SNet, $\theta _ { t + 1 } - \theta _ { t }$ in Eq.(25) is updated by
+
+$$
+\theta _ { t + 1 } = \theta _ { t } - \eta _ { t } \frac { g _ { t , i } } { \sqrt { v _ { t , i } } + \epsilon } ,
+$$
+
+where $g _ { t , i } = \nabla _ { \boldsymbol { \theta } } \mathcal { L } _ { V a l } ^ { i } ( \hat { w } _ { t } ( \boldsymbol { \theta } _ { t } ) )$ . Now, we have
+
+$$
+\begin{array} { l } { { \displaystyle { \mathcal { L } } _ { V a l } \big ( \hat { w } _ { t } ( \theta _ { t + 1 } ) \big ) - { \mathcal { L } } _ { V a l } \big ( \hat { w } _ { t } ( \theta _ { t } ) \big ) } \ ~ } \\ { { \displaystyle \leq - \eta _ { t } \sum _ { i = 1 } ^ { d } \bigg \langle \nabla _ { \theta } { \mathcal L } _ { V a l } ^ { i } \big ( \hat { w } _ { t } ( \theta _ { t } ) \big ) , \frac { g _ { t , i } } { \sqrt { v _ { t , i } } + \epsilon } \bigg \rangle + \frac { L \eta _ { t } ^ { 2 } } { 2 } \sum _ { i = 1 } ^ { d } \frac { g _ { t , i } ^ { 2 } } { ( \sqrt { v _ { t , i } } + \epsilon ) ^ { 2 } } } } \end{array}
+$$
+
+Based on the proof process in (Zaheer et al., 2018) (Eq 4 in p. 13),
+
+$$
+\begin{array} { r l } & { \quad \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) } \\ & { \leq - \frac { \eta _ { t } } { 2 ( \sqrt { \beta _ { 2 } } \rho + \epsilon ) } \| \nabla _ { \theta } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) \| _ { 2 } ^ { 2 } + \left( \frac { \eta _ { t } \rho \sqrt { 1 - \beta _ { 2 } } } { \epsilon ^ { 2 } } + \frac { L \eta ^ { 2 } } { 2 \epsilon ^ { 2 } } \right) \frac { \sigma ^ { 2 } } { M } , } \end{array}
+$$
+
+Now Eq.(21) has become the following:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { V a l } ( \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) \leq \left. \nabla _ { w } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) , - \alpha _ { t } \left[ \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) + \xi ^ { ( t ) } \right] \right. } \\ & { \mathrm { ~ } } \\ & { \mathrm { ~ } \frac { 1 } { 2 } \alpha _ { t } ^ { 2 } \left[ \rho ^ { 2 } + \| \xi ^ { ( t ) } \| _ { 2 } ^ { 2 } - \langle \nabla _ { w } \mathcal { L } _ { T r } ( \hat { w } _ { t } ( \theta _ { t + 1 } ) ) , \xi ^ { ( t ) } \rangle \right] - \frac { \eta _ { t } } { 2 ( \sqrt { \beta _ { 2 } } \rho + \epsilon ) } \| \nabla _ { \theta } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) \| _ { 2 } ^ { 2 } + \left( \frac { \eta _ { t } \rho \sqrt { 1 - \epsilon ^ { 2 } } } { \epsilon ^ { 2 } } - \frac { \eta _ { t } \rho ^ { 2 } } { 2 } \right) } \end{array}
+$$
+
+Taking expectations with respect to $\xi$ on both side of Eq.(29) and rearranging the inequality, we can obtain:
+
+$$
+\begin{array} { r l } & { \quad \mathbb { E } _ { \xi } \left[ \frac { \eta _ { t } } { 2 ( \sqrt { \beta _ { 2 } } \rho + \epsilon ) } \| \nabla _ { \theta } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) \| _ { 2 } ^ { 2 } \right] } \\ & { \le \alpha _ { t } \rho ^ { 2 } + \frac { L } { 2 } \alpha _ { t } ^ { 2 } ( \rho ^ { 2 } + \sigma ^ { 2 } ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t + 1 } ( \theta _ { t + 1 } ) ) + \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) + \left( \frac { \eta _ { t } \rho \sqrt { 1 - \beta _ { 2 } } } { \epsilon ^ { 2 } } + \frac { L \eta ^ { 2 } } { 2 \epsilon ^ { 2 } } \right) \frac { \sigma ^ { 2 } } { M } } \end{array}
+$$
+
+Using telscoping sum, we obtain
+
+$$
+\begin{array} { l } { \displaystyle \sum _ { t = 1 } ^ { T } \frac { \eta _ { t } } { 2 ( \sqrt { \beta _ { 2 } } \rho + \epsilon ) } \mathbb { E } \| \nabla _ { \theta } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) \| _ { 2 } ^ { 2 } } \\ { \displaystyle \le \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { 1 } ) ) - \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { T + 1 } ) ) + \rho ^ { 2 } \sum _ { t = 1 } ^ { T } \alpha _ { t } + \frac { L } { 2 } ( \rho ^ { 2 } + \sigma ^ { 2 } ) \sum _ { t = 1 } ^ { T } \alpha _ { t } ^ { 2 } + \Big ( \frac { \eta _ { t } \rho \sqrt { 1 - \beta _ { 2 } } } { \epsilon ^ { 2 } } + \frac { L \eta ^ { 2 } } { 2 \epsilon ^ { 2 } } \Big ) \frac { \sigma ^ { 2 } } { M } } \\ { \displaystyle \le \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { 1 } ) ) + \rho ^ { 2 } \sum _ { t = 1 } ^ { T } \alpha _ { t } + \frac { L } { 2 } ( \rho ^ { 2 } + \sigma ^ { 2 } ) \sum _ { t = 1 } ^ { T } \alpha _ { t } ^ { 2 } + \Big ( \frac { \eta _ { t } \rho \sqrt { 1 - \beta _ { 2 } } } { \epsilon ^ { 2 } } + \frac { L \eta ^ { 2 } } { 2 \epsilon ^ { 2 } } \Big ) \frac { \sigma ^ { 2 } T } { M } } \end{array}
+$$
+
+Therefore,
+
+$$
+\begin{array} { r l } & { \displaystyle \operatorname* { m i n } _ { t } \mathbb { E } _ { \xi } \left[ \| \nabla _ { \theta } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { t } ) ) \| _ { 2 } ^ { 2 } \right] \leq \frac { \sum _ { t = 1 } ^ { T } \frac { \eta _ { t } } { 2 ( \sqrt { \beta _ { t } } \rho + \epsilon ) } \mathbb { E } _ { \xi } \left\| \nabla _ { \theta } \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta ^ { ( t ) } ) ) \right\| _ { 2 } ^ { 2 } } { \sum _ { t = 1 } ^ { T } \frac { \eta _ { t } } { 2 ( \sqrt { \beta _ { t } } \rho + \epsilon ) } } } \\ &  \displaystyle \leq \frac { \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { 1 } ) ) + \rho ^ { 2 } \sum _ { t = 1 } ^ { T } \alpha _ { t } + \frac { L } { 2 } ( \rho ^ { 2 } + \sigma ^ { 2 } ) \sum _ { t = 1 } ^ { T } \alpha _ { t } ^ { 2 } + \left( \frac { \eta _ { t } \rho \sqrt { \frac { 1 - \beta _ { t } } { \epsilon ^ { 2 } } } + \frac { L \eta ^ { 2 } } { 2 \epsilon ^ { 2 } }  \frac { \sigma ^ { 2 } T } { M } 2 ( \sqrt { \beta _ { 2 } } \rho + \epsilon ) } { \sum _ { t = 1 } ^ { T } \eta _ { t } } } \\ & \right){ \displaystyle \leq \frac { 2 ( \sqrt { \beta _ { 2 } } \rho + \epsilon ) } { T \eta } \left\{ \mathcal { L } _ { V a l } ( \hat { w } _ { t } ( \theta _ { 1 } ) ) + \rho ^ { 2 } \sum _ { t = 1 } ^ { T } \alpha _ { t } + \frac { L } { 2 } ( \rho ^ { 2 } + \sigma ^ { 2 } ) \sum _ { t = 1 } ^ { T } \alpha _ { t } ^ { 2 } + \left( \frac { \eta _ { t } \rho \sqrt { 1 - \beta _ { 2 } } } { \epsilon ^ { 2 } } + \frac { L \eta ^ { 2 } } { 2 \epsilon ^ { 2 } } \right) \frac { \sigma ^ { 2 } T } { M } \right\} } \\ & { \displaystyle \leq \mathcal { O } ( \frac { \ln ( T ) } { T } + \sigma ^ { 2 } ) . } \end{array}
+$$
+
+class LSTMCell(nn.Module):   
+def _init__(self, num_inputs, hidden_size):   
+super(LSTMCell, self).__init__()   
+self.hidden_size $-$ hidden_size   
+self.fc_i2h $-$ nn.Sequential(   
+nn.Linear(num_inputs, hidden_size),   
+nn.ReLU(),   
+nn.Linear(hidden_size, 4 $\star$ hidden_size)   
+)   
+self.fc_h2h $=$ nn.Sequential(   
+nn.Linear(hidden_size, hidden_size),   
+nn.ReLU(),   
+nn.Linear(hidden_size, 4 $\star$ hidden_size)   
+)   
+def forward(self, inputs, state):   
+hx, cx $=$ state   
+$\dot { \iota } 2 \mathrm { h \Omega } =$ self.fc_i2h(inputs)   
+h2h $=$ self.fc_h2h(hx)   
+$\mathrm { ~  ~ { ~ x ~ } ~ } = \mathrm { ~  ~ { ~ i ~ } ~ } 2 \mathrm { h }$ $^ +$ h2h   
+gates $= \times$ .split(self.hidden_size, 1)   
+in_gate $=$ torch.sigmoid(gates[0])   
+forget_gate $-$ torch.sigmoid(gates[1])   
+out_gate $=$ torch.sigmoid(gates[2])   
+in_transform $-$ torch.tanh(gates[3])   
+cx $=$ forget_gate $^ *$ cx $^ +$ in_gate $\star$ in_transform   
+hx $-$ out_gate $\star$ torch.tanh(cx)   
+return hx, cx   
+class MLRNet(nn.Module):   
+def __init__(self, num_layers, hidden_size):   
+super(MLRNet, self).__init__()   
+self.hidden_size $=$ hidden_size   
+self.layer1 $-$ LSTMCell(1, hidden_size)   
+self.layer2 $-$ nn.Linear(hidden_size, 1)   
+def forward(self, x, gamma):   
+self.hx, self.cx $-$ self.layer1(x, (self.hx, self.cx))   
+$\times \quad =$ self.hx   
+$\begin{array} { r l } { \mathbf { x } } & { { } = } \end{array}$ self.layer2(x)   
+out $=$ torch.sigmoid(x)   
+return gamma $\star$ out
+
+# B EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS IN SECTION 4.1
+
+In this section, we attempt to evaluate the capability of MLR-SNet to learn LR schedules compared with baseline methods. Here, we provide implementation details of all experiments.
+
+Datasets. We choose two datasets in image classification (CIFAR-10 and CIFAR-100), and one dataset in text classification (Penn Treebank) to present the efficiency of our method. CIFAR-10 and CIFAR-100 Krizhevsky (2009), consisting of $3 2 \times 3 2$ color images arranged in 10 and 100 classes, respectively. Both datasets contain 50,000 training and 10,000 test images. Penn Treebank Marcus & Marcinkiewicz is composed of $9 2 9 \mathrm { k }$ training words, 73k validation words, and 82k test words, with a 10k vocabulary in total. Our algorithm and RTHO Franceschi et al. (2017) randomly select 1,000 clean images in the training set of CIFAR-10/100 as validation data, and directly use the validation set in Penn Treebank as validation data.
+
+CIFAR-10 & CIFAR-100. We employ ResNet-18 on CIFAR-10 and WideResNet-28-10 Zagoruyko & Komodakis (2016) on CIFAR-100. All compared methods and MLR-SNet are trained for 200 epochs with batch size 128. For baselines involving SGD as base optimizer, we set the initial LR to 0.1, weight decay parameter to $5 e ^ { - 4 }$ and momentum to 0.9 if used. While for Adam, we just follow the default parameter setting. The hyper-parameters of hand-designed LR schedules are listed below: Exponential decay, multiplying LR with 0.95 every epoch; MultiStep decay, decaying LR by 10 every 60 epochs; SGDR, setting $\mathrm { T } \_ 0$ to 10, T_Mult to 2 and minimum LR to $1 e ^ { - 5 }$ . L4, HD and RTHO update LR every data batch, and we use the recommended setting in the original paper of L4 $( \alpha = 0 . 1 5 )$ and search different hyper-lrs from $\{ 1 e ^ { - 3 } , 1 e ^ { - 4 } , 1 e ^ { - 5 } , 1 e ^ { - 6 } , 1 e ^ { - 7 } \}$ for $\mathbf { H D }$ and RTHO, reporting the best performing hyper-lr.
+
+Table 2: Test accuracy $( \% )$ of CIFAR dataset with SGD baselines.   
+
+<table><tr><td rowspan=1 colspan=1>Optimizer</td><td rowspan=1 colspan=2>CIFAR-10 with ResNet18 CIFAR-100 with WRN-28-10</td></tr><tr><td rowspan=1 colspan=1>SGD+Fixed</td><td rowspan=1 colspan=1>92.26 ± 0.12</td><td rowspan=1 colspan=1>70.67±0.34</td></tr><tr><td rowspan=1 colspan=1>SGD+MultiStep</td><td rowspan=1 colspan=1>93.82 ± 0.09</td><td rowspan=1 colspan=1>77.04 ±：0.17</td></tr><tr><td rowspan=1 colspan=1>SGD+Exponential</td><td rowspan=1 colspan=1>90.93士0.11</td><td rowspan=1 colspan=1>72.52 ±0.34</td></tr><tr><td rowspan=1 colspan=1>SGD+SGDR</td><td rowspan=1 colspan=1>93.92 ± 0.11</td><td rowspan=1 colspan=1>72.52 ± 0.34</td></tr><tr><td rowspan=1 colspan=1>Adam</td><td rowspan=1 colspan=1>90.86 ± 0.15</td><td rowspan=1 colspan=1>68.94 ± 0.24</td></tr><tr><td rowspan=1 colspan=1>SGD+L4</td><td rowspan=1 colspan=1>89.15 ± 0.14</td><td rowspan=1 colspan=1>63.61 ± 0.65</td></tr><tr><td rowspan=1 colspan=1>SGD+HD</td><td rowspan=1 colspan=1>92.34 ± 0.09</td><td rowspan=1 colspan=1>72.22 ± 0.30</td></tr><tr><td rowspan=1 colspan=1>SGD+RTHO</td><td rowspan=1 colspan=1>92.60 ± 0.18</td><td rowspan=1 colspan=1>72.32 ± 0.47</td></tr><tr><td rowspan=1 colspan=1>MLR-SNet (Meta-train)</td><td rowspan=1 colspan=1>94.70 ± 0.16</td><td rowspan=1 colspan=1>79.41±0.14</td></tr></table>
+
+Table 3: Test accuracy $( \% )$ of CIFAR dataset with SGDM baselines.   
+
+<table><tr><td rowspan=1 colspan=1>Optimizer</td><td rowspan=1 colspan=1>CIFAR-10 with ResNet18</td><td rowspan=1 colspan=1>CIFAR-100 with WRN-28-10</td></tr><tr><td rowspan=1 colspan=1>SGDM+Fixed</td><td rowspan=1 colspan=1>87.69 ± 0.14</td><td rowspan=1 colspan=1>70.88 ± 0.12</td></tr><tr><td rowspan=1 colspan=1>SGDM+MultiStep</td><td rowspan=1 colspan=1>95.08 ± 0.13</td><td rowspan=1 colspan=1>80.74 ± 0.19</td></tr><tr><td rowspan=1 colspan=1>SGDM+Exponential</td><td rowspan=1 colspan=1>94.64 ± 0.05</td><td rowspan=1 colspan=1>78.87±0.04</td></tr><tr><td rowspan=1 colspan=1>SGDM+SGDR</td><td rowspan=1 colspan=1>95.06 ± 0.17</td><td rowspan=1 colspan=1>80.93 ± 0.05</td></tr><tr><td rowspan=1 colspan=1>Adam</td><td rowspan=1 colspan=1>90.86 ± 0.15</td><td rowspan=1 colspan=1>68.94 ± 0.24</td></tr><tr><td rowspan=1 colspan=1>SGDM+L4</td><td rowspan=1 colspan=1>91.03 ± 0.14</td><td rowspan=1 colspan=1>66.51 ± 2.83</td></tr><tr><td rowspan=1 colspan=1>SGDM+HD</td><td rowspan=1 colspan=1>93.99± 0.12</td><td rowspan=1 colspan=1>76.80 ± 0.19</td></tr><tr><td rowspan=1 colspan=1>SGDM+RTHO</td><td rowspan=1 colspan=1>93.17 ± 0.49</td><td rowspan=1 colspan=1>76.14 ± 0.29</td></tr><tr><td rowspan=1 colspan=1>MLR-SNet (Meta-train)</td><td rowspan=1 colspan=1>94.70 ± 0.16</td><td rowspan=1 colspan=1>79.41±0.14</td></tr></table>
+
+Penn Treebank. We use a 2-layer and 3-layer LSTM network which follows a word-embedding layer and the output is fed into a linear layer to compute the probability of each word in the vocabulary. Hidden size of LSTM cell is set to 512 and so is the word-embedding size. We tie weights of the word-embedding layer and the final linear layer. Dropout is applied to the output of word-embedding layer together with both the first and second LSTM layers with a rate of 0.5. As for training, the LSTM net is trained for 150 epochs with a batch size of 32 and a sequence length of 35. We set the base optimizer SGD to have an initial LR of 20 without momentum, for Adam, the initial LR is set to 0.01 and weight for moving average of gradient is set to 0. We apply a weight decay of $5 e ^ { - 6 }$ to both base optimizers. All experiments involve a 0.25 clipping to the network gradient norm. For both SGD and Adam, we decrease LR by a factor of 4 when performance on validation set shows no progress. For L4, we try different $\alpha$ in $\{ 0 . 1 , 0 . 0 5 , 0 . 0 1 , 0 . 0 0 5 \}$ and reporting the best test perplexity among them. For both HD and RTHO, we search the hyper-lr lying in $\{ 1 , 0 . 5 , 0 . 1 , 0 . 0 5 \}$ , and report the best results.
+
+Table 4: Test perplexity on the Penn Treebank dataset.   
+
+<table><tr><td rowspan=1 colspan=1>Optimizer</td><td rowspan=1 colspan=1>2-layer LSTM</td><td rowspan=1 colspan=1>3-layer LSTM</td></tr><tr><td rowspan=1 colspan=1>SGD+Val Strategy</td><td rowspan=1 colspan=1>74.33 ± 0.23</td><td rowspan=1 colspan=1>76.05 ± 0.39</td></tr><tr><td rowspan=1 colspan=1>Adam+Val Strategy</td><td rowspan=1 colspan=1>71.17 ± 0.23</td><td rowspan=1 colspan=1>74.80 ± 0.73</td></tr><tr><td rowspan=1 colspan=1>SGD+L4</td><td rowspan=1 colspan=1>82.58 ± 1.32</td><td rowspan=1 colspan=1>92.27 ± 0.92</td></tr><tr><td rowspan=1 colspan=1>SGD+HD</td><td rowspan=1 colspan=1>76.90 ± 0.33</td><td rowspan=1 colspan=1>78.63 ± 0.08</td></tr><tr><td rowspan=1 colspan=1>SGD+RTHO</td><td rowspan=1 colspan=1>76.69 ± 0.11</td><td rowspan=1 colspan=1>78.52 ± 0.16</td></tr><tr><td rowspan=1 colspan=1>MLR-SNet (Meta-train)</td><td rowspan=1 colspan=1>72.09 ± 0.72</td><td rowspan=1 colspan=1>72.71±0.17</td></tr></table>
+
+![](images/b2f2e2d708e3639c79a6e1057f102b78f2c67b8124200cac98dab6e947e626c9.jpg)  
+Figure 11: Train loss (perplexity), test accuracy (perplexity) and learned LR schedules of our methods (train) and compared baselines on different tasks.
+
+MLR-SNet architecture and parameter setting. The architecture of MLR-SNet is illustrated in Section 3.2. In our experiment, the size of hidden nodes is set as 40. The initialization of MLR-SNet follows the default setting in Pytorch. The Pytorch implementation of MLR-SNet is listed above.
+
+We employ Adam optimizer to train MLR-SNet, and just set the parameters as originally recommended with a LR of $1 e ^ { - 3 }$ , and a weight decay of $1 e ^ { - 4 }$ , which avoids extra hyper-parameter tuning. For image classification tasks, the input of MLR-SNet is the training loss of a mini batch samples. Every data batch’s LR is predicted by MLR-SNet and we update it twice per epoch according to the loss of the validation data. While for text classification tasks, we take $\frac { \dot { \mathcal { L } } _ { T r } } { \log ( v o c a b u l a r y ~ s i z e ) }$ as input of MLR-SNet to deal with the influence of large scale classes of text. MLR-SNet is updated every 100 batches due to the large number of batches per epoch compared to that in image datasets.
+
+Results. Due to the space limitation, we only present the test accuracy in the main paper. Here, we present the training loss and test accuracy of our method and all compared methods on image and text tasks, as shown in Fig.11. For image tasks, except for Adam and SGD with fixed LR, other methods can decrease the loss to 0 almostly. Though local minima can be reached by these methods, the generalization ability of the these mimimas has a huge difference, which can be summarized from test accuracy curves. As shown in Fig. 11(a),11(b),11(g),11(h), when using SGD to train DNNs, the compared methods SGD with Exponential LR, L4, HD, RTHO fail to find such good solutions to generalize well. Especially, L4 greedily searches LR to decrease loss to 0, making it fairly hard to adapt the complex DNNs training dynamic and obtain a good mimima, while our method can adjust LR to comply with the significant variations of training dynamic, leading to a better generalization solution. As shown in Fig. 11(d),11(e),11(j),11(k), when baseline methods are trained with SGDM, these methods make a great progress in escaping from the bad minimas. In spite of this, our method still shows superiority in finding a solution with better generalization compared with these competitive training strategies.
+
+In the third column in Fig. 11, we plot learned LR schedules of compared methods and our method. As can be seen, our method can learn LR schedules approximating the hand-designed LR schedules while with more locally varying. HD and RTHO often have the same trajectory while producing lower or faster downward trend than ours. This tends to explain our final performances on test set is better than HD and RTHO, since our method can adaptively adjust LR utilizing the past training histories explicitly. L4 greedily searches a LR to decrease the loss. This often leads to a large value causing fluctuations or even divergence (Fig. 11(l)), or a small value causing slow progress (Fig. 11(r)), or both of them (Fig. 11(c) 11(f) 11(i) 11(o)). Such LR schedules often result in bad mimimas. Moreover, all compared methods regard LR as hyper-parameter to learn without a transferable formulation, and the learned LR schedules can not generalize to other learning tasks directly. Generally, they just try to find a proper LR schedule from scratch for new tasks. However, our meta-learned MLR-SNet is plug-and-play and transferrable, which can directly transfer how to schedule LR for SGD to heterogeneous tasks without additional learning.
+
+# Ablation study.
+
+(1) The architecture of MLR-SNet. Fig.12(a) shows the test accuracy on CIFAR-10 with ResNet-18 of different architectures of MLR-SNet. As can be seen, our algorithm is not sensitive to the choose of the MLR-SNet’s architectures. This implies that our algorithm is robust and stable for helping improve DNN training.
+
+(2) The gobal LR of the meta optimizer. To further validate that whether our MLR-SNet behaves robust to the meta optimizer. We adapot Adam optimizer to search the proper LR schedules. Fig. 12(b) shows that our MLR-SNet achieves the similar performance even for different global LRs. This implies our MLR-SNet needs not carefully tune the LR of the meta optimizer, which makes it easy to reproduce and apply to various problems.
+
+(3) The different $\gamma$ values of the MLR-SNet. One important hyperparameter of the MLR-SNet is $\gamma$ , here we verify our method is not sensitive to the choose of $\gamma$ value. We test $\gamma$ values from 0.1 to 10 for the DNNs training. As shown in Fig.12(c), even with different learning scales, our method can still help DNNs achieve almost similar performance. This implies the MLR-SNet is robust to the choose of the $\gamma$ , which makes it easy to be applied into parctice.
+
+# C EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS IN SECTION 4.2
+
+We investigate the transferability of the learned LR schedule when applied to various tasks in Section 4.2 of the main paper. We use the MLR-SNet meta-learned on CIFAR-10 with ResNet-18 in Section 4.1 to directly predict the LR for SGD algorithm to new heterogeneous tasks. We save the learned MLR-SNet at different epochs in the whole one meta-train run. As is shown in Fig.13(a), if we use
+
+![](images/85d6230f2b0ea717e1fbfbb321dd18c4bb0edcf1be2f5212b3416ba7a4a98e89.jpg)  
+(a) Different Architectures of MLR-(b) Different LRs of Meta Optimizer (c) Different $\gamma$ value of MLR-SNet SNet
+
+Figure 12: Ablation study. (a) Test accuracy on CIFAR-10 with ResNet-18 of different architectures of MLR-SNet. ‘a-b’ denotes the configurations of MLR-SNet, where ‘a’ represents the number of layers, and ‘b’ represents the number of hidden nodes. (b)Test accuracy on CIFAR-10 with ResNet18 of different LRs of meta optimizer ’Adam’. (c) Test accuracy on CIFAR-10 with ResNet-18 of different gamma values of MLR-SNet.
+
+![](images/dc3f1b52d846cf74db43c6965fcaf4872c1e3cb17c23f3c810c702211227d39d.jpg)  
+Figure 13: (a) We plot the LR variation curves along iterations with the same input for learned MLR-SNet at different epochs. As is shown, when iteration increases, the LR is almost constant. This means the learned MLR-SNet overfits the short trajectories, while fails for the long trajectories. (b),(c) show the recording train loss and test accuracy with ResNet-18 on CIFAR-100 of different test strategies.
+
+the single learned MLR-SNet at certain epoch, it can be seen that the predicted LR by the learned LR schedules converges after several iterations. This is because that the training trajectories are long in our experiments, and the learned MLR-SNet can not memory all the information since we locally adjust our MLR-SNet according to the validation error. If we directly select one MLR-SNet learned at any epoch, that will raise overfitting issues as shown in Fig.13(a). Thus we should select more than two learned MLR-SNets for test. Here, we propose a heuristic strategy to select MLR-SNets for test. Generally, if we want to select $k$ nets for test, the MLR-SNet learned at $\big [ \frac { 2 0 0 * l } { k - 1 } \big ]$ -th epoch $( l = 0 , 1 , 2 , \cdots , k - 1 )$ should be chosen, where $[ \cdot ]$ denotes ceiling operator. Fig.13(b) and 13(c) show the train loss and test accuracy with ResNet-18 on CIFAR-100 of different test strategies, i.e., choosing different number of nets to transfer. It can be seen that almost choosing more than three nets have similar performance. Therefore, in the following experiments we choose three MLR-SNets to show the transferability.
+
+Transfer to different epochs. We transfer the LR schedules meta-trained with epoch 200 to other different epochs, e.g., 100, 400,1200. All the methods are trained with ResNet-18 on CIFAR-100 with batch size 128 for different epochs. The hyper-parameter setting for compared hand-designed LR schedules is the same with Section 4.1 in the main paper as illustrated above, except for MultiStep LR. For epoch 100, MultiStep LR decays LR by 10 every 30 epochs; For epoch 400, MultiStep LR decays LR by 10 every 120 epochs; For epoch 1200, MultiStep LR decays LR by 10 every 360 epochs. Other hyper-parameters of MultiStep LR keep unchanged. For our method, we use the transferred strategy as below: 1) For epoch 100, we employ the 3 nets at 0-33, 33-67, 67-100 epoch, respectively; 2) For epoch 400, we employ the 3 nets at 0-133, 133-267, 267-400 epoch, respectively; 3) For epoch 1200, we employ the 3 nets at 0-400, 400-800, 800-1200 epoch, respectively.
+
+Table 5: Test accuracy $( \% )$ of CIFAR-10 dataset with different networks.   
+
+<table><tr><td rowspan=1 colspan=1>Optimizer</td><td rowspan=1 colspan=2>ShuffleNetV2 MobileNetV2</td><td rowspan=1 colspan=1>NASNet</td></tr><tr><td rowspan=1 colspan=1>SGD+Fixed</td><td rowspan=1 colspan=1>87.06± 0.33</td><td rowspan=1 colspan=1>90.85 ± 0.52</td><td rowspan=1 colspan=1>89.14± 1.15</td></tr><tr><td rowspan=1 colspan=1>SGD+MultiStep</td><td rowspan=1 colspan=1>88.99 ± 0.11</td><td rowspan=1 colspan=1>92.28 ± 0.28</td><td rowspan=1 colspan=1>94.97 ± 0.10</td></tr><tr><td rowspan=1 colspan=1>SGD+Exponential</td><td rowspan=1 colspan=1>89.45 ± 0.14</td><td rowspan=1 colspan=1>93.47 ± 0.12</td><td rowspan=1 colspan=1>94.59 ± 0.14</td></tr><tr><td rowspan=1 colspan=1>SGD+SGDR</td><td rowspan=1 colspan=1>89.30 ± 0.83</td><td rowspan=1 colspan=1>92.00 ± 1.21</td><td rowspan=1 colspan=1>94.75 ± 0.82</td></tr><tr><td rowspan=1 colspan=1>Adam</td><td rowspan=1 colspan=1>87.95 ± 0.31</td><td rowspan=1 colspan=1>90.64 ± 0.54</td><td rowspan=1 colspan=1>90.60 ± 0.47</td></tr><tr><td rowspan=1 colspan=1>MLR-SNet (Meta-test)</td><td rowspan=1 colspan=1>89.09 ± 0.33</td><td rowspan=1 colspan=1>93.11 ± 0.10</td><td rowspan=1 colspan=1>95.18±0.18</td></tr></table>
+
+Table 6: Validation accuracies on ImageNet dataset.   
+
+<table><tr><td rowspan=1 colspan=1>Optimizer</td><td rowspan=1 colspan=1>Top-1 Accuracy</td><td rowspan=1 colspan=1>Top-5 Accuracy</td></tr><tr><td rowspan=1 colspan=1>SGDM+Fixed</td><td rowspan=1 colspan=1>68.23</td><td rowspan=1 colspan=1>88.47</td></tr><tr><td rowspan=1 colspan=1>SGDM+MultiStep</td><td rowspan=1 colspan=1>75.90</td><td rowspan=1 colspan=1>92.90</td></tr><tr><td rowspan=1 colspan=1>SGDM+Exponential</td><td rowspan=1 colspan=1>75.68</td><td rowspan=1 colspan=1>92.69</td></tr><tr><td rowspan=1 colspan=1>SGDM+SGDR</td><td rowspan=1 colspan=1>75.82</td><td rowspan=1 colspan=1>92.87</td></tr><tr><td rowspan=1 colspan=1>Adam</td><td rowspan=1 colspan=1>63.62</td><td rowspan=1 colspan=1>85.43</td></tr><tr><td rowspan=1 colspan=1>MLR-SNet (Meta-test)</td><td rowspan=1 colspan=1>75.03</td><td rowspan=1 colspan=1>92.39</td></tr></table>
+
+![](images/9096fd3f855516b25560c1fa81850467d4a2a305183a78ab8508822b0f23d83b.jpg)  
+Figure 14: Test accuracy on CIFAR-100 of different DenseNet architectures.
+
+Transfer to different datasets. We transfer the LR schedules meta-learned on CIFAR-10 to SVHN (Netzer et al., 2011), TinyImageNet 6, and Penn Treebank (Marcus & Marcinkiewicz). For image classification, we train a ResNet-18 on SVHN and TinyImageNet, respectively. The hyper-parameters of all compared methods are set the same as those of CIFAR-10. For text classification, we train a 3-layer LSTM on Penn Treebank. The hyper-parameters of all compared methods are with the same setting as introduced in Section 4.1.
+
+Transfer to different net architectures. We transfer the learned LR schedules for different net architectures training. All the methods are trained on CIFAR-10 with different net architectures. The hyper-parameters of all methods are the same with the setting of CIFAR-10 with ResNet-18. We test the meta-learned LR schedule to different configurations of DenseNet Huang et al. (2017). As shown in Fig. 14, our method perform slightly stable than MultiStep strategy at about 75-125 epochs. This tends to show the superiority of adaptive LR to train the DenseNets. Also, we transfer the LR schedules to several novel networks, the results are presented in Fig.8 in the main paper.
+
+Transfer to large scale optimization. We transfer the learned LR schedules for the training of the large scale optimization problems. The predicted LR by MLR-SNet will not substantially increase the complexity compared with hand-designed LR schedules for DNNs training. This makes it feasible and reliable to transfer our meta-learned LR schedules to such large scale optimization problems. We train a ResNet-50 on ImageNet with hand-designed LR schedules and our transferred LR schedules. The training code can be found on https://github.com/pytorch/ examples/tree/master/imagenet, and the parameter setting keeps unchanged except the LR. All compared hand-designed LR schedules are trained by SGDM with a momentum 0.9, a weight decay $5 e ^ { - 4 }$ , an initial learning rate 0.1 for 90 epochs, and batch size 256. Fixed LR uses 0.1 LR during the whole training; Exponential LR multiplies LR with 0.95 every epoch; MultiStep LR decays LR by 10 every 30 epochs; SGDR sets $\mathrm { T } \_ 0$ to 10, T_Mult to 2 and minimum LR to $\bar { 1 } e ^ { - 5 }$ ; Adam just uses the default parameter setting. The results are presented in Fig. 9 in the main paper.
+
+Table 7: Test accuracy $( \% )$ on CIFAR-10 and CIFAR-100 training set of different methods trained on CIFAR-10-C and CIFAR-100-C. Best and Last denote the results of the best and the last epoch. The Bold and Underline Bold denote the first and second best results, respectively.   
+
+<table><tr><td colspan="2">Datasets/Methods</td><td>Fixed</td><td>MultiStep</td><td>Exponential</td><td>SGDR</td><td>Adam</td><td>Ours(Train)</td></tr><tr><td rowspan="2">CIFAR-10-C</td><td>Best</td><td>79.96±4.09</td><td>85.64±1.71</td><td>83.63±1.38</td><td>86.10±1.44</td><td>81.57±1.39</td><td>85.73±1.71</td></tr><tr><td>Last</td><td>77.89±4.05</td><td>85.48±1.71</td><td>83.47±1.37</td><td>78.46±1.92</td><td>80.39±1.65</td><td>85.62±1.76</td></tr><tr><td rowspan="2">CIFAR-100-C</td><td>Best</td><td>46.91±3.08</td><td>52.38±2.43</td><td>49.90±1.93</td><td>52.80±2.39</td><td>45.58±1.95</td><td>52.51±2.38</td></tr><tr><td>Last</td><td>44.81±5.98</td><td>52.28±2.44</td><td>49.75±1.94</td><td>41.68±3.33</td><td>43.94±2.18</td><td>52.35±2.46</td></tr></table>
+
+# D EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS IN SECTION 4.3
+
+The datasets CIFAR-10-C and CIFAR-100-C Hendrycks & Dietterich (2019) can be downloaded at https://zenodo.org/record/2535967#.Xt4mVigzZPY, https://zenodo.org/ record/3555552#.Xt4mdSgzZPY. Each dataset contains 15 types of algorithmically generated corruptions from noise, blur, weather, and digital categories. These corruptions contain Gaussian Noise, Shot Noise, Impulse Noise, Defocus Blur, Frosted Glass Blur, Motion Blur, Zoom Blur, Snow, Frost, Fog, Brightness, Contrast, Elastic, Pixelate and JPEG. All the corruptions are gererated on 10,000 test set images, and each corruption contains 50,000 images since each type of corruption has five levels of severity. We treat CIFAR-10-C or CIFAR-100-C dataset as training set, and train a model with ResNet-18 for each corruption dataset. Finally, we can obtain 15 models for CIFAR-10/100-C. Each corruption can be roughly regarded as a task, and the average accuracy of 15 models on test data 7 is used to evaluate the robust performance of different tasks for each LR schedules strategy.
+
+For experimental setting in Section 4.3, all compared hand-designed LR schedules are trained with a ResNet-18 by SGDM with a momentum 0.9, a weight decay $5 e ^ { - 4 }$ , an initial learning rate 0.1 for 100 epochs, and batch size 128. Fixed LR uses 0.1 LR during the whole training; Exponential LR multiplies LR with 0.95 every epoch; MultiStep LR decays LR by 10 every 30 epochs; SGDR sets T_0 to 10, T_Mult to 2 and minimum LR to ${ \bar { 1 } } e ^ { - 5 }$ ; Adam just uses the default parameter setting. Our method trains the ResNet-18 by SGD with a weight decay $5 e ^ { - 4 }$ , and the MLR-SNet is learned under the guidance of a small set of validation set without corruptions. We randomly choose 10 clean images for each class as validation set. The experimental result is listed in Table 1 in the main paper.
+
+Additional robustness results of transferrable LR schedules on different data corruptions. Furthermore, we want to explore the robust performance of different tasks for our transferrable LR schedules. Different from above experiments where all 15 models are trained under the guidance of a small set of validation set, we just train a ResNet-18 on Gaussian Noise corruption to meta-learn the MLR-SNet, and then transfer the meta-learned LR schedules to other 14 corruptions. We report the average accuracy of 14 models on test data to show the robust performance of our transferred LR schedules. All the methods are meta-tested with a ResNet-18 for 100 epochs with batch size 128. The hyper-parameter setting of hand-designed LR schedules keeps same with above. Table 7 shows the mean test accuracy of 14 models. As can be seen, our transferrable LR schedules obtain the final best performance compared with hand-designed LR schedules. This implies that our transferrable LR schedules can also perform robust and stable than the pre-set LR schedules when the learning tasks are changed. However, our transferrable LR schedules are plug-and-play, and have no additional hyper-parameters to tune when transferred to new heterogeneous tasks.
+
+# E THE PRELIMINARY EXPLORATION OF THE INFLUENCE ON META-TEST TASKS OF THE DIFFERENT META-TRAINING TASKS.
+
+In this section, we study the performance influence on the target task of the different meta-training tasks. Here we fixed the target task as training ResNet-18 on TinyImageNet. We choose three different meta-training task: 1) training ResNet-18 on CIFAR-10; 2) training WideResNet-28-10 on CIFAR-100; 3) training 3-layer LSTM on Penn Treebank. Fig.15 shows the meta-test performance of three different meta-training tasks. It can be seen that the meta-training task more related to the target task would obtain better transferable performance on the target task.
+
+![](images/8671b5d6bf6a4760bf323bf54320aa667739d736c4be2c0b7b65affc4064d47c.jpg)  
+Figure 15: The meta-test performance on TinyImageNet (target task) of different meta-training tasks.
+
+![](images/2d5a15a025d033bbafc7a11eb77704515f9a56b11bb2d15de9cb07f27ae2d0be.jpg)  
+Figure 16: Performance comparion on CIFAR-10 of the MLR-SNet and baselines.
+
+F APPLYING MLR-SNET ON TOP OF ADAM.
+
+To further demostrate the versatility of our method, we apply the MLR-SNet on top of the Adam algorithm. Fig.16 shows that our methods can substantially improve the performance of the original Adam algorithm.
+
+# G EXPERIMENTAL RESULTS OF ADDITIONAL COMPARED METHOD LR CONTROLLER
+
+In this section, we present the experimental results of LR Controller Xu et al. (2019), which is a related work of ours but under the reinforcement learning framework. Due to their learning algorithm is relatively computationally expensive and not very easy to optimize, we will show our method has a superiority in finding such a good LR schedule that scales and generalizes.
+
+To start a fair comparison, we follow all the training settings and structure of LR Controller proposed in Xu et al. (2019) except that we modify the batch size to 128 and increase training steps to cover 200 epochs of data to match our setup in Section $4 . 1 ~ ^ { 8 }$ . Firstly, we train LR Controller on CIFAR-10 with ResNet-18 and CIFAR-100 with WideResNet-28-10 as we do in Section 4.1. As shown in Fig. 17, our method demonstrates evident superiority in finding a solution with better generalization compared with LR Controller strategies. LR Controller performs steadily in the early training phase, but soon fluctuates significantly and fails to progress. This tends to show that the LR Controller suffers from a severe stability issue when training step increases, especially being compared to our MLR-SNet.
+
+Then we transfer the LR schedules learned on CIFAR-10 for our method and LR Controller to CIFAR-100 to verify their transferability. Test settings are the same with those related in Section 4.2. As shown in Fig. 18, LR Controller makes a comparatively slower progress in the whole training process. While our method achieves a competitive performance, which indicates the capability of transferring to other tasks for our method.
+
+![](images/8b093966f11a0a1eaa535d4bb4b3fd13fb9f5190a165a3dc5077c0ed82b47214.jpg)  
+Figure 17: Train loss, test accuracy and learned LR schedules of our method(train) and LR Controller(train) on CIFAR-10 and CIFAR-100.
+
+![](images/f335b99db817b981b8ea9953286155a904218cf23701a59124966081d5206323.jpg)  
+Figure 18: Train loss, test accuracy of our method(test) and LR Controller(test) on CIFAR-100.

parse/train/rJbbOLcex/rJbbOLcex.md

@@ -0,0 +1,269 @@
+# TOPICRNN: A RECURRENT NEURAL NETWORKWITH LONG-RANGE SEMANTIC DEPENDENCY
+
+Adji B. Dieng ∗ Columbia University abd2141@columbia.edu
+
+Chong Wang   
+Deep Learning Technology Center Microsoft Research   
+chowang@microsoft.com   
+Jianfeng Gao   
+Deep Learning Technology Center   
+Microsoft Research   
+jfgao@microsoft.com
+
+John Paisley Columbia University jpaisley@columbia.edu
+
+# ABSTRACT
+
+In this paper, we propose TopicRNN, a recurrent neural network (RNN)-based language model designed to directly capture the global semantic meaning relating words in a document via latent topics. Because of their sequential nature, RNNs are good at capturing the local structure of a word sequence – both semantic and syntactic – but might face difficulty remembering long-range dependencies. Intuitively, these long-range dependencies are of semantic nature. In contrast, latent topic models are able to capture the global semantic structure of a document but do not account for word ordering. The proposed TopicRNN model integrates the merits of RNNs and latent topic models: it captures local (syntactic) dependencies using an RNN and global (semantic) dependencies using latent topics. Unlike previous work on contextual RNN language modeling, our model is learned endto-end. Empirical results on word prediction show that TopicRNN outperforms existing contextual RNN baselines. In addition, TopicRNN can be used as an unsupervised feature extractor for documents. We do this for sentiment analysis on the IMDB movie review dataset and report an error rate of $6 . 2 8 \%$ . This is comparable to the state-of-the-art $5 . 9 1 \%$ resulting from a semi-supervised approach. Finally, TopicRNN also yields sensible topics, making it a useful alternative to document models such as latent Dirichlet allocation.
+
+# 1 INTRODUCTION
+
+When reading a document, short or long, humans have a mechanism that somehow allows them to remember the gist of what they have read so far. Consider the following example:
+
+“The U.S.presidential race isn’t only drawing attention and controversy in the United States – it’s being closely watched across the globe. But what does the rest of the world think about a campaign that has already thrown up one surprise after another? CNN asked 10 journalists for their take on the race so far, and what their country might be hoping for in America’s next —
+
+The missing word in the text above is easily predicted by any human to be either President or Commander in Chief or their synonyms. There have been various language models – from simple ngrams to the most recent RNN-based language models – that aim to solve this problem of predicting correctly the subsequent word in an observed sequence of words.
+
+A good language model should capture at least two important properties of natural language. The first one is correct syntax. In order to do prediction that enjoys this property, we often only need to consider a few preceding words. Therefore, correct syntax is more of a local property. Word order matters in this case. The second property is the semantic coherence of the prediction. To achieve this, we often need to consider many preceding words to understand the global semantic meaning of the sentence or document. The ordering of the words usually matters much less in this case.
+
+Because they only consider a fixed-size context window of preceding words, traditional $n$ -gram and neural probabilistic language models (Bengio et al., 2003) have difficulties in capturing global semantic information. To overcome this, RNN-based language models (Mikolov et al., 2010; 2011) use hidden states to “remember” the history of a word sequence. However, none of these approaches explicitly model the two main properties of language mentioned above, correct syntax and semantic coherence. Previous work by Chelba and Jelinek (2000) and Gao et al. (2004) exploit syntactic or semantic parsers to capture long-range dependencies in language.
+
+In this paper, we propose TopicRNN, a RNN-based language model that is designed to directly capture long-range semantic dependencies via latent topics. These topics provide context to the RNN. Contextual RNNs have received a lot of attention (Mikolov and Zweig, 2012; Mikolov et al., 2014; Ji et al., 2015; Lin et al., 2015; Ji et al., 2016; Ghosh et al., 2016). However, the models closest to ours are the contextual RNN model proposed by Mikolov and Zweig (2012) and its most recent extension to the long-short term memory (LSTM) architecture (Ghosh et al., 2016). These models use pre-trained topic model features as an additional input to the hidden states and/or the output of the RNN. In contrast, TopicRNN does not require pre-trained topic model features and can be learned in an end-to-end fashion. We introduce an automatic way for handling stop words that topic models usually have difficulty dealing with. Under a comparable model size set up, TopicRNN achieves better perplexity scores than the contextual RNN model of Mikolov and Zweig (2012) on the Penn TreeBank dataset 1. Moreover, TopicRNN can be used as an unsupervised feature extractor for downstream applications. For example, we derive document features of the IMDB movie review dataset using TopicRNN for sentiment classification. We reported an error rate of $6 . 2 8 \%$ . This is close to the state-of-the-art $5 . 9 1 \%$ (Miyato et al., 2016) despite that we do not use the labels and adversarial training in the feature extraction stage.
+
+The remainder of the paper is organized as follows: Section 2 provides background on RNN-based language models and probabilistic topic models. Section 3 describes the TopicRNN network architecture, its generative process and how to perform inference for it. Section 4 presents per-word perplexity results on the Penn TreeBank dataset and the classification error rate on the IMDB 100K dataset. Finally, we conclude and provide future research directions in Section 5.
+
+# 2 BACKGROUND
+
+We present the background necessary for building the TopicRNN model. We first review RNN-based language modeling, followed by a discussion on the construction of latent topic models.
+
+# 2.1 RECURRENT NEURAL NETWORK-BASED LANGUAGE MODELS
+
+Language modeling is fundamental to many applications. Examples include speech recognition and machine translation. A language model is a probability distribution over a sequence of words in a predefined vocabulary. More formally, let $V$ be a vocabulary set and $y _ { 1 } , . . . , y _ { T }$ a sequence of $T$ words with each $y _ { t } \in V$ . A language model measures the likelihood of a sequence through a joint probability distribution,
+
+$$
+p ( y _ { 1 } , . . . , y _ { T } ) = p ( y _ { 1 } ) \prod _ { t = 2 } ^ { T } p ( y _ { t } | y _ { 1 : t - 1 } ) .
+$$
+
+Traditional $n$ -gram and feed-forward neural network language models (Bengio et al., 2003) typically make Markov assumptions about the sequential dependencies between words, where the chain rule shown above limits conditioning to a fixed-size context window.
+
+RNN-based language models (Mikolov et al., 2011) sidestep this Markov assumption by defining the conditional probability of each word $y _ { t }$ given all the previous words $y _ { 1 : t - 1 }$ through a hidden
+
+state $h _ { t }$ (typically via a softmax function):
+
+$$
+\begin{array} { r } { p ( y _ { t } | y _ { 1 : t - 1 } ) \triangleq p ( y _ { t } | h _ { t } ) , \quad } \\ { h _ { t } = f ( h _ { t - 1 } , x _ { t } ) . } \end{array}
+$$
+
+The function $f ( \cdot )$ can either be a standard RNN cell or a more complex cell such as GRU (Cho et al., 2014) or LSTM (Hochreiter and Schmidhuber, 1997). The input and target words are related via the relation $x _ { t } \equiv y _ { t - 1 }$ . These RNN-based language models have been quite successful (Mikolov et al., 2011; Chelba et al., 2013; Jozefowicz et al., 2016).
+
+While in principle RNN-based models can “remember” arbitrarily long histories if provided enough capacity, in practice such large-scale neural networks can easily encounter difficulties during optimization (Bengio et al., 1994; Pascanu et al., 2013; Sutskever, 2013) or overfitting issues (Srivastava et al., 2014). Finding better ways to model long-range dependencies in language modeling is therefore an open research challenge. As motivated in the introduction, much of the long-range dependency in language comes from semantic coherence, not from syntactic structure which is more of a local phenomenon. Therefore, models that can capture long-range semantic dependencies in language are complementary to RNNs. In the following section, we describe a family of such models called probabilistic topic models.
+
+# 2.2 PROBABILISTIC TOPIC MODELS
+
+Probabilistic topic models are a family of models that can be used to capture global semantic coherency (Blei and Lafferty, 2009). They provide a powerful tool for summarizing, organizing, and navigating document collections. One basic goal of such models is to find groups of words that tend to co-occur together in the same document. These groups of words are called topics and represent a probability distribution that puts most of its mass on this subset of the vocabulary. Documents are then represented as mixtures over these latent topics. Through posterior inference, the learned topics capture the semantic coherence of the words they cluster together (Mimno et al., 2011).
+
+The simplest topic model is latent Dirichlet allocation (LDA) (Blei et al., 2003). It assumes $K$ underlying topics $\beta = \{ \beta _ { 1 } , \dots , \beta _ { K } \}$ , each of which is a distribution over a fixed vocabulary. The generative process of LDA is as follows:
+
+First generate the $K$ topics, $\beta _ { k } \sim _ { i i d }$ Dirichlet $( \tau )$ . Then for each document containing words $y _ { 1 : T }$ independently generate document-level variables and data:
+
+1. Draw a document-specific topic proportion vector $\theta \sim \mathrm { D i r i c h l e t } ( \alpha )$
+
+2. For the tth word in the document,
+
+(a) Draw topic assignment $z _ { t } \sim \mathrm { D i s c r e t e } ( \theta )$ .   
+(b) Draw word $y _ { t } \sim \mathrm { D i s c r e t e } ( \beta _ { z _ { t } } )$ .
+
+Marginalizing each $z _ { t }$ , we obtain the probability of $y _ { 1 : T }$ via a matrix factorization followed by an integration over the latent variable $\theta$ ,
+
+$$
+p ( \boldsymbol { y } _ { 1 : T } | \beta ) = \int p ( \boldsymbol { \theta } ) \prod _ { t = 1 } ^ { T } \sum _ { z _ { t } } p ( z _ { t } | \boldsymbol { \theta } ) p ( \boldsymbol { y } _ { t } | \boldsymbol { z } _ { t } , \beta ) \mathrm { d } \boldsymbol { \theta } = \int p ( \boldsymbol { \theta } ) \prod _ { t = 1 } ^ { T } ( \beta \boldsymbol { \theta } ) _ { \boldsymbol { y } _ { t } } \mathrm { d } \boldsymbol { \theta } .
+$$
+
+In LDA the prior distribution on the topic proportions is a Dirichlet distribution; it can be replaced by many other distributions. For example, the correlated topic model (Blei and Lafferty, 2006) uses a log-normal distribution. Most topic models are “bag of words” models in that word order is ignored. This makes it easier for topic models to capture global semantic information. However, this is also one of the reasons why topic models do not perform well on general-purpose language modeling applications such as word prediction. While bi-gram topic models have been proposed (Wallach, 2006), higher order models quickly become intractable.
+
+Another issue encountered by topic models is that they do not model stop words well. This is because stop words usually do not carry semantic meaning; their appearance is mainly to make the sentence more readable according to the grammar of the language. They also appear frequently in almost every document and can co-occur with almost any word2. In practice, these stop words are chosen using tf-idf (Blei and Lafferty, 2009).
+
+![](images/066405b85921ee3b7abf10fee80b8ece9ba1fc67a6d4c963781ae3bbc1990f81.jpg)  
+Figure 1: (a) The unrolled TopicRNN architecture: $x _ { 1 } , . . . , x _ { 6 }$ are words in the document, $h _ { t }$ is the state of the RNN at time step $t$ , $x _ { i } \equiv y _ { i - 1 }$ , $l _ { 1 } , . . . , l _ { 6 }$ are stop word indicators, and $\theta$ is the latent representation of the input document and is unshaded by convention. (b) The TopicRNN model architecture in its compact form: $l$ is a binary vector that indicates whether each word in the input document is a stop word or not. Here red indicates stop words and blue indicates content words.
+
+# 3 THE TOPICRNN MODEL
+
+We next describe the proposed TopicRNN model. In TopicRNN, latent topic models are used to capture global semantic dependencies so that the RNN can focus its modeling capacity on the local dynamics of the sequences. With this joint modeling, we hope to achieve better overall performance on downstream applications.
+
+The model. TopicRNN is a generative model. For a document containing the words $y _ { 1 : T }$
+
+1. Draw a topic vector3 $\theta \sim N ( 0 , I )$ .
+
+2. Given word $y _ { 1 : t - 1 }$ , for the tth word $y _ { t }$ in the document, (a) Compute hidden state $h _ { t } = f _ { W } ( x _ { t } , h _ { t - 1 } )$ , where we let $x _ { t } \triangleq y _ { t - 1 }$ . (b) Draw stop word indicator $l _ { t } \sim \mathrm { B e r n o u l l i } ( \sigma ( \Gamma ^ { \top } h _ { t } ) )$ , with $\sigma$ the sigmoid function. (c) Draw word $y _ { t } \sim p ( y _ { t } | h _ { t } , \theta , l _ { t } , B )$ , where
+
+$$
+p ( y _ { t } = i | h _ { t } , \theta , l _ { t } , B ) \propto \exp \left( v _ { i } ^ { \top } h _ { t } + ( 1 - l _ { t } ) b _ { i } ^ { \top } \theta \right) .
+$$
+
+The stop word indicator $l _ { t }$ controls how the topic vector $\theta$ affects the output. If $l _ { t } = 1$ (indicating $y _ { t }$ is a stop word), the topic vector $\theta$ has no contribution to the output. Otherwise, we add a bias to favor those words that are more likely to appear when mixing with $\theta$ , as measured by the dot product between $\theta$ and the latent word vector $b _ { i }$ for the $i$ th vocabulary word. As we can see, the longrange semantic information captured by $\theta$ directly affects the output through an additive procedure. Unlike Mikolov and Zweig (2012), the contextual information is not passed to the hidden layer of the RNN. The main reason behind our choice of using the topic vector as bias instead of passing it into the hidden states of the RNN is because it enables us to have a clear separation of the contributions of global semantics and those of local dynamics. The global semantics come from the topics which are meaningful when stop words are excluded. However these stop words are needed for the local dynamics of the language model. We hence achieve this separation of global vs local via a binary decision model for the stop words. It is unclear how to achieve this if we pass the topics to the hidden states of the RNN. This is because the hidden states of the RNN will account for all words (including stop words) whereas the topics exclude stop words.
+
+We show the unrolled graphical representation of TopicRNN in Figure 1(a). We denote all model parameters as $\Theta = \{ \Gamma , \mathbf { \bar { \it V } } , \mathbf { \bar { \it B } } , W , W _ { c } \}$ (see Appendix A.1 for more details). Parameter $W _ { c }$ is for the inference network, which we will introduce below. The observations are the word sequences $y _ { 1 : T }$ and stop word indicators $l _ { 1 : T }$ .4 The log marginal likelihood of the sequence $y _ { 1 : T }$ is
+
+$$
+\log p ( y _ { 1 : T } , l _ { 1 : T } | h _ { t } ) = \log \int p ( \theta ) \prod _ { t = 1 } ^ { T } p ( y _ { t } | h _ { t } , l _ { t } , \theta ) p ( l _ { t } | h _ { t } ) \mathrm { d } \theta .
+$$
+
+Model inference. Direct optimization of Equation 2 is intractable so we use variational inference for approximating this marginal (Jordan et al., 1999). Let $q ( \theta )$ be the variational distribution on the marginalized variable $\theta$ . We construct the variational objective function, also called the evidence lower bound (ELBO), as follows:
+
+$$
+\begin{array} { r l } { \mathcal { L } ( y _ { 1 : T } , l _ { 1 : T } | q ( \theta ) , \Theta ) \triangleq } & { \mathbb { E } _ { q ( \theta ) } \left[ \displaystyle \sum _ { t = 1 } ^ { T } \log p ( y _ { t } | h _ { t } , l _ { t } , \theta ) + \log p ( l _ { t } | h _ { t } ) + \log p ( \theta ) - \log q ( \theta ) \right] } \\ & { \leq \log p ( y _ { 1 : T } , l _ { 1 : T } | h _ { t } , \Theta ) . } \end{array}
+$$
+
+Following the proposed variational autoencoder technique, we choose the form of $q ( \theta )$ to be an inference network using a feed-forward neural network (Kingma and Welling, 2013; Miao et al., 2015). Let $X _ { c } \in \mathcal { N } _ { + } ^ { | V _ { c } | }$ be the term-frequency representation of $y _ { 1 : T }$ excluding stop words (with $V _ { c }$ the vocabulary size without the stop words). The variational autoencoder inference network $q ( \theta | X _ { c } , W _ { c } )$ with parameter $W _ { c }$ is a feed-forward neural network with ReLU activation units that projects $X _ { c }$ into a $K$ -dimensional latent space. Specifically, we have
+
+$$
+\begin{array} { c } { { q ( \theta | X _ { c } , W _ { c } ) = N ( \theta ; \mu ( X _ { c } ) , \mathrm { d i a g } ( \sigma ^ { 2 } ( X _ { c } ) ) ) , } } \\ { { \mu ( X _ { c } ) = W _ { 1 } g ( X _ { c } ) + a _ { 1 } , } } \\ { { \log \sigma ( X _ { c } ) = W _ { 2 } g ( X _ { c } ) + a _ { 2 } , } } \end{array}
+$$
+
+where $g ( \cdot )$ denotes the feed-forward neural network. The weight matrices $W _ { 1 }$ , $W _ { 2 }$ and biases $a _ { 1 }$ , $a _ { 2 }$ are shared across documents. Each document has its own $\mu ( X _ { c } )$ and $\sigma ( X _ { c } )$ resulting in a unique distribution $q ( \theta | X _ { c } )$ for each document. The output of the inference network is a distribution on $\theta$ , which we regard as the summarization of the semantic information, similar to the topic proportions in latent topic models. We show the role of the inference network in Figure 1(b). During training, the parameters of the inference network and the model are jointly learned and updated via truncated backpropagation through time using the Adam algorithm (Kingma and Ba, 2014). We use stochastic samples from $q ( \theta | X _ { c } )$ and the reparameterization trick towards this end (Kingma and Welling, 2013; Rezende et al., 2014).
+
+Generating sequential text and computing perplexity. Suppose we are given a word sequence $y _ { 1 : t - 1 }$ , from which we have an initial estimation of $q ( \theta | X _ { c } )$ . To generate the next word $y _ { t }$ , we compute the probability distribution of $y _ { t }$ given $y _ { 1 : t - 1 }$ in an online fashion. We choose $\theta$ to be a point estimate $\hat { \theta }$ , the mean of its current distribution $q ( \theta | X _ { c } )$ . Marginalizing over the stop word indicator $l _ { t }$ which is unknown prior to observing $y _ { t }$ , the approximate distribution of $y _ { t }$ is
+
+$$
+p ( y _ { t } | y _ { 1 : t - 1 } ) \approx \sum _ { l _ { t } } p ( y _ { t } | h _ { t } , \hat { \theta } , l _ { t } ) p ( l _ { t } | h _ { t } ) .
+$$
+
+The predicted word $y _ { t }$ is a sample from this predictive distribution. We update $q ( \theta | X _ { c } )$ by including $y _ { t }$ to $X _ { c }$ if $y _ { t }$ is not a stop word. However, updating $q ( \theta | X _ { c } )$ after each word prediction is expensive, so we use a sliding window as was done in Mikolov and Zweig (2012). To compute the perplexity, we use the approximate predictive distribution above.
+
+Model Complexity. TopicRNN has a complexity of $O ( H \times H + H \times ( C + K ) + W _ { c } )$ , where $H$ is the size of the hidden layer of the RNN, $C$ is the vocabulary size, $K$ is the dimension of the topic vector, and $W _ { c }$ is the number of parameters of the inference network. The contextual RNN of Mikolov and Zweig (2012) accounts for $O ( H \times H + H \times ( C + K ) )$ , not including the pre-training process, which might require more parameters than the additional $W _ { c }$ in our complexity.
+
+# 4 EXPERIMENTS
+
+We assess the performance of our proposed TopicRNN model on word prediction and sentiment analysis5. For word prediction we use the Penn TreeBank dataset, a standard benchmark for assessing new language models (Marcus et al., 1993). For sentiment analysis we use the IMDB 100k dataset (Maas et al., 2011), also a common benchmark dataset for this application6. We use RNN, LSTM, and GRU cells in our experiments leading to TopicRNN, TopicLSTM, and TopicGRU.
+
+Table 1: Five Topics from the TopicRNN Model with 100 Neurons and 50 Topics on the PTB Data. (The word $s \& p$ below shows as $s p$ in the data.)   
+
+<table><tr><td rowspan=1 colspan=1>Law</td><td rowspan=1 colspan=1>Company</td><td rowspan=1 colspan=1>Parties</td><td rowspan=1 colspan=1>Trading</td><td rowspan=1 colspan=1>Cars</td></tr><tr><td rowspan=1 colspan=1>law</td><td rowspan=1 colspan=1>spending</td><td rowspan=1 colspan=1>democratic</td><td rowspan=1 colspan=1>stock</td><td rowspan=1 colspan=1>gm</td></tr><tr><td rowspan=2 colspan=1>lawyers judge</td><td rowspan=1 colspan=1>sales</td><td rowspan=1 colspan=1>republicans</td><td rowspan=1 colspan=1>s&amp;p</td><td rowspan=1 colspan=1>auto</td></tr><tr><td rowspan=1 colspan=1>advertising</td><td rowspan=1 colspan=1>gop</td><td rowspan=1 colspan=1>price</td><td rowspan=1 colspan=1>ford</td></tr><tr><td rowspan=3 colspan=1>rightsattorneycourt</td><td rowspan=1 colspan=1>employees</td><td rowspan=1 colspan=1>republican</td><td rowspan=1 colspan=1>investor</td><td rowspan=1 colspan=1> jaguar</td></tr><tr><td rowspan=1 colspan=1>state</td><td rowspan=1 colspan=1>senate</td><td rowspan=1 colspan=1>standard</td><td rowspan=1 colspan=1>car</td></tr><tr><td rowspan=1 colspan=1>taxes</td><td rowspan=1 colspan=1>oakland</td><td rowspan=1 colspan=1>chairman</td><td rowspan=2 colspan=1>carsheadquarters</td></tr><tr><td rowspan=2 colspan=1>generalcommon</td><td rowspan=1 colspan=1>fiscal</td><td rowspan=1 colspan=1>highway</td><td rowspan=1 colspan=1>investors</td></tr><tr><td rowspan=1 colspan=1>appropriation</td><td rowspan=1 colspan=1>democrats</td><td rowspan=1 colspan=1>retirement</td><td rowspan=1 colspan=1>british</td></tr><tr><td rowspan=1 colspan=1>mr</td><td rowspan=1 colspan=1>budget</td><td rowspan=1 colspan=1>bill</td><td rowspan=1 colspan=1>holders</td><td rowspan=1 colspan=1>executives</td></tr><tr><td rowspan=1 colspan=1>insurance</td><td rowspan=1 colspan=1>ad</td><td rowspan=1 colspan=1>district</td><td rowspan=1 colspan=1>merrill</td><td rowspan=1 colspan=1>model</td></tr></table>
+
+![](images/8da45187dfe1c757f0a1bb4330ee5817045258c9a1fb0f190875d6a8a302d04c.jpg)  
+Figure 2: Inferred distributions using TopicGRU on three different documents. The content of these documents is added on the appendix. This shows that some of the topics are being picked up depending on the input document.
+
+# 4.1 WORD PREDICTION
+
+We first tested TopicRNN on the word prediction task using the Penn Treebank (PTB) portion of the Wall Street Journal. We use the standard split, where sections 0-20 (930K tokens) are used for training, sections 21-22 (74K tokens) for validation, and sections 23-24 (82K tokens) for testing (Mikolov et al., 2010). We use a vocabulary of size $1 0 K$ that includes the special token unk for rare words and eos that indicates the end of a sentence. TopicRNN takes documents as inputs. We split the PTB data into blocks of 10 sentences to constitute documents as done by (Mikolov and Zweig, 2012). The inference network takes as input the bag-of-words representation of the input document. For that reason, the vocabulary size of the inference network is reduced to 9551 after excluding 449 pre-defined stop words.
+
+In order to compare with previous work on contextual RNNs we trained TopicRNN using different network sizes. We performed word prediction using a recurrent neural network with 10 neurons,
+
+Table 2: TopicRNN and its counterparts exhibit lower perplexity scores across different network sizes than reported in Mikolov and Zweig (2012). Table 2a shows per-word perplexity scores for 10 neurons. Table 2b and Table 2c correspond to per-word perplexity scores for 100 and 300 neurons respectively. These results prove TopicRNN has more generalization capabilities: for example we only need a TopicGRU with 100 neurons to achieve a better perplexity than stacking 2 LSTMs with 200 neurons each: 112.4 vs 115.9)   
+
+<table><tr><td colspan="3">(a)</td><td colspan="3">(b)</td></tr><tr><td>10 Neurons</td><td>Valid</td><td></td><td>100 Neurons</td><td>Valid</td><td>Test</td></tr><tr><td>RNN (no features)</td><td>239.2</td><td>225.0</td><td colspan="2">RNN (no features)</td><td>150.1 142.1</td></tr><tr><td>RNN (LDA features)</td><td>197.3 187.4</td><td colspan="2">RNN (LDA features)</td><td>132.3</td><td>126.4</td></tr><tr><td>TopicRNN</td><td>184.5 172.2</td><td colspan="2">TopicRNN</td><td>128.5</td><td>122.3</td></tr><tr><td>TopicLSTM</td><td>188.0 175.0</td><td colspan="2">TopicLSTM</td><td>126.0</td><td>118.1</td></tr><tr><td>TopicGRU</td><td>178.3 166.7</td><td colspan="2">TopicGRU</td><td>118.3</td><td>112.4</td></tr><tr><td></td><td>(c)</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>300 Neurons</td><td>Valid</td><td>Test</td><td></td><td></td></tr><tr><td></td><td>RNN (no features)</td><td></td><td>124.7</td><td></td><td></td></tr><tr><td></td><td>RNN (LDA features)</td><td></td><td>113.7</td><td></td><td></td></tr><tr><td></td><td>TopicRNN</td><td>118.3</td><td>112.2</td><td></td><td></td></tr><tr><td></td><td>TopicLSTM</td><td>104.1</td><td>99.5</td><td></td><td></td></tr><tr><td></td><td>TopicGRU</td><td>99.6</td><td>97.3</td><td></td><td></td></tr></table>
+
+100 neurons and 300 neurons. For these experiments, we used a multilayer perceptron with 2 hidden layers and 200 hidden units per layer for the inference network. The number of topics was tuned depending on the size of the RNN. For 10 neurons we used 18 topics. For 100 and 300 neurons we found 50 topics to be optimal. We used the validation set to tune the hyperparameters of the model. We used a maximum of 15 epochs for the experiments and performed early stopping using the validation set. For comparison purposes we did not apply dropout and used 1 layer for the RNN and its counterparts in all the word prediction experiments as reported in Table 2. One epoch for 10 neurons takes 2.5 minutes. For 100 neurons, one epoch is completed in less than 4 minutes. Finally, for 300 neurons one epoch takes less than 6 minutes. These experiments were ran on Microsoft Azure NC12 that has 12 cores, 2 Tesla K80 GPUs, and 112 GB memory. First, we show five randomly drawn topics in Table 1. These results correspond to a network with 100 neurons. We also illustrate some inferred topic distributions for several documents from TopicGRU in Figure 2. Similar to standard topic models, these distributions are also relatively peaky.
+
+Next, we compare the performance of TopicRNN to our baseline contextual RNN using perplexity. Perplexity can be thought of as a measure of surprise for a language model. It is defined as the exponential of the average negative log likelihood. Table 2 summarizes the results for different network sizes. We learn three things from these tables. First, the perplexity is reduced the larger the network size. Second, RNNs with context features perform better than RNNs without context features. Third, we see that TopicRNN gives lower perplexity than the previous baseline result reported by Mikolov and Zweig (2012). Note that to compute these perplexity scores for word prediction we use a sliding window to compute $\theta$ as we move along the sequences. The topic vector $\theta$ that is used from the current batch of words is estimated from the previous batch of words. This enables fair comparison to previously reported results (Mikolov and Zweig, 2012).7
+
+Another aspect of the TopicRNN model we studied is its capacity to generate coherent text. To do this, we randomly drew a document from the test set and used this document as seed input to the inference network to compute $\theta$ . Our expectation is that the topics contained in this seed document are reflected in the generated text. Table 3 shows generated text from models learned on the PTB and IMDB datasets. See Appendix A.3 for more examples.
+
+Table 3: Generated text using the TopicRNN model on the PTB (top) and IMDB (bottom).
+
+they believe that they had senior damages to guarantee and frustration of unk stations eos the rush to minimum effect in composite trading the compound base inflated rate before the common charter ’s report eos wells fargo inc. unk of state control funds without openly scheduling the university ’s exchange rate has been downgraded it ’s unk said eos the united cancer & began critical increasing rate of N N at N N to N N are less for the country to trade rate for more than three months $\$ 1$ workers were mixed eos
+
+lee is head to be watched unk month she eos but the acting surprisingly nothing is very good eos i cant believe that he can unk to a role eos may appear of for the stupid killer really to help with unk unk unk if you wan na go to it fell to the plot clearly eos it gets clear of this movie 70 are so bad mexico direction regarding those films eos then go as unk ’s walk and after unk to see him try to unk before that unk with this film
+
+Table 4: Classification error rate on IMDB 100k dataset. TopicRNN provides the state of the art error rate on this dataset.   
+
+<table><tr><td>Model</td><td>Reported Error rate</td></tr><tr><td>BoW (bnc) (Maas et al.,2011)</td><td>12.20%</td></tr><tr><td>BoW (b△ tc) (Maas et al.,2011)</td><td>11.77%</td></tr><tr><td>LDA (Maas et al.,2011)</td><td>32.58%</td></tr><tr><td>Full + BoW (Maas et al.,2011)</td><td>11.67%</td></tr><tr><td>Full + Unlabelled + BoW (Maas et al.,2011)</td><td>11.11%</td></tr><tr><td>WRRBM (Dahl et al., 2012)</td><td>12.58%</td></tr><tr><td>WRRBM+ BoW (bnc) (Dahl et al.,2012)</td><td>10.77%</td></tr><tr><td>MNB-uni (Wang &amp; Manning,2012)</td><td>16.45%</td></tr><tr><td>MNB-bi (Wang &amp; Manning,2012)</td><td>13.41%</td></tr><tr><td>SVM-uni (Wang&amp;Manning,2012)</td><td>13.05%</td></tr><tr><td>SVM-bi (Wang&amp; Manning,2012)</td><td>10.84%</td></tr><tr><td>NBSVM-uni (Wang&amp; Manning,2012)</td><td>11.71%</td></tr><tr><td>seq2-bown-CNN (Johnson &amp; Zhang,2014)</td><td>14.70%</td></tr><tr><td>NBSVM-bi (Wang &amp; Manning,2012)</td><td>8.78%</td></tr><tr><td>Paragraph Vector (Le &amp; Mikolov,2014)</td><td>7.42%</td></tr><tr><td>SA-LSTM with joint training (Dai &amp; Le,2015)</td><td>14.70%</td></tr><tr><td>LSTM with tuning and dropout (Dai&amp;Le,2015)</td><td>13.50%</td></tr><tr><td>LSTM initialized with word2vec embeddings (Dai &amp; Le,2015)</td><td>10.00%</td></tr><tr><td>SA-LSTM with linear gain (Dai&amp;Le,2015)</td><td>9.17%</td></tr><tr><td>LM-TM (Dai&amp; Le,2015)</td><td>7.64%</td></tr><tr><td>SA-LSTM (Dai&amp; Le,2015)</td><td>7.24%</td></tr><tr><td>Virtual Adversarial (Miyato et al. 2016)</td><td>5.91%</td></tr><tr><td>TopicRNN</td><td>6.28%</td></tr></table>
+
+# 4.2 SENTIMENT ANALYSIS
+
+We performed sentiment analysis using TopicRNN as a feature extractor on the IMDB 100K dataset. This data consists of 100,000 movie reviews from the Internet Movie Database (IMDB) website. The data is split into $7 5 \%$ for training and $2 5 \%$ for testing. Among the 75K training reviews, 50K are unlabelled and 25K are labelled as carrying either a positive or a negative sentiment. All 25K test reviews are labelled. We trained TopicRNN on 65K random training reviews and used the remaining 10K reviews for validation. To learn a classifier, we passed the 25K labelled training reviews through the learned TopicRNN model. We then concatenated the output of the inference network and the last state of the RNN for each of these 25K reviews to compute the feature vectors. We then used these feature vectors to train a neural network with one hidden layer, 50 hidden units, and a sigmoid activation function to predict sentiment, exactly as done in Le and Mikolov (2014).
+
+To train the TopicRNN model, we used a vocabulary of size 5,000 and mapped all other words to the unk token. We took out 439 stop words to create the input of the inference network. We used 500 units and 2 layers for the inference network, and used 2 layers and 300 units per-layer for the
+
+![](images/90b18cb1f6082fedea0158cbe0816998581f5bebf4e8ccb074b500e5acc0de58.jpg)  
+Figure 3: Clusters of a sample of 10000 movie reviews from the IMDB 100K dataset using TopicRNN as feature extractor. We used K-Means to cluster the feature vectors. We then used PCA to reduce the dimension to two for visualization purposes. red is a negative review and green is a positive review.
+
+RNN. We chose a step size of 5 and defined 200 topics. We did not use any regularization such as dropout. We trained the model for 13 epochs and used the validation set to tune the hyperparameters of the model and track perplexity for early stopping. This experiment took close to 78 hours on a MacBook pro quad-core with 16GHz of RAM. See Appendix A.4 for the visualization of some of the topics learned from this data.
+
+Table 4 summarizes sentiment classification results from TopicRNN and other methods. Our error rate is $6 . 2 8 \%$ .8 This is close to the state-of-the-art $5 . 9 1 \%$ (Miyato et al., 2016) despite that we do not use the labels and adversarial training in the feature extraction stage. Our approach is most similar to Le and Mikolov (2014), where the features were extracted in a unsupervised way and then a one-layer neural net was trained for classification.
+
+Figure 3 shows the ability of TopicRNN to cluster documents using the feature vectors as created during the sentiment analysis task. Reviews with positive sentiment are coloured in green while reviews carrying negative sentiment are shown in red. This shows that TopicRNN can be used as an unsupervised feature extractor for downstream applications. Table 3 shows generated text from models learned on the PTB and IMDB datasets. See Appendix A.3 for more examples. The overall generated text from IMDB encodes a negative sentiment.
+
+# 5 DISCUSSION AND FUTURE WORK
+
+In this paper we introduced TopicRNN, a RNN-based language model that combines RNNs and latent topics to capture local (syntactic) and global (semantic) dependencies between words. The global dependencies as captured by the latent topics serve as contextual bias to an RNN-based language model. This contextual information is learned jointly with the RNN parameters by maximizing the evidence lower bound of variational inference. TopicRNN yields competitive per-word perplexity on the Penn Treebank dataset compared to previous contextual RNN models. We have reported a competitive classification error rate for sentiment analysis on the IMDB 100K dataset. We have also illustrated the capacity of TopicRNN to generate sensible topics and text.
+
+In future work, we will study the performance of TopicRNN when stop words are dynamically discovered during training. We will also extend TopicRNN to other applications where capturing context is important such as in dialog modeling. If successful, this will allow us to have a model that performs well across different natural language processing applications.
+
+# REFERENCES
+
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+
+# A APPENDIX
+
+# A.1 DIMENSION OF THE PARAMETERS OF THE MODEL:
+
+We use the following notation: C is the vocabulary size (including stop words), H is the number of hidden units of the RNN, K is the number of topics, and E is the dimension of the inference network hidden layer. Table 5 gives the dimension of each of the parameters of the TopicRNN model (ignoring the biases).
+
+Table 5: Dimensions of the parameters of the model.
+
+$$
+\begin{array} { r }  \frac { \left| \begin{array} { l } { \mathbf { U } } \\ { \mathrm { d i m e n s i o n } } \end{array} \right| \begin{array} { l } { \mathbf { U } } \\ { \mathrm { C \cdot X H ~ \left| ~ H ~ \right| ~ H ~ x ~ H ~ \left| ~ H ~ x ~ C ~ \right| ~ K ~ x ~ C ~ \left| ~ \begin{array} { l } { \mathbf { B } } \\ { \mathrm { K ~ x ~ C ~ } } \end{array} \right| ~ H ~ \left| ~ W _ { 1 } ~ \right| ~ W _ { 2 } } } \end{array} } \end{array}
+$$
+
+# A.2 DOCUMENTS USED TO INFER THE DISTRIBUTIONS ON FIGURE 2
+
+Figure on the left: ’the’, ’market’, ’has’, ’grown’, ’relatively’, ’quiet’, ’since’, ’the’, ’china’, ’crisis’, ’but’, ’if ’, ’the’, ’japanese’, ’return’, ’in’, ’force’, ’their’, ’financial’, ’might’, ’could’, ’compensate’, ’to’, ’some’, ’extent’, ’for’, ’local’, ’investors’, "’", ’<unk>’, ’commitment’, ’another’, ’and’, ’critical’, ’factor’, ’is’, ’the’, ’u.s.’, ’hong’, ’kong’, "’s", ’biggest’, ’export’, ’market’, ’even’, ’before’, ’the’, ’china’, ’crisis’, ’weak’, ’u.s.’, ’demand’, ’was’, ’slowing’, ’local’, ’economic’, ’growth’, ’<unk>’, ’strong’, ’consumer’, ’spending’, ’in’, ’the’, ’u.s.’, ’two’, ’years’, ’ago’, ’helped’, ’<unk>’, ’the’, ’local’, ’economy’, ’at’, ’more’, ’than’, ’twice’, ’its’, ’current’, ’rate’, ’indeed’, ’a’, ’few’, ’economists’, ’maintain’, ’that’, ’global’, ’forces’, ’will’, ’continue’, ’to’, ’govern’, ’hong’, ’kong’, "’s", ’economic’, ’<unk>’, ’once’, ’external’, ’conditions’, ’such’, ’as’, ’u.s.’, ’demand’, ’swing’, ’in’, ’the’, ’territory’, "’s", ’favor’, ’they’, ’argue’, ’local’, ’businessmen’, ’will’, ’probably’, ’overcome’, ’their’, ’N’, ’worries’, ’and’, ’continue’, ’doing’, ’business’, ’as’, ’usual’, ’but’, ’economic’, ’arguments’, ’however’, ’solid’, ’wo’, "n’t", ’necessarily’, ’<unk>’, ’hong’, ’kong’, "’s", ’N’, ’million’, ’people’, ’many’, ’are’, ’refugees’, ’having’, ’fled’, ’china’, "’s", ’<unk>’, ’cycles’, ’of ’, ’political’, ’repression’, ’and’, ’poverty’, ’since’, ’the’, ’communist’, ’party’, ’took’, ’power’, ’in’, ’N’, ’as’, ’a’, ’result’, ’many’, ’of ’, ’those’, ’now’, ’planning’, ’to’, ’leave’, ’hong’, ’kong’, ’ca’, "n’t", ’easily’, ’be’, ’<unk>’, ’by’, ’<unk>’, ’improvements’, ’in’, ’the’, ’colony’, "’s", ’political’, ’and’, ’economic’, ’climate’
+
+Figure on the middle: ’it’, ’said’, ’the’, ’man’, ’whom’, ’it’, ’did’, ’not’, ’name’, ’had’, ’been’, ’found’, ’to’, ’have’, ’the’, ’disease’, ’after’, ’hospital’, ’tests’, ’once’, ’the’, ’disease’, ’was’, ’confirmed’, ’all’, ’the’, ’man’, "’s", ’associates’, ’and’, ’family’, ’were’, ’tested’, ’but’, ’none’, ’have’, ’so’, ’far’, ’been’, ’found’, ’to’, ’have’, ’aids’, ’the’, ’newspaper’, ’said’, ’the’, ’man’, ’had’, ’for’, ’a’, ’long’, ’time’, ’had’, ’a’, ’chaotic’, ’sex’, ’life’, ’including’, ’relations’, ’with’, ’foreign’, ’men’, ’the’, ’newspaper’, ’said’, ’the’, ’polish’, ’government’, ’increased’, ’home’, ’electricity’, ’charges’, ’by’, ’N’, ’N’, ’and’, ’doubled’, ’gas’, ’prices’, ’the’, ’official’, ’news’, ’agency’, ’<unk>’, ’said’, ’the’, ’increases’, ’were’, ’intended’, ’to’, ’bring’, ’<unk>’, ’low’, ’energy’, ’charges’, ’into’, ’line’, ’with’, ’production’, ’costs’, ’and’, ’compensate’, ’for’, ’a’, ’rise’, ’in’, ’coal’, ’prices’, ’in’, ’<unk>’, ’news’, ’south’, ’korea’, ’in’, ’establishing’, ’diplomatic’, ’ties’, ’with’, ’poland’, ’yesterday’, ’announced’, ’\$’, ’N’, ’million’, ’in’, ’loans’, ’to’, ’the’, ’financially’, ’strapped’, ’warsaw’, ’government’, ’in’, ’a’, ’victory’, ’for’, ’environmentalists’, ’hungary’, "’s", ’parliament’, ’terminated’, ’a’, ’multibillion-dollar’, ’river’, ’<unk>’, ’dam’, ’being’, ’built’, ’by’, ’<unk>’, ’firms’, ’the’, ’<unk>’, ’dam’, ’was’, ’designed’, ’to’, ’be’, ’<unk>’, ’with’, ’another’, ’dam’, ’now’, ’nearly’, ’complete’, ’N’, ’miles’, ’<unk>’, ’in’, ’czechoslovakia’, ’in’, ’ending’, ’hungary’, "’s", ’part’, ’of ’, ’the’, ’project’, ’parliament’, ’authorized’, ’prime’, ’minister’, ’<unk>’, ’<unk>’, ’to’, ’modify’, ’a’, ’N’, ’agreement’, ’with’, ’czechoslovakia’, ’which’, ’still’, ’wants’, ’the’, ’dam’, ’to’, ’be’, ’built’, ’mr.’, ’<unk>’, ’said’, ’in’, ’parliament’, ’that’, ’czechoslovakia’, ’and’, ’hungary’, ’would’, ’suffer’, ’environmental’, ’damage’, ’if ’, ’the’, ’<unk>’, ’<unk>’, ’were’, ’built’, ’as’, ’planned’
+
+Figure on the right: ’in’, ’hartford’, ’conn.’, ’the’, ’charter’, ’oak’, ’bridge’, ’will’, ’soon’, ’be’, ’replaced’, ’the’, ’<unk>’, ’<unk>’, ’from’, ’its’, ’<unk>’, ’<unk>’, ’to’, ’a’, ’park’, ’<unk>’, ’are’, ’possible’, ’citizens’, ’in’, ’peninsula’, ’ohio’, ’upset’, ’over’, ’changes’, ’to’, ’a’, ’bridge’, ’negotiated’, ’a’, ’deal’, ’the’, ’bottom’, ’half ’, ’of ’, ’the’, ’<unk>’, ’will’, ’be’, ’type’, ’f ’, ’while’, ’the’, ’top’, ’half ’, ’will’, ’have’, ’the’, ’old’, ’bridge’, "’s", ’<unk>’, ’pattern’, ’similarly’, ’highway’, ’engineers’, ’agreed’, ’to’, ’keep’, ’the’, ’old’, ’<unk>’, ’on’, ’the’, ’key’, ’bridge’, ’in’, ’washington’, ’d.c.’, ’as’, ’long’, ’as’, ’they’, ’could’, ’install’, ’a’, ’crash’, ’barrier’, ’between’, ’the’, ’sidewalk’, ’and’, ’the’, ’road’, ’<unk>’, ’<unk>’, ’drink’, ’carrier’, ’competes’, ’with’, ’<unk>’, ’<unk>’, ’<unk>’, ’just’, ’got’, ’easier’, ’or’, ’so’, ’claims’, ’<unk>’, ’corp.’, ’the’, ’maker’, ’of ’, ’the’, ’<unk>’, ’the’, ’chicago’, ’company’, "’s", ’beverage’, ’carrier’, ’meant’, ’to’, ’replace’, ’<unk>’, ’<unk>’, ’at’, ’<unk>’, ’stands’, ’and’, ’fast-food’, ’outlets’, ’resembles’, ’the’, ’plastic’, ’<unk>’, ’used’, ’on’, ’<unk>’, ’of ’, ’beer’, ’only’, ’the’, ’<unk>’, ’hang’, ’from’, ’a’, ’<unk>’, ’of ’, ’<unk>’, ’the’, ’new’, ’carrier’, ’can’, ’<unk>’, ’as’, ’many’, ’as’, ’four’, ’<unk>’, ’at’, ’once’, ’inventor’, ’<unk>’, ’marvin’, ’says’, ’his’, ’design’, ’virtually’, ’<unk>’, ’<unk>’
+
+# A.3 MORE GENERATED TEXT FROM THE MODEL:
+
+We illustrate below some generated text resulting from training TopicRNN on the PTB dataset. Here we used 50 neurons and 100 topics:
+
+Text1: but the refcorp bond fund might have been unk and unk of the point rate eos house in national unk wall restraint in the property pension fund sold willing to zenith was guaranteed by $\$ 8$ N million at short-term rates maturities around unk products eos deposit posted yields slightly
+
+Text2: it had happened by the treasury ’s clinical fund month were under national disappear institutions but secretary nicholas instruments succeed eos and investors age far compound average new york stock exchange bonds typically sold $\$ 1$ shares in the N but paying yields further an average rate of long-term funds
+
+We illustrate below some generated text resulting from training TopicRNN on the IMDB dataset. The settings are the same as for the sentiment analysis experiment:
+
+the film ’s greatest unk unk and it will likely very nice movies to go to unk why various david proves eos the story were always well scary friend high can be a very strange unk unk is in love with it lacks even perfect for unk for some of the worst movies come on a unk gave a rock unk eos whatever let ’s possible eos that kyle can ’t different reasons about the unk and was not what you ’re not a fan of unk unk us rock which unk still in unk ’s music unk one as
+
+# A.4 TOPICS FROM IMDB:
+
+Below we show some topics resulting from the sentiment analysis on the IMDB dataset. The total number of topics is 200. Note here all the topics turn around movies which is expected since all reviews are about movies.
+
+Table 6: Some Topics from the TopicRNN Model on the IMDB Data.   
+
+<table><tr><td>pitt cameron vicious francisco los revolution refuses cheese</td><td>tarantino dramas cards unbearable catches nonsensical cringe lynch</td><td>producing popcorn practice ninja cruise intimate costs alongside</td><td>ken opera carrey kong hills useless lie repeated</td><td>hudson dragged robinson flight awake rolled easier kurosawa</td><td>campbell africa circumstances burton kubrick friday expression struck</td><td>campbell spots dollar cage freeman murphy 2002 scorcese</td></tr></table>

parse/train/rJbbOLcex/rJbbOLcex_content_list.json

@@ -0,0 +1,1233 @@
+[
+    {
+        "type": "text",
+        "text": "TOPICRNN: A RECURRENT NEURAL NETWORKWITH LONG-RANGE SEMANTIC DEPENDENCY",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adji B. Dieng ∗ Columbia University abd2141@columbia.edu ",
+        "bbox": [
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+            170,
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+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chong Wang   \nDeep Learning Technology Center Microsoft Research   \nchowang@microsoft.com   \nJianfeng Gao   \nDeep Learning Technology Center   \nMicrosoft Research   \njfgao@microsoft.com ",
+        "bbox": [
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+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            247,
+            411,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "John Paisley Columbia University jpaisley@columbia.edu ",
+        "bbox": [
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+            290
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            340,
+            544,
+            354
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose TopicRNN, a recurrent neural network (RNN)-based language model designed to directly capture the global semantic meaning relating words in a document via latent topics. Because of their sequential nature, RNNs are good at capturing the local structure of a word sequence – both semantic and syntactic – but might face difficulty remembering long-range dependencies. Intuitively, these long-range dependencies are of semantic nature. In contrast, latent topic models are able to capture the global semantic structure of a document but do not account for word ordering. The proposed TopicRNN model integrates the merits of RNNs and latent topic models: it captures local (syntactic) dependencies using an RNN and global (semantic) dependencies using latent topics. Unlike previous work on contextual RNN language modeling, our model is learned endto-end. Empirical results on word prediction show that TopicRNN outperforms existing contextual RNN baselines. In addition, TopicRNN can be used as an unsupervised feature extractor for documents. We do this for sentiment analysis on the IMDB movie review dataset and report an error rate of $6 . 2 8 \\%$ . This is comparable to the state-of-the-art $5 . 9 1 \\%$ resulting from a semi-supervised approach. Finally, TopicRNN also yields sensible topics, making it a useful alternative to document models such as latent Dirichlet allocation. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When reading a document, short or long, humans have a mechanism that somehow allows them to remember the gist of what they have read so far. Consider the following example: ",
+        "bbox": [
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+            683,
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+            710
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "“The U.S.presidential race isn’t only drawing attention and controversy in the United States – it’s being closely watched across the globe. But what does the rest of the world think about a campaign that has already thrown up one surprise after another? CNN asked 10 journalists for their take on the race so far, and what their country might be hoping for in America’s next — ",
+        "bbox": [
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+            718,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The missing word in the text above is easily predicted by any human to be either President or Commander in Chief or their synonyms. There have been various language models – from simple ngrams to the most recent RNN-based language models – that aim to solve this problem of predicting correctly the subsequent word in an observed sequence of words. ",
+        "bbox": [
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+            780,
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+            835
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A good language model should capture at least two important properties of natural language. The first one is correct syntax. In order to do prediction that enjoys this property, we often only need to consider a few preceding words. Therefore, correct syntax is more of a local property. Word order matters in this case. The second property is the semantic coherence of the prediction. To achieve this, we often need to consider many preceding words to understand the global semantic meaning of the sentence or document. The ordering of the words usually matters much less in this case. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Because they only consider a fixed-size context window of preceding words, traditional $n$ -gram and neural probabilistic language models (Bengio et al., 2003) have difficulties in capturing global semantic information. To overcome this, RNN-based language models (Mikolov et al., 2010; 2011) use hidden states to “remember” the history of a word sequence. However, none of these approaches explicitly model the two main properties of language mentioned above, correct syntax and semantic coherence. Previous work by Chelba and Jelinek (2000) and Gao et al. (2004) exploit syntactic or semantic parsers to capture long-range dependencies in language. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose TopicRNN, a RNN-based language model that is designed to directly capture long-range semantic dependencies via latent topics. These topics provide context to the RNN. Contextual RNNs have received a lot of attention (Mikolov and Zweig, 2012; Mikolov et al., 2014; Ji et al., 2015; Lin et al., 2015; Ji et al., 2016; Ghosh et al., 2016). However, the models closest to ours are the contextual RNN model proposed by Mikolov and Zweig (2012) and its most recent extension to the long-short term memory (LSTM) architecture (Ghosh et al., 2016). These models use pre-trained topic model features as an additional input to the hidden states and/or the output of the RNN. In contrast, TopicRNN does not require pre-trained topic model features and can be learned in an end-to-end fashion. We introduce an automatic way for handling stop words that topic models usually have difficulty dealing with. Under a comparable model size set up, TopicRNN achieves better perplexity scores than the contextual RNN model of Mikolov and Zweig (2012) on the Penn TreeBank dataset 1. Moreover, TopicRNN can be used as an unsupervised feature extractor for downstream applications. For example, we derive document features of the IMDB movie review dataset using TopicRNN for sentiment classification. We reported an error rate of $6 . 2 8 \\%$ . This is close to the state-of-the-art $5 . 9 1 \\%$ (Miyato et al., 2016) despite that we do not use the labels and adversarial training in the feature extraction stage. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The remainder of the paper is organized as follows: Section 2 provides background on RNN-based language models and probabilistic topic models. Section 3 describes the TopicRNN network architecture, its generative process and how to perform inference for it. Section 4 presents per-word perplexity results on the Penn TreeBank dataset and the classification error rate on the IMDB 100K dataset. Finally, we conclude and provide future research directions in Section 5. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We present the background necessary for building the TopicRNN model. We first review RNN-based language modeling, followed by a discussion on the construction of latent topic models. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 RECURRENT NEURAL NETWORK-BASED LANGUAGE MODELS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Language modeling is fundamental to many applications. Examples include speech recognition and machine translation. A language model is a probability distribution over a sequence of words in a predefined vocabulary. More formally, let $V$ be a vocabulary set and $y _ { 1 } , . . . , y _ { T }$ a sequence of $T$ words with each $y _ { t } \\in V$ . A language model measures the likelihood of a sequence through a joint probability distribution, ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb105d729494a65bc20c2a1abc0e6ced6535856ebbc2b8eb770c089325930a6d.jpg",
+        "text": "$$\np ( y _ { 1 } , . . . , y _ { T } ) = p ( y _ { 1 } ) \\prod _ { t = 2 } ^ { T } p ( y _ { t } | y _ { 1 : t - 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Traditional $n$ -gram and feed-forward neural network language models (Bengio et al., 2003) typically make Markov assumptions about the sequential dependencies between words, where the chain rule shown above limits conditioning to a fixed-size context window. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "RNN-based language models (Mikolov et al., 2011) sidestep this Markov assumption by defining the conditional probability of each word $y _ { t }$ given all the previous words $y _ { 1 : t - 1 }$ through a hidden ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "state $h _ { t }$ (typically via a softmax function): ",
+        "bbox": [
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+            103,
+            450,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f528c70bcc842e4e47a5e9712130c83a249fe8ff140ba7526fda432d1202d9a.jpg",
+        "text": "$$\n\\begin{array} { r } { p ( y _ { t } | y _ { 1 : t - 1 } ) \\triangleq p ( y _ { t } | h _ { t } ) , \\quad } \\\\ { h _ { t } = f ( h _ { t - 1 } , x _ { t } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            127,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The function $f ( \\cdot )$ can either be a standard RNN cell or a more complex cell such as GRU (Cho et al., 2014) or LSTM (Hochreiter and Schmidhuber, 1997). The input and target words are related via the relation $x _ { t } \\equiv y _ { t - 1 }$ . These RNN-based language models have been quite successful (Mikolov et al., 2011; Chelba et al., 2013; Jozefowicz et al., 2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While in principle RNN-based models can “remember” arbitrarily long histories if provided enough capacity, in practice such large-scale neural networks can easily encounter difficulties during optimization (Bengio et al., 1994; Pascanu et al., 2013; Sutskever, 2013) or overfitting issues (Srivastava et al., 2014). Finding better ways to model long-range dependencies in language modeling is therefore an open research challenge. As motivated in the introduction, much of the long-range dependency in language comes from semantic coherence, not from syntactic structure which is more of a local phenomenon. Therefore, models that can capture long-range semantic dependencies in language are complementary to RNNs. In the following section, we describe a family of such models called probabilistic topic models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 PROBABILISTIC TOPIC MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            382,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Probabilistic topic models are a family of models that can be used to capture global semantic coherency (Blei and Lafferty, 2009). They provide a powerful tool for summarizing, organizing, and navigating document collections. One basic goal of such models is to find groups of words that tend to co-occur together in the same document. These groups of words are called topics and represent a probability distribution that puts most of its mass on this subset of the vocabulary. Documents are then represented as mixtures over these latent topics. Through posterior inference, the learned topics capture the semantic coherence of the words they cluster together (Mimno et al., 2011). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The simplest topic model is latent Dirichlet allocation (LDA) (Blei et al., 2003). It assumes $K$ underlying topics $\\beta = \\{ \\beta _ { 1 } , \\dots , \\beta _ { K } \\}$ , each of which is a distribution over a fixed vocabulary. The generative process of LDA is as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "First generate the $K$ topics, $\\beta _ { k } \\sim _ { i i d }$ Dirichlet $( \\tau )$ . Then for each document containing words $y _ { 1 : T }$ independently generate document-level variables and data: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. Draw a document-specific topic proportion vector $\\theta \\sim \\mathrm { D i r i c h l e t } ( \\alpha )$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2. For the tth word in the document, ",
+        "bbox": [
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+            619,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(a) Draw topic assignment $z _ { t } \\sim \\mathrm { D i s c r e t e } ( \\theta )$ .   \n(b) Draw word $y _ { t } \\sim \\mathrm { D i s c r e t e } ( \\beta _ { z _ { t } } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Marginalizing each $z _ { t }$ , we obtain the probability of $y _ { 1 : T }$ via a matrix factorization followed by an integration over the latent variable $\\theta$ , ",
+        "bbox": [
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+            685,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/711e0b913a800889a5f3146ad133afd8e1b85a8d8b267b49eb4f4cfd493b941e.jpg",
+        "text": "$$\np ( \\boldsymbol { y } _ { 1 : T } | \\beta ) = \\int p ( \\boldsymbol { \\theta } ) \\prod _ { t = 1 } ^ { T } \\sum _ { z _ { t } } p ( z _ { t } | \\boldsymbol { \\theta } ) p ( \\boldsymbol { y } _ { t } | \\boldsymbol { z } _ { t } , \\beta ) \\mathrm { d } \\boldsymbol { \\theta } = \\int p ( \\boldsymbol { \\theta } ) \\prod _ { t = 1 } ^ { T } ( \\beta \\boldsymbol { \\theta } ) _ { \\boldsymbol { y } _ { t } } \\mathrm { d } \\boldsymbol { \\theta } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In LDA the prior distribution on the topic proportions is a Dirichlet distribution; it can be replaced by many other distributions. For example, the correlated topic model (Blei and Lafferty, 2006) uses a log-normal distribution. Most topic models are “bag of words” models in that word order is ignored. This makes it easier for topic models to capture global semantic information. However, this is also one of the reasons why topic models do not perform well on general-purpose language modeling applications such as word prediction. While bi-gram topic models have been proposed (Wallach, 2006), higher order models quickly become intractable. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Another issue encountered by topic models is that they do not model stop words well. This is because stop words usually do not carry semantic meaning; their appearance is mainly to make the sentence more readable according to the grammar of the language. They also appear frequently in almost every document and can co-occur with almost any word2. In practice, these stop words are chosen using tf-idf (Blei and Lafferty, 2009). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/066405b85921ee3b7abf10fee80b8ece9ba1fc67a6d4c963781ae3bbc1990f81.jpg",
+        "image_caption": [
+            "Figure 1: (a) The unrolled TopicRNN architecture: $x _ { 1 } , . . . , x _ { 6 }$ are words in the document, $h _ { t }$ is the state of the RNN at time step $t$ , $x _ { i } \\equiv y _ { i - 1 }$ , $l _ { 1 } , . . . , l _ { 6 }$ are stop word indicators, and $\\theta$ is the latent representation of the input document and is unshaded by convention. (b) The TopicRNN model architecture in its compact form: $l$ is a binary vector that indicates whether each word in the input document is a stop word or not. Here red indicates stop words and blue indicates content words. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 THE TOPICRNN MODEL ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We next describe the proposed TopicRNN model. In TopicRNN, latent topic models are used to capture global semantic dependencies so that the RNN can focus its modeling capacity on the local dynamics of the sequences. With this joint modeling, we hope to achieve better overall performance on downstream applications. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The model. TopicRNN is a generative model. For a document containing the words $y _ { 1 : T }$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. Draw a topic vector3 $\\theta \\sim N ( 0 , I )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2. Given word $y _ { 1 : t - 1 }$ , for the tth word $y _ { t }$ in the document, (a) Compute hidden state $h _ { t } = f _ { W } ( x _ { t } , h _ { t - 1 } )$ , where we let $x _ { t } \\triangleq y _ { t - 1 }$ . (b) Draw stop word indicator $l _ { t } \\sim \\mathrm { B e r n o u l l i } ( \\sigma ( \\Gamma ^ { \\top } h _ { t } ) )$ , with $\\sigma$ the sigmoid function. (c) Draw word $y _ { t } \\sim p ( y _ { t } | h _ { t } , \\theta , l _ { t } , B )$ , where ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00b36397011641c5d0cab6d7e1520546c081a0ac3c4670e5cfa56f2545c77bcf.jpg",
+        "text": "$$\np ( y _ { t } = i | h _ { t } , \\theta , l _ { t } , B ) \\propto \\exp \\left( v _ { i } ^ { \\top } h _ { t } + ( 1 - l _ { t } ) b _ { i } ^ { \\top } \\theta \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The stop word indicator $l _ { t }$ controls how the topic vector $\\theta$ affects the output. If $l _ { t } = 1$ (indicating $y _ { t }$ is a stop word), the topic vector $\\theta$ has no contribution to the output. Otherwise, we add a bias to favor those words that are more likely to appear when mixing with $\\theta$ , as measured by the dot product between $\\theta$ and the latent word vector $b _ { i }$ for the $i$ th vocabulary word. As we can see, the longrange semantic information captured by $\\theta$ directly affects the output through an additive procedure. Unlike Mikolov and Zweig (2012), the contextual information is not passed to the hidden layer of the RNN. The main reason behind our choice of using the topic vector as bias instead of passing it into the hidden states of the RNN is because it enables us to have a clear separation of the contributions of global semantics and those of local dynamics. The global semantics come from the topics which are meaningful when stop words are excluded. However these stop words are needed for the local dynamics of the language model. We hence achieve this separation of global vs local via a binary decision model for the stop words. It is unclear how to achieve this if we pass the topics to the hidden states of the RNN. This is because the hidden states of the RNN will account for all words (including stop words) whereas the topics exclude stop words. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We show the unrolled graphical representation of TopicRNN in Figure 1(a). We denote all model parameters as $\\Theta = \\{ \\Gamma , \\mathbf { \\bar { \\it V } } , \\mathbf { \\bar { \\it B } } , W , W _ { c } \\}$ (see Appendix A.1 for more details). Parameter $W _ { c }$ is for the inference network, which we will introduce below. The observations are the word sequences $y _ { 1 : T }$ and stop word indicators $l _ { 1 : T }$ .4 The log marginal likelihood of the sequence $y _ { 1 : T }$ is ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/59f607cdf74d7892aeadb55e6c1184f9997975e435c82181ab4a994f3a0941b1.jpg",
+        "text": "$$\n\\log p ( y _ { 1 : T } , l _ { 1 : T } | h _ { t } ) = \\log \\int p ( \\theta ) \\prod _ { t = 1 } ^ { T } p ( y _ { t } | h _ { t } , l _ { t } , \\theta ) p ( l _ { t } | h _ { t } ) \\mathrm { d } \\theta .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Model inference. Direct optimization of Equation 2 is intractable so we use variational inference for approximating this marginal (Jordan et al., 1999). Let $q ( \\theta )$ be the variational distribution on the marginalized variable $\\theta$ . We construct the variational objective function, also called the evidence lower bound (ELBO), as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea181ae190e40daac2bfd289163bd6f3c651ffc6aadde302c7d83b142658b02c.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathcal { L } ( y _ { 1 : T } , l _ { 1 : T } | q ( \\theta ) , \\Theta ) \\triangleq } & { \\mathbb { E } _ { q ( \\theta ) } \\left[ \\displaystyle \\sum _ { t = 1 } ^ { T } \\log p ( y _ { t } | h _ { t } , l _ { t } , \\theta ) + \\log p ( l _ { t } | h _ { t } ) + \\log p ( \\theta ) - \\log q ( \\theta ) \\right] } \\\\ & { \\leq \\log p ( y _ { 1 : T } , l _ { 1 : T } | h _ { t } , \\Theta ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            306,
+            802,
+            369
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following the proposed variational autoencoder technique, we choose the form of $q ( \\theta )$ to be an inference network using a feed-forward neural network (Kingma and Welling, 2013; Miao et al., 2015). Let $X _ { c } \\in \\mathcal { N } _ { + } ^ { | V _ { c } | }$ be the term-frequency representation of $y _ { 1 : T }$ excluding stop words (with $V _ { c }$ the vocabulary size without the stop words). The variational autoencoder inference network $q ( \\theta | X _ { c } , W _ { c } )$ with parameter $W _ { c }$ is a feed-forward neural network with ReLU activation units that projects $X _ { c }$ into a $K$ -dimensional latent space. Specifically, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0bc9353ede68de1e053b5b76acdb2e8c2300f1f1d2b7a5ff4f751e9799e7f731.jpg",
+        "text": "$$\n\\begin{array} { c } { { q ( \\theta | X _ { c } , W _ { c } ) = N ( \\theta ; \\mu ( X _ { c } ) , \\mathrm { d i a g } ( \\sigma ^ { 2 } ( X _ { c } ) ) ) , } } \\\\ { { \\mu ( X _ { c } ) = W _ { 1 } g ( X _ { c } ) + a _ { 1 } , } } \\\\ { { \\log \\sigma ( X _ { c } ) = W _ { 2 } g ( X _ { c } ) + a _ { 2 } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            457,
+            645,
+            512
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $g ( \\cdot )$ denotes the feed-forward neural network. The weight matrices $W _ { 1 }$ , $W _ { 2 }$ and biases $a _ { 1 }$ , $a _ { 2 }$ are shared across documents. Each document has its own $\\mu ( X _ { c } )$ and $\\sigma ( X _ { c } )$ resulting in a unique distribution $q ( \\theta | X _ { c } )$ for each document. The output of the inference network is a distribution on $\\theta$ , which we regard as the summarization of the semantic information, similar to the topic proportions in latent topic models. We show the role of the inference network in Figure 1(b). During training, the parameters of the inference network and the model are jointly learned and updated via truncated backpropagation through time using the Adam algorithm (Kingma and Ba, 2014). We use stochastic samples from $q ( \\theta | X _ { c } )$ and the reparameterization trick towards this end (Kingma and Welling, 2013; Rezende et al., 2014). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Generating sequential text and computing perplexity. Suppose we are given a word sequence $y _ { 1 : t - 1 }$ , from which we have an initial estimation of $q ( \\theta | X _ { c } )$ . To generate the next word $y _ { t }$ , we compute the probability distribution of $y _ { t }$ given $y _ { 1 : t - 1 }$ in an online fashion. We choose $\\theta$ to be a point estimate $\\hat { \\theta }$ , the mean of its current distribution $q ( \\theta | X _ { c } )$ . Marginalizing over the stop word indicator $l _ { t }$ which is unknown prior to observing $y _ { t }$ , the approximate distribution of $y _ { t }$ is ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3d169b4895cd163400c023b5a6bdcf6d5a9e41115d289e5fd2ef50384ef72437.jpg",
+        "text": "$$\np ( y _ { t } | y _ { 1 : t - 1 } ) \\approx \\sum _ { l _ { t } } p ( y _ { t } | h _ { t } , \\hat { \\theta } , l _ { t } ) p ( l _ { t } | h _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The predicted word $y _ { t }$ is a sample from this predictive distribution. We update $q ( \\theta | X _ { c } )$ by including $y _ { t }$ to $X _ { c }$ if $y _ { t }$ is not a stop word. However, updating $q ( \\theta | X _ { c } )$ after each word prediction is expensive, so we use a sliding window as was done in Mikolov and Zweig (2012). To compute the perplexity, we use the approximate predictive distribution above. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Model Complexity. TopicRNN has a complexity of $O ( H \\times H + H \\times ( C + K ) + W _ { c } )$ , where $H$ is the size of the hidden layer of the RNN, $C$ is the vocabulary size, $K$ is the dimension of the topic vector, and $W _ { c }$ is the number of parameters of the inference network. The contextual RNN of Mikolov and Zweig (2012) accounts for $O ( H \\times H + H \\times ( C + K ) )$ , not including the pre-training process, which might require more parameters than the additional $W _ { c }$ in our complexity. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We assess the performance of our proposed TopicRNN model on word prediction and sentiment analysis5. For word prediction we use the Penn TreeBank dataset, a standard benchmark for assessing new language models (Marcus et al., 1993). For sentiment analysis we use the IMDB 100k dataset (Maas et al., 2011), also a common benchmark dataset for this application6. We use RNN, LSTM, and GRU cells in our experiments leading to TopicRNN, TopicLSTM, and TopicGRU. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bc07ea20d0423b2ed7cf7dec07a782c6d6b4db1ebcb588e2d3ddcafbc9986260.jpg",
+        "table_caption": [
+            "Table 1: Five Topics from the TopicRNN Model with 100 Neurons and 50 Topics on the PTB Data. (The word $s \\& p$ below shows as $s p$ in the data.) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Law</td><td rowspan=1 colspan=1>Company</td><td rowspan=1 colspan=1>Parties</td><td rowspan=1 colspan=1>Trading</td><td rowspan=1 colspan=1>Cars</td></tr><tr><td rowspan=1 colspan=1>law</td><td rowspan=1 colspan=1>spending</td><td rowspan=1 colspan=1>democratic</td><td rowspan=1 colspan=1>stock</td><td rowspan=1 colspan=1>gm</td></tr><tr><td rowspan=2 colspan=1>lawyers judge</td><td rowspan=1 colspan=1>sales</td><td rowspan=1 colspan=1>republicans</td><td rowspan=1 colspan=1>s&amp;p</td><td rowspan=1 colspan=1>auto</td></tr><tr><td rowspan=1 colspan=1>advertising</td><td rowspan=1 colspan=1>gop</td><td rowspan=1 colspan=1>price</td><td rowspan=1 colspan=1>ford</td></tr><tr><td rowspan=3 colspan=1>rightsattorneycourt</td><td rowspan=1 colspan=1>employees</td><td rowspan=1 colspan=1>republican</td><td rowspan=1 colspan=1>investor</td><td rowspan=1 colspan=1> jaguar</td></tr><tr><td rowspan=1 colspan=1>state</td><td rowspan=1 colspan=1>senate</td><td rowspan=1 colspan=1>standard</td><td rowspan=1 colspan=1>car</td></tr><tr><td rowspan=1 colspan=1>taxes</td><td rowspan=1 colspan=1>oakland</td><td rowspan=1 colspan=1>chairman</td><td rowspan=2 colspan=1>carsheadquarters</td></tr><tr><td rowspan=2 colspan=1>generalcommon</td><td rowspan=1 colspan=1>fiscal</td><td rowspan=1 colspan=1>highway</td><td rowspan=1 colspan=1>investors</td></tr><tr><td rowspan=1 colspan=1>appropriation</td><td rowspan=1 colspan=1>democrats</td><td rowspan=1 colspan=1>retirement</td><td rowspan=1 colspan=1>british</td></tr><tr><td rowspan=1 colspan=1>mr</td><td rowspan=1 colspan=1>budget</td><td rowspan=1 colspan=1>bill</td><td rowspan=1 colspan=1>holders</td><td rowspan=1 colspan=1>executives</td></tr><tr><td rowspan=1 colspan=1>insurance</td><td rowspan=1 colspan=1>ad</td><td rowspan=1 colspan=1>district</td><td rowspan=1 colspan=1>merrill</td><td rowspan=1 colspan=1>model</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/8da45187dfe1c757f0a1bb4330ee5817045258c9a1fb0f190875d6a8a302d04c.jpg",
+        "image_caption": [
+            "Figure 2: Inferred distributions using TopicGRU on three different documents. The content of these documents is added on the appendix. This shows that some of the topics are being picked up depending on the input document. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 WORD PREDICTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We first tested TopicRNN on the word prediction task using the Penn Treebank (PTB) portion of the Wall Street Journal. We use the standard split, where sections 0-20 (930K tokens) are used for training, sections 21-22 (74K tokens) for validation, and sections 23-24 (82K tokens) for testing (Mikolov et al., 2010). We use a vocabulary of size $1 0 K$ that includes the special token unk for rare words and eos that indicates the end of a sentence. TopicRNN takes documents as inputs. We split the PTB data into blocks of 10 sentences to constitute documents as done by (Mikolov and Zweig, 2012). The inference network takes as input the bag-of-words representation of the input document. For that reason, the vocabulary size of the inference network is reduced to 9551 after excluding 449 pre-defined stop words. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to compare with previous work on contextual RNNs we trained TopicRNN using different network sizes. We performed word prediction using a recurrent neural network with 10 neurons, ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7c2b5e410c27b39175998703cec163978fcee84084a2048cb025bba6a8c63f4e.jpg",
+        "table_caption": [
+            "Table 2: TopicRNN and its counterparts exhibit lower perplexity scores across different network sizes than reported in Mikolov and Zweig (2012). Table 2a shows per-word perplexity scores for 10 neurons. Table 2b and Table 2c correspond to per-word perplexity scores for 100 and 300 neurons respectively. These results prove TopicRNN has more generalization capabilities: for example we only need a TopicGRU with 100 neurons to achieve a better perplexity than stacking 2 LSTMs with 200 neurons each: 112.4 vs 115.9) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">(a)</td><td colspan=\"3\">(b)</td></tr><tr><td>10 Neurons</td><td>Valid</td><td></td><td>100 Neurons</td><td>Valid</td><td>Test</td></tr><tr><td>RNN (no features)</td><td>239.2</td><td>225.0</td><td colspan=\"2\">RNN (no features)</td><td>150.1 142.1</td></tr><tr><td>RNN (LDA features)</td><td>197.3 187.4</td><td colspan=\"2\">RNN (LDA features)</td><td>132.3</td><td>126.4</td></tr><tr><td>TopicRNN</td><td>184.5 172.2</td><td colspan=\"2\">TopicRNN</td><td>128.5</td><td>122.3</td></tr><tr><td>TopicLSTM</td><td>188.0 175.0</td><td colspan=\"2\">TopicLSTM</td><td>126.0</td><td>118.1</td></tr><tr><td>TopicGRU</td><td>178.3 166.7</td><td colspan=\"2\">TopicGRU</td><td>118.3</td><td>112.4</td></tr><tr><td></td><td>(c)</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>300 Neurons</td><td>Valid</td><td>Test</td><td></td><td></td></tr><tr><td></td><td>RNN (no features)</td><td></td><td>124.7</td><td></td><td></td></tr><tr><td></td><td>RNN (LDA features)</td><td></td><td>113.7</td><td></td><td></td></tr><tr><td></td><td>TopicRNN</td><td>118.3</td><td>112.2</td><td></td><td></td></tr><tr><td></td><td>TopicLSTM</td><td>104.1</td><td>99.5</td><td></td><td></td></tr><tr><td></td><td>TopicGRU</td><td>99.6</td><td>97.3</td><td></td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "100 neurons and 300 neurons. For these experiments, we used a multilayer perceptron with 2 hidden layers and 200 hidden units per layer for the inference network. The number of topics was tuned depending on the size of the RNN. For 10 neurons we used 18 topics. For 100 and 300 neurons we found 50 topics to be optimal. We used the validation set to tune the hyperparameters of the model. We used a maximum of 15 epochs for the experiments and performed early stopping using the validation set. For comparison purposes we did not apply dropout and used 1 layer for the RNN and its counterparts in all the word prediction experiments as reported in Table 2. One epoch for 10 neurons takes 2.5 minutes. For 100 neurons, one epoch is completed in less than 4 minutes. Finally, for 300 neurons one epoch takes less than 6 minutes. These experiments were ran on Microsoft Azure NC12 that has 12 cores, 2 Tesla K80 GPUs, and 112 GB memory. First, we show five randomly drawn topics in Table 1. These results correspond to a network with 100 neurons. We also illustrate some inferred topic distributions for several documents from TopicGRU in Figure 2. Similar to standard topic models, these distributions are also relatively peaky. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next, we compare the performance of TopicRNN to our baseline contextual RNN using perplexity. Perplexity can be thought of as a measure of surprise for a language model. It is defined as the exponential of the average negative log likelihood. Table 2 summarizes the results for different network sizes. We learn three things from these tables. First, the perplexity is reduced the larger the network size. Second, RNNs with context features perform better than RNNs without context features. Third, we see that TopicRNN gives lower perplexity than the previous baseline result reported by Mikolov and Zweig (2012). Note that to compute these perplexity scores for word prediction we use a sliding window to compute $\\theta$ as we move along the sequences. The topic vector $\\theta$ that is used from the current batch of words is estimated from the previous batch of words. This enables fair comparison to previously reported results (Mikolov and Zweig, 2012).7 ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Another aspect of the TopicRNN model we studied is its capacity to generate coherent text. To do this, we randomly drew a document from the test set and used this document as seed input to the inference network to compute $\\theta$ . Our expectation is that the topics contained in this seed document are reflected in the generated text. Table 3 shows generated text from models learned on the PTB and IMDB datasets. See Appendix A.3 for more examples. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Generated text using the TopicRNN model on the PTB (top) and IMDB (bottom). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "they believe that they had senior damages to guarantee and frustration of unk stations eos the rush to minimum effect in composite trading the compound base inflated rate before the common charter ’s report eos wells fargo inc. unk of state control funds without openly scheduling the university ’s exchange rate has been downgraded it ’s unk said eos the united cancer & began critical increasing rate of N N at N N to N N are less for the country to trade rate for more than three months $\\$ 1$ workers were mixed eos ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "lee is head to be watched unk month she eos but the acting surprisingly nothing is very good eos i cant believe that he can unk to a role eos may appear of for the stupid killer really to help with unk unk unk if you wan na go to it fell to the plot clearly eos it gets clear of this movie 70 are so bad mexico direction regarding those films eos then go as unk ’s walk and after unk to see him try to unk before that unk with this film ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b82812e8a881f77ff250a1575adfb0816b4b0a4fd5f1610e70ac36118c2101cf.jpg",
+        "table_caption": [
+            "Table 4: Classification error rate on IMDB 100k dataset. TopicRNN provides the state of the art error rate on this dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Reported Error rate</td></tr><tr><td>BoW (bnc) (Maas et al.,2011)</td><td>12.20%</td></tr><tr><td>BoW (b△ tc) (Maas et al.,2011)</td><td>11.77%</td></tr><tr><td>LDA (Maas et al.,2011)</td><td>32.58%</td></tr><tr><td>Full + BoW (Maas et al.,2011)</td><td>11.67%</td></tr><tr><td>Full + Unlabelled + BoW (Maas et al.,2011)</td><td>11.11%</td></tr><tr><td>WRRBM (Dahl et al., 2012)</td><td>12.58%</td></tr><tr><td>WRRBM+ BoW (bnc) (Dahl et al.,2012)</td><td>10.77%</td></tr><tr><td>MNB-uni (Wang &amp; Manning,2012)</td><td>16.45%</td></tr><tr><td>MNB-bi (Wang &amp; Manning,2012)</td><td>13.41%</td></tr><tr><td>SVM-uni (Wang&amp;Manning,2012)</td><td>13.05%</td></tr><tr><td>SVM-bi (Wang&amp; Manning,2012)</td><td>10.84%</td></tr><tr><td>NBSVM-uni (Wang&amp; Manning,2012)</td><td>11.71%</td></tr><tr><td>seq2-bown-CNN (Johnson &amp; Zhang,2014)</td><td>14.70%</td></tr><tr><td>NBSVM-bi (Wang &amp; Manning,2012)</td><td>8.78%</td></tr><tr><td>Paragraph Vector (Le &amp; Mikolov,2014)</td><td>7.42%</td></tr><tr><td>SA-LSTM with joint training (Dai &amp; Le,2015)</td><td>14.70%</td></tr><tr><td>LSTM with tuning and dropout (Dai&amp;Le,2015)</td><td>13.50%</td></tr><tr><td>LSTM initialized with word2vec embeddings (Dai &amp; Le,2015)</td><td>10.00%</td></tr><tr><td>SA-LSTM with linear gain (Dai&amp;Le,2015)</td><td>9.17%</td></tr><tr><td>LM-TM (Dai&amp; Le,2015)</td><td>7.64%</td></tr><tr><td>SA-LSTM (Dai&amp; Le,2015)</td><td>7.24%</td></tr><tr><td>Virtual Adversarial (Miyato et al. 2016)</td><td>5.91%</td></tr><tr><td>TopicRNN</td><td>6.28%</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 SENTIMENT ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We performed sentiment analysis using TopicRNN as a feature extractor on the IMDB 100K dataset. This data consists of 100,000 movie reviews from the Internet Movie Database (IMDB) website. The data is split into $7 5 \\%$ for training and $2 5 \\%$ for testing. Among the 75K training reviews, 50K are unlabelled and 25K are labelled as carrying either a positive or a negative sentiment. All 25K test reviews are labelled. We trained TopicRNN on 65K random training reviews and used the remaining 10K reviews for validation. To learn a classifier, we passed the 25K labelled training reviews through the learned TopicRNN model. We then concatenated the output of the inference network and the last state of the RNN for each of these 25K reviews to compute the feature vectors. We then used these feature vectors to train a neural network with one hidden layer, 50 hidden units, and a sigmoid activation function to predict sentiment, exactly as done in Le and Mikolov (2014). ",
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+    },
+    {
+        "type": "text",
+        "text": "To train the TopicRNN model, we used a vocabulary of size 5,000 and mapped all other words to the unk token. We took out 439 stop words to create the input of the inference network. We used 500 units and 2 layers for the inference network, and used 2 layers and 300 units per-layer for the ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/90b18cb1f6082fedea0158cbe0816998581f5bebf4e8ccb074b500e5acc0de58.jpg",
+        "image_caption": [
+            "Figure 3: Clusters of a sample of 10000 movie reviews from the IMDB 100K dataset using TopicRNN as feature extractor. We used K-Means to cluster the feature vectors. We then used PCA to reduce the dimension to two for visualization purposes. red is a negative review and green is a positive review. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            130,
+            779,
+            261
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "RNN. We chose a step size of 5 and defined 200 topics. We did not use any regularization such as dropout. We trained the model for 13 epochs and used the validation set to tune the hyperparameters of the model and track perplexity for early stopping. This experiment took close to 78 hours on a MacBook pro quad-core with 16GHz of RAM. See Appendix A.4 for the visualization of some of the topics learned from this data. ",
+        "bbox": [
+            174,
+            388,
+            825,
+            458
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4 summarizes sentiment classification results from TopicRNN and other methods. Our error rate is $6 . 2 8 \\%$ .8 This is close to the state-of-the-art $5 . 9 1 \\%$ (Miyato et al., 2016) despite that we do not use the labels and adversarial training in the feature extraction stage. Our approach is most similar to Le and Mikolov (2014), where the features were extracted in a unsupervised way and then a one-layer neural net was trained for classification. ",
+        "bbox": [
+            174,
+            465,
+            825,
+            535
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the ability of TopicRNN to cluster documents using the feature vectors as created during the sentiment analysis task. Reviews with positive sentiment are coloured in green while reviews carrying negative sentiment are shown in red. This shows that TopicRNN can be used as an unsupervised feature extractor for downstream applications. Table 3 shows generated text from models learned on the PTB and IMDB datasets. See Appendix A.3 for more examples. The overall generated text from IMDB encodes a negative sentiment. ",
+        "bbox": [
+            173,
+            542,
+            825,
+            626
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            657,
+            535,
+            675
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper we introduced TopicRNN, a RNN-based language model that combines RNNs and latent topics to capture local (syntactic) and global (semantic) dependencies between words. The global dependencies as captured by the latent topics serve as contextual bias to an RNN-based language model. This contextual information is learned jointly with the RNN parameters by maximizing the evidence lower bound of variational inference. TopicRNN yields competitive per-word perplexity on the Penn Treebank dataset compared to previous contextual RNN models. We have reported a competitive classification error rate for sentiment analysis on the IMDB 100K dataset. We have also illustrated the capacity of TopicRNN to generate sensible topics and text. ",
+        "bbox": [
+            173,
+            698,
+            825,
+            809
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In future work, we will study the performance of TopicRNN when stop words are dynamically discovered during training. We will also extend TopicRNN to other applications where capturing context is important such as in dialog modeling. If successful, this will allow us to have a model that performs well across different natural language processing applications. ",
+        "bbox": [
+            174,
+            810,
+            825,
+            864
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            287,
+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient descent is difficult. IEEE transactions on neural networks, 5(2):157–166, 1994.   \nY. Bengio, R. Ducharme, P. Vincent, and C. Jauvin. A neural probabilistic language model. journal of machine learning research, 3(Feb):1137–1155, 2003.   \nD. Blei and J. Lafferty. Correlated topic models. Advances in neural information processing systems, 18:147, 2006.   \nD. M. Blei and J. D. Lafferty. Topic models. Text mining: classification, clustering, and applications, 10(71):34, 2009.   \nD. M. Blei, A. Y. Ng, and M. I. Jordan. Latent dirichlet allocation. Journal of machine Learning research, 3(Jan):993–1022, 2003.   \nC. Chelba and F. Jelinek. Structured language modeling. Computer Speech & Language, 14(4): 283–332, 2000.   \nC. Chelba, T. Mikolov, M. Schuster, Q. Ge, T. Brants, P. Koehn, and T. Robinson. One billion word benchmark for measuring progress in statistical language modeling. arXiv preprint arXiv:1312.3005, 2013.   \nK. Cho, B. Van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078, 2014.   \nA. M. Dai and Q. V. Le. Semi-supervised sequence learning. In Advances in Neural Information Processing Systems, pages 3079–3087, 2015.   \nJ. Gao, J.-Y. Nie, G. Wu, and G. Cao. Dependence language model for information retrieval. In Proceedings of the 27th annual international ACM SIGIR conference on Research and development in information retrieval, pages 170–177. ACM, 2004.   \nS. Ghosh, O. Vinyals, B. Strope, S. Roy, T. Dean, and L. Heck. Contextual lstm (clstm) models for large scale nlp tasks. arXiv preprint arXiv:1602.06291, 2016.   \nS. Hochreiter and J. Schmidhuber. Long short-term memory. Neural computation, 9(8):1735–1780, 1997.   \nY. Ji, T. Cohn, L. Kong, C. Dyer, and J. Eisenstein. Document context language models. arXiv preprint arXiv:1511.03962, 2015.   \nY. Ji, G. Haffari, and J. Eisenstein. A latent variable recurrent neural network for discourse relation language models. arXiv preprint arXiv:1603.01913, 2016.   \nM. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. Machine learning, 37(2):183–233, 1999.   \nR. Jozefowicz, O. Vinyals, M. Schuster, N. Shazeer, and Y. Wu. Exploring the limits of language modeling. arXiv preprint arXiv:1602.02410, 2016.   \nD. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \nD. P. Kingma and M. Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   \nQ. V. Le and T. Mikolov. Distributed representations of sentences and documents. In ICML, volume 14, pages 1188–1196, 2014.   \nR. Lin, S. Liu, M. Yang, M. Li, M. Zhou, and S. Li. Hierarchical recurrent neural network for document modeling. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 899–907, 2015.   \nA. L. Maas, R. E. Daly, P. T. Pham, D. Huang, A. Y. Ng, and C. Potts. Learning word vectors for sentiment analysis. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies-Volume 1, pages 142–150. Association for Computational Linguistics, 2011.   \nM. P. Marcus, M. A. Marcinkiewicz, and B. Santorini. Building a large annotated corpus of english: The penn treebank. Computational linguistics, 19(2):313–330, 1993.   \nY. Miao, L. Yu, and P. Blunsom. Neural variational inference for text processing. arXiv preprint arXiv:1511.06038, 2015.   \nT. Mikolov and G. Zweig. Context dependent recurrent neural network language model. In SLT, pages 234–239, 2012.   \nT. Mikolov, M. Karafiát, L. Burget, J. Cernocky, and S. Khudanpur. Recurrent neural network based \\` language model. In Interspeech, volume 2, page 3, 2010.   \nT. Mikolov, S. Kombrink, L. Burget, J. Cernock ˇ y, and S. Khudanpur. Extensions of recurrent neural \\` network language model. In 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 5528–5531. IEEE, 2011.   \nT. Mikolov, A. Joulin, S. Chopra, M. Mathieu, and M. Ranzato. Learning longer memory in recurrent neural networks. arXiv preprint arXiv:1412.7753, 2014.   \nD. Mimno, H. M. Wallach, E. Talley, M. Leenders, and A. McCallum. Optimizing semantic coherence in topic models. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pages 262–272. Association for Computational Linguistics, 2011.   \nT. Miyato, A. M. Dai, and I. Goodfellow. Adversarial training methods for semi-supervised text classification. stat, 1050:7, 2016.   \nR. Pascanu, T. Mikolov, and Y. Bengio. On the difficulty of training recurrent neural networks. ICML (3), 28:1310–1318, 2013.   \nD. J. Rezende, S. Mohamed, and D. Wierstra. Stochastic backpropagation and approximate inference in deep generative models. arXiv preprint arXiv:1401.4082, 2014.   \nN. Srivastava, G. E. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1): 1929–1958, 2014.   \nI. Sutskever. Training recurrent neural networks. PhD thesis, University of Toronto, 2013.   \nH. M. Wallach. Topic modeling: beyond bag-of-words. In Proceedings of the 23rd international conference on Machine learning, pages 977–984. ACM, 2006.   \nH. M. Wallach, D. M. Mimno, and A. McCallum. Rethinking lda: Why priors matter. In Advances in neural information processing systems, pages 1973–1981, 2009. ",
+        "bbox": [
+            171,
+            113,
+            828,
+            925
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            97,
+            826,
+            693
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            710,
+            315,
+            727
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 DIMENSION OF THE PARAMETERS OF THE MODEL: ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            741,
+            560,
+            756
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use the following notation: C is the vocabulary size (including stop words), H is the number of hidden units of the RNN, K is the number of topics, and E is the dimension of the inference network hidden layer. Table 5 gives the dimension of each of the parameters of the TopicRNN model (ignoring the biases). ",
+        "bbox": [
+            173,
+            767,
+            825,
+            824
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "",
+        "table_caption": [
+            "Table 5: Dimensions of the parameters of the model. "
+        ],
+        "table_footnote": [],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/072bbaf4119d10d383fd2a9f3f0fb030bad164e2171c6bb5b384915e3816b592.jpg",
+        "text": "$$\n\\begin{array} { r }  \\frac { \\left| \\begin{array} { l } { \\mathbf { U } } \\\\ { \\mathrm { d i m e n s i o n } } \\end{array} \\right| \\begin{array} { l } { \\mathbf { U } } \\\\ { \\mathrm { C \\cdot X H ~ \\left| ~ H ~ \\right| ~ H ~ x ~ H ~ \\left| ~ H ~ x ~ C ~ \\right| ~ K ~ x ~ C ~ \\left| ~ \\begin{array} { l } { \\mathbf { B } } \\\\ { \\mathrm { K ~ x ~ C ~ } } \\end{array} \\right| ~ H ~ \\left| ~ W _ { 1 } ~ \\right| ~ W _ { 2 } } } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            866,
+            743,
+            901
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 DOCUMENTS USED TO INFER THE DISTRIBUTIONS ON FIGURE 2 ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            104,
+            658,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure on the left: ’the’, ’market’, ’has’, ’grown’, ’relatively’, ’quiet’, ’since’, ’the’, ’china’, ’crisis’, ’but’, ’if ’, ’the’, ’japanese’, ’return’, ’in’, ’force’, ’their’, ’financial’, ’might’, ’could’, ’compensate’, ’to’, ’some’, ’extent’, ’for’, ’local’, ’investors’, \"’\", ’<unk>’, ’commitment’, ’another’, ’and’, ’critical’, ’factor’, ’is’, ’the’, ’u.s.’, ’hong’, ’kong’, \"’s\", ’biggest’, ’export’, ’market’, ’even’, ’before’, ’the’, ’china’, ’crisis’, ’weak’, ’u.s.’, ’demand’, ’was’, ’slowing’, ’local’, ’economic’, ’growth’, ’<unk>’, ’strong’, ’consumer’, ’spending’, ’in’, ’the’, ’u.s.’, ’two’, ’years’, ’ago’, ’helped’, ’<unk>’, ’the’, ’local’, ’economy’, ’at’, ’more’, ’than’, ’twice’, ’its’, ’current’, ’rate’, ’indeed’, ’a’, ’few’, ’economists’, ’maintain’, ’that’, ’global’, ’forces’, ’will’, ’continue’, ’to’, ’govern’, ’hong’, ’kong’, \"’s\", ’economic’, ’<unk>’, ’once’, ’external’, ’conditions’, ’such’, ’as’, ’u.s.’, ’demand’, ’swing’, ’in’, ’the’, ’territory’, \"’s\", ’favor’, ’they’, ’argue’, ’local’, ’businessmen’, ’will’, ’probably’, ’overcome’, ’their’, ’N’, ’worries’, ’and’, ’continue’, ’doing’, ’business’, ’as’, ’usual’, ’but’, ’economic’, ’arguments’, ’however’, ’solid’, ’wo’, \"n’t\", ’necessarily’, ’<unk>’, ’hong’, ’kong’, \"’s\", ’N’, ’million’, ’people’, ’many’, ’are’, ’refugees’, ’having’, ’fled’, ’china’, \"’s\", ’<unk>’, ’cycles’, ’of ’, ’political’, ’repression’, ’and’, ’poverty’, ’since’, ’the’, ’communist’, ’party’, ’took’, ’power’, ’in’, ’N’, ’as’, ’a’, ’result’, ’many’, ’of ’, ’those’, ’now’, ’planning’, ’to’, ’leave’, ’hong’, ’kong’, ’ca’, \"n’t\", ’easily’, ’be’, ’<unk>’, ’by’, ’<unk>’, ’improvements’, ’in’, ’the’, ’colony’, \"’s\", ’political’, ’and’, ’economic’, ’climate’ ",
+        "bbox": [
+            176,
+            147,
+            825,
+            383
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure on the middle: ’it’, ’said’, ’the’, ’man’, ’whom’, ’it’, ’did’, ’not’, ’name’, ’had’, ’been’, ’found’, ’to’, ’have’, ’the’, ’disease’, ’after’, ’hospital’, ’tests’, ’once’, ’the’, ’disease’, ’was’, ’confirmed’, ’all’, ’the’, ’man’, \"’s\", ’associates’, ’and’, ’family’, ’were’, ’tested’, ’but’, ’none’, ’have’, ’so’, ’far’, ’been’, ’found’, ’to’, ’have’, ’aids’, ’the’, ’newspaper’, ’said’, ’the’, ’man’, ’had’, ’for’, ’a’, ’long’, ’time’, ’had’, ’a’, ’chaotic’, ’sex’, ’life’, ’including’, ’relations’, ’with’, ’foreign’, ’men’, ’the’, ’newspaper’, ’said’, ’the’, ’polish’, ’government’, ’increased’, ’home’, ’electricity’, ’charges’, ’by’, ’N’, ’N’, ’and’, ’doubled’, ’gas’, ’prices’, ’the’, ’official’, ’news’, ’agency’, ’<unk>’, ’said’, ’the’, ’increases’, ’were’, ’intended’, ’to’, ’bring’, ’<unk>’, ’low’, ’energy’, ’charges’, ’into’, ’line’, ’with’, ’production’, ’costs’, ’and’, ’compensate’, ’for’, ’a’, ’rise’, ’in’, ’coal’, ’prices’, ’in’, ’<unk>’, ’news’, ’south’, ’korea’, ’in’, ’establishing’, ’diplomatic’, ’ties’, ’with’, ’poland’, ’yesterday’, ’announced’, ’\\$’, ’N’, ’million’, ’in’, ’loans’, ’to’, ’the’, ’financially’, ’strapped’, ’warsaw’, ’government’, ’in’, ’a’, ’victory’, ’for’, ’environmentalists’, ’hungary’, \"’s\", ’parliament’, ’terminated’, ’a’, ’multibillion-dollar’, ’river’, ’<unk>’, ’dam’, ’being’, ’built’, ’by’, ’<unk>’, ’firms’, ’the’, ’<unk>’, ’dam’, ’was’, ’designed’, ’to’, ’be’, ’<unk>’, ’with’, ’another’, ’dam’, ’now’, ’nearly’, ’complete’, ’N’, ’miles’, ’<unk>’, ’in’, ’czechoslovakia’, ’in’, ’ending’, ’hungary’, \"’s\", ’part’, ’of ’, ’the’, ’project’, ’parliament’, ’authorized’, ’prime’, ’minister’, ’<unk>’, ’<unk>’, ’to’, ’modify’, ’a’, ’N’, ’agreement’, ’with’, ’czechoslovakia’, ’which’, ’still’, ’wants’, ’the’, ’dam’, ’to’, ’be’, ’built’, ’mr.’, ’<unk>’, ’said’, ’in’, ’parliament’, ’that’, ’czechoslovakia’, ’and’, ’hungary’, ’would’, ’suffer’, ’environmental’, ’damage’, ’if ’, ’the’, ’<unk>’, ’<unk>’, ’were’, ’built’, ’as’, ’planned’ ",
+        "bbox": [
+            176,
+            397,
+            823,
+            675
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure on the right: ’in’, ’hartford’, ’conn.’, ’the’, ’charter’, ’oak’, ’bridge’, ’will’, ’soon’, ’be’, ’replaced’, ’the’, ’<unk>’, ’<unk>’, ’from’, ’its’, ’<unk>’, ’<unk>’, ’to’, ’a’, ’park’, ’<unk>’, ’are’, ’possible’, ’citizens’, ’in’, ’peninsula’, ’ohio’, ’upset’, ’over’, ’changes’, ’to’, ’a’, ’bridge’, ’negotiated’, ’a’, ’deal’, ’the’, ’bottom’, ’half ’, ’of ’, ’the’, ’<unk>’, ’will’, ’be’, ’type’, ’f ’, ’while’, ’the’, ’top’, ’half ’, ’will’, ’have’, ’the’, ’old’, ’bridge’, \"’s\", ’<unk>’, ’pattern’, ’similarly’, ’highway’, ’engineers’, ’agreed’, ’to’, ’keep’, ’the’, ’old’, ’<unk>’, ’on’, ’the’, ’key’, ’bridge’, ’in’, ’washington’, ’d.c.’, ’as’, ’long’, ’as’, ’they’, ’could’, ’install’, ’a’, ’crash’, ’barrier’, ’between’, ’the’, ’sidewalk’, ’and’, ’the’, ’road’, ’<unk>’, ’<unk>’, ’drink’, ’carrier’, ’competes’, ’with’, ’<unk>’, ’<unk>’, ’<unk>’, ’just’, ’got’, ’easier’, ’or’, ’so’, ’claims’, ’<unk>’, ’corp.’, ’the’, ’maker’, ’of ’, ’the’, ’<unk>’, ’the’, ’chicago’, ’company’, \"’s\", ’beverage’, ’carrier’, ’meant’, ’to’, ’replace’, ’<unk>’, ’<unk>’, ’at’, ’<unk>’, ’stands’, ’and’, ’fast-food’, ’outlets’, ’resembles’, ’the’, ’plastic’, ’<unk>’, ’used’, ’on’, ’<unk>’, ’of ’, ’beer’, ’only’, ’the’, ’<unk>’, ’hang’, ’from’, ’a’, ’<unk>’, ’of ’, ’<unk>’, ’the’, ’new’, ’carrier’, ’can’, ’<unk>’, ’as’, ’many’, ’as’, ’four’, ’<unk>’, ’at’, ’once’, ’inventor’, ’<unk>’, ’marvin’, ’says’, ’his’, ’design’, ’virtually’, ’<unk>’, ’<unk>’ ",
+        "bbox": [
+            176,
+            688,
+            825,
+            895
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 MORE GENERATED TEXT FROM THE MODEL: ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            103,
+            524,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We illustrate below some generated text resulting from training TopicRNN on the PTB dataset. Here we used 50 neurons and 100 topics: ",
+        "bbox": [
+            173,
+            128,
+            821,
+            159
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Text1: but the refcorp bond fund might have been unk and unk of the point rate eos house in national unk wall restraint in the property pension fund sold willing to zenith was guaranteed by $\\$ 8$ N million at short-term rates maturities around unk products eos deposit posted yields slightly ",
+        "bbox": [
+            174,
+            170,
+            825,
+            213
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Text2: it had happened by the treasury ’s clinical fund month were under national disappear institutions but secretary nicholas instruments succeed eos and investors age far compound average new york stock exchange bonds typically sold $\\$ 1$ shares in the N but paying yields further an average rate of long-term funds ",
+        "bbox": [
+            174,
+            226,
+            825,
+            282
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We illustrate below some generated text resulting from training TopicRNN on the IMDB dataset. The settings are the same as for the sentiment analysis experiment: ",
+        "bbox": [
+            174,
+            295,
+            821,
+            324
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "the film ’s greatest unk unk and it will likely very nice movies to go to unk why various david proves eos the story were always well scary friend high can be a very strange unk unk is in love with it lacks even perfect for unk for some of the worst movies come on a unk gave a rock unk eos whatever let ’s possible eos that kyle can ’t different reasons about the unk and was not what you ’re not a fan of unk unk us rock which unk still in unk ’s music unk one as ",
+        "bbox": [
+            174,
+            337,
+            825,
+            406
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 TOPICS FROM IMDB: ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            452,
+            367,
+            465
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Below we show some topics resulting from the sentiment analysis on the IMDB dataset. The total number of topics is 200. Note here all the topics turn around movies which is expected since all reviews are about movies. ",
+        "bbox": [
+            176,
+            478,
+            825,
+            520
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/d4c9087e9aa5bd3a5c9e85907aa4cc622aa9e9be51af2426bc1a9c444771ae61.jpg",
+        "table_caption": [
+            "Table 6: Some Topics from the TopicRNN Model on the IMDB Data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>pitt cameron vicious francisco los revolution refuses cheese</td><td>tarantino dramas cards unbearable catches nonsensical cringe lynch</td><td>producing popcorn practice ninja cruise intimate costs alongside</td><td>ken opera carrey kong hills useless lie repeated</td><td>hudson dragged robinson flight awake rolled easier kurosawa</td><td>campbell africa circumstances burton kubrick friday expression struck</td><td>campbell spots dollar cage freeman murphy 2002 scorcese</td></tr></table>",
+        "bbox": [
+            189,
+            563,
+            816,
+            683
+        ],
+        "page_idx": 12
+    }
+]

parse/train/rkzjUoAcFX/rkzjUoAcFX.md

@@ -0,0 +1,249 @@
+# SAMPLE EFFICIENT ADAPTIVE TEXT-TO-SPEECH
+
+Yutian Chen, Yannis Assael, Brendan Shillingford, David Budden, Scott Reed, Heiga Zen, Quan Wang, Luis C. Cobo, Andrew Trask, Ben Laurie, Caglar Gulcehre, Aäron van den Oord, Oriol Vinyals, Nando de Freitas
+
+DeepMind & Google yutianc@google.com
+
+# ABSTRACT
+
+We present a meta-learning approach for adaptive text-to-speech (TTS) with few data. During training, we learn a multi-speaker model using a shared conditional WaveNet core and independent learned embeddings for each speaker. The aim of training is not to produce a neural network with fixed weights, which is then deployed as a TTS system. Instead, the aim is to produce a network that requires few data at deployment time to rapidly adapt to new speakers. We introduce and benchmark three strategies: (i) learning the speaker embedding while keeping the WaveNet core fixed, (ii) fine-tuning the entire architecture with stochastic gradient descent, and (iii) predicting the speaker embedding with a trained neural network encoder. The experiments show that these approaches are successful at adapting the multi-speaker neural network to new speakers, obtaining state-of-the-art results in both sample naturalness and voice similarity with merely a few minutes of audio data from new speakers.
+
+# 1 INTRODUCTION
+
+Training a large model with lots of data and subsequently deploying this model to carry out classification or regression is an important and common methodology in machine learning. It has been particularly successful in speech recognition (Hinton et al., 2012), machine translation (Wu et al., 2016) and image recognition (Krizhevsky et al., 2012; Szegedy et al., 2015). In this textto-speech (TTS) work, we are instead interested in few-shot meta-learning. Here the objective of training with many data is not to learn a fixed-parameter classifier, but rather to learn a “prior” neural network. This prior TTS network can be adapted rapidly, using few data, to produce TTS systems for new speakers at deployment time. That is, the intention is not to learn a fixed final model, but rather to learn a model prior that harnesses few data at deployment time to learn new behaviours rapidly. The output of training is not longer a fixed model, but rather a fast learner.
+
+Biology provides motivation for this line of research. It may be argued that evolution is a slow adaptation process that has resulted in biological machines with the ability to adapt rapidly to new data during their lifetimes. These machines are born with strong priors that facilitate rapid learning.
+
+We consider a meta-learning approach where the model has two types of parameters: task-dependent parameters and task-independent parameters. During training, we learn all of these parameters but discard the task-dependent parameters for deployment. The goal is to use few data to learn the task-dependent parameters for new tasks rapidly.
+
+Task-dependent parameters play a similar role to latent variables in classical probabilistic graphical models. Intuitively, these variables introduce flexibility, thus making it easier to learn the taskindependent parameters. For example, in classical HMMs, knowing the latent variables results in a simple learning problem of estimating the parameters of an exponential-family distribution. In neural networks, this approach also facilitates learning when there is clear data diversity and categorization. We show this for adaptive TTS (Dutoit, 1997; Taylor, 2009). In this setting, speakers correspond to tasks. During training we have many speakers, and it is therefore helpful to have task-dependent parameters to capture speaker-specific voice styles. At the same time, it is useful to have a large model with shared parameters to capture the generic process of mapping text to speech. To this end, we employ the WaveNet model.
+
+![](images/9f5d981002deb733043323407a9d395f743198a87bba66a17a9ed65b562437be.jpg)  
+Figure 1: Architecture of the WaveNet model for few-shot voice adaptation.
+
+WaveNet (van den Oord et al., 2016) is an autoregressive generative model for audio waveforms that has yielded state-of-art performance in speech synthesis. This model was later modified for real-time speech generation via probability density distillation into a feed-forward model (van den Oord et al., 2017). A fundamental limitation of WaveNet is the need for hours of training data for each speaker. In this paper we describe a new WaveNet training procedure that facilitates adaptation to new speakers, allowing the synthesis of new voices from no more than 10 minutes of data with high sample quality.
+
+We propose several extensions of WaveNet for sample-efficient adaptive TTS. First, we present two non-parametric adaptation methods that involve fine-tuning either the speaker embeddings only or all the model parameters given few data from a new speaker. Second, we present a parametric textindependent approach whereby an auxiliary network is trained to predict new speaker embeddings.
+
+The experiments will show that all the proposed approaches, when provided with just a few seconds or minutes of recording, can generate high-fidelity utterances that closely resemble the vocal tract characteristics of a demonstration speaker, particularly when the entire model is fine-tuned end-to-end. When fine-tuning by first estimating the speaker embedding and subsequently fine-tuning the entire model, we achieve state-of-the-art results in terms of sample naturalness and voice similarity to target speakers. These results are robust across speech datasets recorded under different conditions and, moreover, we demonstrate that the generated samples are capable of confusing the state-of-the-art text-independent speaker verification system (Wan et al., 2018).
+
+TTS techniques require hours of high-quality recordings, collected in controlled environments, for each new voice style. Given this high cost, reducing the length of the training dataset could be valuable. For example, it is likely to be very beneficial when attempting to restore the voices of patients who suffer from voice-impairing medical conditions. In these cases, long high quality recordings are scarce.
+
+# 2 WAVENET ARCHITECTURE
+
+WaveNet is an autoregressive model that factorizes the joint probability distribution of a waveform, $\mathbf { x } = \{ x _ { 1 } , \dots , x _ { T } \}$ , into a product of conditional distributions using the probabilistic chain rule:
+
+$$
+p ( \mathbf { x } | \mathbf { h } ; \mathbf { w } ) = \prod _ { t = 1 } ^ { T } p ( x _ { t } | \mathbf { x } _ { 1 : t - 1 } , \mathbf { h } ; \mathbf { w } ) ,
+$$
+
+where $x _ { t }$ is the $t$ -th timestep sample, and $\mathbf { h }$ and w are respectively the conditioning inputs and parameters of the model. To train a multi-speaker WaveNet, the conditioning inputs $\mathbf { h }$ consist of the speaker identity $s$ , the linguistic features l, and the logarithmic fundamental frequency $\mathbf { f } _ { 0 }$ values. l encodes the sequence of phonemes derived from the input text, and $\mathbf { f } _ { 0 }$ controls the dynamics of the pitch in the generated utterance. Given the speaker identity $s$ for each utterance in the dataset, the
+
+![](images/deb042a920784a5e2ca8a31aa69fc598e17c2847ff8bc9195a8edc4979b3a723.jpg)  
+Figure 2: Training (slow, lots of data), adaptation (fast, few data) and inference stages for the SEAALL architecture. The components with bold pink outlines are fine-tuned during the adaptation phase. The purpose of training is to produce a prior. This prior is combined with few data during adaptation to solve a new task. This adapted model is then deployed in the final inference stage.
+
+model is expressed as:
+
+$$
+p ( \mathbf x | \mathbf I , \mathbf f _ { 0 } ; \mathbf e _ { s } , \mathbf w ) = \prod _ { t = 1 } ^ { T } p ( x _ { t } | \mathbf x _ { 1 : t - 1 } , \mathbf I , \mathbf f _ { 0 } ; \mathbf e _ { s } , \mathbf w ) ,
+$$
+
+where a table of speaker embedding vectors $\mathbf { e } _ { s }$ (Embedding in Figure 1) is learned alongside the standard WaveNet parameters. These vectors capture salient voice characteristics across individual speakers, and provide a convenient mechanism for generalizing WaveNet to the few-shot adaptation setting in this paper. The linguistic features l and fundamental frequency values $\mathbf { f } _ { 0 }$ are both time-series with a lower sampling frequency than the waveform. Thus, to be used as local conditioning variables they are upsampled by a transposed convolutional network. During training, l and $\mathbf { f } _ { 0 }$ are extracted by signal processing methods from pairs of training utterance and transcript, and during testing, those values are predicted from text by existing models (Zen et al., 2016).
+
+# 3 FEW-SHOT ADAPTATION WITH WAVENET
+
+In recent years, a large body of literature uses large datasets to train models to learn an input-output mapping that is then used for inference. In contrast, few-shot meta-learning introduces an additional step, adaptation. In this meta-learning setting, the purpose of training becomes to learn a prior. During adaptation, this prior is combined with few data to rapidly learn a new skill; in this case adapting to a new speakers’ voice style. Finally, the new skill is deployed, which in this paper we are referring to as inference. These three stages — training, adaptation and inference — are illustrated in Figure 2.
+
+We present two multi-speaker WaveNet extensions for few-shot voice adaptation. First, we introduce a non-parametric model fine-tuning approach, which involves adapting either the speaker embeddings or all the model parameters using held-aside demonstration data. Second, and for comparison purposes, we use a parametric approach whereby an auxiliary network is trained to predict the embedding vector of a new speaker using the demonstration data.
+
+# 3.1 NON-PARAMETRIC FEW-SHOT ADAPTATION VIA FINE-TUNING
+
+Inspired by few-shot learning we first pre-train a multi-speaker conditional WaveNet model on a large and diverse dataset, as described in Section 2. Subsequently, we fine-tune the model parameters by retraining with respect to held-aside adaptation data.
+
+Training this WaveNet model to maximize the conditional log-likelihood of the generated audio jointly optimizes both the set of speaker parameters $\left\{ \mathbf { e } _ { s } \right\}$ and the shared WaveNet core parameters w. Next, we extend this method to a new speaker by extracting the l and $\mathbf { f } _ { 0 }$ features from their adaptation data waveforms, and randomly initializing a new embedding vector e. We then optimize e such that the demonstration $\{ \mathbf { x } _ { \mathrm { d e m o } } ^ { ( 1 ) } , \hdots , \mathbf { x } _ { \mathrm { d e m o } } ^ { ( n ) } \}$ , paired with features $\{ ( \mathbf { l } _ { \mathrm { d e m o } } ^ { ( 1 ) } , \mathbf { f } _ { 0 , \mathrm { d e m o } } ^ { ( 1 ) } ) , \dots , ( \mathbf { l } _ { \mathrm { d e m o } } ^ { ( n ) } , \mathbf { f } _ { 0 , \mathrm { d e m o } } ^ { ( n ) } ) \}$ , are likely under the model with w fixed (SEA-EMB):
+
+$$
+\mathbf { e } _ { \mathrm { d e m o } } = \underset { \mathbf { e } } { \arg \operatorname* { m a x } } \sum _ { i } \log p ( \mathbf { x } _ { \mathrm { d e m o } } ^ { ( i ) } | \mathbf { l } _ { \mathrm { d e m o } } ^ { ( i ) } , \mathbf { f } _ { 0 , \mathrm { d e m o } } ^ { ( i ) } ; \mathbf { e } , \mathbf { w } ) .
+$$
+
+Alternatively, all of the model parameters may be additionally fine-tuned (SEA-ALL):
+
+$$
+( \mathbf { e } _ { \mathrm { d e m o } } , \mathbf { w } _ { \mathrm { f i n e t u n e d } } ) = \underset { \mathbf { e } , \mathbf { w } } { \arg \operatorname* { m a x } } \sum _ { i } \log p ( \mathbf { x } _ { \mathrm { d e m o } } ^ { ( i ) } | \mathbf { l } _ { \mathrm { d e m o } } ^ { ( i ) } , \mathbf { f } _ { 0 , \mathrm { d e m o } } ^ { ( i ) } ; \mathbf { e } , \mathbf { w } ) .
+$$
+
+Both methods are non-parametric approaches to few-shot voice adaptation as the number of embedding vectors scales with the number of speakers. However, the training processes are slightly different. Because the SEA-EMB method optimizes only a low-dimensional vector, it is far less prone to overfitting, and we are therefore able to retrain the model to convergence even with mere seconds of adaptation data. By contrast, the SEA-ALL has many more parameters that might overfit to the adaptation data. We therefore hold out $1 0 \%$ of our demonstration data for calculating a standard early termination criterion. We also initialize e with the optimal value from the SEA-EMB method, and we find this initialization significantly improves the generalization performance even with a few seconds of adaptation data.
+
+# 3.2 PARAMETRIC FEW-SHOT ADAPTATION USING AN EMBEDDING ENCODER
+
+In contrast to the non-parametric approach, whereby a different embedding vector is fitted for each speaker, one can train an auxiliary encoder network to predict an embedding vector for a new speaker given their demonstration data. Specifically, we model:
+
+$$
+p ( \mathbf { x } | \mathbf { l } , \mathbf { f } _ { 0 } , \mathbf { x } _ { \mathrm { d e m o } } , \mathbf { l } _ { \mathrm { d e m o } } , \mathbf { f } _ { 0 , \mathrm { d e m o } } ; \mathbf { w } ) = \prod _ { t = 1 } ^ { T } p ( x _ { t } | \mathbf { x } _ { 1 : t - 1 } , \mathbf { l } , \mathbf { f } _ { 0 } ; \mathbf { e } ( \mathbf { x } _ { \mathrm { d e m o } } , \mathbf { l } _ { \mathrm { d e m o } } , \mathbf { f } _ { 0 , \mathrm { d e m o } } ) , \mathbf { w } ) ,
+$$
+
+where for each training example, we include a randomly selected demonstration utterance from that speaker in addition to the regular conditioning inputs. The full WaveNet model and the encoder network $\mathbf { e } ( \cdot )$ are trained together from scratch. We refer the reader to the Appendix for further architectural details. This approach (SEA-ENC) exhibits the advantage of being trained in a transcriptindependent setting given only the input waveform, $\mathbf { e } ( \mathbf { x } _ { \mathrm { d e m o } } )$ , and requires negligible computation at adaptation time. However, the learned encoder can also introduce bias when fitting an embedding due to its limited network capacity. As an example, Li et al. (2017) demonstrated a typical scenario whereby speaker identity information can be very quickly extracted with deep models from audio signals. Nonetheless, that the model is less capable of effectively leveraging additional training than approaches based on statistical methods.
+
+# 3.3 REMOVING IDENTITY-RELATED INFORMATION
+
+The linguistic features and fundamental frequencies which are used as inputs contain information specific to an individual speaker. As an example, the average voice pitch in the fundamental frequency sequence is highly speaker-dependent. Instead, we would like these features to be as speaker-independent as possible such that identity is modeled via global conditioning on the speaker embedding. To achieve this, we normalize the fundamental frequency values to have zero mean and unit variance separately for each speaker during training, denoted as $\hat { \bf f } _ { 0 } : = ( { \bf f } - \mathbb { E } [ { \bf f } _ { s } ] ) / \mathrm { s t d } ( { \bf f } _ { s } )$ . As mentioned earlier, at test time, we use an existing model (Zen et al., 2016) to predict $( 1 , \hat { \bf f } _ { 0 } )$ .
+
+# 4 RELATED WORK
+
+Few-shot learning to build models, where one can rapidly learn using only a small amount of available data, is one of the most important open challenges in machine learning. Recent studies have attempted to address the problem of few-shot learning by using deep neural networks, and they have shown promising results on classification tasks in vision (Santoro et al., 2016; Shyam et al., 2017) and language (Vinyals et al., 2016). Few-shot learning can also be leveraged in reinforcement learning, such as by imitating human Atari gameplay from a single recorded action sequence (Pohlen et al., 2018) or online video (Aytar et al., 2018).
+
+Meta-learning offers a sound framework for addressing few-shot learning. Here, an expensive learning process results in machines with the ability to learn rapidly from few data. Meta-learning has a long history (Harlow, 1949; Thrun and Pratt, 2012), and recent studies include efforts to learn optimization processes (Andrychowicz et al., 2016; Chen et al., 2017) that have been shown to extend naturally to the few-shot setting (Ravi and Larochelle, 2016). An alternative approach is model-agnostic meta learning (MAML) (Finn et al., 2017a), which differs by using a fixed optimizer and learning a set of base parameters that can be adapted to minimize any task loss by few steps of gradient descent. This method has shown promise in robotics (Finn et al., 2017b; Yu et al., 2018).
+
+In generative modeling, few-shot learning has been addressed from several perspectives, including matching networks (Bartunov and Vetrov, 2017) and variable inference for memory addressing (Bornschein et al., 2017). Rezende et al. (2016) developed a sequential generative model that extended the Deep Recurrent Attention Writer (DRAW) model (Gregor et al., 2015), and Reed et al. (2018) extended PixelCNN (Van Oord et al., 2016) with neural attention for few-shot auto-regressive density modeling. Veness et al. (2017) presented a gated linear model able to model complex densities from a single pass of a limited dataset.
+
+Early attempts of few-shot adaptation involved the attention models of Reed et al. (2018) and MAML (Finn et al., 2017a), but we found both of these strategies failed to learn informative speaker embedding in our preliminary experiments.
+
+There is growing interest in developing neural TTS models that can be trained end-to-end without the need for hand-crafted representations. In this study we focus on extending the autoregressive WaveNet model (van den Oord et al., 2016; 2017) to the few-shot learning setting to adapt to speakers that were not presented at training time. Other recent neural TTS models include Tacotron 2 (SkerryRyan et al., 2018) (building on (Wang et al., 2017)) which uses WaveNet as a vocoder to invert mel-spectrograms generated by an attentive sequence-to-sequence model. DeepVoice 2 (Gibiansky et al., 2017) (building on (Arık et al., 2017)) introduced a multi-speaker variation of Tacotron that learns a low-dimensional embedding for each speaker, which was further extended in DeepVoice 3 (Ping et al., 2018) to a 2,400 multi-speaker scenario. Unlike WaveNet and DeepVoice, the Char2Wav (Sotelo et al., 2017) and VoiceLoop (Taigman et al., 2018) models produce World Vocoder Features (Morise et al., 2016) instead of generating raw audio signals.
+
+Although many of these systems have produced high-quality samples for speakers present in the training set, generalizing to new speakers given only a few seconds of audio remains a challenge. There have been several concurrent works to address this few-shot learning problem. The VoiceLoop model introduced a novel memory-based architecture that was extended by Nachmani et al. (2018) to few-shot voice style adaptation, by introducing an auxiliary fitting network that predicts the embedding of a new speaker. Jia et al. (2018) extended the Tacotron model for one-shot speaker adaptation by conditioning on a speaker embedding vector extracted from a pretrained speaker identity model of Wan et al. (2018). The most similar approached to our work was proposed by Arik et al. (2018) for the DeepVoice 3 model. They considered both predicting the embedding with an encoding network and fitting the embedding based on a small amount of adaptation data, but the adaptation was applied to a prediction model for mel-spectrograms with a fixed vocoder.
+
+# 5 EVALUATION
+
+In this section, we evaluate the quality of samples of SEA-ALL, SEA-EMB and SEA-ENC. We first measure the naturalness of the generated utterances using the standard Mean Opinion Score (MOS) procedure. Then, we evaluate the similarity of generated and real samples using the subjective MOS test and objectively using a speaker verification system (Wan et al., 2018). Finally, we study these results varying the size of the adaptation dataset.
+
+# 5.1 EXPERIMENTAL SETUP
+
+We train a WaveNet model for each of our three methods using the same dataset, which combines the high-quality LibriSpeech audiobook corpus (Panayotov et al., 2015) and a proprietary speech corpus. The LibriSpeech dataset consists of 2302 speakers from the train speaker subsets and approximately 500 hours of utterances, sampled at a frequency of $1 6 \mathrm { k H z }$ . The proprietary speech corpus consists of 10 American English speakers and approximately 300 hours of utterances, and we down-sample the recording frequency to $1 6 \mathrm { k H z }$ to match LibriSpeech. The multi-speaker WaveNet model has the same architecture as van den Oord et al. (2016) except that we use a 200-dimensional speaker embedding space to model the large diversity of voices.
+
+Table 1: Naturalness of the adapted voices using a 5-scale MOS score (higher is better) with $9 5 \%$ confidence interval on the LibriSpeech and VCTK held-out adaptation datasets. Numbers in bold are the best few-shot learning results on each dataset without statistically significant difference. van den Oord et al. (2016) was trained with 24-hour production quality data, Nachmani et al. (2018) used all samples of each new speaker, Arik et al. (2018) used 10 samples, and Jia et al. (2018) used 5 seconds.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=2>LibriSpeech</td><td rowspan=1 colspan=2>VCTK</td></tr><tr><td rowspan=1 colspan=1>Real utterance</td><td rowspan=1 colspan=2>4.38±0.04</td><td rowspan=1 colspan=2>4.45 ± 0.04</td></tr><tr><td rowspan=1 colspan=1>van den Oord et al. (2016)</td><td rowspan=1 colspan=4>4.21 ± 0.081</td></tr><tr><td rowspan=1 colspan=1>Nachmani et al. (2018)Arik et al. (2018)adapt embeddingadapt whole-modelencoding + fine-tuningJia et al. (2018)trained on LibriSpeech</td><td rowspan=1 colspan=2>2.53 ± 1.114.12 ± 0.05</td><td rowspan=1 colspan=2>3.66±0.842.67 ± 0.103.16 ± 0.092.99 ± 0.124.01 ± 0.06</td></tr><tr><td rowspan=1 colspan=1>Adaptation data size</td><td rowspan=1 colspan=1>10s</td><td rowspan=1 colspan=1>&lt;5m</td><td rowspan=1 colspan=1>10s</td><td rowspan=1 colspan=1>&lt;10m</td></tr><tr><td rowspan=3 colspan=1>SEA-ALL (ours)SEA-EMB (ours)SEA-ENC (ours)</td><td rowspan=1 colspan=1>3.94± 0.08</td><td rowspan=1 colspan=1>4.13± 0.06</td><td rowspan=1 colspan=1>3.92 ± 0.07</td><td rowspan=1 colspan=1>3.92± 0.07</td></tr><tr><td rowspan=2 colspan=1>3.86 ± 0.073.61 ± 0.06</td><td rowspan=1 colspan=1>3.95 ± 0.07</td><td rowspan=1 colspan=1>3.81 ± 0.07</td><td rowspan=2 colspan=1>3.82 ± 0.073.58 ± 0.06</td></tr><tr><td rowspan=1 colspan=1>3.56 ± 0.06</td><td rowspan=1 colspan=1>3.65 ± 0.06</td></tr></table>
+
+Our few-shot model performance is evaluated using two hold-out datasets. First, the LibriSpeech test corpus consists of 39 speakers, with an average of approximately 52 utterances and 5 minutes of audio per speaker. For every test speaker, we randomly split their demonstration utterances into an adaptation set for adapting our WaveNet models and a test set for evaluation. The subset of utterances used for early termination in Section 3.1 is chosen from the adaptation set. There are about 4.2 utterances on average per speaker in the test set and the rest in the adaptation set. Second, we consider a subset of the CSTR VCTK corpus (Veaux et al., 2017) consisting of 21 American English speakers, with approximately 368 utterances and 12 minutes of audio per speaker. We also apply the adaptation/test split with 10 utterances per speaker for test. We emphasize that no data from VCTK was presented to the model at training time. Since our underlying WaveNet model was trained on data largely from LibriSpeech (which was recorded under noisier conditions than VCTK), one might expect that the generated samples on the VCTK dataset contain characteristic artifacts that make generated samples easier to distinguish from real utterances. However, our evaluation using VCTK indicates that our model generalizes effectively and that such artifacts are not detectable. Synthetic utterances are provided on our demo webpage1.
+
+It is worth mentioning, that SEA-ENC requires no adaptation time. Where for SEA-EMB, it takes $5 \sim 1 0 k$ optimizing steps to fit the embedding vector, and an additional $1 0 0 \sim 2 0 0$ steps to fine-tune the entire model using early stopping for SEA-ALL.
+
+# 5.2 NATURALNESS OF THE GENERATED SAMPLES (MOS)
+
+We measure the quality of the generated samples by conducting a MOS test, whereby subjects are asked to rate the naturalness of generated utterances on a five-point Likert Scale (1: Bad, 2: Poor, 3: Fair, 4: Good, 5: Excellent). Furthermore, we compare with other published few-shot TTS systems systems, that were developed in parallel to this work. However, the literature uses varying combinations of training data and evaluation splits making comparison difficult. The results presented are from the closest experimental setups to ours.
+
+Table 1 presents MOS for the adaptation models compared to real utterances. Two different adaptation dataset sizes are considered; $T = 1 0$ seconds, and $T \leq 5$ minutes for LibriSpeech ( $T \leq 1 0$ minutes for VCTK). For reference on $1 6 \mathrm { k H z }$ data, WaveNet trained on a 24-hour production quality speech dataset (van den Oord et al., 2016) achieves a score of 4.21, while for LibriSpeech our best few-shot model attains an MOS score of 4.13 using only 5 minutes of data given a pre-trained multi-speaker model. We note that both fine-tuning models produce overall “good” samples for both the LibriSpeech and VCTK test sets, with SEA-ALL outperforming SEA-EMB in all cases. SEA-ALL is on par with the state-of-the-art performance on both datasets. The addition of extra adaptation data beyond 10 seconds of audio helps performance on LibriSpeech but not VCTK, and the gap between our best model and the real utterance is also wider on VCTK, possibly due to the different recording conditions.
+
+Table 2: Voice similarity of generated voices using a 5-scale MOS score (higher is better) with $9 5 \%$ confidence interval on the LibriSpeech and VCTK held-out adaptation datasets.   
+
+<table><tr><td>Dataset</td><td colspan="2">LibriSpeech</td><td colspan="2">VCTK</td></tr><tr><td>Real utterance</td><td colspan="2">4.30 ± 0.08</td><td colspan="2">4.59 ± 0.06</td></tr><tr><td>Jia et al. (2018) trained on LibriSpeech</td><td colspan="2"></td><td colspan="2"></td></tr><tr><td>Adaptation data size</td><td>3.03 ± 0.09 10s</td><td>&lt;5m</td><td>2.77 ± 0.08 10s</td><td>&lt;10m</td></tr><tr><td>SEA-ALL (ours)</td><td>3.41 ± 0.10</td><td>3.75 ± 0.09</td><td>3.51±0.10</td><td>3.97± 0.09</td></tr><tr><td>SEA-EMB (ours)</td><td>3.42± 0.10</td><td>3.56 ± 0.10</td><td>3.07 ±0.10</td><td>3.18 ± 0.10</td></tr><tr><td>SEA-ENC (ours)</td><td>2.47 ± 0.09</td><td>2.59 ± 0.09</td><td>2.07± 0.08</td><td>2.19 ±0.09</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# 5.3 VOICE SIMILARITY (MOS)
+
+Beside naturalness, we also measure the similarity of the generated and real voices. The quality of similarity is the main evaluation metric for the voice adaptation problem. We first follow the experiment setup of Jia et al. (2018) to run a MOS test for a subjective assessment and then use a speaker verification model for objective evaluation in the next section.
+
+In every trial of this test a subject is presented with a pair of utterances consisting of a real utterance and another real or generated utterance from the same speaker, and is asked to rate the similarity in voice identity using a five-scale score (1: Not at all similar, 2: Slightly similar, 3: Moderately similar, 4: Very similar, 5: Extremely similar).
+
+Table 2 shows the MOS for real utterances and all the adaptation models under two adaptation data time settings on both datasets. Again, the SEA-ALL model outperforms the other two models, and the improvement over SEA-EMB scales with the amount of adaptation data. Particularly, the learned voices on the VCTK dataset achieve an average score of 3.97, demonstrating the generalization performance on a different dataset. As a rough comparisson, because of varying training setups, the state of the art system of Jia et al. (2018) achieves scores of 3.03 for LibriSpeech and 2.77 for VCTK when trained on LibriSpeech. Their model computes the embedding based on the $d$ -vector, similar to our SEA-ENC approach, and performs competitively for the one-shot learning setting, but its performance saturates with 5 seconds of adaptation data, as explained in Section 3.2. We note the gap of similarity scores between SEA-ALL and real utterances, which suggests that although the generated samples sound similar to the target speakers, humans can still tell the difference from real utterances.
+
+# 5.4 VOICE SIMILARITY (SPEAKER VERIFICATION)
+
+We also apply the state-of-the-art text independent speaker verification (TI-SV) model of (Wan et al., 2018) to objectively assess whether the generated samples preserve the acoustic features of the speakers. We calculate the TI-SV $d$ -vector embeddings for generated and real voices. In Figure 3, we visualize the 2-dimensional projection of the $d$ -vectors for a SEA-ALL model trained on $T \leq 5$ minutes of data on the LibriSpeech dataset, and $T \leq 1 0$ minutes on VCTK. There are clear clusters on both datasets, with a strikingly large inter-cluster distance and low intra-cluster separation. This shows both (1) an ease of correctly identifying the speaker associated with a given generated utterance, and (2) the difficulty in differentiating real from synthetic samples.
+
+A similar figure is presented in (Jia et al., 2018), but there the generated and real samples do not overlap. This indicates that the method presented in this paper generates voices that are more indistinguishable from real ones, when measured with the same verification system. In the following subsections, we further analyze these results.
+
+![](images/25835d9a056887378597e359294a34bd5ae213767dfcfe9fd94b57f428e10b4e.jpg)  
+Figure 3: t-SNE visualization of the $d$ -vector embeddings of real and SEA-ALL-generated utterances, for both the LibriSpeech ( $T \leq 5$ mins) and VCTK ( $T \leq 1 0$ mins) evaluation datasets.
+
+![](images/98f0bc71a273760d499d53be792dbb599763f0ee6acccf593e6c6e3964483cb9.jpg)  
+Figure 4: Detection error trade-off (DET) curve for speaker verification in percentage, using the TI-SV speaker verification model (Wan et al., 2018). The utterances were generated using $T \leq 5$ and $T \leq 1 0$ minute samples from LibriSpeech and VCTK respectively. EER is marked with a dot.
+
+# 5.4.1 DISCERNING DIFFERENT SPEAKERS
+
+We first quantify whether generated utterances are attributed to the correct speaker. Following common practice in speaker verification (Wan et al., 2018), we select the hold-out test set of real utterances from test speakers as the enrollment set and compute the centroid of the $d$ -vectors for each speaker $\mathbf { c } _ { i }$ . We then use the adaptation set of test speakers as the verification set. For every verification utterance, we compute the cosine similarity between its $d$ -vector $\mathbf { v }$ and a randomly chosen centroid $\mathbf { c } _ { i }$ . The utterance is accepted as one from speaker $i$ if the similarity is exceeds a given threshold. We repeat the experiments with the same enrollment set and replace the verification set with samples generated by each adaptation method under different data size settings.
+
+Table 3: Equal error rate (EER) of real and few-shot adapted voice samples for evaluation of voice similarity. Varying adaptation dataset sizes were considered.   
+
+<table><tr><td>Dataset</td><td colspan="3">LibriSpeech</td><td colspan="3">VCTK</td></tr><tr><td>Real utterance</td><td colspan="3"></td><td colspan="3">2.79</td></tr><tr><td>Adaptation data size</td><td>10s</td><td>2.47 1m</td><td>&lt;5m</td><td>10s</td><td>1m</td><td>&lt;10m</td></tr><tr><td>SEA-ALL (ours)</td><td>3.17</td><td>2.47</td><td>1.85</td><td>7.34</td><td>5.02</td><td>4.33</td></tr><tr><td>SEA-EMB (ours)</td><td>3.26</td><td>2.92</td><td>2.74</td><td>10.18</td><td>9.91</td><td>10.24</td></tr><tr><td>SEA-ENC (ours)</td><td>10.73</td><td>9.77</td><td>9.42</td><td>27.20</td><td>25.34</td><td>25.23</td></tr></table>
+
+![](images/98e9962d2247562b56b5bc63c271f368b6c5f06cc98187458e53c3a0d08484cb.jpg)  
+Figure 5: Cosine similarity of real and generated utterances to the real enrollment set.
+
+In our setup we fix the enrollment set together with the speaker verification model from (Wan et al., 2018), and study the performance of different verification sets that are either from real utterances or generated by a TTS system. Table 3 lists the equal error rate (EER) of the verification model with real and generated verification utterances, and Figure 4 shows the detection error trade-off (DET) curves for a more thorough inspection. Figure 4 only shows the adaptation models with the maximum data size setting ( $T \leq 5$ minutes for LibriSpeech and $\leq 1 0$ minutes for VCTK). The results for other data sizes are provided in Appendix B.
+
+We find that SEA-ALL outperforms the other two approaches, and the error rate decreases clearly with the size of demonstration data. Noticeably, the EER of SEA-ALL is even lower than the real utterance on the LibriSpeech dataset with sufficient adaptation data. A possible explanation is that the generated samples might be concentrated closer to the centroid of a speaker’s embeddings than real speech with larger variance across utterances. Our SEA-EMB model performs better than SEA-ENC. Additionally, the benefit of more demonstration data is less significant than for SEA-ALL in both of these models.
+
+# 5.4.2 DISCERNING REAL FROM GENERATED UTTERANCES
+
+In this section, we compare the generated samples and the real utterances of the speaker being imitated. Figure 5 shows the box-plot of the cosine similarity between the embedding centroids of test speakers’ enrollment set and (1) real utterances from the same speaker, (2) real utterances from a different speaker, and (3) generated utterances adapted to the same speaker. Consistent with the observations from the previous subsection, SEA-ALL performs best.
+
+We further consider an adversarial scenario for speaker verification. In contrast to the previous standard speaker verification setup where we now select a verification utterance with either a real utterance from the same speaker or a synthetic sample from a model adapted to the same speaker. Under this setup, the speaker verification system is challenged by synthetic samples and acts as a classifier for real versus generated utterances. The ROC curve of this setup is shown in Figure 6 and the models are using the maximum data size setting. Other data size settings can be found in Appendix C. If the generated samples are indistinguishable from real utterances, the ROC curve approaches the diagonal line (that is, the verification system fails to separate real and generated voices). Importantly, SEA-ALL manages to confuse the verification system especially for the VCTK dataset where the ROC curve is almost inline with the diagonal line with an AUC of 0.56.
+
+# 6 CONCLUSION
+
+This paper studied three variants of meta-learning for sample efficient adaptive TTS. The adaptation method that fine-tunes the entire model, with the speaker embedding vector first optimized, shows impressive performance even with only 10 seconds of audio from new speakers. When adapted with a few minutes of data, our model matches the state-of-the-art performance in sample naturalness. Moreover, it outperforms other recent works in matching the new speaker’s voice. We also demonstrated that the generated samples achieved a similar level of voice similarity to real utterances from the same speaker, when measured by a text independent speaker verification model.
+
+![](images/f0fa25bb0c694d1710032d4d6bcf1673e38d8e8be977b04dbdb18b1488b7e4bf.jpg)  
+Figure 6: ROC curve for real versus generated utterance detection. The utterances were generated using models with 5 and 10 minutes of training data per speaker from LibriSpeech and VCTK respectively. Lower curve indicate that the verification system is having a harder time distinguishing real from generated samples.
+
+Our paper considers the adaptation to new voices with clean, high-quality training data collected in a controlled environment. The few-shot learning of voices with noisy data is beyond the scope of this paper and remains a challenging open research problem.
+
+A requirement for less training data to adapt the model, however, increases the potential for both beneficial and harmful applications of text-to-speech technologies such as the creation of synthesized media. While the requirements for this particular model (including the high-quality training data collected in a controlled environment and equally high quality data from the speakers to which we adapt, as described in Section 5.1) present barriers to misuse, more research must be conducted to mitigate and detect instances of misuse of text-to-speech technologies in general.
+
+# REFERENCES
+
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+T. Pohlen, B. Piot, T. Hester, M. G. Azar, D. Horgan, D. Budden, G. Barth-Maron, H. van Hasselt, J. Quan, M. Vecerík, et al. Observe and look further: Achieving consistent performance on atari. ˇ arXiv preprint arXiv:1805.11593, 2018.   
+Y. Aytar, T. Pfaff, D. Budden, T. L. Paine, Z. Wang, and N. de Freitas. Playing hard exploration games by watching youtube. arXiv preprint arXiv:1805.11592, 2018.   
+H. F. Harlow. The formation of learning sets. Psychological review, 56(1):51, 1949.   
+S. Thrun and L. Pratt. Learning to learn. Springer Science & Business Media, 2012.   
+M. Andrychowicz, M. Denil, S. Gomez, M. W. Hoffman, D. Pfau, T. Schaul, B. Shillingford, and N. De Freitas. Learning to learn by gradient descent by gradient descent. In Advances in Neural Information Processing Systems, pages 3981–3989, 2016.   
+Y. Chen, M. W. Hoffman, S. G. Colmenarejo, M. Denil, T. P. Lillicrap, M. Botvinick, and N. Freitas. Learning to learn without gradient descent by gradient descent. In International Conference on Machine Learning, pages 748–756, 2017.   
+S. Ravi and H. Larochelle. Optimization as a model for few-shot learning. International Conference on Learning Representations, 2016.   
+C. Finn, P. Abbeel, and S. Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In International Conference on Machine Learning, pages 1126–1135, 2017a.   
+C. Finn, T. Yu, T. Zhang, P. Abbeel, and S. Levine. One-shot visual imitation learning via meta-learning. In Conference on Robot Learning, pages 357–368, 2017b.   
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+J. Bornschein, A. Mnih, D. Zoran, and D. J. Rezende. Variational memory addressing in generative models. In Advances in Neural Information Processing Systems, pages 3923–3932, 2017.   
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+K. Gregor, I. Danihelka, A. Graves, D. Rezende, and D. Wierstra. DRAW: A recurrent neural network for image generation. In International Conference on Machine Learning, pages 1462–1471, 2015.   
+S. Reed, Y. Chen, T. Paine, A. van den Oord, S. M. Eslami, D. Rezende, O. Vinyals, and N. de Freitas. Few-shot autoregressive density estimation: Towards learning to learn distributions. In International Conference on Learning Representations, 2018.   
+A. Van Oord, N. Kalchbrenner, and K. Kavukcuoglu. Pixel recurrent neural networks. In International Conference on Machine Learning, pages 1747–1756, 2016.   
+J. Veness, T. Lattimore, A. Bhoopchand, A. Grabska-Barwinska, C. Mattern, and P. Toth. Online learning with gated linear networks. arXiv preprint arXiv:1712.01897, 2017.   
+R. Skerry-Ryan, E. Battenberg, Y. Xiao, Y. Wang, D. Stanton, J. Shor, R. J. Weiss, R. Clark, and R. A. Saurous. Towards end-to-end prosody transfer for expressive speech synthesis with tacotron. arXiv preprint arXiv:1803.09047, 2018.   
+Y. Wang, R. Skerry-Ryan, D. Stanton, Y. Wu, R. J. Weiss, N. Jaitly, Z. Yang, Y. Xiao, Z. Chen, S. Bengio, Q. Le, Y. Agiomyrgiannakis, R. Clark, and R. A. Saurous. Tacotron: Towards end-to-end speech synthesis. In INTERSPEECH, pages 4006–4010, 2017.   
+A. Gibiansky, S. Arik, G. Diamos, J. Miller, K. Peng, W. Ping, J. Raiman, and Y. Zhou. Deep voice 2: Multispeaker neural text-to-speech. In Advances in Neural Information Processing Systems, pages 2962–2970, 2017.   
+S. Ö. Arık, M. Chrzanowski, A. Coates, G. Diamos, A. Gibiansky, Y. Kang, X. Li, J. Miller, A. Ng, J. Raiman, et al. Deep voice: Real-time neural text-to-speech. In International Conference on Machine Learning, pages 195–204, 2017.   
+W. Ping, K. Peng, A. Gibiansky, S. O. Arik, A. Kannan, S. Narang, J. Raiman, and J. Miller. Deep voice 3: 2000-speaker neural text-to-speech. In International Conference on Learning Representations, 2018.   
+J. Sotelo, S. Mehri, K. Kumar, J. F. Santos, K. Kastner, A. Courville, and Y. Bengio. Char2wav: End-to-end speech synthesis. In International Conference on Learning Representations Workshop, 2017.   
+Y. Taigman, L. Wolf, A. Polyak, and E. Nachmani. Voiceloop: Voice fitting and synthesis via a phonological loop. In International Conference on Learning Representations, 2018.   
+M. Morise, F. Yokomori, and K. Ozawa. World: a vocoder-based high-quality speech synthesis system for real-time applications. IEICE transactions on Information and Systems, 99(7):1877–1884, 2016.   
+E. Nachmani, A. Polyak, Y. Taigman, and L. Wolf. Fitting new speakers based on a short untranscribed sample. arXiv preprint arXiv:1802.06984, 2018.   
+Y. Jia, Y. Zhang, R. J. Weiss, Q. Wang, J. Shen, F. Ren, Z. Chen, P. Nguyen, R. Pang, I. L. Moreno, et al. Transfer learning from speaker verification to multispeaker text-to-speech synthesis. arXiv preprint arXiv:1806.04558, 2018.   
+S. O. Arik, J. Chen, K. Peng, W. Ping, and Y. Zhou. Neural voice cloning with a few samples. arXiv preprint arXiv:1802.06006, 2018.   
+V. Panayotov, G. Chen, D. Povey, and S. Khudanpur. Librispeech: an asr corpus based on public domain audio books. In International Conference on Acoustics, Speech and Signal Processing, pages 5206–5210. IEEE, 2015.   
+C. Veaux, J. Yamagishi, K. MacDonald, et al. CSTR VCTK corpus: English multi-speaker corpus for CSTR Voice Cloning Toolkit, 2017.
+
+# A EMBEDDING ENCODER
+
+Our encoding network is illustrated as the summation of two sub-network outputs in Figure 7. The first subnetwork is a pre-trained speaker verification model (TI-SV) (Wan et al., 2018), comprising 3 LSTM layers and a single linear layer. This model maps a waveform sequence of arbitrary length to a fixed 256-dimensional $d$ -vector with a sliding window, and is trained from approximately 36M utterances from 18K speakers extracted from anonymized voice search logs. On top of this we add a shallow MLP to project the output $d$ -vector to the speaker embedding space. The second sub-network comprises 16 1-D convolutional layers. This network reduces the temporal resolution to $2 5 6 \mathrm { m s }$ per frame (for $1 6 \mathrm { k H z }$ audio), then averages across time and projects into the speaker embedding space. The purpose of this network is to extract residual speaker information present in the demonstration waveforms but not captured by the pre-trained TI-SV model.
+
+![](images/638613b970a6c64f4a74f956a7d54690174a5b5f336d824dfaddb8259396a994.jpg)  
+Figure 7: Encoder network architecture for predicting speaker embeddings.
+
+# B DET CURVES VARYING TRAINING DATA SIZES
+
+Here we provide the DET curves of speaker verification problem for models with different training data sizes in addition to those shown in Section 5.4.1.
+
+![](images/4f1d1338c1f0e0f09c422dfa47c76d6b7e03222ce7d8d78699c42da5ff087753.jpg)  
+Figure 8: Detection error trade-off (DET) curve for speaker verification, using the TI-SV speaker verification model (Wan et al., 2018). The utterances were generated using 1 minute or 10 seconds of utterance from LibriSpeech and VCTK. EER is marked with a dot.
+
+# C ROC CURVES VARYING TRAINING DATA SIZES
+
+We provide the ROC curves of the speaker verification problem with adversarial examples from adaptation models with different training data sizes in addition to those shown in Section 5.4.2.
+
+![](images/86029fcc717f345efbaac9ed850be92da33678364e0d7ea289c74d03f8f65d86.jpg)  
+Figure 9: ROC curve for real vs. generated utterance detection. The utterances were generated using 1 minute or 10 seconds of utterance from LibriSpeech and VCTK. Lower curve suggests harder to distinguish real from generated samples.

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+        "text": "SAMPLE EFFICIENT ADAPTIVE TEXT-TO-SPEECH ",
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+        "text": "Yutian Chen, Yannis Assael, Brendan Shillingford, David Budden, Scott Reed, Heiga Zen, Quan Wang, Luis C. Cobo, Andrew Trask, Ben Laurie, Caglar Gulcehre, Aäron van den Oord, Oriol Vinyals, Nando de Freitas ",
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+        "text": "DeepMind & Google yutianc@google.com ",
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+        "text": "ABSTRACT ",
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+        "text": "We present a meta-learning approach for adaptive text-to-speech (TTS) with few data. During training, we learn a multi-speaker model using a shared conditional WaveNet core and independent learned embeddings for each speaker. The aim of training is not to produce a neural network with fixed weights, which is then deployed as a TTS system. Instead, the aim is to produce a network that requires few data at deployment time to rapidly adapt to new speakers. We introduce and benchmark three strategies: (i) learning the speaker embedding while keeping the WaveNet core fixed, (ii) fine-tuning the entire architecture with stochastic gradient descent, and (iii) predicting the speaker embedding with a trained neural network encoder. The experiments show that these approaches are successful at adapting the multi-speaker neural network to new speakers, obtaining state-of-the-art results in both sample naturalness and voice similarity with merely a few minutes of audio data from new speakers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Training a large model with lots of data and subsequently deploying this model to carry out classification or regression is an important and common methodology in machine learning. It has been particularly successful in speech recognition (Hinton et al., 2012), machine translation (Wu et al., 2016) and image recognition (Krizhevsky et al., 2012; Szegedy et al., 2015). In this textto-speech (TTS) work, we are instead interested in few-shot meta-learning. Here the objective of training with many data is not to learn a fixed-parameter classifier, but rather to learn a “prior” neural network. This prior TTS network can be adapted rapidly, using few data, to produce TTS systems for new speakers at deployment time. That is, the intention is not to learn a fixed final model, but rather to learn a model prior that harnesses few data at deployment time to learn new behaviours rapidly. The output of training is not longer a fixed model, but rather a fast learner. ",
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+    {
+        "type": "text",
+        "text": "Biology provides motivation for this line of research. It may be argued that evolution is a slow adaptation process that has resulted in biological machines with the ability to adapt rapidly to new data during their lifetimes. These machines are born with strong priors that facilitate rapid learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider a meta-learning approach where the model has two types of parameters: task-dependent parameters and task-independent parameters. During training, we learn all of these parameters but discard the task-dependent parameters for deployment. The goal is to use few data to learn the task-dependent parameters for new tasks rapidly. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Task-dependent parameters play a similar role to latent variables in classical probabilistic graphical models. Intuitively, these variables introduce flexibility, thus making it easier to learn the taskindependent parameters. For example, in classical HMMs, knowing the latent variables results in a simple learning problem of estimating the parameters of an exponential-family distribution. In neural networks, this approach also facilitates learning when there is clear data diversity and categorization. We show this for adaptive TTS (Dutoit, 1997; Taylor, 2009). In this setting, speakers correspond to tasks. During training we have many speakers, and it is therefore helpful to have task-dependent parameters to capture speaker-specific voice styles. At the same time, it is useful to have a large model with shared parameters to capture the generic process of mapping text to speech. To this end, we employ the WaveNet model. ",
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+        "image_caption": [
+            "Figure 1: Architecture of the WaveNet model for few-shot voice adaptation. "
+        ],
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+    {
+        "type": "text",
+        "text": "WaveNet (van den Oord et al., 2016) is an autoregressive generative model for audio waveforms that has yielded state-of-art performance in speech synthesis. This model was later modified for real-time speech generation via probability density distillation into a feed-forward model (van den Oord et al., 2017). A fundamental limitation of WaveNet is the need for hours of training data for each speaker. In this paper we describe a new WaveNet training procedure that facilitates adaptation to new speakers, allowing the synthesis of new voices from no more than 10 minutes of data with high sample quality. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose several extensions of WaveNet for sample-efficient adaptive TTS. First, we present two non-parametric adaptation methods that involve fine-tuning either the speaker embeddings only or all the model parameters given few data from a new speaker. Second, we present a parametric textindependent approach whereby an auxiliary network is trained to predict new speaker embeddings. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The experiments will show that all the proposed approaches, when provided with just a few seconds or minutes of recording, can generate high-fidelity utterances that closely resemble the vocal tract characteristics of a demonstration speaker, particularly when the entire model is fine-tuned end-to-end. When fine-tuning by first estimating the speaker embedding and subsequently fine-tuning the entire model, we achieve state-of-the-art results in terms of sample naturalness and voice similarity to target speakers. These results are robust across speech datasets recorded under different conditions and, moreover, we demonstrate that the generated samples are capable of confusing the state-of-the-art text-independent speaker verification system (Wan et al., 2018). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "TTS techniques require hours of high-quality recordings, collected in controlled environments, for each new voice style. Given this high cost, reducing the length of the training dataset could be valuable. For example, it is likely to be very beneficial when attempting to restore the voices of patients who suffer from voice-impairing medical conditions. In these cases, long high quality recordings are scarce. ",
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+        "type": "text",
+        "text": "2 WAVENET ARCHITECTURE ",
+        "text_level": 1,
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+        "text": "WaveNet is an autoregressive model that factorizes the joint probability distribution of a waveform, $\\mathbf { x } = \\{ x _ { 1 } , \\dots , x _ { T } \\}$ , into a product of conditional distributions using the probabilistic chain rule: ",
+        "bbox": [
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+        "img_path": "images/9e53e45896942f70b51e2a9ec83b374879e319d72bef4afd0082229a1dac204a.jpg",
+        "text": "$$\np ( \\mathbf { x } | \\mathbf { h } ; \\mathbf { w } ) = \\prod _ { t = 1 } ^ { T } p ( x _ { t } | \\mathbf { x } _ { 1 : t - 1 } , \\mathbf { h } ; \\mathbf { w } ) ,\n$$",
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+        "type": "text",
+        "text": "where $x _ { t }$ is the $t$ -th timestep sample, and $\\mathbf { h }$ and w are respectively the conditioning inputs and parameters of the model. To train a multi-speaker WaveNet, the conditioning inputs $\\mathbf { h }$ consist of the speaker identity $s$ , the linguistic features l, and the logarithmic fundamental frequency $\\mathbf { f } _ { 0 }$ values. l encodes the sequence of phonemes derived from the input text, and $\\mathbf { f } _ { 0 }$ controls the dynamics of the pitch in the generated utterance. Given the speaker identity $s$ for each utterance in the dataset, the ",
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+        "image_caption": [
+            "Figure 2: Training (slow, lots of data), adaptation (fast, few data) and inference stages for the SEAALL architecture. The components with bold pink outlines are fine-tuned during the adaptation phase. The purpose of training is to produce a prior. This prior is combined with few data during adaptation to solve a new task. This adapted model is then deployed in the final inference stage. "
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+        "type": "text",
+        "text": "model is expressed as: ",
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+        "img_path": "images/e0a9d1cb1b6c97dfa272b8fda95d9c346ca558cc89278112b4d95a33062d0b9d.jpg",
+        "text": "$$\np ( \\mathbf x | \\mathbf I , \\mathbf f _ { 0 } ; \\mathbf e _ { s } , \\mathbf w ) = \\prod _ { t = 1 } ^ { T } p ( x _ { t } | \\mathbf x _ { 1 : t - 1 } , \\mathbf I , \\mathbf f _ { 0 } ; \\mathbf e _ { s } , \\mathbf w ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where a table of speaker embedding vectors $\\mathbf { e } _ { s }$ (Embedding in Figure 1) is learned alongside the standard WaveNet parameters. These vectors capture salient voice characteristics across individual speakers, and provide a convenient mechanism for generalizing WaveNet to the few-shot adaptation setting in this paper. The linguistic features l and fundamental frequency values $\\mathbf { f } _ { 0 }$ are both time-series with a lower sampling frequency than the waveform. Thus, to be used as local conditioning variables they are upsampled by a transposed convolutional network. During training, l and $\\mathbf { f } _ { 0 }$ are extracted by signal processing methods from pairs of training utterance and transcript, and during testing, those values are predicted from text by existing models (Zen et al., 2016). ",
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+        "type": "text",
+        "text": "3 FEW-SHOT ADAPTATION WITH WAVENET ",
+        "text_level": 1,
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+        "text": "In recent years, a large body of literature uses large datasets to train models to learn an input-output mapping that is then used for inference. In contrast, few-shot meta-learning introduces an additional step, adaptation. In this meta-learning setting, the purpose of training becomes to learn a prior. During adaptation, this prior is combined with few data to rapidly learn a new skill; in this case adapting to a new speakers’ voice style. Finally, the new skill is deployed, which in this paper we are referring to as inference. These three stages — training, adaptation and inference — are illustrated in Figure 2. ",
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+        "text": "We present two multi-speaker WaveNet extensions for few-shot voice adaptation. First, we introduce a non-parametric model fine-tuning approach, which involves adapting either the speaker embeddings or all the model parameters using held-aside demonstration data. Second, and for comparison purposes, we use a parametric approach whereby an auxiliary network is trained to predict the embedding vector of a new speaker using the demonstration data. ",
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+        "type": "text",
+        "text": "3.1 NON-PARAMETRIC FEW-SHOT ADAPTATION VIA FINE-TUNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Inspired by few-shot learning we first pre-train a multi-speaker conditional WaveNet model on a large and diverse dataset, as described in Section 2. Subsequently, we fine-tune the model parameters by retraining with respect to held-aside adaptation data. ",
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+        "text": "Training this WaveNet model to maximize the conditional log-likelihood of the generated audio jointly optimizes both the set of speaker parameters $\\left\\{ \\mathbf { e } _ { s } \\right\\}$ and the shared WaveNet core parameters w. Next, we extend this method to a new speaker by extracting the l and $\\mathbf { f } _ { 0 }$ features from their adaptation data waveforms, and randomly initializing a new embedding vector e. We then optimize e such that the demonstration $\\{ \\mathbf { x } _ { \\mathrm { d e m o } } ^ { ( 1 ) } , \\hdots , \\mathbf { x } _ { \\mathrm { d e m o } } ^ { ( n ) } \\}$ , paired with features $\\{ ( \\mathbf { l } _ { \\mathrm { d e m o } } ^ { ( 1 ) } , \\mathbf { f } _ { 0 , \\mathrm { d e m o } } ^ { ( 1 ) } ) , \\dots , ( \\mathbf { l } _ { \\mathrm { d e m o } } ^ { ( n ) } , \\mathbf { f } _ { 0 , \\mathrm { d e m o } } ^ { ( n ) } ) \\}$ , are likely under the model with w fixed (SEA-EMB): ",
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+    {
+        "type": "equation",
+        "img_path": "images/86de3e25710e841d481feb582ffcccae9eeebc2cbf34daa2fe7bcc9254459e9a.jpg",
+        "text": "$$\n\\mathbf { e } _ { \\mathrm { d e m o } } = \\underset { \\mathbf { e } } { \\arg \\operatorname* { m a x } } \\sum _ { i } \\log p ( \\mathbf { x } _ { \\mathrm { d e m o } } ^ { ( i ) } | \\mathbf { l } _ { \\mathrm { d e m o } } ^ { ( i ) } , \\mathbf { f } _ { 0 , \\mathrm { d e m o } } ^ { ( i ) } ; \\mathbf { e } , \\mathbf { w } ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Alternatively, all of the model parameters may be additionally fine-tuned (SEA-ALL): ",
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+    {
+        "type": "equation",
+        "img_path": "images/0b5350e891667241cead504bcf6f484314ae91b18ce03b1c53de2aaf5abff5da.jpg",
+        "text": "$$\n( \\mathbf { e } _ { \\mathrm { d e m o } } , \\mathbf { w } _ { \\mathrm { f i n e t u n e d } } ) = \\underset { \\mathbf { e } , \\mathbf { w } } { \\arg \\operatorname* { m a x } } \\sum _ { i } \\log p ( \\mathbf { x } _ { \\mathrm { d e m o } } ^ { ( i ) } | \\mathbf { l } _ { \\mathrm { d e m o } } ^ { ( i ) } , \\mathbf { f } _ { 0 , \\mathrm { d e m o } } ^ { ( i ) } ; \\mathbf { e } , \\mathbf { w } ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Both methods are non-parametric approaches to few-shot voice adaptation as the number of embedding vectors scales with the number of speakers. However, the training processes are slightly different. Because the SEA-EMB method optimizes only a low-dimensional vector, it is far less prone to overfitting, and we are therefore able to retrain the model to convergence even with mere seconds of adaptation data. By contrast, the SEA-ALL has many more parameters that might overfit to the adaptation data. We therefore hold out $1 0 \\%$ of our demonstration data for calculating a standard early termination criterion. We also initialize e with the optimal value from the SEA-EMB method, and we find this initialization significantly improves the generalization performance even with a few seconds of adaptation data. ",
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+    {
+        "type": "text",
+        "text": "3.2 PARAMETRIC FEW-SHOT ADAPTATION USING AN EMBEDDING ENCODER",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In contrast to the non-parametric approach, whereby a different embedding vector is fitted for each speaker, one can train an auxiliary encoder network to predict an embedding vector for a new speaker given their demonstration data. Specifically, we model: ",
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+        "img_path": "images/4a43fd8120af53c6f975f67174d46b28f6fce5969ab232e04f996ac2d13721b4.jpg",
+        "text": "$$\np ( \\mathbf { x } | \\mathbf { l } , \\mathbf { f } _ { 0 } , \\mathbf { x } _ { \\mathrm { d e m o } } , \\mathbf { l } _ { \\mathrm { d e m o } } , \\mathbf { f } _ { 0 , \\mathrm { d e m o } } ; \\mathbf { w } ) = \\prod _ { t = 1 } ^ { T } p ( x _ { t } | \\mathbf { x } _ { 1 : t - 1 } , \\mathbf { l } , \\mathbf { f } _ { 0 } ; \\mathbf { e } ( \\mathbf { x } _ { \\mathrm { d e m o } } , \\mathbf { l } _ { \\mathrm { d e m o } } , \\mathbf { f } _ { 0 , \\mathrm { d e m o } } ) , \\mathbf { w } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where for each training example, we include a randomly selected demonstration utterance from that speaker in addition to the regular conditioning inputs. The full WaveNet model and the encoder network $\\mathbf { e } ( \\cdot )$ are trained together from scratch. We refer the reader to the Appendix for further architectural details. This approach (SEA-ENC) exhibits the advantage of being trained in a transcriptindependent setting given only the input waveform, $\\mathbf { e } ( \\mathbf { x } _ { \\mathrm { d e m o } } )$ , and requires negligible computation at adaptation time. However, the learned encoder can also introduce bias when fitting an embedding due to its limited network capacity. As an example, Li et al. (2017) demonstrated a typical scenario whereby speaker identity information can be very quickly extracted with deep models from audio signals. Nonetheless, that the model is less capable of effectively leveraging additional training than approaches based on statistical methods. ",
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+    {
+        "type": "text",
+        "text": "3.3 REMOVING IDENTITY-RELATED INFORMATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The linguistic features and fundamental frequencies which are used as inputs contain information specific to an individual speaker. As an example, the average voice pitch in the fundamental frequency sequence is highly speaker-dependent. Instead, we would like these features to be as speaker-independent as possible such that identity is modeled via global conditioning on the speaker embedding. To achieve this, we normalize the fundamental frequency values to have zero mean and unit variance separately for each speaker during training, denoted as $\\hat { \\bf f } _ { 0 } : = ( { \\bf f } - \\mathbb { E } [ { \\bf f } _ { s } ] ) / \\mathrm { s t d } ( { \\bf f } _ { s } )$ . As mentioned earlier, at test time, we use an existing model (Zen et al., 2016) to predict $( 1 , \\hat { \\bf f } _ { 0 } )$ . ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Few-shot learning to build models, where one can rapidly learn using only a small amount of available data, is one of the most important open challenges in machine learning. Recent studies have attempted to address the problem of few-shot learning by using deep neural networks, and they have shown promising results on classification tasks in vision (Santoro et al., 2016; Shyam et al., 2017) and language (Vinyals et al., 2016). Few-shot learning can also be leveraged in reinforcement learning, such as by imitating human Atari gameplay from a single recorded action sequence (Pohlen et al., 2018) or online video (Aytar et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "Meta-learning offers a sound framework for addressing few-shot learning. Here, an expensive learning process results in machines with the ability to learn rapidly from few data. Meta-learning has a long history (Harlow, 1949; Thrun and Pratt, 2012), and recent studies include efforts to learn optimization processes (Andrychowicz et al., 2016; Chen et al., 2017) that have been shown to extend naturally to the few-shot setting (Ravi and Larochelle, 2016). An alternative approach is model-agnostic meta learning (MAML) (Finn et al., 2017a), which differs by using a fixed optimizer and learning a set of base parameters that can be adapted to minimize any task loss by few steps of gradient descent. This method has shown promise in robotics (Finn et al., 2017b; Yu et al., 2018). ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In generative modeling, few-shot learning has been addressed from several perspectives, including matching networks (Bartunov and Vetrov, 2017) and variable inference for memory addressing (Bornschein et al., 2017). Rezende et al. (2016) developed a sequential generative model that extended the Deep Recurrent Attention Writer (DRAW) model (Gregor et al., 2015), and Reed et al. (2018) extended PixelCNN (Van Oord et al., 2016) with neural attention for few-shot auto-regressive density modeling. Veness et al. (2017) presented a gated linear model able to model complex densities from a single pass of a limited dataset. ",
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+    {
+        "type": "text",
+        "text": "Early attempts of few-shot adaptation involved the attention models of Reed et al. (2018) and MAML (Finn et al., 2017a), but we found both of these strategies failed to learn informative speaker embedding in our preliminary experiments. ",
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+    {
+        "type": "text",
+        "text": "There is growing interest in developing neural TTS models that can be trained end-to-end without the need for hand-crafted representations. In this study we focus on extending the autoregressive WaveNet model (van den Oord et al., 2016; 2017) to the few-shot learning setting to adapt to speakers that were not presented at training time. Other recent neural TTS models include Tacotron 2 (SkerryRyan et al., 2018) (building on (Wang et al., 2017)) which uses WaveNet as a vocoder to invert mel-spectrograms generated by an attentive sequence-to-sequence model. DeepVoice 2 (Gibiansky et al., 2017) (building on (Arık et al., 2017)) introduced a multi-speaker variation of Tacotron that learns a low-dimensional embedding for each speaker, which was further extended in DeepVoice 3 (Ping et al., 2018) to a 2,400 multi-speaker scenario. Unlike WaveNet and DeepVoice, the Char2Wav (Sotelo et al., 2017) and VoiceLoop (Taigman et al., 2018) models produce World Vocoder Features (Morise et al., 2016) instead of generating raw audio signals. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Although many of these systems have produced high-quality samples for speakers present in the training set, generalizing to new speakers given only a few seconds of audio remains a challenge. There have been several concurrent works to address this few-shot learning problem. The VoiceLoop model introduced a novel memory-based architecture that was extended by Nachmani et al. (2018) to few-shot voice style adaptation, by introducing an auxiliary fitting network that predicts the embedding of a new speaker. Jia et al. (2018) extended the Tacotron model for one-shot speaker adaptation by conditioning on a speaker embedding vector extracted from a pretrained speaker identity model of Wan et al. (2018). The most similar approached to our work was proposed by Arik et al. (2018) for the DeepVoice 3 model. They considered both predicting the embedding with an encoding network and fitting the embedding based on a small amount of adaptation data, but the adaptation was applied to a prediction model for mel-spectrograms with a fixed vocoder. ",
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+    {
+        "type": "text",
+        "text": "5 EVALUATION ",
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+        "type": "text",
+        "text": "In this section, we evaluate the quality of samples of SEA-ALL, SEA-EMB and SEA-ENC. We first measure the naturalness of the generated utterances using the standard Mean Opinion Score (MOS) procedure. Then, we evaluate the similarity of generated and real samples using the subjective MOS test and objectively using a speaker verification system (Wan et al., 2018). Finally, we study these results varying the size of the adaptation dataset. ",
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+        "type": "text",
+        "text": "5.1 EXPERIMENTAL SETUP ",
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+        "type": "text",
+        "text": "We train a WaveNet model for each of our three methods using the same dataset, which combines the high-quality LibriSpeech audiobook corpus (Panayotov et al., 2015) and a proprietary speech corpus. The LibriSpeech dataset consists of 2302 speakers from the train speaker subsets and approximately 500 hours of utterances, sampled at a frequency of $1 6 \\mathrm { k H z }$ . The proprietary speech corpus consists of 10 American English speakers and approximately 300 hours of utterances, and we down-sample the recording frequency to $1 6 \\mathrm { k H z }$ to match LibriSpeech. The multi-speaker WaveNet model has the same architecture as van den Oord et al. (2016) except that we use a 200-dimensional speaker embedding space to model the large diversity of voices. ",
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+    {
+        "type": "table",
+        "img_path": "images/4921a10ea8d363f0cdff19a1af76038c9c75a884391ca719bf28d041d06b4b4d.jpg",
+        "table_caption": [
+            "Table 1: Naturalness of the adapted voices using a 5-scale MOS score (higher is better) with $9 5 \\%$ confidence interval on the LibriSpeech and VCTK held-out adaptation datasets. Numbers in bold are the best few-shot learning results on each dataset without statistically significant difference. van den Oord et al. (2016) was trained with 24-hour production quality data, Nachmani et al. (2018) used all samples of each new speaker, Arik et al. (2018) used 10 samples, and Jia et al. (2018) used 5 seconds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=2>LibriSpeech</td><td rowspan=1 colspan=2>VCTK</td></tr><tr><td rowspan=1 colspan=1>Real utterance</td><td rowspan=1 colspan=2>4.38±0.04</td><td rowspan=1 colspan=2>4.45 ± 0.04</td></tr><tr><td rowspan=1 colspan=1>van den Oord et al. (2016)</td><td rowspan=1 colspan=4>4.21 ± 0.081</td></tr><tr><td rowspan=1 colspan=1>Nachmani et al. (2018)Arik et al. (2018)adapt embeddingadapt whole-modelencoding + fine-tuningJia et al. (2018)trained on LibriSpeech</td><td rowspan=1 colspan=2>2.53 ± 1.114.12 ± 0.05</td><td rowspan=1 colspan=2>3.66±0.842.67 ± 0.103.16 ± 0.092.99 ± 0.124.01 ± 0.06</td></tr><tr><td rowspan=1 colspan=1>Adaptation data size</td><td rowspan=1 colspan=1>10s</td><td rowspan=1 colspan=1>&lt;5m</td><td rowspan=1 colspan=1>10s</td><td rowspan=1 colspan=1>&lt;10m</td></tr><tr><td rowspan=3 colspan=1>SEA-ALL (ours)SEA-EMB (ours)SEA-ENC (ours)</td><td rowspan=1 colspan=1>3.94± 0.08</td><td rowspan=1 colspan=1>4.13± 0.06</td><td rowspan=1 colspan=1>3.92 ± 0.07</td><td rowspan=1 colspan=1>3.92± 0.07</td></tr><tr><td rowspan=2 colspan=1>3.86 ± 0.073.61 ± 0.06</td><td rowspan=1 colspan=1>3.95 ± 0.07</td><td rowspan=1 colspan=1>3.81 ± 0.07</td><td rowspan=2 colspan=1>3.82 ± 0.073.58 ± 0.06</td></tr><tr><td rowspan=1 colspan=1>3.56 ± 0.06</td><td rowspan=1 colspan=1>3.65 ± 0.06</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Our few-shot model performance is evaluated using two hold-out datasets. First, the LibriSpeech test corpus consists of 39 speakers, with an average of approximately 52 utterances and 5 minutes of audio per speaker. For every test speaker, we randomly split their demonstration utterances into an adaptation set for adapting our WaveNet models and a test set for evaluation. The subset of utterances used for early termination in Section 3.1 is chosen from the adaptation set. There are about 4.2 utterances on average per speaker in the test set and the rest in the adaptation set. Second, we consider a subset of the CSTR VCTK corpus (Veaux et al., 2017) consisting of 21 American English speakers, with approximately 368 utterances and 12 minutes of audio per speaker. We also apply the adaptation/test split with 10 utterances per speaker for test. We emphasize that no data from VCTK was presented to the model at training time. Since our underlying WaveNet model was trained on data largely from LibriSpeech (which was recorded under noisier conditions than VCTK), one might expect that the generated samples on the VCTK dataset contain characteristic artifacts that make generated samples easier to distinguish from real utterances. However, our evaluation using VCTK indicates that our model generalizes effectively and that such artifacts are not detectable. Synthetic utterances are provided on our demo webpage1. ",
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+    {
+        "type": "text",
+        "text": "It is worth mentioning, that SEA-ENC requires no adaptation time. Where for SEA-EMB, it takes $5 \\sim 1 0 k$ optimizing steps to fit the embedding vector, and an additional $1 0 0 \\sim 2 0 0$ steps to fine-tune the entire model using early stopping for SEA-ALL. ",
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+    {
+        "type": "text",
+        "text": "5.2 NATURALNESS OF THE GENERATED SAMPLES (MOS) ",
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+        "type": "text",
+        "text": "We measure the quality of the generated samples by conducting a MOS test, whereby subjects are asked to rate the naturalness of generated utterances on a five-point Likert Scale (1: Bad, 2: Poor, 3: Fair, 4: Good, 5: Excellent). Furthermore, we compare with other published few-shot TTS systems systems, that were developed in parallel to this work. However, the literature uses varying combinations of training data and evaluation splits making comparison difficult. The results presented are from the closest experimental setups to ours. ",
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+        "type": "text",
+        "text": "Table 1 presents MOS for the adaptation models compared to real utterances. Two different adaptation dataset sizes are considered; $T = 1 0$ seconds, and $T \\leq 5$ minutes for LibriSpeech ( $T \\leq 1 0$ minutes for VCTK). For reference on $1 6 \\mathrm { k H z }$ data, WaveNet trained on a 24-hour production quality speech dataset (van den Oord et al., 2016) achieves a score of 4.21, while for LibriSpeech our best few-shot model attains an MOS score of 4.13 using only 5 minutes of data given a pre-trained multi-speaker model. We note that both fine-tuning models produce overall “good” samples for both the LibriSpeech and VCTK test sets, with SEA-ALL outperforming SEA-EMB in all cases. SEA-ALL is on par with the state-of-the-art performance on both datasets. The addition of extra adaptation data beyond 10 seconds of audio helps performance on LibriSpeech but not VCTK, and the gap between our best model and the real utterance is also wider on VCTK, possibly due to the different recording conditions. ",
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+        "table_caption": [
+            "Table 2: Voice similarity of generated voices using a 5-scale MOS score (higher is better) with $9 5 \\%$ confidence interval on the LibriSpeech and VCTK held-out adaptation datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"2\">LibriSpeech</td><td colspan=\"2\">VCTK</td></tr><tr><td>Real utterance</td><td colspan=\"2\">4.30 ± 0.08</td><td colspan=\"2\">4.59 ± 0.06</td></tr><tr><td>Jia et al. (2018) trained on LibriSpeech</td><td colspan=\"2\"></td><td colspan=\"2\"></td></tr><tr><td>Adaptation data size</td><td>3.03 ± 0.09 10s</td><td>&lt;5m</td><td>2.77 ± 0.08 10s</td><td>&lt;10m</td></tr><tr><td>SEA-ALL (ours)</td><td>3.41 ± 0.10</td><td>3.75 ± 0.09</td><td>3.51±0.10</td><td>3.97± 0.09</td></tr><tr><td>SEA-EMB (ours)</td><td>3.42± 0.10</td><td>3.56 ± 0.10</td><td>3.07 ±0.10</td><td>3.18 ± 0.10</td></tr><tr><td>SEA-ENC (ours)</td><td>2.47 ± 0.09</td><td>2.59 ± 0.09</td><td>2.07± 0.08</td><td>2.19 ±0.09</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "type": "text",
+        "text": "5.3 VOICE SIMILARITY (MOS) ",
+        "text_level": 1,
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+        "text": "Beside naturalness, we also measure the similarity of the generated and real voices. The quality of similarity is the main evaluation metric for the voice adaptation problem. We first follow the experiment setup of Jia et al. (2018) to run a MOS test for a subjective assessment and then use a speaker verification model for objective evaluation in the next section. ",
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+        "text": "In every trial of this test a subject is presented with a pair of utterances consisting of a real utterance and another real or generated utterance from the same speaker, and is asked to rate the similarity in voice identity using a five-scale score (1: Not at all similar, 2: Slightly similar, 3: Moderately similar, 4: Very similar, 5: Extremely similar). ",
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+        "type": "text",
+        "text": "Table 2 shows the MOS for real utterances and all the adaptation models under two adaptation data time settings on both datasets. Again, the SEA-ALL model outperforms the other two models, and the improvement over SEA-EMB scales with the amount of adaptation data. Particularly, the learned voices on the VCTK dataset achieve an average score of 3.97, demonstrating the generalization performance on a different dataset. As a rough comparisson, because of varying training setups, the state of the art system of Jia et al. (2018) achieves scores of 3.03 for LibriSpeech and 2.77 for VCTK when trained on LibriSpeech. Their model computes the embedding based on the $d$ -vector, similar to our SEA-ENC approach, and performs competitively for the one-shot learning setting, but its performance saturates with 5 seconds of adaptation data, as explained in Section 3.2. We note the gap of similarity scores between SEA-ALL and real utterances, which suggests that although the generated samples sound similar to the target speakers, humans can still tell the difference from real utterances. ",
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+        "type": "text",
+        "text": "5.4 VOICE SIMILARITY (SPEAKER VERIFICATION) ",
+        "text_level": 1,
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+        "text": "We also apply the state-of-the-art text independent speaker verification (TI-SV) model of (Wan et al., 2018) to objectively assess whether the generated samples preserve the acoustic features of the speakers. We calculate the TI-SV $d$ -vector embeddings for generated and real voices. In Figure 3, we visualize the 2-dimensional projection of the $d$ -vectors for a SEA-ALL model trained on $T \\leq 5$ minutes of data on the LibriSpeech dataset, and $T \\leq 1 0$ minutes on VCTK. There are clear clusters on both datasets, with a strikingly large inter-cluster distance and low intra-cluster separation. This shows both (1) an ease of correctly identifying the speaker associated with a given generated utterance, and (2) the difficulty in differentiating real from synthetic samples. ",
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+        "text": "A similar figure is presented in (Jia et al., 2018), but there the generated and real samples do not overlap. This indicates that the method presented in this paper generates voices that are more indistinguishable from real ones, when measured with the same verification system. In the following subsections, we further analyze these results. ",
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+        "image_caption": [
+            "Figure 3: t-SNE visualization of the $d$ -vector embeddings of real and SEA-ALL-generated utterances, for both the LibriSpeech ( $T \\leq 5$ mins) and VCTK ( $T \\leq 1 0$ mins) evaluation datasets. "
+        ],
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+        "image_caption": [
+            "Figure 4: Detection error trade-off (DET) curve for speaker verification in percentage, using the TI-SV speaker verification model (Wan et al., 2018). The utterances were generated using $T \\leq 5$ and $T \\leq 1 0$ minute samples from LibriSpeech and VCTK respectively. EER is marked with a dot. "
+        ],
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+        "type": "text",
+        "text": "5.4.1 DISCERNING DIFFERENT SPEAKERS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first quantify whether generated utterances are attributed to the correct speaker. Following common practice in speaker verification (Wan et al., 2018), we select the hold-out test set of real utterances from test speakers as the enrollment set and compute the centroid of the $d$ -vectors for each speaker $\\mathbf { c } _ { i }$ . We then use the adaptation set of test speakers as the verification set. For every verification utterance, we compute the cosine similarity between its $d$ -vector $\\mathbf { v }$ and a randomly chosen centroid $\\mathbf { c } _ { i }$ . The utterance is accepted as one from speaker $i$ if the similarity is exceeds a given threshold. We repeat the experiments with the same enrollment set and replace the verification set with samples generated by each adaptation method under different data size settings. ",
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+        "table_caption": [
+            "Table 3: Equal error rate (EER) of real and few-shot adapted voice samples for evaluation of voice similarity. Varying adaptation dataset sizes were considered. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"3\">LibriSpeech</td><td colspan=\"3\">VCTK</td></tr><tr><td>Real utterance</td><td colspan=\"3\"></td><td colspan=\"3\">2.79</td></tr><tr><td>Adaptation data size</td><td>10s</td><td>2.47 1m</td><td>&lt;5m</td><td>10s</td><td>1m</td><td>&lt;10m</td></tr><tr><td>SEA-ALL (ours)</td><td>3.17</td><td>2.47</td><td>1.85</td><td>7.34</td><td>5.02</td><td>4.33</td></tr><tr><td>SEA-EMB (ours)</td><td>3.26</td><td>2.92</td><td>2.74</td><td>10.18</td><td>9.91</td><td>10.24</td></tr><tr><td>SEA-ENC (ours)</td><td>10.73</td><td>9.77</td><td>9.42</td><td>27.20</td><td>25.34</td><td>25.23</td></tr></table>",
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+    {
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+        "img_path": "images/98e9962d2247562b56b5bc63c271f368b6c5f06cc98187458e53c3a0d08484cb.jpg",
+        "image_caption": [
+            "Figure 5: Cosine similarity of real and generated utterances to the real enrollment set. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In our setup we fix the enrollment set together with the speaker verification model from (Wan et al., 2018), and study the performance of different verification sets that are either from real utterances or generated by a TTS system. Table 3 lists the equal error rate (EER) of the verification model with real and generated verification utterances, and Figure 4 shows the detection error trade-off (DET) curves for a more thorough inspection. Figure 4 only shows the adaptation models with the maximum data size setting ( $T \\leq 5$ minutes for LibriSpeech and $\\leq 1 0$ minutes for VCTK). The results for other data sizes are provided in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "We find that SEA-ALL outperforms the other two approaches, and the error rate decreases clearly with the size of demonstration data. Noticeably, the EER of SEA-ALL is even lower than the real utterance on the LibriSpeech dataset with sufficient adaptation data. A possible explanation is that the generated samples might be concentrated closer to the centroid of a speaker’s embeddings than real speech with larger variance across utterances. Our SEA-EMB model performs better than SEA-ENC. Additionally, the benefit of more demonstration data is less significant than for SEA-ALL in both of these models. ",
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+    {
+        "type": "text",
+        "text": "5.4.2 DISCERNING REAL FROM GENERATED UTTERANCES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we compare the generated samples and the real utterances of the speaker being imitated. Figure 5 shows the box-plot of the cosine similarity between the embedding centroids of test speakers’ enrollment set and (1) real utterances from the same speaker, (2) real utterances from a different speaker, and (3) generated utterances adapted to the same speaker. Consistent with the observations from the previous subsection, SEA-ALL performs best. ",
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+    {
+        "type": "text",
+        "text": "We further consider an adversarial scenario for speaker verification. In contrast to the previous standard speaker verification setup where we now select a verification utterance with either a real utterance from the same speaker or a synthetic sample from a model adapted to the same speaker. Under this setup, the speaker verification system is challenged by synthetic samples and acts as a classifier for real versus generated utterances. The ROC curve of this setup is shown in Figure 6 and the models are using the maximum data size setting. Other data size settings can be found in Appendix C. If the generated samples are indistinguishable from real utterances, the ROC curve approaches the diagonal line (that is, the verification system fails to separate real and generated voices). Importantly, SEA-ALL manages to confuse the verification system especially for the VCTK dataset where the ROC curve is almost inline with the diagonal line with an AUC of 0.56. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This paper studied three variants of meta-learning for sample efficient adaptive TTS. The adaptation method that fine-tunes the entire model, with the speaker embedding vector first optimized, shows impressive performance even with only 10 seconds of audio from new speakers. When adapted with a few minutes of data, our model matches the state-of-the-art performance in sample naturalness. Moreover, it outperforms other recent works in matching the new speaker’s voice. We also demonstrated that the generated samples achieved a similar level of voice similarity to real utterances from the same speaker, when measured by a text independent speaker verification model. ",
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+    },
+    {
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+        "img_path": "images/f0fa25bb0c694d1710032d4d6bcf1673e38d8e8be977b04dbdb18b1488b7e4bf.jpg",
+        "image_caption": [
+            "Figure 6: ROC curve for real versus generated utterance detection. The utterances were generated using models with 5 and 10 minutes of training data per speaker from LibriSpeech and VCTK respectively. Lower curve indicate that the verification system is having a harder time distinguishing real from generated samples. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Our paper considers the adaptation to new voices with clean, high-quality training data collected in a controlled environment. The few-shot learning of voices with noisy data is beyond the scope of this paper and remains a challenging open research problem. ",
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+    {
+        "type": "text",
+        "text": "A requirement for less training data to adapt the model, however, increases the potential for both beneficial and harmful applications of text-to-speech technologies such as the creation of synthesized media. While the requirements for this particular model (including the high-quality training data collected in a controlled environment and equally high quality data from the speakers to which we adapt, as described in Section 5.1) present barriers to misuse, more research must be conducted to mitigate and detect instances of misuse of text-to-speech technologies in general. ",
+        "bbox": [
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+    {
+        "type": "text",
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Fitting new speakers based on a short untranscribed sample. arXiv preprint arXiv:1802.06984, 2018.   \nY. Jia, Y. Zhang, R. J. Weiss, Q. Wang, J. Shen, F. Ren, Z. Chen, P. Nguyen, R. Pang, I. L. Moreno, et al. Transfer learning from speaker verification to multispeaker text-to-speech synthesis. arXiv preprint arXiv:1806.04558, 2018.   \nS. O. Arik, J. Chen, K. Peng, W. Ping, and Y. Zhou. Neural voice cloning with a few samples. arXiv preprint arXiv:1802.06006, 2018.   \nV. Panayotov, G. Chen, D. Povey, and S. Khudanpur. Librispeech: an asr corpus based on public domain audio books. In International Conference on Acoustics, Speech and Signal Processing, pages 5206–5210. IEEE, 2015.   \nC. Veaux, J. Yamagishi, K. MacDonald, et al. CSTR VCTK corpus: English multi-speaker corpus for CSTR Voice Cloning Toolkit, 2017. ",
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+        "type": "text",
+        "text": "A EMBEDDING ENCODER ",
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+        "text": "Our encoding network is illustrated as the summation of two sub-network outputs in Figure 7. The first subnetwork is a pre-trained speaker verification model (TI-SV) (Wan et al., 2018), comprising 3 LSTM layers and a single linear layer. This model maps a waveform sequence of arbitrary length to a fixed 256-dimensional $d$ -vector with a sliding window, and is trained from approximately 36M utterances from 18K speakers extracted from anonymized voice search logs. On top of this we add a shallow MLP to project the output $d$ -vector to the speaker embedding space. The second sub-network comprises 16 1-D convolutional layers. This network reduces the temporal resolution to $2 5 6 \\mathrm { m s }$ per frame (for $1 6 \\mathrm { k H z }$ audio), then averages across time and projects into the speaker embedding space. The purpose of this network is to extract residual speaker information present in the demonstration waveforms but not captured by the pre-trained TI-SV model. ",
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+    {
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+        "img_path": "images/638613b970a6c64f4a74f956a7d54690174a5b5f336d824dfaddb8259396a994.jpg",
+        "image_caption": [
+            "Figure 7: Encoder network architecture for predicting speaker embeddings. "
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+    {
+        "type": "text",
+        "text": "B DET CURVES VARYING TRAINING DATA SIZES ",
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+    {
+        "type": "text",
+        "text": "Here we provide the DET curves of speaker verification problem for models with different training data sizes in addition to those shown in Section 5.4.1. ",
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+    {
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+        "img_path": "images/4f1d1338c1f0e0f09c422dfa47c76d6b7e03222ce7d8d78699c42da5ff087753.jpg",
+        "image_caption": [
+            "Figure 8: Detection error trade-off (DET) curve for speaker verification, using the TI-SV speaker verification model (Wan et al., 2018). The utterances were generated using 1 minute or 10 seconds of utterance from LibriSpeech and VCTK. EER is marked with a dot. "
+        ],
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+    {
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+        "text": "C ROC CURVES VARYING TRAINING DATA SIZES ",
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+    {
+        "type": "text",
+        "text": "We provide the ROC curves of the speaker verification problem with adversarial examples from adaptation models with different training data sizes in addition to those shown in Section 5.4.2. ",
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+        "image_caption": [
+            "Figure 9: ROC curve for real vs. generated utterance detection. The utterances were generated using 1 minute or 10 seconds of utterance from LibriSpeech and VCTK. Lower curve suggests harder to distinguish real from generated samples. "
+        ],
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