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+# Dirichlet Energy Constrained Learning for Deep Graph Neural Networks
+
+Kaixiong Zhou Rice University Kaixiong.Zhou@rice.edu
+
+Xiao Huang The Hong Kong Polytechnic University xiaohuang@comp.polyu.edu.hk
+
+Daochen Zha Rice University Daochen.Zha@rice.edu
+
+Rui Chen Samsung Research America rui.chen1@samsung.com
+
+Li Li Samsung Research America li.li1@samsung.com
+
+Soo-Hyun Choi∗ Samsung Electronics soohyunc@gmail.com
+
+Xia Hu Rice University xia.hu@rice.edu
+
+# Abstract
+
+Graph neural networks (GNNs) integrate deep architectures and topological structure modeling in an effective way. However, the performance of existing GNNs would decrease significantly when they stack many layers, because of the oversmoothing issue. Node embeddings tend to converge to similar vectors when GNNs keep recursively aggregating the representations of neighbors. To enable deep GNNs, several methods have been explored recently. But they are developed from either techniques in convolutional neural networks or heuristic strategies. There is no generalizable and theoretical principle to guide the design of deep GNNs. To this end, we analyze the bottleneck of deep GNNs by leveraging the Dirichlet energy of node embeddings, and propose a generalizable principle to guide the training of deep GNNs. Based on it, a novel deep GNN framework – Energetic Graph Neural Networks (EGNN) is designed. It could provide lower and upper constraints in terms of Dirichlet energy at each layer to avoid over-smoothing. Experimental results demonstrate that EGNN achieves state-of-the-art performance by using deep layers.
+
+# 1 Introduction
+
+Graph neural networks (GNNs) [1] are promising deep learning tools to analyze networked data, such as social networks [2, 3, 4], academic networks [5, 6, 7], and molecular graphs [8, 9, 10, 11]. Based on spatial graph convolutions, GNNs apply a recursive aggregation mechanism to update the representation of each node by incorporating representations of itself and its neighbors [12]. A variety of GNN variations have been explored for different real-world networks and applications [13, 14].
+
+A key limitation of GNNs is that when we stack many layers, the performance would decrease significantly. Experiments show that GNNs often achieve the best performance with less than 3 layers [15, 13]. As the layer number increases, the node representations will converge to indistinguishable vectors due to the recursive neighborhood aggregation and non-linear activation [16, 17].
+
+Such phenomenon is recognized as over-smoothing issue [18, 19, 20, 21, 22]. It prevents the stacking of many layers and modeling the dependencies to high-order neighbors.
+
+A number of algorithms have been proposed to alleviate the over-smoothing issue and construct deep GNNs, including embedding normalization [23, 24, 25], residual connection [26, 27, 28] and random data augmentation [29, 30, 31]. However, some of them are motivated directly by techniques in convolutional neural networks (CNNs) [32], such as the embedding normalization and residual connection. Others are based on heuristic strategies, such as random embedding propagation [30] and dropping edge [29]. Most of them only achieve comparable or even worse performance compared to their shallow models. Recently, a metric of Dirichlet energy has been applied to quantify the over-smoothing [33], which is based on measuring node pair distances. With the increasing of layers, the Dirichlet energy converges to zero since node embeddings become close to each other. But there is a lack of empirical methods to leverage this metric to overcome the over-smoothing issue.
+
+Therefore, it remains a non-trivial task to train a deep GNN architecture due to three challenges. First, the existing efforts are developed from diverse perspectives, without a generalizable principle and analysis. The abundance of these components also makes the design of deep GNNs challenging, i.e., how should we choose a suitable one or combinations for real-world scenarios? Second, even if an effective indicator of over-smoothing is given, it is hard to theoretically analyze the bottleneck and propose a generalizable principle to guide the training of deep GNNs. Third, even if theoretical guidance is given, it may be difficult to be utilized and implemented to train GNNs empirically.
+
+To this end, in this paper, we target to develop a generalizable framework with a theoretical basis, to handle the over-smoothing issue and enable effective deep GNN architectures. In particular, we will investigate two research questions. 1) Is there a theoretical and generalizable principle to guide the architecture design and training of deep GNNs? 2) How can we develop an effective architecture to achieve state-of-the-art performance by stacking a large number of layers? Following these questions, we make three major contributions as follows.
+
+• We propose a generalizable principle – Dirichlet energy constrained learning, to guide the training of deep GNNs by regularizing Dirichlet energy. Without proper training, the Dirichlet energy would be either too small due to the over-smoothing issue, or too large when the node embeddings are over-separating. Our principle carefully defines an appropriate range of Dirichlet energy at each layer. Being regularized within this range, a deep GNN model could be trained by jointly optimizing the task loss and energy value.
+
+• We design a novel deep architecture – Energetic Graph Neural Networks (EGNN). It follows the proposed principle and could efficiently learn an optimal Dirichlet energy. It consists of three components, i.e., orthogonal weight controlling, lower-bounded residual connection, and shifted ReLU (SReLU) activation. The trainable weights at graph convolutional layers are orthogonally initialized as diagonal matrices, whose diagonal values are regularized to meet the upper energy limit and eliminate the over-separating. The residual connection strength is determined by the lower energy limit to avoid the over-smoothing. While the widely-used ReLU activation causes the extra loss of Dirichlet energy, the linear mapping worsens the learning ability of GNNs. We apply SReLU with a trainable shift to provide a trade-off between the non-linear and linear mappings.
+
+• We show that the proposed principle and EGNN can well explain most of the existing techniques for deep GNNs. Empirical results demonstrate that EGNN could be easily trained to reach 64 layers and achieves surprisingly competitive performance on benchmarks.
+
+# 2 Problem Statement
+
+Notations. Given an undirected graph consisting of $n$ nodes, it is represented as $G = ( A , X )$ , where $A \in \mathbb { R } ^ { n \times n }$ denotes the adjacency matrix and $\boldsymbol { X } \in \mathbb { R } ^ { n \times d }$ denotes the feature matrix. Let ${ \tilde { A } } : = A + I _ { n }$ and $\tilde { D } : = D + I _ { n }$ be the adjacency and degree matrix of the graph augmented with selfloops. The augmented normalized Laplacian is then given by $\tilde { \Delta } : = I _ { n } - \tilde { P }$ , where $\bar { \tilde { P } } : = \tilde { D } ^ { - \frac { 1 } { 2 } } \tilde { A } \tilde { D } ^ { - \frac { 1 } { 2 } }$ is an augmented normalized adjacency matrix used for the neighborhood aggregation in GNN models.
+
+Node classification task. GNNs have been adopted in many applications [6, 11, 34]. Without loss of generality, we take node classification as an example. Given a graph $G = ( A , X )$ and a set of its nodes with labels for training, the goal is to predict the labels of nodes in a test set.
+
+We now use the graph convolutional network (GCN) [15] as a typical example, to illustrate how traditional GNNs perform the network analysis task. Formally, the layer-wise forward-propagation operation in GCN at the $k$ -th layer is defined as:
+
+$$
+X ^ { ( k ) } = \sigma ( \tilde { P } X ^ { ( k - 1 ) } W ^ { ( k ) } ) .
+$$
+
+$X ^ { ( k ) }$ and $X ^ { ( k - 1 ) }$ are node embedding matrices at layers $k$ and $k - 1$ , respectively; $W ^ { ( k ) } \in \mathbb { R } ^ { d \times d }$ denotes trainable weights used for feature transformation; $\sigma$ denotes an activation function such as ReLU; $X ^ { ( 0 ) } = X$ at the initial layer of GCN. The embeddings at the final layer are optimized with a node classification loss function, e.g., cross-entropy loss. The recursive neighborhood aggregation in Eq. (1) will make node embeddings similar to each other as the number of layer $k$ increases. This property, i.e., over-smoothing, prevents traditional GNNs from exploring neighbors many hops away. In practice, the dependencies to high-order neighbors are important to the node classification. The traditional shallow GNNs may have sub-optimal performances in the downstream tasks [16, 28].
+
+# 3 Dirichlet Energy Constrained Learning
+
+In this paper, we aim to develop an effective principle to alleviate the over-smoothing issue and enable deep GNNs to leverage the high-order neighbors. We first theoretically analyze the over-smoothing issue, and then provide a principle to explain the key constraint in training deep GNNs.
+
+Node pair distance has been widely adopted to quantify the over-smoothing based on embedding similarities [19, 23]. Among the series of distance metrics, Dirichlet energy is simple and expressive for the over-smoothing analysis [33]. Thus, we adopt Dirichlet energy and formally define it as below.
+
+Definition 1. Given node embedding matrix $X ^ { ( k ) } = [ x _ { 1 } ^ { ( k ) } , \cdots , x _ { n } ^ { ( k ) } ] ^ { \top } \in \mathbb { R } ^ { n \times d }$ learned from GCN at the $k$ -th layer, the Dirichlet energy $E ( X ^ { ( k ) } )$ is defined as follows:
+
+$$
+E ( { \boldsymbol { X } } ^ { ( k ) } ) = \operatorname { t r } ( { \boldsymbol { X } } ^ { ( k ) } ^ { \top } \tilde { \Delta } { \boldsymbol { X } } ^ { ( k ) } ) = \frac { 1 } { 2 } \sum a _ { i j } | | \frac { \boldsymbol { x } _ { i } ^ { ( k ) } } { \sqrt { 1 + d _ { i } } } - \frac { { \boldsymbol { x } } _ { j } ^ { ( k ) } } { \sqrt { 1 + d _ { j } } } | | _ { 2 } ^ { 2 } ,
+$$
+
+where $\operatorname { t r } ( \cdot )$ denotes trace of a matrix; $a _ { i j }$ is edge weight given by the $( i , j )$ -th element in matrix $A$ ; $d _ { i }$ is node degree given by the $i$ -th diagonal element in matrix $D$ . Dirichlet energy reveals the embedding smoothness with the weighted node pair distance. While a smaller value of $E ( X ^ { ( k ) } )$ is highly related to the over-smoothing, a larger one indicates that the node embeddings are over-separating even for those nodes with the same label. Considering the node classification task, one would prefer to have an appropriate Dirichlet energy at each layer to separate the nodes of different classes while keeping those of the same class close. However, under some conditions, the upper bound of Dirichlet energy is theoretically proved to converge to 0 in the limit of infinite layers [33]. In other words, all nodes converge to a trivial fixed point in the embedding space.
+
+Based on the previous analysis, we derive the corresponding lower bound and revisit the oversmoothing/separating problem from the model design and training perspectives. To simplify the derivation process, we remove the non-linear activation $\sigma$ , and re-express GCN as: $X ^ { ( \bar { k } ) } =$ $P \cdot \cdot \cdot P X W ^ { ( 1 ) } \cdot \cdot \cdot W ^ { ( k ) }$ . The impact of non-linear function will be considered in the model design.
+
+Lemma 1. The Dirichlet energy at the $k$ -th layer is bounded as follows:
+
+$$
+( 1 - \lambda _ { 1 } ) ^ { 2 } s _ { \operatorname* { m i n } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) \leq E ( X ^ { ( k ) } ) \leq ( 1 - \lambda _ { 0 } ) ^ { 2 } s _ { \operatorname* { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) .
+$$
+
+The detailed proof is provided in the Appendix. $\lambda _ { 1 }$ and $\lambda _ { 0 }$ are the non-zero eigenvalues of matrix $\tilde { \Delta }$ that are most close to values 1 and 0, respectively. $s _ { \mathrm { m i n } } ^ { ( k ) }$ and $s _ { \mathrm { m a x } } ^ { ( k ) }$ are the squares of minimum and maximum singular values of weight $W ^ { ( k ) }$ , respectively. Note that the eigenvalues of $\tilde { \Delta }$ vary with the real-world graphs, and locate within range $[ 0 , 2 )$ . We relax the above bounds as below.
+
+Lemma 2. The lower and upper bounds of Dirichlet energy at the $k$ -th layer could be relaxed as:
+
+$$
+0 \leq E ( X ^ { ( k ) } ) \leq s _ { \operatorname* { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) .
+$$
+
+Besides the uncontrollable eigenvalues determined by the underlying graph, it is shown that the Dirichlet energy can be either too small or too large without proper design and training on weight $W ^ { ( k ) }$ . On one hand, based on the common Glorot initialization [35] and L2 regularization, we empirically find that some of the weight matrices approximate to zero in a deep GCN. The corresponding square singular values are hence close to zero in these intermediate layers. That means the Dirichlet energy will become zero at the higher layers of GCN and causes the over-smoothing issue. On the other hand, without the proper weight initialization and regularization, a large $s _ { \mathrm { m a x } } ^ { ( k ) }$ may lead to the energy explosion and the over-separating.
+
+The Dirichlet energy plays a key role in training a deep GNN model. However, the optimal value of Dirichlet energy varies in the different layers and applications. It is hard to be specified ahead and then enforces the node representation learning. Therefore, we propose a principle – Dirichlet energy constrained learning, defined in Proposition 1. It provides appropriate lower and upper limits of Dirichlet energy. Regularized by such a given range, a deep GNN model could be trained by jointly optimizing the node classification loss and Dirichlet energy at each layer.
+
+Proposition 1. Dirichlet energy constrained learning defines the lower & upper limits at layer $k$ as:
+
+$$
+c _ { \operatorname* { m i n } } E ( X ^ { ( k - 1 ) } ) \leq E ( X ^ { ( k ) } ) \leq c _ { \operatorname* { m a x } } E ( X ^ { ( 0 ) } ) .
+$$
+
+We apply the transformed initial feature through trainable function $f$ $\because X ^ { ( 0 ) } = f ( X ) \in \mathbb { R } ^ { n \times d }$ . Both $c _ { \mathrm { m i n } }$ and $c _ { \mathrm { m a x } }$ are positive hyperparameters. From value interval $( 0 , 1 )$ , hyperparameter $c _ { \mathrm { m i n } }$ is selected by satisfying constraint of $E ( X ^ { ( k ) } ) \ge c _ { \mathrm { m i n } } ^ { k } E ( X ^ { ( 0 ) } ) > 0$ . In such a way, the over-smoothing is overcome since the Dirichlet energies of all the layers are larger than appropriate limits related to ckmin. Compared with the initial transformed feature $X ^ { ( 0 ) }$ , the intermediate node embeddings of the same class are expected to be merged closely to have a smaller Dirichlet energy and facilitate the downstream applications. Therefore, we exploit the upper limit $c _ { \mathrm { m a x } } E ( X ^ { ( 0 ) } )$ to avoid overseparating, where $c _ { \mathrm { m a x } }$ is usually selected from $( 0 , 1 ]$ . In the experiment part, we show that the optimal energy accompanied with the minimized classification loss locates within the above range at each layer. Furthermore, hyperparameters $c _ { \mathrm { m i n } }$ and $c _ { \mathrm { m a x } }$ could be easily selected from the large and appropriate value scopes, which do not affect the model performance.
+
+Given both the low and upper limits, an intuitive solution to search the optimal energy is to train GNNs by optimizing the following constrained problem:
+
+$$
+\begin{array} { r l } & { \operatorname* { m i n } \quad \mathcal { L } _ { \mathrm { t a s k } } + \gamma \sum _ { k } | | W ^ { ( k ) } | | _ { F } , } \\ & { \mathrm { s . t . } \quad c _ { \operatorname* { m i n } } E \big ( X ^ { ( k - 1 ) } \big ) \leq E \big ( X ^ { ( k ) } \big ) \leq c _ { \operatorname* { m a x } } E \big ( X ^ { ( 0 ) } \big ) , \mathrm { f o r } k = 1 , \cdots , K . } \end{array}
+$$
+
+$\mathcal { L } _ { \mathrm { t a s k } }$ denotes the cross-entropy loss of node classification task; $K$ is layer number of GNN; $| | \cdot | | _ { F }$ denotes Frobenius norm of a matrix; and $\gamma$ is loss hyperparameter. Note that Dirichlet energy has also been adopted to regularize the node representation learning in shallow neural networks [36, 37, 38]. We instead focus on optimizing deep GNNs as shown in Eq. (6), where $K$ is often large.
+
+# 4 Energetic Graph Neural Networks - EGNN
+
+It is non-trivial to optimize Problem (6) due to the expensive computation of $E ( X ^ { ( k ) } )$ . Furthermore, the numerous constraints make the problem a very complex optimization hyper-planes, at which the raw task objective tends to fall into local optimums. Instead of directly optimizing Problem (6), we propose an efficient model EGNN to satisfy the constrained learning from three perspectives: weight controlling, residual connection and activation function. We introduce them one by one as follows.
+
+# 4.1 Orthogonal Weight Controlling
+
+According to Lemma 2, without regularizing the maximum square singular value $s _ { \mathrm { m a x } } ^ { ( k ) }$ of matrix $W ^ { ( k ) }$ , the upper bound of Dirichlet energy can be larger than the upper limit, i.e., $s _ { \mathrm { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) >$ $c _ { \mathrm { m a x } } E ( X ^ { ( 0 ) } )$ . That means the Dirichlet energy of a layer may break the upper limit of constrained learning, and makes Problem (6) infeasible. In this section, we show how to satisfy such limit by controlling the singular values during weight initialization and model regularization.
+
+Orthogonal initialization. Since the widely-used initialization methods (e.g., Glorot initialization) fail to restrict the scopes of singular values, we adopt the orthogonal approach that initializes trainable weight $W ^ { ( k ) }$ as a diagonal matrix with explicit singular values [39]. To restrict $s _ { \mathrm { m a x } } ^ { ( k ) }$ and meet the constrained learning, we apply an equality constraint of $s _ { \mathrm { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) = c _ { \mathrm { m a x } } E ( X ^ { ( 0 ) } )$ at each layer. Based on this condition, we derive Proposition 2 to initialize those weights $W ^ { ( k ) }$ and their square singular values for all the layers of EGNN, and give Lemma 3 to show how we can satisfy the upper limit of constrained learning. The detailed derivation and proof are listed in Appendix.
+
+Proposition 2. At the first layer, weight $W ^ { ( 1 ) }$ is initialized as a diagonal matrix $\sqrt { c _ { \operatorname* { m a x } } } \cdot I _ { d }$ , where $I _ { d }$ is identity matrix with dimension $d$ and the square singular values are $c _ { \mathrm { m a x } }$ . At the higher layer $k > 1$ , weight $W ^ { ( k ) }$ is initialized with an identity matrix $I _ { d }$ , where the square singular values are 1.
+
+Lemma 3. Based on the above orthogonal initialization, at the starting point of training, the Dirichlet energy of EGNN satisfies the upper limit at each layer $k$ : $E ( X ^ { ( k ) } ) \le c _ { \mathrm { { m a x } } } E ( X ^ { ( 0 ) } )$ .
+
+Orthogonal regularization. However, without proper regularization, the initialized weights cannot guarantee they will still satisfy the constrained learning during model training. Therefore, we propose a training loss that penalizes the distances between the trainable weights and initialized weights $\sqrt { c _ { \operatorname* { m a x } } } I _ { d }$ or $I _ { d }$ . To be specific, we modify the optimization problem (6) as follows:
+
+$$
+\operatorname* { m i n } \mathcal { L } _ { \mathrm { t a s k } } + \gamma | | W ^ { ( 1 ) } - \sqrt { c _ { \operatorname* { m a x } } } I _ { d } | | _ { F } + \gamma \sum _ { k = 2 } ^ { K } | | W ^ { ( k ) } - I _ { d } | | _ { F } .
+$$
+
+Comparing with the original problem (6), we instead use the weight penalization to meet the upper limit of constrained learning, and make the model training efficient. While a larger $\gamma$ highly regularizes the trainable weights around the initialized ones to satisfy the constrained learning, a smaller $\gamma$ assigns the model more freedom to adapt to task data and optimize the node classification loss. Considering the above orthogonal initialization where weight $W ^ { ( k ) }$ is diagonal and sparse, we use the simplest distance constraint in Eq. (7) to update weight at the vicinity of its initialization. The singular values of updated sparse weight will be mainly determined by the dominant diagonal values, which are potentially close to the initialized ones. Therefore, we are able to control the singular values and regularize the upper limit of Dirichlet energy even at the model training phase. In the future work, more the advanced orthogonal initialization and regularization approaches could be explored to further boost performance of deep GNNs [40, 41, 42].
+
+# 4.2 Lower-bounded Residual Connection
+
+Although the square singular values are initialized and regularized properly, we may still fail to guarantee the lower limit of constrained learning in some specific graphs. According to Lemma 1, the lower bound of Dirichlet energy is $( 1 - \lambda _ { 1 } ) ^ { 2 } s _ { \operatorname* { m i n } } ^ { ( k ) } E ( \bar { X ^ { ( k - 1 ) } } )$ . In the real-world applications, may exactly equal to 1 and relaxes the lower bound as zero as shown in Lemma 2. For example, in Erdos–Rényi graph with dense connections [ ˝ 43], the eigenvalues of matrix $\tilde { \Delta }$ converge to 1 with high probability [17]. Even though $s _ { \mathrm { m i n } } ^ { ( k ) } > 0$ , the Dirichlet energy can be smaller than the lower limit and leads to the over-smoothing. To tackle this problem, we adopt residual connections to the initial layer $X ^ { ( 0 ) }$ and the previous layer $X ^ { ( k - 1 ) }$ . To be specific, we define the residual graph convolutions as:
+
+$$
+X ^ { ( k ) } = \sigma ( [ ( 1 - c _ { \operatorname * { m i n } } ) \tilde { P } X ^ { ( k - 1 ) } + \alpha X ^ { ( k - 1 ) } + \beta X ^ { ( 0 ) } ] W ^ { ( k ) } ) .
+$$
+
+$\alpha$ and $\beta$ are residual connection strengths determined by the lower limit of constrained learning, i.e., $\alpha + \beta = c _ { \operatorname* { m i n } }$ . We are aware that the residual technique has been used before to set up deep GNNs [26, 44, 28]. However, they either apply the whole residual components, or combine an arbitrary fraction without theoretical insight. Instead, we use an appropriate residual connection according to the lower limit of Dirichlet energy. In the experiment part, we show that while a strong residual connection overwhelms information in the higher layers and reduces the classification performance, a weak one will lead to the over-smoothing. In the following, we justify that both the lower and upper limits in the constrained learning can be satisfied with the proposed lower-bounded residual connection. The detailed proofs are provided in Appendix.
+
+Lemma 4. Suppose that $c _ { \operatorname* { m a x } } \geq c _ { \operatorname* { m i n } } / ( 2 c _ { \operatorname* { m i n } } - 1 ) ^ { 2 }$ . Based upon the orthogonal controlling and residual connection, the Dirichlet energy of initialized EGNN is larger than the lower limit at each layer $k$ , i.e., $E ( X ^ { ( k ) } ) \geq c _ { \operatorname* { m i n } } E ( X ^ { ( k - 1 ) } )$ .
+
+Lemma 5. Suppose that $\begin{array} { r } { \sqrt { c _ { \mathrm { m a x } } } \ge \frac { \beta } { ( 1 - c _ { \mathrm { m i n } } ) \lambda _ { 0 } + \beta } } \end{array}$ . Being augmented with the orthogonal controlling and residual connection, the Dirichlet energy of initialized EGNN is smaller than the upper limit at each layer $k$ , i.e., $E ( X ^ { ( k ) } ) \le c _ { \mathrm { { m a x } } } E ( X ^ { ( 0 ) } )$ .
+
+# 4.3 SReLU Activation
+
+Note that the previous theoretical analysis and model design are conducted by ignoring the activation function, which is usually given by ReLU in GNN. In this section, we first theoretically discuss the impact of ReLU on the Dirichlet energy, and then demonstrate the appropriate choice of activation.
+
+Lemma 6. We have $E ( \sigma ( X ^ { ( k ) } ) ) \leq E ( X ^ { ( k ) } )$ if activation function $\sigma$ is ReLU or Leaky-ReLU [33].
+
+It is shown that the application of ReLU further reduces the Dirichlet energy, since the negative embeddings are non-linearly mapped to zero. Although the trainable weights and residual connections are properly designed, the declining Dirichlet energy may violate the lower limit. On the other hand, a simplified GNN with linear identity activation will have limited model learning ability although it does not change the energy value. For example, simple graph convolution (SGC) model achieves comparable performance with the traditional GCN only with careful hyperparameter tuning [45]. We propose to apply SReLU to achieve a good trade-off between the non-linear and linear activations [46, 47]. SReLU is defined element-wisely as:
+
+$$
+\sigma ( X ^ { ( k ) } ) = \operatorname* { m a x } ( b , X ^ { ( k ) } ) ,
+$$
+
+where $b$ is a trainable shift shared for each feature dimension of $X ^ { ( k ) }$ . SReLU interpolates between the non-linearity and linearity depending on shift $b$ . While the linear identity activation is approximated if $b$ is close to $\infty$ , the non-linear mapping is activated if node embedding is smaller than the specific $b$ . In our experiments, we initialize $b$ with a negative value to provide an initial trade-off, and adapt it to the given task by back-propagating the training loss.
+
+# 4.4 Connections to Previous Work
+
+Recently, various techniques have been explored to enable deep GNNs [16, 24, 30]. Some of them are designed heuristically from diverse perspectives, and others are analogous to CNN components without theoretical insight tailored to graph analytics. In the following, we show how our principle and EGNN explain the existing algorithms, and expect to provide reliable theoretical guidance to the future design of deep GNNs.
+
+Embedding normalization. The general normalization layers, such as pair [23], batch [25] and group [24] normalizations, have been used to set up deep GNNs. The pair normalization (PairNorm) aims to keep the node pair distances as a constant in the different layers, and hence relieves the oversmoothing. Motivated from CNNs, the batch and group normalizations re-scale the node embeddings of a batch and a group, respectively. Similar to the operation in PairNorm, they learn to maintain the node pair distance in the node batch or group. The adopted Dirichlet energy is also a variant of the node pair distance. The existing normalization methods can be regarded as training GNN model with a constant energy constraint. However, this will prevent GNN from optimizing the energy as analyzed in Section 3. We instead regularize it within the lower and upper energy limits, and let model discover the optimum.
+
+Dropping edge. As a data augmentation method, dropping edge (DropEdge) randomly masks a fraction of edges at each epoch [29]. It makes graph connections sparse and relieves the oversmoothing by reducing information propagation. Specially, the contribution of DropEdge could be explained from the perspective of Dirichlet energy. In Erdos–Rényi graph, eigenvalue ˝ $\lambda _ { 0 }$ converges to 1 if the graph connections are more and more dense [17]. DropEdge reduces the value of $\lambda _ { 0 }$ , and helps improve the upper bound of Dirichlet energy $( 1 - \lambda _ { 0 } ) ^ { 2 } s _ { \mathrm { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } )$ to slow down the energy decreasing speed. In the extreme case where all the edges are dropped in any a graph,
+
+Laplacian $\tilde { \Delta }$ becomes a zero matrix. As a result, we have eigenvalue $\lambda _ { 0 }$ of zero and maximize the upper bound. In practice, the dropping rate has to be determined carefully depending on various tasks. Instead, our principle assigns model freedom to optimize the Dirichlet energy within a large and appropriate range.
+
+Residual connection. Motivated from CNNs, residual connection has been applied to preserve the previous node embeddings and relieve the over-smoothing. Especially, the embedding from the last layer is reused and combined completely in related work [26, 48, 49]. A fraction of the initial embedding is preserved in model GCNII [28] and APPNP [50]. Networks JKNet [27] and DAGNN [51] aggregate all the previous embeddings at the final layers. The existing work uses the residua connection empirically. In this work, we derive and explain the residual connection to guarantee the lower limit of Dirichlet energy. By modifying hyperparameter $c _ { \mathrm { m i n } }$ , our EGNN can easily evolve to the existing deep residual GNNs, such as GCNII and APPNP.
+
+Model simplification. Model SGC [45] removes all the activation and trainable weights to avoid over-fitting issue, and simplifies the training of deep GNNs. It is equivalent to EGNN with $c _ { m a x } = 1$ and $b = - \infty$ , where weights $W ^ { ( k ) }$ and shifts $b$ are remained as constants. Such simplification will reduce the model learning ability. As shown in Eq. (7), we adopt loss hyperparameter $\gamma$ to learn the trade-off between maintaining the orthogonal weights or updating them to model data characteristics.
+
+# 5 Experiments
+
+In this section, we empirically evaluate the effectiveness of EGNN on real-world datasets. We aim to answer the following questions. Q1: How does our EGNN compare with the state-of-the-art deep GNN models? Q2: Whether or not the Dirichlet energy at each layer of EGNN satisfies the constrained learning? Q3: How does each component of EGNN affect the model performance? Q4: How do the model hyperparameters impact the performance of EGNN?
+
+# 5.1 Experiment Setup
+
+Datasets. Following the practice of previous work, we evaluate EGNN by performing node classification on four benchmark datasets: Cora, Pubmed [52], Coauthor-Physics [53] and Ogbn-arxiv [54]. The detailed statistics are listed in Appendix.
+
+Baselines. We consider seven state-of-the-art baselines: GCN [15], PairNorm [23], DropEdge [29], SGC [45], JKNet [27], APPNP [50], and GCNII [28]. They are implemented based on their open repositories. The detailed descriptions of these baselines are provided in Appendix.
+
+Implementation. We implement all the baselines using Pytorch Geometric [55] based on their official implementations. The model hyperparameters are reused according to the public papers or are fine-tuned by ourselves if the classification accuracy could be further improved. Specially, we apply max-pooling to obtain the final node representation at the last layer of JKNet. In Ogbn-arxiv, we additionally include batch normalization between the successive layers in all the considered GNN models except PairNorm. Although more tricks (e.g., label reusing and linear transformation as listed in leader board) could be applied to improve node classification in Ogbn-arxiv, we focus on comparing the original GNN models in enabling deep layer stacking. The training hyperparameters are carefully set by following the previous common setting and are listed in Appendix.
+
+We implement our EGNN upon GCN, except for the components of weight initialization and regularization, lower-bounded residual connection and SReLU. We choose hyperparameters $c _ { \mathrm { m a x } }$ $c _ { \mathrm { m i n } }$ , $\gamma$ and $b$ based on the validation set. For the weight initialization, we set $c _ { \mathrm { m a x } }$ to be 1 for all the datasets; that is, the trainable weights are initialized as identity matrices at all the graph convolutional layers. The loss hyperparameter $\gamma$ is 20 in Cora, Pubmed and Coauthor-Physics to strictly regularize towards the orthogonal matrix; and it is $1 0 ^ { - 4 }$ in Ogbn-arxiv to improve the model’s learning ability. For the lower-bounded residual connection, we choose residual strength $c _ { \mathrm { m i n } }$ from range [0.1, 0.75] and list the details in Appendix. The trainable shift $b$ is initialized with $- 1 0$ in Cora and Pubmed; it is initialized to $- 5$ and $- 1$ in Coauthor-Physics and Ogbn-arxiv, respectively. We also study these hyperparameters in the following experiments. All the experiment results are the averages of 10 runs.
+
+Table 1: Node classification accuracies in percentage with various depths: 2, 16, 32/64. The highest accuracy at each column is in bold.   
+
+<table><tr><td>Datasets</td><td colspan="3">Cora</td><td colspan="3">Pubmed</td><td colspan="3">Coauthors-Physics</td><td colspan="3">Ogbn-arxiv</td></tr><tr><td>Layer Num</td><td>2</td><td>16</td><td>64</td><td>2</td><td>16</td><td>64</td><td>2</td><td>16</td><td>32</td><td>2</td><td>16</td><td>32</td></tr><tr><td>GCN</td><td>82.5</td><td>22.0</td><td>21.9</td><td>79.7</td><td>37.9</td><td>38.4</td><td>92.4</td><td>13.5</td><td>13.1</td><td>70.4</td><td>70.6</td><td>68.5</td></tr><tr><td>PairNorm</td><td>74.5</td><td>44.2</td><td>14.2</td><td>73.8</td><td>68.6</td><td>60.0</td><td>86.3</td><td>84.0</td><td>83.6</td><td>67.6</td><td>70.4</td><td>69.6</td></tr><tr><td>DropEdge</td><td>82.7</td><td>23.6</td><td>25.2</td><td>79.6</td><td>45.9</td><td>40.0</td><td>92.5</td><td>85.1</td><td>35.2</td><td>70.5</td><td>70.4</td><td>67.1</td></tr><tr><td>SGC</td><td>75.7</td><td>72.1</td><td>24.1</td><td>76.1</td><td>70.2</td><td>38.2</td><td>92.2</td><td>91.7</td><td>84.8</td><td>69.2</td><td>64.0</td><td>59.5</td></tr><tr><td>JKNet</td><td>80.8</td><td>74.5</td><td>70.0</td><td>77.2</td><td>70.0</td><td>66.1</td><td>92.7</td><td>92.2</td><td>91.6</td><td>70.6</td><td>71.8</td><td>71.4</td></tr><tr><td>APPNP</td><td>82.9</td><td>79.4</td><td>79.5</td><td>79.3</td><td>77.1</td><td>76.8</td><td>92.3</td><td>92.7</td><td>92.6</td><td>68.3</td><td>65.5</td><td>60.7</td></tr><tr><td>GCNII</td><td>82.4</td><td>84.6</td><td>85.4</td><td>77.5</td><td>79.8</td><td>79.9</td><td>92.5</td><td>92.9</td><td>92.9</td><td>70.1</td><td>71.5</td><td>70.5</td></tr><tr><td>EGNN</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2</td><td>80.0</td><td>80.1</td><td>92.6</td><td>93.1</td><td>93.3</td><td>68.4</td><td>72.7</td><td>72.7</td></tr></table>
+
+# 5.2 Experiment Results
+
+Node classification results. To answer research question Q1, Table 1 summarizes the test classification accuracies. Each accuracy is averaged over 10 random trials. We report the results with $2 / 1 6 / 6 4$ layers for Cora and Pubmed, and $2 / \bar { 1 } 6 / 3 2$ layers for Coauthor-Physics and Ogbn-arxiv. Due to space limit, we report the detailed results of mean accuracy and standard deviation in Appendix.
+
+We observe that our EGNN generally outperforms all the baselines across the four datasets, especially in the deep cases $K \geq 1 6$ ). Notably, the node classification accuracy is consistently improved with the layer stacking in EGNN until $K = 3 2$ or 64, which demonstrates the benefits of deep graph neural architecture to leverage neighbors multiple hops away. While the state-of-the-art models PairNorm, DropEdge, SGC, JKNet, and APPNP alleviate the over-smoothing issue to some extend, their performances still drop with the increasing of layers. Most of their 32/64-layer models are even worse than their corresponding shallow versions. As the most competitive deep architecture in literature, GCNII augments the transformation matrix as $( 1 - \phi ) I _ { d } + \phi W ^ { ( k ) }$ , where $0 < \phi < 1$ is a hyperparameter to preserve the identity mapping and enhance the minimum singular value of the augmented weight. Instead of explicitly defining the strength of identity mapping, we propose the orthogonal weight initialization based on the upper limit of Dirichlet energy and apply the orthogonal weight regularization. Based on Eq. (7), EGNN automatically learns the optimal trade-off between identity mapping and task adaption. Furthermore, we use SReLU activation and the residual connection to theoretically control the lower limit of Dirichlet energy. The experimental results show that EGNN not only outperforms GCNII in the small graphs Cora, Pubmed and Coauthor-Physics, but also delivers significantly superior performance in the large graph Obgn-arxiv, achieving ${ \mathrm { { 3 . 1 \% } } }$ improvement over GCNII with 32 layers.
+
+Dirichelet energy visualization. To answer research question Q2, we show the Dirichlet energy at each layer of a 64-layer EGNN in Cora and Pubmed datasets in Figure 1. To have better visualization purposes, by keeping other default hyperparameters unchanged, EGNN is trained with $c _ { \mathrm { m a x } } / c _ { \mathrm { m i n } } \quad = \quad 0 . 4 / 0 . 1 5$ and $c _ { \mathrm { m a x } } / c _ { \mathrm { m i n } } ~ = ~ 0 . 4 / 0 . 1 1$ in Cora and Pubmed, respectively. We only plot and
+
+![](images/dce42cd4d2dabb4d48968d4eb24a0fba4c484473f3441181ed5f2cfce3b2dee9.jpg)  
+Figure 1: Dirichelet energy variation with layers in Cora (Left) and Pubmed (Right). The upper and lower denotes the energy limits.
+
+compare with the baseline approaches of GCN and GCNII due to space limit. For other methods, the Dirichlet energy is either close to zero or overly large due to the over-smoothing issue or over-separating issue of node embeddings, respectively.
+
+It is shown that the Dirichlet energies of EGNN are strictly constrained within the range determined by the lower and upper limits of the constrained learning. Due to the over-smoothing issue in GCN, all the node embeddings converge to zero vectors. GCNII has comparable or smaller Dirichlet energy by carefully and explicitly designing both the initial connection and identity mapping strengths. In contrast, our EGNN only gives the appropriate limits of Dirichlet energy, and let the model learn the optimal energy at each layer for a specific task. The following hyperparameter studies will show that the values of $c _ { \mathrm { m i n } }$ and $c _ { \mathrm { m a x } }$ could be easily selected from a large appropriate range.
+
+Table 2: Ablation studies on weight initialization, lower limit $c _ { \mathrm { m i n } }$ and activation function of EGNN.   
+
+<table><tr><td rowspan="2">Component</td><td rowspan="2">Type</td><td colspan="3">Cora</td><td colspan="2">Pubmed</td><td colspan="2">Coauthors-Physics|</td><td colspan="3">Ogbn-arxiv</td></tr><tr><td>2</td><td>16</td><td>64 2</td><td>16</td><td>64 2</td><td>16</td><td>32</td><td>2</td><td>16</td><td>32</td></tr><tr><td>Weight</td><td>Glorot</td><td>77.8</td><td>40.2</td><td>23.6</td><td>68.4 62.6</td><td>60.2</td><td>92.6 81.7</td><td>73.4</td><td></td><td>68.472.8</td><td>72.7</td></tr><tr><td rowspan="3">initialization Lower limit setting</td><td>Orthogonal</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2 80.0 80.1</td><td></td><td>92.6 93.1</td><td>93.3</td><td></td><td>68.4 72.7 72.7</td><td></td></tr><tr><td>0.</td><td>83.6</td><td>68.6</td><td>12.9</td><td>78.9 77.1</td><td>44.1</td><td>92.8 91.4</td><td>79.7</td><td>70.9</td><td>69.4</td><td>62.4</td></tr><tr><td>0.1~ 0.75</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2 80.0 80.1</td><td></td><td>92.6 93.1</td><td>93.3</td><td>68.4</td><td>72.7 72.7</td><td></td></tr><tr><td rowspan="3">Cmin Activation</td><td>0.95</td><td>65.4</td><td>72.0</td><td>71.5</td><td>74.0 75.3</td><td>75.7</td><td>89.4 90.4</td><td>90.5</td><td>56.5</td><td>66.869.5</td><td></td></tr><tr><td>Linear</td><td>83.1</td><td>85.6 85.5</td><td></td><td>79.2 79.9</td><td>79.9</td><td>92.6 93.1</td><td>93.1</td><td></td><td>64.872.5</td><td>71.0</td></tr><tr><td>SReLU</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2 80.0 80.1</td><td></td><td>92.6 93.1</td><td>93.3</td><td>68.4</td><td>72.7 72.7</td><td></td></tr><tr><td rowspan="2"></td><td>ReLU</td><td>83.1</td><td>85.2</td><td>85.0</td><td>79.1 79.7 79.9</td><td></td><td>92.6 93.1</td><td>93.1</td><td></td><td>68.6 72.4 72.4</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+![](images/99e58173f858d76b515bb8cac7d2c1655dd74a6660d3b26194b469d96a922a77.jpg)  
+Figure 2: The impacts of hyperparameters $b$ , $\gamma$ , $c _ { \mathrm { m i n } }$ and $c _ { \mathrm { m a x } }$ on 64-layer EGNN trained in Cora. Y-axis is test accuracy in percent.
+
+Ablation studies of EGNN components. To demonstrate how each component affects the training of graph neural architecture and answer research question Q3, we perform the ablation experiments with EGNN on all the datasets. For the component of orthogonal weight initialization and regularization, we compare and replace them with the traditional Glorot initialization and Frobenius norm regularization as shown in Eq. (6). Considering the component of lower-bounded residual connection, we vary the lower limit hyperparameter $c _ { \mathrm { m i n } }$ from 0, $0 . 1 \sim 0 . 7 5$ and 0.95. Within the range of $0 . 1 \sim 0 . 7 5$ , the adoption of specific values is specified for each dataset in Appendix. The component of the activation function is studied from candidates of linear identity activation, SReLU, and ReLU. Table 2 reports the results of the above ablation studies.
+
+The orthogonal weight initialization and regularization are crucial to train the deep graph neural architecture. In Cora, Pubmed, and Coauthor-Physics, Glorot initialization and Frobenius norm regularization fail to control the singular values of trainable weights, which may lead to overly large or small Dirichlet energy and affect the node classification performance. In Ogbn-arxiv, the input node features are described by dense word embeddings of a paper [56], where the trainable weights in GNN are required to capture data statistics and optimize the classification task. EGNN applies a small loss hyperparameter $\gamma$ of $1 0 ^ { - 4 }$ to let the model adapt to the given task, which is equivalent to the traditional regularization. Therefore, the two approaches have comparable performances.
+
+An appropriate lower limit could enable the deep EGNN. While the Dirichlet energy may approach zero without the residual connection, the overwhelming residual information with $c _ { \operatorname* { m i n } } = 0 . 9 5$ prevents the higher layer from learning the new neighborhood information. Within the large and appropriate range of [0.1, 0.75], $c _ { \mathrm { m i n } }$ could be easily selected to achieve superior performance.
+
+Activation SReLU performs slightly better than the linear identity activation and ReLU. This is because SReLU could automatically learn the trade-off between linear and non-linear activations, which prevents the significant dropping of Dirichlet energy and ensures the model learning ability.
+
+Hyperparameter analysis. To understand the hyperparameter impacts on a 64-layer EGNN and answer research question Q4, we conduct experiments with different values of initial shift $b$ , loss factor $\gamma$ , lower limit factor $c _ { \mathrm { m i n } }$ and upper one $c _ { \mathrm { m a x } }$ . We present the hyperparameter study in Figure 2 for Cora, and show the others with similar tendencies in Appendix.
+
+We observe that our method is not sensitive to the choices of $b$ , $\gamma$ , $c _ { \mathrm { m i n } }$ and $c _ { \mathrm { m a x } }$ in a wide range: (i) The initial shift value should be $b \leq 0$ , in order to avoid the overly nonlinear mapping and Dirichlet energy damage. (ii) It is shown that EGNN approximates the optimal performance once the loss factor $\gamma$ is larger than a specific threshold. The thresholds are 0.3 in Cora, 0.1 in Pubmed and Coauthor-Physics, and 1es-4 in Ogbn-arxiv, respectively. The threshold depends on the specific dataset: while a larger potentially works in the small dataset to strictly regularize Dirichlet energy, a smaller one would be preferred for the large dataset to capture the complex data manifold. (iii) $c _ { \mathrm { m i n } }$ within the appropriate range [0.1, 0.75] allows the model to expand neighborhood size and preserve residual information to avoid the over-smoothing. (iv) As shown in Figure 1, since energy $E ( X ^ { ( k ) } )$ at the hidden layer is much smaller than $E ( X ^ { ( 0 ) } )$ from the input layer, we could easily satisfy the upper limit with $c _ { \mathrm { m a x } }$ in a large range [0.2, 1]. Given these large hyperparameter ranges, EGNN could be easily trained with deep layers.
+
+# 6 Conclusions
+
+In this paper, we propose a Dirichlet energy constrained learning principle to show the importance of regularizing the Dirichlet energy at each layer within reasonable lower and upper limits. Such energy constraint is theoretically proved to help avoid the over-smoothing and over-separating issues. We then design EGNN based on our theoretical results and empirically demonstrate that the constrained learning plays a key role in guiding the design and training of deep graph neural architecture. The detailed analysis is presented to illustrate how our principle connects and combines the previous deep methods. The experiments on benchmarks show that EGNN could be easily trained to achieve superior node classification performances with deep layer stacking. We believe that the constrained learning principle will help discover deeper and more powerful GNNs in the future.
+
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+    {
+        "type": "text",
+        "text": "Rui Chen Samsung Research America rui.chen1@samsung.com ",
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+    {
+        "type": "text",
+        "text": "Li Li Samsung Research America li.li1@samsung.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Soo-Hyun Choi∗ Samsung Electronics soohyunc@gmail.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Xia Hu Rice University xia.hu@rice.edu ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Graph neural networks (GNNs) integrate deep architectures and topological structure modeling in an effective way. However, the performance of existing GNNs would decrease significantly when they stack many layers, because of the oversmoothing issue. Node embeddings tend to converge to similar vectors when GNNs keep recursively aggregating the representations of neighbors. To enable deep GNNs, several methods have been explored recently. But they are developed from either techniques in convolutional neural networks or heuristic strategies. There is no generalizable and theoretical principle to guide the design of deep GNNs. To this end, we analyze the bottleneck of deep GNNs by leveraging the Dirichlet energy of node embeddings, and propose a generalizable principle to guide the training of deep GNNs. Based on it, a novel deep GNN framework – Energetic Graph Neural Networks (EGNN) is designed. It could provide lower and upper constraints in terms of Dirichlet energy at each layer to avoid over-smoothing. Experimental results demonstrate that EGNN achieves state-of-the-art performance by using deep layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Graph neural networks (GNNs) [1] are promising deep learning tools to analyze networked data, such as social networks [2, 3, 4], academic networks [5, 6, 7], and molecular graphs [8, 9, 10, 11]. Based on spatial graph convolutions, GNNs apply a recursive aggregation mechanism to update the representation of each node by incorporating representations of itself and its neighbors [12]. A variety of GNN variations have been explored for different real-world networks and applications [13, 14]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A key limitation of GNNs is that when we stack many layers, the performance would decrease significantly. Experiments show that GNNs often achieve the best performance with less than 3 layers [15, 13]. As the layer number increases, the node representations will converge to indistinguishable vectors due to the recursive neighborhood aggregation and non-linear activation [16, 17]. ",
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+    {
+        "type": "text",
+        "text": "Such phenomenon is recognized as over-smoothing issue [18, 19, 20, 21, 22]. It prevents the stacking of many layers and modeling the dependencies to high-order neighbors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A number of algorithms have been proposed to alleviate the over-smoothing issue and construct deep GNNs, including embedding normalization [23, 24, 25], residual connection [26, 27, 28] and random data augmentation [29, 30, 31]. However, some of them are motivated directly by techniques in convolutional neural networks (CNNs) [32], such as the embedding normalization and residual connection. Others are based on heuristic strategies, such as random embedding propagation [30] and dropping edge [29]. Most of them only achieve comparable or even worse performance compared to their shallow models. Recently, a metric of Dirichlet energy has been applied to quantify the over-smoothing [33], which is based on measuring node pair distances. With the increasing of layers, the Dirichlet energy converges to zero since node embeddings become close to each other. But there is a lack of empirical methods to leverage this metric to overcome the over-smoothing issue. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Therefore, it remains a non-trivial task to train a deep GNN architecture due to three challenges. First, the existing efforts are developed from diverse perspectives, without a generalizable principle and analysis. The abundance of these components also makes the design of deep GNNs challenging, i.e., how should we choose a suitable one or combinations for real-world scenarios? Second, even if an effective indicator of over-smoothing is given, it is hard to theoretically analyze the bottleneck and propose a generalizable principle to guide the training of deep GNNs. Third, even if theoretical guidance is given, it may be difficult to be utilized and implemented to train GNNs empirically. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "To this end, in this paper, we target to develop a generalizable framework with a theoretical basis, to handle the over-smoothing issue and enable effective deep GNN architectures. In particular, we will investigate two research questions. 1) Is there a theoretical and generalizable principle to guide the architecture design and training of deep GNNs? 2) How can we develop an effective architecture to achieve state-of-the-art performance by stacking a large number of layers? Following these questions, we make three major contributions as follows. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a generalizable principle – Dirichlet energy constrained learning, to guide the training of deep GNNs by regularizing Dirichlet energy. Without proper training, the Dirichlet energy would be either too small due to the over-smoothing issue, or too large when the node embeddings are over-separating. Our principle carefully defines an appropriate range of Dirichlet energy at each layer. Being regularized within this range, a deep GNN model could be trained by jointly optimizing the task loss and energy value. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We design a novel deep architecture – Energetic Graph Neural Networks (EGNN). It follows the proposed principle and could efficiently learn an optimal Dirichlet energy. It consists of three components, i.e., orthogonal weight controlling, lower-bounded residual connection, and shifted ReLU (SReLU) activation. The trainable weights at graph convolutional layers are orthogonally initialized as diagonal matrices, whose diagonal values are regularized to meet the upper energy limit and eliminate the over-separating. The residual connection strength is determined by the lower energy limit to avoid the over-smoothing. While the widely-used ReLU activation causes the extra loss of Dirichlet energy, the linear mapping worsens the learning ability of GNNs. We apply SReLU with a trainable shift to provide a trade-off between the non-linear and linear mappings. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "• We show that the proposed principle and EGNN can well explain most of the existing techniques for deep GNNs. Empirical results demonstrate that EGNN could be easily trained to reach 64 layers and achieves surprisingly competitive performance on benchmarks. ",
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+    {
+        "type": "text",
+        "text": "2 Problem Statement ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Notations. Given an undirected graph consisting of $n$ nodes, it is represented as $G = ( A , X )$ , where $A \\in \\mathbb { R } ^ { n \\times n }$ denotes the adjacency matrix and $\\boldsymbol { X } \\in \\mathbb { R } ^ { n \\times d }$ denotes the feature matrix. Let ${ \\tilde { A } } : = A + I _ { n }$ and $\\tilde { D } : = D + I _ { n }$ be the adjacency and degree matrix of the graph augmented with selfloops. The augmented normalized Laplacian is then given by $\\tilde { \\Delta } : = I _ { n } - \\tilde { P }$ , where $\\bar { \\tilde { P } } : = \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }$ is an augmented normalized adjacency matrix used for the neighborhood aggregation in GNN models. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Node classification task. GNNs have been adopted in many applications [6, 11, 34]. Without loss of generality, we take node classification as an example. Given a graph $G = ( A , X )$ and a set of its nodes with labels for training, the goal is to predict the labels of nodes in a test set. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "We now use the graph convolutional network (GCN) [15] as a typical example, to illustrate how traditional GNNs perform the network analysis task. Formally, the layer-wise forward-propagation operation in GCN at the $k$ -th layer is defined as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5174a7cf4ad97e0fbed41146d143f9246c453b33dbe398ebdbb3f12a564213e9.jpg",
+        "text": "$$\nX ^ { ( k ) } = \\sigma ( \\tilde { P } X ^ { ( k - 1 ) } W ^ { ( k ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$X ^ { ( k ) }$ and $X ^ { ( k - 1 ) }$ are node embedding matrices at layers $k$ and $k - 1$ , respectively; $W ^ { ( k ) } \\in \\mathbb { R } ^ { d \\times d }$ denotes trainable weights used for feature transformation; $\\sigma$ denotes an activation function such as ReLU; $X ^ { ( 0 ) } = X$ at the initial layer of GCN. The embeddings at the final layer are optimized with a node classification loss function, e.g., cross-entropy loss. The recursive neighborhood aggregation in Eq. (1) will make node embeddings similar to each other as the number of layer $k$ increases. This property, i.e., over-smoothing, prevents traditional GNNs from exploring neighbors many hops away. In practice, the dependencies to high-order neighbors are important to the node classification. The traditional shallow GNNs may have sub-optimal performances in the downstream tasks [16, 28]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Dirichlet Energy Constrained Learning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we aim to develop an effective principle to alleviate the over-smoothing issue and enable deep GNNs to leverage the high-order neighbors. We first theoretically analyze the over-smoothing issue, and then provide a principle to explain the key constraint in training deep GNNs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Node pair distance has been widely adopted to quantify the over-smoothing based on embedding similarities [19, 23]. Among the series of distance metrics, Dirichlet energy is simple and expressive for the over-smoothing analysis [33]. Thus, we adopt Dirichlet energy and formally define it as below. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Definition 1. Given node embedding matrix $X ^ { ( k ) } = [ x _ { 1 } ^ { ( k ) } , \\cdots , x _ { n } ^ { ( k ) } ] ^ { \\top } \\in \\mathbb { R } ^ { n \\times d }$ learned from GCN at the $k$ -th layer, the Dirichlet energy $E ( X ^ { ( k ) } )$ is defined as follows: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b24a8fb1436cd938879fb09e14b056b3430df72952cc609a12845be764999f6d.jpg",
+        "text": "$$\nE ( { \\boldsymbol { X } } ^ { ( k ) } ) = \\operatorname { t r } ( { \\boldsymbol { X } } ^ { ( k ) } ^ { \\top } \\tilde { \\Delta } { \\boldsymbol { X } } ^ { ( k ) } ) = \\frac { 1 } { 2 } \\sum a _ { i j } | | \\frac { \\boldsymbol { x } _ { i } ^ { ( k ) } } { \\sqrt { 1 + d _ { i } } } - \\frac { { \\boldsymbol { x } } _ { j } ^ { ( k ) } } { \\sqrt { 1 + d _ { j } } } | | _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            535,
+            725,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { t r } ( \\cdot )$ denotes trace of a matrix; $a _ { i j }$ is edge weight given by the $( i , j )$ -th element in matrix $A$ ; $d _ { i }$ is node degree given by the $i$ -th diagonal element in matrix $D$ . Dirichlet energy reveals the embedding smoothness with the weighted node pair distance. While a smaller value of $E ( X ^ { ( k ) } )$ is highly related to the over-smoothing, a larger one indicates that the node embeddings are over-separating even for those nodes with the same label. Considering the node classification task, one would prefer to have an appropriate Dirichlet energy at each layer to separate the nodes of different classes while keeping those of the same class close. However, under some conditions, the upper bound of Dirichlet energy is theoretically proved to converge to 0 in the limit of infinite layers [33]. In other words, all nodes converge to a trivial fixed point in the embedding space. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Based on the previous analysis, we derive the corresponding lower bound and revisit the oversmoothing/separating problem from the model design and training perspectives. To simplify the derivation process, we remove the non-linear activation $\\sigma$ , and re-express GCN as: $X ^ { ( \\bar { k } ) } =$ $P \\cdot \\cdot \\cdot P X W ^ { ( 1 ) } \\cdot \\cdot \\cdot W ^ { ( k ) }$ . The impact of non-linear function will be considered in the model design. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. The Dirichlet energy at the $k$ -th layer is bounded as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b8d826bc40899e936fb16025ca7db59a326d77ee8110b360179fbbf3858e47b3.jpg",
+        "text": "$$\n( 1 - \\lambda _ { 1 } ) ^ { 2 } s _ { \\operatorname* { m i n } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) \\leq E ( X ^ { ( k ) } ) \\leq ( 1 - \\lambda _ { 0 } ) ^ { 2 } s _ { \\operatorname* { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The detailed proof is provided in the Appendix. $\\lambda _ { 1 }$ and $\\lambda _ { 0 }$ are the non-zero eigenvalues of matrix $\\tilde { \\Delta }$ that are most close to values 1 and 0, respectively. $s _ { \\mathrm { m i n } } ^ { ( k ) }$ and $s _ { \\mathrm { m a x } } ^ { ( k ) }$ are the squares of minimum and maximum singular values of weight $W ^ { ( k ) }$ , respectively. Note that the eigenvalues of $\\tilde { \\Delta }$ vary with the real-world graphs, and locate within range $[ 0 , 2 )$ . We relax the above bounds as below. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. The lower and upper bounds of Dirichlet energy at the $k$ -th layer could be relaxed as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3860dcb8cb460210c19c324069f4ec35cbc1af399e927816bab36ae1a435b65d.jpg",
+        "text": "$$\n0 \\leq E ( X ^ { ( k ) } ) \\leq s _ { \\operatorname* { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Besides the uncontrollable eigenvalues determined by the underlying graph, it is shown that the Dirichlet energy can be either too small or too large without proper design and training on weight $W ^ { ( k ) }$ . On one hand, based on the common Glorot initialization [35] and L2 regularization, we empirically find that some of the weight matrices approximate to zero in a deep GCN. The corresponding square singular values are hence close to zero in these intermediate layers. That means the Dirichlet energy will become zero at the higher layers of GCN and causes the over-smoothing issue. On the other hand, without the proper weight initialization and regularization, a large $s _ { \\mathrm { m a x } } ^ { ( k ) }$ may lead to the energy explosion and the over-separating. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Dirichlet energy plays a key role in training a deep GNN model. However, the optimal value of Dirichlet energy varies in the different layers and applications. It is hard to be specified ahead and then enforces the node representation learning. Therefore, we propose a principle – Dirichlet energy constrained learning, defined in Proposition 1. It provides appropriate lower and upper limits of Dirichlet energy. Regularized by such a given range, a deep GNN model could be trained by jointly optimizing the node classification loss and Dirichlet energy at each layer. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Dirichlet energy constrained learning defines the lower & upper limits at layer $k$ as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9089a5c73ca732636f8ebb625e7530dfd3b464dfeba25673d130f045d32d6a7.jpg",
+        "text": "$$\nc _ { \\operatorname* { m i n } } E ( X ^ { ( k - 1 ) } ) \\leq E ( X ^ { ( k ) } ) \\leq c _ { \\operatorname* { m a x } } E ( X ^ { ( 0 ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We apply the transformed initial feature through trainable function $f$ $\\because X ^ { ( 0 ) } = f ( X ) \\in \\mathbb { R } ^ { n \\times d }$ . Both $c _ { \\mathrm { m i n } }$ and $c _ { \\mathrm { m a x } }$ are positive hyperparameters. From value interval $( 0 , 1 )$ , hyperparameter $c _ { \\mathrm { m i n } }$ is selected by satisfying constraint of $E ( X ^ { ( k ) } ) \\ge c _ { \\mathrm { m i n } } ^ { k } E ( X ^ { ( 0 ) } ) > 0$ . In such a way, the over-smoothing is overcome since the Dirichlet energies of all the layers are larger than appropriate limits related to ckmin. Compared with the initial transformed feature $X ^ { ( 0 ) }$ , the intermediate node embeddings of the same class are expected to be merged closely to have a smaller Dirichlet energy and facilitate the downstream applications. Therefore, we exploit the upper limit $c _ { \\mathrm { m a x } } E ( X ^ { ( 0 ) } )$ to avoid overseparating, where $c _ { \\mathrm { m a x } }$ is usually selected from $( 0 , 1 ]$ . In the experiment part, we show that the optimal energy accompanied with the minimized classification loss locates within the above range at each layer. Furthermore, hyperparameters $c _ { \\mathrm { m i n } }$ and $c _ { \\mathrm { m a x } }$ could be easily selected from the large and appropriate value scopes, which do not affect the model performance. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given both the low and upper limits, an intuitive solution to search the optimal energy is to train GNNs by optimizing the following constrained problem: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/985eb571e69cd179112f65d767b476f0c404c0a30927b3724fb099de444bb7d8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\operatorname* { m i n } \\quad \\mathcal { L } _ { \\mathrm { t a s k } } + \\gamma \\sum _ { k } | | W ^ { ( k ) } | | _ { F } , } \\\\ & { \\mathrm { s . t . } \\quad c _ { \\operatorname* { m i n } } E \\big ( X ^ { ( k - 1 ) } \\big ) \\leq E \\big ( X ^ { ( k ) } \\big ) \\leq c _ { \\operatorname* { m a x } } E \\big ( X ^ { ( 0 ) } \\big ) , \\mathrm { f o r } k = 1 , \\cdots , K . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
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+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { L } _ { \\mathrm { t a s k } }$ denotes the cross-entropy loss of node classification task; $K$ is layer number of GNN; $| | \\cdot | | _ { F }$ denotes Frobenius norm of a matrix; and $\\gamma$ is loss hyperparameter. Note that Dirichlet energy has also been adopted to regularize the node representation learning in shallow neural networks [36, 37, 38]. We instead focus on optimizing deep GNNs as shown in Eq. (6), where $K$ is often large. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Energetic Graph Neural Networks - EGNN ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is non-trivial to optimize Problem (6) due to the expensive computation of $E ( X ^ { ( k ) } )$ . Furthermore, the numerous constraints make the problem a very complex optimization hyper-planes, at which the raw task objective tends to fall into local optimums. Instead of directly optimizing Problem (6), we propose an efficient model EGNN to satisfy the constrained learning from three perspectives: weight controlling, residual connection and activation function. We introduce them one by one as follows. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Orthogonal Weight Controlling ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "According to Lemma 2, without regularizing the maximum square singular value $s _ { \\mathrm { m a x } } ^ { ( k ) }$ of matrix $W ^ { ( k ) }$ , the upper bound of Dirichlet energy can be larger than the upper limit, i.e., $s _ { \\mathrm { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) >$ $c _ { \\mathrm { m a x } } E ( X ^ { ( 0 ) } )$ . That means the Dirichlet energy of a layer may break the upper limit of constrained learning, and makes Problem (6) infeasible. In this section, we show how to satisfy such limit by controlling the singular values during weight initialization and model regularization. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Orthogonal initialization. Since the widely-used initialization methods (e.g., Glorot initialization) fail to restrict the scopes of singular values, we adopt the orthogonal approach that initializes trainable weight $W ^ { ( k ) }$ as a diagonal matrix with explicit singular values [39]. To restrict $s _ { \\mathrm { m a x } } ^ { ( k ) }$ and meet the constrained learning, we apply an equality constraint of $s _ { \\mathrm { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } ) = c _ { \\mathrm { m a x } } E ( X ^ { ( 0 ) } )$ at each layer. Based on this condition, we derive Proposition 2 to initialize those weights $W ^ { ( k ) }$ and their square singular values for all the layers of EGNN, and give Lemma 3 to show how we can satisfy the upper limit of constrained learning. The detailed derivation and proof are listed in Appendix. ",
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+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. At the first layer, weight $W ^ { ( 1 ) }$ is initialized as a diagonal matrix $\\sqrt { c _ { \\operatorname* { m a x } } } \\cdot I _ { d }$ , where $I _ { d }$ is identity matrix with dimension $d$ and the square singular values are $c _ { \\mathrm { m a x } }$ . At the higher layer $k > 1$ , weight $W ^ { ( k ) }$ is initialized with an identity matrix $I _ { d }$ , where the square singular values are 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3. Based on the above orthogonal initialization, at the starting point of training, the Dirichlet energy of EGNN satisfies the upper limit at each layer $k$ : $E ( X ^ { ( k ) } ) \\le c _ { \\mathrm { { m a x } } } E ( X ^ { ( 0 ) } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Orthogonal regularization. However, without proper regularization, the initialized weights cannot guarantee they will still satisfy the constrained learning during model training. Therefore, we propose a training loss that penalizes the distances between the trainable weights and initialized weights $\\sqrt { c _ { \\operatorname* { m a x } } } I _ { d }$ or $I _ { d }$ . To be specific, we modify the optimization problem (6) as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d784ec0b0637333d8da70a0936ed80abfa3934ada94eda68d1797d850973b84a.jpg",
+        "text": "$$\n\\operatorname* { m i n } \\mathcal { L } _ { \\mathrm { t a s k } } + \\gamma | | W ^ { ( 1 ) } - \\sqrt { c _ { \\operatorname* { m a x } } } I _ { d } | | _ { F } + \\gamma \\sum _ { k = 2 } ^ { K } | | W ^ { ( k ) } - I _ { d } | | _ { F } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Comparing with the original problem (6), we instead use the weight penalization to meet the upper limit of constrained learning, and make the model training efficient. While a larger $\\gamma$ highly regularizes the trainable weights around the initialized ones to satisfy the constrained learning, a smaller $\\gamma$ assigns the model more freedom to adapt to task data and optimize the node classification loss. Considering the above orthogonal initialization where weight $W ^ { ( k ) }$ is diagonal and sparse, we use the simplest distance constraint in Eq. (7) to update weight at the vicinity of its initialization. The singular values of updated sparse weight will be mainly determined by the dominant diagonal values, which are potentially close to the initialized ones. Therefore, we are able to control the singular values and regularize the upper limit of Dirichlet energy even at the model training phase. In the future work, more the advanced orthogonal initialization and regularization approaches could be explored to further boost performance of deep GNNs [40, 41, 42]. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Lower-bounded Residual Connection ",
+        "text_level": 1,
+        "bbox": [
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+            597,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although the square singular values are initialized and regularized properly, we may still fail to guarantee the lower limit of constrained learning in some specific graphs. According to Lemma 1, the lower bound of Dirichlet energy is $( 1 - \\lambda _ { 1 } ) ^ { 2 } s _ { \\operatorname* { m i n } } ^ { ( k ) } E ( \\bar { X ^ { ( k - 1 ) } } )$ . In the real-world applications, may exactly equal to 1 and relaxes the lower bound as zero as shown in Lemma 2. For example, in Erdos–Rényi graph with dense connections [ ˝ 43], the eigenvalues of matrix $\\tilde { \\Delta }$ converge to 1 with high probability [17]. Even though $s _ { \\mathrm { m i n } } ^ { ( k ) } > 0$ , the Dirichlet energy can be smaller than the lower limit and leads to the over-smoothing. To tackle this problem, we adopt residual connections to the initial layer $X ^ { ( 0 ) }$ and the previous layer $X ^ { ( k - 1 ) }$ . To be specific, we define the residual graph convolutions as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/035fd85430f2ca097b696ec1a0b82ffec610d4ee322ddf014eacdaf2e27dccdc.jpg",
+        "text": "$$\nX ^ { ( k ) } = \\sigma ( [ ( 1 - c _ { \\operatorname * { m i n } } ) \\tilde { P } X ^ { ( k - 1 ) } + \\alpha X ^ { ( k - 1 ) } + \\beta X ^ { ( 0 ) } ] W ^ { ( k ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\alpha$ and $\\beta$ are residual connection strengths determined by the lower limit of constrained learning, i.e., $\\alpha + \\beta = c _ { \\operatorname* { m i n } }$ . We are aware that the residual technique has been used before to set up deep GNNs [26, 44, 28]. However, they either apply the whole residual components, or combine an arbitrary fraction without theoretical insight. Instead, we use an appropriate residual connection according to the lower limit of Dirichlet energy. In the experiment part, we show that while a strong residual connection overwhelms information in the higher layers and reduces the classification performance, a weak one will lead to the over-smoothing. In the following, we justify that both the lower and upper limits in the constrained learning can be satisfied with the proposed lower-bounded residual connection. The detailed proofs are provided in Appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 4. Suppose that $c _ { \\operatorname* { m a x } } \\geq c _ { \\operatorname* { m i n } } / ( 2 c _ { \\operatorname* { m i n } } - 1 ) ^ { 2 }$ . Based upon the orthogonal controlling and residual connection, the Dirichlet energy of initialized EGNN is larger than the lower limit at each layer $k$ , i.e., $E ( X ^ { ( k ) } ) \\geq c _ { \\operatorname* { m i n } } E ( X ^ { ( k - 1 ) } )$ . ",
+        "bbox": [
+            173,
+            90,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5. Suppose that $\\begin{array} { r } { \\sqrt { c _ { \\mathrm { m a x } } } \\ge \\frac { \\beta } { ( 1 - c _ { \\mathrm { m i n } } ) \\lambda _ { 0 } + \\beta } } \\end{array}$ . Being augmented with the orthogonal controlling and residual connection, the Dirichlet energy of initialized EGNN is smaller than the upper limit at each layer $k$ , i.e., $E ( X ^ { ( k ) } ) \\le c _ { \\mathrm { { m a x } } } E ( X ^ { ( 0 ) } )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 SReLU Activation ",
+        "text_level": 1,
+        "bbox": [
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+            213,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the previous theoretical analysis and model design are conducted by ignoring the activation function, which is usually given by ReLU in GNN. In this section, we first theoretically discuss the impact of ReLU on the Dirichlet energy, and then demonstrate the appropriate choice of activation. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 6. We have $E ( \\sigma ( X ^ { ( k ) } ) ) \\leq E ( X ^ { ( k ) } )$ if activation function $\\sigma$ is ReLU or Leaky-ReLU [33]. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "It is shown that the application of ReLU further reduces the Dirichlet energy, since the negative embeddings are non-linearly mapped to zero. Although the trainable weights and residual connections are properly designed, the declining Dirichlet energy may violate the lower limit. On the other hand, a simplified GNN with linear identity activation will have limited model learning ability although it does not change the energy value. For example, simple graph convolution (SGC) model achieves comparable performance with the traditional GCN only with careful hyperparameter tuning [45]. We propose to apply SReLU to achieve a good trade-off between the non-linear and linear activations [46, 47]. SReLU is defined element-wisely as: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/34ccce6bb8818a438a799d56ece4b542158db7d96e29c17f0a471cc3a8357268.jpg",
+        "text": "$$\n\\sigma ( X ^ { ( k ) } ) = \\operatorname* { m a x } ( b , X ^ { ( k ) } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            435,
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+            454
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $b$ is a trainable shift shared for each feature dimension of $X ^ { ( k ) }$ . SReLU interpolates between the non-linearity and linearity depending on shift $b$ . While the linear identity activation is approximated if $b$ is close to $\\infty$ , the non-linear mapping is activated if node embedding is smaller than the specific $b$ . In our experiments, we initialize $b$ with a negative value to provide an initial trade-off, and adapt it to the given task by back-propagating the training loss. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4 Connections to Previous Work ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            547,
+            423,
+            563
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Recently, various techniques have been explored to enable deep GNNs [16, 24, 30]. Some of them are designed heuristically from diverse perspectives, and others are analogous to CNN components without theoretical insight tailored to graph analytics. In the following, we show how our principle and EGNN explain the existing algorithms, and expect to provide reliable theoretical guidance to the future design of deep GNNs. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Embedding normalization. The general normalization layers, such as pair [23], batch [25] and group [24] normalizations, have been used to set up deep GNNs. The pair normalization (PairNorm) aims to keep the node pair distances as a constant in the different layers, and hence relieves the oversmoothing. Motivated from CNNs, the batch and group normalizations re-scale the node embeddings of a batch and a group, respectively. Similar to the operation in PairNorm, they learn to maintain the node pair distance in the node batch or group. The adopted Dirichlet energy is also a variant of the node pair distance. The existing normalization methods can be regarded as training GNN model with a constant energy constraint. However, this will prevent GNN from optimizing the energy as analyzed in Section 3. We instead regularize it within the lower and upper energy limits, and let model discover the optimum. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Dropping edge. As a data augmentation method, dropping edge (DropEdge) randomly masks a fraction of edges at each epoch [29]. It makes graph connections sparse and relieves the oversmoothing by reducing information propagation. Specially, the contribution of DropEdge could be explained from the perspective of Dirichlet energy. In Erdos–Rényi graph, eigenvalue ˝ $\\lambda _ { 0 }$ converges to 1 if the graph connections are more and more dense [17]. DropEdge reduces the value of $\\lambda _ { 0 }$ , and helps improve the upper bound of Dirichlet energy $( 1 - \\lambda _ { 0 } ) ^ { 2 } s _ { \\mathrm { m a x } } ^ { ( k ) } E ( X ^ { ( k - 1 ) } )$ to slow down the energy decreasing speed. In the extreme case where all the edges are dropped in any a graph, ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Laplacian $\\tilde { \\Delta }$ becomes a zero matrix. As a result, we have eigenvalue $\\lambda _ { 0 }$ of zero and maximize the upper bound. In practice, the dropping rate has to be determined carefully depending on various tasks. Instead, our principle assigns model freedom to optimize the Dirichlet energy within a large and appropriate range. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Residual connection. Motivated from CNNs, residual connection has been applied to preserve the previous node embeddings and relieve the over-smoothing. Especially, the embedding from the last layer is reused and combined completely in related work [26, 48, 49]. A fraction of the initial embedding is preserved in model GCNII [28] and APPNP [50]. Networks JKNet [27] and DAGNN [51] aggregate all the previous embeddings at the final layers. The existing work uses the residua connection empirically. In this work, we derive and explain the residual connection to guarantee the lower limit of Dirichlet energy. By modifying hyperparameter $c _ { \\mathrm { m i n } }$ , our EGNN can easily evolve to the existing deep residual GNNs, such as GCNII and APPNP. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Model simplification. Model SGC [45] removes all the activation and trainable weights to avoid over-fitting issue, and simplifies the training of deep GNNs. It is equivalent to EGNN with $c _ { m a x } = 1$ and $b = - \\infty$ , where weights $W ^ { ( k ) }$ and shifts $b$ are remained as constants. Such simplification will reduce the model learning ability. As shown in Eq. (7), we adopt loss hyperparameter $\\gamma$ to learn the trade-off between maintaining the orthogonal weights or updating them to model data characteristics. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we empirically evaluate the effectiveness of EGNN on real-world datasets. We aim to answer the following questions. Q1: How does our EGNN compare with the state-of-the-art deep GNN models? Q2: Whether or not the Dirichlet energy at each layer of EGNN satisfies the constrained learning? Q3: How does each component of EGNN affect the model performance? Q4: How do the model hyperparameters impact the performance of EGNN? ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Experiment Setup ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets. Following the practice of previous work, we evaluate EGNN by performing node classification on four benchmark datasets: Cora, Pubmed [52], Coauthor-Physics [53] and Ogbn-arxiv [54]. The detailed statistics are listed in Appendix. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines. We consider seven state-of-the-art baselines: GCN [15], PairNorm [23], DropEdge [29], SGC [45], JKNet [27], APPNP [50], and GCNII [28]. They are implemented based on their open repositories. The detailed descriptions of these baselines are provided in Appendix. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implementation. We implement all the baselines using Pytorch Geometric [55] based on their official implementations. The model hyperparameters are reused according to the public papers or are fine-tuned by ourselves if the classification accuracy could be further improved. Specially, we apply max-pooling to obtain the final node representation at the last layer of JKNet. In Ogbn-arxiv, we additionally include batch normalization between the successive layers in all the considered GNN models except PairNorm. Although more tricks (e.g., label reusing and linear transformation as listed in leader board) could be applied to improve node classification in Ogbn-arxiv, we focus on comparing the original GNN models in enabling deep layer stacking. The training hyperparameters are carefully set by following the previous common setting and are listed in Appendix. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We implement our EGNN upon GCN, except for the components of weight initialization and regularization, lower-bounded residual connection and SReLU. We choose hyperparameters $c _ { \\mathrm { m a x } }$ $c _ { \\mathrm { m i n } }$ , $\\gamma$ and $b$ based on the validation set. For the weight initialization, we set $c _ { \\mathrm { m a x } }$ to be 1 for all the datasets; that is, the trainable weights are initialized as identity matrices at all the graph convolutional layers. The loss hyperparameter $\\gamma$ is 20 in Cora, Pubmed and Coauthor-Physics to strictly regularize towards the orthogonal matrix; and it is $1 0 ^ { - 4 }$ in Ogbn-arxiv to improve the model’s learning ability. For the lower-bounded residual connection, we choose residual strength $c _ { \\mathrm { m i n } }$ from range [0.1, 0.75] and list the details in Appendix. The trainable shift $b$ is initialized with $- 1 0$ in Cora and Pubmed; it is initialized to $- 5$ and $- 1$ in Coauthor-Physics and Ogbn-arxiv, respectively. We also study these hyperparameters in the following experiments. All the experiment results are the averages of 10 runs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/eb6a3160746032ad0c13bd91cf02274f7cf80e6f3775585a563e31529fe811bd.jpg",
+        "table_caption": [
+            "Table 1: Node classification accuracies in percentage with various depths: 2, 16, 32/64. The highest accuracy at each column is in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td colspan=\"3\">Cora</td><td colspan=\"3\">Pubmed</td><td colspan=\"3\">Coauthors-Physics</td><td colspan=\"3\">Ogbn-arxiv</td></tr><tr><td>Layer Num</td><td>2</td><td>16</td><td>64</td><td>2</td><td>16</td><td>64</td><td>2</td><td>16</td><td>32</td><td>2</td><td>16</td><td>32</td></tr><tr><td>GCN</td><td>82.5</td><td>22.0</td><td>21.9</td><td>79.7</td><td>37.9</td><td>38.4</td><td>92.4</td><td>13.5</td><td>13.1</td><td>70.4</td><td>70.6</td><td>68.5</td></tr><tr><td>PairNorm</td><td>74.5</td><td>44.2</td><td>14.2</td><td>73.8</td><td>68.6</td><td>60.0</td><td>86.3</td><td>84.0</td><td>83.6</td><td>67.6</td><td>70.4</td><td>69.6</td></tr><tr><td>DropEdge</td><td>82.7</td><td>23.6</td><td>25.2</td><td>79.6</td><td>45.9</td><td>40.0</td><td>92.5</td><td>85.1</td><td>35.2</td><td>70.5</td><td>70.4</td><td>67.1</td></tr><tr><td>SGC</td><td>75.7</td><td>72.1</td><td>24.1</td><td>76.1</td><td>70.2</td><td>38.2</td><td>92.2</td><td>91.7</td><td>84.8</td><td>69.2</td><td>64.0</td><td>59.5</td></tr><tr><td>JKNet</td><td>80.8</td><td>74.5</td><td>70.0</td><td>77.2</td><td>70.0</td><td>66.1</td><td>92.7</td><td>92.2</td><td>91.6</td><td>70.6</td><td>71.8</td><td>71.4</td></tr><tr><td>APPNP</td><td>82.9</td><td>79.4</td><td>79.5</td><td>79.3</td><td>77.1</td><td>76.8</td><td>92.3</td><td>92.7</td><td>92.6</td><td>68.3</td><td>65.5</td><td>60.7</td></tr><tr><td>GCNII</td><td>82.4</td><td>84.6</td><td>85.4</td><td>77.5</td><td>79.8</td><td>79.9</td><td>92.5</td><td>92.9</td><td>92.9</td><td>70.1</td><td>71.5</td><td>70.5</td></tr><tr><td>EGNN</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2</td><td>80.0</td><td>80.1</td><td>92.6</td><td>93.1</td><td>93.3</td><td>68.4</td><td>72.7</td><td>72.7</td></tr></table>",
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Experiment Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            301,
+            349,
+            318
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Node classification results. To answer research question Q1, Table 1 summarizes the test classification accuracies. Each accuracy is averaged over 10 random trials. We report the results with $2 / 1 6 / 6 4$ layers for Cora and Pubmed, and $2 / \\bar { 1 } 6 / 3 2$ layers for Coauthor-Physics and Ogbn-arxiv. Due to space limit, we report the detailed results of mean accuracy and standard deviation in Appendix. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We observe that our EGNN generally outperforms all the baselines across the four datasets, especially in the deep cases $K \\geq 1 6$ ). Notably, the node classification accuracy is consistently improved with the layer stacking in EGNN until $K = 3 2$ or 64, which demonstrates the benefits of deep graph neural architecture to leverage neighbors multiple hops away. While the state-of-the-art models PairNorm, DropEdge, SGC, JKNet, and APPNP alleviate the over-smoothing issue to some extend, their performances still drop with the increasing of layers. Most of their 32/64-layer models are even worse than their corresponding shallow versions. As the most competitive deep architecture in literature, GCNII augments the transformation matrix as $( 1 - \\phi ) I _ { d } + \\phi W ^ { ( k ) }$ , where $0 < \\phi < 1$ is a hyperparameter to preserve the identity mapping and enhance the minimum singular value of the augmented weight. Instead of explicitly defining the strength of identity mapping, we propose the orthogonal weight initialization based on the upper limit of Dirichlet energy and apply the orthogonal weight regularization. Based on Eq. (7), EGNN automatically learns the optimal trade-off between identity mapping and task adaption. Furthermore, we use SReLU activation and the residual connection to theoretically control the lower limit of Dirichlet energy. The experimental results show that EGNN not only outperforms GCNII in the small graphs Cora, Pubmed and Coauthor-Physics, but also delivers significantly superior performance in the large graph Obgn-arxiv, achieving ${ \\mathrm { { 3 . 1 \\% } } }$ improvement over GCNII with 32 layers. ",
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+        "text": "Dirichelet energy visualization. To answer research question Q2, we show the Dirichlet energy at each layer of a 64-layer EGNN in Cora and Pubmed datasets in Figure 1. To have better visualization purposes, by keeping other default hyperparameters unchanged, EGNN is trained with $c _ { \\mathrm { m a x } } / c _ { \\mathrm { m i n } } \\quad = \\quad 0 . 4 / 0 . 1 5$ and $c _ { \\mathrm { m a x } } / c _ { \\mathrm { m i n } } ~ = ~ 0 . 4 / 0 . 1 1$ in Cora and Pubmed, respectively. We only plot and ",
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+        "img_path": "images/dce42cd4d2dabb4d48968d4eb24a0fba4c484473f3441181ed5f2cfce3b2dee9.jpg",
+        "image_caption": [
+            "Figure 1: Dirichelet energy variation with layers in Cora (Left) and Pubmed (Right). The upper and lower denotes the energy limits. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "compare with the baseline approaches of GCN and GCNII due to space limit. For other methods, the Dirichlet energy is either close to zero or overly large due to the over-smoothing issue or over-separating issue of node embeddings, respectively. ",
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+        "type": "text",
+        "text": "It is shown that the Dirichlet energies of EGNN are strictly constrained within the range determined by the lower and upper limits of the constrained learning. Due to the over-smoothing issue in GCN, all the node embeddings converge to zero vectors. GCNII has comparable or smaller Dirichlet energy by carefully and explicitly designing both the initial connection and identity mapping strengths. In contrast, our EGNN only gives the appropriate limits of Dirichlet energy, and let the model learn the optimal energy at each layer for a specific task. The following hyperparameter studies will show that the values of $c _ { \\mathrm { m i n } }$ and $c _ { \\mathrm { m a x } }$ could be easily selected from a large appropriate range. ",
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+        "type": "table",
+        "img_path": "images/939d87987c8a2858260640ac81b2729fb8ec688c09d8bffc22b675280f869965.jpg",
+        "table_caption": [
+            "Table 2: Ablation studies on weight initialization, lower limit $c _ { \\mathrm { m i n } }$ and activation function of EGNN. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Component</td><td rowspan=\"2\">Type</td><td colspan=\"3\">Cora</td><td colspan=\"2\">Pubmed</td><td colspan=\"2\">Coauthors-Physics|</td><td colspan=\"3\">Ogbn-arxiv</td></tr><tr><td>2</td><td>16</td><td>64 2</td><td>16</td><td>64 2</td><td>16</td><td>32</td><td>2</td><td>16</td><td>32</td></tr><tr><td>Weight</td><td>Glorot</td><td>77.8</td><td>40.2</td><td>23.6</td><td>68.4 62.6</td><td>60.2</td><td>92.6 81.7</td><td>73.4</td><td></td><td>68.472.8</td><td>72.7</td></tr><tr><td rowspan=\"3\">initialization Lower limit setting</td><td>Orthogonal</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2 80.0 80.1</td><td></td><td>92.6 93.1</td><td>93.3</td><td></td><td>68.4 72.7 72.7</td><td></td></tr><tr><td>0.</td><td>83.6</td><td>68.6</td><td>12.9</td><td>78.9 77.1</td><td>44.1</td><td>92.8 91.4</td><td>79.7</td><td>70.9</td><td>69.4</td><td>62.4</td></tr><tr><td>0.1~ 0.75</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2 80.0 80.1</td><td></td><td>92.6 93.1</td><td>93.3</td><td>68.4</td><td>72.7 72.7</td><td></td></tr><tr><td rowspan=\"3\">Cmin Activation</td><td>0.95</td><td>65.4</td><td>72.0</td><td>71.5</td><td>74.0 75.3</td><td>75.7</td><td>89.4 90.4</td><td>90.5</td><td>56.5</td><td>66.869.5</td><td></td></tr><tr><td>Linear</td><td>83.1</td><td>85.6 85.5</td><td></td><td>79.2 79.9</td><td>79.9</td><td>92.6 93.1</td><td>93.1</td><td></td><td>64.872.5</td><td>71.0</td></tr><tr><td>SReLU</td><td>83.2</td><td>85.4</td><td>85.7</td><td>79.2 80.0 80.1</td><td></td><td>92.6 93.1</td><td>93.3</td><td>68.4</td><td>72.7 72.7</td><td></td></tr><tr><td rowspan=\"2\"></td><td>ReLU</td><td>83.1</td><td>85.2</td><td>85.0</td><td>79.1 79.7 79.9</td><td></td><td>92.6 93.1</td><td>93.1</td><td></td><td>68.6 72.4 72.4</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "image_caption": [
+            "Figure 2: The impacts of hyperparameters $b$ , $\\gamma$ , $c _ { \\mathrm { m i n } }$ and $c _ { \\mathrm { m a x } }$ on 64-layer EGNN trained in Cora. Y-axis is test accuracy in percent. "
+        ],
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+        "type": "text",
+        "text": "Ablation studies of EGNN components. To demonstrate how each component affects the training of graph neural architecture and answer research question Q3, we perform the ablation experiments with EGNN on all the datasets. For the component of orthogonal weight initialization and regularization, we compare and replace them with the traditional Glorot initialization and Frobenius norm regularization as shown in Eq. (6). Considering the component of lower-bounded residual connection, we vary the lower limit hyperparameter $c _ { \\mathrm { m i n } }$ from 0, $0 . 1 \\sim 0 . 7 5$ and 0.95. Within the range of $0 . 1 \\sim 0 . 7 5$ , the adoption of specific values is specified for each dataset in Appendix. The component of the activation function is studied from candidates of linear identity activation, SReLU, and ReLU. Table 2 reports the results of the above ablation studies. ",
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+    {
+        "type": "text",
+        "text": "The orthogonal weight initialization and regularization are crucial to train the deep graph neural architecture. In Cora, Pubmed, and Coauthor-Physics, Glorot initialization and Frobenius norm regularization fail to control the singular values of trainable weights, which may lead to overly large or small Dirichlet energy and affect the node classification performance. In Ogbn-arxiv, the input node features are described by dense word embeddings of a paper [56], where the trainable weights in GNN are required to capture data statistics and optimize the classification task. EGNN applies a small loss hyperparameter $\\gamma$ of $1 0 ^ { - 4 }$ to let the model adapt to the given task, which is equivalent to the traditional regularization. Therefore, the two approaches have comparable performances. ",
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+        "type": "text",
+        "text": "An appropriate lower limit could enable the deep EGNN. While the Dirichlet energy may approach zero without the residual connection, the overwhelming residual information with $c _ { \\operatorname* { m i n } } = 0 . 9 5$ prevents the higher layer from learning the new neighborhood information. Within the large and appropriate range of [0.1, 0.75], $c _ { \\mathrm { m i n } }$ could be easily selected to achieve superior performance. ",
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+        "text": "Activation SReLU performs slightly better than the linear identity activation and ReLU. This is because SReLU could automatically learn the trade-off between linear and non-linear activations, which prevents the significant dropping of Dirichlet energy and ensures the model learning ability. ",
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+        "type": "text",
+        "text": "Hyperparameter analysis. To understand the hyperparameter impacts on a 64-layer EGNN and answer research question Q4, we conduct experiments with different values of initial shift $b$ , loss factor $\\gamma$ , lower limit factor $c _ { \\mathrm { m i n } }$ and upper one $c _ { \\mathrm { m a x } }$ . We present the hyperparameter study in Figure 2 for Cora, and show the others with similar tendencies in Appendix. ",
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+        "text": "We observe that our method is not sensitive to the choices of $b$ , $\\gamma$ , $c _ { \\mathrm { m i n } }$ and $c _ { \\mathrm { m a x } }$ in a wide range: (i) The initial shift value should be $b \\leq 0$ , in order to avoid the overly nonlinear mapping and Dirichlet energy damage. (ii) It is shown that EGNN approximates the optimal performance once the loss factor $\\gamma$ is larger than a specific threshold. The thresholds are 0.3 in Cora, 0.1 in Pubmed and Coauthor-Physics, and 1es-4 in Ogbn-arxiv, respectively. The threshold depends on the specific dataset: while a larger potentially works in the small dataset to strictly regularize Dirichlet energy, a smaller one would be preferred for the large dataset to capture the complex data manifold. (iii) $c _ { \\mathrm { m i n } }$ within the appropriate range [0.1, 0.75] allows the model to expand neighborhood size and preserve residual information to avoid the over-smoothing. (iv) As shown in Figure 1, since energy $E ( X ^ { ( k ) } )$ at the hidden layer is much smaller than $E ( X ^ { ( 0 ) } )$ from the input layer, we could easily satisfy the upper limit with $c _ { \\mathrm { m a x } }$ in a large range [0.2, 1]. Given these large hyperparameter ranges, EGNN could be easily trained with deep layers. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
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+        "text": "In this paper, we propose a Dirichlet energy constrained learning principle to show the importance of regularizing the Dirichlet energy at each layer within reasonable lower and upper limits. Such energy constraint is theoretically proved to help avoid the over-smoothing and over-separating issues. We then design EGNN based on our theoretical results and empirically demonstrate that the constrained learning plays a key role in guiding the design and training of deep graph neural architecture. The detailed analysis is presented to illustrate how our principle connects and combines the previous deep methods. The experiments on benchmarks show that EGNN could be easily trained to achieve superior node classification performances with deep layer stacking. We believe that the constrained learning principle will help discover deeper and more powerful GNNs in the future. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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parse/train/BJlo91BYPr/BJlo91BYPr.md

@@ -0,0 +1,336 @@
+# IRRATIONALITY CAN HELP REWARD INFERENCE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Specifying reward functions is difficult, which motivates the area of reward inference: learning rewards from human behavior. The starting assumption in the area is that human behavior is optimal given the desired reward function, but in reality people have many different forms of irrationality, from noise to myopia to risk aversion and beyond. This fact seems like it will be strictly harmful to reward inference: it is already hard to infer the reward from rational behavior, and noise and systematic biases make actions have less direct of a relationship with the reward. Our insight in this work is that, contrary to expectations, irrationality can actually help rather than hinder reward inference. For some types and amounts of irrationality, the expert now produces more varied policies compared to rational behavior, which help disambiguate among different reward parameters – those that otherwise correspond to the same rational behavior. We put this to the test in a systematic analysis of the effect of irrationality on reward inference. We start by covering the space of irrationalities as deviations from the Bellman update, simulate expert behavior, and measure the accuracy of inference to contrast the different types and study the gains and losses. We provide a mutual informationbased analysis of our findings, and wrap up by discussing the need to accurately model irrationality, as well as to what extent we might expect (or be able to train) real people to exhibit helpful irrationalities when teaching rewards to learners.
+
+# 1 INTRODUCTION
+
+The application of reinforcement learning (RL) in increasingly complex environments has been most successful for problems that are already represented by a specified reward function (Lillicrap et al., 2015; Mnih et al., 2015; 2016; Silver et al., 2016). Unfortunately, not only do real-world tasks usually lack an explicit exogenously-specified reward function, but attempting to specify one tends to lead to unexpected side-effects as the agent is faced with new situations (Lehman et al., 2018).
+
+This has motivated the area of reward inference: the process of estimating a reward function from human inputs. The inputs are traditionally demonstrations, leading to inverse reinforcement learning (IRL) $\mathrm { N g }$ et al., 2000; Abbeel & $\mathrm { N g }$ , 2004) or inverse optimal control (IOC) (Kalman, 1964; Jameson & Kreindler, 1973; Mombaur et al., 2010; Finn et al., 2016). Recent work has expanded the range of inputs significantly,to comparisons (Wirth et al., 2017; Sadigh et al., 2017; Christiano et al., 2017), natural language instructions (MacGlashan et al., 2015; Fu et al., 2019), physical corrections (Jain et al., 2015; Bajcsy et al., 2017), proxy rewards (Hadfield-Menell et al., 2017; Ratner et al., 2018), or scalar reward values (Griffith et al., 2013; Loftin et al., 2014).
+
+The central assumption behind these methods is that human behavior is rational, i.e. optimal with respect to the desired reward (cumulative, in expectation). Unfortunately, decades of research in behavioral economics and cognitive science Chipman (2014) has unearthed a deluge of irrationalities, i.e. of ways in which people deviate from optimal decision making: hyperbolic discounting, scope insensitivity, optimism bias, decision noise, certainty effects, loss aversion, status quo bias, etc.
+
+Work on reward inference has predominantly used one model of irrationality: decision-making noise, where the probability of an action relates to the value that action has. The most widely used model by far is a Bolzmann distribution stemming from the Luce-Sherpard rule (Luce, 1959; Shepard, 1957; Lucas et al., 2009) and the principle of maximum (causal) entropy in (Ziebart et al., 2008; 2010), which we will refer to as Bolzmann-rationality (Fisac et al., 2017). Recent work has started to incorporate systematic biases though, like risk-aversion (Singh et al., 2017), having the wrong dynamics belief (Reddy et al., 2018), and myopia and hyperbolic discounting (Evans & Goodman, 2015; Evans et al., 2016).
+
+Learning from irrational experts feels like daunting task: reward inference is already hard with rational behavior, but now a learner needs to make sense of behavior that is noisy or systematically biased. Our goal in this work is to characterize just how muddied the waters are – how (and how much) do different irrationalities affect reward inference?
+
+Our insight is that, contrary to expectations, irrationality can actually help, rather than hinder, reward inference.
+
+Our explanation is that how good reward inference is depends on the mutual information between the policies produced by the expert and the reward parameters to be inferred. While it is often possible for two reward parameters to produce the same rational behavior, irrationalities can sometimes produce different behaviors that disambiguate between those same two reward parameters. For instance, noise can help when it is related to the value function, as Boltzmann noise is, because it distinguishes the difference in values even when the optimal action stays the same. Optimism can be helpful because the expert takes fewer risk-avoiding actions and acts more directly on their goal.
+
+Overall, we contribute 1) an analysis and comparison of the effects of different biases on reward inference testing our insight, 2) a way to systematically formalize and cover the space of irrationalities in order to conduct such an analysis, and 3) evidence for the importance of assuming the right type of irrationality during inference.
+
+Our good news is that irrationalities can indeed be an ally for inference. Of course, this is not always true – the details of which irrationality type and how much of it also matter. We see these results as opening the door to a better understanding of reward inference, as well as to practical ways of making inference easier by asking for the right kind of expert demonstrations – after all, in some cases it might be easier for people to act optimistically or myopically than to act rationally. Our results reinforce that optimal teaching is different from optimal doing, but point out that some forms of teaching might actually be easier than doing.
+
+# 2 METHOD
+
+# 2.1 EXPLORING IRRATIONALITY THROUGH SIMULATION
+
+Our goal is to explore the effect irrationalities have on reward inference if the learner knows about them – we explore the need for the learner to accurately model irrationalities in section 4.2. While ideally we would recruit human subjects with different irrationalities and measure how well we can learn rewards, this is prohibitive because we do not get to dictate someone’s irrationality type: people exhibit a mix of them, some yet to be discovered. Further, measuring accuracy of inference is complicated by the fact that we do not have ground truth access to the desired reward: the learner can measure agreement with some test set, but the test set itself is produced subject to the same irrationalities that produced the training data. As experimenters, we would remain deluded about the human’s true intentions and preferences.
+
+To address this issue, we simulate expert behavior subject to different irrationalities based on ground truth reward functions, run reward inference, and measure the performance against the ground truth, i.e. the accuracy of a Bayesian posterior on the reward function given the (simulated) expert’s inputs.
+
+# 2.2 TYPES AND DEGREES OF IRRATIONALITY
+
+There are many possible irrationalities that people exhibit (Chipman, 2014), far more than what we could study in one paper. They come with varying degrees of mathematical formalization and replication across human studies. To provide good coverage of this space, we start from the Bellman update, and systematically manipulate its terms and operators to produce a variety of different irrationalities that deviate from the optimal MDP policy in complementary ways. For instance, operating on the discount factor can model more myopic behavior, while operating on the transition function can model optimism or the illusion of control. Figure 1 summarizes our approach, which we detail below.
+
+![](images/de748af0803fffaa2f2131dd7e3efd0f4b9524736cbffd30556b5dc1bf796a92.jpg)  
+Figure 1: We modify the components of the Bellman update to cover different types of irrationalities: changing the max into a softmax to capture noise, changing the transition function to capture optimism/pessimism or the illusion of control, changing the reward values to capture the nonlinear perception of gains and losses (prospect theory), changing the average reward over time into a maximum (extremal), and changing the discounting to capture more myopic decision-making.
+
+# 2.2.1 RATIONAL EXPERT
+
+The rational expert does value iteration using the Bellman update from figure 1. Our models change this update to produce different types of non-rational behavior.
+
+# 2.2.2 MODIFYING THE MAX OPERATOR: BOLZMANN
+
+Boltzmann-rationality modifies the maximum over actions $\mathrm { m a x } _ { a }$ with a Boltzmann operator with parameter $\beta$ :
+
+$$
+V _ { i + 1 } ( s ) = { \bf B o l t z } _ { a } ^ { \beta } \sum _ { s ^ { \prime } \in S } T ( s ^ { \prime } | s , a ) \left( r ( s , a , s ^ { \prime } ) + \gamma V _ { i } ( s ) \right)
+$$
+
+Where $\begin{array} { r } { \mathbf { B o l t z } ^ { \beta } ( \mathbf { x } ) = \sum _ { i } x _ { i } e ^ { \beta x _ { i } } / \sum _ { i } e ^ { \beta x _ { i } } } \end{array}$ (Ziebart et al., 2010; Asadi $\&$ Littman, 2017) This models that people will not be perfect, but rather noisily pick actions in a way that is related to the Qvalue of those actions. The constant $\beta$ is called the rationality constant, because as $\beta  \infty$ , the human choices approach perfect rationality (optimality), whereas $\beta = 0$ produces uniformly random choices. This is the standard assumption for reward inference that does not assume perfect rationality, because it easily transforms the rationality assumption into a probability distribution over actions, enabling learners to make sense of imperfect demonstrations that otherwise do not match up with any reward parameters.
+
+# 2.2.3 MODIFYING THE TRANSITION FUNCTION
+
+Our next set of irrationalities manipulate the transition function away from reality.
+
+Illusion of Control. Humans often overestimate their ability to control random events. To model this, we consider experts that use the Bellman update:
+
+$$
+V _ { i + 1 } ( s ) = \operatorname* { m a x } _ { a } \sum _ { s ^ { \prime } \in S } T ^ { n } ( s ^ { \prime } | s , a ) \left( r ( s , a , s ^ { \prime } ) + \gamma V _ { i } ( s ) \right)
+$$
+
+where $T ^ { n } ( s ^ { \prime } | s , a ) \propto \left( T ( s ^ { \prime } | s , a ) \right) ^ { n }$ . As $n \to \infty$ , the demonstrator acts as if it exists in a deterministic environment. As $n  0$ , the expert acts as if it had an equal chance of transitioning to every possible successor state. At $n = 1$ , the expert is the rational expert.
+
+Optimism/Pessimism. Humans tend to systematically overestimate their chance experiencing of positive over negative events. We model this using experts that modify the probability they get outcomes based on the value of those outcomes:
+
+$$
+V _ { i + 1 } ( s ) = \operatorname* { m a x } _ { a } \sum _ { s ^ { \prime } \in S } T ^ { 1 / \tau } ( s ^ { \prime } | s , a ) \left( r ( s , a , s ^ { \prime } ) + \gamma V _ { i } ( s ) \right)
+$$
+
+where $T ^ { 1 / \tau } ( s ^ { \prime } | s , a ) \propto T ( s ^ { \prime } | s , a ) e ^ { \big ( r ( s , a , s ^ { \prime } ) + \gamma V _ { i } ( s ) \big ) / \tau }$ . $1 / \tau$ controls how pessimistic or optimistic the expert is. As $1 / \tau \to + \infty$ , the expert becomes increasingly certain that good transitions will happen. As $1 / \tau \to - \infty$ , the expert becomes increasingly certain that bad transitions will happen. As $1 / \tau \to 0$ , the expert approaches the rational expert.
+
+# 2.2.4 MODIFYING THE REWARD: PROSPECT THEORY
+
+Next, we consider experts that use the modified Bellman update:
+
+$$
+V _ { i + 1 } ( s ) = \operatorname* { m a x } _ { a } \sum _ { s ^ { \prime } \in S } T ( s ^ { \prime } | s , a ) \left( f ( r ( s , a , s ^ { \prime } ) ) + \gamma V _ { i } ( s ) \right)
+$$
+
+where $f : \mathbb { R } \to \mathbb { R }$ is some scalar function. This is equivalent to solving the MDP with reward $f \circ r$ This allows us to model human behavior such as loss aversion and scope insensitivity.
+
+Prospect Theory Kahneman & Tversky (2013) inspires us to consider a particular family of reward transforms:
+
+$$
+f _ { c } ( r ) = \left\{ \begin{array} { l l } { \log ( 1 + | r | ) } & { r > 0 } \\ { 0 } & { r = 0 } \\ { - c \log ( 1 + | r | ) } & { r < 0 } \end{array} \right.
+$$
+
+$c$ controls how loss averse the expert is. As $c  \infty$ , the expert primarily focuses on avoiding negative rewards. As $c \to 0$ , the expert focuses on maximizing positive rewards and
+
+2.2.5 MODIFYING THE SUM BETWEEN REWARD AND FUTURE VALUE: EXTREMAL
+
+Extremal. Humans seem to exhibit duration neglect, sometimes only caring about the maximum intensity of an experiennce (Do et al., 2008). We model this using experts that use the Bellman step:
+
+$$
+V _ { i + 1 } ( s ) = \operatorname* { m a x } _ { a } \sum _ { s ^ { \prime } \in S } T ( s ^ { \prime } | s , a ) \left( \operatorname* { m a x } \left[ r ( s , a , s ^ { \prime } ) , ( 1 - \alpha ) r ( s , a , s ^ { \prime } ) + \alpha V _ { i } ( s ) \right] \right)
+$$
+
+These experts maximize the expected maximum reward along a trajectory, instead of the expected sum of rewards. As $\alpha  1$ , the expert maximizes the expected maximum reward they achieve along their full trajectory. As $\alpha  0$ , the expert becomes greedy, and only cares about the reward they achieve in the next timestep.
+
+# 2.2.6 MODIFYING THE DISCOUNTING
+
+Myopic Discount. In practice, humans are often myopic, only considering immediate rewards. One way to model this is to decrease gamma in the Bellman update. At $\gamma = 1$ , this is the rational expert. As $\gamma  0$ , the expert becomes greedy and only acts to maximize immediate reward.
+
+Myopic VI. As another way to model human myopia, we consider a expert that performs only $h$ steps of Bellman updates. That is, this expert cares equally about rewards for horizon $h$ , and discount to 0 reward after that. As $h \to \infty$ , this expert becomes rational. If $h = 1$ , this expert only cares about the immediate reward.
+
+Hyperbolic Discounting. Human also exhibit hyperbolic discounting, with a high discount rate for the immediate future and a low discount rate for the far future. Alexander & Brown (2010) formulate this as the following Bellman update:
+
+$$
+V _ { i + 1 } ( s ) = \operatorname* { m a x } _ { a } \sum _ { s ^ { \prime } \in S } T ( s ^ { \prime } | s , a ) \left( r ( s , a , s ^ { \prime } ) + V _ { i } ( s ) \right) / ( 1 + k V _ { i } ( s ) )
+$$
+
+$k$ modulates how much the expert prefers rewards now versus the future. As $k  0$ , this expert becomes the rational expert.
+
+# 3 IMPACT OF IRRATIONALITIES ON REWARD INFERENCE
+
+# 3.1 EXPERIMENTAL DESIGN
+
+Simulation Environment. To reduce possible confounding from our choice of environment, we used a small 5x5 gridworld where the irrationalities nonetheless cause experts to exhibit different behavior. Our gridworld consists of three types of cells: ice, holes, and rewards. The expert can start in any ice cell. At each ice cell, the expert can move in one of the four cardinal directions. With probability 0.8, they will go in that direction. With probability 0.2, they will instead go in one of the two adjacent directions. Holes and rewards are terminal states, and return the expert back to their start state. They receive a penalty of $- 1 0$ for falling into a hole and $\theta _ { i } \in [ 0 , 4 ]$ for entering into the ith reward cell.
+
+![](images/1c54b2a5f7b51a0f6ed5ef0da25e969304cdb510afc805fbbef04a4e75a138f3.jpg)  
+Figure 2: The log loss (lower $=$ better) of the posterior as a function of the parameter we vary for each irrationality type. These six irrationalities all have parameter settings that outperform rational experts. For the models that interpolate to rational expert, we denote the value that is closest to rational using a dashed vertical line.
+
+Dependent Measures. To separate the inference difficulty caused by suboptimal inference from the difficulty caused by expert irrationality, we perform the exact Bayesian update on the trajectory $\theta$ (Ramachandran & Amir, 2007), which gives us the posterior on $\theta$ given $\xi$ :
+
+$$
+P ( \theta | \xi ) = \frac { P ( \xi | \theta ) P ( \theta ) } { \int _ { \theta ^ { \prime } } P ( \xi | \theta ^ { \prime } ) P ( \theta ^ { \prime } ) }
+$$
+
+We use two metrics to measure the difficulty of inference The first is the expected log loss of this posterior, or negative log-likelihood:
+
+$$
+\operatorname { L o g } \operatorname { L o s s } ( \theta | \xi ) = E _ { \theta , \xi \sim \pi _ { \theta } } \left[ - \log P ( \theta | \xi ) \right] .
+$$
+
+A low log loss implies that we are assigning a high likelihood to the true $\theta$ . As we are performing exact Bayesian inference with the true model $P ( \xi | \theta )$ and prior $P ( \theta )$ , the log loss is equal to the entropy of the posterior $H ( \theta | \xi )$ .
+
+The second metric is the ${ \bf L } ^ { 2 }$ -distance between the mean posterior $\theta$ and the actual theta:
+
+$$
+L ^ { 2 } ( \theta | \xi ) = E _ { \theta ^ { * } , \xi \sim \pi _ { \theta ^ { * } } } \left[ | | E [ \theta | \xi ] - \theta ^ { * } | | ^ { 2 } \right]
+$$
+
+The closer the inferred posterior mean of $\theta$ is to the actual value $\theta ^ { * }$ , the lower the loss.
+
+For each irrationality type, we calculate the performance of reward inference on trajectories of a fixed length $T$ , with respect to the two metrics above. To sample a trajectory of length $T$ from a expert, we fix $\theta ^ { * }$ and start state $s$ . Then, we perform the expert’s (possibly modified) Bellman updates until convergence to recover the policy $\pi _ { \theta ^ { \ast } }$ . Finally, we generate rollouts starting from state $s$ until $T$ state, action pairs have been sampled from $\pi _ { \theta ^ { \ast } }$ .
+
+![](images/91bd9cad2fdde456d85cd2c875735c23c953b0a1fd13d3111241ce67dd0be7fd.jpg)  
+Figure 3: A best case analysis for each irrationality type: the log $\mathrm { { l o s s } } / L ^ { 2 }$ distance from mean (lowe $\fallingdotseq$ better) for experts, as a function of the length of trajectory observed. Each irrationality uses the parameter value that is most informative. As discussed in section 3.2, different irrationality types have different slopes and converge to different values. In addition, the best performing irrationality type according to log loss is not the best performing type according to $L ^ { 2 }$ loss.
+
+# 3.2 ANALYSIS
+
+Impact of Each Irrationality. We found that of the 8 irrationalities we studied, 6 had parameter settings that lead to lower log loss than the rational expert. We report how the parameter influences the log loss for each of these experts in figure 2.1 For $\bar { T ^ { \mathrm { ~ } } } = 3 0$ , Optimism with $1 \bar { / } \tau = 3 . 1 6$ performed the best, followed by Boltzmann with $\beta = 1 0 0$ and Hyperbolic with $k = 0 . 1$ . Both forms of Myopia also outperformed the rational expert, with best performance occurring at $\gamma = 0 . 9$ and $h = 5$ . Finally, the Extremal expert also slightly outperformed the rational expert, with best performance at $\alpha = 0 . 9$ . Notably, in every case, neither the most irrational expert nor the perfectly rational expert was the most informative.
+
+Impact of Data for Different Irrationalities. Next, we investigate how the quality of inference varies as we increase the length of the observed trajectory $T$ . We report our results for the best performing parameter for each irrationality type in figure 3. Interestingly, while both metrics decrease monotonically regardless of irrationality type, the rate at which they decrease differs by the irrationality type, and the best performing irrationality type according to log loss (Optimism) is not the best performing type according to $L ^ { 2 }$ distance (Boltzmann).
+
+What is behind these differences? To explain these results, we use the notion of mutual information $\mathbf { I } ( X ; Y )$ between two variables, defined as:
+
+$$
+\mathbf { I } ( X ; Y ) = E _ { X , Y } \left[ \log \left( { \frac { P ( X , Y ) } { P ( X ) P ( Y ) } } \right) \right] = H ( X ) - H ( X | Y )
+$$
+
+The mutual information measures how much our uncertainty about $X$ decreases by observing $Y$
+
+For reward inference, the term we care about is the mutual information between the expert’s trajectory and the reward parameters
+
+$$
+\mathbf { I } ( \theta ; \xi ) = E _ { \theta , \xi \sim \theta } \left[ \log \left( \frac { P ( \theta , \xi ) } { P ( \theta ) P ( \xi ) } \right) \right] = H ( \theta ) - H ( \theta | \xi )
+$$
+
+The mutual information $\mathbf { I } ( \theta ; \xi )$ is equal to a constant minus the posterior log loss under the true model. A expert with mutual information will cause the learner to have a lower posterior log loss.
+
+![](images/dd6570023cc2d9989e5b6db0126e6119217fe0a74a5724cdce37feaf15820ea9.jpg)  
+Figure 4: (a) Optimism bias produces different actions for $\theta ^ { * } = ( 4 , 1 )$ vs. $\theta ^ { * } = ( 1 , 4 )$ in the states shown: the rational policy is to go away from the hole regardless of $\theta$ , but an optimistic expert takes the chance and goes for the larger reward – up in the first case, down in the second. (b) Pessimism bias produces different actions for $\theta ^ { * } = ( 1 , 1 )$ vs. $\theta ^ { * } = ( 4 , 4 )$ : when the reward is sufficiently large, the expert becomes convinced that no action it takes will lead to the reward, leading it to perform random actions.
+
+![](images/f25101b72d105b95a6d3b13a71a54dde5d4bf814053246b04cf619ed1876a3b9.jpg)  
+Figure 5: (a) Boltzmann-rationality produces different policies for $\theta ^ { * } = ( 1 , 1 )$ vs. $\theta ^ { * } = ( 4 , 4 )$ : when $| | \theta | |$ is larger, the policy becomes closer to that of the rational expert. (b) A Myopic expert produces different policies for $\theta ^ { * } = ( 4 , 1 )$ vs. $\theta ^ { * } = ( 4 , 0 )$ : while the rational expert always detours around the hole and attempts to reach the larger reward, myopia causes the myopic expert to go for the smaller source of reward when it is non-zero.
+
+By the information processing inequality, we have the bound $\mathbf { I } ( \theta ; \xi ) \le \mathbf { I } ( \theta ; \pi )$ .
+
+To have higher mutual information, different $\theta \mathrm { s }$ should be mapped to different policies $\pi \mathbf { S }$ . Indeed, we found that the experts that were able to outperform the rational expert were able to disambiguate between $\theta \mathrm { s }$ that the rational expert could not. To visualize this, we show examples of how the policy of several irrational experts differ when the rational expert’s policies are identical in figures 4 and 5.
+
+We plot the correlation between $\mathbf { I } ( \theta ; \xi )$ and $\mathbf { I } ( \theta ; \pi )$ in figure 6. Experts that have more informative policies tend to have more informative trajectories, but the correlation is not perfect. Notably, the Optimism expert has the most informative trajectories of length 30, but has less informative policies than the Boltzmann expert.
+
+In the limit of infinite data from every state, we would have $\mathbf { I } ( \theta ; \xi ) \to \mathbf { I } ( \theta ; \pi )$ . However, as each trajectory begins from the same start state, and not every state is reachable with every policy, the bound is not achievable in general, even if we observe an arbitrarily large number of trajectories. This highlights the need for off-policy data in reward inference tasks.
+
+# 4 DISCUSSION
+
+# 4.1 SUMMARY
+
+We show that, contrary to what we might expect, suboptimal experts can actually help an agent learn the reward function. Optimism bias, myopia (via heavier discounting or hyperbolic discounting), and noise via Boltzmann rationality were the most informative irrationalities in our environments, far surpassing the performance of the rational expert for their ideal settings. Our contribution overall was to identify a systematic set of irrationalities by looking at deviations in the terms of the Bellman update, and show that being irrational is not automatically harmful to inference by quantifying and comparing the inference performance for these different types.
+
+![](images/c54a08ac7fb09b2c6111870adf576bd40971e85bd59c0180107a839953fe4b18.jpg)  
+Figure 6: The informativeness of policies correlates with the informativeness of trajectories of length 30, as discussed in section 3.2
+
+# 4.2 LIMITATIONS AND FUTURE WORK.
+
+Estimating expert irrationality. One major limitation of our work is that our findings hold for when the learner knows the type and parameter value of the irrationality. In practice, reward inference will require solving the difficult task of estimating the irrationality type and degree (Armstrong & Mindermann, 2018; Shah et al., 2019). We still need to quantify to what extent these results still hold given uncertainty about the irrationality model. It does, however, seem crucial to reward inference that learners do reason explicitly about irrationality – not only is the learner unable to take advantage of the irrationality to make better inference if it does not model it, but actually reward inference in general suffers tremendously if the learner assumes the wrong type.
+
+In figure 10 in the Appendix, we compare inference with the true model vs. with assuming a Boltzmann model as default. The results are quite striking: not knowing the irrationality harms inference tremendously. Whether irrationalities help, this means that it is really important to model them.
+
+Generalization to other environments. A second limitation of our work is that we only tested these models in a limited range of environments. Further work is needed to test generalization of our findings across different MDPs of interest. Our analysis of mutual information lends credence to the Boltzmann rationality result generalizing well: these policies are much more varied with the reward parameters. In contrast, how useful the optimism bias is depends on the task: if we know about what to avoid already, as was the case for our learner, the bias is useful; if, on the other hand, we would know the goal but do not know what to avoid, the bias can hinder inference. Overall, this paper merely points out that there is a lot of richness to the ways in which these biases affect inference, and provides a quantitative comparison for a starting domain – much more is needed to gain a deeper understanding of this phenomenon.
+
+Applications to real humans. A third limitation is that we do not know where real humans lie. Do they have the helpful irrationality types? Do they fall in the range of parameters for these types that help inference? And what happens when types combine? While these questions are daunting, there is also a hidden opportunity here: what if we could influence humans to exhibit helpful types of irrationality? It might be much easier for them, for instance, to act myopically than to act rationally. In the end, reward inference is the confluence of two factors: how well the robot learns, and how well the teacher teaches. Our results point out that it might be easier than previously thought to be a good teacher – even easier than being a rational expert.
+
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+
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+
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+
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+
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+
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+
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+<table><tr><td rowspan=1 colspan=1>Policy</td><td rowspan=1 colspan=1>Parameter</td><td rowspan=1 colspan=3>Values</td></tr><tr><td rowspan=1 colspan=1>Rational</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=3>[0.99]</td></tr><tr><td rowspan=1 colspan=1>Boltzmann</td><td rowspan=1 colspan=1>β</td><td rowspan=1 colspan=3>[1,1.78,3.16, 5.62,10,17.8,31.6,56.2,100,178,316, 562,1000,1780,3160,5620,10000]</td></tr><tr><td rowspan=1 colspan=1>Optimism</td><td rowspan=1 colspan=1>1/T</td><td rowspan=1 colspan=3>[-10,-3.16,-1,-0.316,-0.1,0.1, 0.316,1,3.16, 10]</td></tr><tr><td rowspan=1 colspan=1>Illusion of Control</td><td rowspan=1 colspan=1>n</td><td rowspan=1 colspan=3>[0.1,0.178,0.316, 0.562,1, 1.78,3.16,5.62,10]</td></tr><tr><td rowspan=1 colspan=1>Prospect Theory</td><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=3>[0.1,0.178,0.316,0.562,1., 1.78, 3.16, 5.62, 10]</td></tr><tr><td rowspan=1 colspan=1>Extremal</td><td rowspan=1 colspan=1>a</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0.5,0.7,0.8,0.9,0.99,0.999]</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Myopicγ</td><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0.5, 0.7, 0.8, 0.9, 0.99,0.999]</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Myopic h</td><td rowspan=1 colspan=1>h</td><td rowspan=1 colspan=3>[1,2,3,4,5,6]</td></tr><tr><td rowspan=1 colspan=1>Hyperbolic</td><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=3>[0.01,0.1,0.178,0.316,0.562, 1, 1.78,3.16, 5.62, 10]</td></tr></table>
+
+![](images/34c5463d3898ee1084653976217e13ccd9d4b7b6a9afba64601d69807d34fdd6.jpg)  
+Table 1: The parameter values we search over for each policy.   
+Figure 7: Log loss for the posterior on $\theta$ , given trajectories from the Prospect Theory expert and the Illusion of Control expert.
+
+# A MORE EXPERIMENTAL DETAILS
+
+To enable exact inference, we discretized $\theta$ , using 5 evenly spaced points for each $\theta _ { i }$ . Our specific grid is included in figures 4 and 5 As there are two reward cells, this gives us 25 possible distinct reward parameters. We assumed a uniform prior on the reward parameter.
+
+We list the parameter values we search over for each policy in table 1. Except for myopic $\gamma$ and myopic $h$ , we use $\gamma = 0 . 9 9$ . For myopic $h$ , we use $\gamma = 1$ .
+
+From each start state, we sample 10 trajectories of each length for each reward parameter, policy combination.
+
+# B ADDITIONAL RESULTS
+
+We include the plots for the log loss of trajectories from the Prospect Theory and Illusion of Control experts in 7
+
+In addition, we include the plots for the $L ^ { 2 }$ loss for all 8 irrationalities in figures 8 and figure 9.
+
+# C MODEL MISSPECIFICATION GREATLY IMPAIRS INFERENCE
+
+Given that several types of irrationality can help inference when the form of irrationality is known, a natural question to ask is how important is it to known the irrationality exactly. To investigate this, we plot the log loss of the posterior of a learner who falsely assumes that the expert is Boltzmann
+
+![](images/08818adf3c14a488a0f1be0aaaad3d2b9a7083a76910a26f67fc85002d077f0e.jpg)  
+Figure 8: The $L ^ { 2 }$ distance (lower $=$ better) of posterior mean of $\theta$ to the true $\theta ^ { * }$ ,s as a function of the parameter we vary for each irrationality type. These six irrationalities all have parameter settings that outperform rational experts. For the models that interpolate to rational expert, we denote the value that is closest to rational using a dashed vertical line.
+
+![](images/4a0ff6d1de63578c4cc214e0df6e10a7e7d4b73b970eeb11a58bd5c1f64006e7.jpg)  
+Figure 9: The $L ^ { 2 }$ distance (lower=better) of the posterior mean $\theta$ to th true $\theta ^ { * }$ , given trajectories from the Prospect Theory expert and the Illusion of Control expert.
+
+![](images/2c024f3d652c29266ded497a2aa9877d66396cab5c249b37fd0d11cf65cbeb1e.jpg)  
+Inference performance under model misspecification   
+Figure 10: A comparison of reward inference using a correct model of the irrationality type, versus always using a Boltzman model. (Lower log loss $=$ better.) The inference impairment from using the misspecified irrationality model (Boltzmann) greatly outweighs the variation in inference performance caused by the various irrationality types themselves. Hence, compared to using a misspecified model of irrationality, expert irrationality is not in itself a major impairment to reward inference, and sometimes expert irrationality can even helps when a model of the irrationality is known.
+
+rational with $\beta = 1 0 0$ . Where applicable, the log loss is averaged over possible hyperparameter settings for the expert.
+
+We report the results in figure 10. The log loss of the posterior if we wrongly imagine the expert is Boltzmann-rational far outweighs differences between particular irrationality types.
+
+# C.1 WHY IS USING A MISSPECIFIED IRRATIONALITY TYPE FOR INFERENCE SO BAD?
+
+Fundamentally, misspecification is bad for inference because different experts might exhibit the same action only under different reward parameters. For example, consider figure the case where the actual expert is myopic, with small $n$ . Then the myopic agent might go toward a closer reward even if it is much smaller, as shown in figure 11. This would cause the learner to falsely infer that the closer reward is quite large, leading to a posterior with extremely high log loss when the reward is actually smaller.
+
+![](images/fd5993c5792e438c89ac621f6c049c8578b51048e417886124512fcce976f926.jpg)  
+Figure 11: An example of why assuming Boltzmann is bad for a myopic agent - the Boltzmann rational agent would take this trajectory only if the reward at the bottom was not much less than the reward at the top. The myopic agent with $n \leq 4$ , however, only ”sees” the reward at the bottom. Consequently, inferring the preferences of the myopic agent as if it were Boltzmann leads to poor performance in this case.

parse/train/BJlo91BYPr/BJlo91BYPr_content_list.json

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+    {
+        "type": "text",
+        "text": "IRRATIONALITY CAN HELP REWARD INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Specifying reward functions is difficult, which motivates the area of reward inference: learning rewards from human behavior. The starting assumption in the area is that human behavior is optimal given the desired reward function, but in reality people have many different forms of irrationality, from noise to myopia to risk aversion and beyond. This fact seems like it will be strictly harmful to reward inference: it is already hard to infer the reward from rational behavior, and noise and systematic biases make actions have less direct of a relationship with the reward. Our insight in this work is that, contrary to expectations, irrationality can actually help rather than hinder reward inference. For some types and amounts of irrationality, the expert now produces more varied policies compared to rational behavior, which help disambiguate among different reward parameters – those that otherwise correspond to the same rational behavior. We put this to the test in a systematic analysis of the effect of irrationality on reward inference. We start by covering the space of irrationalities as deviations from the Bellman update, simulate expert behavior, and measure the accuracy of inference to contrast the different types and study the gains and losses. We provide a mutual informationbased analysis of our findings, and wrap up by discussing the need to accurately model irrationality, as well as to what extent we might expect (or be able to train) real people to exhibit helpful irrationalities when teaching rewards to learners. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The application of reinforcement learning (RL) in increasingly complex environments has been most successful for problems that are already represented by a specified reward function (Lillicrap et al., 2015; Mnih et al., 2015; 2016; Silver et al., 2016). Unfortunately, not only do real-world tasks usually lack an explicit exogenously-specified reward function, but attempting to specify one tends to lead to unexpected side-effects as the agent is faced with new situations (Lehman et al., 2018). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This has motivated the area of reward inference: the process of estimating a reward function from human inputs. The inputs are traditionally demonstrations, leading to inverse reinforcement learning (IRL) $\\mathrm { N g }$ et al., 2000; Abbeel & $\\mathrm { N g }$ , 2004) or inverse optimal control (IOC) (Kalman, 1964; Jameson & Kreindler, 1973; Mombaur et al., 2010; Finn et al., 2016). Recent work has expanded the range of inputs significantly,to comparisons (Wirth et al., 2017; Sadigh et al., 2017; Christiano et al., 2017), natural language instructions (MacGlashan et al., 2015; Fu et al., 2019), physical corrections (Jain et al., 2015; Bajcsy et al., 2017), proxy rewards (Hadfield-Menell et al., 2017; Ratner et al., 2018), or scalar reward values (Griffith et al., 2013; Loftin et al., 2014). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The central assumption behind these methods is that human behavior is rational, i.e. optimal with respect to the desired reward (cumulative, in expectation). Unfortunately, decades of research in behavioral economics and cognitive science Chipman (2014) has unearthed a deluge of irrationalities, i.e. of ways in which people deviate from optimal decision making: hyperbolic discounting, scope insensitivity, optimism bias, decision noise, certainty effects, loss aversion, status quo bias, etc. ",
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+        "type": "text",
+        "text": "Work on reward inference has predominantly used one model of irrationality: decision-making noise, where the probability of an action relates to the value that action has. The most widely used model by far is a Bolzmann distribution stemming from the Luce-Sherpard rule (Luce, 1959; Shepard, 1957; Lucas et al., 2009) and the principle of maximum (causal) entropy in (Ziebart et al., 2008; 2010), which we will refer to as Bolzmann-rationality (Fisac et al., 2017). Recent work has started to incorporate systematic biases though, like risk-aversion (Singh et al., 2017), having the wrong dynamics belief (Reddy et al., 2018), and myopia and hyperbolic discounting (Evans & Goodman, 2015; Evans et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Learning from irrational experts feels like daunting task: reward inference is already hard with rational behavior, but now a learner needs to make sense of behavior that is noisy or systematically biased. Our goal in this work is to characterize just how muddied the waters are – how (and how much) do different irrationalities affect reward inference? ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our insight is that, contrary to expectations, irrationality can actually help, rather than hinder, reward inference. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our explanation is that how good reward inference is depends on the mutual information between the policies produced by the expert and the reward parameters to be inferred. While it is often possible for two reward parameters to produce the same rational behavior, irrationalities can sometimes produce different behaviors that disambiguate between those same two reward parameters. For instance, noise can help when it is related to the value function, as Boltzmann noise is, because it distinguishes the difference in values even when the optimal action stays the same. Optimism can be helpful because the expert takes fewer risk-avoiding actions and acts more directly on their goal. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overall, we contribute 1) an analysis and comparison of the effects of different biases on reward inference testing our insight, 2) a way to systematically formalize and cover the space of irrationalities in order to conduct such an analysis, and 3) evidence for the importance of assuming the right type of irrationality during inference. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our good news is that irrationalities can indeed be an ally for inference. Of course, this is not always true – the details of which irrationality type and how much of it also matter. We see these results as opening the door to a better understanding of reward inference, as well as to practical ways of making inference easier by asking for the right kind of expert demonstrations – after all, in some cases it might be easier for people to act optimistically or myopically than to act rationally. Our results reinforce that optimal teaching is different from optimal doing, but point out that some forms of teaching might actually be easier than doing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.1 EXPLORING IRRATIONALITY THROUGH SIMULATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our goal is to explore the effect irrationalities have on reward inference if the learner knows about them – we explore the need for the learner to accurately model irrationalities in section 4.2. While ideally we would recruit human subjects with different irrationalities and measure how well we can learn rewards, this is prohibitive because we do not get to dictate someone’s irrationality type: people exhibit a mix of them, some yet to be discovered. Further, measuring accuracy of inference is complicated by the fact that we do not have ground truth access to the desired reward: the learner can measure agreement with some test set, but the test set itself is produced subject to the same irrationalities that produced the training data. As experimenters, we would remain deluded about the human’s true intentions and preferences. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To address this issue, we simulate expert behavior subject to different irrationalities based on ground truth reward functions, run reward inference, and measure the performance against the ground truth, i.e. the accuracy of a Bayesian posterior on the reward function given the (simulated) expert’s inputs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.2 TYPES AND DEGREES OF IRRATIONALITY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are many possible irrationalities that people exhibit (Chipman, 2014), far more than what we could study in one paper. They come with varying degrees of mathematical formalization and replication across human studies. To provide good coverage of this space, we start from the Bellman update, and systematically manipulate its terms and operators to produce a variety of different irrationalities that deviate from the optimal MDP policy in complementary ways. For instance, operating on the discount factor can model more myopic behavior, while operating on the transition function can model optimism or the illusion of control. Figure 1 summarizes our approach, which we detail below. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/de748af0803fffaa2f2131dd7e3efd0f4b9524736cbffd30556b5dc1bf796a92.jpg",
+        "image_caption": [
+            "Figure 1: We modify the components of the Bellman update to cover different types of irrationalities: changing the max into a softmax to capture noise, changing the transition function to capture optimism/pessimism or the illusion of control, changing the reward values to capture the nonlinear perception of gains and losses (prospect theory), changing the average reward over time into a maximum (extremal), and changing the discounting to capture more myopic decision-making. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2.1 RATIONAL EXPERT ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The rational expert does value iteration using the Bellman update from figure 1. Our models change this update to produce different types of non-rational behavior. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2.2 MODIFYING THE MAX OPERATOR: BOLZMANN",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Boltzmann-rationality modifies the maximum over actions $\\mathrm { m a x } _ { a }$ with a Boltzmann operator with parameter $\\beta$ : ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e57712ebd91e1cca26a04b03475c9d2b9db3995031c515591396737920f9070b.jpg",
+        "text": "$$\nV _ { i + 1 } ( s ) = { \\bf B o l t z } _ { a } ^ { \\beta } \\sum _ { s ^ { \\prime } \\in S } T ( s ^ { \\prime } | s , a ) \\left( r ( s , a , s ^ { \\prime } ) + \\gamma V _ { i } ( s ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            419,
+            679,
+            453
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Where $\\begin{array} { r } { \\mathbf { B o l t z } ^ { \\beta } ( \\mathbf { x } ) = \\sum _ { i } x _ { i } e ^ { \\beta x _ { i } } / \\sum _ { i } e ^ { \\beta x _ { i } } } \\end{array}$ (Ziebart et al., 2010; Asadi $\\&$ Littman, 2017) This models that people will not be perfect, but rather noisily pick actions in a way that is related to the Qvalue of those actions. The constant $\\beta$ is called the rationality constant, because as $\\beta  \\infty$ , the human choices approach perfect rationality (optimality), whereas $\\beta = 0$ produces uniformly random choices. This is the standard assumption for reward inference that does not assume perfect rationality, because it easily transforms the rationality assumption into a probability distribution over actions, enabling learners to make sense of imperfect demonstrations that otherwise do not match up with any reward parameters. ",
+        "bbox": [
+            173,
+            457,
+            825,
+            570
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2.3 MODIFYING THE TRANSITION FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            585,
+            517,
+            601
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our next set of irrationalities manipulate the transition function away from reality. ",
+        "bbox": [
+            176,
+            609,
+            710,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Illusion of Control. Humans often overestimate their ability to control random events. To model this, we consider experts that use the Bellman update: ",
+        "bbox": [
+            173,
+            640,
+            823,
+            670
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9fd0cd76731fca511043446482e21f22e2cdf41a7610323a5edea25056f03da7.jpg",
+        "text": "$$\nV _ { i + 1 } ( s ) = \\operatorname* { m a x } _ { a } \\sum _ { s ^ { \\prime } \\in S } T ^ { n } ( s ^ { \\prime } | s , a ) \\left( r ( s , a , s ^ { \\prime } ) + \\gamma V _ { i } ( s ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            675,
+            678,
+            710
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $T ^ { n } ( s ^ { \\prime } | s , a ) \\propto \\left( T ( s ^ { \\prime } | s , a ) \\right) ^ { n }$ . As $n \\to \\infty$ , the demonstrator acts as if it exists in a deterministic environment. As $n  0$ , the expert acts as if it had an equal chance of transitioning to every possible successor state. At $n = 1$ , the expert is the rational expert. ",
+        "bbox": [
+            174,
+            717,
+            825,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Optimism/Pessimism. Humans tend to systematically overestimate their chance experiencing of positive over negative events. We model this using experts that modify the probability they get outcomes based on the value of those outcomes: ",
+        "bbox": [
+            173,
+            776,
+            826,
+            818
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c7fcd0edb3c69419827e504d242f3ec710cee111f68956182aec8efa85d7a500.jpg",
+        "text": "$$\nV _ { i + 1 } ( s ) = \\operatorname* { m a x } _ { a } \\sum _ { s ^ { \\prime } \\in S } T ^ { 1 / \\tau } ( s ^ { \\prime } | s , a ) \\left( r ( s , a , s ^ { \\prime } ) + \\gamma V _ { i } ( s ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            821,
+            684,
+            856
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $T ^ { 1 / \\tau } ( s ^ { \\prime } | s , a ) \\propto T ( s ^ { \\prime } | s , a ) e ^ { \\big ( r ( s , a , s ^ { \\prime } ) + \\gamma V _ { i } ( s ) \\big ) / \\tau }$ . $1 / \\tau$ controls how pessimistic or optimistic the expert is. As $1 / \\tau \\to + \\infty$ , the expert becomes increasingly certain that good transitions will happen. As $1 / \\tau \\to - \\infty$ , the expert becomes increasingly certain that bad transitions will happen. As $1 / \\tau \\to 0$ , the expert approaches the rational expert. ",
+        "bbox": [
+            174,
+            863,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2.4 MODIFYING THE REWARD: PROSPECT THEORY ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            558,
+            118
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next, we consider experts that use the modified Bellman update: ",
+        "bbox": [
+            174,
+            127,
+            596,
+            142
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d01680d23a36ad5ba5faa48aca0e9066cf11b628a443169e90b5d2c0245c00a.jpg",
+        "text": "$$\nV _ { i + 1 } ( s ) = \\operatorname* { m a x } _ { a } \\sum _ { s ^ { \\prime } \\in S } T ( s ^ { \\prime } | s , a ) \\left( f ( r ( s , a , s ^ { \\prime } ) ) + \\gamma V _ { i } ( s ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            148,
+            684,
+            183
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $f : \\mathbb { R } \\to \\mathbb { R }$ is some scalar function. This is equivalent to solving the MDP with reward $f \\circ r$ This allows us to model human behavior such as loss aversion and scope insensitivity. ",
+        "bbox": [
+            169,
+            189,
+            821,
+            218
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Prospect Theory Kahneman & Tversky (2013) inspires us to consider a particular family of reward transforms: ",
+        "bbox": [
+            173,
+            223,
+            825,
+            251
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b104e23cd2936133e4295feb6a97e84ac414191aa8582d6e2b83bbb46a26703.jpg",
+        "text": "$$\nf _ { c } ( r ) = \\left\\{ \\begin{array} { l l } { \\log ( 1 + | r | ) } & { r > 0 } \\\\ { 0 } & { r = 0 } \\\\ { - c \\log ( 1 + | r | ) } & { r < 0 } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            250,
+            617,
+            301
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$c$ controls how loss averse the expert is. As $c  \\infty$ , the expert primarily focuses on avoiding negative rewards. As $c \\to 0$ , the expert focuses on maximizing positive rewards and ",
+        "bbox": [
+            173,
+            304,
+            825,
+            334
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2.5 MODIFYING THE SUM BETWEEN REWARD AND FUTURE VALUE: EXTREMAL ",
+        "bbox": [
+            171,
+            347,
+            758,
+            363
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Extremal. Humans seem to exhibit duration neglect, sometimes only caring about the maximum intensity of an experiennce (Do et al., 2008). We model this using experts that use the Bellman step: ",
+        "bbox": [
+            173,
+            372,
+            826,
+            401
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eda8d27dc5348e8b9be14615414bf12c741b2323a5ac73085aa9219de941e665.jpg",
+        "text": "$$\nV _ { i + 1 } ( s ) = \\operatorname* { m a x } _ { a } \\sum _ { s ^ { \\prime } \\in S } T ( s ^ { \\prime } | s , a ) \\left( \\operatorname* { m a x } \\left[ r ( s , a , s ^ { \\prime } ) , ( 1 - \\alpha ) r ( s , a , s ^ { \\prime } ) + \\alpha V _ { i } ( s ) \\right] \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            407,
+            756,
+            441
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "These experts maximize the expected maximum reward along a trajectory, instead of the expected sum of rewards. As $\\alpha  1$ , the expert maximizes the expected maximum reward they achieve along their full trajectory. As $\\alpha  0$ , the expert becomes greedy, and only cares about the reward they achieve in the next timestep. ",
+        "bbox": [
+            173,
+            446,
+            825,
+            505
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2.6 MODIFYING THE DISCOUNTING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            518,
+            449,
+            534
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Myopic Discount. In practice, humans are often myopic, only considering immediate rewards. One way to model this is to decrease gamma in the Bellman update. At $\\gamma = 1$ , this is the rational expert. As $\\gamma  0$ , the expert becomes greedy and only acts to maximize immediate reward. ",
+        "bbox": [
+            174,
+            542,
+            825,
+            585
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Myopic VI. As another way to model human myopia, we consider a expert that performs only $h$ steps of Bellman updates. That is, this expert cares equally about rewards for horizon $h$ , and discount to 0 reward after that. As $h \\to \\infty$ , this expert becomes rational. If $h = 1$ , this expert only cares about the immediate reward. ",
+        "bbox": [
+            173,
+            601,
+            825,
+            657
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Hyperbolic Discounting. Human also exhibit hyperbolic discounting, with a high discount rate for the immediate future and a low discount rate for the far future. Alexander & Brown (2010) formulate this as the following Bellman update: ",
+        "bbox": [
+            173,
+            671,
+            825,
+            714
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82de072e6fcb88441633d876b901ebf66656f189c6531fa9ff78d9b732c7b847.jpg",
+        "text": "$$\nV _ { i + 1 } ( s ) = \\operatorname* { m a x } _ { a } \\sum _ { s ^ { \\prime } \\in S } T ( s ^ { \\prime } | s , a ) \\left( r ( s , a , s ^ { \\prime } ) + V _ { i } ( s ) \\right) / ( 1 + k V _ { i } ( s ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            720,
+            717,
+            756
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$k$ modulates how much the expert prefers rewards now versus the future. As $k  0$ , this expert becomes the rational expert. ",
+        "bbox": [
+            173,
+            762,
+            823,
+            791
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 IMPACT OF IRRATIONALITIES ON REWARD INFERENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            809,
+            658,
+            827
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 EXPERIMENTAL DESIGN ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            840,
+            385,
+            856
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Simulation Environment. To reduce possible confounding from our choice of environment, we used a small 5x5 gridworld where the irrationalities nonetheless cause experts to exhibit different behavior. Our gridworld consists of three types of cells: ice, holes, and rewards. The expert can start in any ice cell. At each ice cell, the expert can move in one of the four cardinal directions. With probability 0.8, they will go in that direction. With probability 0.2, they will instead go in one of the two adjacent directions. Holes and rewards are terminal states, and return the expert back to their start state. They receive a penalty of $- 1 0$ for falling into a hole and $\\theta _ { i } \\in [ 0 , 4 ]$ for entering into the ith reward cell. ",
+        "bbox": [
+            173,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/1c54b2a5f7b51a0f6ed5ef0da25e969304cdb510afc805fbbef04a4e75a138f3.jpg",
+        "image_caption": [
+            "Figure 2: The log loss (lower $=$ better) of the posterior as a function of the parameter we vary for each irrationality type. These six irrationalities all have parameter settings that outperform rational experts. For the models that interpolate to rational expert, we denote the value that is closest to rational using a dashed vertical line. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            99,
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+            421
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Dependent Measures. To separate the inference difficulty caused by suboptimal inference from the difficulty caused by expert irrationality, we perform the exact Bayesian update on the trajectory $\\theta$ (Ramachandran & Amir, 2007), which gives us the posterior on $\\theta$ given $\\xi$ : ",
+        "bbox": [
+            173,
+            585,
+            825,
+            628
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/207bbd0b0ddd9d886314031db9576680c26cfdf931b1311ddafac91908330f6e.jpg",
+        "text": "$$\nP ( \\theta | \\xi ) = \\frac { P ( \\xi | \\theta ) P ( \\theta ) } { \\int _ { \\theta ^ { \\prime } } P ( \\xi | \\theta ^ { \\prime } ) P ( \\theta ^ { \\prime } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            645,
+            593,
+            681
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use two metrics to measure the difficulty of inference The first is the expected log loss of this posterior, or negative log-likelihood: ",
+        "bbox": [
+            173,
+            688,
+            823,
+            717
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71eb8d950ec618375e446939ae68a922d827e17ed956cb24694a4a9eb5648ea1.jpg",
+        "text": "$$\n\\operatorname { L o g } \\operatorname { L o s s } ( \\theta | \\xi ) = E _ { \\theta , \\xi \\sim \\pi _ { \\theta } } \\left[ - \\log P ( \\theta | \\xi ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            722,
+            637,
+            739
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A low log loss implies that we are assigning a high likelihood to the true $\\theta$ . As we are performing exact Bayesian inference with the true model $P ( \\xi | \\theta )$ and prior $P ( \\theta )$ , the log loss is equal to the entropy of the posterior $H ( \\theta | \\xi )$ . ",
+        "bbox": [
+            174,
+            742,
+            825,
+            786
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The second metric is the ${ \\bf L } ^ { 2 }$ -distance between the mean posterior $\\theta$ and the actual theta: ",
+        "bbox": [
+            174,
+            791,
+            753,
+            806
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ebbef57e26a946b7947dac4dd7152b4a8b9270e126960203bea9f04b2cf4bdfe.jpg",
+        "text": "$$\nL ^ { 2 } ( \\theta | \\xi ) = E _ { \\theta ^ { * } , \\xi \\sim \\pi _ { \\theta ^ { * } } } \\left[ | | E [ \\theta | \\xi ] - \\theta ^ { * } | | ^ { 2 } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            810,
+            630,
+            829
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The closer the inferred posterior mean of $\\theta$ is to the actual value $\\theta ^ { * }$ , the lower the loss. ",
+        "bbox": [
+            171,
+            832,
+            738,
+            848
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For each irrationality type, we calculate the performance of reward inference on trajectories of a fixed length $T$ , with respect to the two metrics above. To sample a trajectory of length $T$ from a expert, we fix $\\theta ^ { * }$ and start state $s$ . Then, we perform the expert’s (possibly modified) Bellman updates until convergence to recover the policy $\\pi _ { \\theta ^ { \\ast } }$ . Finally, we generate rollouts starting from state $s$ until $T$ state, action pairs have been sampled from $\\pi _ { \\theta ^ { \\ast } }$ . ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/91bd9cad2fdde456d85cd2c875735c23c953b0a1fd13d3111241ce67dd0be7fd.jpg",
+        "image_caption": [
+            "Figure 3: A best case analysis for each irrationality type: the log $\\mathrm { { l o s s } } / L ^ { 2 }$ distance from mean (lowe $\\fallingdotseq$ better) for experts, as a function of the length of trajectory observed. Each irrationality uses the parameter value that is most informative. As discussed in section 3.2, different irrationality types have different slopes and converge to different values. In addition, the best performing irrationality type according to log loss is not the best performing type according to $L ^ { 2 }$ loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            102,
+            758,
+            332
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            444,
+            285,
+            458
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Impact of Each Irrationality. We found that of the 8 irrationalities we studied, 6 had parameter settings that lead to lower log loss than the rational expert. We report how the parameter influences the log loss for each of these experts in figure 2.1 For $\\bar { T ^ { \\mathrm { ~ } } } = 3 0$ , Optimism with $1 \\bar { / } \\tau = 3 . 1 6$ performed the best, followed by Boltzmann with $\\beta = 1 0 0$ and Hyperbolic with $k = 0 . 1$ . Both forms of Myopia also outperformed the rational expert, with best performance occurring at $\\gamma = 0 . 9$ and $h = 5$ . Finally, the Extremal expert also slightly outperformed the rational expert, with best performance at $\\alpha = 0 . 9$ . Notably, in every case, neither the most irrational expert nor the perfectly rational expert was the most informative. ",
+        "bbox": [
+            173,
+            468,
+            825,
+            580
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Impact of Data for Different Irrationalities. Next, we investigate how the quality of inference varies as we increase the length of the observed trajectory $T$ . We report our results for the best performing parameter for each irrationality type in figure 3. Interestingly, while both metrics decrease monotonically regardless of irrationality type, the rate at which they decrease differs by the irrationality type, and the best performing irrationality type according to log loss (Optimism) is not the best performing type according to $L ^ { 2 }$ distance (Boltzmann). ",
+        "bbox": [
+            173,
+            595,
+            825,
+            681
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "What is behind these differences? To explain these results, we use the notion of mutual information $\\mathbf { I } ( X ; Y )$ between two variables, defined as: ",
+        "bbox": [
+            173,
+            695,
+            821,
+            724
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/351c8665fde16167341bad25d6ab25b9f516c2848314c48edc9e38f058eb91d3.jpg",
+        "text": "$$\n\\mathbf { I } ( X ; Y ) = E _ { X , Y } \\left[ \\log \\left( { \\frac { P ( X , Y ) } { P ( X ) P ( Y ) } } \\right) \\right] = H ( X ) - H ( X | Y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            731,
+            705,
+            767
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The mutual information measures how much our uncertainty about $X$ decreases by observing $Y$ ",
+        "bbox": [
+            169,
+            772,
+            805,
+            787
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For reward inference, the term we care about is the mutual information between the expert’s trajectory and the reward parameters ",
+        "bbox": [
+            171,
+            794,
+            823,
+            823
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a79410fb0bfc69f5f86d0bfb8624e24dfe9374838665bcdfb2455afea72c998a.jpg",
+        "text": "$$\n\\mathbf { I } ( \\theta ; \\xi ) = E _ { \\theta , \\xi \\sim \\theta } \\left[ \\log \\left( \\frac { P ( \\theta , \\xi ) } { P ( \\theta ) P ( \\xi ) } \\right) \\right] = H ( \\theta ) - H ( \\theta | \\xi )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            829,
+            689,
+            864
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The mutual information $\\mathbf { I } ( \\theta ; \\xi )$ is equal to a constant minus the posterior log loss under the true model. A expert with mutual information will cause the learner to have a lower posterior log loss. ",
+        "bbox": [
+            176,
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+        "type": "image",
+        "img_path": "images/dd6570023cc2d9989e5b6db0126e6119217fe0a74a5724cdce37feaf15820ea9.jpg",
+        "image_caption": [
+            "Figure 4: (a) Optimism bias produces different actions for $\\theta ^ { * } = ( 4 , 1 )$ vs. $\\theta ^ { * } = ( 1 , 4 )$ in the states shown: the rational policy is to go away from the hole regardless of $\\theta$ , but an optimistic expert takes the chance and goes for the larger reward – up in the first case, down in the second. (b) Pessimism bias produces different actions for $\\theta ^ { * } = ( 1 , 1 )$ vs. $\\theta ^ { * } = ( 4 , 4 )$ : when the reward is sufficiently large, the expert becomes convinced that no action it takes will lead to the reward, leading it to perform random actions. "
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+            "Figure 5: (a) Boltzmann-rationality produces different policies for $\\theta ^ { * } = ( 1 , 1 )$ vs. $\\theta ^ { * } = ( 4 , 4 )$ : when $| | \\theta | |$ is larger, the policy becomes closer to that of the rational expert. (b) A Myopic expert produces different policies for $\\theta ^ { * } = ( 4 , 1 )$ vs. $\\theta ^ { * } = ( 4 , 0 )$ : while the rational expert always detours around the hole and attempts to reach the larger reward, myopia causes the myopic expert to go for the smaller source of reward when it is non-zero. "
+        ],
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+    {
+        "type": "text",
+        "text": "By the information processing inequality, we have the bound $\\mathbf { I } ( \\theta ; \\xi ) \\le \\mathbf { I } ( \\theta ; \\pi )$ . ",
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+    {
+        "type": "text",
+        "text": "To have higher mutual information, different $\\theta \\mathrm { s }$ should be mapped to different policies $\\pi \\mathbf { S }$ . Indeed, we found that the experts that were able to outperform the rational expert were able to disambiguate between $\\theta \\mathrm { s }$ that the rational expert could not. To visualize this, we show examples of how the policy of several irrational experts differ when the rational expert’s policies are identical in figures 4 and 5. ",
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+    {
+        "type": "text",
+        "text": "We plot the correlation between $\\mathbf { I } ( \\theta ; \\xi )$ and $\\mathbf { I } ( \\theta ; \\pi )$ in figure 6. Experts that have more informative policies tend to have more informative trajectories, but the correlation is not perfect. Notably, the Optimism expert has the most informative trajectories of length 30, but has less informative policies than the Boltzmann expert. ",
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+    {
+        "type": "text",
+        "text": "In the limit of infinite data from every state, we would have $\\mathbf { I } ( \\theta ; \\xi ) \\to \\mathbf { I } ( \\theta ; \\pi )$ . However, as each trajectory begins from the same start state, and not every state is reachable with every policy, the bound is not achievable in general, even if we observe an arbitrarily large number of trajectories. This highlights the need for off-policy data in reward inference tasks. ",
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+    {
+        "type": "text",
+        "text": "4 DISCUSSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 SUMMARY ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We show that, contrary to what we might expect, suboptimal experts can actually help an agent learn the reward function. Optimism bias, myopia (via heavier discounting or hyperbolic discounting), and noise via Boltzmann rationality were the most informative irrationalities in our environments, far surpassing the performance of the rational expert for their ideal settings. Our contribution overall was to identify a systematic set of irrationalities by looking at deviations in the terms of the Bellman update, and show that being irrational is not automatically harmful to inference by quantifying and comparing the inference performance for these different types. ",
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+        "img_path": "images/c54a08ac7fb09b2c6111870adf576bd40971e85bd59c0180107a839953fe4b18.jpg",
+        "image_caption": [
+            "Figure 6: The informativeness of policies correlates with the informativeness of trajectories of length 30, as discussed in section 3.2 "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 LIMITATIONS AND FUTURE WORK. ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Estimating expert irrationality. One major limitation of our work is that our findings hold for when the learner knows the type and parameter value of the irrationality. In practice, reward inference will require solving the difficult task of estimating the irrationality type and degree (Armstrong & Mindermann, 2018; Shah et al., 2019). We still need to quantify to what extent these results still hold given uncertainty about the irrationality model. It does, however, seem crucial to reward inference that learners do reason explicitly about irrationality – not only is the learner unable to take advantage of the irrationality to make better inference if it does not model it, but actually reward inference in general suffers tremendously if the learner assumes the wrong type. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In figure 10 in the Appendix, we compare inference with the true model vs. with assuming a Boltzmann model as default. The results are quite striking: not knowing the irrationality harms inference tremendously. Whether irrationalities help, this means that it is really important to model them. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Generalization to other environments. A second limitation of our work is that we only tested these models in a limited range of environments. Further work is needed to test generalization of our findings across different MDPs of interest. Our analysis of mutual information lends credence to the Boltzmann rationality result generalizing well: these policies are much more varied with the reward parameters. In contrast, how useful the optimism bias is depends on the task: if we know about what to avoid already, as was the case for our learner, the bias is useful; if, on the other hand, we would know the goal but do not know what to avoid, the bias can hinder inference. Overall, this paper merely points out that there is a lot of richness to the ways in which these biases affect inference, and provides a quantitative comparison for a starting domain – much more is needed to gain a deeper understanding of this phenomenon. ",
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+    {
+        "type": "text",
+        "text": "Applications to real humans. A third limitation is that we do not know where real humans lie. Do they have the helpful irrationality types? Do they fall in the range of parameters for these types that help inference? And what happens when types combine? While these questions are daunting, there is also a hidden opportunity here: what if we could influence humans to exhibit helpful types of irrationality? It might be much easier for them, for instance, to act myopically than to act rationally. In the end, reward inference is the confluence of two factors: how well the robot learns, and how well the teacher teaches. Our results point out that it might be easier than previously thought to be a good teacher – even easier than being a rational expert. ",
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+    {
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+        "text": "Christian Wirth, Riad Akrour, Gerhard Neumann, and Johannes Furnkranz. A survey of preference- ¨ based reinforcement learning methods. The Journal of Machine Learning Research, 18(1):4945– 4990, 2017. ",
+        "bbox": [
+            171,
+            412,
+            825,
+            455
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, and Anind K Dey. Maximum entropy inverse reinforcement learning. In Aaai, volume 8, pp. 1433–1438. Chicago, IL, USA, 2008. ",
+        "bbox": [
+            171,
+            464,
+            823,
+            494
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Brian D Ziebart, J Andrew Bagnell, and Anind K Dey. Modeling interaction via the principle of maximum causal entropy. 2010. ",
+        "bbox": [
+            173,
+            502,
+            823,
+            531
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/7714a52ca38940c3c4897693f40ed7b32aaa1106bbff6b94cccf43a9ec4f2e0b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Policy</td><td rowspan=1 colspan=1>Parameter</td><td rowspan=1 colspan=3>Values</td></tr><tr><td rowspan=1 colspan=1>Rational</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=3>[0.99]</td></tr><tr><td rowspan=1 colspan=1>Boltzmann</td><td rowspan=1 colspan=1>β</td><td rowspan=1 colspan=3>[1,1.78,3.16, 5.62,10,17.8,31.6,56.2,100,178,316, 562,1000,1780,3160,5620,10000]</td></tr><tr><td rowspan=1 colspan=1>Optimism</td><td rowspan=1 colspan=1>1/T</td><td rowspan=1 colspan=3>[-10,-3.16,-1,-0.316,-0.1,0.1, 0.316,1,3.16, 10]</td></tr><tr><td rowspan=1 colspan=1>Illusion of Control</td><td rowspan=1 colspan=1>n</td><td rowspan=1 colspan=3>[0.1,0.178,0.316, 0.562,1, 1.78,3.16,5.62,10]</td></tr><tr><td rowspan=1 colspan=1>Prospect Theory</td><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=3>[0.1,0.178,0.316,0.562,1., 1.78, 3.16, 5.62, 10]</td></tr><tr><td rowspan=1 colspan=1>Extremal</td><td rowspan=1 colspan=1>a</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0.5,0.7,0.8,0.9,0.99,0.999]</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Myopicγ</td><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0.5, 0.7, 0.8, 0.9, 0.99,0.999]</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Myopic h</td><td rowspan=1 colspan=1>h</td><td rowspan=1 colspan=3>[1,2,3,4,5,6]</td></tr><tr><td rowspan=1 colspan=1>Hyperbolic</td><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=3>[0.01,0.1,0.178,0.316,0.562, 1, 1.78,3.16, 5.62, 10]</td></tr></table>",
+        "bbox": [
+            261,
+            101,
+            735,
+            332
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/34c5463d3898ee1084653976217e13ccd9d4b7b6a9afba64601d69807d34fdd6.jpg",
+        "image_caption": [
+            "Table 1: The parameter values we search over for each policy. ",
+            "Figure 7: Log loss for the posterior on $\\theta$ , given trajectories from the Prospect Theory expert and the Illusion of Control expert. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            372,
+            745,
+            517
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A MORE EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            585,
+            472,
+            601
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To enable exact inference, we discretized $\\theta$ , using 5 evenly spaced points for each $\\theta _ { i }$ . Our specific grid is included in figures 4 and 5 As there are two reward cells, this gives us 25 possible distinct reward parameters. We assumed a uniform prior on the reward parameter. ",
+        "bbox": [
+            174,
+            616,
+            823,
+            659
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We list the parameter values we search over for each policy in table 1. Except for myopic $\\gamma$ and myopic $h$ , we use $\\gamma = 0 . 9 9$ . For myopic $h$ , we use $\\gamma = 1$ . ",
+        "bbox": [
+            174,
+            665,
+            823,
+            694
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "From each start state, we sample 10 trajectories of each length for each reward parameter, policy combination. ",
+        "bbox": [
+            173,
+            699,
+            823,
+            728
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            750,
+            397,
+            765
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We include the plots for the log loss of trajectories from the Prospect Theory and Illusion of Control experts in 7 ",
+        "bbox": [
+            174,
+            780,
+            825,
+            809
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In addition, we include the plots for the $L ^ { 2 }$ loss for all 8 irrationalities in figures 8 and figure 9. ",
+        "bbox": [
+            176,
+            814,
+            795,
+            830
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C MODEL MISSPECIFICATION GREATLY IMPAIRS INFERENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            851,
+            689,
+            866
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Given that several types of irrationality can help inference when the form of irrationality is known, a natural question to ask is how important is it to known the irrationality exactly. To investigate this, we plot the log loss of the posterior of a learner who falsely assumes that the expert is Boltzmann",
+        "bbox": [
+            176,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/08818adf3c14a488a0f1be0aaaad3d2b9a7083a76910a26f67fc85002d077f0e.jpg",
+        "image_caption": [
+            "Figure 8: The $L ^ { 2 }$ distance (lower $=$ better) of posterior mean of $\\theta$ to the true $\\theta ^ { * }$ ,s as a function of the parameter we vary for each irrationality type. These six irrationalities all have parameter settings that outperform rational experts. For the models that interpolate to rational expert, we denote the value that is closest to rational using a dashed vertical line. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            156,
+            790,
+            478
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/4a0ff6d1de63578c4cc214e0df6e10a7e7d4b73b970eeb11a58bd5c1f64006e7.jpg",
+        "image_caption": [
+            "Figure 9: The $L ^ { 2 }$ distance (lower=better) of the posterior mean $\\theta$ to th true $\\theta ^ { * }$ , given trajectories from the Prospect Theory expert and the Illusion of Control expert. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            674,
+            746,
+            819
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c024f3d652c29266ded497a2aa9877d66396cab5c249b37fd0d11cf65cbeb1e.jpg",
+        "image_caption": [
+            "Inference performance under model misspecification ",
+            "Figure 10: A comparison of reward inference using a correct model of the irrationality type, versus always using a Boltzman model. (Lower log loss $=$ better.) The inference impairment from using the misspecified irrationality model (Boltzmann) greatly outweighs the variation in inference performance caused by the various irrationality types themselves. Hence, compared to using a misspecified model of irrationality, expert irrationality is not in itself a major impairment to reward inference, and sometimes expert irrationality can even helps when a model of the irrationality is known. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            123,
+            787,
+            376
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "rational with $\\beta = 1 0 0$ . Where applicable, the log loss is averaged over possible hyperparameter settings for the expert. ",
+        "bbox": [
+            176,
+            515,
+            823,
+            544
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We report the results in figure 10. The log loss of the posterior if we wrongly imagine the expert is Boltzmann-rational far outweighs differences between particular irrationality types. ",
+        "bbox": [
+            174,
+            549,
+            823,
+            579
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 WHY IS USING A MISSPECIFIED IRRATIONALITY TYPE FOR INFERENCE SO BAD? ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            595,
+            766,
+            609
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Fundamentally, misspecification is bad for inference because different experts might exhibit the same action only under different reward parameters. For example, consider figure the case where the actual expert is myopic, with small $n$ . Then the myopic agent might go toward a closer reward even if it is much smaller, as shown in figure 11. This would cause the learner to falsely infer that the closer reward is quite large, leading to a posterior with extremely high log loss when the reward is actually smaller. ",
+        "bbox": [
+            173,
+            621,
+            825,
+            705
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd5993c5792e438c89ac621f6c049c8578b51048e417886124512fcce976f926.jpg",
+        "image_caption": [
+            "Figure 11: An example of why assuming Boltzmann is bad for a myopic agent - the Boltzmann rational agent would take this trajectory only if the reward at the bottom was not much less than the reward at the top. The myopic agent with $n \\leq 4$ , however, only ”sees” the reward at the bottom. Consequently, inferring the preferences of the myopic agent as if it were Boltzmann leads to poor performance in this case. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            341,
+            352,
+            656,
+            588
+        ],
+        "page_idx": 14
+    }
+]

parse/train/EmeWbcWORRg/EmeWbcWORRg.md

@@ -0,0 +1,242 @@
+# CHIP: CHannel Independence-based Pruning for Compact Neural Networks
+
+Yang Sui Miao Yin Yi Xie Huy Phan Saman Zonouz Bo Yuan
+
+Department of Electrical and Computer Engineering Rutgers University Piscataway, NJ 08854, USA {yang.sui, miao.yin, yi.xie, huy.phan, saman.zonouz}@rutgers.edu, bo.yuan@soe.rutgers.edu
+
+# Abstract
+
+Filter pruning has been widely used for neural network compression because of its enabled practical acceleration. To date, most of the existing filter pruning works explore the importance of filters via using intra-channel information. In this paper, starting from an inter-channel perspective, we propose to perform efficient filter pruning using channel independence, a metric that measures the correlations among different feature maps. The less independent feature map is interpreted as containing less useful information/knowledge, and hence its corresponding filter can be pruned without affecting model capacity. We systematically investigate the quantification metric, measuring scheme and sensitiveness/reliability of channel independence in the context of filter pruning. Our evaluation results for different models on various datasets show the superior performance of our approach. Notably, on CIFAR-10 dataset our solution can bring $0 . 9 0 \%$ and $0 . 9 4 \%$ accuracy increase over baseline ResNet-56 and ResNet-110 models, respectively, and meanwhile the model size and FLOPs are reduced by $4 2 . 8 \%$ and $4 7 . 4 \%$ (for ResNet-56) and $4 8 . 3 \%$ and $5 2 . 1 \%$ (for ResNet-110), respectively. On ImageNet dataset, our approach can achieve $4 0 . 8 \%$ and $4 4 . 8 \%$ storage and computation reductions, respectively, with $0 . 1 5 \%$ accuracy increase over the baseline ResNet-50 model. The code is available at https://github.com/Eclipsess/CHIP_NeurIPS2021.
+
+# 1 Introduction
+
+Convolutional neural networks (CNNs) have obtained widespread adoptions in numerous important AI applications [17, 48, 47, 12, 11, 44, 35]. However, CNNs are inherently computation intensive and storage intensive, thereby posing severe challenges for their efficient deployment on resourceconstrained embedded platforms. To address these challenges, model compression is widely used to accelerate and compress CNN models on edge devices. To date, various types of compression strategies, such as network pruning [15, 16, 37, 60, 28, 14, 1, 49, 10, 63, 36, 18, 13, 3, 2, 25, 53, 50, 9, 39, 33], quantization [15, 55, 43, 8], low-rank approximation [56, 40, 58, 57], knowledge distillation [22, 41] and structured matrix-based construction [45, 29, 6], have been proposed and explored. Among them, network pruning is the most popular and extensively studied model compression technique in both academia and industry.
+
+Based on their differences in pruning granularity, pruning approaches can be roughly categorized to weight pruning [16, 15] and filter pruning [54, 27, 21, 38, 34]. Weight pruning focuses on the proper selection of the to-be-pruned weights within the filters. Although enabling a high compression ratio, this strategy meanwhile causes unstructured sparsity patterns, which are not well supported by the general-purpose hardware in practice. On the other hand, filter pruning emphasizes the removal of (a) Feature information-based filter pruning from an intra-channel perspective.
+
+![](images/c9ba9593b7e7aea9838bfc444990f50c41332473f661bc5ee3b7d8e95f1de553.jpg)
+
+![](images/87ec15d20874f76ea8ffa5caf51b96ba844175dab0889b2682f0399320562a8c.jpg)  
+(b) Feature information-based filter pruning from an inter-channel perspective.   
+Figure 1: Intra-channel vs Inter-channel perspectives for filter pruning.
+
+the entire selected filters. The resulting structured sparsity patterns can be then properly leveraged by the off-the-shelf CPUs/GPUs to achieve acceleration in the real-world scenario.
+
+Existing Filter Pruning Methods. Motivated by the potential practical speedup offered by filter pruning, to date numerous research efforts have been conducted to study how to determine the important filters – the key component of efficient filter pruning. A well-known strategy is to utilize the norms of different filters to evaluate their importance. Such a "smaller-norm-less-important" hypothesis is adopted in several pioneering filter pruning works [27, 19]. Later, considering the limitations of norm-based criterion in real scenarios, [20] proposes to utilize geometric medianbased criterion. More recently, first determining those important feature maps and then preserving the corresponding filters, instead of directly selecting the filters, become a popular strategy for filter pruning. As indicated in [31], the features, by their natures, reflect and capture rich and important information and characteristics of both input data and filters, and hence measuring the importance of features can provide a better guideline to determine the important filters. Built on this pruning philosophy, several feature-guided filter pruning approaches [31, 51] have been proposed and developed, and the evaluation results show their superior performance over the state-of-the-art filter-guided counterparts with respect to both task performance (e.g., accuracy) and compression performance (e.g., model size and floating-point operations (FLOPs) reductions).
+
+Determining Importance: Intra-channel & Inter-channel Perspectives. These recent advancements on filter pruning indeed show the huge benefits of leveraging feature information to determine the importance of filters. To date, some feature-guided approaches measure the importance from the intra-channel perspective. In other words, no matter which importance metric is used, the importance of one feature map (and its corresponding filter), is measured only upon the information of this feature map in its own channel. On the other aspect, the inter-channel perspective, which essentially determines the filter importance via using cross-channel information [20, 42, 46, 52, 26], is still being further explored. To be specific, [20] and [42] adopt cross-channel geometric median and Hessian, respectively, to measure the channel importance. However, such measurement is based on filter instead of feature map information, and hence the rich and important feature characteristics are not properly identified and extracted. [52, 26] also explore inter-channel-based filter pruning via introducing budget constraints across channels. However, such exploration and utilization of the inter-channel information are implicit and indirect, thereby limiting the practical pruning performance.
+
+Benefits of Inter-channel Perspective. In principle, the feature information across multiple channels, if being leveraged properly, can potentially provide richer knowledge for filter pruning than the intra-channel information. Specifically, this is because: 1) the importance of one filter, if being solely determined by its corresponding feature map, may be sensitive to input data; while the cross-channel feature information can bring more stable and reliable measurement; and 2) consider the essential mission of pruning is to remove the unnecessary redundancy, the inter-channel strategy can inherently better identify and capture the potential unnecessary correlations among different feature maps (and the corresponding filters), and thereby unlocking the new opportunity of achieving better task and compression performance.
+
+Technical Preview and Contributions. Motivated by these promising potential benefits, in this paper we propose to explore and leverage the cross-channel feature information for efficient filter pruning. To be specific, we propose Channel Independence, a cross-channel correlation-based metric to measure the importance of filters. Channel independence can be intuitively understood as the measurement of "replaceability": when the feature map of one filter is measured as exhibiting lower independence, it means this feature map tends to be more linearly dependent on other feature maps of other channels. In such a scenario, the contained information of this low-independence feature map is believed to have already been implicitly encoded in other feature maps – in other words, it does not contain useful information or knowledge. Therefore the corresponding filter, which outputs this low-independence feature map, is viewed as unimportant and can be safely removed without affecting the model capacity. Overall, the contributions of this paper are summarized as:
+
+• We propose channel independence, a metric that measures the correlation of multiple feature maps, to determine the importance of filters. Built from an inter-channel perspective, channel independence can identify and capture the filter importance in a more global and precise way, thereby providing a better guideline for filter pruning.   
+• We systematically investigate and analyze the suitable quantification metric, the complexity of the measuring scheme and the sensitiveness & reliability of channel independence, and then we develop a low-cost fine-grained high-robustness channel independence calculation scheme for efficient filter pruning.   
+• We empirically apply the channel independence-based importance determination in different filter pruning tasks. The evaluation results show that our proposed approach brings very high pruning performance with preserving high accuracy. Notably, on CIFAR-10 dataset our solution can bring $0 . 9 0 \%$ and $0 . 9 4 \%$ accuracy increase over baseline ResNet-56 and ResNet-110 models, respectively, and meanwhile the model size and FLOPs are reduced by $4 2 . 8 \%$ and $4 7 . 4 \%$ (for ResNet-56) and $4 8 . 3 \%$ and $5 2 . 1 \%$ (for ResNet-110), respectively. On ImageNet dataset, our approach can achieve $4 0 . 8 \%$ and $4 4 . 8 \%$ storage and computation reductions, respectively, with $0 . 1 5 \%$ accuracy increase over the baseline ResNet-50 model.
+
+# 2 Preliminaries
+
+Filter Pruning. For a CNN model with $L$ layers, its $l$ -th convolutional layer $\begin{array} { r l } { \boldsymbol { w ^ { l } } } & { { } = } \end{array}$ $\{ \mathcal { F } _ { 1 } ^ { l } , \mathcal { F } _ { 2 } ^ { l } , \cdot \cdot \cdot , \mathcal { F } _ { c ^ { l } } ^ { l } \}$ contains $c ^ { l }$ filters $\mathcal { F } _ { i } ^ { l } \in \mathbb { R } ^ { c ^ { l - 1 } \times k ^ { l } \times k ^ { l } }$ , where $c ^ { l }$ , $c ^ { l - 1 }$ and $k ^ { l }$ denote the number of output channels, the number of input channels and the kernel size, respectively. In general, network pruning can be formulated as the following optimization problem:
+
+$$
+\operatorname* { m i n } _ { \{ \mathscr { W } ^ { l } \} _ { l = 1 } ^ { L } } \mathscr { L } ( \pmb { \mathscr { V } } , f ( \pmb { \mathscr { X } } , \pmb { \mathscr { W } } ^ { l } ) ) , \mathrm { s . t . } \| \pmb { \mathscr { W } } ^ { l } \| _ { 0 } \leq \kappa ^ { l } ,
+$$
+
+where $\mathcal L ( \cdot , \cdot )$ is the loss function, $_ { \mathscr { p } }$ is the ground-truth labels, $_ { x }$ is the input data, and $f ( \cdot , \cdot )$ is the output function of CNN model $\{ \mathcal { W } ^ { l } \} _ { l = 1 } ^ { L }$ . Besides, $\Vert \cdot \Vert _ { 0 }$ is the $\ell _ { 0 }$ -norm that measures the number of non-zero filters in the set, and $\kappa ^ { l }$ is the number of filters to be preserved in the $l$ -th layer.
+
+Feature-guided Filter Pruning. Consider the feature maps, in principle, contain rich and important information of both filters and input data, approaches using feature information have become popular and achieved the state-of-the-art performance for filter pruning. To be specific, unlike the filter-guided methods that directly minimize the loss function involved with filters (as Eq. 1), the objective of feature-guided filter pruning is to minimize the following loss function:
+
+$$
+\operatorname* { m i n } _ { \{ \pmb { A } ^ { l } \} _ { l = 1 } ^ { L } } \mathcal { L } ( \pmb { \mathscr { V } } , \pmb { A } ^ { l } ) , \mathrm { s . t . } \| \pmb { A } ^ { l } \| _ { 0 } \leq \kappa ^ { l } ,
+$$
+
+where $\pmb { \mathcal { A } } ^ { l } = \{ \pmb { A } _ { 1 } ^ { l } , \pmb { A } _ { 2 } ^ { l } , \pmb { \cdot \cdot } \cdot , \pmb { A } _ { c ^ { l } } ^ { l } \} \in \mathbb { R } ^ { c ^ { l } \times h \times w }$ is a set of feature maps output from the $l$ -th layer, and $\mathbf { \Delta } A _ { i } ^ { l } \in \mathbb { R } ^ { h \times w }$ is the feature map corresponds to the $i$ -th channel. In general, after the $\kappa ^ { l }$ important feature maps are identified and selected, their corresponding $\kappa ^ { l }$ filters are preserved after pruning.
+
+# 3 The Proposed Method
+
+# 3.1 Motivation
+
+As formulated in Eq. 2, feature-guided filter pruning leverages the generated feature maps in each layer to identify the important filters. To achieve that, various types of feature information, such as the high ranks [31] and the scaling factors [51], have been proposed and utilized to select the proper feature maps and the corresponding filters. A common point for these state-of-the-art approaches is that all of them focus on measuring the importance via using the information contained in each feature map. On the other hand, the correlation among different feature maps, as another type of rich information provided by the neural networks, is little exploited in the existing filter pruning works.
+
+Why Inter-channel Perspective? We argue that the feature information across multiple channels is of significant importance and richness, and it can be leveraged towards efficient filter pruning. Such an inter-channel perspective is motivated by two promising benefits. First, filter pruning is essentially a data-driven strategy. When the importance of one filter solely depends on the information represented by its own generated feature map, the measurement of the importance may be unstable and sensitive to the slight change of input data. On the other hand, determining the importance built upon information contained in the multiple feature maps, if performed properly, can reduce the potential disturbance incurred by the change of input data, and thereby making the importance ranking more reliable and stable. Second, the inter-channel strategy, by its nature, can better model and capture the cross-channel correlation. In the context of model compression, these identified correlations can be interpreted as a type of architecture-level redundancy, which is exactly what filter pruning aims to remove. Therefore, inter-channel strategy can enable more aggressive pruning while still preserving high accuracy.
+
+# 3.2 Channel Independence: A New Lens for Filter Importance
+
+Key Idea. Motivated by these promising benefits, we propose to explore the filter importance from the inter-channel perspective. Our key idea is to use channel independence to represent the importance of each feature map (and its corresponding filter). To be specific, when one feature map of one channel is highly linearly dependent on other feature maps of other channels, it implies that its contained information has already been largely encoded in other feature maps. Consequently, even we remove the corresponding filter, the represented information and knowledge of its generated low-independence feature map can still be largely preserved and approximately reconstructed by other feature maps of other filters after the fine-tuning procedure. In other words, the filters that generate low-independence feature maps tend to exhibit more "replaceability", which can be interpreted as lower importance. Therefore, removing those filters with low channel-independence feature maps will be safe while still preserving high model capacity.
+
+How to Measure Channel Independence? Next we discuss how to properly measure the independence of one feature map from others. To that end, four important questions need to be answered.
+
+Question #1: Which mathematical metric should be adopted to quantify the independence of one feature map from other feature maps?
+
+Analysis. Considering the entire set of feature maps generated from one layer is a 3-D tensor, we propose to extract the linear dependence information of each feature map within the framework of linear algebra. To be specific, given output feature map set of the $l$ -th layer $\mathbf { \mathcal { A } } ^ { l }$ , we first matricize $\mathbf { \mathcal { A } } ^ { l }$ $\begin{array} { r } { \bar { \mathbf { A } ^ { l } } = [ \bar { \mathbf { a } _ { 1 } ^ { l } } ^ { T } , \bar { \mathbf { a } _ { 2 } ^ { l } } ^ { \hat { T } } , \cdot \cdot \cdot , \bar { \mathbf { a } _ { c ^ { l } } ^ { l } } ^ { T } ] ^ { T } \in \mathbb { R } ^ { c ^ { l } \times h w } } \end{array}$ , where a row vector $\mathbf { \pmb { a } } _ { i } ^ { l } \in \mathbb { R } ^ { h w }$ is the vectorized $A _ { i } ^ { l }$ . In such a scenario, the linear independence of each vectorized feature map $\mathbf { \Delta } _ { \mathbf { \alpha } \mathbf { \beta } _ { i } } ^ { \mathbf { \alpha } _ { i } }$ , as a row of the matricized entire set of feature maps $A ^ { l }$ , can be measured via the existing matrix analysis tool.
+
+![](images/bdedde3b59005664ff3eb98fde5f577fd78c3340addb81074a5448bea585b5f9.jpg)  
+Figure 2: The change of (a) rank and (b) nuclear norm of the entire set of feature maps $( A ^ { l } )$ when one feature map $( \pmb { a } _ { i } ^ { l } )$ is removed. The $\mathbf { X }$ -axis represents the index of the feature map that is removed. The y-axis represents the corresponding rank/nuclear norm change of the entire set of feature maps. The feature maps are output from one layer of the ResNet-50 model with input as ImageNet image. It is seen that change of nuclear norm can better reveal the impact of the deleted feature map on the entire set of feature maps.
+
+The most straightforward solution is to use rank to determine the independence of $\pmb { a } _ { i } ^ { l }$ , since rank mathematically represents the maximum number of linearly independent rows/columns of the matrix. For instance, we can remove one row from the matrix, and calculate the rank change of the matrix, and then identify the impact and the importance of the deleted row – the less rank change, the less independence (and the importance) of the removed row.
+
+Our Proposal. However, in the context of filter pruning, we believe, the change of nuclear norm of the entire set of feature maps, is a better metric to quantify the independence of each feature map. This is because, as the $\overline { { \ell _ { 1 } } }$ -norm of singular values of the matrix, the nuclear norm can reveal richer "soft" information on the impact of the deleted row on the matrix; while the rank, as the $\ell _ { 0 }$ -norm of the singular values, is too "hard" to reflect such change. For instance, as shown in Fig. 2, when we select $\mathbf { \bar { \mathbf { a } } } _ { i } ^ { l }$ , as one row of $A ^ { l }$ , to be removed, the rank change of $A ^ { l }$ is almost the same regardless of our selection of $\mathbf { \Delta } \mathbf { a } _ { i } ^ { l }$ ; while the corresponding changes of nuclear norm vary significantly when different $\mathbf { \Delta } \mathbf { a } _ { i } ^ { l }$ are deleted. Therefore, the change of nuclear norm can be viewed as a more precise metric to measure the linear independence of one feature map in a more fine-grained way. In general, the channel independence of one feature map is defined and calculated as below:
+
+Definition 1 (Channel independence of single feature map) For the i-th layer with output feature maps $\pmb { \mathcal { A } } ^ { l } = \{ \pmb { A } _ { 1 } ^ { l } , \pmb { A } _ { 2 } ^ { l } , \pmb { \cdot \cdot } \cdot , \pmb { A } _ { c ^ { l } } ^ { l } \} \in \mathbb { R } ^ { c ^ { l } \times h \times w }$ , the Channel Independence $( C I )$ of one feature map $\pmb { A } _ { i } ^ { l } \in \mathbb { R } ^ { h \times w }$ in the $i$ -th channel is defined and calculated as:
+
+$$
+C I ( \boldsymbol { A } _ { i } ^ { l } ) \triangleq \| \boldsymbol { A } ^ { l } \| _ { * } - \| \boldsymbol { M } _ { i } ^ { l } \odot \boldsymbol { A } ^ { l } \| _ { * } ,
+$$
+
+where $\pmb { A } ^ { l } \in \mathbb { R } ^ { c ^ { l } \times h w }$ is the matricized $\mathbf { \mathcal { A } } ^ { l } , \parallel \cdot \parallel _ { * }$ is the nuclear norm, $\odot$ is the Hadamard product, and $M _ { i } ^ { l } \in \mathbb { R } ^ { c ^ { l } \times h w }$ is the row mask matrix whose $i$ -th row entries are zeros and other entries are ones.
+
+Question #2: What is the proper scheme to quantify the independence of multiple feature maps?
+
+Analysis. Eq. 3 describes the measurement of channel independence for a single feature map. However, in practice filter pruning typically aims to remove multiple filters, which means the independence of the combination of multiple feature maps needs to be calculated. In the case of pruning $m$ filters, such scenario corresponds to checking the changes of nuclear norm of the original $c ^ { l }$ -row $A ^ { l }$ after removing $m$ rows $( \pmb { a } _ { i } ^ { l } )$ . A straightforward solution is to just calculate $C _ { m } ^ { c ^ { l } }$ changes of nuclear norms for all the possible $m$ -row removal choices, and then select the one which corresponds to the smallest change. However, this strategy is very computationally expensive, and sometimes even intractable when $c ^ { l }$ is large. For instance, in order to identify the smallest nuclear norm change for pruning $5 0 \%$ filters of a 256-output channel ResNet-50 layer, such brutal-force measurement requires more than $5 \times 1 0 ^ { 7 5 }$ times of nuclear norm calculation.
+
+Our Proposal. To address this computational challenge, we propose to leverage the independence of individual feature map to approximate the independence of their combination. To be specific, in order to determine $m$ least independent rows in the $A ^ { l }$ , we first iteratively remove one row $\mathbf { \bar { \rho } } ( \mathbf { \pmb { a } } _ { i } ^ { l } )$ from $A ^ { l }$ and calculate the corresponding nuclear norm change between the remaining $( c ^ { l } - 1 )$ -row matrix and the original $c ^ { l }$ -row $A ^ { l }$ . Then, among the $c ^ { l }$ calculated changes, we identify the $m$ smallest ones and the corresponding removed $\mathbf { \Delta } \mathbf { a } _ { i } ^ { l }$ . Those selected $m$ vectorized feature maps $\mathbf { \Delta } \mathbf { a } _ { i } ^ { l }$ are interpreted as the less independent from other feature maps, and hence their corresponding filters $\mathcal { F } _ { i } ^ { l }$ are less important ones that should be pruned. In general, this individual independence-based measurement can closely approximate the combined independence of multiple feature maps (see Definition 2). Such approximation requires much less computational complexity (reduction from $\mathcal { O } ( C ( N , \kappa ) )$ to $\mathcal { O } ( N ) )$ ; while still achieving superior filter pruning performance (see Section 4 for evaluation results).
+
+![](images/addd57364a5979c6538f101cbab3bac8a73465404aeaf4b35049b07786699385.jpg)  
+Figure 3: Example of filter pruning process using the change of nuclear norm-based channel independence (CI) criterion.
+
+Definition 2 (Approximated channel independence of combined multiple feature maps) For the $i$ -th layer with output feature maps $\pmb { \mathcal { A } } ^ { l } \in \mathbb { R } ^ { c ^ { l } \times h \times w }$ , the channel independence of combined $m$ feature maps $\{ A _ { b _ { i } } ^ { l } \} _ { i = 1 } ^ { m }$ , where $A _ { b _ { i } } ^ { l } \in \mathbb { R } ^ { h \times w }$ is in the $b _ { i }$ -th channel, is defined and approximated as:
+
+$$
+C I ( \{ A _ { b _ { i } } ^ { l } \} _ { i = 1 } ^ { m } ) \triangleq \| A ^ { l } \| _ { * } - \| M _ { b _ { 1 } , \cdots , b _ { m } } ^ { l } \odot A ^ { l } \| _ { * } \approx \sum _ { i = 1 } ^ { m } C I ( A _ { b _ { i } } ^ { l } ) ,
+$$
+
+where $M _ { b _ { 1 } , \cdots , b _ { m } } ^ { l }$ is the multi-row mask matrix, in which the $b _ { 1 } , \cdots , b _ { m }$ -th row entries are zeros and all the other entries are ones.
+
+Question #3: How is the sensitiveness of channel independence related to the distribution of input data?
+
+Our Observation. Consider our proposed channel independence-based filter pruning is a data-driven approach, its reliability with different distributions of input data should be carefully ensured and examined. To that end, we perform empirical evaluations on channel independence with respect to multiple input images. We observe that the average channel independence of each feature map is very stable at the batch level. In other words, we can simply input small batches of image samples, and calculate the average channel independence, and then such averaged channel independence with a small number of input data can be used to estimate the channel independence with all the input data. As illustrated in Fig. 4, for the same feature map, the average channel independence in different batches remains very similar, thereby indicating that our channel independence-based approach is robust against different input data.
+
+Question #4: Is this one-shot importance determination scheme good enough? Do we need to further learn and adjust the pruning mask from the data?
+
+Our Observation. As described above, our proposed scheme calculates the channel independence to identify the filter importance. Considering our approach is built on one-shot calculation, a natural extension is to further adjust the importance ranking via additional learning. To be specific, if we interpret the filter pruning is a channel-wise masking operation over the entire weight tensor, the
+
+![](images/2b60805cf00b69baba7e728bce7dc09ad7a7b7adfe70f2772e4030fda785d574.jpg)  
+Figure 4: The channel independence of feature maps for one layer in ResNet-50. Here the channel independence is averaged for one batch of input images. The $\mathbf { X }$ -axis is the index of the feature map. The y-axis is the index of batches of input images. Here the batch size is 128. Different colors denote the different values of channel independence. It is seen that the average channel independence is very stable regardless of different input data batches.
+
+Algorithm 1 CHannel Independence-based Pruning (CHIP) procedure for the $l$ -th layer
+
+Input: Pre-trained weight tensor $\boldsymbol { w ^ { l } }$ , $N$ sets of feature maps $\pmb { \mathcal { A } } ^ { l } = \{ \pmb { A } _ { 1 } ^ { l } , \pmb { A } _ { 2 } ^ { l } , \cdot \cdot \cdot , \pmb { A } _ { c ^ { l } } ^ { l } \} \in \mathbb { R } ^ { c ^ { l } \times h \times w }$ from $N$ input samples, and the desired number of filters to be preserved $\kappa ^ { l }$ .
+
+Output: Pruned weight tensor $\boldsymbol { \mathcal { W } } _ { p r u n e } ^ { l }$ .   
+1: for each input sample do   
+2: Flatten feature maps: $\pmb { A } ^ { l } : = \mathrm { r e s h a p e } ( \pmb { A } ^ { l } , [ c ^ { l } , h w ] ) ;$ ;   
+3: for $i = 1$ to $c ^ { l }$ do   
+4: CI calculation: Calculate $C I ( A _ { i } ^ { l } )$ via Equation 3;   
+5: end for   
+6: end for   
+7: Averaging: Average $C I ( A _ { i } ^ { l } )$ under all $N$ input samples;   
+8: Sorting: Sort $\{ C I ( A _ { i } ^ { l } ) \} _ { i = 1 } ^ { c ^ { l } }$ in ascending order;   
+9: Pruning: Prune $c ^ { l } - \kappa ^ { l }$ filters in $\boldsymbol { w ^ { l } }$ corresponding to the $c ^ { l } - \kappa ^ { l }$ smallest $C I ( A _ { i } ^ { l } )$ ;   
+10: Fine-tuning: Obtain final $\boldsymbol { \mathcal { W } } _ { p r u n e } ^ { l }$ via fine-tuning $\boldsymbol { w ^ { l } }$ with removing the pruned filter channels.
+
+selection of channel mask can be learned from data, and such learning process can use the pruning mask determined by our approach as the initialization. Though in principle this learning-based strategy is expected to enable additional performance improvement, our empirical evaluations show that the consecutive learning procedure does not easily bring further accuracy increase (with the target compression ratio) or compression ratio increase (with the target accuracy) – more experimental details are reported in Supplementary Material. We hypothesize the reason for such phenomenon is that, our proposed nuclear norm change-based channel independence, though only requires one-time calculation, already identifies and captures the importance of feature maps (and its corresponding filters) with high quality, and hence further learning-based adjustment of pruning mask does not easily provide additional improvement.
+
+The Overall Algorithm. After addressing the above four problems, we can then integrate our proposals and observations to develop the entire filter pruning procedure from the inter-channel perspective. Algorithm 1 describes and summarizes the overall scheme for our proposed CHannel Independence-based filter Pruning (CHIP) algorithm.
+
+# 4 Experiments
+
+# 4.1 Experimental Settings
+
+Baselines Models and Datasets. To demonstrate the effectiveness and generality of our proposed channel independence-based approach, we evaluate its pruning performance for various baseline models on different image classification datasets. To be specific, we conduct experiments for three CNN models (ResNet-56, ResNet-110 and VGG-16) on CIFAR-10 dataset [24]. Also, we further evaluate our approach and compare its performance with other state-of-the-art pruning methods for ResNet-50 model on large-scale ImageNet dataset [5].
+
+Pruning and Fine-tuning Configurations. We conduct our empirical evaluations on Nvidia Tesla V100 GPUs with PyTorch 1.7 framework. To determine the importance of each filter, we randomly sample 5 batches (640 input images) to calculate the average channel independence of each feature map in all the experiments. After performing the channel independence-based filter pruning, we then perform fine-tuning on the pruned models with Stochastic Gradient Descent (SGD) as the optimizer. To be specific, we perform the fine-tuning for 300 epochs on CIFAR-10 datasets with the batch size, momentum, weight decay and initial learning rate as 128, 0.9, 0.05 and 0.01, respectively. On the ImageNet dataset, fine-tuning is performed for 180 epochs with the batch size, momentum, weight decay and initial learning rate as 256, 0.99, 0.0001 and 0.1, respectively.
+
+Table 1: Experimental results on CIFAR-10 dataset.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Top-1 Accuracy (%) △</td><td rowspan="2">#Params. (↓%)</td><td rowspan="2">FLOPs (↓%)</td></tr><tr><td>Baseline</td><td>Pruned</td></tr><tr><td colspan="6">ResNet-56</td></tr><tr><td>l1-norm (2016)[27]</td><td>93.04</td><td>93.06</td><td>+0.02</td><td>0.73M(13.7)</td><td>90.90M(27.6)</td></tr><tr><td>NISP (2018) [59]</td><td>93.04</td><td>93.01</td><td>-0.03</td><td>0.49M(42.4)</td><td>81.00M(35.5)</td></tr><tr><td>GAL (2019) [32]</td><td>93.26</td><td>93.38</td><td>+0.12</td><td>0.75M(11.8)</td><td>78.30M(37.6)</td></tr><tr><td>HRank (2020) [31]</td><td>93.26</td><td>93.52</td><td>+0.26</td><td>0.71M(16.8)</td><td>88.72M(29.3)</td></tr><tr><td>CHIP (Ours)</td><td>93.26</td><td>94.16</td><td>+0.90</td><td>0.48M(42.8)</td><td>65.94M(47.4)</td></tr><tr><td>GAL(2019)[32]</td><td>93.26</td><td>91.58</td><td>-1.68</td><td>0.29M(65.9)</td><td>49.99M(60.2)</td></tr><tr><td>LASSO (2017) [21]</td><td>92.80</td><td>91.80</td><td>-1.00</td><td>N/A</td><td>62.00M(50.6)</td></tr><tr><td>HRank (2020)[31]</td><td>93.26</td><td>90.72</td><td>-2.54</td><td>0.27M(68.1)</td><td>32.52M(74.1)</td></tr><tr><td>CHIP (Ours)</td><td>93.26</td><td>92.05</td><td>-1.21</td><td>0.24M(71.8)</td><td>34.79M(72.3)</td></tr><tr><td colspan="6">ResNet-110</td></tr><tr><td>l1-norm (2016) [27]</td><td>93.53</td><td>93.30</td><td>-0.23</td><td>1.16M(32.4)</td><td>155.00M(38.7)</td></tr><tr><td>HRank (2020)[31]</td><td>93.50</td><td>94.23</td><td>+0.73</td><td>1.04M(39.4)</td><td>148.70M(41.2)</td></tr><tr><td>CHIP (Ours)</td><td>93.50</td><td>94.44</td><td>+0.94</td><td>0.89M(48.3)</td><td>121.09M(52.1)</td></tr><tr><td>GAL (2019)[32]</td><td>93.50</td><td>92.74</td><td>-0.76</td><td>0.95M(44.8)</td><td>130.20M(48.5)</td></tr><tr><td>HRank (2020)[31]</td><td>93.50</td><td>92.65</td><td>-0.85</td><td>0.53M(68.7)</td><td>79.30M(68.6)</td></tr><tr><td>CHIP (Ours)</td><td>93.50</td><td>93.63</td><td>+0.13</td><td>0.54M(68.3)</td><td>71.69M(71.6)</td></tr><tr><td colspan="6">VGG-16</td></tr><tr><td>SSS (2018) [23]</td><td>93.96</td><td>93.02</td><td>-0.94</td><td>3.93M(73.8)</td><td>183.13M(41.6)</td></tr><tr><td>GAL (2019) [32]</td><td>93.96</td><td>93.77</td><td>-0.19</td><td>3.36M(77.6)</td><td>189.49M(39.6)</td></tr><tr><td>HRank (2020)[31]</td><td>93.96</td><td>93.43</td><td>-0.53</td><td>2.51M(82.9)</td><td>145.61M(53.5)</td></tr><tr><td>CHIP (Ours)</td><td>93.96</td><td>93.86</td><td>-0.10</td><td>2.76M(81.6)</td><td>131.17M(58.1)</td></tr><tr><td>GAL (2019)[32]</td><td>93.96</td><td>93.42</td><td>-0.54</td><td>2.67M(82.2)</td><td>171.89M(45.2)</td></tr><tr><td>HRank (2020) [31]</td><td>93.96</td><td>92.34</td><td>-1.62</td><td>2.64M(82.1)</td><td>108.61M(65.3)</td></tr><tr><td>CHIP (Ours)</td><td>93.96</td><td>93.72</td><td>-0.24</td><td>2.50M(83.3)</td><td>104.78M(66.6)</td></tr><tr><td>HRank (2020) [31]</td><td>93.96</td><td>91.23</td><td>-2.73</td><td>1.78M(92.0)</td><td>73.70M(76.5)</td></tr><tr><td>CHIP (Ours)</td><td>93.96</td><td>93.18</td><td>-0.78</td><td>1.90M(87.3)</td><td>66.95M(78.6)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# 4.2 Evaluation and Comparison on CIFAR-10 Dataset
+
+Table 1 shows the evaluation results of the pruned ResNet-56, ResNet-110 and VGG-16 models on CIFAR-10 dataset. To be consistent with prior works, we evaluate the performance for two scenarios: targeting high accuracy and targeting high model size and FLOPs reductions.
+
+ResNet-56. For ResNet-56 model, our channel independence-based approach can bring $0 . 9 0 \%$ accuracy increase over the baseline model with $4 2 . 8 \%$ and $4 7 . 4 \%$ model size and FLOPs reductions, respectively. When we adopt aggressive compression with $7 1 . 8 \%$ and $7 2 . 3 \%$ model size and FLOPs reductions, we can still achieve high performance – our solution enables $1 . 3 3 \%$ higher accuracy than HRank [31] with the similar model size and computational costs.
+
+ResNet-110. For ResNet-110 model, our approach can bring $0 . 9 4 \%$ accuracy increase over the baseline model with $4 8 . 3 \%$ and $5 2 . 1 \%$ model size and FLOPs reductions, respectively. When we perform aggressive pruning with $6 8 . 3 \%$ and $7 1 . 6 \%$ model size and FLOPs reductions, our pruned model can still achieve $0 . 1 \hat { 3 } \%$ higher accuracy over the baseline model.
+
+VGG-16. For VGG-16 model, our approach can bring $8 1 . 6 \%$ and $5 8 . 1 \%$ model size and FLOPs reductions, respectively, with only $0 . { \bar { 1 } } \%$ accuracy drop. Moreover, with $8 3 . 3 \%$ and $6 6 . 6 \%$ storage and computational cost reductions, our pruned model can achieve $1 . 3 8 \%$ higher accuracy than the model using other pruning approaches under a similar compression ratio. For even higher FLOPs reduction $( 7 8 . 6 \% )$ ), our method can bring nearly $2 \%$ accuracy increase over the prior works.
+
+# 4.3 Evaluation and Comparison on ImageNet Dataset
+
+Table 2 summarizes the pruning performance of our approach for ResNet-50 on ImageNet dataset. It is seen that when targeting a moderate compression ratio, our approach can achieve $4 0 . 8 \%$ and $4 4 . 8 \%$ storage and computation reductions, respectively, with $0 . 1 5 \%$ accuracy increase over the baseline model. When we further increase the compression ratio, our approach still achieves superior performance than state-of-the-art works. For instance, compared with SCOP [51], our approach shows higher accuracy $\left( 0 . 1 2 \% \right)$ in moderate compression region and the same accuracy in high compress region; while meanwhile enjoying a much smaller model size and fewer FLOPs.
+
+Table 2: Experimental results on ImageNet dataset.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Top-1 Accuracy (%)</td><td colspan="5">Top-5 Accuracy (%) Params.FLOPs</td></tr><tr><td></td><td>Baseline Pruned</td><td>△</td><td>Baseline</td><td>Pruned</td><td>△</td><td>↓(%)</td><td>↓(%)</td></tr><tr><td colspan="10">ResNet-50</td></tr><tr><td>ThiNet (2017) [37]</td><td>72.88</td><td></td><td>72.04-0.84</td><td>91.14</td><td>90.67</td><td>-0.47</td><td>33.7</td><td>36.8</td></tr><tr><td>SFP (2018)[19]</td><td>76.15</td><td>74.61</td><td>-1.54</td><td>92.87</td><td>92.06</td><td>-0.81</td><td>N/A</td><td>41.8</td></tr><tr><td>Autopruner (2020) [36]</td><td>76.15</td><td>74.76</td><td>-1.39</td><td>92.87</td><td>92.15</td><td>-0.72</td><td>N/A</td><td>48.7</td></tr><tr><td>FPGM (2019)[20]</td><td>76.15</td><td>75.59</td><td>-0.56</td><td>92.87</td><td>92.63</td><td>-0.24</td><td>37.5</td><td>42.2</td></tr><tr><td>Taylor (2019) [38]</td><td>76.18</td><td>74.50</td><td>-1.68</td><td>N/A</td><td>N/A</td><td>N/A</td><td>44.5</td><td>44.9</td></tr><tr><td>C-SGD (2019) [7]</td><td>75.33</td><td>74.93</td><td>-0.40</td><td>92.56</td><td>92.27</td><td>-0.29</td><td>N/A</td><td>46.2</td></tr><tr><td>GAL (2019) [32]</td><td>76.15</td><td>71.95</td><td>-4.20</td><td>92.87</td><td>90.94</td><td>-1.93</td><td>16.9</td><td>43</td></tr><tr><td>RRBP (2019)[61]</td><td>76.10</td><td>73.00</td><td>-3.10</td><td>92.90</td><td>91.00</td><td>-1.90</td><td>N/A</td><td>54.5</td></tr><tr><td>PFP (2020)[30]</td><td>76.13</td><td>75.91</td><td>-0.22</td><td>92.87</td><td>92.81</td><td>-0.06</td><td>18.1</td><td>10.8</td></tr><tr><td>HRank (2020) [31]</td><td>76.15</td><td>74.98</td><td>-1.17</td><td>92.87</td><td>92.33</td><td>-0.54</td><td>36.6</td><td>43.7</td></tr><tr><td>SCOP (2020) [51]</td><td>76.15</td><td>75.95</td><td>-0.20</td><td>92.87</td><td>92.79</td><td>-0.08</td><td>42.8</td><td>45.3</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>76.30</td><td>+0.15</td><td>92.87</td><td>93.02</td><td>+0.15</td><td>40.8</td><td>44.8</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>76.15</td><td>0.00</td><td>92.87</td><td>92.91</td><td>+0.04</td><td>44.2</td><td>48.7</td></tr><tr><td>PFP(2020)[30]</td><td>76.13</td><td>75.21</td><td>-0.92</td><td>92.87</td><td>92.43</td><td>-0.44</td><td>30.1</td><td>44</td></tr><tr><td>SCOP (2020) [51]</td><td>76.15</td><td>75.26</td><td>-0.89</td><td>92.87</td><td>92.53</td><td>-0.34</td><td>51.8</td><td>54.6</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>75.26</td><td>-0.89</td><td>92.87</td><td>92.53</td><td>-0.34</td><td>56.7</td><td>62.8</td></tr><tr><td>HRank(2020)[31]</td><td>76.15</td><td>71.98</td><td>-4.17</td><td>92.87</td><td>91.01</td><td>--1.86</td><td>46.0</td><td>62.1</td></tr><tr><td>HRank (2020) [31]</td><td>76.15</td><td>69.10</td><td>-7.05</td><td>92.87</td><td>89.58</td><td>-3.29</td><td>67.5</td><td>76.0</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>73.30</td><td>-2.85</td><td>92.87</td><td>91.48</td><td>-1.39</td><td>68.6</td><td>76.7</td></tr></table>
+
+# 5 Conclusion
+
+In this paper, we propose to use channel independence, an inter-channel perspective-motivated metric, to evaluate the importance of filters for network pruning. By systematically exploring the quantification metric, measuring scheme, and sensitiveness and reliability of channel independence, we develop CHIP, a CHannel Independence-based filter pruning for neural network compression. Extensive evaluation results on different datasets show our proposed approach brings significant storage and computational cost reductions while still preserving high model accuracy.
+
+# Broader Impact
+
+As technology advances, cell phones, laptops, wearable gadgets and intelligent connected vehicles with specific chips are required to handle more complicated tasks by deploying neural networks. However, more powerful networks will cost more memory size and running time. Network pruning is the main strategy to reduce the memory size and accelerate the run-time during the inference stage. Benefiting from pruning techniques and specific designs for hardware [62, 4], IoT (Internet of Things) devices are able to execute complex projects based on small and efficient models.
+
+# Acknowledgements and Funding Disclosure
+
+Bo Yuan would like to thank the support from National Science Foundation (NSF) award CCF1937403. Saman Zonouz would like to thank the support from NSF CPS and SATC programs.
+
+# References
+
+[1] Davis Blalock, Jose Javier Gonzalez Ortiz, Jonathan Frankle, and John Guttag. What is the state of neural network pruning? In I. Dhillon, D. Papailiopoulos, and V. Sze, editors, Proceedings of Machine Learning and Systems, volume 2, pages 129–146, 2020.   
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+        "text": "CHIP: CHannel Independence-based Pruning for Compact Neural Networks ",
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+        "text": "Yang Sui Miao Yin Yi Xie Huy Phan Saman Zonouz Bo Yuan ",
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+        "text": "Department of Electrical and Computer Engineering Rutgers University Piscataway, NJ 08854, USA {yang.sui, miao.yin, yi.xie, huy.phan, saman.zonouz}@rutgers.edu, bo.yuan@soe.rutgers.edu ",
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+        "text": "Abstract ",
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+        "text": "Filter pruning has been widely used for neural network compression because of its enabled practical acceleration. To date, most of the existing filter pruning works explore the importance of filters via using intra-channel information. In this paper, starting from an inter-channel perspective, we propose to perform efficient filter pruning using channel independence, a metric that measures the correlations among different feature maps. The less independent feature map is interpreted as containing less useful information/knowledge, and hence its corresponding filter can be pruned without affecting model capacity. We systematically investigate the quantification metric, measuring scheme and sensitiveness/reliability of channel independence in the context of filter pruning. Our evaluation results for different models on various datasets show the superior performance of our approach. Notably, on CIFAR-10 dataset our solution can bring $0 . 9 0 \\%$ and $0 . 9 4 \\%$ accuracy increase over baseline ResNet-56 and ResNet-110 models, respectively, and meanwhile the model size and FLOPs are reduced by $4 2 . 8 \\%$ and $4 7 . 4 \\%$ (for ResNet-56) and $4 8 . 3 \\%$ and $5 2 . 1 \\%$ (for ResNet-110), respectively. On ImageNet dataset, our approach can achieve $4 0 . 8 \\%$ and $4 4 . 8 \\%$ storage and computation reductions, respectively, with $0 . 1 5 \\%$ accuracy increase over the baseline ResNet-50 model. The code is available at https://github.com/Eclipsess/CHIP_NeurIPS2021. ",
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+        "text": "1 Introduction ",
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+        "text": "Convolutional neural networks (CNNs) have obtained widespread adoptions in numerous important AI applications [17, 48, 47, 12, 11, 44, 35]. However, CNNs are inherently computation intensive and storage intensive, thereby posing severe challenges for their efficient deployment on resourceconstrained embedded platforms. To address these challenges, model compression is widely used to accelerate and compress CNN models on edge devices. To date, various types of compression strategies, such as network pruning [15, 16, 37, 60, 28, 14, 1, 49, 10, 63, 36, 18, 13, 3, 2, 25, 53, 50, 9, 39, 33], quantization [15, 55, 43, 8], low-rank approximation [56, 40, 58, 57], knowledge distillation [22, 41] and structured matrix-based construction [45, 29, 6], have been proposed and explored. Among them, network pruning is the most popular and extensively studied model compression technique in both academia and industry. ",
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+        "text": "Based on their differences in pruning granularity, pruning approaches can be roughly categorized to weight pruning [16, 15] and filter pruning [54, 27, 21, 38, 34]. Weight pruning focuses on the proper selection of the to-be-pruned weights within the filters. Although enabling a high compression ratio, this strategy meanwhile causes unstructured sparsity patterns, which are not well supported by the general-purpose hardware in practice. On the other hand, filter pruning emphasizes the removal of (a) Feature information-based filter pruning from an intra-channel perspective. ",
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+            "(b) Feature information-based filter pruning from an inter-channel perspective. ",
+            "Figure 1: Intra-channel vs Inter-channel perspectives for filter pruning. "
+        ],
+        "image_footnote": [],
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+        "text": "the entire selected filters. The resulting structured sparsity patterns can be then properly leveraged by the off-the-shelf CPUs/GPUs to achieve acceleration in the real-world scenario. ",
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+        "text": "Existing Filter Pruning Methods. Motivated by the potential practical speedup offered by filter pruning, to date numerous research efforts have been conducted to study how to determine the important filters – the key component of efficient filter pruning. A well-known strategy is to utilize the norms of different filters to evaluate their importance. Such a \"smaller-norm-less-important\" hypothesis is adopted in several pioneering filter pruning works [27, 19]. Later, considering the limitations of norm-based criterion in real scenarios, [20] proposes to utilize geometric medianbased criterion. More recently, first determining those important feature maps and then preserving the corresponding filters, instead of directly selecting the filters, become a popular strategy for filter pruning. As indicated in [31], the features, by their natures, reflect and capture rich and important information and characteristics of both input data and filters, and hence measuring the importance of features can provide a better guideline to determine the important filters. Built on this pruning philosophy, several feature-guided filter pruning approaches [31, 51] have been proposed and developed, and the evaluation results show their superior performance over the state-of-the-art filter-guided counterparts with respect to both task performance (e.g., accuracy) and compression performance (e.g., model size and floating-point operations (FLOPs) reductions). ",
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+        "text": "Determining Importance: Intra-channel & Inter-channel Perspectives. These recent advancements on filter pruning indeed show the huge benefits of leveraging feature information to determine the importance of filters. To date, some feature-guided approaches measure the importance from the intra-channel perspective. In other words, no matter which importance metric is used, the importance of one feature map (and its corresponding filter), is measured only upon the information of this feature map in its own channel. On the other aspect, the inter-channel perspective, which essentially determines the filter importance via using cross-channel information [20, 42, 46, 52, 26], is still being further explored. To be specific, [20] and [42] adopt cross-channel geometric median and Hessian, respectively, to measure the channel importance. However, such measurement is based on filter instead of feature map information, and hence the rich and important feature characteristics are not properly identified and extracted. [52, 26] also explore inter-channel-based filter pruning via introducing budget constraints across channels. However, such exploration and utilization of the inter-channel information are implicit and indirect, thereby limiting the practical pruning performance. ",
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+    {
+        "type": "text",
+        "text": "Benefits of Inter-channel Perspective. In principle, the feature information across multiple channels, if being leveraged properly, can potentially provide richer knowledge for filter pruning than the intra-channel information. Specifically, this is because: 1) the importance of one filter, if being solely determined by its corresponding feature map, may be sensitive to input data; while the cross-channel feature information can bring more stable and reliable measurement; and 2) consider the essential mission of pruning is to remove the unnecessary redundancy, the inter-channel strategy can inherently better identify and capture the potential unnecessary correlations among different feature maps (and the corresponding filters), and thereby unlocking the new opportunity of achieving better task and compression performance. ",
+        "bbox": [
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+    {
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+        "text": "Technical Preview and Contributions. Motivated by these promising potential benefits, in this paper we propose to explore and leverage the cross-channel feature information for efficient filter pruning. To be specific, we propose Channel Independence, a cross-channel correlation-based metric to measure the importance of filters. Channel independence can be intuitively understood as the measurement of \"replaceability\": when the feature map of one filter is measured as exhibiting lower independence, it means this feature map tends to be more linearly dependent on other feature maps of other channels. In such a scenario, the contained information of this low-independence feature map is believed to have already been implicitly encoded in other feature maps – in other words, it does not contain useful information or knowledge. Therefore the corresponding filter, which outputs this low-independence feature map, is viewed as unimportant and can be safely removed without affecting the model capacity. Overall, the contributions of this paper are summarized as: ",
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+        "text": "• We propose channel independence, a metric that measures the correlation of multiple feature maps, to determine the importance of filters. Built from an inter-channel perspective, channel independence can identify and capture the filter importance in a more global and precise way, thereby providing a better guideline for filter pruning.   \n• We systematically investigate and analyze the suitable quantification metric, the complexity of the measuring scheme and the sensitiveness & reliability of channel independence, and then we develop a low-cost fine-grained high-robustness channel independence calculation scheme for efficient filter pruning.   \n• We empirically apply the channel independence-based importance determination in different filter pruning tasks. The evaluation results show that our proposed approach brings very high pruning performance with preserving high accuracy. Notably, on CIFAR-10 dataset our solution can bring $0 . 9 0 \\%$ and $0 . 9 4 \\%$ accuracy increase over baseline ResNet-56 and ResNet-110 models, respectively, and meanwhile the model size and FLOPs are reduced by $4 2 . 8 \\%$ and $4 7 . 4 \\%$ (for ResNet-56) and $4 8 . 3 \\%$ and $5 2 . 1 \\%$ (for ResNet-110), respectively. On ImageNet dataset, our approach can achieve $4 0 . 8 \\%$ and $4 4 . 8 \\%$ storage and computation reductions, respectively, with $0 . 1 5 \\%$ accuracy increase over the baseline ResNet-50 model. ",
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+        "text": "2 Preliminaries ",
+        "text_level": 1,
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+        "text": "Filter Pruning. For a CNN model with $L$ layers, its $l$ -th convolutional layer $\\begin{array} { r l } { \\boldsymbol { w ^ { l } } } & { { } = } \\end{array}$ $\\{ \\mathcal { F } _ { 1 } ^ { l } , \\mathcal { F } _ { 2 } ^ { l } , \\cdot \\cdot \\cdot , \\mathcal { F } _ { c ^ { l } } ^ { l } \\}$ contains $c ^ { l }$ filters $\\mathcal { F } _ { i } ^ { l } \\in \\mathbb { R } ^ { c ^ { l - 1 } \\times k ^ { l } \\times k ^ { l } }$ , where $c ^ { l }$ , $c ^ { l - 1 }$ and $k ^ { l }$ denote the number of output channels, the number of input channels and the kernel size, respectively. In general, network pruning can be formulated as the following optimization problem: ",
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+        "img_path": "images/8e50c25ad462122863e7f1e861d40fed2311ecf3126061bab1e2bab7ec7c2534.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\{ \\mathscr { W } ^ { l } \\} _ { l = 1 } ^ { L } } \\mathscr { L } ( \\pmb { \\mathscr { V } } , f ( \\pmb { \\mathscr { X } } , \\pmb { \\mathscr { W } } ^ { l } ) ) , \\mathrm { s . t . } \\| \\pmb { \\mathscr { W } } ^ { l } \\| _ { 0 } \\leq \\kappa ^ { l } ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal L ( \\cdot , \\cdot )$ is the loss function, $_ { \\mathscr { p } }$ is the ground-truth labels, $_ { x }$ is the input data, and $f ( \\cdot , \\cdot )$ is the output function of CNN model $\\{ \\mathcal { W } ^ { l } \\} _ { l = 1 } ^ { L }$ . Besides, $\\Vert \\cdot \\Vert _ { 0 }$ is the $\\ell _ { 0 }$ -norm that measures the number of non-zero filters in the set, and $\\kappa ^ { l }$ is the number of filters to be preserved in the $l$ -th layer. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Feature-guided Filter Pruning. Consider the feature maps, in principle, contain rich and important information of both filters and input data, approaches using feature information have become popular and achieved the state-of-the-art performance for filter pruning. To be specific, unlike the filter-guided methods that directly minimize the loss function involved with filters (as Eq. 1), the objective of feature-guided filter pruning is to minimize the following loss function: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/0f3c475203931b02fe6dfc0572e70ed07bee3f080585dfc5a481e1f7697b013b.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\{ \\pmb { A } ^ { l } \\} _ { l = 1 } ^ { L } } \\mathcal { L } ( \\pmb { \\mathscr { V } } , \\pmb { A } ^ { l } ) , \\mathrm { s . t . } \\| \\pmb { A } ^ { l } \\| _ { 0 } \\leq \\kappa ^ { l } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            131
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { \\mathcal { A } } ^ { l } = \\{ \\pmb { A } _ { 1 } ^ { l } , \\pmb { A } _ { 2 } ^ { l } , \\pmb { \\cdot \\cdot } \\cdot , \\pmb { A } _ { c ^ { l } } ^ { l } \\} \\in \\mathbb { R } ^ { c ^ { l } \\times h \\times w }$ is a set of feature maps output from the $l$ -th layer, and $\\mathbf { \\Delta } A _ { i } ^ { l } \\in \\mathbb { R } ^ { h \\times w }$ is the feature map corresponds to the $i$ -th channel. In general, after the $\\kappa ^ { l }$ important feature maps are identified and selected, their corresponding $\\kappa ^ { l }$ filters are preserved after pruning. ",
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+    {
+        "type": "text",
+        "text": "3 The Proposed Method ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "3.1 Motivation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As formulated in Eq. 2, feature-guided filter pruning leverages the generated feature maps in each layer to identify the important filters. To achieve that, various types of feature information, such as the high ranks [31] and the scaling factors [51], have been proposed and utilized to select the proper feature maps and the corresponding filters. A common point for these state-of-the-art approaches is that all of them focus on measuring the importance via using the information contained in each feature map. On the other hand, the correlation among different feature maps, as another type of rich information provided by the neural networks, is little exploited in the existing filter pruning works. ",
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+    {
+        "type": "text",
+        "text": "Why Inter-channel Perspective? We argue that the feature information across multiple channels is of significant importance and richness, and it can be leveraged towards efficient filter pruning. Such an inter-channel perspective is motivated by two promising benefits. First, filter pruning is essentially a data-driven strategy. When the importance of one filter solely depends on the information represented by its own generated feature map, the measurement of the importance may be unstable and sensitive to the slight change of input data. On the other hand, determining the importance built upon information contained in the multiple feature maps, if performed properly, can reduce the potential disturbance incurred by the change of input data, and thereby making the importance ranking more reliable and stable. Second, the inter-channel strategy, by its nature, can better model and capture the cross-channel correlation. In the context of model compression, these identified correlations can be interpreted as a type of architecture-level redundancy, which is exactly what filter pruning aims to remove. Therefore, inter-channel strategy can enable more aggressive pruning while still preserving high accuracy. ",
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+    {
+        "type": "text",
+        "text": "3.2 Channel Independence: A New Lens for Filter Importance ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Key Idea. Motivated by these promising benefits, we propose to explore the filter importance from the inter-channel perspective. Our key idea is to use channel independence to represent the importance of each feature map (and its corresponding filter). To be specific, when one feature map of one channel is highly linearly dependent on other feature maps of other channels, it implies that its contained information has already been largely encoded in other feature maps. Consequently, even we remove the corresponding filter, the represented information and knowledge of its generated low-independence feature map can still be largely preserved and approximately reconstructed by other feature maps of other filters after the fine-tuning procedure. In other words, the filters that generate low-independence feature maps tend to exhibit more \"replaceability\", which can be interpreted as lower importance. Therefore, removing those filters with low channel-independence feature maps will be safe while still preserving high model capacity. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "How to Measure Channel Independence? Next we discuss how to properly measure the independence of one feature map from others. To that end, four important questions need to be answered. ",
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+    },
+    {
+        "type": "text",
+        "text": "Question #1: Which mathematical metric should be adopted to quantify the independence of one feature map from other feature maps? ",
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+    {
+        "type": "text",
+        "text": "Analysis. Considering the entire set of feature maps generated from one layer is a 3-D tensor, we propose to extract the linear dependence information of each feature map within the framework of linear algebra. To be specific, given output feature map set of the $l$ -th layer $\\mathbf { \\mathcal { A } } ^ { l }$ , we first matricize $\\mathbf { \\mathcal { A } } ^ { l }$ $\\begin{array} { r } { \\bar { \\mathbf { A } ^ { l } } = [ \\bar { \\mathbf { a } _ { 1 } ^ { l } } ^ { T } , \\bar { \\mathbf { a } _ { 2 } ^ { l } } ^ { \\hat { T } } , \\cdot \\cdot \\cdot , \\bar { \\mathbf { a } _ { c ^ { l } } ^ { l } } ^ { T } ] ^ { T } \\in \\mathbb { R } ^ { c ^ { l } \\times h w } } \\end{array}$ , where a row vector $\\mathbf { \\pmb { a } } _ { i } ^ { l } \\in \\mathbb { R } ^ { h w }$ is the vectorized $A _ { i } ^ { l }$ . In such a scenario, the linear independence of each vectorized feature map $\\mathbf { \\Delta } _ { \\mathbf { \\alpha } \\mathbf { \\beta } _ { i } } ^ { \\mathbf { \\alpha } _ { i } }$ , as a row of the matricized entire set of feature maps $A ^ { l }$ , can be measured via the existing matrix analysis tool. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/bdedde3b59005664ff3eb98fde5f577fd78c3340addb81074a5448bea585b5f9.jpg",
+        "image_caption": [
+            "Figure 2: The change of (a) rank and (b) nuclear norm of the entire set of feature maps $( A ^ { l } )$ when one feature map $( \\pmb { a } _ { i } ^ { l } )$ is removed. The $\\mathbf { X }$ -axis represents the index of the feature map that is removed. The y-axis represents the corresponding rank/nuclear norm change of the entire set of feature maps. The feature maps are output from one layer of the ResNet-50 model with input as ImageNet image. It is seen that change of nuclear norm can better reveal the impact of the deleted feature map on the entire set of feature maps. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "The most straightforward solution is to use rank to determine the independence of $\\pmb { a } _ { i } ^ { l }$ , since rank mathematically represents the maximum number of linearly independent rows/columns of the matrix. For instance, we can remove one row from the matrix, and calculate the rank change of the matrix, and then identify the impact and the importance of the deleted row – the less rank change, the less independence (and the importance) of the removed row. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our Proposal. However, in the context of filter pruning, we believe, the change of nuclear norm of the entire set of feature maps, is a better metric to quantify the independence of each feature map. This is because, as the $\\overline { { \\ell _ { 1 } } }$ -norm of singular values of the matrix, the nuclear norm can reveal richer \"soft\" information on the impact of the deleted row on the matrix; while the rank, as the $\\ell _ { 0 }$ -norm of the singular values, is too \"hard\" to reflect such change. For instance, as shown in Fig. 2, when we select $\\mathbf { \\bar { \\mathbf { a } } } _ { i } ^ { l }$ , as one row of $A ^ { l }$ , to be removed, the rank change of $A ^ { l }$ is almost the same regardless of our selection of $\\mathbf { \\Delta } \\mathbf { a } _ { i } ^ { l }$ ; while the corresponding changes of nuclear norm vary significantly when different $\\mathbf { \\Delta } \\mathbf { a } _ { i } ^ { l }$ are deleted. Therefore, the change of nuclear norm can be viewed as a more precise metric to measure the linear independence of one feature map in a more fine-grained way. In general, the channel independence of one feature map is defined and calculated as below: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Channel independence of single feature map) For the i-th layer with output feature maps $\\pmb { \\mathcal { A } } ^ { l } = \\{ \\pmb { A } _ { 1 } ^ { l } , \\pmb { A } _ { 2 } ^ { l } , \\pmb { \\cdot \\cdot } \\cdot , \\pmb { A } _ { c ^ { l } } ^ { l } \\} \\in \\mathbb { R } ^ { c ^ { l } \\times h \\times w }$ , the Channel Independence $( C I )$ of one feature map $\\pmb { A } _ { i } ^ { l } \\in \\mathbb { R } ^ { h \\times w }$ in the $i$ -th channel is defined and calculated as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efe290a6627c2699e2d675a1211bca006c03df5c5dc92f6555b817cf0f6f3e44.jpg",
+        "text": "$$\nC I ( \\boldsymbol { A } _ { i } ^ { l } ) \\triangleq \\| \\boldsymbol { A } ^ { l } \\| _ { * } - \\| \\boldsymbol { M } _ { i } ^ { l } \\odot \\boldsymbol { A } ^ { l } \\| _ { * } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            618,
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+            637
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { A } ^ { l } \\in \\mathbb { R } ^ { c ^ { l } \\times h w }$ is the matricized $\\mathbf { \\mathcal { A } } ^ { l } , \\parallel \\cdot \\parallel _ { * }$ is the nuclear norm, $\\odot$ is the Hadamard product, and $M _ { i } ^ { l } \\in \\mathbb { R } ^ { c ^ { l } \\times h w }$ is the row mask matrix whose $i$ -th row entries are zeros and other entries are ones. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Question #2: What is the proper scheme to quantify the independence of multiple feature maps? ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Analysis. Eq. 3 describes the measurement of channel independence for a single feature map. However, in practice filter pruning typically aims to remove multiple filters, which means the independence of the combination of multiple feature maps needs to be calculated. In the case of pruning $m$ filters, such scenario corresponds to checking the changes of nuclear norm of the original $c ^ { l }$ -row $A ^ { l }$ after removing $m$ rows $( \\pmb { a } _ { i } ^ { l } )$ . A straightforward solution is to just calculate $C _ { m } ^ { c ^ { l } }$ changes of nuclear norms for all the possible $m$ -row removal choices, and then select the one which corresponds to the smallest change. However, this strategy is very computationally expensive, and sometimes even intractable when $c ^ { l }$ is large. For instance, in order to identify the smallest nuclear norm change for pruning $5 0 \\%$ filters of a 256-output channel ResNet-50 layer, such brutal-force measurement requires more than $5 \\times 1 0 ^ { 7 5 }$ times of nuclear norm calculation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our Proposal. To address this computational challenge, we propose to leverage the independence of individual feature map to approximate the independence of their combination. To be specific, in order to determine $m$ least independent rows in the $A ^ { l }$ , we first iteratively remove one row $\\mathbf { \\bar { \\rho } } ( \\mathbf { \\pmb { a } } _ { i } ^ { l } )$ from $A ^ { l }$ and calculate the corresponding nuclear norm change between the remaining $( c ^ { l } - 1 )$ -row matrix and the original $c ^ { l }$ -row $A ^ { l }$ . Then, among the $c ^ { l }$ calculated changes, we identify the $m$ smallest ones and the corresponding removed $\\mathbf { \\Delta } \\mathbf { a } _ { i } ^ { l }$ . Those selected $m$ vectorized feature maps $\\mathbf { \\Delta } \\mathbf { a } _ { i } ^ { l }$ are interpreted as the less independent from other feature maps, and hence their corresponding filters $\\mathcal { F } _ { i } ^ { l }$ are less important ones that should be pruned. In general, this individual independence-based measurement can closely approximate the combined independence of multiple feature maps (see Definition 2). Such approximation requires much less computational complexity (reduction from $\\mathcal { O } ( C ( N , \\kappa ) )$ to $\\mathcal { O } ( N ) )$ ; while still achieving superior filter pruning performance (see Section 4 for evaluation results). ",
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+    {
+        "type": "image",
+        "img_path": "images/addd57364a5979c6538f101cbab3bac8a73465404aeaf4b35049b07786699385.jpg",
+        "image_caption": [
+            "Figure 3: Example of filter pruning process using the change of nuclear norm-based channel independence (CI) criterion. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Definition 2 (Approximated channel independence of combined multiple feature maps) For the $i$ -th layer with output feature maps $\\pmb { \\mathcal { A } } ^ { l } \\in \\mathbb { R } ^ { c ^ { l } \\times h \\times w }$ , the channel independence of combined $m$ feature maps $\\{ A _ { b _ { i } } ^ { l } \\} _ { i = 1 } ^ { m }$ , where $A _ { b _ { i } } ^ { l } \\in \\mathbb { R } ^ { h \\times w }$ is in the $b _ { i }$ -th channel, is defined and approximated as: ",
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+        "type": "equation",
+        "img_path": "images/e9148721fa32fafe3497155cb0b996e7e9fe12f54d6f038748b2db67361fd50c.jpg",
+        "text": "$$\nC I ( \\{ A _ { b _ { i } } ^ { l } \\} _ { i = 1 } ^ { m } ) \\triangleq \\| A ^ { l } \\| _ { * } - \\| M _ { b _ { 1 } , \\cdots , b _ { m } } ^ { l } \\odot A ^ { l } \\| _ { * } \\approx \\sum _ { i = 1 } ^ { m } C I ( A _ { b _ { i } } ^ { l } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $M _ { b _ { 1 } , \\cdots , b _ { m } } ^ { l }$ is the multi-row mask matrix, in which the $b _ { 1 } , \\cdots , b _ { m }$ -th row entries are zeros and all the other entries are ones. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Question #3: How is the sensitiveness of channel independence related to the distribution of input data? ",
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+    {
+        "type": "text",
+        "text": "Our Observation. Consider our proposed channel independence-based filter pruning is a data-driven approach, its reliability with different distributions of input data should be carefully ensured and examined. To that end, we perform empirical evaluations on channel independence with respect to multiple input images. We observe that the average channel independence of each feature map is very stable at the batch level. In other words, we can simply input small batches of image samples, and calculate the average channel independence, and then such averaged channel independence with a small number of input data can be used to estimate the channel independence with all the input data. As illustrated in Fig. 4, for the same feature map, the average channel independence in different batches remains very similar, thereby indicating that our channel independence-based approach is robust against different input data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Question #4: Is this one-shot importance determination scheme good enough? Do we need to further learn and adjust the pruning mask from the data? ",
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+    {
+        "type": "text",
+        "text": "Our Observation. As described above, our proposed scheme calculates the channel independence to identify the filter importance. Considering our approach is built on one-shot calculation, a natural extension is to further adjust the importance ranking via additional learning. To be specific, if we interpret the filter pruning is a channel-wise masking operation over the entire weight tensor, the ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2b60805cf00b69baba7e728bce7dc09ad7a7b7adfe70f2772e4030fda785d574.jpg",
+        "image_caption": [
+            "Figure 4: The channel independence of feature maps for one layer in ResNet-50. Here the channel independence is averaged for one batch of input images. The $\\mathbf { X }$ -axis is the index of the feature map. The y-axis is the index of batches of input images. Here the batch size is 128. Different colors denote the different values of channel independence. It is seen that the average channel independence is very stable regardless of different input data batches. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 CHannel Independence-based Pruning (CHIP) procedure for the $l$ -th layer ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Input: Pre-trained weight tensor $\\boldsymbol { w ^ { l } }$ , $N$ sets of feature maps $\\pmb { \\mathcal { A } } ^ { l } = \\{ \\pmb { A } _ { 1 } ^ { l } , \\pmb { A } _ { 2 } ^ { l } , \\cdot \\cdot \\cdot , \\pmb { A } _ { c ^ { l } } ^ { l } \\} \\in \\mathbb { R } ^ { c ^ { l } \\times h \\times w }$ from $N$ input samples, and the desired number of filters to be preserved $\\kappa ^ { l }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Output: Pruned weight tensor $\\boldsymbol { \\mathcal { W } } _ { p r u n e } ^ { l }$ .   \n1: for each input sample do   \n2: Flatten feature maps: $\\pmb { A } ^ { l } : = \\mathrm { r e s h a p e } ( \\pmb { A } ^ { l } , [ c ^ { l } , h w ] ) ;$ ;   \n3: for $i = 1$ to $c ^ { l }$ do   \n4: CI calculation: Calculate $C I ( A _ { i } ^ { l } )$ via Equation 3;   \n5: end for   \n6: end for   \n7: Averaging: Average $C I ( A _ { i } ^ { l } )$ under all $N$ input samples;   \n8: Sorting: Sort $\\{ C I ( A _ { i } ^ { l } ) \\} _ { i = 1 } ^ { c ^ { l } }$ in ascending order;   \n9: Pruning: Prune $c ^ { l } - \\kappa ^ { l }$ filters in $\\boldsymbol { w ^ { l } }$ corresponding to the $c ^ { l } - \\kappa ^ { l }$ smallest $C I ( A _ { i } ^ { l } )$ ;   \n10: Fine-tuning: Obtain final $\\boldsymbol { \\mathcal { W } } _ { p r u n e } ^ { l }$ via fine-tuning $\\boldsymbol { w ^ { l } }$ with removing the pruned filter channels. ",
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+    },
+    {
+        "type": "text",
+        "text": "selection of channel mask can be learned from data, and such learning process can use the pruning mask determined by our approach as the initialization. Though in principle this learning-based strategy is expected to enable additional performance improvement, our empirical evaluations show that the consecutive learning procedure does not easily bring further accuracy increase (with the target compression ratio) or compression ratio increase (with the target accuracy) – more experimental details are reported in Supplementary Material. We hypothesize the reason for such phenomenon is that, our proposed nuclear norm change-based channel independence, though only requires one-time calculation, already identifies and captures the importance of feature maps (and its corresponding filters) with high quality, and hence further learning-based adjustment of pruning mask does not easily provide additional improvement. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Overall Algorithm. After addressing the above four problems, we can then integrate our proposals and observations to develop the entire filter pruning procedure from the inter-channel perspective. Algorithm 1 describes and summarizes the overall scheme for our proposed CHannel Independence-based filter Pruning (CHIP) algorithm. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 Experimental Settings ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Baselines Models and Datasets. To demonstrate the effectiveness and generality of our proposed channel independence-based approach, we evaluate its pruning performance for various baseline models on different image classification datasets. To be specific, we conduct experiments for three CNN models (ResNet-56, ResNet-110 and VGG-16) on CIFAR-10 dataset [24]. Also, we further evaluate our approach and compare its performance with other state-of-the-art pruning methods for ResNet-50 model on large-scale ImageNet dataset [5]. ",
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+    {
+        "type": "text",
+        "text": "Pruning and Fine-tuning Configurations. We conduct our empirical evaluations on Nvidia Tesla V100 GPUs with PyTorch 1.7 framework. To determine the importance of each filter, we randomly sample 5 batches (640 input images) to calculate the average channel independence of each feature map in all the experiments. After performing the channel independence-based filter pruning, we then perform fine-tuning on the pruned models with Stochastic Gradient Descent (SGD) as the optimizer. To be specific, we perform the fine-tuning for 300 epochs on CIFAR-10 datasets with the batch size, momentum, weight decay and initial learning rate as 128, 0.9, 0.05 and 0.01, respectively. On the ImageNet dataset, fine-tuning is performed for 180 epochs with the batch size, momentum, weight decay and initial learning rate as 256, 0.99, 0.0001 and 0.1, respectively. ",
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+        "table_caption": [
+            "Table 1: Experimental results on CIFAR-10 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Top-1 Accuracy (%) △</td><td rowspan=\"2\">#Params. (↓%)</td><td rowspan=\"2\">FLOPs (↓%)</td></tr><tr><td>Baseline</td><td>Pruned</td></tr><tr><td colspan=\"6\">ResNet-56</td></tr><tr><td>l1-norm (2016)[27]</td><td>93.04</td><td>93.06</td><td>+0.02</td><td>0.73M(13.7)</td><td>90.90M(27.6)</td></tr><tr><td>NISP (2018) [59]</td><td>93.04</td><td>93.01</td><td>-0.03</td><td>0.49M(42.4)</td><td>81.00M(35.5)</td></tr><tr><td>GAL (2019) [32]</td><td>93.26</td><td>93.38</td><td>+0.12</td><td>0.75M(11.8)</td><td>78.30M(37.6)</td></tr><tr><td>HRank (2020) [31]</td><td>93.26</td><td>93.52</td><td>+0.26</td><td>0.71M(16.8)</td><td>88.72M(29.3)</td></tr><tr><td>CHIP (Ours)</td><td>93.26</td><td>94.16</td><td>+0.90</td><td>0.48M(42.8)</td><td>65.94M(47.4)</td></tr><tr><td>GAL(2019)[32]</td><td>93.26</td><td>91.58</td><td>-1.68</td><td>0.29M(65.9)</td><td>49.99M(60.2)</td></tr><tr><td>LASSO (2017) [21]</td><td>92.80</td><td>91.80</td><td>-1.00</td><td>N/A</td><td>62.00M(50.6)</td></tr><tr><td>HRank (2020)[31]</td><td>93.26</td><td>90.72</td><td>-2.54</td><td>0.27M(68.1)</td><td>32.52M(74.1)</td></tr><tr><td>CHIP (Ours)</td><td>93.26</td><td>92.05</td><td>-1.21</td><td>0.24M(71.8)</td><td>34.79M(72.3)</td></tr><tr><td colspan=\"6\">ResNet-110</td></tr><tr><td>l1-norm (2016) [27]</td><td>93.53</td><td>93.30</td><td>-0.23</td><td>1.16M(32.4)</td><td>155.00M(38.7)</td></tr><tr><td>HRank (2020)[31]</td><td>93.50</td><td>94.23</td><td>+0.73</td><td>1.04M(39.4)</td><td>148.70M(41.2)</td></tr><tr><td>CHIP (Ours)</td><td>93.50</td><td>94.44</td><td>+0.94</td><td>0.89M(48.3)</td><td>121.09M(52.1)</td></tr><tr><td>GAL (2019)[32]</td><td>93.50</td><td>92.74</td><td>-0.76</td><td>0.95M(44.8)</td><td>130.20M(48.5)</td></tr><tr><td>HRank (2020)[31]</td><td>93.50</td><td>92.65</td><td>-0.85</td><td>0.53M(68.7)</td><td>79.30M(68.6)</td></tr><tr><td>CHIP (Ours)</td><td>93.50</td><td>93.63</td><td>+0.13</td><td>0.54M(68.3)</td><td>71.69M(71.6)</td></tr><tr><td colspan=\"6\">VGG-16</td></tr><tr><td>SSS (2018) [23]</td><td>93.96</td><td>93.02</td><td>-0.94</td><td>3.93M(73.8)</td><td>183.13M(41.6)</td></tr><tr><td>GAL (2019) [32]</td><td>93.96</td><td>93.77</td><td>-0.19</td><td>3.36M(77.6)</td><td>189.49M(39.6)</td></tr><tr><td>HRank (2020)[31]</td><td>93.96</td><td>93.43</td><td>-0.53</td><td>2.51M(82.9)</td><td>145.61M(53.5)</td></tr><tr><td>CHIP (Ours)</td><td>93.96</td><td>93.86</td><td>-0.10</td><td>2.76M(81.6)</td><td>131.17M(58.1)</td></tr><tr><td>GAL (2019)[32]</td><td>93.96</td><td>93.42</td><td>-0.54</td><td>2.67M(82.2)</td><td>171.89M(45.2)</td></tr><tr><td>HRank (2020) [31]</td><td>93.96</td><td>92.34</td><td>-1.62</td><td>2.64M(82.1)</td><td>108.61M(65.3)</td></tr><tr><td>CHIP (Ours)</td><td>93.96</td><td>93.72</td><td>-0.24</td><td>2.50M(83.3)</td><td>104.78M(66.6)</td></tr><tr><td>HRank (2020) [31]</td><td>93.96</td><td>91.23</td><td>-2.73</td><td>1.78M(92.0)</td><td>73.70M(76.5)</td></tr><tr><td>CHIP (Ours)</td><td>93.96</td><td>93.18</td><td>-0.78</td><td>1.90M(87.3)</td><td>66.95M(78.6)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "type": "text",
+        "text": "4.2 Evaluation and Comparison on CIFAR-10 Dataset ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Table 1 shows the evaluation results of the pruned ResNet-56, ResNet-110 and VGG-16 models on CIFAR-10 dataset. To be consistent with prior works, we evaluate the performance for two scenarios: targeting high accuracy and targeting high model size and FLOPs reductions. ",
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+        "type": "text",
+        "text": "ResNet-56. For ResNet-56 model, our channel independence-based approach can bring $0 . 9 0 \\%$ accuracy increase over the baseline model with $4 2 . 8 \\%$ and $4 7 . 4 \\%$ model size and FLOPs reductions, respectively. When we adopt aggressive compression with $7 1 . 8 \\%$ and $7 2 . 3 \\%$ model size and FLOPs reductions, we can still achieve high performance – our solution enables $1 . 3 3 \\%$ higher accuracy than HRank [31] with the similar model size and computational costs. ",
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+    {
+        "type": "text",
+        "text": "ResNet-110. For ResNet-110 model, our approach can bring $0 . 9 4 \\%$ accuracy increase over the baseline model with $4 8 . 3 \\%$ and $5 2 . 1 \\%$ model size and FLOPs reductions, respectively. When we perform aggressive pruning with $6 8 . 3 \\%$ and $7 1 . 6 \\%$ model size and FLOPs reductions, our pruned model can still achieve $0 . 1 \\hat { 3 } \\%$ higher accuracy over the baseline model. ",
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+    {
+        "type": "text",
+        "text": "VGG-16. For VGG-16 model, our approach can bring $8 1 . 6 \\%$ and $5 8 . 1 \\%$ model size and FLOPs reductions, respectively, with only $0 . { \\bar { 1 } } \\%$ accuracy drop. Moreover, with $8 3 . 3 \\%$ and $6 6 . 6 \\%$ storage and computational cost reductions, our pruned model can achieve $1 . 3 8 \\%$ higher accuracy than the model using other pruning approaches under a similar compression ratio. For even higher FLOPs reduction $( 7 8 . 6 \\% )$ ), our method can bring nearly $2 \\%$ accuracy increase over the prior works. ",
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+    {
+        "type": "text",
+        "text": "4.3 Evaluation and Comparison on ImageNet Dataset ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 2 summarizes the pruning performance of our approach for ResNet-50 on ImageNet dataset. It is seen that when targeting a moderate compression ratio, our approach can achieve $4 0 . 8 \\%$ and $4 4 . 8 \\%$ storage and computation reductions, respectively, with $0 . 1 5 \\%$ accuracy increase over the baseline model. When we further increase the compression ratio, our approach still achieves superior performance than state-of-the-art works. For instance, compared with SCOP [51], our approach shows higher accuracy $\\left( 0 . 1 2 \\% \\right)$ in moderate compression region and the same accuracy in high compress region; while meanwhile enjoying a much smaller model size and fewer FLOPs. ",
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+        "img_path": "images/47f84a66630f3cc912e6386edbcbf57d166e3fa647873a50409dba22ae36a83e.jpg",
+        "table_caption": [
+            "Table 2: Experimental results on ImageNet dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Top-1 Accuracy (%)</td><td colspan=\"5\">Top-5 Accuracy (%) Params.FLOPs</td></tr><tr><td></td><td>Baseline Pruned</td><td>△</td><td>Baseline</td><td>Pruned</td><td>△</td><td>↓(%)</td><td>↓(%)</td></tr><tr><td colspan=\"10\">ResNet-50</td></tr><tr><td>ThiNet (2017) [37]</td><td>72.88</td><td></td><td>72.04-0.84</td><td>91.14</td><td>90.67</td><td>-0.47</td><td>33.7</td><td>36.8</td></tr><tr><td>SFP (2018)[19]</td><td>76.15</td><td>74.61</td><td>-1.54</td><td>92.87</td><td>92.06</td><td>-0.81</td><td>N/A</td><td>41.8</td></tr><tr><td>Autopruner (2020) [36]</td><td>76.15</td><td>74.76</td><td>-1.39</td><td>92.87</td><td>92.15</td><td>-0.72</td><td>N/A</td><td>48.7</td></tr><tr><td>FPGM (2019)[20]</td><td>76.15</td><td>75.59</td><td>-0.56</td><td>92.87</td><td>92.63</td><td>-0.24</td><td>37.5</td><td>42.2</td></tr><tr><td>Taylor (2019) [38]</td><td>76.18</td><td>74.50</td><td>-1.68</td><td>N/A</td><td>N/A</td><td>N/A</td><td>44.5</td><td>44.9</td></tr><tr><td>C-SGD (2019) [7]</td><td>75.33</td><td>74.93</td><td>-0.40</td><td>92.56</td><td>92.27</td><td>-0.29</td><td>N/A</td><td>46.2</td></tr><tr><td>GAL (2019) [32]</td><td>76.15</td><td>71.95</td><td>-4.20</td><td>92.87</td><td>90.94</td><td>-1.93</td><td>16.9</td><td>43</td></tr><tr><td>RRBP (2019)[61]</td><td>76.10</td><td>73.00</td><td>-3.10</td><td>92.90</td><td>91.00</td><td>-1.90</td><td>N/A</td><td>54.5</td></tr><tr><td>PFP (2020)[30]</td><td>76.13</td><td>75.91</td><td>-0.22</td><td>92.87</td><td>92.81</td><td>-0.06</td><td>18.1</td><td>10.8</td></tr><tr><td>HRank (2020) [31]</td><td>76.15</td><td>74.98</td><td>-1.17</td><td>92.87</td><td>92.33</td><td>-0.54</td><td>36.6</td><td>43.7</td></tr><tr><td>SCOP (2020) [51]</td><td>76.15</td><td>75.95</td><td>-0.20</td><td>92.87</td><td>92.79</td><td>-0.08</td><td>42.8</td><td>45.3</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>76.30</td><td>+0.15</td><td>92.87</td><td>93.02</td><td>+0.15</td><td>40.8</td><td>44.8</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>76.15</td><td>0.00</td><td>92.87</td><td>92.91</td><td>+0.04</td><td>44.2</td><td>48.7</td></tr><tr><td>PFP(2020)[30]</td><td>76.13</td><td>75.21</td><td>-0.92</td><td>92.87</td><td>92.43</td><td>-0.44</td><td>30.1</td><td>44</td></tr><tr><td>SCOP (2020) [51]</td><td>76.15</td><td>75.26</td><td>-0.89</td><td>92.87</td><td>92.53</td><td>-0.34</td><td>51.8</td><td>54.6</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>75.26</td><td>-0.89</td><td>92.87</td><td>92.53</td><td>-0.34</td><td>56.7</td><td>62.8</td></tr><tr><td>HRank(2020)[31]</td><td>76.15</td><td>71.98</td><td>-4.17</td><td>92.87</td><td>91.01</td><td>--1.86</td><td>46.0</td><td>62.1</td></tr><tr><td>HRank (2020) [31]</td><td>76.15</td><td>69.10</td><td>-7.05</td><td>92.87</td><td>89.58</td><td>-3.29</td><td>67.5</td><td>76.0</td></tr><tr><td>CHIP (Ours)</td><td>76.15</td><td>73.30</td><td>-2.85</td><td>92.87</td><td>91.48</td><td>-1.39</td><td>68.6</td><td>76.7</td></tr></table>",
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+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we propose to use channel independence, an inter-channel perspective-motivated metric, to evaluate the importance of filters for network pruning. By systematically exploring the quantification metric, measuring scheme, and sensitiveness and reliability of channel independence, we develop CHIP, a CHannel Independence-based filter pruning for neural network compression. Extensive evaluation results on different datasets show our proposed approach brings significant storage and computational cost reductions while still preserving high model accuracy. ",
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+    {
+        "type": "text",
+        "text": "Broader Impact ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As technology advances, cell phones, laptops, wearable gadgets and intelligent connected vehicles with specific chips are required to handle more complicated tasks by deploying neural networks. However, more powerful networks will cost more memory size and running time. Network pruning is the main strategy to reduce the memory size and accelerate the run-time during the inference stage. Benefiting from pruning techniques and specific designs for hardware [62, 4], IoT (Internet of Things) devices are able to execute complex projects based on small and efficient models. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements and Funding Disclosure ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Bo Yuan would like to thank the support from National Science Foundation (NSF) award CCF1937403. Saman Zonouz would like to thank the support from NSF CPS and SATC programs. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "type": "text",
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A systematic dnn weight pruning framework using alternating direction method of multipliers. In Proceedings of the European Conference on Computer Vision (ECCV), pages 184–199, 2018.   \n[61] Yuefu Zhou, Ya Zhang, Yanfeng Wang, and Qi Tian. Accelerate cnn via recursive bayesian pruning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 3306–3315, 2019.   \n[62] Chaoyang Zhu, Kejie Huang, Shuyuan Yang, Ziqi Zhu, Hejia Zhang, and Haibin Shen. An efficient hardware accelerator for structured sparse convolutional neural networks on fpgas. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 28(9):1953–1965, 2020.   \n[63] Zhuangwei Zhuang, Mingkui Tan, Bohan Zhuang, Jing Liu, Yong Guo, Qingyao Wu, Junzhou Huang, and Jinhui Zhu. Discrimination-aware channel pruning for deep neural networks. In Advances in Neural Information Processing Systems, pages 875–886, 2018. ",
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parse/train/rylztAEYvr/rylztAEYvr.md

@@ -0,0 +1,269 @@
+# ITERATIVE TARGET AUGMENTATION FOR EFFECTIVE CONDITIONAL GENERATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Many challenging prediction problems, from molecular optimization to program synthesis, involve creating complex structured objects as outputs. However, available training data may not be sufficient for a generative model to learn all possible complex transformations. By leveraging the idea that evaluation is easier than generation, we show how a simple, broadly applicable, iterative target augmentation scheme can be surprisingly effective in guiding the training and use of such models. Our scheme views the generative model as a prior distribution, and employs a separately trained filter as the likelihood. In each augmentation step, we filter the model’s outputs to obtain additional prediction targets for the next training epoch. Our method is applicable in the supervised as well as semi-supervised settings. We demonstrate that our approach yields significant gains over strong baselines both in molecular optimization and program synthesis. In particular, our augmented model outperforms the previous state-of-the-art in molecular optimization by over $10 \%$ in absolute gain.
+
+# 1 INTRODUCTION
+
+Deep architectures are becoming increasingly adept at generating complex objects such as images, text, molecules, or programs. Many useful generation problems can be seen as translation tasks, where the goal is to take a source (precursor) object such as a molecule and turn it into a target satisfying given design characteristics. Indeed, molecular optimization of this kind is a key step in drug development, though the adoption of automated tools remains limited due to accuracy concerns. We propose here a simple, broadly applicable meta-algorithm to improve translation quality.
+
+Translation is a challenging task for many reasons. Objects are complex and the available training data pairs do not fully exemplify the intricate ways in which valid targets can be created from the precursors. Moreover, precursors provided at test time may differ substantially from those available during training — a scenario common in drug development. While data augmentation and semisupervised methods have been used to address some of these challenges, the focus has been on either simple prediction tasks (e.g., classification) or augmenting data primarily on the source side. We show, in contrast, that iteratively augmenting translation targets significantly improves performance on complex generation tasks in which each precursor corresponds to multiple possible outputs.
+
+Our iterative target augmentation approach builds on the idea that it is easier to evaluate candidate objects than to generate them. Thus a learned predictor of target object quality (a filter) can be used to effectively guide the generation process. To this end, we construct an external filter and apply it to the complex generative model’s sampled translations of training set precursors. Candidate translations that pass the filter criteria become part of the training data for the next training epoch. The translation model is therefore iteratively guided to generate candidates that pass the filter. The generative model can be viewed as an adaptively tuned prior distribution over complex objects, with the filter as the likelihood. For this reason, it is helpful to apply the filter at test time as well, or to use the approach transductively1 to adapt the generation process to novel test cases. The approach is reminiscent of self-training or reranking approaches employed with some success for parsing (McClosky et al., 2006; Charniak et al., 2016). However, in our case, it is the candidate generator that is complex while the filter is relatively simple and remains fixed during the iterative process.
+
+We demonstrate that our meta-algorithm is quite effective and consistent in its ability to improve translation quality in the supervised setting. On a program synthesis task (Bunel et al., 2018), under the same neural architecture, our augmented model outperforms their MLE baseline by $8 \%$ and their RL model by $3 \%$ in top-1 generalization accuracy (in absolute measure). On molecular optimization (Jin et al., 2019a), their sequence to sequence translation baseline, when combined with our target data augmentation, achieves a new state-of-the-art result and outperforms their graph based approach by over $10 \%$ in success rate. Their graph based methods are also improved by iterative target augmentation with more than $10 \%$ absolute gain. The results reflect the difficulty of generation in comparison to evaluation; indeed, the gains persist even if the filter quality is reduced somewhat. Source side augmentation with unlabeled precursors (the semi-supervised setting) can further improve results, but only when combined with the filter in the target data augmentation framework. We provide ablation experiments to empirically highlight the effect of our method and also offer some theoretical insights for why it is effective.
+
+# 2 RELATED WORK
+
+Molecular Optimization The goal of molecular optimization is to learn to modify compounds so as to improve their chemical properties. Jaques et al. (2017); You et al. (2018); Popova et al. (2018) used reinforcement learning approaches, while Jin et al. (2019a;b) formulated this problem as graphto-graph translation and significantly outperformed previous methods. However, their performance remains imperfect due to the limited size of given training sets. Our work uses property prediction models to check whether generated molecules have desired chemical properties. Recent advances in graph convolutional networks (Duvenaud et al., 2015; Gilmer et al., 2017) have provided effective solutions to predict those properties in silico. In this work, we use an off-the-shelf property prediction model (Yang et al., 2019) to filter proposed translation pairs during data augmentation.
+
+Program Synthesis Program synthesis is the task of generating a program (using domain-specific language) based on given input-output specifications (Bunel et al., 2018; Gulwani, 2011; Devlin et al., 2017). One can check a generated program’s correctness by simply executing it on each input and verifying its output. Indeed, Zhang et al. (2018); Chen et al. (2019) leverage this idea in their respective decoding procedures, while also using structural constraints on valid programs.
+
+Semi-supervised Learning Our method is related to various approaches in semi-supervised learning. In image and text classification, data augmentation and label guessing (Berthelot et al., 2019; Xie et al., 2019) are commonly applied to obtain artificial labels for unlabeled data. In machine translation, Norouzi et al. (2016) sample new targets from a stationary distribution in order to match the model distribution to the exponentiated payoff distribution centered at a single target sentence. Back-translation (Sennrich et al., 2015; Edunov et al., 2018) creates extra translation pairs by using a backward translation system to translate unlabeled sentences from a target language into a source language. In contrast, our method works in the forward direction because many translation tasks are not symmetric. Moreover, our data augmentation is carried out over multiple iterations, in which we use the augmented model to generate new data for the next iteration.
+
+In syntactic parsing, our method is closely related to self-training (McClosky et al., 2006). They generate new parse trees from unlabeled sentences by applying an existing parser followed by a reranker, and then treat the resulting parse trees as new training targets. However, their method is not iterative, and their reranker is explicitly trained to operate over the top $k$ outputs of the parser; in contrast, our filter is independent of the generative model. In addition we show that our approach, which can be viewed as iteratively combining reranking and self-training, is theoretically motivated and can improve the performance of highly complex neural models in multiple domains. Co-training (Blum & Mitchell, 1998) and tri-training (Zhou & Li, 2005; Charniak et al., 2016) also augment a parsing dataset by adding targets on which multiple baseline models agree. Instead of using multiple learners, our method uses task-specific constraints to select correct outputs.
+
+# 3 ITERATIVE TARGET AUGMENTATION
+
+Our iterative target augmentation framework can be applied to any conditional generation task with task-specific constraints. For example, molecular optimization (Jin et al., 2019a;b) is the task of transforming a given molecule $X$ into another compound $Y$ with improved chemical properties, while constraining $Y$ to remain similar to $X$ . Program synthesis (Bunel et al., 2018; Chen et al., 2019) is the task of generating a program $Y$ satisfying input specification $X$ ; for example, $X$ may be a set of input-output test cases which $Y$ must pass.
+
+![](images/f33dc5c1c6d59e0a2c35da3ef721ec2824870ac6cd676d78a5bdb93341fc7263.jpg)  
+Figure 1: Illustration of our data generation process in the program synthesis setting. Given an input-output specification, we first use our generation model to generate candidate programs, and then select correct programs using our external filter. Images of input-output specification and the program A are from Bunel et al. (2018).
+
+<table><tr><td>Algorithm1 Augmentation by iterative target augmentation</td><td></td></tr><tr><td colspan="3">Input: Original training set D = [(X1, Yi),...,(Xn,Yn)]</td></tr><tr><td></td><td>1:procedure AUGMENTDATASET(D,Mt)</td><td></td></tr><tr><td>2:</td><td>Dt+1=D</td><td>&gt; Initialize augmented dataset.</td></tr><tr><td>3:</td><td>for (Xi,Yi) in D do</td><td></td></tr><tr><td>4:</td><td>for attempt in 1,.., C do</td><td></td></tr><tr><td>5:</td><td>Apply model Mt to Xi to sample candidate Y&#x27;</td><td></td></tr><tr><td>6:</td><td>if Y&#x27; passes external filter then</td><td></td></tr><tr><td>7:</td><td>Add (Xi,Y&#x27;) to Dt+1</td><td></td></tr><tr><td>8:</td><td>if K successful translations added then</td><td></td></tr><tr><td>9:</td><td>break from loop</td><td></td></tr><tr><td>10:</td><td>return augmented dataset Dt+1</td><td></td></tr><tr><td colspan="2">11: procedure TRAIN(D)</td><td></td></tr><tr><td>12:</td><td>for epoch in 1,..., n1 do</td><td> Regular training</td></tr><tr><td>13:</td><td>Train model on D.</td><td></td></tr><tr><td>14:</td><td>for epoch in 1,..., n2 do</td><td>&gt; Iterative target augmentation</td></tr><tr><td>15:</td><td>Dt+1 = AUGMENTDATASET(D,Mt)</td><td></td></tr><tr><td>16:</td><td>Mt+1 ← Train model Mt on Dt+1·</td><td></td></tr></table>
+
+Without loss of generality, we formulate the generation task as a translation problem. For a given input $X$ , the model learns to generate an output $Y$ satisfying the constraint $^ c$ . The proposed augmentation framework can be applied to any translation model $\mathcal { M }$ trained on an existing dataset $\boldsymbol { \mathcal { D } } = \{ ( X _ { i } , Y _ { i } ) \}$ . As illustrated in Figure 1, our method is an iterative procedure in which each iteration consists of the following two steps:
+
+• Augmentation Step: Let $\mathcal { D } _ { t }$ be the training set at iteration $t$ . To construct each next training set $\mathcal { D } _ { t + 1 }$ , we feed each input $X _ { i } \in \mathcal { D }$ (the original training set, not $\mathcal { D } _ { t }$ ) into the translation model up to $C$ times to sample $C$ candidate translations $Y _ { i } ^ { 1 } \ldots Y _ { i } ^ { \overline { { C } } }$ .2 We take the first $K$ distinct translations for each $X _ { i }$ satisfying the constraint $^ c$ and add them to $\mathcal { D } _ { t + 1 }$ . When we do not find $K$ distinct valid translations, we simply add the original translation $Y _ { i }$ to $\mathcal { D } _ { t + 1 }$ .
+
+• Training Step: We continue to train the model $\mathcal { M } _ { t }$ over the new training set $\mathcal { D } _ { t + 1 }$ for one epoch.
+
+The above training procedure is summarized in Algorithm 1. As the constraint $^ c$ is known a priori, we can construct an external filter to remove generated outputs that violate $^ c$ during the augmentation step. At test time, we also use this filter to screen predicted outputs. To propose the final translation of a given input $X$ , we have the model generate up to $L$ outputs until we find one satisfying the constraint $^ c$ . If all $L$ attempts fail for a particular input, we just output the first of the failed attempts.
+
+Finally, as an additional improvement, we observe that the augmentation step can be carried out for unlabeled inputs $X$ that have no corresponding $Y$ . Thus we can further augment our training dataset in the transductive setting by including test set inputs during the augmentation step, or in the semi-supervised setting by simply including unlabeled inputs.
+
+# 4 MOTIVATION FOR ITERATIVE TARGET AUGMENTATION
+
+We provide here some theoretical motivation for our iterative target augmentation framework. For simplicity, we consider an external filter $_ { c _ { X , Y } }$ that is a binary indicator function representing whether output $Y$ satisfies the desired constraint in relation to input $X$ . In other words, we would like to generate $Y$ such that $Y \in B ( X ) = \{ Y ^ { \prime } | c _ { X , Y ^ { \prime } } = 1 \}$ . If the initial translation model $P ^ { ( 0 ) } ( Y | X )$ serves as a reasonable prior distribution over outputs, we could simply “invert” the filter and use
+
+$$
+P ^ { ( * ) } ( Y | X ) \propto P ^ { ( 0 ) } ( Y | X ) \cdot c _ { X , Y }
+$$
+
+as the ideal translation model. While this posterior calculation is typically not feasible but could be approximated through samples, it relies heavily on the appropriateness of the prior (model prior to augmentation). Instead, we go a step further and iteratively optimize our parametrically defined prior translation model $P _ { \theta } ( Y | X )$ . Note that the resulting prior can become much more concentrated around acceptable translations.
+
+We maximize the log-likelihood that candidate translations satisfy the constraints implicitly encoded in the filter
+
+$$
+\mathbb { E } _ { X } \left[ \log P _ { \theta } ( \pmb { c } _ { X , Y } = 1 \mid X ) \right]
+$$
+
+In many cases there are multiple viable outputs for any given input $X$ . The training data may provide only one (or none) of them. Therefore, we treat the output structure $Y$ as a latent variable, and expand the inner term of Eq.(2) as
+
+$$
+\begin{array} { l l l } { \log P _ { \theta } ( \boldsymbol { c } _ { X , Y } = 1 \mid X ) } & { = } & { \displaystyle \log \sum _ { Y } P _ { \theta } ( Y , \boldsymbol { c } _ { X , Y } = 1 \mid X ) } \\ & { = } & { \displaystyle \log \sum _ { Y } P ( \boldsymbol { c } _ { X , Y } = 1 \mid Y , X ) P _ { \theta } ( Y \mid X ) } \\ & { = } & { \displaystyle \log \sum _ { Y } \boldsymbol { c } _ { X , Y } \cdot P _ { \theta } ( Y \mid X ) } \end{array}
+$$
+
+Since the above objective involves discrete latent variables $Y$ , we propose to maximize Eq.(5) using the standard EM algorithm (Dempster et al., 1977), especially its incremental, approximate variant. The target augmentation step in our approach is a sampled version of the $\mathrm { E }$ -step where the posterior samples are drawn with rejection sampling guided by the filter. The number of samples $K$ controls the quality of approximation to the posterior.3 The additional training step based on the augmented targets corresponds to a generalized M-step. More precisely, let $P _ { \theta } ^ { ( t ) } ( Y | X )$ be the current translation model after epochs of augmentation training. In epoch , the augmentation step first samples $C$ different candidates for each input $X$ using the old model $P ^ { ( t ) }$ parameterized by $\theta ^ { ( t ) }$ , and then removes those which violate the constraint $^ c$ , interpretable as samples from the current posterior $Q ^ { ( t ) } ( Y | X ) \propto P _ { \theta ^ { ( t ) } } ( Y | X ) c _ { X , Y }$ . As a result, the training step maximizes the EM auxiliary objective via stochastic gradient descent:
+
+$$
+J ( \theta \mid \theta ^ { ( t ) } ) = \mathbb { E } _ { X } \left[ \sum _ { Y } Q ^ { ( t ) } ( Y | X ) \log P _ { \theta } ( Y | X ) \right]
+$$
+
+We train the model with multiple iterations and show empirically that model performance indeed keeps improving as we add more iterations. The EM approach is likely to converge to a different and better-performing translation model than the initial posterior calculation discussed above.
+
+# 5 EXPERIMENTS
+
+We demonstrate the broad applicability of iterative target augmentation by applying it to two tasks of different domains: molecular optimization and program synthesis.
+
+![](images/1d6bfac9d14ffa73e428aa3b82e76d9eea78cf0494f4ad32d88c0ac75deafb3d.jpg)  
+Figure 2: Illustration of molecular optimization. Molecules can be modeled as graphs, with atoms as nodes and bonds as edges. Here, the task is to train a translation model to modify a given input molecule into a target molecule with higher drug-likeness (QED) score. The constraint has two components: the output $Y$ must be highly drug-like, and must be sufficiently similar to the input $X$ .
+
+# 5.1 MOLECULAR OPTIMIZATION
+
+The goal of molecular optimization is to learn to modify molecules so as to improve their chemical properties. As illustrated in Figure 2, this task is formulated as a graph-to-graph translation problem. Similar to machine translation, the training set is a set of molecular pairs $\{ ( X , Y ) \}$ . $X$ is the input molecule (precursor) and $Y$ is a similar molecule with improved chemical properties. Each molecule in the training set $\mathcal { D }$ is further labeled with its property score. Our method is well-suited to this task because the target molecule is not unique: each precursor molecule can be modified in many different ways to optimize its properties.
+
+External Filter The constraint for this task contains two parts: 1) the chemical property of $Y$ must exceed a certain threshold $\beta$ , and 2) the molecular similarity between $X$ and $Y$ must exceed a certain threshold $\delta$ . The molecular similarity $\sin ( X , Y )$ is defined as Tanimoto similarity on Morgan fingerprints (Rogers & Hahn, 2010), which measures structural overlap between two molecules.
+
+In real world settings, ground truth values of chemical properties are often evaluated through experimental assays, which are too expensive and time-consuming to run for iterative target augmentation. Therefore, we construct an in silico property predictor $F _ { 1 }$ to approximate the true property evaluator $F _ { 0 }$ . To train this property prediction model, we use the molecules in the training set and their labeled property values. The predictor $F _ { 1 }$ is parameterized as a graph convolutional network and trained using the Chemprop package (Yang et al., 2019). During data augmentation, we use $F _ { 1 }$ to filter out molecules whose predicted property is under the threshold $\beta$ .
+
+# 5.1.1 EXPERIMENTAL SETUP
+
+We follow the evaluation setup of Jin et al. (2019b) for two molecular optimization tasks:
+
+1. QED Optimization: The task is to improve the drug-likeness (QED) of a given compound $X$ . The similarity constraint is $\sin ( X , Y ) \bar { \geq } 0 . 4$ and the property constraint is $\bar { \mathrm { Q E D } } ( Y ) \stackrel { - } { = } 0 . 9$ , with $\mathrm { Q E D } ( Y ) \in \left[ 0 , 1 \right]$ defined by the system of Bickerton et al. (2012).
+
+2. DRD2 Optimization: The task is to optimize biological activity against the dopamine type 2 receptor (DRD2). The similarity constraint is $\sin ( \bar { X } , Y ) \ge 0 . 4 $ and the property constraint is $\mathrm { D R D 2 } ( Y ) \ge 0 . 5$ , where $\mathrm { D R D 2 } ( Y ) \in [ 0 , 1 ]$ is the predicted probability of biological activity given by the model from Olivecrona et al. (2017).
+
+We treat the output of the in silico evaluators from Bickerton et al. (2012) and Olivecrona et al. (2017) as ground truth, and we use them only during test-time evaluation.4
+
+Evaluation Metrics. During evaluation, we are interested both in the probability that the model will find a successful modification for a given molecule, as well as the diversity of the successful modifications when there are multiple. We translate each molecule in the test set $Z = 2 0$ times, resulting in candidate modifications $Y _ { 1 } \ldots Y _ { Z }$ (not necessarily distinct). We use the following two evaluation metrics:
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>QED Succ.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq</td><td rowspan=1 colspan=1>58.5</td><td rowspan=1 colspan=1>0.331</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>0.176</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+ (Ours)</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+, semi-supervised (Ours)</td><td rowspan=1 colspan=1>95.0</td><td rowspan=1 colspan=1>0.471</td><td rowspan=1 colspan=1>99.6</td><td rowspan=1 colspan=1>0.408</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+,transductive (Ours)</td><td rowspan=1 colspan=1>92.6</td><td rowspan=1 colspan=1>0.451</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>0.358</td></tr><tr><td rowspan=1 colspan=1>HierGNN</td><td rowspan=1 colspan=1>76.6</td><td rowspan=1 colspan=1>0.477</td><td rowspan=1 colspan=1>85.9</td><td rowspan=1 colspan=1>0.192</td></tr><tr><td rowspan=1 colspan=1>HierGNN+ (Ours)</td><td rowspan=1 colspan=1>93.1</td><td rowspan=1 colspan=1>0.514</td><td rowspan=1 colspan=1>97.6</td><td rowspan=1 colspan=1>0.418</td></tr></table>
+
+Table 1: Performance of different models on QED and DRD2 optimization tasks. Italicized models with $^ +$ are augmented with iterative target augmentation. We emphasize that iterative target augmentation remains critical to performance in the semi-supervised and transductive settings; data augmentation without an external filter instead decreases performance.
+
+1. Success: The fraction of molecules $X$ for which any of the outputs $Y _ { 1 } \ldots Y _ { Z }$ meet the required similarity and property constraints (specified previously for each task). This is our main metric.
+
+2. Diversity: For each molecule $X$ , we measure the average Tanimoto distance (defined as $1 -$ $\mathrm { s i m } ( Y _ { i } , Y _ { j } ) ,$ ) between pairs within the set of successfully translated compounds among $Y _ { 1 } \ldots Y _ { Z }$ . If there are one or fewer successful translations then the diversity is 0. We average this quantity across all test molecules.
+
+Models and Baselines. We consider the following two model architectures from Jin et al. (2019a) to show that our augmentation scheme is not tied to specific neural architectures.
+
+1. VSeq2Seq, a sequence-to-sequence translation model generating molecules by their SMILES string (Weininger, 1988). 2. HierGNN, a hierarchical graph-to-graph architecture that achieves state-of-the-art performance on the QED and DRD2 tasks, outperforming VSeq2Seq by a wide margin.
+
+We apply our iterative augmentation procedure to the above two models, generating up to $K = 4$ new targets per precursor during each epoch of iterative target augmentation. Additionally, we evaluate our augmentation of VSeq2Seq in a transductive setting, as well as in a semi-supervised setting where we provide 100K additional source-side precursors from the ZINC database (Sterling & Irwin, 2015). Full hyperparameters are in Appendix A.
+
+# 5.1.2 RESULTS
+
+As shown in Table 1, our iterative augmentation paradigm significantly improves the performance of VSeq2Seq and HierGNN. On both datasets, the translation success rate increases by over $10 \%$ in absolute terms for both models. In fact, ${ \tt V S e q 2 S e q + }$ , our augmentation of the simple VSeq2Seq model, outperforms the non-augmented version of HierGNN. This result strongly confirms our hypothesis about the inherent challenge of learning translation models in data sparse scenarios. Moreover, we find that adding more precursors during data augmentation further improves the VSeq2Seq model. On the QED dataset, the translation success rate improves from $8 9 . 0 \%$ to $9 2 . 6 \%$ by just adding test set molecules as precursors $( \mathrm { V S e q 2 S e q + }$ , transductive). When instead adding 100K presursors from the external ZINC database, the performance further increases to $9 5 . 0 \%$ $( \mathrm { V S e q 2 S e q + }$ , semisupervised). We observe similar improvements for the DRD2 task as well. Beyond accuracy gain, our augmentation strategy also improves the diversity of generated molecules. For instance, on the DRD2 dataset, our approach yields $100 \%$ relative gain in terms of output diversity.
+
+Importance of Property Predictor Although the property predictor used in data augmentation is different from the ground truth property evaluator used at test time, the difference in evaluators does not derail the overall training process. Here we analyze the influence of the quality of the property predictor used in data augmentation. Specifically, we rerun our experiments using less accurate predictors in the property-predicting component of our external filter. We obtain these less accurate predictors by undertraining Chemprop and decreasing its hidden dimension. For comparison, we also report results with the oracle property predictor which is the ground truth property evaluator.
+
+As shown in Figure 3, on the DRD2 dataset, we are able to maintain strong performance despite using predictors that deviate significantly from the ground truth. This implies that our framework can potentially be applied to other properties that are harder to predict. On the QED dataset, our
+
+![](images/999d43365de7d71b2b6c8124c3a40c734e5b26914b76bfcadc281fdbea734f9d.jpg)  
+Figure 3: Left: QED success rate vs. Chemprop predictor’s RMSE with respect to ground truth on test set. The red line shows the performance of the (unaugmented) VSeq2Seq baseline. Right: Same plot for DRD2. In each plot, the far left point with zero RMSE is obtained by reusing the ground truth predictor, while the second-from-left point is the Chemprop predictor we use to obtain our main results. Points further to the right are weaker predictors trained for fewer epochs and with less capacity, simulating a scenario where the property is more difficult to model.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>QED Succ.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>58.5</td><td rowspan=1 colspan=1>0.331</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>0.176</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq(test)</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>77.4</td><td rowspan=1 colspan=1>0.471</td><td rowspan=1 colspan=1>87.2</td><td rowspan=1 colspan=1>0.200</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq(train)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>81.8</td><td rowspan=1 colspan=1>0.430</td><td rowspan=1 colspan=1>92.2</td><td rowspan=1 colspan=1>0.321</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+</td><td rowspan=1 colspan=1>了</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq(no-filter)</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>47.5</td><td rowspan=1 colspan=1>0.297</td><td rowspan=1 colspan=1>51.0</td><td rowspan=1 colspan=1>0.185</td></tr></table>
+
+Table 2: Ablation analysis of filtering at training and test time. “Train” indicates a model whose training process uses data augmentation according to our framework. “Test” indicates a model that uses the external filter at prediction time to discard candidate outputs which fail to pass the filter. The evaluation for VSeq2Seq(no-filter) is conducted after 10 augmentation epochs, as the best validation set performance only decreases over the course of training.
+
+method is less tolerant to inaccurate property prediction because the property constraint is much tighter — it requires the QED score of an output $Y$ to be in the range [0.9, 1.0].
+
+Importance of External Filtering Our full model $( { \mathrm { V S e q } } 2 { \mathrm { S e q } } +$ ) uses the external filter during both training and testing. We further experiment with Vseq2seq(test), a version of our model trained without data augmentation but which uses the external filter to remove invalid outputs at test time. As shown in Table 2, VSeq2Seq(test) performs significantly worse than our full model trained under data augmentation. Similarly, a model VSeq2Seq(train) trained with the data augmentation but without the prediction time filtering also performs much worse than the full model.
+
+In addition, we run an augmentation-only version of the model without an external filter. This model (referred to as VSeq2Seq(no-filter) in Table 2) augments the data in each epoch by simply using the first $K$ distinct candidate translations for each precursor $X$ in the training set, without using the external filter at all. In addition, we provide this model with the 100K unlabeled precursors from the semi-supervised setting. Nevertheless, we find that the performance of this model steadily declines from that of the bootstrapped starting point with each data augmentation epoch. Thus the external filter is necessary to prevent poor targets from leading the model training astray.
+
+# 5.2 PROGRAM SYNTHESIS
+
+In program synthesis, the source is a set of input-output specifications for the program, and the target is a program that passes all test cases. Our method is suitable for this task because the target program is not unique. Multiple programs may be consistent with the given input-output specifications. The external filter is straightforward for this task: we simply check whether the generated output passes all test cases. Note that at evaluation time, each instance contains extra held-out input-output test cases; the program must pass these in addition to the given test cases in order to be considered correct. When we perform prediction time filtering, we do not use held-out test cases in our filter.
+
+Table 3: Model performance on Karel program synthesis task. $\mathrm { M L E + }$ is our augmented version of the MLE model (Bunel et al., 2018).   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Top-1Generalization</td></tr><tr><td rowspan=1 colspan=1>MLE (Bunel et al.,2018)</td><td rowspan=1 colspan=1>71.91</td></tr><tr><td rowspan=1 colspan=1>MLE+RL+BeamSearch(Bunel et al., 2018)</td><td rowspan=1 colspan=1>77.12</td></tr><tr><td rowspan=1 colspan=1>MLE+ (Ours)</td><td rowspan=1 colspan=1>80.17</td></tr></table>
+
+![](images/8cba7a23895f93e56d42a79eaa18a931a6a0954468ed34f6aabd4f2fba87f5b4.jpg)  
+Figure 4: Top-1 generalization accuracy of $\mathrm { M L E + }$ model on validation set of Karel task across different epochs.
+
+# 5.2.1 EXPERIMENTAL SETUP
+
+Our task is based on the educational Karel programming language (Pattis, 1981) used for evaluation in Bunel et al. (2018) and Chen et al. (2019). Commands in the Karel language guide a robot’s actions in a 2D grid, and may include for loops, while loops, and conditionals. Figure 1 contains an example. We follow the experiment setup of Bunel et al. (2018).
+
+Evaluation Metrics. The evaluation metric is top-1 generalization. This metric measures how often the model can generate a program that passes the input-output test cases on the test set. At test time, we use our model to generate up to $L$ candidate programs and select the first one to pass the input-output specifications (not including held-out test cases).
+
+Models and Baselines. Our main baseline is the MLE baseline from Bunel et al. (2018). This model consists of a CNN encoder for the input-output grids and a LSTM decoder along with a handcoded syntax checker. It is trained to maximize the likelihood of the provided target program. Our model is the augmentation of this MLE baseline by our iterative target augmentation framework. As with molecular optimization, we generate up to $K = 4$ new targets per precursor during each augmentation step. Additionally, we compare against the best model from Bunel et al. (2018), which finetunes the same MLE architecture using an RL method with beam search to estimate gradients.5 We use the same hyperparameters as the original MLE baseline; see Appendix A for details.
+
+# 5.2.2 RESULTS
+
+Table 3 shows the performance of our model in comparison to previous work. Our model (MLE+) outperforms the base MLE model in Bunel et al. (2018) model by a wide margin. Moreover, our model outperforms the best reinforcement learning model (RL $^ +$ Beam Search) in Bunel et al. (2018), which was trained to directly maximize the generalization metric. This demonstrates the efficacy of our approach in the program synthesis domain. Since our augmentation framework is complementary to architectural improvements, we hypothesize that other techniques, such as execution based synthesis (Chen et al., 2019), can benefit from our approach as well.
+
+# 6 CONCLUSION
+
+In this work, we have presented an iterative target augmentation framework for generation tasks with multiple possible outputs. Our approach is theoretically motivated, and we demonstrate strong empirical results on both the molecular optimization and program synthesis tasks, significantly outperforming baseline models on each task. Moreover, we find that iterative target augmentation is complementary to architectural improvements, and that its effect can be quite robust to the quality of the external filter. Finally, in principle our approach is applicable to other domains as well.
+
+# REFERENCES
+
+David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin Raffel. Mixmatch: A holistic approach to semi-supervised learning. arXiv preprint arXiv:1905.02249, 2019.
+
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+Mohammad Norouzi, Samy Bengio, Navdeep Jaitly, Mike Schuster, Yonghui Wu, Dale Schuurmans, et al. Reward augmented maximum likelihood for neural structured prediction. In Advances In Neural Information Processing Systems, pp. 1723–1731, 2016.
+
+Marcus Olivecrona, Thomas Blaschke, Ola Engkvist, and Hongming Chen. Molecular de-novo design through deep reinforcement learning. Journal of cheminformatics, 9(1):48, 2017.   
+Richard E. Pattis. Karel the Robot: A Gentle Introduction to the Art of Programming. John Wiley & Sons, Inc., New York, NY, USA, 1st edition, 1981. ISBN 0471089281.   
+Mariya Popova, Olexandr Isayev, and Alexander Tropsha. Deep reinforcement learning for de novo drug design. Science advances, 4(7):eaap7885, 2018.   
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+
+# A MODEL HYPERPARAMETERS
+
+Our augmented models share the same hyperparameters as their baseline counterparts in all cases.
+
+# A.1 MOLECULAR OPTIMIZATION
+
+For the VSeq2Seq model we use batch size 64, embedding and hidden dimension 300, VAE latent dimension 30, and an LSTM with depth 1 (bidirectional in the encoder, unidirectional in the decoder). For models using iterative target augmentation, $n _ { 1 }$ is set to 5 and $n _ { 2 }$ is set to 10, while for the baseline models we train for 20 epochs (corresponding to $n _ { 1 } = 2 0 , n _ { 2 } = 0 \rangle$ ). The HierGNN model shares the same hyperparameters as in Jin et al. (2019a).
+
+For the training time and prediction time filtering parameters, we set $K = 4$ , $C = 2 0 0$ , and $L = 1 0$ for both the QED and DRD2 tasks.
+
+# A.2 PROGRAM SYNTHESIS
+
+For the Karel program synthesis task, we use the same hyperparameters as the MLE baseline model in Bunel et al. (2018). We use a beam size of 64 at test time, the same as the MLE baseline, but simply sample programs from the decoder distribution when running iterative target augmentation during training. The baseline model is trained for 100 epochs, while for the model employing iterative target augmentation we train as normal for $n _ { 1 } = 1 5$ epochs followed by $n _ { 2 } = 5 0$ epochs of iterative target augmentation. Due to the large size of the full training dataset, in each epoch of iterative augmentation we use $\textstyle { \frac { 1 } { 1 0 } }$ of the dataset, so in total we make 5 passes over the entire dataset.
+
+For the training time and prediction time filtering parameters, we set $K = 4$ , $C = 5 0$ , and $L = 1 0$ .
+
+# B ADDITIONAL EXPERIMENTAL DETAILS
+
+# B.1 DATASET SIZES
+
+In Table 4 we provide the training, validation, and test set sizes for all of our tasks. For each task we use the same splits as our baselines.
+
+<table><tr><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>Training Set</td><td rowspan=1 colspan=1>Validation Set</td><td rowspan=1 colspan=1>Test Set</td></tr><tr><td rowspan=1 colspan=1>QED</td><td rowspan=1 colspan=1>88306</td><td rowspan=1 colspan=1>360</td><td rowspan=1 colspan=1>800</td></tr><tr><td rowspan=1 colspan=1>DRD2</td><td rowspan=1 colspan=1>34404</td><td rowspan=1 colspan=1>500</td><td rowspan=1 colspan=1>1000</td></tr><tr><td rowspan=1 colspan=1>Karel</td><td rowspan=1 colspan=1>1116854</td><td rowspan=1 colspan=1>2500</td><td rowspan=1 colspan=1>2500</td></tr></table>
+
+Table 4: Number of source-target pairs in training, validation, and test sets for each task.
+
+# B.2 MOLECULAR OPTIMIZATION LEARNING CURVES
+
+In Figure 5, we provide the validation set performance per iterative target augmentation epoch for our ${ \tt V S e q 2 S e q + }$ model on both the QED and DRD2 tasks. The corresponding figure for the $\mathrm { M L E + }$ model on the Karel task is in the main text in Figure 4.
+
+![](images/c76462e112f67c90e5c6d87ab74ce7b21534d5a38fda935423ac0a645343971c.jpg)  
+Figure 5: Left: QED success rate for ${ \tt V S e q 2 S e q + }$ on validation set for each epoch of iterative target augmentation. Right: Same plot for DRD2. For each plot, the far left point indicates the performance of the bootstrapped model.
+
+# B.3 FURTHER MOLECULAR OPTIMIZATION EXPERIMENTS
+
+In our molecular optimization tasks, we experiment with the effect of modifying $K$ , the number of new targets added per precursor during each training epoch. In all other experiments we have used $K = 4$ . Since taking $K = 0$ corresponds to the base non-augmented model, it is unsurprising that performance may suffer when $K$ is too small. However, as shown in Table 5, at least in molecular optimization there is relatively little change in performance for $K$ much larger than 4.
+
+Table 5: Performance of our model ${ \tt V S e q 2 S e q + }$ with different values of $K$ . All other experiments use $K = 4$ .   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>QED Succ.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+,K=2</td><td rowspan=1 colspan=1>85.1</td><td rowspan=1 colspan=1>0.453</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>0.327</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+, K=4</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+, K=8</td><td rowspan=1 colspan=1>88.4</td><td rowspan=1 colspan=1>0.480</td><td rowspan=1 colspan=1>97.6</td><td rowspan=1 colspan=1>0.373</td></tr></table>
+
+We also experiment with a version of our method which continually grows the training dataset by keeping all augmented targets, instead of discarding new targets at the end of each epoch. We chose the latter version for our main experiments due to its closer alignment to our EM motivation. However, we demonstrate in Table 6 that performance gains from continually growing the dataset are small to insignificant in our molecular optimization tasks.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>QEDSucc.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+,keep-targets</td><td rowspan=1 colspan=1>89.8</td><td rowspan=1 colspan=1>0.465</td><td rowspan=1 colspan=1>97.6</td><td rowspan=1 colspan=1>0.363</td></tr></table>
+
+Table 6: Performance of our proposed augmentation scheme, ${ \tt V S e q 2 S e q + }$ , compared to an alternative version $( \mathrm { V S e q 2 S e q + }$ , keep-targets) which keeps all generated targets and continually grows the training dataset.
+
+# B.4 PROGRAM SYNTHESIS ABLATIONS
+
+In Table 7 we provide the same ablation analysis that we provided in the main text for molecular optimization, demonstrating that both training time iterative target augmentation as well as prediction time filtering are beneficial to model performance. However, we note that even MLE(train), our model without prediction time filtering, outperforms the best RL method from Bunel et al. (2018).
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Top-1 Generalization</td></tr><tr><td rowspan=1 colspan=1>MLE*</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>70.91</td></tr><tr><td rowspan=1 colspan=1>MLE(test)*</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>74.12</td></tr><tr><td rowspan=1 colspan=1>MLE(train)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>77.92</td></tr><tr><td rowspan=1 colspan=1>MLE+</td><td rowspan=1 colspan=1>【</td><td rowspan=1 colspan=1>【</td><td rowspan=1 colspan=1>80.17</td></tr></table>
+
+Table 7: Ablation analysis of filtering at training and test time. “Train” indicates a model whose training process uses data augmentation according to our framework. “Test” indicates a model that uses the external filter at prediction time to discard candidate outputs which fail to pass the filter. Note that MLE and MLE(test) are based on an MLE checkpoint which underperforms the published result from Bunel et al. (2018) by 1 point, due to training for fewer epochs.

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+[
+    {
+        "type": "text",
+        "text": "ITERATIVE TARGET AUGMENTATION FOR EFFECTIVE CONDITIONAL GENERATION ",
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+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Many challenging prediction problems, from molecular optimization to program synthesis, involve creating complex structured objects as outputs. However, available training data may not be sufficient for a generative model to learn all possible complex transformations. By leveraging the idea that evaluation is easier than generation, we show how a simple, broadly applicable, iterative target augmentation scheme can be surprisingly effective in guiding the training and use of such models. Our scheme views the generative model as a prior distribution, and employs a separately trained filter as the likelihood. In each augmentation step, we filter the model’s outputs to obtain additional prediction targets for the next training epoch. Our method is applicable in the supervised as well as semi-supervised settings. We demonstrate that our approach yields significant gains over strong baselines both in molecular optimization and program synthesis. In particular, our augmented model outperforms the previous state-of-the-art in molecular optimization by over $10 \\%$ in absolute gain. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Deep architectures are becoming increasingly adept at generating complex objects such as images, text, molecules, or programs. Many useful generation problems can be seen as translation tasks, where the goal is to take a source (precursor) object such as a molecule and turn it into a target satisfying given design characteristics. Indeed, molecular optimization of this kind is a key step in drug development, though the adoption of automated tools remains limited due to accuracy concerns. We propose here a simple, broadly applicable meta-algorithm to improve translation quality. ",
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+        "type": "text",
+        "text": "Translation is a challenging task for many reasons. Objects are complex and the available training data pairs do not fully exemplify the intricate ways in which valid targets can be created from the precursors. Moreover, precursors provided at test time may differ substantially from those available during training — a scenario common in drug development. While data augmentation and semisupervised methods have been used to address some of these challenges, the focus has been on either simple prediction tasks (e.g., classification) or augmenting data primarily on the source side. We show, in contrast, that iteratively augmenting translation targets significantly improves performance on complex generation tasks in which each precursor corresponds to multiple possible outputs. ",
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+        "type": "text",
+        "text": "Our iterative target augmentation approach builds on the idea that it is easier to evaluate candidate objects than to generate them. Thus a learned predictor of target object quality (a filter) can be used to effectively guide the generation process. To this end, we construct an external filter and apply it to the complex generative model’s sampled translations of training set precursors. Candidate translations that pass the filter criteria become part of the training data for the next training epoch. The translation model is therefore iteratively guided to generate candidates that pass the filter. The generative model can be viewed as an adaptively tuned prior distribution over complex objects, with the filter as the likelihood. For this reason, it is helpful to apply the filter at test time as well, or to use the approach transductively1 to adapt the generation process to novel test cases. The approach is reminiscent of self-training or reranking approaches employed with some success for parsing (McClosky et al., 2006; Charniak et al., 2016). However, in our case, it is the candidate generator that is complex while the filter is relatively simple and remains fixed during the iterative process. ",
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+        "type": "text",
+        "text": "We demonstrate that our meta-algorithm is quite effective and consistent in its ability to improve translation quality in the supervised setting. On a program synthesis task (Bunel et al., 2018), under the same neural architecture, our augmented model outperforms their MLE baseline by $8 \\%$ and their RL model by $3 \\%$ in top-1 generalization accuracy (in absolute measure). On molecular optimization (Jin et al., 2019a), their sequence to sequence translation baseline, when combined with our target data augmentation, achieves a new state-of-the-art result and outperforms their graph based approach by over $10 \\%$ in success rate. Their graph based methods are also improved by iterative target augmentation with more than $10 \\%$ absolute gain. The results reflect the difficulty of generation in comparison to evaluation; indeed, the gains persist even if the filter quality is reduced somewhat. Source side augmentation with unlabeled precursors (the semi-supervised setting) can further improve results, but only when combined with the filter in the target data augmentation framework. We provide ablation experiments to empirically highlight the effect of our method and also offer some theoretical insights for why it is effective. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Molecular Optimization The goal of molecular optimization is to learn to modify compounds so as to improve their chemical properties. Jaques et al. (2017); You et al. (2018); Popova et al. (2018) used reinforcement learning approaches, while Jin et al. (2019a;b) formulated this problem as graphto-graph translation and significantly outperformed previous methods. However, their performance remains imperfect due to the limited size of given training sets. Our work uses property prediction models to check whether generated molecules have desired chemical properties. Recent advances in graph convolutional networks (Duvenaud et al., 2015; Gilmer et al., 2017) have provided effective solutions to predict those properties in silico. In this work, we use an off-the-shelf property prediction model (Yang et al., 2019) to filter proposed translation pairs during data augmentation. ",
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+    {
+        "type": "text",
+        "text": "Program Synthesis Program synthesis is the task of generating a program (using domain-specific language) based on given input-output specifications (Bunel et al., 2018; Gulwani, 2011; Devlin et al., 2017). One can check a generated program’s correctness by simply executing it on each input and verifying its output. Indeed, Zhang et al. (2018); Chen et al. (2019) leverage this idea in their respective decoding procedures, while also using structural constraints on valid programs. ",
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+    {
+        "type": "text",
+        "text": "Semi-supervised Learning Our method is related to various approaches in semi-supervised learning. In image and text classification, data augmentation and label guessing (Berthelot et al., 2019; Xie et al., 2019) are commonly applied to obtain artificial labels for unlabeled data. In machine translation, Norouzi et al. (2016) sample new targets from a stationary distribution in order to match the model distribution to the exponentiated payoff distribution centered at a single target sentence. Back-translation (Sennrich et al., 2015; Edunov et al., 2018) creates extra translation pairs by using a backward translation system to translate unlabeled sentences from a target language into a source language. In contrast, our method works in the forward direction because many translation tasks are not symmetric. Moreover, our data augmentation is carried out over multiple iterations, in which we use the augmented model to generate new data for the next iteration. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In syntactic parsing, our method is closely related to self-training (McClosky et al., 2006). They generate new parse trees from unlabeled sentences by applying an existing parser followed by a reranker, and then treat the resulting parse trees as new training targets. However, their method is not iterative, and their reranker is explicitly trained to operate over the top $k$ outputs of the parser; in contrast, our filter is independent of the generative model. In addition we show that our approach, which can be viewed as iteratively combining reranking and self-training, is theoretically motivated and can improve the performance of highly complex neural models in multiple domains. Co-training (Blum & Mitchell, 1998) and tri-training (Zhou & Li, 2005; Charniak et al., 2016) also augment a parsing dataset by adding targets on which multiple baseline models agree. Instead of using multiple learners, our method uses task-specific constraints to select correct outputs. ",
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+    {
+        "type": "text",
+        "text": "3 ITERATIVE TARGET AUGMENTATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our iterative target augmentation framework can be applied to any conditional generation task with task-specific constraints. For example, molecular optimization (Jin et al., 2019a;b) is the task of transforming a given molecule $X$ into another compound $Y$ with improved chemical properties, while constraining $Y$ to remain similar to $X$ . Program synthesis (Bunel et al., 2018; Chen et al., 2019) is the task of generating a program $Y$ satisfying input specification $X$ ; for example, $X$ may be a set of input-output test cases which $Y$ must pass. ",
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+    {
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+        "img_path": "images/f33dc5c1c6d59e0a2c35da3ef721ec2824870ac6cd676d78a5bdb93341fc7263.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of our data generation process in the program synthesis setting. Given an input-output specification, we first use our generation model to generate candidate programs, and then select correct programs using our external filter. Images of input-output specification and the program A are from Bunel et al. (2018). "
+        ],
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+    {
+        "type": "table",
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+        "table_body": "<table><tr><td>Algorithm1 Augmentation by iterative target augmentation</td><td></td></tr><tr><td colspan=\"3\">Input: Original training set D = [(X1, Yi),...,(Xn,Yn)]</td></tr><tr><td></td><td>1:procedure AUGMENTDATASET(D,Mt)</td><td></td></tr><tr><td>2:</td><td>Dt+1=D</td><td>&gt; Initialize augmented dataset.</td></tr><tr><td>3:</td><td>for (Xi,Yi) in D do</td><td></td></tr><tr><td>4:</td><td>for attempt in 1,.., C do</td><td></td></tr><tr><td>5:</td><td>Apply model Mt to Xi to sample candidate Y&#x27;</td><td></td></tr><tr><td>6:</td><td>if Y&#x27; passes external filter then</td><td></td></tr><tr><td>7:</td><td>Add (Xi,Y&#x27;) to Dt+1</td><td></td></tr><tr><td>8:</td><td>if K successful translations added then</td><td></td></tr><tr><td>9:</td><td>break from loop</td><td></td></tr><tr><td>10:</td><td>return augmented dataset Dt+1</td><td></td></tr><tr><td colspan=\"2\">11: procedure TRAIN(D)</td><td></td></tr><tr><td>12:</td><td>for epoch in 1,..., n1 do</td><td> Regular training</td></tr><tr><td>13:</td><td>Train model on D.</td><td></td></tr><tr><td>14:</td><td>for epoch in 1,..., n2 do</td><td>&gt; Iterative target augmentation</td></tr><tr><td>15:</td><td>Dt+1 = AUGMENTDATASET(D,Mt)</td><td></td></tr><tr><td>16:</td><td>Mt+1 ← Train model Mt on Dt+1·</td><td></td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Without loss of generality, we formulate the generation task as a translation problem. For a given input $X$ , the model learns to generate an output $Y$ satisfying the constraint $^ c$ . The proposed augmentation framework can be applied to any translation model $\\mathcal { M }$ trained on an existing dataset $\\boldsymbol { \\mathcal { D } } = \\{ ( X _ { i } , Y _ { i } ) \\}$ . As illustrated in Figure 1, our method is an iterative procedure in which each iteration consists of the following two steps: ",
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+    {
+        "type": "text",
+        "text": "• Augmentation Step: Let $\\mathcal { D } _ { t }$ be the training set at iteration $t$ . To construct each next training set $\\mathcal { D } _ { t + 1 }$ , we feed each input $X _ { i } \\in \\mathcal { D }$ (the original training set, not $\\mathcal { D } _ { t }$ ) into the translation model up to $C$ times to sample $C$ candidate translations $Y _ { i } ^ { 1 } \\ldots Y _ { i } ^ { \\overline { { C } } }$ .2 We take the first $K$ distinct translations for each $X _ { i }$ satisfying the constraint $^ c$ and add them to $\\mathcal { D } _ { t + 1 }$ . When we do not find $K$ distinct valid translations, we simply add the original translation $Y _ { i }$ to $\\mathcal { D } _ { t + 1 }$ . ",
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+    {
+        "type": "text",
+        "text": "• Training Step: We continue to train the model $\\mathcal { M } _ { t }$ over the new training set $\\mathcal { D } _ { t + 1 }$ for one epoch. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The above training procedure is summarized in Algorithm 1. As the constraint $^ c$ is known a priori, we can construct an external filter to remove generated outputs that violate $^ c$ during the augmentation step. At test time, we also use this filter to screen predicted outputs. To propose the final translation of a given input $X$ , we have the model generate up to $L$ outputs until we find one satisfying the constraint $^ c$ . If all $L$ attempts fail for a particular input, we just output the first of the failed attempts. ",
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+    {
+        "type": "text",
+        "text": "Finally, as an additional improvement, we observe that the augmentation step can be carried out for unlabeled inputs $X$ that have no corresponding $Y$ . Thus we can further augment our training dataset in the transductive setting by including test set inputs during the augmentation step, or in the semi-supervised setting by simply including unlabeled inputs. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4 MOTIVATION FOR ITERATIVE TARGET AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We provide here some theoretical motivation for our iterative target augmentation framework. For simplicity, we consider an external filter $_ { c _ { X , Y } }$ that is a binary indicator function representing whether output $Y$ satisfies the desired constraint in relation to input $X$ . In other words, we would like to generate $Y$ such that $Y \\in B ( X ) = \\{ Y ^ { \\prime } | c _ { X , Y ^ { \\prime } } = 1 \\}$ . If the initial translation model $P ^ { ( 0 ) } ( Y | X )$ serves as a reasonable prior distribution over outputs, we could simply “invert” the filter and use ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5ee89673a9fa415684745c4087db0b1547fc157cad5a53d7ff180a893537477b.jpg",
+        "text": "$$\nP ^ { ( * ) } ( Y | X ) \\propto P ^ { ( 0 ) } ( Y | X ) \\cdot c _ { X , Y }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "as the ideal translation model. While this posterior calculation is typically not feasible but could be approximated through samples, it relies heavily on the appropriateness of the prior (model prior to augmentation). Instead, we go a step further and iteratively optimize our parametrically defined prior translation model $P _ { \\theta } ( Y | X )$ . Note that the resulting prior can become much more concentrated around acceptable translations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We maximize the log-likelihood that candidate translations satisfy the constraints implicitly encoded in the filter ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36da1a4ad910f6df76742a8a3e4151beacc239353edd8029c3c918fa7b219fe6.jpg",
+        "text": "$$\n\\mathbb { E } _ { X } \\left[ \\log P _ { \\theta } ( \\pmb { c } _ { X , Y } = 1 \\mid X ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In many cases there are multiple viable outputs for any given input $X$ . The training data may provide only one (or none) of them. Therefore, we treat the output structure $Y$ as a latent variable, and expand the inner term of Eq.(2) as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ecb991487f389ad31bff73e53baff04f0d2d322490440f478cd50c18d1d64a2.jpg",
+        "text": "$$\n\\begin{array} { l l l } { \\log P _ { \\theta } ( \\boldsymbol { c } _ { X , Y } = 1 \\mid X ) } & { = } & { \\displaystyle \\log \\sum _ { Y } P _ { \\theta } ( Y , \\boldsymbol { c } _ { X , Y } = 1 \\mid X ) } \\\\ & { = } & { \\displaystyle \\log \\sum _ { Y } P ( \\boldsymbol { c } _ { X , Y } = 1 \\mid Y , X ) P _ { \\theta } ( Y \\mid X ) } \\\\ & { = } & { \\displaystyle \\log \\sum _ { Y } \\boldsymbol { c } _ { X , Y } \\cdot P _ { \\theta } ( Y \\mid X ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since the above objective involves discrete latent variables $Y$ , we propose to maximize Eq.(5) using the standard EM algorithm (Dempster et al., 1977), especially its incremental, approximate variant. The target augmentation step in our approach is a sampled version of the $\\mathrm { E }$ -step where the posterior samples are drawn with rejection sampling guided by the filter. The number of samples $K$ controls the quality of approximation to the posterior.3 The additional training step based on the augmented targets corresponds to a generalized M-step. More precisely, let $P _ { \\theta } ^ { ( t ) } ( Y | X )$ be the current translation model after epochs of augmentation training. In epoch , the augmentation step first samples $C$ different candidates for each input $X$ using the old model $P ^ { ( t ) }$ parameterized by $\\theta ^ { ( t ) }$ , and then removes those which violate the constraint $^ c$ , interpretable as samples from the current posterior $Q ^ { ( t ) } ( Y | X ) \\propto P _ { \\theta ^ { ( t ) } } ( Y | X ) c _ { X , Y }$ . As a result, the training step maximizes the EM auxiliary objective via stochastic gradient descent: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/42d824130df2e6784b931ba2eb8b81879e303e1b423b386f618d13a2cf0677f6.jpg",
+        "text": "$$\nJ ( \\theta \\mid \\theta ^ { ( t ) } ) = \\mathbb { E } _ { X } \\left[ \\sum _ { Y } Q ^ { ( t ) } ( Y | X ) \\log P _ { \\theta } ( Y | X ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We train the model with multiple iterations and show empirically that model performance indeed keeps improving as we add more iterations. The EM approach is likely to converge to a different and better-performing translation model than the initial posterior calculation discussed above. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate the broad applicability of iterative target augmentation by applying it to two tasks of different domains: molecular optimization and program synthesis. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1d6bfac9d14ffa73e428aa3b82e76d9eea78cf0494f4ad32d88c0ac75deafb3d.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of molecular optimization. Molecules can be modeled as graphs, with atoms as nodes and bonds as edges. Here, the task is to train a translation model to modify a given input molecule into a target molecule with higher drug-likeness (QED) score. The constraint has two components: the output $Y$ must be highly drug-like, and must be sufficiently similar to the input $X$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1 MOLECULAR OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The goal of molecular optimization is to learn to modify molecules so as to improve their chemical properties. As illustrated in Figure 2, this task is formulated as a graph-to-graph translation problem. Similar to machine translation, the training set is a set of molecular pairs $\\{ ( X , Y ) \\}$ . $X$ is the input molecule (precursor) and $Y$ is a similar molecule with improved chemical properties. Each molecule in the training set $\\mathcal { D }$ is further labeled with its property score. Our method is well-suited to this task because the target molecule is not unique: each precursor molecule can be modified in many different ways to optimize its properties. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "External Filter The constraint for this task contains two parts: 1) the chemical property of $Y$ must exceed a certain threshold $\\beta$ , and 2) the molecular similarity between $X$ and $Y$ must exceed a certain threshold $\\delta$ . The molecular similarity $\\sin ( X , Y )$ is defined as Tanimoto similarity on Morgan fingerprints (Rogers & Hahn, 2010), which measures structural overlap between two molecules. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In real world settings, ground truth values of chemical properties are often evaluated through experimental assays, which are too expensive and time-consuming to run for iterative target augmentation. Therefore, we construct an in silico property predictor $F _ { 1 }$ to approximate the true property evaluator $F _ { 0 }$ . To train this property prediction model, we use the molecules in the training set and their labeled property values. The predictor $F _ { 1 }$ is parameterized as a graph convolutional network and trained using the Chemprop package (Yang et al., 2019). During data augmentation, we use $F _ { 1 }$ to filter out molecules whose predicted property is under the threshold $\\beta$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We follow the evaluation setup of Jin et al. (2019b) for two molecular optimization tasks: ",
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+        "text": "1. QED Optimization: The task is to improve the drug-likeness (QED) of a given compound $X$ . The similarity constraint is $\\sin ( X , Y ) \\bar { \\geq } 0 . 4$ and the property constraint is $\\bar { \\mathrm { Q E D } } ( Y ) \\stackrel { - } { = } 0 . 9$ , with $\\mathrm { Q E D } ( Y ) \\in \\left[ 0 , 1 \\right]$ defined by the system of Bickerton et al. (2012). ",
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+    {
+        "type": "text",
+        "text": "2. DRD2 Optimization: The task is to optimize biological activity against the dopamine type 2 receptor (DRD2). The similarity constraint is $\\sin ( \\bar { X } , Y ) \\ge 0 . 4 $ and the property constraint is $\\mathrm { D R D 2 } ( Y ) \\ge 0 . 5$ , where $\\mathrm { D R D 2 } ( Y ) \\in [ 0 , 1 ]$ is the predicted probability of biological activity given by the model from Olivecrona et al. (2017). ",
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+    {
+        "type": "text",
+        "text": "We treat the output of the in silico evaluators from Bickerton et al. (2012) and Olivecrona et al. (2017) as ground truth, and we use them only during test-time evaluation.4 ",
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+    {
+        "type": "text",
+        "text": "Evaluation Metrics. During evaluation, we are interested both in the probability that the model will find a successful modification for a given molecule, as well as the diversity of the successful modifications when there are multiple. We translate each molecule in the test set $Z = 2 0$ times, resulting in candidate modifications $Y _ { 1 } \\ldots Y _ { Z }$ (not necessarily distinct). We use the following two evaluation metrics: ",
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+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>QED Succ.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq</td><td rowspan=1 colspan=1>58.5</td><td rowspan=1 colspan=1>0.331</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>0.176</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+ (Ours)</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+, semi-supervised (Ours)</td><td rowspan=1 colspan=1>95.0</td><td rowspan=1 colspan=1>0.471</td><td rowspan=1 colspan=1>99.6</td><td rowspan=1 colspan=1>0.408</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+,transductive (Ours)</td><td rowspan=1 colspan=1>92.6</td><td rowspan=1 colspan=1>0.451</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>0.358</td></tr><tr><td rowspan=1 colspan=1>HierGNN</td><td rowspan=1 colspan=1>76.6</td><td rowspan=1 colspan=1>0.477</td><td rowspan=1 colspan=1>85.9</td><td rowspan=1 colspan=1>0.192</td></tr><tr><td rowspan=1 colspan=1>HierGNN+ (Ours)</td><td rowspan=1 colspan=1>93.1</td><td rowspan=1 colspan=1>0.514</td><td rowspan=1 colspan=1>97.6</td><td rowspan=1 colspan=1>0.418</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Performance of different models on QED and DRD2 optimization tasks. Italicized models with $^ +$ are augmented with iterative target augmentation. We emphasize that iterative target augmentation remains critical to performance in the semi-supervised and transductive settings; data augmentation without an external filter instead decreases performance. ",
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+        "text": "1. Success: The fraction of molecules $X$ for which any of the outputs $Y _ { 1 } \\ldots Y _ { Z }$ meet the required similarity and property constraints (specified previously for each task). This is our main metric. ",
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+        "type": "text",
+        "text": "2. Diversity: For each molecule $X$ , we measure the average Tanimoto distance (defined as $1 -$ $\\mathrm { s i m } ( Y _ { i } , Y _ { j } ) ,$ ) between pairs within the set of successfully translated compounds among $Y _ { 1 } \\ldots Y _ { Z }$ . If there are one or fewer successful translations then the diversity is 0. We average this quantity across all test molecules. ",
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+        "type": "text",
+        "text": "Models and Baselines. We consider the following two model architectures from Jin et al. (2019a) to show that our augmentation scheme is not tied to specific neural architectures. ",
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+    {
+        "type": "text",
+        "text": "1. VSeq2Seq, a sequence-to-sequence translation model generating molecules by their SMILES string (Weininger, 1988). 2. HierGNN, a hierarchical graph-to-graph architecture that achieves state-of-the-art performance on the QED and DRD2 tasks, outperforming VSeq2Seq by a wide margin. ",
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+        "type": "text",
+        "text": "We apply our iterative augmentation procedure to the above two models, generating up to $K = 4$ new targets per precursor during each epoch of iterative target augmentation. Additionally, we evaluate our augmentation of VSeq2Seq in a transductive setting, as well as in a semi-supervised setting where we provide 100K additional source-side precursors from the ZINC database (Sterling & Irwin, 2015). Full hyperparameters are in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "5.1.2 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As shown in Table 1, our iterative augmentation paradigm significantly improves the performance of VSeq2Seq and HierGNN. On both datasets, the translation success rate increases by over $10 \\%$ in absolute terms for both models. In fact, ${ \\tt V S e q 2 S e q + }$ , our augmentation of the simple VSeq2Seq model, outperforms the non-augmented version of HierGNN. This result strongly confirms our hypothesis about the inherent challenge of learning translation models in data sparse scenarios. Moreover, we find that adding more precursors during data augmentation further improves the VSeq2Seq model. On the QED dataset, the translation success rate improves from $8 9 . 0 \\%$ to $9 2 . 6 \\%$ by just adding test set molecules as precursors $( \\mathrm { V S e q 2 S e q + }$ , transductive). When instead adding 100K presursors from the external ZINC database, the performance further increases to $9 5 . 0 \\%$ $( \\mathrm { V S e q 2 S e q + }$ , semisupervised). We observe similar improvements for the DRD2 task as well. Beyond accuracy gain, our augmentation strategy also improves the diversity of generated molecules. For instance, on the DRD2 dataset, our approach yields $100 \\%$ relative gain in terms of output diversity. ",
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+        "type": "text",
+        "text": "Importance of Property Predictor Although the property predictor used in data augmentation is different from the ground truth property evaluator used at test time, the difference in evaluators does not derail the overall training process. Here we analyze the influence of the quality of the property predictor used in data augmentation. Specifically, we rerun our experiments using less accurate predictors in the property-predicting component of our external filter. We obtain these less accurate predictors by undertraining Chemprop and decreasing its hidden dimension. For comparison, we also report results with the oracle property predictor which is the ground truth property evaluator. ",
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+        "type": "text",
+        "text": "As shown in Figure 3, on the DRD2 dataset, we are able to maintain strong performance despite using predictors that deviate significantly from the ground truth. This implies that our framework can potentially be applied to other properties that are harder to predict. On the QED dataset, our ",
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+        "img_path": "images/999d43365de7d71b2b6c8124c3a40c734e5b26914b76bfcadc281fdbea734f9d.jpg",
+        "image_caption": [
+            "Figure 3: Left: QED success rate vs. Chemprop predictor’s RMSE with respect to ground truth on test set. The red line shows the performance of the (unaugmented) VSeq2Seq baseline. Right: Same plot for DRD2. In each plot, the far left point with zero RMSE is obtained by reusing the ground truth predictor, while the second-from-left point is the Chemprop predictor we use to obtain our main results. Points further to the right are weaker predictors trained for fewer epochs and with less capacity, simulating a scenario where the property is more difficult to model. "
+        ],
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>QED Succ.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>58.5</td><td rowspan=1 colspan=1>0.331</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>0.176</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq(test)</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>77.4</td><td rowspan=1 colspan=1>0.471</td><td rowspan=1 colspan=1>87.2</td><td rowspan=1 colspan=1>0.200</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq(train)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>81.8</td><td rowspan=1 colspan=1>0.430</td><td rowspan=1 colspan=1>92.2</td><td rowspan=1 colspan=1>0.321</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+</td><td rowspan=1 colspan=1>了</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq(no-filter)</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>47.5</td><td rowspan=1 colspan=1>0.297</td><td rowspan=1 colspan=1>51.0</td><td rowspan=1 colspan=1>0.185</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Ablation analysis of filtering at training and test time. “Train” indicates a model whose training process uses data augmentation according to our framework. “Test” indicates a model that uses the external filter at prediction time to discard candidate outputs which fail to pass the filter. The evaluation for VSeq2Seq(no-filter) is conducted after 10 augmentation epochs, as the best validation set performance only decreases over the course of training. ",
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+    {
+        "type": "text",
+        "text": "method is less tolerant to inaccurate property prediction because the property constraint is much tighter — it requires the QED score of an output $Y$ to be in the range [0.9, 1.0]. ",
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+    {
+        "type": "text",
+        "text": "Importance of External Filtering Our full model $( { \\mathrm { V S e q } } 2 { \\mathrm { S e q } } +$ ) uses the external filter during both training and testing. We further experiment with Vseq2seq(test), a version of our model trained without data augmentation but which uses the external filter to remove invalid outputs at test time. As shown in Table 2, VSeq2Seq(test) performs significantly worse than our full model trained under data augmentation. Similarly, a model VSeq2Seq(train) trained with the data augmentation but without the prediction time filtering also performs much worse than the full model. ",
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+    {
+        "type": "text",
+        "text": "In addition, we run an augmentation-only version of the model without an external filter. This model (referred to as VSeq2Seq(no-filter) in Table 2) augments the data in each epoch by simply using the first $K$ distinct candidate translations for each precursor $X$ in the training set, without using the external filter at all. In addition, we provide this model with the 100K unlabeled precursors from the semi-supervised setting. Nevertheless, we find that the performance of this model steadily declines from that of the bootstrapped starting point with each data augmentation epoch. Thus the external filter is necessary to prevent poor targets from leading the model training astray. ",
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+    {
+        "type": "text",
+        "text": "5.2 PROGRAM SYNTHESIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In program synthesis, the source is a set of input-output specifications for the program, and the target is a program that passes all test cases. Our method is suitable for this task because the target program is not unique. Multiple programs may be consistent with the given input-output specifications. The external filter is straightforward for this task: we simply check whether the generated output passes all test cases. Note that at evaluation time, each instance contains extra held-out input-output test cases; the program must pass these in addition to the given test cases in order to be considered correct. When we perform prediction time filtering, we do not use held-out test cases in our filter. ",
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+    {
+        "type": "table",
+        "img_path": "images/1e1aba09d90571e49a3c61d6282cf09bee176b38fc6dc981ed130f3730b0ecc9.jpg",
+        "table_caption": [
+            "Table 3: Model performance on Karel program synthesis task. $\\mathrm { M L E + }$ is our augmented version of the MLE model (Bunel et al., 2018). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Top-1Generalization</td></tr><tr><td rowspan=1 colspan=1>MLE (Bunel et al.,2018)</td><td rowspan=1 colspan=1>71.91</td></tr><tr><td rowspan=1 colspan=1>MLE+RL+BeamSearch(Bunel et al., 2018)</td><td rowspan=1 colspan=1>77.12</td></tr><tr><td rowspan=1 colspan=1>MLE+ (Ours)</td><td rowspan=1 colspan=1>80.17</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/8cba7a23895f93e56d42a79eaa18a931a6a0954468ed34f6aabd4f2fba87f5b4.jpg",
+        "image_caption": [
+            "Figure 4: Top-1 generalization accuracy of $\\mathrm { M L E + }$ model on validation set of Karel task across different epochs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our task is based on the educational Karel programming language (Pattis, 1981) used for evaluation in Bunel et al. (2018) and Chen et al. (2019). Commands in the Karel language guide a robot’s actions in a 2D grid, and may include for loops, while loops, and conditionals. Figure 1 contains an example. We follow the experiment setup of Bunel et al. (2018). ",
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+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics. The evaluation metric is top-1 generalization. This metric measures how often the model can generate a program that passes the input-output test cases on the test set. At test time, we use our model to generate up to $L$ candidate programs and select the first one to pass the input-output specifications (not including held-out test cases). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Models and Baselines. Our main baseline is the MLE baseline from Bunel et al. (2018). This model consists of a CNN encoder for the input-output grids and a LSTM decoder along with a handcoded syntax checker. It is trained to maximize the likelihood of the provided target program. Our model is the augmentation of this MLE baseline by our iterative target augmentation framework. As with molecular optimization, we generate up to $K = 4$ new targets per precursor during each augmentation step. Additionally, we compare against the best model from Bunel et al. (2018), which finetunes the same MLE architecture using an RL method with beam search to estimate gradients.5 We use the same hyperparameters as the original MLE baseline; see Appendix A for details. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2.2 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3 shows the performance of our model in comparison to previous work. Our model (MLE+) outperforms the base MLE model in Bunel et al. (2018) model by a wide margin. Moreover, our model outperforms the best reinforcement learning model (RL $^ +$ Beam Search) in Bunel et al. (2018), which was trained to directly maximize the generalization metric. This demonstrates the efficacy of our approach in the program synthesis domain. Since our augmentation framework is complementary to architectural improvements, we hypothesize that other techniques, such as execution based synthesis (Chen et al., 2019), can benefit from our approach as well. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we have presented an iterative target augmentation framework for generation tasks with multiple possible outputs. Our approach is theoretically motivated, and we demonstrate strong empirical results on both the molecular optimization and program synthesis tasks, significantly outperforming baseline models on each task. Moreover, we find that iterative target augmentation is complementary to architectural improvements, and that its effect can be quite robust to the quality of the external filter. Finally, in principle our approach is applicable to other domains as well. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "Marcus Olivecrona, Thomas Blaschke, Ola Engkvist, and Hongming Chen. Molecular de-novo design through deep reinforcement learning. Journal of cheminformatics, 9(1):48, 2017.   \nRichard E. Pattis. Karel the Robot: A Gentle Introduction to the Art of Programming. John Wiley & Sons, Inc., New York, NY, USA, 1st edition, 1981. ISBN 0471089281.   \nMariya Popova, Olexandr Isayev, and Alexander Tropsha. Deep reinforcement learning for de novo drug design. Science advances, 4(7):eaap7885, 2018.   \nDavid Rogers and Mathew Hahn. Extended-connectivity fingerprints. J. Chem. Inf. Model., 50(5): 742–754, 2010.   \nRico Sennrich, Barry Haddow, and Alexandra Birch. Improving neural machine translation models with monolingual data. arXiv preprint arXiv:1511.06709, 2015.   \nTeague Sterling and John J Irwin. Zinc 15–ligand discovery for everyone. Journal of chemical information and modeling, 55(11):2324–2337, 2015.   \nDavid Weininger. Smiles, a chemical language and information system. 1. introduction to methodology and encoding rules. J. Chem. Inf. Model., 28(1):31–36, 1988.   \nQizhe Xie, Zihang Dai, Eduard Hovy, Minh-Thang Luong, and Quoc V Le. Unsupervised data augmentation. arXiv preprint arXiv:1904.12848, 2019.   \nKevin Yang, Kyle Swanson, Wengong Jin, Connor W Coley, Philipp Eiden, Hua Gao, Angel Guzman-Perez, Tim Hopper, Brian Kelley, Miriam Mathea, et al. Analyzing learned molecular representations for property prediction. Journal of chemical information and modeling, 2019.   \nJiaxuan You, Bowen Liu, Zhitao Ying, Vijay Pande, and Jure Leskovec. Graph convolutional policy network for goal-directed molecular graph generation. In Advances in Neural Information Processing Systems, pp. 6410–6421, 2018.   \nLisa Zhang, Gregory Rosenblatt, Ethan Fetaya, Renjie Liao, William E Byrd, Raquel Urtasun, and Richard Zemel. Leveraging constraint logic programming for neural guided program synthesis. 2018.   \nZhi-Hua Zhou and Ming Li. Tri-training: Exploiting unlabeled data using three classifiers. IEEE Transactions on Knowledge & Data Engineering, (11):1529–1541, 2005. ",
+        "bbox": [
+            171,
+            89,
+            826,
+            589
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A MODEL HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            449,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Our augmented models share the same hyperparameters as their baseline counterparts in all cases. ",
+        "bbox": [
+            173,
+            133,
+            812,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 MOLECULAR OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            165,
+            418,
+            179
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the VSeq2Seq model we use batch size 64, embedding and hidden dimension 300, VAE latent dimension 30, and an LSTM with depth 1 (bidirectional in the encoder, unidirectional in the decoder). For models using iterative target augmentation, $n _ { 1 }$ is set to 5 and $n _ { 2 }$ is set to 10, while for the baseline models we train for 20 epochs (corresponding to $n _ { 1 } = 2 0 , n _ { 2 } = 0 \\rangle$ ). The HierGNN model shares the same hyperparameters as in Jin et al. (2019a). ",
+        "bbox": [
+            174,
+            190,
+            825,
+            261
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the training time and prediction time filtering parameters, we set $K = 4$ , $C = 2 0 0$ , and $L = 1 0$ for both the QED and DRD2 tasks. ",
+        "bbox": [
+            173,
+            267,
+            823,
+            296
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 PROGRAM SYNTHESIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            313,
+            372,
+            327
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the Karel program synthesis task, we use the same hyperparameters as the MLE baseline model in Bunel et al. (2018). We use a beam size of 64 at test time, the same as the MLE baseline, but simply sample programs from the decoder distribution when running iterative target augmentation during training. The baseline model is trained for 100 epochs, while for the model employing iterative target augmentation we train as normal for $n _ { 1 } = 1 5$ epochs followed by $n _ { 2 } = 5 0$ epochs of iterative target augmentation. Due to the large size of the full training dataset, in each epoch of iterative augmentation we use $\\textstyle { \\frac { 1 } { 1 0 } }$ of the dataset, so in total we make 5 passes over the entire dataset. ",
+        "bbox": [
+            174,
+            338,
+            825,
+            438
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the training time and prediction time filtering parameters, we set $K = 4$ , $C = 5 0$ , and $L = 1 0$ . ",
+        "bbox": [
+            173,
+            443,
+            818,
+            458
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            478,
+            534,
+            494
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B.1 DATASET SIZES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            510,
+            326,
+            525
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In Table 4 we provide the training, validation, and test set sizes for all of our tasks. For each task we use the same splits as our baselines. ",
+        "bbox": [
+            174,
+            535,
+            825,
+            564
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/872b884518b5e633402244e65cfc6337d5ba944ab5fcbf58bd166b43b8e341eb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>Training Set</td><td rowspan=1 colspan=1>Validation Set</td><td rowspan=1 colspan=1>Test Set</td></tr><tr><td rowspan=1 colspan=1>QED</td><td rowspan=1 colspan=1>88306</td><td rowspan=1 colspan=1>360</td><td rowspan=1 colspan=1>800</td></tr><tr><td rowspan=1 colspan=1>DRD2</td><td rowspan=1 colspan=1>34404</td><td rowspan=1 colspan=1>500</td><td rowspan=1 colspan=1>1000</td></tr><tr><td rowspan=1 colspan=1>Karel</td><td rowspan=1 colspan=1>1116854</td><td rowspan=1 colspan=1>2500</td><td rowspan=1 colspan=1>2500</td></tr></table>",
+        "bbox": [
+            321,
+            577,
+            674,
+            638
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Number of source-target pairs in training, validation, and test sets for each task. ",
+        "bbox": [
+            207,
+            648,
+            787,
+            664
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B.2 MOLECULAR OPTIMIZATION LEARNING CURVES ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            689,
+            558,
+            704
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we provide the validation set performance per iterative target augmentation epoch for our ${ \\tt V S e q 2 S e q + }$ model on both the QED and DRD2 tasks. The corresponding figure for the $\\mathrm { M L E + }$ model on the Karel task is in the main text in Figure 4. ",
+        "bbox": [
+            173,
+            715,
+            825,
+            758
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/c76462e112f67c90e5c6d87ab74ce7b21534d5a38fda935423ac0a645343971c.jpg",
+        "image_caption": [
+            "Figure 5: Left: QED success rate for ${ \\tt V S e q 2 S e q + }$ on validation set for each epoch of iterative target augmentation. Right: Same plot for DRD2. For each plot, the far left point indicates the performance of the bootstrapped model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            103,
+            815,
+            250
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.3 FURTHER MOLECULAR OPTIMIZATION EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            328,
+            593,
+            343
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In our molecular optimization tasks, we experiment with the effect of modifying $K$ , the number of new targets added per precursor during each training epoch. In all other experiments we have used $K = 4$ . Since taking $K = 0$ corresponds to the base non-augmented model, it is unsurprising that performance may suffer when $K$ is too small. However, as shown in Table 5, at least in molecular optimization there is relatively little change in performance for $K$ much larger than 4. ",
+        "bbox": [
+            174,
+            353,
+            825,
+            424
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/e3a839d65fb50efbaccd012eeb22b4fc99124c2d8d9b37cfd0c79f78cbf9f17f.jpg",
+        "table_caption": [
+            "Table 5: Performance of our model ${ \\tt V S e q 2 S e q + }$ with different values of $K$ . All other experiments use $K = 4$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>QED Succ.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+,K=2</td><td rowspan=1 colspan=1>85.1</td><td rowspan=1 colspan=1>0.453</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>0.327</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+, K=4</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+, K=8</td><td rowspan=1 colspan=1>88.4</td><td rowspan=1 colspan=1>0.480</td><td rowspan=1 colspan=1>97.6</td><td rowspan=1 colspan=1>0.373</td></tr></table>",
+        "bbox": [
+            241,
+            438,
+            754,
+            500
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We also experiment with a version of our method which continually grows the training dataset by keeping all augmented targets, instead of discarding new targets at the end of each epoch. We chose the latter version for our main experiments due to its closer alignment to our EM motivation. However, we demonstrate in Table 6 that performance gains from continually growing the dataset are small to insignificant in our molecular optimization tasks. ",
+        "bbox": [
+            176,
+            553,
+            825,
+            623
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/454abde5dbafec2dcc93b3a084c51ef94eab0e0ddf8794b43570f920ca828433.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>QEDSucc.</td><td rowspan=1 colspan=1>QED Div.</td><td rowspan=1 colspan=1>DRD2 Succ.</td><td rowspan=1 colspan=1>DRD2 Div.</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>0.470</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>0.361</td></tr><tr><td rowspan=1 colspan=1>VSeq2Seq+,keep-targets</td><td rowspan=1 colspan=1>89.8</td><td rowspan=1 colspan=1>0.465</td><td rowspan=1 colspan=1>97.6</td><td rowspan=1 colspan=1>0.363</td></tr></table>",
+        "bbox": [
+            218,
+            636,
+            779,
+            684
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Performance of our proposed augmentation scheme, ${ \\tt V S e q 2 S e q + }$ , compared to an alternative version $( \\mathrm { V S e q 2 S e q + }$ , keep-targets) which keeps all generated targets and continually grows the training dataset. ",
+        "bbox": [
+            176,
+            694,
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+            736
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.4 PROGRAM SYNTHESIS ABLATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            761,
+            459,
+            776
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Table 7 we provide the same ablation analysis that we provided in the main text for molecular optimization, demonstrating that both training time iterative target augmentation as well as prediction time filtering are beneficial to model performance. However, we note that even MLE(train), our model without prediction time filtering, outperforms the best RL method from Bunel et al. (2018). ",
+        "bbox": [
+            173,
+            787,
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+            844
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/322e08c75c950840b5c9cfe0dd107ab1b2a0b28b97614fe745300fc2eaabd9bd.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Top-1 Generalization</td></tr><tr><td rowspan=1 colspan=1>MLE*</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>70.91</td></tr><tr><td rowspan=1 colspan=1>MLE(test)*</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>74.12</td></tr><tr><td rowspan=1 colspan=1>MLE(train)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>77.92</td></tr><tr><td rowspan=1 colspan=1>MLE+</td><td rowspan=1 colspan=1>【</td><td rowspan=1 colspan=1>【</td><td rowspan=1 colspan=1>80.17</td></tr></table>",
+        "bbox": [
+            318,
+            433,
+            679,
+            508
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Ablation analysis of filtering at training and test time. “Train” indicates a model whose training process uses data augmentation according to our framework. “Test” indicates a model that uses the external filter at prediction time to discard candidate outputs which fail to pass the filter. Note that MLE and MLE(test) are based on an MLE checkpoint which underperforms the published result from Bunel et al. (2018) by 1 point, due to training for fewer epochs. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    }
+]