Add files using upload-large-folder tool

parse/train/BJgklhAcK7/BJgklhAcK7.md

@@ -0,0 +1,426 @@
+# META-LEARNING WITH LATENT EMBEDDING OPTIMIZATION
+
+Andrei A. Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu, Simon Osindero & Raia Hadsell
+
+DeepMind, London, UK {andreirusu, dushyantr, sygi, vinyals, razp, osindero, raia}@google.com
+
+# ABSTRACT
+
+Gradient-based meta-learning techniques are both widely applicable and proficient at solving challenging few-shot learning and fast adaptation problems. However, they have practical difficulties when operating on high-dimensional parameter spaces in extreme low-data regimes. We show that it is possible to bypass these limitations by learning a data-dependent latent generative representation of model parameters, and performing gradient-based meta-learning in this lowdimensional latent space. The resulting approach, latent embedding optimization (LEO), decouples the gradient-based adaptation procedure from the underlying high-dimensional space of model parameters. Our evaluation shows that LEO can achieve state-of-the-art performance on the competitive miniImageNet and tieredImageNet few-shot classification tasks. Further analysis indicates LEO is able to capture uncertainty in the data, and can perform adaptation more effectively by optimizing in latent space.
+
+# 1 INTRODUCTION
+
+Humans have a remarkable ability to quickly grasp new concepts from a very small number of examples or a limited amount of experience, leveraging prior knowledge and context. In contrast, traditional deep learning approaches (LeCun et al., 2015; Schmidhuber, 2015) treat each task independently and hence are often data inefficient – despite providing significant performance improvements across the board, such as for image classification (Simonyan & Zisserman, 2014; He et al., 2016), reinforcement learning (Mnih et al., 2015; Silver et al., 2017), and machine translation (Cho et al., 2014; Sutskever et al., 2014). Just as humans can efficiently learn new tasks, it is desirable for learning algorithms to quickly adapt to and incorporate new and unseen information.
+
+Few-shot learning tasks challenge models to learn a new concept or behaviour with very few examples or limited experience (Fei-Fei et al., 2006; Lake et al., 2011). One approach to address this class of problems is meta-learning, a broad family of techniques focused on learning how to learn or to quickly adapt to new information. More specifically, optimization-based meta-learning approaches (Ravi & Larochelle, 2017; Finn et al., 2017) aim to find a single set of model parameters that can be adapted with a few steps of gradient descent to individual tasks. However, using only a few samples (typically 1 or 5) to compute gradients in a high-dimensional parameter space could make generalization difficult, especially under the constraint of a shared starting point for task-specific adaptation.
+
+In this work we propose a new approach, named Latent Embedding Optimization (LEO), which learns a low-dimensional latent embedding of model parameters and performs optimization-based meta-learning in this space. Intuitively, the approach provides two advantages. First, the initial parameters for a new task are conditioned on the training data, which enables a task-specific starting point for adaptation. By incorporating a relation network into the encoder, this initialization can better consider the joint relationship between all of the input data. Second, by optimizing in the lower-dimensional latent space, the approach can adapt the behaviour of the model more effectively. Further, by allowing this process to be stochastic, the ambiguities present in the few-shot data regime can be expressed.
+
+We demonstrate that LEO achieves state-of-the-art results on both the miniImageNet and tieredImageNet datasets, and run an ablation study and further analysis to show that both conditional parameter generation and optimization in latent space are critical for the success of the method. Source code for our experiments is available at https://github.com/deepmind/leo.
+
+# 2 MODEL
+
+# 2.1 PROBLEM DEFINITION
+
+We define the $N$ -way $K$ -shot problem using the episodic formulation of Vinyals et al. (2016). Each task instance $\mathcal { T } _ { i }$ is a classification problem sampled from a task distribution $p ( \mathcal { T } )$ . The tasks are divided into a training meta-set $\mathcal { S } ^ { t \bar { r } }$ , validation meta-set $S ^ { v a l }$ , and test meta-set $\mathcal { S } ^ { t e s t }$ , each with a disjoint set of target classes (i.e., a class seen during testing is not seen during training). The validation meta-set is used for model selection, and the testing meta-set is used only for final evaluation.
+
+Each task instance $\begin{array} { r } { T _ { i } \sim p \left( T \right) } \end{array}$ is composed of a training set $\mathcal { D } ^ { t r }$ and validation set $\mathcal { D } ^ { v a l }$ , and only contains $N$ classes randomly selected from the appropriate meta-set (e.g. for a task instance in the training meta-set, the classes are a subset of those available in ${ \mathcal { S } } ^ { t r }$ ). In most setups, the training set $\check { \mathcal { D } ^ { t r } } = \left\{ ( \mathbf { x } _ { n } ^ { k } , y _ { n } ^ { k } ) \ | \ k = 1 \ldots K ; n = 1 \ldots N \right\}$ contains $K$ samples for each class. The validation set $\mathcal { D } ^ { v a l }$ can contain several other samples from the same classes, providing an estimate of generalization performance on the $N$ classes for this problem instance. We note that the validation set of a problem instance $\mathcal { D } ^ { v a l }$ (used to optimize a meta-learning objective) should not be confused with the held-out validation meta-set $S ^ { v a \hat { l } }$ (used for model selection).
+
+# 2.2 MODEL-AGNOSTIC META-LEARNING
+
+Model-agnostic meta-learning (MAML) (Finn et al., 2017) is an approach to optimization-based meta-learning that is related to our work. For some parametric model $f _ { \theta }$ , MAML aims to find a single set of parameters $\theta$ which, using a few optimization steps, can be successfully adapted to any novel task sampled from the same distribution. For a particular task instance $\mathcal { T } _ { i } \dot { ~ } = ( \dot { \mathcal { D } } ^ { t r } , \mathcal { D } ^ { v a l } )$ , the parameters are adapted to task-specific model parameters $\theta _ { i } ^ { \prime }$ by applying some differentiable function, typically an update rule of the form:
+
+$$
+\begin{array} { r } { \theta _ { i } ^ { \prime } = \mathcal { G } \left( \boldsymbol { \theta } , \mathcal { D } ^ { t r } \right) , } \end{array}
+$$
+
+where $\mathcal { G }$ is typically implemented as a step of gradient descent on the few-shot training set $\mathcal { D } ^ { t r }$ , $\theta _ { i } ^ { \prime } = \theta - \alpha \nabla _ { \theta } \mathcal { L } _ { T _ { i } } ^ { t r } \left( \mathbf { \bar { \it f } } _ { \theta } \right)$ . Generally, multiple sequential adaptation steps can be applied. The learning rate $\alpha$ can also be meta-learned concurrently, in which case we refer to this algorithm as MetaSGD (Li et al., 2017). During meta-training, the parameters $\theta$ are updated by back-propagating through the adaptation procedure, in order to reduce errors on the validation set $\dot { \mathcal { D } } ^ { v a l }$ :
+
+$$
+\theta  \theta - \eta \nabla _ { \theta } \sum _ { \mathcal { T } _ { i } \sim p ( \mathcal { T } ) } \mathcal { L } _ { \mathcal { T } _ { i } } ^ { v a l } ( f _ { \theta _ { i } ^ { \prime } } )
+$$
+
+The approach includes the main ingredients of optimization-based meta-learning with neural networks: initialization is done by maintaining an explicit set of model parameters $\theta$ ; the adaptation procedure, or “inner loop”, takes $\theta$ as input and returns $\theta _ { i } ^ { \prime }$ adapted specifically for task instance $\mathcal { T } _ { i }$ , by iteratively using gradient descent (Eq. 1); and termination, which is handled simply by choosing a fixed number of optimization steps in the “inner loop”. MAML updates $\theta$ by differentiating through the “inner loop” in order to minimize errors of instance-specific adapted models $f _ { \theta _ { i } ^ { \prime } }$ on the corresponding validation set (Eq. 2). We refer to this process as the “outer loop” of meta-learning. In the next section we use the same stages to describe Latent Embedding Optimization (LEO).
+
+# 2.3 LATENT EMBEDDING OPTIMIZATION FOR META-LEARNING
+
+The primary contribution of this paper is to show that it is possible, and indeed beneficial, to decouple optimization-based meta-learning techniques from the high-dimensional space of model parameters. We achieve this by learning a stochastic latent space with an information bottleneck, conditioned on the input data, from which the high-dimensional parameters are generated.
+
+# Algorithm 1 Latent Embedding Optimization
+
+Require: Training meta-set $S ^ { t r } \in \mathcal { T }$   
+Require: Learning rates $\alpha$ , $\eta$   
+1: Randomly initialize $\phi _ { e } , \phi _ { r } , \phi _ { d }$   
+2: Let $\phi = \{ \phi _ { e } , \phi _ { r } , \phi _ { d } , \alpha \}$   
+3: while not converged do   
+4: for number of tasks in batch do   
+5: Sample task instance $\mathcal { T } _ { i } \sim \mathcal { S } ^ { t r }$   
+6: Let $\left( \hat { \mathcal { D } } ^ { t r } , \mathcal { D } ^ { v a l } \right) = \mathcal { T } _ { i }$   
+7: Encode Dtr to z using gφe and gφr   
+8: Decode z to initial params θi using gφd   
+9: Initialize $\mathbf { z } ^ { \prime } = \mathbf { z }$ , $\theta _ { i } ^ { \prime } = \theta _ { i }$   
+10: for number of adaptation steps do   
+11: Compute training loss $\mathcal { L } _ { T _ { i } } ^ { t \hat { r } } \left( f _ { \theta _ { i } ^ { \prime } } \right)$   
+12: Perform gradient step w.r.t. $\mathbf { z } ^ { \prime }$ :   
+$\mathbf { z } ^ { \prime }  \mathbf { z } ^ { \prime } - \alpha \nabla _ { \mathbf { z } ^ { \prime } } \mathcal { L } _ { T _ { i } } ^ { t r } ( f _ { \boldsymbol { \theta } _ { i } ^ { \prime } } )$   
+13: Decode $\mathbf { z } ^ { \prime }$ to obtain $\theta _ { i } ^ { \prime }$ using $g _ { \phi _ { d } }$   
+14: end for   
+15: Compute validation loss $\mathcal { L } _ { \mathcal { T } _ { i } } ^ { v a l } \left( f _ { \theta _ { i } ^ { \prime } } \right)$   
+16: end for   
+17: Perform gradient step w.r.t $\phi$ :   
+$\begin{array} { r l } { \phi  \phi - \eta \nabla _ { \phi } \sum _ { T _ { i } } \hat { \mathcal { L } } _ { T _ { i } } ^ { v a l } \big ( \dot { f } _ { \theta _ { i } ^ { \prime } } \big ) } & { { } } \end{array}$
+
+18: end while
+
+![](images/d36a3535f06afea22a8ecc199ec0938e1a5a71ef6a8dc8c7865c22888cfb693a.jpg)  
+Figure 1: High-level intuition for LEO. While MAML operates directly in a high dimensional parameter space $\Theta$ , LEO performs meta-learning within a low-dimensional latent space $\mathcal { Z }$ , from which the parameters are generated.
+
+![](images/92776299aaa677b30d18985db80f35833fe6593d7b2c854d63d506c65b362899.jpg)  
+Figure 2: Overview of the architecture of LEO.
+
+Instead of explicitly instantiating and maintaining a unique set of model parameters $\theta$ , as in MAML, we learn a generative distribution of model parameters which serves the same purpose. This is a natural extension: we relax the requirement of finding a single optimal $\theta ^ { \ast } \in \Theta$ to that of approximating a data-dependent conditional probability distribution over $\Theta$ , which can be more expressive. The choice of architecture, composed of an encoding process, and decoding (or parameter generation) process, enables us to perform the MAML gradient-based adaptation steps (or “inner loop”) in the learned, low-dimensional embedding space of the parameter generative model (Figure 1).
+
+# 2.3.1 MODEL OVERVIEW
+
+The high-level operation is then as follows (Algorithm 1). First, given a task instance $\mathcal { T } _ { i }$ , the inputs $\{ \mathbf { x } _ { n } ^ { k } \}$ are passed through a stochastic encoder to produce a latent code $\mathbf { z }$ , which is then decoded to parameters $\theta _ { i }$ using a parameter generator1. Given these instantiated model parameters, one or more adaptation steps are applied in the latent space, by differentiating the loss with respect to $\mathbf { z }$ , taking a gradient step to get $\mathbf { z } ^ { \prime }$ , decoding new model parameters, and obtaining the new loss. Finally, optimized codes are decoded to produce the final adapted parameters $\theta _ { i } ^ { \prime }$ , which can be used to perform the task, or compute the task-specific meta-loss. In this way, LEO incorporates aspects of model-based and optimization-based meta-learning, producing parameters that are first conditioned on the input data and then adapted by gradient descent.
+
+Figure 2 shows the architecture of the resulting network. Intuitively, the decoder is akin to a generative model, mapping from a low-dimensional latent code to a distribution over model parameters. The encoding process ensures that the initial latent code and parameters before gradient-based adaptation are already data-dependent. This encoding process also exploits a relation network that allows the latent code to be context-dependent, considering the pairwise relationship between all classes in the problem instance. In the following sections, we explain the LEO procedure more formally.
+
+# 2.3.2 INITIALIZATION: GENERATING PARAMETERS CONDITIONED ON A FEW EXAMPLES
+
+The first stage is to instantiate the model parameters that will be adapted to each task instance. Whereas MAML explicitly maintains a single set of model parameters, LEO utilises a datadependent latent encoding which is then decoded to generate the actual initial parameters. In what follows, we describe an encoding scheme which leverages a relation network to map the few-shot examples into a single latent vector. This design choice allows the approach to consider context when producing a parameter initialization. Intuitively, decision boundaries required for fine-grained distinctions between similar classes might need to be different from those for broader classification.
+
+Encoding The encoding process involves a simple feed-forward mapping of each data point, followed by a relation network that considers the pair-wise relationship between the data in the problem instance. The overall encoding process is defined in Eq. 3, and proceeds as follows. First, each example from a problem instance $\mathcal { T } _ { i } = \left( \mathcal { D } ^ { t r } , \mathcal { D } ^ { v a l } \right) \sim \dot { \mathbf { \sigma } } p \left( \mathcal { T } \right)$ is processed by an encoder network $g _ { \phi _ { e } } : \mathcal { R } ^ { n _ { x } }  \mathcal { R } ^ { n _ { h } }$ , which maps from input space to a code in an intermediate hidden-layer code space $\mathcal { H }$ . Then, codes in $\mathcal { H }$ corresponding to different training examples are concatenated pair-wise (resulting in $( N K ) ^ { 2 }$ pairs in the case of $K$ -shot classification) and processed by a relation network $g _ { \phi _ { r } }$ , in a similar fashion to Oreshkin et al. (2018) and Sung et al. (2017). The $( N K ) ^ { 2 }$ outputs are grouped by class and averaged within each group to obtain the $( 2 \times N )$ parameters of a probability distribution in a low-dimensional space $\mathcal { Z } = \mathcal { R } ^ { n _ { z } }$ , where $n _ { z } \ll \dim ( \theta )$ , for each of the $N$ classes.
+
+Thus, given the $K$ -shot training samples corresponding to a class $n$ : $\mathcal { D } _ { n } ^ { t r } ~ = ~ \left\{ \left( \mathbf { x } _ { n } ^ { k } , y _ { n } ^ { k } \right) ~ \vert ~ k ~ = ~ \right.$ $1 \ldots K \}$ the encoder $g _ { \phi _ { e } }$ and relation network $g _ { \phi _ { r } }$ together parameterize a class-conditional multivariate Gaussian distribution with a diagonal covariance, which we can sample from in order to output a class-dependent latent code $\mathbf { z } _ { n } \in { \mathcal { Z } }$ as follows:
+
+$$
+\begin{array} { c } { { \displaystyle { \pmb { \mu } } _ { n } ^ { e } , { \pmb \sigma } _ { n } ^ { e } = \frac 1 { N K ^ { 2 } } \sum _ { k _ { n } = 1 } ^ { K } \sum _ { m = 1 } ^ { N } \sum _ { k _ { m } = 1 } ^ { K } g _ { \phi _ { r } } \left( g _ { \phi _ { e } } \left( \mathbf { x } _ { n } ^ { k _ { n } } \right) , g _ { \phi _ { e } } \left( \mathbf { x } _ { m } ^ { k _ { m } } \right) \right) } } \\ { { \mathbf { z } _ { n } \sim q \left( \mathbf { z } _ { n } | \mathcal { D } _ { n } ^ { t r } \right) = \mathcal { N } \left( \mu _ { n } ^ { e } , d i a g ( { \pmb \sigma } _ { n } ^ { e 2 } ) \right) } } \end{array}
+$$
+
+Intuitively, the encoder and relation network define a stochastic mapping from one or more class examples to a single code in the latent embedding space $\mathcal { Z }$ corresponding to that class. The final latent code can be obtained as the concatenation of class-dependent codes: ${ \bf z } = [ { \bf z } _ { 1 } , { \bf z } _ { 2 } , \ldots , { \bf z } _ { N } ]$ .
+
+Decoding Without loss of generality, for few-shot classification, we can use the class-specific latent codes to instantiate just the top layer weights of the classifier. This allows the meta-learning in latent space to modulate the important high-level parameters of the classifier, without requiring the generator to produce very high-dimensional parameters. In this case, $f _ { \theta _ { i } ^ { \prime } }$ is a $N$ -way linear softmax classifier, with model parameters $\theta _ { i } ^ { \prime } = \left\{ \mathbf { w } _ { n } \mid n = 1 \ldots N \right\}$ , and each $\mathbf { x } _ { n } ^ { k }$ can be either the raw input or some learned representation2. Then, given the latent codes $\mathbf { z } _ { n } \in \mathcal { Z } , n = 1 \ldots N$ , the decoder function $g _ { \phi _ { d } } \colon { \mathcal { Z } } \to \Theta$ is used to parameterize a Gaussian distribution with diagonal covariance in model parameter space $\Theta$ , from which we can sample class-dependent parameters ${ \bf w } _ { n }$ :
+
+$$
+\begin{array} { c } { \mu _ { n } ^ { d } , { \pmb \sigma } _ { n } ^ { d } = g _ { \phi _ { d } } \left( \mathbf { z } _ { n } \right) } \\ { \mathbf { w } _ { n } \sim p \left( \mathbf { w } | \mathbf { z } _ { n } \right) = \mathcal { N } \left( \pmb { \mu } _ { n } ^ { d } , d i a g ( { \pmb \sigma } _ { n } ^ { d } ^ { 2 } ) \right) } \end{array}
+$$
+
+In other words, codes $\mathbf { z } _ { n }$ are mapped independently to the top-layer parameters $\theta _ { i }$ of a softmax classifier using the decoder $g _ { \phi _ { d } }$ , which is essentially a stochastic generator of model parameters.
+
+# 2.3.3 ADAPTATION BY LATENT EMBEDDING OPTIMIZATION (LEO) (THE “INNER LOOP”)
+
+Given the decoded parameters, we can then define the “inner loop” classification loss using the cross-entropy function, as follows:
+
+$$
+\mathcal { L } _ { \mathcal { T } _ { i } } ^ { t r } \left( f _ { \theta _ { i } } \right) = \sum _ { ( \mathbf { x } , y ) \in \mathcal { D } ^ { t r } } \bigg [ - \mathbf { w } _ { y } \cdot \mathbf { x } + \log \Big ( \sum _ { j = 1 } ^ { N } e ^ { \mathbf { w } _ { j } \cdot \mathbf { x } } \Big ) \bigg ]
+$$
+
+It is important to note that the decoder $g _ { \phi _ { d } }$ is a differentiable mapping between the latent space $\mathcal { Z }$ and the higher-dimensional model parameter space $\Theta$ . Primarily, this allows gradient-based optimization of the latent codes with respect to the training loss, with ${ \bf z } _ { n } ^ { \prime } = { \bf z } _ { n } - \alpha \nabla _ { { \bf z } _ { n } } \mathcal { L } _ { T _ { i } } ^ { t r }$ . The decoder $g _ { \phi _ { d } }$ will convert adapted latent codes $\mathbf { z } _ { n } ^ { \prime }$ to effective model parameters $\theta _ { i } ^ { \prime }$ for each adaptation step, which can be repeated several times, as in Algorithm 1. In addition, by backpropagating errors through the decoder, the encoder and relation net can learn to provide a data-conditioned latent encoding $\mathbf { z }$ that produces an appropriate initialization point $\theta _ { i }$ for the classifier model.
+
+# 2.3.4 META-TRAINING STRATEGY (THE “OUTER LOOP”)
+
+For each task instance $\mathcal { T } _ { i }$ , the initialization and adaptation procedure produce a new classifier $f _ { \theta _ { i } ^ { \prime } }$ tailored to the training set $\mathcal { D } ^ { t r }$ of the instance, which we can then evaluate on the validation set of that instance $\mathcal { D } ^ { v a l }$ . During meta-training we use that evaluation to differentiate through the “inner loop” and update the encoder, relation, and decoder network parameters: $\phi _ { e }$ , $\phi _ { r }$ , and $\phi _ { d }$ . Meta-training is performed by minimizing the following objective:
+
+$$
+\operatorname* { m i n } _ { \phi _ { c } , \phi _ { r } , \phi _ { d } } \sum _ { T _ { i } \sim p ( T ) } \left[ \mathcal { L } _ { T _ { i } } ^ { v a l } \left( f _ { \theta _ { i } ^ { \prime } } \right) + \beta D _ { K L } \left( q ( \mathbf { z } _ { n } | \mathcal { D } _ { n } ^ { t r } ) | | p ( \mathbf { z } _ { n } ) \right) + \gamma | | \mathrm { s t o p g r a d } ( \mathbf { z } _ { n } ^ { \prime } ) - \mathbf { z } _ { n } | | _ { 2 } ^ { 2 } \right] + R ( \mathbf { z } _ { n } | \mathbf { z } _ { n } ) ,
+$$
+
+where $p ( \mathbf { z } _ { n } ) = \mathcal { N } ( 0 , \mathcal { T } )$ . Similar to the loss defined in (Higgins et al., 2017) we use a weighted KL-divergence term to regularize the latent space and encourage the generative model to learn a disentangled embedding, which should also simplify the LEO “inner loop” by removing correlations between latent space gradient dimensions. The third term in Eq. (6) encourages the encoder and relation net to output a parameter initialization that is close to the adapted code, thereby reducing the load of the adaptation procedure if possible.
+
+$L _ { 2 }$ regularization was used with all weights of the model, as well as a soft, layer-wise orthogonality constraint on decoder network weights, which encourages the dimensions of the latent code as well as the decoder network to be maximally expressive. In the case of linear encoder, relation, and decoder networks, and assuming that $\mathcal { C } _ { d }$ is the correlation matrix between rows of $\phi _ { d }$ , then the regularization term takes the following form:
+
+$$
+R = \lambda _ { 1 } \Big ( | | \phi _ { e } | | _ { 2 } ^ { 2 } + | | \phi _ { r } | | _ { 2 } ^ { 2 } + | | \phi _ { d } | | _ { 2 } ^ { 2 } \Big ) + \lambda _ { 2 } | | \mathcal { C } _ { d } - \mathcal { T } | | _ { 2 }
+$$
+
+# 2.3.5 BEYOND CLASSIFICATION AND LINEAR OUTPUT LAYERS
+
+Thus far we have used few-shot classification as a working example to highlight our proposed method, and in this domain we generate only a single linear output layer. However, our approach can be applied to any model $f _ { \theta _ { i } }$ which maps observations to outputs, e.g. a nonlinear MLP or LSTM, by using a single latent code $\mathbf { z }$ to generate the entire parameter vector $\theta _ { i }$ with an appropriate decoder. In the general case, $\mathbf { z }$ is conditioned on $\mathcal { D } ^ { t r }$ by passing both inputs and labels to the encoder. Furthermore, the loss $\mathcal { L } _ { T _ { i } }$ is not restricted to be a classification loss, and can be replaced by any differentiable loss function which can be computed on $\mathcal { D } ^ { t r }$ and $\mathcal { D } ^ { v a l }$ sets of a task instance $\mathcal { T } _ { i }$ .
+
+# 3 RELATED WORK
+
+The problem of few-shot adaptation has been approached in the context of fast weights (Hinton & Plaut, 1987; Ba et al., 2016), learning-to-learn (Schmidhuber, 1987; Thrun & Pratt, 1998; Hochreiter et al., 2001; Andrychowicz et al., 2016), and through meta-learning. Many recent approaches to meta-learning can be broadly categorized as metric-based methods, which focus on learning similarity metrics for members of the same class (e.g. Koch et al., 2015; Vinyals et al., 2016; Snell et al.,
+
+2017); memory-based methods, which exploit memory architectures to store key training examples or directly encode fast adaptation algorithms (e.g. Santoro et al., 2016; Ravi & Larochelle, 2017); and optimization-based methods, which search for parameters that are conducive to fast gradientbased adaptation to new tasks (e.g. Finn et al., 2017; 2018).
+
+Related work has also explored the use of one neural network to produce (some fraction of) the parameters of another (Ha et al., 2016; Krueger et al., 2017), with some approaches focusing on the goal of fast adaptation. Munkhdalai et al. (2017) meta-learn an algorithm to change additive biases across deep networks conditioned on the few-shot training samples. In contrast, Gidaris & Komodakis (2018) use an attention kernel to output class conditional mixing of linear output weights for novel categories, starting from a pre-trained deep model. Qiao et al. (2017) learn to output top linear layer parameters from the activations provided by a pre-trained feature embedding, but they do not make use of gradient-based adaptation. None of the aforementioned approaches to fast adaptation explicitly learn a probability distribution over model parameters, or make use of latent variable generative models to characterize it.
+
+Approaches which use optimization-based meta-learning include MAML (Finn et al., 2017) and REPTILE (Nichol & Schulman, 2018). While MAML backpropagates the meta-loss through the “inner loop”, REPTILE simplifies the computation by incorporating an $L _ { 2 }$ loss which updates the meta-model parameters towards the instance-specific adapted models. These approaches use the full, high-dimensional set of model parameters within the “inner loop”, while Lee & Choi (2018) learn a layer-wise subspace in which to use gradient-based adaptation. However, it is not clear how these methods scale to large expressive models such as residual networks (especially given the uncertainty in the few-shot data regime), since MAML is prone to overfitting (Mishra et al., 2018). Recognizing this issue, Zhou et al. (2018) train a deep input representation, or “concept space”, and use it as input to an MLP meta-learner, but perform gradient-based adaptation directly in its parameter space, which is still comparatively high-dimensional. As we will show, performing adaptation in latent space to generate a simple linear layer can lead to superior generalization.
+
+Probabilistic meta-learning approaches such as those of Bauer et al. (2017) and Grant et al. (2018) have shown the advantages of learning Gaussian posteriors over model parameters. Concurrently with our work, Kim et al. (2018) and Finn et al. (2018) propose probabilistic extensions to MAML that are trained using a variational approximation, using simple posteriors. However, it is not immediately clear how to extend them to more complex distributions with a more diverse set of tasks. Other concurrent works have introduced deep parameter generators (Lacoste et al., 2018; Wu et al., 2018) that can better capture a wider distribution of model parameters, but do not employ gradientbased adaptation. In contrast, our approach employs both a generative model of parameters, and adaptation in a low-dimensional latent space, aided by a data-dependent initialization.
+
+Finally, recently proposed Neural Processes (Garnelo et al., 2018a;b) bear similarity to our work: they also learn a mapping to and from a latent space that can be used for few-shot function estimation. However, coming from a Gaussian processes perspective, their work does not perform “inner loop” adaptation and is trained by optimizing a variational objective.
+
+# 4 EVALUATION
+
+We evaluate the proposed approach on few-shot regression and classification tasks. This evaluation aims to answer the following key questions: (1) Is LEO capable of modeling a distribution over model parameters when faced with uncertainty? (2) Can LEO learn from multimodal task distributions and is this reflected in ambiguous problem instances, where multiple distinct solutions are possible? (3) Is LEO competitive on large-scale few-shot learning benchmarks?
+
+# 4.1 FEW-SHOT REGRESSION
+
+To answer the first two questions we adopt the simple regression task of Finn et al. (2018). 1D regression problems are generated in equal proportions using either a sine wave with random amplitude and phase, or a line with random slope and intercept. Inputs are sampled randomly, creating a multimodal task distribution. Crucially, random Gaussian noise with standard deviation 0.3 is added to regression targets. Coupled with the small number of training samples (5-shot), the task is challenging for 2 main reasons: (1) learning a distribution over models becomes necessary, in order to account for the uncertainty introduced by noisy labels; (2) problem instances may be likely under both modes: in some cases a sine wave may fit the data as well as a line. Faced with such ambiguity, learning a generative distribution of model parameters should allow several different likely models to be sampled, in a similar way to how generative models such as VAEs can capture different modes of a multimodal data distribution.
+
+We used a 3-layer MLP as the underlying model architecture of $f _ { \theta }$ , and we produced the entire parameter tensor $\theta$ with the LEO generator, conditionally on $D ^ { t r }$ , the few-shot training inputs concatenated with noisy labels. For further details, see Appendix A.
+
+![](images/e0603af67139ec9240c2466cd41f1a1a3ff104a664a7aa1c98cf4f6fa7025ac2.jpg)  
+Figure 3: Meta-learning with LEO of a multimodal task distribution with sines and lines, using 5-shot regression with noisy targets. Our model outputs a distribution of possible solutions, which is also multimodal in ambiguous cases. True regression targets are plotted in black, while the 5 training examples are highlighted with red circles and vertical dashed lines. Several samples from our model are plotted with dotted lines (best seen in color).
+
+In Figure 3 we show samples from a single model trained on noisy sines and lines, with true regression targets in black and training samples marked with red circles and vertical dashed lines. Plots (a) and (b) illustrate how LEO captures some of the uncertainty in ambiguous problem instances within each mode, especially in parts of the input space far from any training samples. Conversely, in parts which contain data, models fit the regression target well. Interestingly, when both sines and lines could explain the data, as shown in panels (c) and (d), we see that LEO can sample very different models, from both families, reflecting its ability to represent parametric uncertainty appropriately.
+
+# 4.2 FEW-SHOT CLASSIFICATION
+
+In order to answer the final question we scale up our approach to 1-shot and 5-shot classification problems defined using two commonly used ImageNet subsets.
+
+# 4.2.1 DATASETS
+
+The miniImageNet dataset (Vinyals et al., 2016) is a subset of 100 classes selected randomly from the ILSVRC-12 dataset (Russakovsky et al., 2014) with 600 images sampled from each class. Following the split proposed by Ravi & Larochelle (2017), the dataset is divided into training, validation, and test meta-sets, with 64, 16, and 20 classes respectively.
+
+The tieredImageNet dataset (Ren et al., 2018) is a larger subset of ILSVRC-12 with 608 classes (779,165 images) grouped into 34 higher-level nodes in the ImageNet human-curated hierarchy (Deng et al., 2009a). This set of nodes is partitioned into 20, 6, and 8 disjoint sets of training, validation, and testing nodes, and the corresponding classes form the respective meta-sets. As argued in Ren et al. (2018), this split near the root of the ImageNet hierarchy results in a more challenging, yet realistic regime with test classes that are less similar to training classes.
+
+# 4.2.2 PRE-TRAINED FEATURES
+
+Two potential difficulties of using LEO to instantiate parameters with a generator network are: (1) modeling distributions over very high-dimensional parameter spaces; and (2) requiring metalearning (and hence, gradient computation in the inner loop) to be performed with respect to a high-dimensional input space. We address these issues by pre-training a visual representation of the data and then using the generator to instantiate the parameters for the final layer - a linear softmax classifier operating on this representation. We train a 28-layer Wide Residual Network (WRN-28- 10) (Zagoruyko & Komodakis, 2016a) with supervised classification using only data and classes from the training meta-set. Recent state-of-the-art approaches use the penultimate layer representation (Zhou et al., 2018; Qiao et al., 2017; Bauer et al., 2017; Gidaris & Komodakis, 2018); however, we choose the intermediate feature representation in layer 21, given that higher layers tend to specialize to the training distribution (Yosinski et al., 2014). For details regarding the training, evaluation, and network architectures, see Appendix B.
+
+# 4.2.3 FINE-TUNING
+
+Following the LEO adaptation procedure (Algorithm 1) we also use fine-tuning3 by performing a few steps of gradient-based adaptation directly in parameter space using the few-shot set $\mathcal { D } ^ { t r }$ . This is similar to the adaptation procedure of MAML, or Meta-SGD (Li et al., 2017) when the learning rates are learned, with the important difference that starting points of fine-tuning are custom generated by LEO for every task instance $\mathcal { T } _ { i }$ . Empirically, we find that fine-tuning applies a very small change to the parameters with only a slight improvement in performance on supervised classification tasks.
+
+# 4.3 RESULTS
+
+<table><tr><td>Model</td><td colspan="2">miniImageNet test accuracy</td></tr><tr><td></td><td>1-shot</td><td>5-shot</td></tr><tr><td>Matching networks (Vinyals et al.,2016) Meta-learnerLSTM(Ravi &amp;Larochelle,2017)</td><td>43.56± 0.84% 43.44±0.77%</td><td>55.31±0.73%</td></tr><tr><td>MAML (Finn et al.,2017)</td><td></td><td>60.60 ± 0.71%</td></tr><tr><td></td><td>48.70±1.84%</td><td>63.11 ± 0.92%</td></tr><tr><td>LLAMA (Grant et al.,2018) REPTILE (Nichol &amp; Schulman,2018)</td><td>49.40 ± 1.83%</td><td></td></tr><tr><td>PLATIPUS (Finn et al., 2018)</td><td>49.97 ±0.32% 50.13 ±1.86%</td><td>65.99 ± 0.58%</td></tr><tr><td>Meta-SGD (our features)</td><td>54.24±0.03%</td><td></td></tr><tr><td>SNAIL (Mishra et al.,2018)</td><td>55.71 ±0.99%</td><td>70.86 ± 0.04%</td></tr><tr><td>(Gidaris &amp; Komodakis,2018)</td><td></td><td>68.88 ±0.92%</td></tr><tr><td>(Bauer et al.,2017)</td><td>56.20±0.86%</td><td>73.00 ± 0.64%</td></tr><tr><td>(Munkhdalai et al.,2017)</td><td>56.30 ±0.40%</td><td>73.90 ± 0.30%</td></tr><tr><td>DEML+Meta-SGD (Zhou et al.,2018) 4</td><td>57.10±0.70%</td><td>70.04± 0.63%</td></tr><tr><td></td><td>58.49 ± 0.91%</td><td>71.28 ± 0.69%</td></tr><tr><td>TADAM(Oreshkin et al.,2018)</td><td>58.50±0.30%</td><td>76.70±0.30%</td></tr><tr><td>(Qiao et al., 2017)</td><td>59.60 ± 0.41%</td><td>73.74± 0.19%</td></tr><tr><td>LEO (ours)</td><td>61.76 ±0.08%</td><td>77.59 ±0.12%</td></tr><tr><td rowspan="2">Model</td><td>tieredImageNet test accuracy</td><td></td></tr><tr><td>1-shot</td><td>5-shot</td></tr><tr><td rowspan="4">MAML(deeper net,evaluated in Liu et al.(2018)) Prototypical Nets (Ren et al.,2018) Relation Net (evaluated in Liu et al. (2018)) Transductive Prop.Nets (Liu et al.,2018)</td><td>51.67 ± 1.81%</td><td>70.30 ± 0.08%</td></tr><tr><td>53.31± 0.89%</td><td>72.69 ±0.74%</td></tr><tr><td>54.48 ± 0.93%</td><td>71.32 ± 0.78%</td></tr><tr><td>57.41 ± 0.94%</td><td>71.55 ± 0.74%</td></tr><tr><td rowspan="2">Meta-SGD (our features) LEO (ours)</td><td>62.95±0.03%</td><td>79.34± 0.06%</td></tr><tr><td>66.33 ± 0.05%</td><td>81.44 ±0.09%</td></tr></table>
+
+Table 1: Test accuracies on miniImageNet and tieredImageNet. For each dataset, the first set of results use convolutional networks, while the second use much deeper residual networks, predominantly in conjuction with pre-training.
+
+The classification accuracies for LEO and other baselines are shown in Table 1. LEO sets the new state-of-the-art performance on the 1-shot and 5-shot tasks for both miniImageNet and tieredImageNet datasets. We also evaluated LEO on the “multi-view” feature representation used by Qiao et al. (2017) with miniImageNet, which involves significant data augmentation compared to the approaches in Table 1. LEO is state-of-the-art using these features as well, with $6 3 . 9 7 \pm 0 . 2 0 \%$ and $7 9 . 4 9 \pm 0 . 7 0 \%$ test accuracies on the 1-shot and 5-shot tasks respectively.
+
+# 4.4 ABLATION STUDY
+
+To assess the effects of different components, we also performed an ablation study, with detailed results in Table 2. To ensure a fair comparison, all approaches begin with the same pre-trained features (Section 4.2.2). The Meta-SGD case performs gradient-based adaption directly in the parameter space in the same way as MAML, but also meta-learns the inner loop learning rate (as we do for LEO). The main approach, labeled as LEO in the table, uses a stochastic parameter generator for several steps of latent embedding optimization, followed by fine-tuning steps in parameter space (see subsection 4.2.3). All versions of LEO are at or above the previous state-of-the-art on all tasks.
+
+Table 2: Ablation study and comparison to Meta-SGD. Unless otherwise specified, LEO stands for using the stochastic generator for latent embedding optimization followed by fine-tuning.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">miniImageNet test accuracy</td><td colspan="2">tieredImageNet test accuracy</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Meta-SGD (our features)</td><td>54.24 ±0.03%</td><td>70.86 ±0.04%</td><td>62.95 ±0.03%</td><td>79.34± 0.06%</td></tr><tr><td>Conditional generator only</td><td>60.33 ± 0.11%</td><td>74.53 ± 0.11%</td><td>65.17±0.15%</td><td>78.77 ± 0.03%</td></tr><tr><td>Conditional generator + fine-tuning</td><td>60.62 ± 0.31%</td><td>76.42 ±0.09%</td><td>65.74±0.28%</td><td>80.65 ±0.07%</td></tr><tr><td>Previous SOTA</td><td>59.60 ± 0.41%</td><td>76.70±0.30%</td><td>57.41 ± 0.94%</td><td>72.69 ± 0.74%</td></tr><tr><td>LEO (random prior)</td><td>61.01±0.12%</td><td>77.27±0.05%</td><td>65.39±0.10%</td><td>80.83±0.13%</td></tr><tr><td>LEO (deterministic)</td><td>61.48 ± 0.05%</td><td>76.53 ± 0.24%</td><td>66.18 ± 0.17%</td><td>82.06 ±0.08%</td></tr><tr><td>LEO (no fine-tuning)</td><td>61.62 ±0.15%</td><td>77.46 ± 0.12%</td><td>66.14± 0.17%</td><td>80.89 ± 0.11%</td></tr><tr><td>LEO (ours)</td><td>61.76 ± 0.08%</td><td>77.59±0.12%</td><td>66.33 ± 0.05%</td><td>81.44± 0.09%</td></tr></table>
+
+The largest difference in performance is between Meta-SGD and the other cases (all of which exploit a latent representation of model parameters), indicating that the low-dimensional bottleneck is critical for this application. The “conditional generator only” case (without adaptation in latent space) yields a poorer result than LEO, and even adding fine-tuning in parameter space does not recover performance; this illustrates the efficacy of the latent adaptation procedure. The importance of the data-dependent encoding is highlighted by the “random prior” case, in which the encoding process is replaced by the prior $p ( \mathbf { z } _ { n } )$ , and performance decreases. We also find that incorporating stochasticity can be important for miniImageNet, but not for tieredImageNet, which we hypothesize is because the latter is much larger. Finally, the fine-tuning steps only yield a statistically significant improvement on the 5-shot tieredImageNet task. Thus, both the data-conditional encoding and latent space adaptation are critical to the performance of LEO.
+
+![](images/d5347d4c964c5b780322b9a7bc2beca8ecdb0546997db2269ed57395bb263277.jpg)  
+Figure 4: t-SNE plot of latent space codes before and after adaptation: (a) Initial codes $\mathbf { z } _ { n }$ (blue) and adapted codes $\mathbf { z } _ { n } ^ { \prime }$ (orange); (b) Same as (a) but colored by class; (c) Same as (a) but highlighting codes $\mathbf { z } _ { n }$ for validation class “Jellyfish” (left) and corresponding adapted codes $ { \mathbf { z } } _ { n } ^ { \prime }$ (right).
+
+# 4.5 LATENT EMBEDDING VISUALIZATION
+
+To qualitatively characterize the learnt embedding space, we plot codes produced by the relational encoder before and after the LEO procedure, using a 5-way 1-shot model and 1000 task instances from the validation meta-set of miniImageNet. Figure 4 shows a t-SNE projection of class conditional encoder outputs $\mathbf { z } _ { n }$ as well as their respective final adapted versions $ { \mathbf { z } } _ { n } ^ { \prime }$ . If the effect of LEO were minimal, we would expect latent codes to have roughly the same structure before and after adaptation. In contrast, Figure 4(a) clearly shows that latent codes change substantially during LEO, since encoder output codes form a large cluster (blue) to which adapted codes (orange) do not belong. Figure 4(b) shows the same t-SNE embedding as (a) colored by class label. Note that encoder outputs, on the left side of plot (b), have a lower degree of class conditional separation compared to $ { \mathbf { z } } _ { n } ^ { \prime }$ clusters on the right, suggesting that qualitatively different structure is introduced by the LEO procedure. We further illustrate this point by highlighting latent codes for the “Jellyfish” validation class in Figure 4(c), which are substantially different before and after adaptation.
+
+![](images/13fabddbc95af362f29bb99ed5df2cca409bfbeafcd52da7edb50211e49d5f86.jpg)  
+Figure 5: Curvature and coverage metrics for a number of different models, computed over 1000 problem instances drawn uniformly from the test meta-set. For all plots, the whiskers span from the $\mathrm { { \bar { 5 } ^ { t h } } }$ to $9 5 ^ { \mathrm { t h } }$ percentile of the observed quantities.
+
+The additional structure of adapted codes $\mathbf { z } _ { n } ^ { \prime }$ may explain LEO’s superior performance over approaches predicting parameters directly from inputs, since the decoder may not be able to produce sufficiently different weights for different classes given very similar latent codes, especially when the decoder is linear. Conversely, LEO can reduce the uncertainty of the encoder mapping, which is inherent in the few-shot regime, by adapting latent codes with a generic, gradient-based procedure.
+
+# 4.6 CURVATURE AND COVERAGE ANALYSIS
+
+We hypothesize that by performing the inner-loop optimization in a lower-dimensional latent space, the adapted solutions do not need to be close together in parameter space, as each latent step can cover a larger region of parameter space and effect a greater change on the underlying function. To support this intuition, we compute a number of curvature and coverage measures, shown in Figure 5.
+
+The curvature provides a measure of the sensitivity of a function with respect to some space. If adapting in latent space allows as much control over the function as in parameter space, one would expect similar curvatures. However, as demonstrated in Figure 5(a), the curvature for LEO in $\mathbf { z }$ space (the absolute eigenvalues of the Hessian of the loss) is 2 orders of magnitude higher than in $\theta$ , indicating that a fixed step in $\mathbf { z }$ will change the function more drastically than taking the same step directly in $\theta$ . This is also observed in the “gen+ft” case, where the latent embedding is still used, but adaptation is performed directly in $\theta$ space. This suggests that the latent bottleneck is responsible for this effect. Figure 5(b) shows that this is due to the expansion of space caused by the decoder. In this case the decoder is linear, and the singular values describe how much a vector projected through this decoder grows along different directions, with a value of one preserving volume. We observe that the decoder is expanding the space by at least one order of magnitude. Finally, Figure 5(c) demonstrates this effect along the specific gradient directions used in the inner loop adaptation: the small gradient steps in $\mathbf { z }$ taken by LEO induce much larger steps in $\theta$ space, larger than the gradient steps taken by Meta-SGD in $\theta$ space directly. Thus, the results support the intuition that LEO is able to ‘transport’ models further during adaptation by performing meta-learning in the latent space.
+
+# 5 CONCLUSIONS AND FUTURE WORK
+
+We have introduced Latent Embedding Optimization (LEO), a meta-learning technique which uses a parameter generative model to capture the diverse range of parameters useful for a distribution over tasks, and demonstrated a new state-of-the-art result on the challenging 5-way 1- and 5-shot miniImageNet and tieredImageNet classification problems. LEO achieves this by learning a lowdimensional data-dependent latent embedding, and performing gradient-based adaptation in this space, which means that it allows for a task-specific parameter initialization and can perform adaptation more effectively.
+
+Future work could focus on replacing the pre-trained feature extractor with one learned jointly through meta-learning, or using LEO for tasks in reinforcement learning or with sequential data.
+
+# REFERENCES
+
+Marcin Andrychowicz, Misha Denil, Sergio Gomez, Matthew W Hoffman, David Pfau, Tom Schaul, Brendan Shillingford, and Nando De Freitas. Learning to learn by gradient descent by gradient descent. In Advances in Neural Information Processing Systems, pp. 3981–3989, 2016.
+
+Jimmy Ba, Geoffrey E Hinton, Volodymyr Mnih, Joel Z Leibo, and Catalin Ionescu. Using fast weights to attend to the recent past. In Advances in Neural Information Processing Systems, pp. 4331–4339, 2016.
+
+M. Bauer, M. Rojas-Carulla, J. Bartłomiej Swia¸tkowski, B. Sch ´ olkopf, and R. E. Turner. Discrimi- ¨ native k-shot learning using probabilistic models. ArXiv e-prints, June 2017.
+
+Kyunghyun Cho, Bart van Merrienboer, C¸ aglar Gulc¸ehre, Fethi Bougares, Holger Schwenk, and ¨ Yoshua Bengio. Learning phrase representations using RNN encoder-decoder for statistical machine translation. CoRR, abs/1406.1078, 2014. URL http://arxiv.org/abs/1406. 1078.
+
+J. Deng, W. Dong, R. Socher, L. Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248–255, June 2009a.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 248–255. IEEE, 2009b.
+
+Li Fei-Fei, Rob Fergus, and Pietro Perona. One-shot learning of object categories. IEEE transactions on pattern analysis and machine intelligence, 28(4):594–611, 2006.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In International Conference on Machine Learning, pp. 1126–1135, 2017.
+
+Chelsea Finn, Kelvin Xu, and Sergey Levine. Probabilistic model-agnostic meta-learning. arXiv preprint arXiv:1806.02817, 2018.
+
+Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and S. M. Ali Eslami. Conditional neural processes. In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 1704–1713, Stockholmsmssan, Stockholm Sweden, 10–15 Jul 2018a. PMLR. URL http://proceedings.mlr.press/v80/garnelo18a.html.
+
+Marta Garnelo, J. Schwarz, D. Rosenbaum, F. Viola, D. J. Rezende, S. M. A. Eslami, and Y. Whye Teh. Neural Processes. ArXiv e-prints, July 2018b.
+
+Spyros Gidaris and Nikos Komodakis. Dynamic few-shot visual learning without forgetting. CoRR, abs/1804.09458, 2018. URL http://arxiv.org/abs/1804.09458.
+
+Erin Grant, Chelsea Finn, Sergey Levine, Trevor Darrell, and Thomas Griffiths. Recasting gradientbased meta-learning as hierarchical bayes. In Proceedings of the 6th International Conference on Learning Representations (ICLR), 2018.
+
+David Ha, Andrew Dai, and Quoc V Le. Hypernetworks. arXiv preprint arXiv:1609.09106, 2016.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. Beta-VAE: Learning basic visual concepts with a constrained variational framework. ICLR, 2017.
+
+Geoffrey E Hinton and David C Plaut. Using fast weights to deblur old memories. In Proceedings of the ninth annual conference of the Cognitive Science Society, pp. 177–186, 1987.
+
+Sepp Hochreiter, A. Steven Younger, and Peter R. Conwell. Learning to learn using gradient descent. In Proceedings of the International Conference on Artificial Neural Networks, ICANN ’01, pp. 87–94, London, UK, UK, 2001. Springer-Verlag. ISBN 3-540-42486-5. URL http://dl. acm.org/citation.cfm?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ 646258.684281.
+
+T. Kim, J. Yoon, O. Dia, S. Kim, Y. Bengio, and S. Ahn. Bayesian Model-Agnostic Meta-Learning. ArXiv e-prints, June 2018.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. CoRR, abs/1412.6980, 2014. URL http://arxiv.org/abs/1412.6980.
+
+Gregory Koch, Richard Zemel, and Ruslan Salakhutdinov. Siamese neural networks for one-shot image recognition. In ICML Deep Learning Workshop, volume 2, 2015.
+
+D. Krueger, C.-W. Huang, R. Islam, R. Turner, A. Lacoste, and A. Courville. Bayesian Hypernetworks. ArXiv e-prints, October 2017.
+
+A. Lacoste, B. Oreshkin, W. Chung, T. Boquet, N. Rostamzadeh, and D. Krueger. Uncertainty in Multitask Transfer Learning. ArXiv e-prints, June 2018.
+
+Brenden Lake, Ruslan Salakhutdinov, Jason Gross, and Joshua Tenenbaum. One shot learning of simple visual concepts. In Proceedings of the Annual Meeting of the Cognitive Science Society, volume 33, 2011.
+
+Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. nature, 521(7553):436, 2015.
+
+Y. Lee and S. Choi. Gradient-Based Meta-Learning with Learned Layerwise Metric and Subspace. ArXiv e-prints, January 2018.
+
+Zhenguo Li, Fengwei Zhou, Fei Chen, and Hang Li. Meta-sgd: Learning to learn quickly for few shot learning. CoRR, abs/1707.09835, 2017. URL http://arxiv.org/abs/1707. 09835.
+
+Yanbin Liu, Juho Lee, Minseop Park, Saehoon Kim, and Yi Yang. Transductive propagation network for few-shot learning. arXiv preprint arXiv:1805.10002, 2018.
+
+Nikhil Mishra, Mostafa Rohaninejad, Xi Chen, and Pieter Abbeel. A simple neural attentive metalearner. In Proceedings of the 6th International Conference on Learning Representations (ICLR), 2018.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, February 2015. ISSN 00280836. URL http://dx.doi.org/10.1038/nature14236.
+
+Tsendsuren Munkhdalai, Xingdi Yuan, Soroush Mehri, Tong Wang, and Adam Trischler. Learning rapid-temporal adaptations. CoRR, abs/1712.09926, 2017. URL http://arxiv.org/abs/ 1712.09926.
+
+Alex Nichol and John Schulman. Reptile: a scalable metalearning algorithm. arXiv preprint arXiv:1803.02999, 2018.
+
+B. N. Oreshkin, P. Rodriguez, and A. Lacoste. TADAM: Task dependent adaptive metric for improved few-shot learning. ArXiv e-prints, May 2018.
+
+Siyuan Qiao, Chenxi Liu, Wei Shen, and Alan Yuille. Few-shot image recognition by predicting parameters from activations. arXiv preprint arXiv:1706.03466, 2017.
+
+Sachin Ravi and Hugo Larochelle. Optimization as a model for few-shot learning. In Proceedings of the 5th International Conference on Learning Representations (ICLR), 2017.
+
+Mengye Ren, Sachin Ravi, Eleni Triantafillou, Jake Snell, Kevin Swersky, Josh B. Tenenbaum, Hugo Larochelle, and Richard S. Zemel. Meta-learning for semi-supervised few-shot classification. In International Conference on Learning Representations, 2018. URL https://openreview. net/forum?id ${ . } = { }$ HJcSzz-CZ.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael S. Bernstein, Alexander C. Berg, and FeiFei Li. Imagenet large scale visual recognition challenge. CoRR, abs/1409.0575, 2014. URL http://arxiv.org/abs/1409.0575.
+
+Adam Santoro, Sergey Bartunov, Matthew Botvinick, Daan Wierstra, and Timothy Lillicrap. Metalearning with memory-augmented neural networks. In International conference on machine learning, pp. 1842–1850, 2016.
+
+Jurgen Schmidhuber. ¨ Evolutionary principles in self-referential learning, or on learning how to learn: the meta-meta-... hook. PhD thesis, Technische Universitat M¨ unchen, 1987. ¨
+
+Jurgen Schmidhuber. Deep learning in neural networks: An overview. ¨ Neural networks, 61:85–117, 2015.
+
+David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, Yutian Chen, Timothy Lillicrap, Fan Hui, Laurent Sifre, George van den Driessche, Thore Graepel, and Demis Hassabis. Mastering the game of go without human knowledge. Nature, 550:354–, October 2017. URL http: //dx.doi.org/10.1038/nature24270.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
+
+Jake Snell, Kevin Swersky, and Richard S. Zemel. Prototypical networks for few-shot learning. CoRR, abs/1703.05175, 2017. URL http://arxiv.org/abs/1703.05175.
+
+Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip H. S. Torr, and Timothy M. Hospedales. Learning to compare: Relation network for few-shot learning. CoRR, abs/1711.06025, 2017. URL http://arxiv.org/abs/1711.06025.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V. Le. Sequence to sequence learning with neural networks. CoRR, abs/1409.3215, 2014. URL http://arxiv.org/abs/1409.3215.
+
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).
+
+Sebastian Thrun and Lorien Pratt. Learning to learn: Introduction and overview. In Learning to learn, pp. 3–17. Springer, 1998.
+
+Oriol Vinyals, Charles Blundell, Tim Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in Neural Information Processing Systems, pp. 3630–3638, 2016.
+
+T. Wu, J. Peurifoy, I. L. Chuang, and M. Tegmark. Meta-learning autoencoders for few-shot prediction. ArXiv e-prints, July 2018.
+
+Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? CoRR, abs/1411.1792, 2014. URL http://arxiv.org/abs/1411. 1792.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. In British Machine Vision Conference, 2016a.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. CoRR, abs/1605.07146, 2016b. URL http://arxiv.org/abs/1605.07146.
+
+Fengwei Zhou, Bin Wu, and Zhenguo Li. Deep meta-learning: Learning to learn in the concept space. CoRR, abs/1802.03596, 2018. URL http://arxiv.org/abs/1802.03596.
+
+# A EXPERIMENTAL SETUP - REGRESSION
+
+# A.1 REGRESSION TASK DESCRIPTION
+
+We used the experimental setup of Finn et al. (2018) for 1D 5-shot noisy regression tasks. Inputs were sampled uniformly from $[ - 5 , 5 ]$ . A multimodal task distribution was used. Half of the problem instances were sinusoids with amplitude and phase sampled uniformly from [0.1, 5] and $[ 0 , \pi ]$ respectively. The other half were lines with slope and intercept sampled uniformly from the interval $[ - 3 , 3 ]$ . Gaussian noise with standard deviation 0.3 was added to regression targets.
+
+# A.2 LEO NETWORK ARCHITECTURE
+
+As Table 3 shows, the underlying model $f _ { \theta }$ (for which parameters $\theta$ were generated) was a 3-layer MLP with 40 units in all hidden layers and rectifier nonlinearities. A single code $\mathbf { z }$ was used to generate $\theta$ with the decoder, conditioned on concatenated inputs and regression targets from $\mathcal { D } ^ { t r }$ which were passed as inputs to the encoder. Sampling of latent codes and parameters was used both during training and evaluation.
+
+The encoder was a 3-layer MLP with 32 units per layer and rectifier nonlinearities; the bottleneck embedding space size was: $n _ { z } = 1 6$ . The relation network and decoder were both 3-layer MLPs with 32 units per layer. For simplicity we did not use biases in any layer of the encoder, decoder nor the relation network. Note that the last dimension of the relation network and decoder outputs are two times larger than $n _ { z }$ and $\dim ( \theta )$ respectively, as they are used to parameterize both the means and variances of the corresponding Gaussian distributions.
+
+Table 3: Architecture details for 5-way 1-shot miniImageNet and tieredImageNet. The shapes correspond to the meta-training phase. We used a meta-batch of 12 task instances in parallel.   
+
+<table><tr><td rowspan=1 colspan=1>Part of the model</td><td rowspan=1 colspan=1>Architecture</td><td rowspan=1 colspan=1> Hidden layer size|Shape of the output</td><td rowspan=1 colspan=1> Hidden layer size|Shape of the output</td></tr><tr><td rowspan=1 colspan=1>Inference model (fe)</td><td rowspan=1 colspan=1> 3-layer MLP with ReLU</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>(12,5,1)</td></tr><tr><td rowspan=1 colspan=1>Encoder</td><td rowspan=1 colspan=1> 3-layer MLP with ReLU</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>(12,5,16)</td></tr><tr><td rowspan=1 colspan=1>Relation network</td><td rowspan=1 colspan=1> 3-layer MLP with ReLU</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>(12,2 × 16)</td></tr><tr><td rowspan=1 colspan=1>Decoder</td><td rowspan=1 colspan=1>3-layer MLP with ReLU</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>(12,2 × 1761)</td></tr></table>
+
+# B EXPERIMENTAL SETUP - CLASSIFICATION
+
+# B.1 DATA PREPARATION
+
+We used the standard 5-way 1-shot and 5-shot classification setups, where each task instance involves classifying images from 5 different categories sampled randomly from one of the meta-sets, and $\mathcal { D } ^ { t r }$ contains 1 or 5 training examples respectively. $\mathcal { D } ^ { v a l }$ contains 15 samples during metatraining, as decribed in Finn et al. (2017), and all the remaining examples during validation and testing, following Qiao et al. (2017).
+
+We did not employ any data augmentation or feature averaging during meta-learning, or any other data apart from the corresponding training and validation meta-sets. The only exception is the special case of “multi-view” embedding results, where features were averaged over representations of 4 corner and central crops and their horizontal mirrored versions, which we provide for full comparison with Qiao et al. (2017). Apart from the differences described here, the feature training pipeline closely followed that of Qiao et al. (2017).
+
+# B.2 FEATURE PRE-TRAINING
+
+As described in Section 4.2.2, we trained dataset specific feature embeddings before meta-learning, in a similar fashion to Qiao et al. (2017) and Bauer et al. (2017). A Wide Residual Network WRN28-10 (Zagoruyko & Komodakis, 2016b) with 3 steps of dimensionality reduction was used to classify images of $8 0 \times 8 0$ pixels from only the meta-training set into the corresponding training classes (64 in case of miniImageNet and 351 for tieredImageNet). We used dropout $( p _ { k e e p } = 0 . 5 )$ inside residual blocks, as described in (Zagoruyko & Komodakis, 2016b), which is turned off during evaluation and for dataset export. An L2 regularization term of $5 e ^ { - 4 }$ was used, 0.9 Nesterov momentum, and SGD with a learning rate schedule. The initial learning rate was 0.1 and it was multiplied with 0.2 at the steps given in Table 4. Mini-batches were of size of 1024. Data augmentation for pre-training was similar to the inception pipeline (Szegedy et al.), with color distortions and image deformations and scaling in training mode. For 64-way evaluation accuracy and dataset export we used only the center crop (with a ratio of $\frac { 8 0 } { 9 2 }$ : about $8 5 . 9 5 \%$ of the image) which was then resized to $8 0 \times 8 0$ and passed to the network.
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Step 1</td><td rowspan=1 colspan=1>Step 2</td><td rowspan=1 colspan=1>Step 3</td><td rowspan=1 colspan=1>Step 4</td><td rowspan=1 colspan=1>Step 5</td><td rowspan=1 colspan=1>Total Steps</td></tr><tr><td rowspan=1 colspan=1>miniImageNet</td><td rowspan=1 colspan=1>3×103</td><td rowspan=1 colspan=1>5×10³</td><td rowspan=1 colspan=1>7×103</td><td rowspan=1 colspan=1>8×103</td><td rowspan=1 colspan=1>9×103</td><td rowspan=1 colspan=1>1×104</td></tr><tr><td rowspan=1 colspan=1>tieredImageNet</td><td rowspan=1 colspan=1>2×104</td><td rowspan=1 colspan=1>2.5×104</td><td rowspan=1 colspan=1>3×104</td><td rowspan=1 colspan=1>3.5×104</td><td rowspan=1 colspan=1>4×104</td><td rowspan=1 colspan=1>5×104</td></tr></table>
+
+Table 4: Learning rate annealing schedules used to train feature extractors for miniImageNet and tieredImageNet.
+
+Activations in layer 21, with average pooling over spatial dimensions, were precomputed and saved as feature embeddings with $n _ { x } = \dim ( \mathbf { x } ) = 6 4 0$ , which substantially simplified the meta-learning process.
+
+# B.3 LEO NETWORK ARCHITECTURE
+
+We used the same network architecture of parameter generator for all datasets and tasks. The encoder and decoder networks were linear with the bottleneck embedding space of size $n _ { z } ~ = ~ 6 4$ . The relation network was a 3-layer fully connected network with 128 units per layer and rectifier nonlinearities. For simplicity we did not use biases in any layer of the encoder, decoder nor the relation network. Table 5 summarizes this information. Note that the last dimension of the relation network and decoder outputs are two times larger than $n _ { z }$ and $\dim ( \mathbf { x } )$ respectively, as they are used to parameterize both the means and variances of the corresponding Gaussian distributions.
+
+The “Meta-SGD (our features)” baseline used the same one-layer softmax classifier as base model.
+
+Table 5: Architecture details for 5-way 1-shot miniImageNet and tieredImageNet. The shapes correspond to the meta-training phase. We used a meta-batch of 12 task instances in parallel.   
+
+<table><tr><td rowspan=1 colspan=1>Part of the model</td><td rowspan=1 colspan=1>Architecture</td><td rowspan=1 colspan=1>Shape of the output</td><td rowspan=1 colspan=1>When trained?</td></tr><tr><td rowspan=1 colspan=1>Feature extractor</td><td rowspan=1 colspan=1>WRN-28-10</td><td rowspan=1 colspan=1>(12,5,1,640)</td><td rowspan=1 colspan=1>before LEO</td></tr><tr><td rowspan=1 colspan=1>Encoder</td><td rowspan=1 colspan=1>linear</td><td rowspan=1 colspan=1>(12,5,1,64)</td><td rowspan=1 colspan=1>during outer loop</td></tr><tr><td rowspan=1 colspan=1>Relation network</td><td rowspan=1 colspan=1> 3-layer MLP with ReLU</td><td rowspan=1 colspan=1>(12,5V,2 × 64)</td><td rowspan=1 colspan=1>during outer loop</td></tr><tr><td rowspan=1 colspan=1>Decoder</td><td rowspan=1 colspan=1>linear</td><td rowspan=1 colspan=1>(12,2 × 640)</td><td rowspan=1 colspan=1>during outer loop</td></tr></table>
+
+# B.4 OPTIMIZATION
+
+We used a parallel implementation similar to that of Finn et al. (2017), where the “inner loop” is performed in parallel on a batch 12 problem instances for every meta-update. Using a relation network in the encoder has negligible computational cost given that $k ^ { 2 }$ is small in typical $k$ -shot learning domains, and the relation network is only used once per problem instance, to get the initial model parameters before adaptation. Within the LEO “inner loop” we perform 5 steps of adaptation in latent space, followed by 5 steps of fine-tuning in parameter space. The learning rates for these spaces were meta-learned in a similar fashion to Meta-SGD (Li et al., 2017), after being initialized to 1 and
+
+0.001 for the latent and parameter spaces respectively. We applied dropout independently on the feature embedding in every step, with the probability of not being dropped out $p _ { k e e p }$ chosen (together with other hyperparameters) using random search based on the validation meta-set accuracy.
+
+Parameters of the encoder, relation, and decoder networks as well as per-parameter learning rates in latent and parameter spaces were optimized jointly using Adam (Kingma & Ba, 2014) to minimize the meta-learning objective (Eq. 6) over problem instances from the training meta-set, iterating for up to 100 000 steps, with early stopping using validation accuracy.
+
+Meta-learning objectives can lead to difficult optimization processes in practice, specifically when coupled with stochastic sampling in latent and parameters spaces. For ease of experimentation we clip the meta-gradient, as well as its norm, at an absolute value of 0.1. Please note this was only done for the encoder, relation, decoder networks and learning rates, not the inner loop latent space adaptation gradients.
+
+B.5 HYPER-PARAMETERS   
+
+<table><tr><td rowspan="2">Hyperparameter</td><td colspan="2">miniImageNet</td><td colspan="2">tieredImageNet</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>η (Algorithm 1)</td><td>0.00043653954</td><td>0.00117573555</td><td>0.00040645397</td><td>0.00073469522</td></tr><tr><td>γ (Eq. (6))</td><td>1.33365371e-9</td><td>5.39245830e-6</td><td>1.24305386e-8</td><td></td></tr><tr><td>β (Eq. (6))</td><td>0.124171967</td><td></td><td></td><td>3.05077069e-6</td></tr><tr><td>入1 (Eq. (7))</td><td>0.000108982953</td><td>0.0440372182</td><td>7.10800960e-6</td><td>0.00188644980</td></tr><tr><td>入2 (Eq. (7))</td><td></td><td>3.75922509e-6</td><td>3.10725285e-8</td><td>4.90658551e-8</td></tr><tr><td></td><td>303.216647</td><td>0.00844225971</td><td>5180.09554</td><td>0.0081711619</td></tr><tr><td>Pkeep</td><td>0.711524088</td><td>0.755402644</td><td>0.644395979</td><td>0.628325359</td></tr></table>
+
+Table 6: Values of hyperparameters chosen to maximize meta-validation accuracy during random search.
+
+To find the best values of hyperparameters, we performed a random grid search and we choose the set which lead to highest validation meta-set accuracy. The reported performance of our models is an average $\pm$ a standard deviation) over 5 independent runs (using different random seeds) with the best hyperparameters kept fixed. The result of a single run is an average accuracy over 50000 task instances. After choosing hyperparameters (given in Table 6) we used both meta-training and meta-validation sets for training, in line with recent state-of-the-art approaches, e.g. Qiao et al. (2017).
+
+The evaluation of each of the LEO baselines follow the same procedure; in particular, we perform a separate random search for each of them.
+
+# B.6 TRAINING TIME
+
+Training of LEO took 1-2 hours for miniImageNet and around 5 hours for tieredImageNet on a multi-core CPU (for each of the 5 independent runs). Our approach allows for caching the feature embeddings before training LEO, which leads to a very efficient meta-learning process.
+
+Training of the image extractor was more compute-intensive, taking 5 hours for miniImageNet and around a day for tieredImageNet using 32 GPUs.
+
+# B.7 OVERVIEW OF THE TRAINING PROCEDURE
+
+In summary, there are three stages in our approach to meta-training:
+
+1. In the first stage we use 64-way classification to pre-train the feature embedding only on the meta-training set, hence without the meta-validation classes.   
+2. In the second stage we train LEO on the meta-training set with early stopping on metavalidation, and we choose the best hyperparameters using random grid search.
+
+3. In the third stage we train LEO again from scratch 5 times using the embedding trained in stage 1 and the chosen set of hyperparameters from stage 2. However, in this stage we metalearn on embeddings from both meta-train and meta-validation sets, with early-stopping on meta-validation.
+
+While it may not be intuitive to use early stopping on meta-validation in stage 3, it is still a proxy for good generalization since it favors models with high performance on classes excluded during feature embedding pre-training.
+
+# B.8 OVERVIEW OF THE EVALUATION PROCEDURE
+
+The procedure for evaluation is similar to meta-training, except that we disable stochasticity and dropout. Naturally, instead of computing the meta-training loss, the parameters (adapted based on $\mathcal { L } _ { \mathcal { T } _ { i } } ^ { t r }$ ) are only used for inference on that particular task. That is:
+
+1. A problem instance is drawn from the evaluation meta-set.   
+2. The few-shot samples are encoded to latent space, then decoded; the means are used to initialize the parameters of the inference model.   
+3. A few steps of adaptation are performed in latent space, followed (optionally) by a few steps of adaptation in parameter space.   
+4. The resulting parameters are used as the final adapted model for that particular problem instance.

parse/train/BJlBSkHtDS/BJlBSkHtDS.md

@@ -0,0 +1,298 @@
+# PADÉ ACTIVATION UNITS: END-TO-END LEARNING OF FLEXIBLE ACTIVATION FUNCTIONS IN DEEP NETWORKS
+
+Alejandro Molina1, Patrick Schramowski1, Kristian Kersting1,2 1 AI and Machine Learning Group, CS Department, TU Darmstadt, Germany 2 Centre for Cognitive Science, TU Darmstadt, Germany {molina,schramowski,kersting}@cs.tu-darmstadt.de
+
+# ABSTRACT
+
+The performance of deep network learning strongly depends on the choice of the non-linear activation function associated with each neuron. However, deciding on the best activation is non-trivial, and the choice depends on the architecture, hyper-parameters, and even on the dataset. Typically these activations are fixed by hand before training. Here, we demonstrate how to eliminate the reliance on first picking fixed activation functions by using flexible parametric rational functions instead. The resulting Padé Activation Units (PAUs) can both approximate common activation functions and also learn new ones while providing compact representations. Our empirical evidence shows that end-to-end learning deep networks with PAUs can increase the predictive performance. Moreover, PAUs pave the way to approximations with provable robustness.
+
+https://github.com/ml-research/pau
+
+# 1 INTRODUCTION
+
+An important building block of deep learning is the non-linearities introduced by the activation functions $f ( x )$ . They play a major role in the success of training deep neural networks, both in terms of training time and predictive performance. Consider e.g. Rectified Linear Unit (ReLU) due to Nair and Hinton (2010). The demonstrated benefits in training deep networks, see e.g. (Glorot et al., 2011), brought renewed attention to the development of new activation functions. Since then, several ReLU variations with different properties have been introduced such as LeakyReLUs (Maas et al., 2013), ELUs (Clevert et al., 2016), RReLUs (Xu et al., 2015), among others. Another line of research, such as (Ramachandran et al., 2018) automatically searches for activation functions. It identified the Swish unit empirically as a good candidate. However, for a given dataset, there are no guarantees that Swish unit behaves well and the proposed search algorithm is computationally quite demanding.
+
+The activation functions are traditionally fixed and, in turn, impose a set of inductive biases on the network. One attempt to relax this bias are for instance PReLUs (He et al., 2015), where the negative slope is subject to optimization allowing for more flexibility than other ReLU variants. Learnable activation functions generalize this idea. They exploit parameterizations of the activation functions, adapted in an end-to-end fashion to different network architectures and datasets during training. For instance, Maxout (Goodfellow et al., 2013) and Mixout (Zhao et al., 2017) use a fixed set of piecewise linear components and optimized their (hyper-)parameters. Although they are theoretically universal function approximators, they heavily increase the number of parameters of the network and strongly depend on hyper-parameters such as the number of components to realize this potential. Vercellino and Wang (2017) used a meta-learning approach for learning task-specific activation functions (hyperactivations). However, as Vercellino and Wang admit, the implementation of hyperactivations, while easy to express notationally, can be frustrating to implement for generalizability over any given activation network. Recently, Goyal et al. (2019) proposed a learnable activation function based on Taylor approximation and suggested a transformation strategy to avoid exploding gradients. However, relying on polynomials suffers from well-known limitations such as exploding values and a tendency to oscillate (Trefethen, 2012).
+
+![](images/5efe39ebb8cdcf63f9d4aeee3ded84ca98bdf438899c138706e518a9c851ef62.jpg)  
+Figure 1: Approximations of common activation functions (ReLU, Sigmoid, Tanh, Swish and Leaky ReLU $\mathrm { \Delta } \alpha = 0 . 2 0 )$ ) using PAUs (marked with $^ *$ ). As one can see, PAUs can encode common activation functions very well. (Best viewed in color)
+
+As an alternative, we here introduce a learnable activation function based on the Padé approximation, i.e., rational functions. In contrast to approximations for high accuracy hardware implementation of the hyperbolic tangent and the sigmoid activation functions (Hajduk, 2018), we do not assume fixed coefficients. The resulting Padé Activation Units (PAU) can be learned using standard stochastic gradient and, hence, seamlessly integrated into the deep learning stack. PAUs provide more flexibility and increase the predictive performance of deep neural networks, as we demonstrate.
+
+We proceed as follows. We start off by introducing PAUs. Then introduce Padé networks and show that they are universal approximators. Before concluding, we present our empirical evaluation.
+
+# 2 PADÉ ACTIVATION UNITS (PAU)
+
+Our starting point is the set of intuitive assumptions activation functions should ultimately fulfill shown in Tab. 1. The assumptions (i,v) concern the ability of neural networks to approximate functions. Rational functions can fulfill assumptions (i,iv), and our experimental evaluation demonstrates that assumptions (ii,iii,v) also hold.
+
+(i) They must allow the networks to be universal function approximators.   
+(ii) They should ameliorate gradient vanishing.   
+(iii) They should be stable.   
+(iv) They should be parsimonious on the number of parameters.   
+(v) They should provide networks with high predictive performance.
+
+<table><tr><td colspan="2"></td><td colspan="4">Assumptions</td></tr><tr><td>Activation</td><td>Learnable</td><td>i ii</td><td>ii</td><td>iv</td><td>V</td></tr><tr><td>ReLU</td><td>N</td><td></td><td>Y</td><td></td><td>Y</td></tr><tr><td>ReLU6</td><td>N</td><td></td><td>Y</td><td></td><td></td></tr><tr><td>RReLU</td><td>N</td><td></td><td></td><td></td><td></td></tr><tr><td>LReLU</td><td>N</td><td></td><td></td><td></td><td></td></tr><tr><td>ELU</td><td>N</td><td></td><td>YYY</td><td></td><td></td></tr><tr><td>CELU</td><td>N</td><td>YYYYYYY</td><td>YYYYYY Y</td><td></td><td></td></tr><tr><td>Swish</td><td>N</td><td></td><td>？ Y</td><td></td><td></td></tr><tr><td>PReLU</td><td>Y</td><td>Y</td><td>Y</td><td></td><td></td></tr><tr><td>Maxout</td><td>Y</td><td></td><td>Y</td><td>YN</td><td></td></tr><tr><td>Mixture</td><td>Y</td><td></td><td></td><td></td><td>Y</td></tr><tr><td>APL</td><td>Y</td><td></td><td>YY</td><td>YYY</td><td>Y</td></tr><tr><td>SReLU</td><td>Y</td><td>YYYYYY</td><td>YYYY Y</td><td></td><td>Y</td></tr><tr><td>SLAF</td><td>Y</td><td></td><td>Y</td><td>N Y</td><td>？</td></tr><tr><td>PAU</td><td>Y</td><td>Y</td><td>Y</td><td>Y</td><td>Y Y</td></tr></table>
+
+Table 1: (Left) The intuitive assumptions activation functions (AFs) should ultimately fulfill. (Right) Existing ELU, CELU and ReLU like AFS do not fulfill them. Only learnable AFs allow one to tune their shape at training time, ignoring hyper-parameters such as $\alpha$ for LReLU. For Swish, our experimental results do not indicate problems with vanishing gradients. SLAF (Goyal et al., 2019) showed undefined values (iii), and we could not judge their performance (v).
+
+# 2.1 PADÉ APPROXIMATION OF ACTIVATION FUNCTIONS
+
+Let us now formally introduce PAUs. Assume for the moment that we start with a fixed activation function $f ( x )$ . The Padé approximant (Brezinski and Van Iseghem, 1994) is the “best” approximation of $f ( x )$ by a rational function of given orders $m$ and $n$ . Applied to typical actication functions, Fig. 1 shows that they can be approximated well using rational functions.
+
+More precisely, given $f ( x )$ , the Padé approximant is the rational function $F ( x )$ over polynomials $P ( x ) , Q ( x )$ of order $m$ , $n$ of the form
+
+$$
+F ( x ) = { \frac { P ( x ) } { Q ( x ) } } = { \frac { \sum _ { j = 0 } ^ { m } a _ { j } x ^ { j } } { 1 + \sum _ { k = 1 } ^ { n } b _ { k } x ^ { k } } } = { \frac { a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \cdot \cdot \cdot + a _ { m } x ^ { m } } { 1 + b _ { 1 } x + b _ { 2 } x ^ { 2 } + \cdot \cdot \cdot + b _ { n } x ^ { n } } } ~ ,
+$$
+
+which agrees with $f ( x )$ the best. The Padé approximant often gives a better approximation of a function $f ( x )$ than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons, it has been used before in the context of graph convolutional networks (Chen et al., 2018). However, they have not been considered so far for general deep networks. Padé Activation Units (PAUs) go one step further, instead of fixing the coefficients $a _ { j } , b _ { k }$ to approximate a particular activation function, we allow them to be free parameters that can be optimized end-to-end with the rest of the neural network. This allows the optimization process to find the activation function needed at each layer automatically.
+
+The flexibility of Padé is not only a blessing but might also be a curse: it can model processes that contain poles. For a learnable activation function, however, a pole may produce undefined values depending on the input as well as instabilities at learning and inference time. Therefore we consider a restriction, called safe $P A U$ , that guarantees that the polynomial $Q ( x )$ is not 0, i.e., we avoid poles. In general, restricting $Q ( x )$ implies that either $Q ( x ) \mapsto \mathbb { R } _ { > 0 }$ or $Q ( \dot { x } ) \mapsto \mathbb R _ { < 0 }$ , but as $P ( x ) \mapsto \mathbb { R }$ we can focus on $Q ( x ) \mapsto \mathbb { R } _ { > 0 }$ wlog. However, as $\begin{array} { r } { \operatorname* { l i m } _ { Q ( x ) \to 0 ^ { + } } F ( x ) \to \infty } \end{array}$ learning and inference become unstable. To fix this, we impose a stronger constraint, namely $Q ( x ) \geq q \gg 0$ . In this work, $q = 1$ , i.e., $\forall x : Q ( x ) \geq 1$ , preventing poles and allowing for safe computation on $\mathbb { R }$ :
+
+$$
+F ( x ) = { \frac { P ( x ) } { Q ( x ) } } = { \frac { \sum _ { j = 0 } ^ { m } a _ { j } x ^ { j } } { 1 + | \sum _ { k = 1 } ^ { n } b _ { k } x ^ { k } | } } = { \frac { a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \dotsb + a _ { m } x ^ { m } } { 1 + | b _ { 1 } x + b _ { 2 } x ^ { 2 } + \dotsb + b _ { n } x ^ { n } | } } \ .
+$$
+
+Other values for $q \in ( 0 , 1 )$ might still be interesting, as they could provide gradient amplification due to the partial derivatives having $Q ( X )$ in the denominator. However, we leave this for future work.
+
+# 2.2 LEARNING SAFE PADÉ APPROXIMATIONS USING BACKPROPAGATION
+
+In contrast to the standard way of fitting Padé approximants where the coefficients are found via derivatives and algebraic manipulation against a given function, we optimize their polynomials via backpropagation and (stochastic) gradient descent. To do this, we have to compute the gradients with respect to the parameters $\begin{array} { r } { \frac { \partial F } { \partial a _ { j } } , \frac { \partial \check { F } } { \partial b _ { k } } } \end{array}$ as well as the gradient for the input $\textstyle { \frac { \partial F } { \partial x } }$ . A simple alternative is to implement the forward pass as described in Eq. 2, and let automatic differentiation do the job. To be more efficient, however, we can also implement PAUs directly in CUDA (Nickolls et al. (2008)), and for this we need to compute the gradients ourselves:
+
+$$
+\frac { \partial F } { \partial x } = \frac { \partial P ( x ) } { \partial x } \frac { 1 } { Q ( x ) } - \frac { \partial Q ( x ) } { \partial x } \frac { P ( x ) } { Q ( x ) ^ { 2 } } , \quad \frac { \partial F } { \partial a _ { j } } = \frac { x ^ { j } } { Q ( x ) } \quad \mathrm { a n d } \quad \frac { \partial F } { \partial b _ { k } } = - x ^ { k } \frac { A ( X ) } { | A ( X ) | } \frac { P ( X ) } { Q ( x ) ^ { 2 } } ,
+$$
+
+where ) = a1 + 2a2x + · · · + mamxm−1 , ∂Q(x) $\begin{array} { r } { \frac { \partial Q ( x ) } { \partial x } = \frac { A ( X ) } { | A ( X ) | } \left( b _ { 1 } + 2 b _ { 2 } x + \cdot \cdot \cdot + n b _ { n } x ^ { n - 1 } \right) } \end{array}$ , $A ( X ) = b _ { 1 } x + b _ { 2 } x ^ { 2 } + \cdot \cdot \cdot + b _ { n } x ^ { n }$ , and $Q ( x ) = 1 + | A ( X ) |$ . Here we reuse the expressions to reduce computations. To avoid divisions by zero when computing the gradients, we define $\frac { z } { | z | }$ as the sign of $z$ . With the gradients at hand, PAUs can seamlessly be placed together with other modules onto the differentiable programming stack.
+
+# 3 PADÉ NETWORKS
+
+Having PAUs at hand, one can define Padé networks as follows: Padé networks are feedforward networks with PAU activation functions that may include convolutional and residual architectures with pooling layers. To use Padé networks effectively, one simply replaces the standard activation functions in a neural network by PAUs and then proceed to optimize all the parameters and use the network as usual. However, even if every PAU contains a low number of parameters (coefficients $a _ { j } , b _ { k } )$ , in the extreme case, learning one PAU per neuron may considerably increase the complexity of the networks and in turn the learning time. To ameliorate this and inspired by the idea of weightsharing as introduced by Teh and Hinton (2001), we propose to learn one PAU per layer. Therefore we only add $\phi$ many parameters, where $\phi = L * ( m + n )$ and $L$ is the number of activation layers in the network. In our experiments we set $\phi = 1 0 L$ , a rather small number of parameters (iv).
+
+The last step missing before we can start the optimization process is to initialize the coefficients of the PAUs. Surely, one can do random initialization of the coefficients and allow the optimizer to train the network end-to-end. However, we obtained better results after initializing all PAUs with coefficients that approximate standard activation functions. For a discussion on how to obtain different PAU coefficients, we refer to Sec. A.1.
+
+Before evaluating Padé Networks empirically, let us touch upon their expressivity and how to sparsify them.
+
+# 3.1 PADÉ NETWORKS ARE UNIVERSAL FUNCTION APPROXIMATORS
+
+A standard multi-layer perceptron (MLP) with enough hidden units and non-polynomial activation functions is a universal approximator, see e.g. (Hornik et al., 1989; Leshno et al., 1993). Padé Networks are also universal approximators.
+
+Theorem 1. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be a PAU activation function. Let ${ \mathcal { N } } ^ { \rho }$ represent the class of neural networks with activation function $\rho$ . Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then ${ \mathcal { N } } ^ { \rho }$ is dense in $C ( K )$ .
+
+The proof holds for both PAUs and safe PAUs as it makes no assumptions on the form of the denominator $Q ( x )$ , and is a direct application of the following propositions:
+
+Proposition 1. (From Theorem 1.1 in (Kidger and Lyons, 2019)) Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous function. Let $\mathcal { N } _ { n } ^ { \rho }$ represent the class of neural networks with activation function $\rho _ { i }$ , with n neurons in the input layer, one neuron in the output layer, and one hidden layer with an arbitrary number of neurons. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N } _ { n } ^ { \rho }$ is dense in $C ( K )$ if and only if $\rho$ is non-polynomial.
+
+Proposition 2. (From Theorem 3.2 in (Kidger and Lyons, 2019)) Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous point. Let function which is continuously differentiable at at least one point, with nonzero derivative at that $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 2 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+Proof. Let $\rho ( x ) = P ( x ) / Q ( x )$ , we have to consider the following two cases:
+
+Case 1: $Q ( x ) \neq 1$ , by definition, $\rho ( x )$ is non-polynomial. Then by proposition 1, we get that $\mathcal { N } _ { n } ^ { \rho }$ is dense in $C ( K )$ .
+
+Case 2: $Q ( x ) = 1$ , here $\textstyle \rho ( x ) = \sum _ { j = 0 } ^ { m } a _ { j } x ^ { j }$ , is polynomial, continuous and continuously differentiable in propositi $\mathbb { R }$ . Let any  2, we get $a _ { j > 0 } > 0$ re exists a p is dense in $\alpha \in \mathbb { R }$ such that $\rho ^ { \prime } ( \alpha ) \neq 0$ and then by $\mathcal { N N } _ { n , m , n + m + 2 } ^ { \rho }$ $C ( K ; \mathbb { R } ^ { m } )$
+
+# 3.2 SPARSE PADÉ NETWORKS AND RANDOMIZED PAUS
+
+Padé Networks can $\epsilon$ -approximate neural networks with ReLU activations (Telgarsky, 2017). This implies that by using PAUs, we are embedding a virtual network into the networks we want to use. This, in turn, is the operating assumption of the lottery ticket hypothesis due to Frankle and Carbin (2019). Thus, we expect that lottery ticket pruning can find well-performing Padé networks that are smaller than their original counterparts while reducing inference time and potentially improving the predictive performance.
+
+Generally, overfitting is an important practical concern when training neural networks. Usually, we can apply regularization techniques such as Dropout (Srivastava et al., 2014). Unfortunately, although each PAU approximates a small ReLU network section, we do not have access to the internal representation of this virtual network. Therefore, we can not regularize the activation function via standard Dropout. An alternative for regularizing activation functions was introduced in Randomized Leaky ReLUs (RReLUs, Xu et al. (2015)), where the negative slope parameter is sampled uniformly on a range. This makes the activation function behave differently at training time for every input $x$ , forwarding and backpropagating according to $x$ and the sampled noise.
+
+![](images/d7a905d82b9dffce065aa2a8b5f3b5fdcf7afadbd9826edbf48199e7800c7121.jpg)  
+Figure 2: PAU compared to baseline activation function units over 5 runs on Fashion-MNIST using the LeNet architecture: (left) mean test-accuracy (the higher, the better) and (right) mean train-loss (the lower, the better). As one can see, PAU outperforms all baseline activations and enable the networks to achieve a lower loss during training compared to all baselines. (Best viewed in color)
+
+We can employ a similar technique to make PAUs resistant to overfitting. Consider a PAU with coefficients $\mathbf { C } = \{ a _ { 0 } , \cdot \cdot \cdot , a _ { m } , b _ { 0 } , \cdot \cdot \cdot , b _ { n } \}$ . We can introduce additive noise during training into each coefficient $c _ { i } \in \mathbf { C }$ for every input $x _ { j }$ via $c _ { i , j } = c _ { i } + z _ { i , j }$ where $z _ { i , j } \sim U ( l _ { i } , u _ { i } )$ , $l _ { i } = ( 1 - \alpha \% ) * c _ { i }$ and reciprocally $u _ { i } = \left( 1 + \alpha \% \right) * { \check { c } } _ { i }$ . This results in Randomized PAU (RPAU):
+
+$$
+R ( x _ { j } ) = \frac { c _ { 0 , j } + c _ { 1 , j } x + c _ { 2 , j } x ^ { 2 } + \cdot \cdot \cdot + c _ { m , x } x ^ { m } } { 1 + | c _ { m + 1 } x + c _ { m + 2 } x ^ { 2 } + \cdot \cdot \cdot + c _ { m + n } x ^ { n } | } .
+$$
+
+We compute the gradients as before and simply replace the coefficients by their noisy counterparts.
+
+# 4 EXPERIMENTAL EVALUATION
+
+Our intention here is to investigate the behavior and performance of PAUs as well as to compare them to other activation functions using standard deep neural networks. All our experiments are implemented in PyTorch with PAU implemented in CUDA, and were executed on an NVIDIA DGX-2 system. In all experiments, we initialized PAUs with coefficients that approximate LeakyReLUs for a rational function of order $m = 5 , n = 4$ . In all experiments except for ImageNet, we report the mean of 5 runs initialized with different seeds for the accuracy on the test-set after training. And, we compared PAU to the following activation functions: ReLU, ReLU6, Leaky ReLU (LReLU), Random ReLU (RReLU), ELU, CELU, Swish, Parametric ReLU (PReLU), Maxout, Mixture of activations (Mixture), SLAF, APL and SReLU. For details on all the activation functions, we refer to Appendix A.2. As datasets we considered MNIST, Fahion-MNIST, CIFAR-10 and ImageNet.
+
+4.1 EMPIRICAL RESULTS ON MNIST AND FASHION-MNIST BENCHMARKS
+
+First we evaluated PAUs on MNIST (LeCun et al., 2010) and Fashion-MNIST (Xiao et al., 2017) using two different architectures: LeNet (LeCun et al., 1998) and VGG-8 (Simonyan and Zisserman, 2015). For more details on the architectures, learning settings, and results, we refer to Sec. A.3.
+
+As can be seen in Fig. 2 and Tab. 2, PAU outperformed on average the baseline activation functions on every network in terms of predictive performance. Moreover, the results are stable on different runs (c.f. mean $\pm$ std). PAUs also enable the networks to achieve a lower loss during training compared to all baselines on all networks. Actually, PAU achieved the best results on both datasets and on Fashion-MNIST it provides the best results for both architectures. As expected, reducing the bias is beneficial in this experiment. Comparing the baseline activation functions on the MNIST dataset and the different architectures, there is no clear choice of activation that achieves the best performance. However, PAU always matches or even outperforms the best performing baseline activation function. This shows that a learnable activation function relieves the network designer of having to commit to a potentially suboptimal choice. Moreover, Fig. 2 also shows that PAU is more stable than SLAF. This is not unexpected as Taylor approximations tend to oscillate and overshoot (Trefethen, 2012). We also observed undefined values at training time for SLAF; therefore, we do not compare against it in the following experiments. Finally, when considering the number of parameters used by PAU, we can see that they are very efficient. The VGG-8 network uses 9.2 million parameters, PAU here uses 50 parameters, and for LeNet, the network uses 0.5 million parameters while PAU uses only 40.
+
+Table 2: Performance comparison of activation functions on MNIST and Fashion-MNIST (the higher, the better) on two common deep architectures. Shown are the results averaged over 5 reruns as well as the top result among these 5 runs. The best $( \vphantom { \sqrt [ 4 ] { 3 } } \cdots )$ and runner-up $( ^ { 6 6 } \circ ^ { 7 3 } )$ results per architecture are bold. As one can see, PAUs consistently outperform the other activation functions on average and yields the top performance on each dataset.   
+
+<table><tr><td rowspan="3"></td><td colspan="2">VGG-8 best</td><td colspan="2">LeNet</td><td colspan="2">VGG-8</td><td colspan="2">LeNet</td></tr><tr><td colspan="2">mean ± std</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td></tr><tr><td colspan="9"></td></tr><tr><td colspan="9"></td></tr><tr><td>ReLU</td><td>99.17 ±0.10</td><td>MNIST 99.30</td><td>99.17 ± 0.05</td><td>99.25</td><td>89.11 ± 0.43</td><td>Fashion-MNIST 89.69</td><td>89.86 ± 0.32</td><td>90.48</td></tr><tr><td>ReLU6</td><td>99.28 ± 0.04</td><td>99.31</td><td>99.09 ±0.09</td><td>99.22</td><td>89.87 ±0.62</td><td>90.38</td><td>89.74±0.27</td><td>89.96</td></tr><tr><td>LReLU</td><td>99.13 ± 0.11</td><td>99.27</td><td>99.10 ± 0.06</td><td>99.22</td><td>89.37 ± 0.30</td><td>89.74</td><td>89.74± 0.24</td><td>90.02</td></tr><tr><td>RReLU</td><td>99.16 ±0.13</td><td>99.28</td><td>99.20±0.13 099.38</td><td></td><td>88.46±0.85</td><td>89.32</td><td>89.74±0.19</td><td>89.88</td></tr><tr><td>ELU</td><td>99.15 ± 0.09</td><td>99.28</td><td>99.15 ±0.06</td><td>99.22</td><td>89.65 ± 0.33</td><td>90.06</td><td>89.84± 0.47</td><td>90.25</td></tr><tr><td>CELU</td><td>99.15 ± 0.09</td><td>99.28</td><td>99.15 ±0.06</td><td>99.22</td><td>89.65 ± 0.33</td><td>90.06</td><td>89.84± 0.47</td><td>90.25</td></tr><tr><td>Swish</td><td>99.10 ±0.06</td><td>99.20</td><td>99.19 ±0.09</td><td>99.29</td><td>88.54±0.59</td><td>89.36</td><td>89.54± 0.22</td><td>89.89</td></tr><tr><td>PReLU</td><td>99.16 ± 0.09</td><td>99.25</td><td>99.14±0.09</td><td>99.24</td><td>88.82 ±0.51</td><td>89.54</td><td>90.09 ±0.22090.29</td><td></td></tr><tr><td>SLAF</td><td></td><td></td><td></td><td></td><td>90.60 ± 0.00</td><td>90.60</td><td>89.33 ± 0.28</td><td>89.80</td></tr><tr><td>APL</td><td>099.35±0.11·99.50</td><td></td><td>99.18±0.10</td><td>99.33</td><td>91.41±0.48·92.25</td><td></td><td>89.72 ± 0.30</td><td>90.01</td></tr><tr><td>SReLU</td><td>99.15 ± 0.03</td><td>99.20</td><td>99.13±0.14</td><td>99.27</td><td>89.65 ±0.42</td><td>90.31</td><td>89.83 ±0.30</td><td>90.28</td></tr><tr><td>PAU</td><td>99.30±0.05 099.40</td><td></td><td>099.21±0.04</td><td>99.26</td><td>091.25±0.18091.56</td><td></td><td>90.33±0.15·90.62</td><td></td></tr><tr><td>RPAU</td><td>·99.35±0.04  99.38 99.26±0.11·99.42</td><td></td><td></td><td></td><td></td><td></td><td>91.23±0.15 91.41090.20±0.11 090.29</td><td></td></tr></table>
+
+In summary, this shows that PAUs are stable, parsimonious and can improve the predictive performance of deep neural networks (iii,iv,v).
+
+# 4.2 LEARNED ACTIVATION FUNCTIONS ON MNIST AND FASHION-MNIST
+
+When looking at the activation functions learned from the data, we can see that the PAU family is flexible yet presents similarities to standard functions. In particular, Fig. 3 illustrates that some of the learned activations seem to be smoothed versions of Leaky ReLUs, since V-shaped activations are simply Leaky ReLUs with negative $\alpha$ values. This is not surprising as we initialize PAUs with coefficients that match Leaky ReLUs. Finding different initialization and optimization parameters is left as future work. In contrast, when learning piecewise approximations of the same activations using Maxout, we would require a high $k$ . This significantly increases the number of parameters of the network. This again provides more evidence in favor of PAUs being flexible and parsimonious (iv). SLAF produced undefined values during training on all networks except Fashion-MNIST where LeNet finished 4 runs and VGG only one run.
+
+# 4.3 EMPIRICAL RESULTS ON CIFAR-10
+
+After investigating PAU on MNIST and Fashion-MNIST, we considered a more challenging setting: CIFAR-10 (Krizhevsky et al. (2009)). We also considered other learnable activation functions, namely Maxout $( \mathrm { k } \mathrm { = } 2 )$ and Mixture of activations (Id and ReLU) as well as another popular deep network architectures: MobileNetV2 (Sandler et al., 2018), ResNet101 (He et al., 2016) and DenseNet121 (Huang et al., 2017). For more details on the learning settings and results, we refer to Sec. A.4.
+
+Let us start by considering the results for VGG-8 and MobileNetV2 on CIFAR-10. These networks are the smallest of this round of experiments and, therefore, could benefit more from bias reduction. Indeed, we can see in Tab. 3 that both networks take advantage of learnable activation functions, i.e., Maxout, PAU, and RPAU. As expected, adding more capacity to VGG-8 helps and this is what Maxout is doing. Moreover, even if Mixtures do not seem to provide a significant benefit on VGG-8, they do help in MobileNetV2. Here, we see again that PAU and RPAU are either in the lead or close to the best when it comes to predictive performance, without having to make a choice apriori.
+
+Table 3: Performance comparison of activation functions on CIFAR-10 (the higher, the better) on four state-of-the-art deep neural architectures. Shown are the results averaged over 5 reruns as well as the top result among these 5 runs. The best $( \vphantom { \sqrt [ 4 ] { 3 } } \cdots )$ and runner-up (“◦”) results per architecture are bold. As one can see, PAUs are either in the lead or close to the best. $( ^ { 6 6 } { } ^ { \ast } { } ^ { \ast } { } ^ { \ast } { } ^ { \ast } { } ^ { \ast } { } ^ { \ast } )$ are experiments that did not finish on time.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">VGG-8</td><td colspan="2">MobileNetV2</td><td colspan="2">ResNet101</td><td colspan="2">DenseNet121</td></tr><tr><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td></tr><tr><td></td><td colspan="6"></td><td></td><td></td></tr><tr><td>ReLU</td><td>92.32 ± 0.16</td><td>92.58</td><td>91.51 ± 0.28</td><td>91.82</td><td>95.07 ± 0.17</td><td></td><td>95.36095.36±0.18 095.63</td><td></td></tr><tr><td>ReLU6</td><td>92.36 ±0.06</td><td>92.47</td><td>91.30 ± 0.23</td><td>91.57</td><td>95.11 ± 0.24</td><td>95.29</td><td>95.33 ± 0.14</td><td>95.46</td></tr><tr><td>LReLU</td><td>92.43 ± 0.14</td><td>92.65</td><td>91.94 ± 0.12</td><td>92.08</td><td>95.08 ±0.19</td><td>95.29</td><td>·95.42±0.17·95.65</td><td></td></tr><tr><td>RReLU</td><td>92.32 ± 0.07</td><td>92.42</td><td>094.66±0.16</td><td>6094.94</td><td>95.21±0.23</td><td>095.51</td><td>95.00 ±0.12</td><td>95.14</td></tr><tr><td>ELU</td><td>91.24 ± 0.09</td><td>91.33</td><td>90.43 ±0.14</td><td>90.61</td><td>94.04± 0.14</td><td>94.24</td><td>90.78 ±0.29</td><td>91.23</td></tr><tr><td>CELU</td><td>91.24 ± 0.09</td><td>91.33</td><td>90.69 ±0.27</td><td>90.97</td><td>93.80 ±0.36</td><td>94.25</td><td>90.88 ±0.19</td><td>91.08</td></tr><tr><td>PReLU</td><td>92.22 ±0.26</td><td>92.51</td><td>93.54± 0.45</td><td>93.95</td><td>94.15 ± 0.39</td><td>94.50</td><td>94.98 ±0.16</td><td>95.15</td></tr><tr><td>Swish</td><td>91.58 ±0.18</td><td>91.86</td><td>92.04±0.13</td><td>92.21</td><td>91.83 ± 1.61</td><td>92.84</td><td>93.04±0.16</td><td>93.32</td></tr><tr><td>Maxout</td><td>·93.03±0.11</td><td>·93.23</td><td>94.41 ±0.10</td><td>94.54</td><td>95.11 ± 0.13</td><td>95.23</td><td>***</td><td>***</td></tr><tr><td>Mixture</td><td>91.86 ±0.14</td><td>92.06</td><td>94.06 ± 0.16</td><td>94.25</td><td>94.50± 0.25</td><td>94.71</td><td>93.33 ±0.17</td><td>93.59</td></tr><tr><td>APL</td><td>91.63 ± 0.13</td><td>91.82</td><td>93.62 ±0.64</td><td>94.50</td><td>94.12 ± 0.36</td><td>94.50</td><td>94.45 ±0.23</td><td>94.78</td></tr><tr><td>SReLU</td><td>092.66± 0.27 093.13</td><td></td><td>94.03 ± 0.11</td><td>94.25</td><td>095.24±0.13</td><td>95.38</td><td>94.77 ±0.24</td><td>95.20</td></tr><tr><td>PAU</td><td>92.51± 0.16</td><td>92.70</td><td>94.57±0.21</td><td>94.90</td><td>95.16±0.13</td><td>95.28</td><td>95.03±0.07</td><td>95.16</td></tr><tr><td>RPAU</td><td>92.50 ±0.09</td><td>92.62</td><td>·94.82±0.21·95.13</td><td></td><td>·95.34±0.13·95.54</td><td></td><td>95.27±0.10</td><td>95.41</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Now, let us have a look at the performance of PAU and RPAU on the larger networks ResNet101 and DenseNet121. As these networks are so expressive, we do not expect the flexibility of the learnable activation functions to have a big impact on the performance. Tab. 3 confirms this. Nevertheless, they are still competitive and their performance is stable as shown by the standard deviation. On ResNet101, PAUs actually provided the top performance.
+
+# 4.4 FINDING SPARSE PADÉ NETWORKS
+
+As discussed in Sec. 3.2, using PAUs in a network is equivalent to introducing virtual networks of ReLUs in the middle of the network, effectively adding virtual depth to the networks. Therefore, we also investigated whether pruning can help one to unmask smaller sub-networks whose performance is similar to the original network. In a sense, we are removing blocks of the real network as they get replaced by the virtual network. Here, we only do pruning on the convolutional layers. For details about the algorithm and hyper-parameters, wee refer to Sec. A.4.3.
+
+Specifically, we compared PAU against the best activation functions for the different architectures. However, we discarded Maxout, as instead of pruning it introduces many more parameters into the network defeating the original purpose. As one can see in Fig. 4, pruning on the already sizeoptimized networks VGG-8 and MobileNetV2 has an effect on the predictive performance. However, the performance of PAU remains above the other activation functions despite the increase in pruning pressure. In contrast, when we look at ResNet101, we see that the performance of PAU is not influenced by pruning, showing that indeed we can find sparse Padé network without major loss in accuracy. And what is more, PAU enables the ResNet101 subnetwork, pruned by $30 \%$ , to achieve a higher accuracy compared to all pruned and not pruned networks.
+
+![](images/724d775b800a1442799e3b498a9a937bef334b7d16fc37353526bee8b541b3f5.jpg)  
+Figure 3: Estimated activation functions after training the VGG-8 network with RPAU on FashionMNIST. The center line indicates the PAU while the surrounding area indicates the space of the additive noise in RPAUs. As one can see, the PAU family differs from common activation functions but capture characteristics of them. (Best viewed in color)
+
+![](images/f6f8ba30cac791aa66edbb46332514c5a26cf85b13001207739147175f3818e7.jpg)  
+Figure 4: Comparison of the predictive accuracy (higher is better) for the architectures VGG-8, MobileNetV2 and ResNet101 between PAU and the best activation functions according to Tab. 3. PAU is consistently better. On ResNet101 PAU is not affected by the increase pruning pressure. Furthermore, PAU enables the ResNet101 subnetwork, pruned by $30 \%$ , to achieve a higher accuracy compared to all pruned and not pruned networks. (Best viewed in color)
+
+# 4.5 EMPIRICAL RESULTS ON IMAGENET
+
+Finally, we investigated the performance on a much larger dataset, namely ImageNet (Russakovsky et al. (2015)) used to train MobileNetV2. As can be seen in Fig. 5 and Tab. 4, PAU and Swish clearly dominate in performance (v). PAU leads in top-1 accuracy and Swish in top-5 accuracy. Moreover, both PAU and Swish show faster learning compared to the other activation functions.
+
+Furthermore, we argue that the rapid learning rate of PAU in all the experiments indicate that they do not exhibit vanishing gradient issues (ii).
+
+![](images/2488f2ddb65dd7f31205283870493875f1e64c1aec1a49687f7811a78eff62a9.jpg)  
+Figure 5: MobileNetV2 top-1 test accuracy on the left (higher is better) and training loss on the right (lower is better) for multiple activation functions in ImageNet. PAU achieves higher accuracy and lower loss values in fewer epochs. (Best viewed in color)
+
+Table 4: MobileNetV2 top-1 and top-5 accuracies in ImageNet (higher is better) for different activations. Best $( \mathbf { \ddot { \sigma } } \mathbf { \dot { \sigma } } )$ and runner-up (“◦”) are shown in bold. PAU is the best in top-1 accuracy and runner-up for top-5.   
+
+<table><tr><td>MobileNetV2</td><td>ReLU</td><td>ReLU6</td><td>LReLU</td><td>RReLU</td><td>ELU</td><td>CELU</td><td>PReLU</td><td>Swish</td><td>SReLU</td><td>PAU</td></tr><tr><td>Acc@1</td><td>69.65</td><td>69.83</td><td>70.03</td><td>69.12</td><td>69.13</td><td>69.17</td><td>68.61</td><td>071.24</td><td>70.62</td><td>·71.35</td></tr><tr><td>Acc@5</td><td>89.09</td><td>89.34</td><td>89.26</td><td>88.80</td><td>88.46</td><td>88.59</td><td>88.51</td><td>·89.95</td><td>89.59</td><td>089.85</td></tr></table>
+
+# 4.6 SUMMARIZED RESULTS
+
+Finally, we compare the PAU family of activation functions to all the other activation functions and we aggregate the number of occurrences where PAU performed better or worse in comparison. The aggregate results can be found in Tab. 5. These results are the aggregates of all the experiments on all datasets and all architectures. As we can see, the PAU family is very competitive.
+
+<table><tr><td>Baselines</td><td></td><td>ReLUReLU6LReLURReLUELU</td><td></td><td></td><td></td><td>CELUPReLU|</td><td></td><td>SwishMaxoutMixturel</td><td></td><td></td><td>APL</td><td>SReLU</td></tr><tr><td>PAU/RPAU&gt;= Baseline</td><td>34</td><td>35</td><td>34</td><td>33</td><td>40</td><td>40</td><td>39</td><td>41</td><td>9</td><td>20</td><td>32</td><td>33</td></tr><tr><td>PAU/RPAU&lt;Baseline</td><td>8</td><td>7</td><td>8</td><td>9</td><td>2</td><td>2</td><td>3</td><td>1</td><td>6</td><td>0</td><td>7</td><td>8</td></tr></table>
+
+Table 5: The number of models on which PAU and RPAU outperforms or underperforms each baseline activation function we compared against in our experiments.
+
+To summarize, PAUs satisfy all the assumptions (i-v). They allow the network to be universal function approximators (i) as shown by theorem 1. They present a fast and stable learning behavior (ii, iii) as shown in Figs. (5,6,7,8). The number of parameters introduced by PAUs is minimal in comparison to the size of the networks. In our experiments, we add 10 parameters per layer, showing that PAUs are parsimonious (iv). Finally, they allow deep neural networks to provide high predictive performance (v) as shown in Tab. 5.
+
+# 5 CONCLUSIONS
+
+We have presented a novel learnable activation function, called Padé Activation Unit (PAU). PAUs encode activation functions as rational functions, trainable in an end-to-end fashion using backpropagation. This makes it easy for practitioners to replace standard activation functions with PAU units in any neural network. The results of our empirical evaluation for image classification demonstrate that PAUs can indeed learn new activation functions and in turn novel neural networks that are competitive to state-of-the-art networks with fixed and learned activation functions. Actually, across all activation functions and architectures, Padé networks are among the top performing networks. This clearly shows that the reliance on first picking fixed, hand-engineered activation functions can be eliminated and that learning activation functions is actually beneficial and simple. Moreover, our results provide the first empirically evidence that the open question “Can rational functions be used to design algorithms for training neural networks?” raised by Telgarsky (2017) can be answered affirmatively for common deep architectures.
+
+Our work provides several interesting avenues for future work. One should explore more the space between safe and unsafe PAUs, in order to gain even more predictive power. Most interestingly, since Padé networks can be reduced to ReLU networks. one should explore globally optimal training (Arora et al., 2018) as well as provable robustness (Croce et al., 2019) of Padé approximations of general deep networks.
+
+Acknowledgments. PS and KK were supported by funds of the German Federal Ministry of Food and Agriculture (BMEL) based on a decision of the Parliament of the Federal Republic of Germany via the Federal Office for Agriculture and Food (BLE) under the innovation support program, project “DePhenS” (FKZ 2818204715).
+
+REFERENCES   
+F. Agostinelli, M. D. Hoffman, P. J. Sadowski, and P. Baldi. Learning activation functions to improve deep neural networks. In Workshop Track Proceedings of the International Conference on Learning Representations, 2015.   
+R. Arora, A. Basu, P. Mianjy, and A. Mukherjee. Understanding deep neural networks with rectified linear units. In International Conference on Learning Representations, 2018.   
+J. T. Barron. Continuously differentiable exponential linear units. arXiv preprint arXiv:1704.07483, 2017.   
+C. Brezinski and J. Van Iseghem. Padé approximations. Handbook of numerical analysis, 3:47–222, 1994.   
+Z. Chen, F. Chen, R. Lai, X. Zhang, and C.-T. Lu. Rational neural networks for approximating graph convolution operator on jump discontinuities. In 2018 IEEE International Conference on Data Mining (ICDM). IEEE, 2018.   
+D. Clevert, T. Unterthiner, and S. Hochreiter. Fast and accurate deep network learning by exponential linear units (elus). In 4th International Conference on Learning Representations (ICLR), 2016.   
+F. Croce, M. Andriushchenko, and M. Hein. Provable robustness of relu networks via maximization of linear regions. In The 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), pages 2057–2066, 2019.   
+J. Frankle and M. Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. In ICLR, 2019.   
+X. Glorot, A. Bordes, and Y. Bengio. Deep sparse rectifier neural networks. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics (AISTATS), pages 315–323, 2011.   
+I. J. Goodfellow, D. Warde-Farley, M. Mirza, A. C. Courville, and Y. Bengio. Maxout networks. In Proceedings of the 30th International Conference on Machine Learning (ICML), pages 1319–1327, 2013.   
+M. Goyal, R. Goyal, and B. Lall. Learning activation functions: A new paradigm of understanding neural networks. arXiv preprint arXiv:1906.09529, 2019.   
+Z. Hajduk. Hardware implementation of hyperbolic tangent and sigmoid activation functions. Bulletin of the Polish Academy of Sciences. Technical Sciences, 66(5), 2018.   
+K. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pages 1026–1034, 2015.   
+K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   
+K. Hornik, M. B. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359–366, 1989.   
+G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017.   
+X. Jin, C. Xu, J. Feng, Y. Wei, J. Xiong, and S. Yan. Deep learning with s-shaped rectified linear activation units. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 2016.   
+P. Kidger and T. Lyons. Universal approximation with deep narrow networks. arXiv preprint arXiv:1905.08539, 2019.   
+D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, (ICLR), 2015.   
+A. Krizhevsky and G. Hinton. Convolutional deep belief networks on cifar-10. 2010.   
+A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. Technical report, Computer Science Department, University of Toronto, 2009.   
+Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of IEEE, 86(11):2278–2324, 1998.   
+Y. LeCun, C. Cortes, and C. Burges. Mnist handwritten digit database. at&t labs, 2010.   
+M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6):861–867, 1993.   
+A. L. Maas, A. Y. Hannun, and A. Y. Ng. Rectifier nonlinearities improve neural network acoustic models. In in ICML Workshop on Deep Learning for Audio, Speech and Language Processing, 2013.   
+F. Manessi and A. Rozza. Learning combinations of activation functions. In 2018 24th International Conference on Pattern Recognition (ICPR), pages 61–66. IEEE, 2018.   
+V. Nair and G. E. Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th international conference on machine learning (ICML-10), pages 807–814, 2010.   
+J. Nickolls, I. Buck, and M. Garland. Scalable parallel programming. In 2008 IEEE Hot Chips 20 Symposium (HCS), pages 40–53. IEEE, 2008.   
+N. Qian. On the momentum term in gradient descent learning algorithms. Neural networks, 12(1): 145–151, 1999.   
+P. Ramachandran, B. Zoph, and Q. V. Le. Searching for activation functions. In Proceedings of the Workshop Track of the 6th International Conference on Learning Representations (ICLR), 2018.   
+O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. S. Bernstein, A. C. Berg, and F. Li. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.   
+M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L.-C. Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4510–4520, 2018.   
+K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In Proceedings of the 3rd International Conference on Learning Representations(ICLR), 2015.   
+N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. The journal of machine learning research, 15(1): 1929–1958, 2014.   
+Y. W. Teh and G. E. Hinton. Rate-coded restricted boltzmann machines for face recognition. In Prcoeedings of Neural Information Processing Systems (NIPS), pages 908–914, 2001.   
+M. Telgarsky. Neural networks and rational functions. In Proceedings of the 34th International Conference on Machine Learning (ICML), pages 3387–3393, 2017.   
+L. N. Trefethen. Approximation Theory and Approximation Practice. SIAM, 2012. ISBN 978-1-611- 97239-9.   
+C. J. Vercellino and W. Y. Wang. Hyperactivations for activation function exploration. In 31st Conference on Neural Information Processing Systems (NIPS 2017), Workshop on Meta-learning. Long Beach, USA, 2017.   
+H. Xiao, K. Rasul, and R. Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. CoRR, 2017.   
+B. Xu, N. Wang, T. Chen, and M. Li. Empirical evaluation of rectified activations in convolutional network. CoRR, 2015.   
+H.-Z. Zhao, F.-X. Liu, and L.-Y. Li. Improving deep convolutional neural networks with mixed maxout units. PloS one, 12(7), 2017.
+
+# A APPENDIX
+
+# A.1 INITIALIZATION COEFFICIENTS
+
+As show in Table 6 we compute initial coefficients for PAU approximations to different known activation functions. We predefined the orders to be [5,4] and for Sigmoid, Tanh and Swish, we have computed the Padé approximant using the standard techniques. For the different variants of PRelu, LeakyRelu and Relu we optimized the coefficients using least squares over the line range between [-3,3] in steps of 0.000001.
+
+<table><tr><td></td><td>Sigmoid</td><td>Tanh</td><td>Swish</td><td>ReLU</td><td>LReLU(0.01)</td><td>LReLU(0.20)</td><td>LReLU(0.25)</td><td>LReLU(0.30)</td><td>LReLU(-0.5)</td></tr><tr><td>ao</td><td>1/2</td><td>0</td><td>0</td><td>0.02996348</td><td>0.02979246</td><td>0.02557776</td><td>0.02423485</td><td>0.02282366</td><td>0.02650441</td></tr><tr><td>α1</td><td>1/4</td><td>1</td><td>1/2</td><td>0.61690165</td><td>0.61837738</td><td>0.66182815</td><td>0.67709718</td><td>0.69358438</td><td>0.80772912</td></tr><tr><td>a2</td><td>1/18</td><td>0</td><td>b/4</td><td>2.37539147</td><td>2.32335207</td><td>1.58182975</td><td>1.43858363</td><td>1.30847432</td><td>13.56611639</td></tr><tr><td>a3</td><td>1/144</td><td>1/9</td><td>362/56</td><td>3.06608078</td><td>3.05202660</td><td>2.94478759</td><td>2.95497990</td><td>2.97681599</td><td>7.00217900</td></tr><tr><td>a4</td><td>1/2016</td><td>0</td><td>63/168</td><td>1.52474449</td><td>1.48548002</td><td>0.95287794</td><td>0.85679722</td><td>0.77165297</td><td>11.61477781</td></tr><tr><td>a5</td><td>1/60480</td><td>1/945</td><td>b4/3360</td><td>0.25281987</td><td>0.25103717</td><td>0.23319681</td><td>0.23229612</td><td>0.23252265</td><td>0.68720375</td></tr><tr><td>b1</td><td>0</td><td>0</td><td>0</td><td>1.19160814</td><td>1.14201226</td><td>0.50962605</td><td>0.41014746</td><td>0.32849543</td><td>13.70648993</td></tr><tr><td>b2</td><td>1/9</td><td>4/9</td><td>362/28</td><td>4.40811795</td><td>4.39322834</td><td>4.18376890</td><td>4.14691964</td><td>4.11557902</td><td>6.07781733</td></tr><tr><td>b</td><td>0</td><td>0</td><td>0</td><td>0.91111034</td><td>0.87154450</td><td>0.37832090</td><td>0.30292546</td><td>0.24155603</td><td>12.32535229</td></tr><tr><td>b4</td><td>1/10008</td><td>1/63</td><td>64 /1680</td><td>0.34885983</td><td>0.34720652</td><td>0.32407314</td><td>0.32002850</td><td>0.31659365</td><td>0.54006880</td></tr></table>
+
+Table 6: Initial coefficients to approximate different activation functions.
+
+# A.2 LIST OF ACTIVATION FUNCTIONS
+
+For our experiments, we compare against the following activation functions with their respective parameters.
+
+• ReLU (Nair and Hinton, 2010): $y = \operatorname* { m a x } ( x , 0 )$   
+• ReLU6 (Krizhevsky and Hinton, 2010): $y = \operatorname* { m i n } ( \operatorname* { m a x } ( x , 0 ) , 6 )$ , a variation of ReLU with an upper bound.   
+• Leaky ReLU (Maas et al., 2013): $y = \operatorname* { m a x } ( 0 , x ) + \alpha * \operatorname* { m i n } ( 0 , x )$ with the negative slope, which is defined by the parameter $\alpha$ . Leaky ReLU enables a small amount of information to flow when $x < 0$ . Random ReLU ( $\mathrm { X u }$ et al., 2015): a randomized variation of Leaky ReLU. ELU (Clevert et al., 2016): $y = \operatorname* { m a x } ( 0 , x ) + \operatorname* { m i n } ( 0 , \alpha * ( \exp ( x ) - 1 ) ) .$   
+• CELU (Barron, 2017): $y = \operatorname* { m a x } ( 0 , x ) + \operatorname* { m i n } ( 0 , \alpha * ( \exp ( x / \alpha ) - 1 ) )$ . Swish (Ramachandran et al., 2018): $y = x * \mathrm { s i g m o i d } ( x )$ , which tends to work better than ReLU on deeper models across a number of challenging datasets.   
+• Parametric ReLU (PReLU) (He et al., 2015) $y = \operatorname* { m a x } ( 0 , x ) + \alpha * \operatorname* { m i n } ( 0 , x )$ , where the leaky parameter $\alpha$ is a learn-able parameter of the network.   
+• Maxout (Goodfellow et al., 2013): $y = \operatorname* { m a x } ( z _ { i j } )$ , where $z _ { i j } = x ^ { T } W _ { \dots i j } + b _ { i j }$ , and $W \in R ^ { d \times m \times k }$ and $b \in R ^ { m \times k }$ are learned parameters.   
+Mixture of activations (Manessi and Rozza, 2018): a combination of weighted activation functions e.g. {id, ReLU}, where the weight is a learnable parameter of the network. SLAF (Goyal et al., 2019): a learnable activation function based on a Taylor approximation.   
+• APL (Agostinelli et al., 2015): a learnable piecewise linear activation function.   
+• SReLU (Jin et al., 2016): a learnable S-shaped rectified linear activation function.
+
+A.3 DETAILS OF THE MNIST AND FASHION-MNIST EXPERIMENT
+
+# A.3.1 NETWORK ARCHITECTURES
+
+Here we describe the architectures for the networks VGG and LeNet, along with the number of trainable parameters. The number of parameters of the activation function is reported for using PAU. Common not trainable activation functions do not have trainable parameters. PReLU has one
+
+trainable parameter. In total the VGG network as 9224508 parameters with 50 for PAU, and the LeNet network has 61746 parameters with 40 for PAU.
+
+Table 7: Architecture of Simple Convolutional Neural Networks   
+
+<table><tr><td>No.</td><td>VGG</td><td># params</td><td>LeNet # params</td></tr><tr><td>1</td><td>Convolutional</td><td>Convolutional 640 5x5x6</td><td>156</td></tr><tr><td>2</td><td>3x3x64 Activation</td><td>10 Activation</td><td>10</td></tr><tr><td>3</td><td>Max-Pooling</td><td>0</td><td>Max-Pooling 0</td></tr><tr><td>4</td><td>Convolutional 3x3x128</td><td>73856 5x5x16</td><td>Convolutional 2416</td></tr><tr><td>5</td><td>Activation</td><td>10</td><td>Activation 10</td></tr><tr><td>6</td><td>Max-Pooling</td><td>0</td><td>Max-Pooling 0</td></tr><tr><td>7</td><td>Convolutional 3x3x256</td><td>295168 5x5x120</td><td>Convolutional 48120</td></tr><tr><td>8</td><td>Convolutional 3x3x256</td><td>590080</td><td>Activation 10</td></tr><tr><td>9</td><td>Activation</td><td>Linear 10 84</td><td>10164</td></tr><tr><td>10</td><td>Max-Pooling</td><td>0</td><td>Activation 10</td></tr><tr><td>11</td><td>Convolutional 3x3x512</td><td>Linear 1180160 10</td><td>850</td></tr><tr><td>12</td><td>Convolutional 3x3x512</td><td>2359808</td><td>Softmax 0</td></tr><tr><td>13</td><td>Activation</td><td>10</td><td></td></tr><tr><td>14</td><td>Max-Pooling</td><td>0</td><td></td></tr><tr><td>15</td><td>Convolutional 3x3x512</td><td>2359808</td><td></td></tr><tr><td>16</td><td>Convolutional 3x3x512</td><td>2359808</td><td></td></tr><tr><td>17</td><td>Activation</td><td>10</td><td></td></tr><tr><td>18</td><td>Max-Pooling</td><td>0</td><td></td></tr><tr><td>19</td><td>Linear 10</td><td>5130</td><td></td></tr><tr><td>20</td><td>Softmax</td><td>0</td><td></td></tr></table>
+
+# A.3.2 LEARNING PARAMETERS
+
+The parameters of the networks, both the layer weights and the coefficients of the PAUs, were trained over 100 epochs using Adam (Kingma and Ba, 2015) with a learning rate of 0.002 or SGD (Qian, 1999) with a learning rate of 0.01, momentum set to 0.5, and without weight decay. In all experiments we used a batch size of 256 samples. The weights of the networks were initialized randomly and the coefficients of the PAUs were initialized with the initialization constants of Leaky ReLU, see Tab. 6. We report the mean of 5 different runs for both the accuracy on the test-set and the loss on the train-set after each training epoch.
+
+# A.3.3 PREDICTIVE PERFORMANCE
+
+![](images/b696c34891530e3535f22a331a8c5996169901dfd1780460e695de2131474de8.jpg)  
+MNIST   
+Figure 6: PAU compared to baseline activation function units on 5 runs of MNIST using the VGG and LeNet: first column mean test-accuracy, second column mean train-loss. PAU consistently outperforms or matches the best performances of the baseline activations. Moreover, PAUs enable the networks to achieve a lower loss during training compared to all baselines.
+
+![](images/48f8c9a6cc6dfacc1e6b29afebea8fb99b41c24b027dccbfc4962a4f70de7cde.jpg)  
+Fashion-MNIST  
+Figure 7: PAU compared to baseline activation function units on 5 runs of Fashion-MNIST using the VGG and LeNet architectures: first column mean test-accuracy, second column mean train-loss. PAU consistently outperforms the baselines activation functions in terms of performance and training time, especially on the VGG.
+
+# A.4 DETAILS OF THE CIFAR10 AND IMAGENET EXPERIMENT
+
+# A.4.1 LEARNING PARAMETERS
+
+The parameters of the networks, both the layer weights and the coefficients of the PAUs, were trained over 400 epochs using SGD with momentum set to 0.9. On the Cifar10 dataset we have different optimizer setups for PAU layers and the rest of the network. For the PAU layers we use constant learning rates per networks and no weight decay. For updating the rest of the network we use initial learning rate of 0.1, and learning rate decay of 0.985 per epoch and set weight decay to $5 e - 4$ . In all experiments we used a batch size of 64 samples. The weights of the networks were initialized randomly and the coefficients of the PAUs were initialized with the initialization constants of Leaky ReLU, see Tab. A.1. The additive noise of the Randomized PAUs is set to $\alpha = 1 \%$ training the networks VGG8 and MobileNetV2, respectively $\alpha = 1 0 \%$ for ResNet101.
+
+On the Imagenet dataset we use the same optimizer for PAU and the rest of the network. We follow the default setup provided by Pytorch and use an initial learning rate of 0.1, and decay the learning rate by $10 \%$ after 30, 60 and 90 epochs.
+
+# A.4.2 NETWORK ARCHITECTURES
+
+The network architectures were taken from reference implementations in PyTorch and we modified them to use PAUs. All architectures are the same among the different activation functions except for Maxout. The default amount of trainable parameters of VGG8 (Simonyan and Zisserman, 2015) is 3,918,858. Using PAU 50 additional parameters are introduced. Maxout is extending the VGG8 network to a total number of 7,829,642 parameters. MobileNetV2 (Sandler et al., 2018) is contains by default 2,296,922 trainable parameters. PAU adds 360 additional parameters. The Maxout activation function is results in a total number of 3,524,506 parameters. With respect to the number of parameters ResNet101 (He et al., 2016) is the largest network we train. By default it contains 42,512,970 trainable parameters, we introduce 100 PAUs and therefore add 1000 additional parameters to the network. If one is replacing each activation function using Maxout the resulting ResNet101 network contains 75,454,090 trainable parameters. The default DenseNet121 (Huang et al., 2017) network has 6,956,298 parameters. Replacing the activation functions with PAU adds 1200 parameters to the network.
+
+# A.4.3 PRUNING EXPERIMENT
+
+For the pruning experiment, we implement the "Lottery ticket hypothesis" (Frankle and Carbin (2019)) in PyTorch. We compare PAUs against the best activation for the network architecture according to the average predictive accuracy from Tab. 3. More precisely, we compare the predictive performance under pruning for the networks $N _ { 1 } = \{ \mathrm { V G G - } 8 _ { \mathrm { p a u } }$ , MobileNetV2pau, $\mathrm { R e s N e t 1 0 1 _ { p a u } } \big \}$ against the networks $N _ { 2 } = \mathrm { \{ V G G - 8 _ { L R e L U } } $ , MobileNetV2RReLU, $\mathrm { R e s N e t 1 0 1 _ { p a u } } \}$ . Here we avoided Maxout as it heavily increases the parameters in the model, defeating the purpose of pruning. Unlike the original paper, we compress the convolutions using a fixed pruning parameter per iteration of $p \% = \bar { 1 } 0 , 2 \bar { 0 } , \bar { 3 0 } , 4 0 , 5 0 , 6 0$ and evaluated once per network. After each training iteration we remove $p \%$ of filters in every convolution and the filters we remove are the ones where the sum of its weights is lowest. After pruning, we proceed to re-initialize the network and repeat the training and pruning proceedure with the next $p \%$ parameter.
+
+# A.4.4 PREDICTIVE PERFORMANCE CIFAR10
+
+![](images/d5c9c2701a29d4ea7aac7bcf065caac2074bea75336a7df8e3dc03913c875285.jpg)  
+Figure 8: PAU compared to baseline activation function units on 5 runs of CIFAR-10. Accuracy on the left column and loss on the right one. (Best viewed in color)

parse/train/BJlBSkHtDS/BJlBSkHtDS_content_list.json

@@ -0,0 +1,1378 @@
+[
+    {
+        "type": "text",
+        "text": "PADÉ ACTIVATION UNITS: END-TO-END LEARNING OF FLEXIBLE ACTIVATION FUNCTIONS IN DEEP NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            99,
+            826,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alejandro Molina1, Patrick Schramowski1, Kristian Kersting1,2 1 AI and Machine Learning Group, CS Department, TU Darmstadt, Germany 2 Centre for Cognitive Science, TU Darmstadt, Germany {molina,schramowski,kersting}@cs.tu-darmstadt.de ",
+        "bbox": [
+            184,
+            195,
+            691,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            290,
+            544,
+            305
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The performance of deep network learning strongly depends on the choice of the non-linear activation function associated with each neuron. However, deciding on the best activation is non-trivial, and the choice depends on the architecture, hyper-parameters, and even on the dataset. Typically these activations are fixed by hand before training. Here, we demonstrate how to eliminate the reliance on first picking fixed activation functions by using flexible parametric rational functions instead. The resulting Padé Activation Units (PAUs) can both approximate common activation functions and also learn new ones while providing compact representations. Our empirical evidence shows that end-to-end learning deep networks with PAUs can increase the predictive performance. Moreover, PAUs pave the way to approximations with provable robustness. ",
+        "bbox": [
+            233,
+            321,
+            766,
+            474
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "https://github.com/ml-research/pau ",
+        "bbox": [
+            235,
+            479,
+            566,
+            492
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            522,
+            336,
+            539
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An important building block of deep learning is the non-linearities introduced by the activation functions $f ( x )$ . They play a major role in the success of training deep neural networks, both in terms of training time and predictive performance. Consider e.g. Rectified Linear Unit (ReLU) due to Nair and Hinton (2010). The demonstrated benefits in training deep networks, see e.g. (Glorot et al., 2011), brought renewed attention to the development of new activation functions. Since then, several ReLU variations with different properties have been introduced such as LeakyReLUs (Maas et al., 2013), ELUs (Clevert et al., 2016), RReLUs (Xu et al., 2015), among others. Another line of research, such as (Ramachandran et al., 2018) automatically searches for activation functions. It identified the Swish unit empirically as a good candidate. However, for a given dataset, there are no guarantees that Swish unit behaves well and the proposed search algorithm is computationally quite demanding. ",
+        "bbox": [
+            174,
+            556,
+            825,
+            695
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The activation functions are traditionally fixed and, in turn, impose a set of inductive biases on the network. One attempt to relax this bias are for instance PReLUs (He et al., 2015), where the negative slope is subject to optimization allowing for more flexibility than other ReLU variants. Learnable activation functions generalize this idea. They exploit parameterizations of the activation functions, adapted in an end-to-end fashion to different network architectures and datasets during training. For instance, Maxout (Goodfellow et al., 2013) and Mixout (Zhao et al., 2017) use a fixed set of piecewise linear components and optimized their (hyper-)parameters. Although they are theoretically universal function approximators, they heavily increase the number of parameters of the network and strongly depend on hyper-parameters such as the number of components to realize this potential. Vercellino and Wang (2017) used a meta-learning approach for learning task-specific activation functions (hyperactivations). However, as Vercellino and Wang admit, the implementation of hyperactivations, while easy to express notationally, can be frustrating to implement for generalizability over any given activation network. Recently, Goyal et al. (2019) proposed a learnable activation function based on Taylor approximation and suggested a transformation strategy to avoid exploding gradients. However, relying on polynomials suffers from well-known limitations such as exploding values and a tendency to oscillate (Trefethen, 2012). ",
+        "bbox": [
+            174,
+            702,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/5efe39ebb8cdcf63f9d4aeee3ded84ca98bdf438899c138706e518a9c851ef62.jpg",
+        "image_caption": [
+            "Figure 1: Approximations of common activation functions (ReLU, Sigmoid, Tanh, Swish and Leaky ReLU $\\mathrm { \\Delta } \\alpha = 0 . 2 0 )$ ) using PAUs (marked with $^ *$ ). As one can see, PAUs can encode common activation functions very well. (Best viewed in color) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            104,
+            820,
+            270
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As an alternative, we here introduce a learnable activation function based on the Padé approximation, i.e., rational functions. In contrast to approximations for high accuracy hardware implementation of the hyperbolic tangent and the sigmoid activation functions (Hajduk, 2018), we do not assume fixed coefficients. The resulting Padé Activation Units (PAU) can be learned using standard stochastic gradient and, hence, seamlessly integrated into the deep learning stack. PAUs provide more flexibility and increase the predictive performance of deep neural networks, as we demonstrate. ",
+        "bbox": [
+            173,
+            366,
+            825,
+            450
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We proceed as follows. We start off by introducing PAUs. Then introduce Padé networks and show that they are universal approximators. Before concluding, we present our empirical evaluation. ",
+        "bbox": [
+            173,
+            457,
+            823,
+            486
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PADÉ ACTIVATION UNITS (PAU) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            512,
+            477,
+            529
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our starting point is the set of intuitive assumptions activation functions should ultimately fulfill shown in Tab. 1. The assumptions (i,v) concern the ability of neural networks to approximate functions. Rational functions can fulfill assumptions (i,iv), and our experimental evaluation demonstrates that assumptions (ii,iii,v) also hold. ",
+        "bbox": [
+            173,
+            547,
+            826,
+            604
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "(i) They must allow the networks to be universal function approximators.   \n(ii) They should ameliorate gradient vanishing.   \n(iii) They should be stable.   \n(iv) They should be parsimonious on the number of parameters.   \n(v) They should provide networks with high predictive performance. ",
+        "bbox": [
+            204,
+            651,
+            444,
+            824
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/477e35b0fbd63954eb0ae0e4f34f673c5d881286c706ad16bb71c6901fe1dc88.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"4\">Assumptions</td></tr><tr><td>Activation</td><td>Learnable</td><td>i ii</td><td>ii</td><td>iv</td><td>V</td></tr><tr><td>ReLU</td><td>N</td><td></td><td>Y</td><td></td><td>Y</td></tr><tr><td>ReLU6</td><td>N</td><td></td><td>Y</td><td></td><td></td></tr><tr><td>RReLU</td><td>N</td><td></td><td></td><td></td><td></td></tr><tr><td>LReLU</td><td>N</td><td></td><td></td><td></td><td></td></tr><tr><td>ELU</td><td>N</td><td></td><td>YYY</td><td></td><td></td></tr><tr><td>CELU</td><td>N</td><td>YYYYYYY</td><td>YYYYYY Y</td><td></td><td></td></tr><tr><td>Swish</td><td>N</td><td></td><td>？ Y</td><td></td><td></td></tr><tr><td>PReLU</td><td>Y</td><td>Y</td><td>Y</td><td></td><td></td></tr><tr><td>Maxout</td><td>Y</td><td></td><td>Y</td><td>YN</td><td></td></tr><tr><td>Mixture</td><td>Y</td><td></td><td></td><td></td><td>Y</td></tr><tr><td>APL</td><td>Y</td><td></td><td>YY</td><td>YYY</td><td>Y</td></tr><tr><td>SReLU</td><td>Y</td><td>YYYYYY</td><td>YYYY Y</td><td></td><td>Y</td></tr><tr><td>SLAF</td><td>Y</td><td></td><td>Y</td><td>N Y</td><td>？</td></tr><tr><td>PAU</td><td>Y</td><td>Y</td><td>Y</td><td>Y</td><td>Y Y</td></tr></table>",
+        "bbox": [
+            473,
+            633,
+            790,
+            838
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Table 1: (Left) The intuitive assumptions activation functions (AFs) should ultimately fulfill. (Right) Existing ELU, CELU and ReLU like AFS do not fulfill them. Only learnable AFs allow one to tune their shape at training time, ignoring hyper-parameters such as $\\alpha$ for LReLU. For Swish, our experimental results do not indicate problems with vanishing gradients. SLAF (Goyal et al., 2019) showed undefined values (iii), and we could not judge their performance (v). ",
+        "bbox": [
+            173,
+            849,
+            826,
+            920
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 PADÉ APPROXIMATION OF ACTIVATION FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            575,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let us now formally introduce PAUs. Assume for the moment that we start with a fixed activation function $f ( x )$ . The Padé approximant (Brezinski and Van Iseghem, 1994) is the “best” approximation of $f ( x )$ by a rational function of given orders $m$ and $n$ . Applied to typical actication functions, Fig. 1 shows that they can be approximated well using rational functions. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            185
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "More precisely, given $f ( x )$ , the Padé approximant is the rational function $F ( x )$ over polynomials $P ( x ) , Q ( x )$ of order $m$ , $n$ of the form ",
+        "bbox": [
+            173,
+            191,
+            823,
+            220
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/766de660fe5e756e38caccb3a522e97b259a0859c6fb81482f004e47439d7414.jpg",
+        "text": "$$\nF ( x ) = { \\frac { P ( x ) } { Q ( x ) } } = { \\frac { \\sum _ { j = 0 } ^ { m } a _ { j } x ^ { j } } { 1 + \\sum _ { k = 1 } ^ { n } b _ { k } x ^ { k } } } = { \\frac { a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \\cdot \\cdot \\cdot + a _ { m } x ^ { m } } { 1 + b _ { 1 } x + b _ { 2 } x ^ { 2 } + \\cdot \\cdot \\cdot + b _ { n } x ^ { n } } } ~ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            224,
+            738,
+            263
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "which agrees with $f ( x )$ the best. The Padé approximant often gives a better approximation of a function $f ( x )$ than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons, it has been used before in the context of graph convolutional networks (Chen et al., 2018). However, they have not been considered so far for general deep networks. Padé Activation Units (PAUs) go one step further, instead of fixing the coefficients $a _ { j } , b _ { k }$ to approximate a particular activation function, we allow them to be free parameters that can be optimized end-to-end with the rest of the neural network. This allows the optimization process to find the activation function needed at each layer automatically. ",
+        "bbox": [
+            173,
+            266,
+            825,
+            378
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The flexibility of Padé is not only a blessing but might also be a curse: it can model processes that contain poles. For a learnable activation function, however, a pole may produce undefined values depending on the input as well as instabilities at learning and inference time. Therefore we consider a restriction, called safe $P A U$ , that guarantees that the polynomial $Q ( x )$ is not 0, i.e., we avoid poles. In general, restricting $Q ( x )$ implies that either $Q ( x ) \\mapsto \\mathbb { R } _ { > 0 }$ or $Q ( \\dot { x } ) \\mapsto \\mathbb R _ { < 0 }$ , but as $P ( x ) \\mapsto \\mathbb { R }$ we can focus on $Q ( x ) \\mapsto \\mathbb { R } _ { > 0 }$ wlog. However, as $\\begin{array} { r } { \\operatorname* { l i m } _ { Q ( x ) \\to 0 ^ { + } } F ( x ) \\to \\infty } \\end{array}$ learning and inference become unstable. To fix this, we impose a stronger constraint, namely $Q ( x ) \\geq q \\gg 0$ . In this work, $q = 1$ , i.e., $\\forall x : Q ( x ) \\geq 1$ , preventing poles and allowing for safe computation on $\\mathbb { R }$ : ",
+        "bbox": [
+            173,
+            383,
+            826,
+            498
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea53cbc0ec96e723011a56e6a127c0ecbd5c3e99d88a601984198626355947b4.jpg",
+        "text": "$$\nF ( x ) = { \\frac { P ( x ) } { Q ( x ) } } = { \\frac { \\sum _ { j = 0 } ^ { m } a _ { j } x ^ { j } } { 1 + | \\sum _ { k = 1 } ^ { n } b _ { k } x ^ { k } | } } = { \\frac { a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \\dotsb + a _ { m } x ^ { m } } { 1 + | b _ { 1 } x + b _ { 2 } x ^ { 2 } + \\dotsb + b _ { n } x ^ { n } | } } \\ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            501,
+            743,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Other values for $q \\in ( 0 , 1 )$ might still be interesting, as they could provide gradient amplification due to the partial derivatives having $Q ( X )$ in the denominator. However, we leave this for future work. ",
+        "bbox": [
+            173,
+            550,
+            826,
+            579
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 LEARNING SAFE PADÉ APPROXIMATIONS USING BACKPROPAGATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            594,
+            692,
+            611
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In contrast to the standard way of fitting Padé approximants where the coefficients are found via derivatives and algebraic manipulation against a given function, we optimize their polynomials via backpropagation and (stochastic) gradient descent. To do this, we have to compute the gradients with respect to the parameters $\\begin{array} { r } { \\frac { \\partial F } { \\partial a _ { j } } , \\frac { \\partial \\check { F } } { \\partial b _ { k } } } \\end{array}$ as well as the gradient for the input $\\textstyle { \\frac { \\partial F } { \\partial x } }$ . A simple alternative is to implement the forward pass as described in Eq. 2, and let automatic differentiation do the job. To be more efficient, however, we can also implement PAUs directly in CUDA (Nickolls et al. (2008)), and for this we need to compute the gradients ourselves: ",
+        "bbox": [
+            173,
+            619,
+            826,
+            722
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af2cdf4c6539ae3358613562ce683e8bca70c2ca0460ced617348fd1f9a81414.jpg",
+        "text": "$$\n\\frac { \\partial F } { \\partial x } = \\frac { \\partial P ( x ) } { \\partial x } \\frac { 1 } { Q ( x ) } - \\frac { \\partial Q ( x ) } { \\partial x } \\frac { P ( x ) } { Q ( x ) ^ { 2 } } , \\quad \\frac { \\partial F } { \\partial a _ { j } } = \\frac { x ^ { j } } { Q ( x ) } \\quad \\mathrm { a n d } \\quad \\frac { \\partial F } { \\partial b _ { k } } = - x ^ { k } \\frac { A ( X ) } { | A ( X ) | } \\frac { P ( X ) } { Q ( x ) ^ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            724,
+            802,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ) = a1 + 2a2x + · · · + mamxm−1 , ∂Q(x) $\\begin{array} { r } { \\frac { \\partial Q ( x ) } { \\partial x } = \\frac { A ( X ) } { | A ( X ) | } \\left( b _ { 1 } + 2 b _ { 2 } x + \\cdot \\cdot \\cdot + n b _ { n } x ^ { n - 1 } \\right) } \\end{array}$ , $A ( X ) = b _ { 1 } x + b _ { 2 } x ^ { 2 } + \\cdot \\cdot \\cdot + b _ { n } x ^ { n }$ , and $Q ( x ) = 1 + | A ( X ) |$ . Here we reuse the expressions to reduce computations. To avoid divisions by zero when computing the gradients, we define $\\frac { z } { | z | }$ as the sign of $z$ . With the gradients at hand, PAUs can seamlessly be placed together with other modules onto the differentiable programming stack. ",
+        "bbox": [
+            173,
+            763,
+            825,
+            844
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PADÉ NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            863,
+            354,
+            880
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Having PAUs at hand, one can define Padé networks as follows: Padé networks are feedforward networks with PAU activation functions that may include convolutional and residual architectures with pooling layers. To use Padé networks effectively, one simply replaces the standard activation functions in a neural network by PAUs and then proceed to optimize all the parameters and use the network as usual. However, even if every PAU contains a low number of parameters (coefficients $a _ { j } , b _ { k } )$ , in the extreme case, learning one PAU per neuron may considerably increase the complexity of the networks and in turn the learning time. To ameliorate this and inspired by the idea of weightsharing as introduced by Teh and Hinton (2001), we propose to learn one PAU per layer. Therefore we only add $\\phi$ many parameters, where $\\phi = L * ( m + n )$ and $L$ is the number of activation layers in the network. In our experiments we set $\\phi = 1 0 L$ , a rather small number of parameters (iv). ",
+        "bbox": [
+            173,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The last step missing before we can start the optimization process is to initialize the coefficients of the PAUs. Surely, one can do random initialization of the coefficients and allow the optimizer to train the network end-to-end. However, we obtained better results after initializing all PAUs with coefficients that approximate standard activation functions. For a discussion on how to obtain different PAU coefficients, we refer to Sec. A.1. ",
+        "bbox": [
+            174,
+            222,
+            825,
+            291
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Before evaluating Padé Networks empirically, let us touch upon their expressivity and how to sparsify them. ",
+        "bbox": [
+            176,
+            299,
+            823,
+            327
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 PADÉ NETWORKS ARE UNIVERSAL FUNCTION APPROXIMATORS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            343,
+            660,
+            358
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A standard multi-layer perceptron (MLP) with enough hidden units and non-polynomial activation functions is a universal approximator, see e.g. (Hornik et al., 1989; Leshno et al., 1993). Padé Networks are also universal approximators. ",
+        "bbox": [
+            174,
+            369,
+            825,
+            412
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $\\rho \\colon \\mathbb { R }  \\mathbb { R }$ be a PAU activation function. Let ${ \\mathcal { N } } ^ { \\rho }$ represent the class of neural networks with activation function $\\rho$ . Let $K \\subseteq \\mathbb { R } ^ { n }$ be compact. Then ${ \\mathcal { N } } ^ { \\rho }$ is dense in $C ( K )$ . ",
+        "bbox": [
+            173,
+            416,
+            823,
+            445
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proof holds for both PAUs and safe PAUs as it makes no assumptions on the form of the denominator $Q ( x )$ , and is a direct application of the following propositions: ",
+        "bbox": [
+            174,
+            457,
+            821,
+            486
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. (From Theorem 1.1 in (Kidger and Lyons, 2019)) Let $\\rho \\colon \\mathbb { R }  \\mathbb { R }$ be any continuous function. Let $\\mathcal { N } _ { n } ^ { \\rho }$ represent the class of neural networks with activation function $\\rho _ { i }$ , with n neurons in the input layer, one neuron in the output layer, and one hidden layer with an arbitrary number of neurons. Let $K \\subseteq \\mathbb { R } ^ { n }$ be compact. Then $\\mathcal { N } _ { n } ^ { \\rho }$ is dense in $C ( K )$ if and only if $\\rho$ is non-polynomial. ",
+        "bbox": [
+            174,
+            489,
+            826,
+            546
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. (From Theorem 3.2 in (Kidger and Lyons, 2019)) Let $\\rho \\colon \\mathbb { R }  \\mathbb { R }$ be any continuous point. Let function which is continuously differentiable at at least one point, with nonzero derivative at that $K \\subseteq \\mathbb { R } ^ { n }$ be compact. Then $\\mathcal { N N } _ { n , m , n + m + 2 } ^ { \\rho }$ is dense in $C ( K ; \\mathbb { R } ^ { m } )$ . ",
+        "bbox": [
+            173,
+            550,
+            825,
+            594
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $\\rho ( x ) = P ( x ) / Q ( x )$ , we have to consider the following two cases: ",
+        "bbox": [
+            176,
+            609,
+            681,
+            626
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Case 1: $Q ( x ) \\neq 1$ , by definition, $\\rho ( x )$ is non-polynomial. Then by proposition 1, we get that $\\mathcal { N } _ { n } ^ { \\rho }$ is dense in $C ( K )$ . ",
+        "bbox": [
+            173,
+            630,
+            823,
+            660
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Case 2: $Q ( x ) = 1$ , here $\\textstyle \\rho ( x ) = \\sum _ { j = 0 } ^ { m } a _ { j } x ^ { j }$ , is polynomial, continuous and continuously differentiable in propositi $\\mathbb { R }$ . Let any  2, we get $a _ { j > 0 } > 0$ re exists a p is dense in $\\alpha \\in \\mathbb { R }$ such that $\\rho ^ { \\prime } ( \\alpha ) \\neq 0$ and then by $\\mathcal { N N } _ { n , m , n + m + 2 } ^ { \\rho }$ $C ( K ; \\mathbb { R } ^ { m } )$ ",
+        "bbox": [
+            174,
+            661,
+            825,
+            709
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 SPARSE PADÉ NETWORKS AND RANDOMIZED PAUS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            751,
+            576,
+            765
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Padé Networks can $\\epsilon$ -approximate neural networks with ReLU activations (Telgarsky, 2017). This implies that by using PAUs, we are embedding a virtual network into the networks we want to use. This, in turn, is the operating assumption of the lottery ticket hypothesis due to Frankle and Carbin (2019). Thus, we expect that lottery ticket pruning can find well-performing Padé networks that are smaller than their original counterparts while reducing inference time and potentially improving the predictive performance. ",
+        "bbox": [
+            174,
+            776,
+            825,
+            861
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Generally, overfitting is an important practical concern when training neural networks. Usually, we can apply regularization techniques such as Dropout (Srivastava et al., 2014). Unfortunately, although each PAU approximates a small ReLU network section, we do not have access to the internal representation of this virtual network. Therefore, we can not regularize the activation function via standard Dropout. An alternative for regularizing activation functions was introduced in Randomized Leaky ReLUs (RReLUs, Xu et al. (2015)), where the negative slope parameter is sampled uniformly on a range. This makes the activation function behave differently at training time for every input $x$ , forwarding and backpropagating according to $x$ and the sampled noise. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/d7a905d82b9dffce065aa2a8b5f3b5fdcf7afadbd9826edbf48199e7800c7121.jpg",
+        "image_caption": [
+            "Figure 2: PAU compared to baseline activation function units over 5 runs on Fashion-MNIST using the LeNet architecture: (left) mean test-accuracy (the higher, the better) and (right) mean train-loss (the lower, the better). As one can see, PAU outperforms all baseline activations and enable the networks to achieve a lower loss during training compared to all baselines. (Best viewed in color) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            101,
+            821,
+            238
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            338,
+            825,
+            395
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can employ a similar technique to make PAUs resistant to overfitting. Consider a PAU with coefficients $\\mathbf { C } = \\{ a _ { 0 } , \\cdot \\cdot \\cdot , a _ { m } , b _ { 0 } , \\cdot \\cdot \\cdot , b _ { n } \\}$ . We can introduce additive noise during training into each coefficient $c _ { i } \\in \\mathbf { C }$ for every input $x _ { j }$ via $c _ { i , j } = c _ { i } + z _ { i , j }$ where $z _ { i , j } \\sim U ( l _ { i } , u _ { i } )$ , $l _ { i } = ( 1 - \\alpha \\% ) * c _ { i }$ and reciprocally $u _ { i } = \\left( 1 + \\alpha \\% \\right) * { \\check { c } } _ { i }$ . This results in Randomized PAU (RPAU): ",
+        "bbox": [
+            174,
+            401,
+            825,
+            458
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2c19ea1aeb8e662cf5fefbbb7fb4edab8cbfae2923d1448f82e739e44654fb4.jpg",
+        "text": "$$\nR ( x _ { j } ) = \\frac { c _ { 0 , j } + c _ { 1 , j } x + c _ { 2 , j } x ^ { 2 } + \\cdot \\cdot \\cdot + c _ { m , x } x ^ { m } } { 1 + | c _ { m + 1 } x + c _ { m + 2 } x ^ { 2 } + \\cdot \\cdot \\cdot + c _ { m + n } x ^ { n } | } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            462,
+            673,
+            498
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We compute the gradients as before and simply replace the coefficients by their noisy counterparts. ",
+        "bbox": [
+            173,
+            502,
+            816,
+            517
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            536,
+            450,
+            553
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our intention here is to investigate the behavior and performance of PAUs as well as to compare them to other activation functions using standard deep neural networks. All our experiments are implemented in PyTorch with PAU implemented in CUDA, and were executed on an NVIDIA DGX-2 system. In all experiments, we initialized PAUs with coefficients that approximate LeakyReLUs for a rational function of order $m = 5 , n = 4$ . In all experiments except for ImageNet, we report the mean of 5 runs initialized with different seeds for the accuracy on the test-set after training. And, we compared PAU to the following activation functions: ReLU, ReLU6, Leaky ReLU (LReLU), Random ReLU (RReLU), ELU, CELU, Swish, Parametric ReLU (PReLU), Maxout, Mixture of activations (Mixture), SLAF, APL and SReLU. For details on all the activation functions, we refer to Appendix A.2. As datasets we considered MNIST, Fahion-MNIST, CIFAR-10 and ImageNet. ",
+        "bbox": [
+            173,
+            566,
+            826,
+            707
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 EMPIRICAL RESULTS ON MNIST AND FASHION-MNIST BENCHMARKS ",
+        "bbox": [
+            173,
+            722,
+            710,
+            738
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "First we evaluated PAUs on MNIST (LeCun et al., 2010) and Fashion-MNIST (Xiao et al., 2017) using two different architectures: LeNet (LeCun et al., 1998) and VGG-8 (Simonyan and Zisserman, 2015). For more details on the architectures, learning settings, and results, we refer to Sec. A.3. ",
+        "bbox": [
+            174,
+            748,
+            826,
+            791
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As can be seen in Fig. 2 and Tab. 2, PAU outperformed on average the baseline activation functions on every network in terms of predictive performance. Moreover, the results are stable on different runs (c.f. mean $\\pm$ std). PAUs also enable the networks to achieve a lower loss during training compared to all baselines on all networks. Actually, PAU achieved the best results on both datasets and on Fashion-MNIST it provides the best results for both architectures. As expected, reducing the bias is beneficial in this experiment. Comparing the baseline activation functions on the MNIST dataset and the different architectures, there is no clear choice of activation that achieves the best performance. However, PAU always matches or even outperforms the best performing baseline activation function. This shows that a learnable activation function relieves the network designer of having to commit to a potentially suboptimal choice. Moreover, Fig. 2 also shows that PAU is more stable than SLAF. This is not unexpected as Taylor approximations tend to oscillate and overshoot (Trefethen, 2012). We also observed undefined values at training time for SLAF; therefore, we do not compare against it in the following experiments. Finally, when considering the number of parameters used by PAU, we can see that they are very efficient. The VGG-8 network uses 9.2 million parameters, PAU here uses 50 parameters, and for LeNet, the network uses 0.5 million parameters while PAU uses only 40. ",
+        "bbox": [
+            173,
+            797,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/999bb2e8c46dc83873443e20940f7e1fcf8321ad8a515a8606123b2ee225b61d.jpg",
+        "table_caption": [
+            "Table 2: Performance comparison of activation functions on MNIST and Fashion-MNIST (the higher, the better) on two common deep architectures. Shown are the results averaged over 5 reruns as well as the top result among these 5 runs. The best $( \\vphantom { \\sqrt [ 4 ] { 3 } } \\cdots )$ and runner-up $( ^ { 6 6 } \\circ ^ { 7 3 } )$ results per architecture are bold. As one can see, PAUs consistently outperform the other activation functions on average and yields the top performance on each dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">VGG-8 best</td><td colspan=\"2\">LeNet</td><td colspan=\"2\">VGG-8</td><td colspan=\"2\">LeNet</td></tr><tr><td colspan=\"2\">mean ± std</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td></tr><tr><td colspan=\"9\"></td></tr><tr><td colspan=\"9\"></td></tr><tr><td>ReLU</td><td>99.17 ±0.10</td><td>MNIST 99.30</td><td>99.17 ± 0.05</td><td>99.25</td><td>89.11 ± 0.43</td><td>Fashion-MNIST 89.69</td><td>89.86 ± 0.32</td><td>90.48</td></tr><tr><td>ReLU6</td><td>99.28 ± 0.04</td><td>99.31</td><td>99.09 ±0.09</td><td>99.22</td><td>89.87 ±0.62</td><td>90.38</td><td>89.74±0.27</td><td>89.96</td></tr><tr><td>LReLU</td><td>99.13 ± 0.11</td><td>99.27</td><td>99.10 ± 0.06</td><td>99.22</td><td>89.37 ± 0.30</td><td>89.74</td><td>89.74± 0.24</td><td>90.02</td></tr><tr><td>RReLU</td><td>99.16 ±0.13</td><td>99.28</td><td>99.20±0.13 099.38</td><td></td><td>88.46±0.85</td><td>89.32</td><td>89.74±0.19</td><td>89.88</td></tr><tr><td>ELU</td><td>99.15 ± 0.09</td><td>99.28</td><td>99.15 ±0.06</td><td>99.22</td><td>89.65 ± 0.33</td><td>90.06</td><td>89.84± 0.47</td><td>90.25</td></tr><tr><td>CELU</td><td>99.15 ± 0.09</td><td>99.28</td><td>99.15 ±0.06</td><td>99.22</td><td>89.65 ± 0.33</td><td>90.06</td><td>89.84± 0.47</td><td>90.25</td></tr><tr><td>Swish</td><td>99.10 ±0.06</td><td>99.20</td><td>99.19 ±0.09</td><td>99.29</td><td>88.54±0.59</td><td>89.36</td><td>89.54± 0.22</td><td>89.89</td></tr><tr><td>PReLU</td><td>99.16 ± 0.09</td><td>99.25</td><td>99.14±0.09</td><td>99.24</td><td>88.82 ±0.51</td><td>89.54</td><td>90.09 ±0.22090.29</td><td></td></tr><tr><td>SLAF</td><td></td><td></td><td></td><td></td><td>90.60 ± 0.00</td><td>90.60</td><td>89.33 ± 0.28</td><td>89.80</td></tr><tr><td>APL</td><td>099.35±0.11·99.50</td><td></td><td>99.18±0.10</td><td>99.33</td><td>91.41±0.48·92.25</td><td></td><td>89.72 ± 0.30</td><td>90.01</td></tr><tr><td>SReLU</td><td>99.15 ± 0.03</td><td>99.20</td><td>99.13±0.14</td><td>99.27</td><td>89.65 ±0.42</td><td>90.31</td><td>89.83 ±0.30</td><td>90.28</td></tr><tr><td>PAU</td><td>99.30±0.05 099.40</td><td></td><td>099.21±0.04</td><td>99.26</td><td>091.25±0.18091.56</td><td></td><td>90.33±0.15·90.62</td><td></td></tr><tr><td>RPAU</td><td>·99.35±0.04  99.38 99.26±0.11·99.42</td><td></td><td></td><td></td><td></td><td></td><td>91.23±0.15 91.41090.20±0.11 090.29</td><td></td></tr></table>",
+        "bbox": [
+            174,
+            181,
+            823,
+            400
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            426,
+            825,
+            511
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In summary, this shows that PAUs are stable, parsimonious and can improve the predictive performance of deep neural networks (iii,iv,v). ",
+        "bbox": [
+            174,
+            517,
+            825,
+            546
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 LEARNED ACTIVATION FUNCTIONS ON MNIST AND FASHION-MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            564,
+            705,
+            579
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When looking at the activation functions learned from the data, we can see that the PAU family is flexible yet presents similarities to standard functions. In particular, Fig. 3 illustrates that some of the learned activations seem to be smoothed versions of Leaky ReLUs, since V-shaped activations are simply Leaky ReLUs with negative $\\alpha$ values. This is not surprising as we initialize PAUs with coefficients that match Leaky ReLUs. Finding different initialization and optimization parameters is left as future work. In contrast, when learning piecewise approximations of the same activations using Maxout, we would require a high $k$ . This significantly increases the number of parameters of the network. This again provides more evidence in favor of PAUs being flexible and parsimonious (iv). SLAF produced undefined values during training on all networks except Fashion-MNIST where LeNet finished 4 runs and VGG only one run. ",
+        "bbox": [
+            174,
+            592,
+            825,
+            731
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 EMPIRICAL RESULTS ON CIFAR-10 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            751,
+            464,
+            765
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After investigating PAU on MNIST and Fashion-MNIST, we considered a more challenging setting: CIFAR-10 (Krizhevsky et al. (2009)). We also considered other learnable activation functions, namely Maxout $( \\mathrm { k } \\mathrm { = } 2 )$ and Mixture of activations (Id and ReLU) as well as another popular deep network architectures: MobileNetV2 (Sandler et al., 2018), ResNet101 (He et al., 2016) and DenseNet121 (Huang et al., 2017). For more details on the learning settings and results, we refer to Sec. A.4. ",
+        "bbox": [
+            174,
+            776,
+            825,
+            847
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Let us start by considering the results for VGG-8 and MobileNetV2 on CIFAR-10. These networks are the smallest of this round of experiments and, therefore, could benefit more from bias reduction. Indeed, we can see in Tab. 3 that both networks take advantage of learnable activation functions, i.e., Maxout, PAU, and RPAU. As expected, adding more capacity to VGG-8 helps and this is what Maxout is doing. Moreover, even if Mixtures do not seem to provide a significant benefit on VGG-8, they do help in MobileNetV2. Here, we see again that PAU and RPAU are either in the lead or close to the best when it comes to predictive performance, without having to make a choice apriori. ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/56559abc7ef80f02c90b5a007aa1c26991bd46b414931f931c88c2e03e1b8475.jpg",
+        "table_caption": [
+            "Table 3: Performance comparison of activation functions on CIFAR-10 (the higher, the better) on four state-of-the-art deep neural architectures. Shown are the results averaged over 5 reruns as well as the top result among these 5 runs. The best $( \\vphantom { \\sqrt [ 4 ] { 3 } } \\cdots )$ and runner-up (“◦”) results per architecture are bold. As one can see, PAUs are either in the lead or close to the best. $( ^ { 6 6 } { } ^ { \\ast } { } ^ { \\ast } { } ^ { \\ast } { } ^ { \\ast } { } ^ { \\ast } { } ^ { \\ast } )$ are experiments that did not finish on time. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">VGG-8</td><td colspan=\"2\">MobileNetV2</td><td colspan=\"2\">ResNet101</td><td colspan=\"2\">DenseNet121</td></tr><tr><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td><td>mean ± std</td><td>best</td></tr><tr><td></td><td colspan=\"6\"></td><td></td><td></td></tr><tr><td>ReLU</td><td>92.32 ± 0.16</td><td>92.58</td><td>91.51 ± 0.28</td><td>91.82</td><td>95.07 ± 0.17</td><td></td><td>95.36095.36±0.18 095.63</td><td></td></tr><tr><td>ReLU6</td><td>92.36 ±0.06</td><td>92.47</td><td>91.30 ± 0.23</td><td>91.57</td><td>95.11 ± 0.24</td><td>95.29</td><td>95.33 ± 0.14</td><td>95.46</td></tr><tr><td>LReLU</td><td>92.43 ± 0.14</td><td>92.65</td><td>91.94 ± 0.12</td><td>92.08</td><td>95.08 ±0.19</td><td>95.29</td><td>·95.42±0.17·95.65</td><td></td></tr><tr><td>RReLU</td><td>92.32 ± 0.07</td><td>92.42</td><td>094.66±0.16</td><td>6094.94</td><td>95.21±0.23</td><td>095.51</td><td>95.00 ±0.12</td><td>95.14</td></tr><tr><td>ELU</td><td>91.24 ± 0.09</td><td>91.33</td><td>90.43 ±0.14</td><td>90.61</td><td>94.04± 0.14</td><td>94.24</td><td>90.78 ±0.29</td><td>91.23</td></tr><tr><td>CELU</td><td>91.24 ± 0.09</td><td>91.33</td><td>90.69 ±0.27</td><td>90.97</td><td>93.80 ±0.36</td><td>94.25</td><td>90.88 ±0.19</td><td>91.08</td></tr><tr><td>PReLU</td><td>92.22 ±0.26</td><td>92.51</td><td>93.54± 0.45</td><td>93.95</td><td>94.15 ± 0.39</td><td>94.50</td><td>94.98 ±0.16</td><td>95.15</td></tr><tr><td>Swish</td><td>91.58 ±0.18</td><td>91.86</td><td>92.04±0.13</td><td>92.21</td><td>91.83 ± 1.61</td><td>92.84</td><td>93.04±0.16</td><td>93.32</td></tr><tr><td>Maxout</td><td>·93.03±0.11</td><td>·93.23</td><td>94.41 ±0.10</td><td>94.54</td><td>95.11 ± 0.13</td><td>95.23</td><td>***</td><td>***</td></tr><tr><td>Mixture</td><td>91.86 ±0.14</td><td>92.06</td><td>94.06 ± 0.16</td><td>94.25</td><td>94.50± 0.25</td><td>94.71</td><td>93.33 ±0.17</td><td>93.59</td></tr><tr><td>APL</td><td>91.63 ± 0.13</td><td>91.82</td><td>93.62 ±0.64</td><td>94.50</td><td>94.12 ± 0.36</td><td>94.50</td><td>94.45 ±0.23</td><td>94.78</td></tr><tr><td>SReLU</td><td>092.66± 0.27 093.13</td><td></td><td>94.03 ± 0.11</td><td>94.25</td><td>095.24±0.13</td><td>95.38</td><td>94.77 ±0.24</td><td>95.20</td></tr><tr><td>PAU</td><td>92.51± 0.16</td><td>92.70</td><td>94.57±0.21</td><td>94.90</td><td>95.16±0.13</td><td>95.28</td><td>95.03±0.07</td><td>95.16</td></tr><tr><td>RPAU</td><td>92.50 ±0.09</td><td>92.62</td><td>·94.82±0.21·95.13</td><td></td><td>·95.34±0.13·95.54</td><td></td><td>95.27±0.10</td><td>95.41</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            174,
+            181,
+            825,
+            400
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            426,
+            823,
+            455
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Now, let us have a look at the performance of PAU and RPAU on the larger networks ResNet101 and DenseNet121. As these networks are so expressive, we do not expect the flexibility of the learnable activation functions to have a big impact on the performance. Tab. 3 confirms this. Nevertheless, they are still competitive and their performance is stable as shown by the standard deviation. On ResNet101, PAUs actually provided the top performance. ",
+        "bbox": [
+            174,
+            462,
+            825,
+            532
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 FINDING SPARSE PADÉ NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            551,
+            460,
+            565
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Sec. 3.2, using PAUs in a network is equivalent to introducing virtual networks of ReLUs in the middle of the network, effectively adding virtual depth to the networks. Therefore, we also investigated whether pruning can help one to unmask smaller sub-networks whose performance is similar to the original network. In a sense, we are removing blocks of the real network as they get replaced by the virtual network. Here, we only do pruning on the convolutional layers. For details about the algorithm and hyper-parameters, wee refer to Sec. A.4.3. ",
+        "bbox": [
+            174,
+            577,
+            825,
+            662
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we compared PAU against the best activation functions for the different architectures. However, we discarded Maxout, as instead of pruning it introduces many more parameters into the network defeating the original purpose. As one can see in Fig. 4, pruning on the already sizeoptimized networks VGG-8 and MobileNetV2 has an effect on the predictive performance. However, the performance of PAU remains above the other activation functions despite the increase in pruning pressure. In contrast, when we look at ResNet101, we see that the performance of PAU is not influenced by pruning, showing that indeed we can find sparse Padé network without major loss in accuracy. And what is more, PAU enables the ResNet101 subnetwork, pruned by $30 \\%$ , to achieve a higher accuracy compared to all pruned and not pruned networks. ",
+        "bbox": [
+            173,
+            669,
+            823,
+            698
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/724d775b800a1442799e3b498a9a937bef334b7d16fc37353526bee8b541b3f5.jpg",
+        "image_caption": [
+            "Figure 3: Estimated activation functions after training the VGG-8 network with RPAU on FashionMNIST. The center line indicates the PAU while the surrounding area indicates the space of the additive noise in RPAUs. As one can see, the PAU family differs from common activation functions but capture characteristics of them. (Best viewed in color) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            723,
+            821,
+            852
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/f6f8ba30cac791aa66edbb46332514c5a26cf85b13001207739147175f3818e7.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of the predictive accuracy (higher is better) for the architectures VGG-8, MobileNetV2 and ResNet101 between PAU and the best activation functions according to Tab. 3. PAU is consistently better. On ResNet101 PAU is not affected by the increase pruning pressure. Furthermore, PAU enables the ResNet101 subnetwork, pruned by $30 \\%$ , to achieve a higher accuracy compared to all pruned and not pruned networks. (Best viewed in color) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            106,
+            815,
+            238
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            358,
+            825,
+            457
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.5 EMPIRICAL RESULTS ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            479,
+            467,
+            493
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, we investigated the performance on a much larger dataset, namely ImageNet (Russakovsky et al. (2015)) used to train MobileNetV2. As can be seen in Fig. 5 and Tab. 4, PAU and Swish clearly dominate in performance (v). PAU leads in top-1 accuracy and Swish in top-5 accuracy. Moreover, both PAU and Swish show faster learning compared to the other activation functions. ",
+        "bbox": [
+            174,
+            507,
+            825,
+            563
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we argue that the rapid learning rate of PAU in all the experiments indicate that they do not exhibit vanishing gradient issues (ii). ",
+        "bbox": [
+            174,
+            569,
+            823,
+            599
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2488f2ddb65dd7f31205283870493875f1e64c1aec1a49687f7811a78eff62a9.jpg",
+        "image_caption": [
+            "Figure 5: MobileNetV2 top-1 test accuracy on the left (higher is better) and training loss on the right (lower is better) for multiple activation functions in ImageNet. PAU achieves higher accuracy and lower loss values in fewer epochs. (Best viewed in color) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            614,
+            785,
+            852
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/d68c9026f495a0cc8357fc0ebc2b75257af0d85609e155f31248856121ab0ef9.jpg",
+        "table_caption": [
+            "Table 4: MobileNetV2 top-1 and top-5 accuracies in ImageNet (higher is better) for different activations. Best $( \\mathbf { \\ddot { \\sigma } } \\mathbf { \\dot { \\sigma } } )$ and runner-up (“���”) are shown in bold. PAU is the best in top-1 accuracy and runner-up for top-5. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MobileNetV2</td><td>ReLU</td><td>ReLU6</td><td>LReLU</td><td>RReLU</td><td>ELU</td><td>CELU</td><td>PReLU</td><td>Swish</td><td>SReLU</td><td>PAU</td></tr><tr><td>Acc@1</td><td>69.65</td><td>69.83</td><td>70.03</td><td>69.12</td><td>69.13</td><td>69.17</td><td>68.61</td><td>071.24</td><td>70.62</td><td>·71.35</td></tr><tr><td>Acc@5</td><td>89.09</td><td>89.34</td><td>89.26</td><td>88.80</td><td>88.46</td><td>88.59</td><td>88.51</td><td>·89.95</td><td>89.59</td><td>089.85</td></tr></table>",
+        "bbox": [
+            181,
+            99,
+            818,
+            142
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.6 SUMMARIZED RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            222,
+            379,
+            236
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, we compare the PAU family of activation functions to all the other activation functions and we aggregate the number of occurrences where PAU performed better or worse in comparison. The aggregate results can be found in Tab. 5. These results are the aggregates of all the experiments on all datasets and all architectures. As we can see, the PAU family is very competitive. ",
+        "bbox": [
+            173,
+            247,
+            825,
+            304
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/ab9408f7b3b01794375f88dba83a6b03fbf797e15690cdd745ff7705a22c8d65.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Baselines</td><td></td><td>ReLUReLU6LReLURReLUELU</td><td></td><td></td><td></td><td>CELUPReLU|</td><td></td><td>SwishMaxoutMixturel</td><td></td><td></td><td>APL</td><td>SReLU</td></tr><tr><td>PAU/RPAU&gt;= Baseline</td><td>34</td><td>35</td><td>34</td><td>33</td><td>40</td><td>40</td><td>39</td><td>41</td><td>9</td><td>20</td><td>32</td><td>33</td></tr><tr><td>PAU/RPAU&lt;Baseline</td><td>8</td><td>7</td><td>8</td><td>9</td><td>2</td><td>2</td><td>3</td><td>1</td><td>6</td><td>0</td><td>7</td><td>8</td></tr></table>",
+        "bbox": [
+            173,
+            315,
+            823,
+            358
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 5: The number of models on which PAU and RPAU outperforms or underperforms each baseline activation function we compared against in our experiments. ",
+        "bbox": [
+            174,
+            367,
+            823,
+            396
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To summarize, PAUs satisfy all the assumptions (i-v). They allow the network to be universal function approximators (i) as shown by theorem 1. They present a fast and stable learning behavior (ii, iii) as shown in Figs. (5,6,7,8). The number of parameters introduced by PAUs is minimal in comparison to the size of the networks. In our experiments, we add 10 parameters per layer, showing that PAUs are parsimonious (iv). Finally, they allow deep neural networks to provide high predictive performance (v) as shown in Tab. 5. ",
+        "bbox": [
+            174,
+            410,
+            825,
+            494
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            515,
+            328,
+            531
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented a novel learnable activation function, called Padé Activation Unit (PAU). PAUs encode activation functions as rational functions, trainable in an end-to-end fashion using backpropagation. This makes it easy for practitioners to replace standard activation functions with PAU units in any neural network. The results of our empirical evaluation for image classification demonstrate that PAUs can indeed learn new activation functions and in turn novel neural networks that are competitive to state-of-the-art networks with fixed and learned activation functions. Actually, across all activation functions and architectures, Padé networks are among the top performing networks. This clearly shows that the reliance on first picking fixed, hand-engineered activation functions can be eliminated and that learning activation functions is actually beneficial and simple. Moreover, our results provide the first empirically evidence that the open question “Can rational functions be used to design algorithms for training neural networks?” raised by Telgarsky (2017) can be answered affirmatively for common deep architectures. ",
+        "bbox": [
+            173,
+            546,
+            825,
+            713
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our work provides several interesting avenues for future work. One should explore more the space between safe and unsafe PAUs, in order to gain even more predictive power. Most interestingly, since Padé networks can be reduced to ReLU networks. one should explore globally optimal training (Arora et al., 2018) as well as provable robustness (Croce et al., 2019) of Padé approximations of general deep networks. ",
+        "bbox": [
+            174,
+            720,
+            825,
+            790
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments. PS and KK were supported by funds of the German Federal Ministry of Food and Agriculture (BMEL) based on a decision of the Parliament of the Federal Republic of Germany via the Federal Office for Agriculture and Food (BLE) under the innovation support program, project “DePhenS” (FKZ 2818204715). ",
+        "bbox": [
+            174,
+            805,
+            825,
+            861
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nF. Agostinelli, M. D. Hoffman, P. J. Sadowski, and P. Baldi. Learning activation functions to improve deep neural networks. In Workshop Track Proceedings of the International Conference on Learning Representations, 2015.   \nR. Arora, A. Basu, P. Mianjy, and A. Mukherjee. Understanding deep neural networks with rectified linear units. In International Conference on Learning Representations, 2018.   \nJ. T. Barron. Continuously differentiable exponential linear units. arXiv preprint arXiv:1704.07483, 2017.   \nC. Brezinski and J. Van Iseghem. Padé approximations. Handbook of numerical analysis, 3:47–222, 1994.   \nZ. Chen, F. Chen, R. Lai, X. Zhang, and C.-T. Lu. Rational neural networks for approximating graph convolution operator on jump discontinuities. In 2018 IEEE International Conference on Data Mining (ICDM). IEEE, 2018.   \nD. Clevert, T. Unterthiner, and S. Hochreiter. Fast and accurate deep network learning by exponential linear units (elus). In 4th International Conference on Learning Representations (ICLR), 2016.   \nF. Croce, M. Andriushchenko, and M. Hein. Provable robustness of relu networks via maximization of linear regions. In The 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), pages 2057–2066, 2019.   \nJ. Frankle and M. Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. In ICLR, 2019.   \nX. Glorot, A. Bordes, and Y. Bengio. Deep sparse rectifier neural networks. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics (AISTATS), pages 315–323, 2011.   \nI. J. Goodfellow, D. Warde-Farley, M. Mirza, A. C. Courville, and Y. Bengio. Maxout networks. In Proceedings of the 30th International Conference on Machine Learning (ICML), pages 1319–1327, 2013.   \nM. Goyal, R. Goyal, and B. Lall. Learning activation functions: A new paradigm of understanding neural networks. arXiv preprint arXiv:1906.09529, 2019.   \nZ. Hajduk. Hardware implementation of hyperbolic tangent and sigmoid activation functions. Bulletin of the Polish Academy of Sciences. Technical Sciences, 66(5), 2018.   \nK. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pages 1026–1034, 2015.   \nK. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   \nK. Hornik, M. B. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359–366, 1989.   \nG. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017.   \nX. Jin, C. Xu, J. Feng, Y. Wei, J. Xiong, and S. Yan. Deep learning with s-shaped rectified linear activation units. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 2016.   \nP. Kidger and T. Lyons. Universal approximation with deep narrow networks. arXiv preprint arXiv:1905.08539, 2019.   \nD. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, (ICLR), 2015.   \nA. Krizhevsky and G. Hinton. Convolutional deep belief networks on cifar-10. 2010.   \nA. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. Technical report, Computer Science Department, University of Toronto, 2009.   \nY. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of IEEE, 86(11):2278–2324, 1998.   \nY. LeCun, C. Cortes, and C. Burges. Mnist handwritten digit database. at&t labs, 2010.   \nM. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6):861–867, 1993.   \nA. L. Maas, A. Y. Hannun, and A. Y. Ng. Rectifier nonlinearities improve neural network acoustic models. In in ICML Workshop on Deep Learning for Audio, Speech and Language Processing, 2013.   \nF. Manessi and A. Rozza. Learning combinations of activation functions. In 2018 24th International Conference on Pattern Recognition (ICPR), pages 61–66. IEEE, 2018.   \nV. Nair and G. E. Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th international conference on machine learning (ICML-10), pages 807–814, 2010.   \nJ. Nickolls, I. Buck, and M. Garland. Scalable parallel programming. In 2008 IEEE Hot Chips 20 Symposium (HCS), pages 40–53. IEEE, 2008.   \nN. Qian. On the momentum term in gradient descent learning algorithms. Neural networks, 12(1): 145–151, 1999.   \nP. Ramachandran, B. Zoph, and Q. V. Le. Searching for activation functions. In Proceedings of the Workshop Track of the 6th International Conference on Learning Representations (ICLR), 2018.   \nO. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. S. Bernstein, A. C. Berg, and F. Li. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.   \nM. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L.-C. Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4510–4520, 2018.   \nK. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In Proceedings of the 3rd International Conference on Learning Representations(ICLR), 2015.   \nN. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. The journal of machine learning research, 15(1): 1929–1958, 2014.   \nY. W. Teh and G. E. Hinton. Rate-coded restricted boltzmann machines for face recognition. In Prcoeedings of Neural Information Processing Systems (NIPS), pages 908–914, 2001.   \nM. Telgarsky. Neural networks and rational functions. In Proceedings of the 34th International Conference on Machine Learning (ICML), pages 3387–3393, 2017.   \nL. N. Trefethen. Approximation Theory and Approximation Practice. SIAM, 2012. ISBN 978-1-611- 97239-9.   \nC. J. Vercellino and W. Y. Wang. Hyperactivations for activation function exploration. In 31st Conference on Neural Information Processing Systems (NIPS 2017), Workshop on Meta-learning. Long Beach, USA, 2017.   \nH. Xiao, K. Rasul, and R. Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. CoRR, 2017.   \nB. Xu, N. Wang, T. Chen, and M. Li. Empirical evaluation of rectified activations in convolutional network. CoRR, 2015.   \nH.-Z. Zhao, F.-X. Liu, and L.-Y. Li. Improving deep convolutional neural networks with mixed maxout units. PloS one, 12(7), 2017. ",
+        "bbox": [
+            168,
+            89,
+            828,
+            931
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            74,
+            828,
+            929
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            103,
+            826,
+            170
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 INITIALIZATION COEFFICIENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            431,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As show in Table 6 we compute initial coefficients for PAU approximations to different known activation functions. We predefined the orders to be [5,4] and for Sigmoid, Tanh and Swish, we have computed the Padé approximant using the standard techniques. For the different variants of PRelu, LeakyRelu and Relu we optimized the coefficients using least squares over the line range between [-3,3] in steps of 0.000001. ",
+        "bbox": [
+            173,
+            159,
+            825,
+            229
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/4e489ad5b0775262d906f71fc463993eba6b7144b654b75adbe0c13680923dd0.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Initial coefficients to approximate different activation functions. "
+        ],
+        "table_body": "<table><tr><td></td><td>Sigmoid</td><td>Tanh</td><td>Swish</td><td>ReLU</td><td>LReLU(0.01)</td><td>LReLU(0.20)</td><td>LReLU(0.25)</td><td>LReLU(0.30)</td><td>LReLU(-0.5)</td></tr><tr><td>ao</td><td>1/2</td><td>0</td><td>0</td><td>0.02996348</td><td>0.02979246</td><td>0.02557776</td><td>0.02423485</td><td>0.02282366</td><td>0.02650441</td></tr><tr><td>α1</td><td>1/4</td><td>1</td><td>1/2</td><td>0.61690165</td><td>0.61837738</td><td>0.66182815</td><td>0.67709718</td><td>0.69358438</td><td>0.80772912</td></tr><tr><td>a2</td><td>1/18</td><td>0</td><td>b/4</td><td>2.37539147</td><td>2.32335207</td><td>1.58182975</td><td>1.43858363</td><td>1.30847432</td><td>13.56611639</td></tr><tr><td>a3</td><td>1/144</td><td>1/9</td><td>362/56</td><td>3.06608078</td><td>3.05202660</td><td>2.94478759</td><td>2.95497990</td><td>2.97681599</td><td>7.00217900</td></tr><tr><td>a4</td><td>1/2016</td><td>0</td><td>63/168</td><td>1.52474449</td><td>1.48548002</td><td>0.95287794</td><td>0.85679722</td><td>0.77165297</td><td>11.61477781</td></tr><tr><td>a5</td><td>1/60480</td><td>1/945</td><td>b4/3360</td><td>0.25281987</td><td>0.25103717</td><td>0.23319681</td><td>0.23229612</td><td>0.23252265</td><td>0.68720375</td></tr><tr><td>b1</td><td>0</td><td>0</td><td>0</td><td>1.19160814</td><td>1.14201226</td><td>0.50962605</td><td>0.41014746</td><td>0.32849543</td><td>13.70648993</td></tr><tr><td>b2</td><td>1/9</td><td>4/9</td><td>362/28</td><td>4.40811795</td><td>4.39322834</td><td>4.18376890</td><td>4.14691964</td><td>4.11557902</td><td>6.07781733</td></tr><tr><td>b</td><td>0</td><td>0</td><td>0</td><td>0.91111034</td><td>0.87154450</td><td>0.37832090</td><td>0.30292546</td><td>0.24155603</td><td>12.32535229</td></tr><tr><td>b4</td><td>1/10008</td><td>1/63</td><td>64 /1680</td><td>0.34885983</td><td>0.34720652</td><td>0.32407314</td><td>0.32002850</td><td>0.31659365</td><td>0.54006880</td></tr></table>",
+        "bbox": [
+            174,
+            241,
+            825,
+            359
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 LIST OF ACTIVATION FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            409,
+            450,
+            424
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For our experiments, we compare against the following activation functions with their respective parameters. ",
+        "bbox": [
+            176,
+            435,
+            825,
+            464
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• ReLU (Nair and Hinton, 2010): $y = \\operatorname* { m a x } ( x , 0 )$   \n• ReLU6 (Krizhevsky and Hinton, 2010): $y = \\operatorname* { m i n } ( \\operatorname* { m a x } ( x , 0 ) , 6 )$ , a variation of ReLU with an upper bound.   \n• Leaky ReLU (Maas et al., 2013): $y = \\operatorname* { m a x } ( 0 , x ) + \\alpha * \\operatorname* { m i n } ( 0 , x )$ with the negative slope, which is defined by the parameter $\\alpha$ . Leaky ReLU enables a small amount of information to flow when $x < 0$ . Random ReLU ( $\\mathrm { X u }$ et al., 2015): a randomized variation of Leaky ReLU. ELU (Clevert et al., 2016): $y = \\operatorname* { m a x } ( 0 , x ) + \\operatorname* { m i n } ( 0 , \\alpha * ( \\exp ( x ) - 1 ) ) .$   \n• CELU (Barron, 2017): $y = \\operatorname* { m a x } ( 0 , x ) + \\operatorname* { m i n } ( 0 , \\alpha * ( \\exp ( x / \\alpha ) - 1 ) )$ . Swish (Ramachandran et al., 2018): $y = x * \\mathrm { s i g m o i d } ( x )$ , which tends to work better than ReLU on deeper models across a number of challenging datasets.   \n• Parametric ReLU (PReLU) (He et al., 2015) $y = \\operatorname* { m a x } ( 0 , x ) + \\alpha * \\operatorname* { m i n } ( 0 , x )$ , where the leaky parameter $\\alpha$ is a learn-able parameter of the network.   \n• Maxout (Goodfellow et al., 2013): $y = \\operatorname* { m a x } ( z _ { i j } )$ , where $z _ { i j } = x ^ { T } W _ { \\dots i j } + b _ { i j }$ , and $W \\in R ^ { d \\times m \\times k }$ and $b \\in R ^ { m \\times k }$ are learned parameters.   \nMixture of activations (Manessi and Rozza, 2018): a combination of weighted activation functions e.g. {id, ReLU}, where the weight is a learnable parameter of the network. SLAF (Goyal et al., 2019): a learnable activation function based on a Taylor approximation.   \n• APL (Agostinelli et al., 2015): a learnable piecewise linear activation function.   \n• SReLU (Jin et al., 2016): a learnable S-shaped rectified linear activation function. ",
+        "bbox": [
+            214,
+            476,
+            825,
+            816
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 DETAILS OF THE MNIST AND FASHION-MNIST EXPERIMENT ",
+        "bbox": [
+            174,
+            830,
+            651,
+            845
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3.1 NETWORK ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            857,
+            429,
+            872
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here we describe the architectures for the networks VGG and LeNet, along with the number of trainable parameters. The number of parameters of the activation function is reported for using PAU. Common not trainable activation functions do not have trainable parameters. PReLU has one ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "trainable parameter. In total the VGG network as 9224508 parameters with 50 for PAU, and the LeNet network has 61746 parameters with 40 for PAU. ",
+        "bbox": [
+            169,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/a3f29e904bb7e4148b853bcef79548909100c942617948261d34a2a50662ab13.jpg",
+        "table_caption": [
+            "Table 7: Architecture of Simple Convolutional Neural Networks "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>No.</td><td>VGG</td><td># params</td><td>LeNet # params</td></tr><tr><td>1</td><td>Convolutional</td><td>Convolutional 640 5x5x6</td><td>156</td></tr><tr><td>2</td><td>3x3x64 Activation</td><td>10 Activation</td><td>10</td></tr><tr><td>3</td><td>Max-Pooling</td><td>0</td><td>Max-Pooling 0</td></tr><tr><td>4</td><td>Convolutional 3x3x128</td><td>73856 5x5x16</td><td>Convolutional 2416</td></tr><tr><td>5</td><td>Activation</td><td>10</td><td>Activation 10</td></tr><tr><td>6</td><td>Max-Pooling</td><td>0</td><td>Max-Pooling 0</td></tr><tr><td>7</td><td>Convolutional 3x3x256</td><td>295168 5x5x120</td><td>Convolutional 48120</td></tr><tr><td>8</td><td>Convolutional 3x3x256</td><td>590080</td><td>Activation 10</td></tr><tr><td>9</td><td>Activation</td><td>Linear 10 84</td><td>10164</td></tr><tr><td>10</td><td>Max-Pooling</td><td>0</td><td>Activation 10</td></tr><tr><td>11</td><td>Convolutional 3x3x512</td><td>Linear 1180160 10</td><td>850</td></tr><tr><td>12</td><td>Convolutional 3x3x512</td><td>2359808</td><td>Softmax 0</td></tr><tr><td>13</td><td>Activation</td><td>10</td><td></td></tr><tr><td>14</td><td>Max-Pooling</td><td>0</td><td></td></tr><tr><td>15</td><td>Convolutional 3x3x512</td><td>2359808</td><td></td></tr><tr><td>16</td><td>Convolutional 3x3x512</td><td>2359808</td><td></td></tr><tr><td>17</td><td>Activation</td><td>10</td><td></td></tr><tr><td>18</td><td>Max-Pooling</td><td>0</td><td></td></tr><tr><td>19</td><td>Linear 10</td><td>5130</td><td></td></tr><tr><td>20</td><td>Softmax</td><td>0</td><td></td></tr></table>",
+        "bbox": [
+            289,
+            142,
+            702,
+            683
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.2 LEARNING PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            729,
+            406,
+            744
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The parameters of the networks, both the layer weights and the coefficients of the PAUs, were trained over 100 epochs using Adam (Kingma and Ba, 2015) with a learning rate of 0.002 or SGD (Qian, 1999) with a learning rate of 0.01, momentum set to 0.5, and without weight decay. In all experiments we used a batch size of 256 samples. The weights of the networks were initialized randomly and the coefficients of the PAUs were initialized with the initialization constants of Leaky ReLU, see Tab. 6. We report the mean of 5 different runs for both the accuracy on the test-set and the loss on the train-set after each training epoch. ",
+        "bbox": [
+            173,
+            753,
+            826,
+            853
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.3 PREDICTIVE PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            429,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b696c34891530e3535f22a331a8c5996169901dfd1780460e695de2131474de8.jpg",
+        "image_caption": [
+            "MNIST ",
+            "Figure 6: PAU compared to baseline activation function units on 5 runs of MNIST using the VGG and LeNet: first column mean test-accuracy, second column mean train-loss. PAU consistently outperforms or matches the best performances of the baseline activations. Moreover, PAUs enable the networks to achieve a lower loss during training compared to all baselines. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            175,
+            823,
+            428
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/48f8c9a6cc6dfacc1e6b29afebea8fb99b41c24b027dccbfc4962a4f70de7cde.jpg",
+        "image_caption": [
+            "Fashion-MNIST",
+            "Figure 7: PAU compared to baseline activation function units on 5 runs of Fashion-MNIST using the VGG and LeNet architectures: first column mean test-accuracy, second column mean train-loss. PAU consistently outperforms the baselines activation functions in terms of performance and training time, especially on the VGG. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            574,
+            823,
+            825
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.4 DETAILS OF THE CIFAR10 AND IMAGENET EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            609,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.4.1 LEARNING PARAMETERS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            128,
+            406,
+            143
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The parameters of the networks, both the layer weights and the coefficients of the PAUs, were trained over 400 epochs using SGD with momentum set to 0.9. On the Cifar10 dataset we have different optimizer setups for PAU layers and the rest of the network. For the PAU layers we use constant learning rates per networks and no weight decay. For updating the rest of the network we use initial learning rate of 0.1, and learning rate decay of 0.985 per epoch and set weight decay to $5 e - 4$ . In all experiments we used a batch size of 64 samples. The weights of the networks were initialized randomly and the coefficients of the PAUs were initialized with the initialization constants of Leaky ReLU, see Tab. A.1. The additive noise of the Randomized PAUs is set to $\\alpha = 1 \\%$ training the networks VGG8 and MobileNetV2, respectively $\\alpha = 1 0 \\%$ for ResNet101. ",
+        "bbox": [
+            174,
+            154,
+            825,
+            279
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "On the Imagenet dataset we use the same optimizer for PAU and the rest of the network. We follow the default setup provided by Pytorch and use an initial learning rate of 0.1, and decay the learning rate by $10 \\%$ after 30, 60 and 90 epochs. ",
+        "bbox": [
+            176,
+            286,
+            825,
+            328
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.4.2 NETWORK ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            343,
+            428,
+            358
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The network architectures were taken from reference implementations in PyTorch and we modified them to use PAUs. All architectures are the same among the different activation functions except for Maxout. The default amount of trainable parameters of VGG8 (Simonyan and Zisserman, 2015) is 3,918,858. Using PAU 50 additional parameters are introduced. Maxout is extending the VGG8 network to a total number of 7,829,642 parameters. MobileNetV2 (Sandler et al., 2018) is contains by default 2,296,922 trainable parameters. PAU adds 360 additional parameters. The Maxout activation function is results in a total number of 3,524,506 parameters. With respect to the number of parameters ResNet101 (He et al., 2016) is the largest network we train. By default it contains 42,512,970 trainable parameters, we introduce 100 PAUs and therefore add 1000 additional parameters to the network. If one is replacing each activation function using Maxout the resulting ResNet101 network contains 75,454,090 trainable parameters. The default DenseNet121 (Huang et al., 2017) network has 6,956,298 parameters. Replacing the activation functions with PAU adds 1200 parameters to the network. ",
+        "bbox": [
+            174,
+            367,
+            825,
+            547
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.4.3 PRUNING EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            564,
+            393,
+            578
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For the pruning experiment, we implement the \"Lottery ticket hypothesis\" (Frankle and Carbin (2019)) in PyTorch. We compare PAUs against the best activation for the network architecture according to the average predictive accuracy from Tab. 3. More precisely, we compare the predictive performance under pruning for the networks $N _ { 1 } = \\{ \\mathrm { V G G - } 8 _ { \\mathrm { p a u } }$ , MobileNetV2pau, $\\mathrm { R e s N e t 1 0 1 _ { p a u } } \\big \\}$ against the networks $N _ { 2 } = \\mathrm { \\{ V G G - 8 _ { L R e L U } } $ , MobileNetV2RReLU, $\\mathrm { R e s N e t 1 0 1 _ { p a u } } \\}$ . Here we avoided Maxout as it heavily increases the parameters in the model, defeating the purpose of pruning. Unlike the original paper, we compress the convolutions using a fixed pruning parameter per iteration of $p \\% = \\bar { 1 } 0 , 2 \\bar { 0 } , \\bar { 3 0 } , 4 0 , 5 0 , 6 0$ and evaluated once per network. After each training iteration we remove $p \\%$ of filters in every convolution and the filters we remove are the ones where the sum of its weights is lowest. After pruning, we proceed to re-initialize the network and repeat the training and pruning proceedure with the next $p \\%$ parameter. ",
+        "bbox": [
+            173,
+            588,
+            825,
+            741
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.4.4 PREDICTIVE PERFORMANCE CIFAR10 ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            495,
+            117
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/d5c9c2701a29d4ea7aac7bcf065caac2074bea75336a7df8e3dc03913c875285.jpg",
+        "image_caption": [
+            "Figure 8: PAU compared to baseline activation function units on 5 runs of CIFAR-10. Accuracy on the left column and loss on the right one. (Best viewed in color) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            127,
+            821,
+            834
+        ],
+        "page_idx": 16
+    }
+]

parse/train/IOqr2ZyXHz1/IOqr2ZyXHz1.md

@@ -0,0 +1,246 @@
+# CONTINUAL LIFELONG CAUSAL EFFECT INFERENCEWITH REAL WORLD EVIDENCE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The era of real world evidence has witnessed an increasing availability of observational data, which much facilitates the development of causal effect inference. Although significant advances have been made to overcome the challenges in causal effect estimation, such as missing counterfactual outcomes and selection bias, they only focus on source-specific and stationary observational data. In this paper, we investigate a new research problem of causal effect inference from incrementally available observational data, and present three new evaluation criteria accordingly, including extensibility, adaptability, and accessibility. We propose a Continual Causal Effect Representation Learning method for estimating causal effect with observational data, which are incrementally available from non-stationary data distributions. Instead of having access to all seen observational data, our method only stores a limited subset of feature representations learned from previous data. Combining the selective and balanced representation learning, feature representation distillation, and feature transformation, our method achieves the continual causal effect estimation for new data without compromising the estimation capability for original data. Extensive experiments demonstrate the significance of continual causal effect inference and the effectiveness of our method.
+
+# 1 INTRODUCTION
+
+Causal effect inference is a critical research topic across many domains, such as statistics, computer science, public policy, and economics. Randomized controlled trials (RCT) are usually considered as the gold-standard for causal effect inference, which randomly assigns participants into a treatment or control group. As the RCT is conducted, the only expected difference between the treatment and control groups is the outcome variable being studied. However, in reality, randomized controlled trials are always time-consuming and expensive, and thus the study cannot involve many subjects, which may be not representative of the real-world population the intervention would eventually target. Nowadays, estimating causal effects from observational data has become an appealing research direction owing to a large amount of available data and low budget requirements, compared with RCT (Yao et al., 2020). Researchers have developed various strategies for causal effect inference with observational data, such as tree-based methods (Chipman et al., 2010; Wager & Athey, 2018), representation learning methods (Johansson et al., 2016; Li & Fu, 2017; Shalit et al., 2017; Chu et al., 2020), adapting Bayesian algorithms (Alaa & van der Schaar, 2017), generative adversarial nets (Yoon et al., 2018), variational autoencoders (Louizos et al., 2017) and so on.
+
+Although significant advances have been made to overcome the challenges in causal effect estimation with observational data, such as missing counterfactual outcomes and selection bias between treatment and control groups, the existing methods only focus on source-specific and stationary observational data. Such learning strategies assume that all observational data are already available during the training phase and from the only one source. This assumption is unsubstantial in practice due to two reasons. The first one is based on the characteristics of observational data, which are incrementally available from non-stationary data distributions. For instance, the number of electronic medical records in one hospital is growing every day, or the electronic medical records for one disease may be from different hospitals or even different countries. This characteristic implies that one cannot have access to all observational data at one time point and from one single source. The second reason is based on the realistic consideration of accessibility. For example, when the new observational are available, if we want to refine the model previously trained by original data, maybe the original training data are no longer accessible due to a variety of reasons, e.g., legacy data may be unrecorded, proprietary, too large to store, or subject to privacy constraint (Zhang et al., 2020). This practical concern of accessibility is ubiquitous in various academic and industrial applications. That’s what it boiled down to: in the era of big data, we face the new challenges in causal inference with observational data: the extensibility for incrementally available observational data, the adaptability for extra domain adaptation problem except for the imbalance between treatment and control groups in one source, and the accessibility for a huge amount of data.
+
+Existing causal effect inference methods, however, are unable to deal with the aforementioned new challenges, i.e., extensibility, adaptability, and accessibility. Although it is possible to adapt existing causal inference methods to address the new challenges, these adapted methods still have inevitable defects. Three straightforward adaptation strategies are described as follows. (1) If we directly apply the model previously trained based on original data to new observational data, the performance on new task will be very poor due to the domain shift issues among different data sources; (2) If we utilize newly available data to re-train the previously learned model, adapting changes in the data distribution, old knowledge will be completely or partially overwritten by the new one, which can result in severe performance degradation on old tasks. This is the well-known catastrophic forgetting problem (McCloskey & Cohen, 1989; French, 1999); (3) To overcome the catastrophic forgetting problem, we may rely on the storage of old data and combine the old and new data together, and then re-train the model from scratch. However, this strategy is memory inefficient and time-consuming, and it brings practical concerns such as copyright or privacy issues when storing data for a long time (Samet et al., 2013). Our empirical evaluations in Section 4 demonstrate that any of these three strategies in combination with the existing causal effect inference methods is deficient.
+
+To address the above issues, we propose a Continual Causal Effect Representation Learning method (CERL) for estimating causal effect with incrementally available observational data. Instead of having access to all previous observational data, we only store a limited subset of feature representations learned from previous data. Combining the selective and balanced representation learning, feature representation distillation, and feature transformation, our method preserves the knowledge learned from previous data and update the knowledge by leveraging new data, so that it can achieve the continual causal effect estimation for new data without compromising the estimation capability for previous data. To summarize, our main contributions include:
+
+• Our work is the first to introduce the continual lifelong causal effect inference problem for the incrementally available observational data and three corresponding evaluation criteria, i.e., extensibility, adaptability, and accessibility.   
+• We propose a new framework for continual lifelong causal effect inference based on deep representation learning and continual learning.   
+• Extensive experiments demonstrate the deficiency of existing methods when facing the incrementally available observational data and our model’s outstanding performance.
+
+# 2 BACKGROUND AND PROBLEM STATEMENT
+
+Suppose that the observational data contain $\cdot$ units collected from $\cdot$ different domains and the $\cdot$ -th dataset $D _ { d }$ contains the data $-$ collected from $d$ -th domain, which contains $\cdot$ units. Let $X$ denote all observed variables, $Y$ denote the outcomes in the observational data, and $\cdot$ is a binary variable. Let $D _ { 1 : d } = \{ D _ { 1 } , D _ { 2 } , . . . , D _ { d } \}$ be the set of combination of $\cdot$ dataset, separately collected from $d$ different domains. For $d$ datasets $\_$ , they have the common observed variables but due to the fact that they are collected from different domains, they have different distributions with respect to $X$ , $\cdot$ , and $T$ in each dataset. Each unit in the observational data received one of two treatments. Let $\cdot$ denote the treatment assignment for unit $\cdot$ ; $\cdot$ . For binary treatments, $t _ { i } = 1$ is for the treatment group, and $\cdot$ for the control group. The outcome for unit $i$ is denoted by $y _ { t } ^ { \ i }$ when treatment $t$ is applied to unit $i$ ; that is, $\cdot$ is the potential outcome of unit $\cdot$ in the treatment group and $\cdot$ is the potential outcome of unit $i$ in the control group. For observational data, only one of the potential outcomes is observed. The observed outcome is called the factual outcome and the remaining unobserved potential outcomes are called counterfactual outcomes.
+
+In this paper, we follow the potential outcome framework for estimating treatment effects (Rubin, 1974; Splawa-Neyman et al., 1990). The individual treatment effect (ITE) for unit $i$ is the difference between the potential treated and control outcomes, and is defined as $\mathrm { I T E } _ { i } = y _ { 1 } ^ { i } - y _ { 0 } ^ { i }$ . The average treatment effect (ATE) is the difference between the mean potential treated and control outcomes, which is defined as $\begin{array} { r } { \mathrm { A } \dot { \mathrm { T E } } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( y _ { 1 } ^ { i } - y _ { 0 } ^ { i } ) } \end{array}$ .
+
+The success of the potential outcome framework is based on the following assumptions (Imbens & Rubin, 2015), which ensure that the treatment effect can be identified. Stable Unit Treatment Value Assumption (SUTVA): The potential outcomes for any units do not vary with the treatments assigned to other units, and, for each unit, there are no different forms or versions of each treatment level, which lead to different potential outcomes. Consistency: The potential outcome of treatment $t$ is equal to the observed outcome if the actual treatment received is $t$ . Positivity: For any value of $x$ , treatment assignment is not deterministic, i.e., $P ( T = t | X = x ) > 0$ , for all $t$ and $x$ . Ignorability: Given covariates, treatment assignment is independent to the potential outcomes, i.e., $( y _ { 1 } , y _ { 0 } ) \perp t | x$ .
+
+The goal of our work is to develop a novel continual causal inference framework, given new available observational data $D _ { d }$ , to estimate the causal effect for newly available data $D _ { d }$ as well as the previous data $D _ { 1 : ( d - 1 ) }$ without having access to previous training data in $D _ { 1 : ( d - 1 ) }$ .
+
+# 3 THE PROPOSED FRAMEWORK
+
+The availability of “real world evidence” is expected to facilitate the development of causal effect inference models for various academic and industrial applications. How to achieve continual learning from incrementally available observational data from non-stationary data domains is a new direction in causal effect inference. Rather than only focusing on handling the selection bias problem, we also need to take into comprehensive consideration three aspects of the model, i.e., the extensibility for incrementally available observational data, the adaptability for various data sources, and the accessibility for a huge amount of data.
+
+We propose the Continual Causal Effect Representation Learning method (CERL) for estimating causal effect with incrementally available observational data. Based on selective and balanced representation learning for treatment effect estimation, CERL incorporates feature representation distillation to preserve the knowledge learned from previous observational data. Besides, aiming at adapting the updated model to original and new data without having access to the original data, and solving the selection bias between treatment and control groups, we propose one representation transformation function, which maps partial original feature representations into new feature representation space and makes the global feature representation space balanced with respect to treatment and control groups. Therefore, CERL can achieve the continual causal effect estimation for new data and meanwhile preserve the estimation capability for previous data, without the aid of original data.
+
+# 3.1 MODEL ARCHITECTURE
+
+To estimate the incrementally available observational data, the framework of CERL is mainly composed of two components: (1) the baseline causal effect learning model is only for the first available observational data, and thus we don’t need to consider the domain shift issue among different data sources. This component is equivalent to the traditional causal effect estimation problem; (2) the continual causal effect learning model is for the sequentially available observational data, where we need to handle more complex issues, such as knowledge transfer, catastrophic forgetting, global representation balance, and memory constraint. We present the details of each component as follows.
+
+# 3.1.1 THE BASELINE CAUSAL EFFECT LEARNING MODEL
+
+We first describe the baseline causal effect learning model for the initial observational dataset and then bring in subsequent datasets. For causal effect estimation in the initial dataset, it can be transformed into the traditional causal effect estimation problem. Motivated by the empirical success of deep representation learning for counterfactual inference (Shalit et al., 2017; Chu et al., 2020), we propose to learn the selective and balanced feature representations for treated and control units, and then infer the potential outcomes based on learned representation space.
+
+Learning Selective and Balanced Representation. Firstly, we adopt a deep feature selection model that enables variable selection in one deep neural network, i.e., $g _ { w _ { 1 } } : X \to R$ , where $X$ denotes the original covariate space, $R$ denotes the representation space, and $w _ { 1 }$ are the learnable parameters in function $g$ . The elastic net regularization term (Zou & Hastie, 2005) is adopted in our model, i.e., $L _ { w _ { 1 } } = \| \dot { w } _ { 1 } \| _ { 2 } ^ { 2 } + \| w _ { 1 } \| _ { 1 }$ . Elastic net throughout the fully connected representation layers assigns larger weights to important features. This strategy can effectively filter out the irrelevant variables and highlight the important variables.
+
+Due to the selection bias between treatment and control groups and among the sequential different data sources, the magnitudes of confounders may be significantly different. To effectively eliminate the imbalance caused by the significant difference in magnitudes between treatment and control groups and among different data sources, we propose to use cosine normalization in the last representation layer. In the multi-layer neural networks, we traditionally use dot products between the output vector of the previous layer and the incoming weight vector, and then input the products to the activation function. The result of dot product is unbounded. Cosine normalization uses cosine similarity instead of simple dot products in neural networks, which can bound the pre-activation between $- 1$ and 1. The result could be even smaller when the dimension is high. As a result, the fined as variance can be controlled within a very narrow range (Luo et al., 2018). Cosine normalization is de- $\begin{array} { r } { r = \sigma ( r _ { n o r m } ) = \sigma \big ( \cos ( w , x ) \big ) = \sigma ( \frac { w \cdot x } { | w | | x | } ) } \end{array}$ , where $r _ { n o r m }$ is the normalized pre-activation, $w$ is the incoming weight vector, $x$ is the input vector, and $\sigma$ is nonlinear activation function.
+
+Motivated by Shalit et al. (2017), we adopt integral probability metrics (IPM) when learning the representation space to balance the treatment and control groups. The IPM measures the divergence between the representation distributions of treatment and control groups, so we want to minimize the IPM to make two distributions more similar. Let $P ( g ( x ) | t = \bar { 1 } )$ ) and $Q ( g ( x ) | t = 0 )$ ) denote the empirical distributions of the representation vectors for the treatment and control groups, respectively. We adopt the IPM defined in the family of 1-Lipschitz functions, which leads to IPM being the Wasserstein distance (Sriperumbudur et al., 2012; Shalit et al., 2017). In particular, the IPM term with Wasserstein distance is defined as $\begin{array} { r } { { \bf W } { \ a s s } ( P , Q ) = \operatorname* { i n f } _ { k \in { \mathcal K } } \int _ { g ( x ) } \| k \bar { ( g ( x ) ) } - } \end{array}$ $g ( x ) \| P ( g ( x ) ) d ( g ( x ) )$ , where $\gamma$ denotes the hyper-parameter controlling the trade-off between $\mathrm { W a s s } ( P , Q )$ and other terms in the final objective function. $\begin{array} { r } { \mathcal { K } = \{ k | Q ( k ( g ( x ) ) ) = P ( g ( x ) ) \} } \end{array}$ defines the set of push-forward functions that transform the representation distribution of the treatment distribution $P$ to that of the control $Q$ and $g ( x ) \in \{ g ( x ) _ { i } \} _ { i : t _ { i } = 1 }$ .
+
+Inferring Potential Outcomes. We aim to learn a function $h _ { \theta _ { 1 } } : R \times T  Y$ that maps the representation vectors and treatment assignment to the corresponding observed outcomes, and it can be parameterized by deep neural networks. To overcome the risk of losing the influence of $T$ on $R$ ${ \bf \xi } , h _ { \theta _ { 1 } } ( g _ { w _ { 1 } } ( x ) , t )$ is partitioned into two separate tasks for treatment and control groups, respectively. Each unit is only updated in the task corresponding to its observed treatment. Let $\hat { y } _ { i } =$ $h _ { \theta _ { 1 } } ( g _ { w _ { 1 } } ( x ) , t )$ denote the inferred observed outcome of unit he mean squared error in predicting factual ou $i$ correspcomes: t . $t _ { i }$ . $\begin{array} { r } { L _ { Y } = \frac { 1 } { n _ { 1 } } \sum _ { i = 1 } ^ { n _ { 1 } } ( \hat { y } _ { i } - y _ { i } ) ^ { 2 } } \end{array}$
+
+Putting all the above together, the objective function of our baseline causal effect learning model is: $L \doteq L _ { Y } + \alpha W a s s ( \bar { P } , Q ) + \lambda L _ { w _ { 1 } }$ , where $\alpha$ and $\lambda$ denote the hyper-parameters controlling the trade-off among $W a s s ( P , Q )$ , $L _ { w }$ , and $L _ { Y }$ in the objective function.
+
+# 3.1.2 THE SUSTAINABILITY OF MODEL LEARNING
+
+By far, we have built the baseline model for causal effect estimation with observational data from a single source. To avoid catastrophic forgetting when learning new data, we propose to preserve a subset of lower-dimensional feature representations rather than all original covariates. We also can adjust the number of preserved feature representations according to the memory constraint.
+
+After the completion of baseline model training, we store a subset of feature representations $R _ { 1 } = \{ g _ { w _ { 1 } } ( x ) \bar { | } x \in D _ { 1 } \}$ and the corresponding $\{ Y , T \} \in D _ { 1 }$ as memory $M _ { 1 }$ . The size of stored representation vectors can be reduced to satisfy the pre-specified memory constraint by a herding algorithm (Welling, 2009; Rebuffi et al., 2017). The herding algorithm can create a representative set of samples from distribution and requires fewer samples to achieve a high approximation quality than random subsampling. We run the herding algorithm separately for treatment and control groups to store the same number of feature representations from treatment and control groups. At this point, we only store the memory set $M _ { 1 }$ and model $g _ { w 1 }$ , without the original data $( D _ { 1 } )$ .
+
+# 3.1.3 THE CONTINUAL CAUSAL EFFECT LEARNING MODEL
+
+For now, we have stored memory $M _ { 1 }$ and baseline model. To continually estimate the causal effect for incrementally available observational data, we incorporate feature representation distillation and feature representation transformation to estimate causal effect for all seen data based on balanced global feature representation space. The framework of CERL is shown in Fig. 1.
+
+Feature Representation Distillation. For next available dataset $D _ { 2 } \ = \ \{ ( x , y , t ) | x \ \in \ X , y \ \in$ $Y , t \in T \}$ collected from second domain, we adopt the same selective representation learning $g _ { w _ { 2 } } : X \to R _ { 2 }$ with elastic net regularization $( L _ { w _ { 2 } } )$ on new parameters $w _ { 2 }$ . Because we expect our model can estimate causal effect for both previous and new data, we want the new model to inherit some knowledge from previous model. In continual learning, knowledge distillation (Hinton et al., 2015; Li & Hoiem, 2017) is commonly adopted to alleviate the catastrophic forgetting, where knowledge is transferred from one network to another network by encouraging the outputs of the original and new network to be similar. However, for the continual causal effect estimation problem, we focus more on the feature representations, which are required to be balanced between treatment and control, and among different data domains. Inspired by Hou et al. (2019); Dhar et al. (2019); Iscen et al. (2020), we propose feature representation distillation to encourage the representation vector $\{ g _ { w _ { 1 } } ( x ) | x \in D _ { 2 } \}$ based on baseline model to be similar to the representation vector $\{ g _ { w _ { 2 } } ( x ) | \dot { x } \in D _ { 2 } \}$ based on new model by Euclidean distance. This feature distillation can help prevent the learned representations from drifting too much in the new feature representation space. Because we apply the cosine normalization to feature representations and $\| { \dot { A } } - B \| ^ { 2 } = ( { \dot { A } } - B ) ^ { \tau } ( A - B ) { \stackrel { \sim } { = } } \| A \| ^ { 2 } + \| B \| ^ { 2 } - 2 A ^ { \tau } B = 2 { \bigl ( } 1 - c o s ( { \dot { A , } } B ) { \bigr ) }$ , the feature representation distillation is defined as $L _ { F D } ( x ) = 1 - c o s \bigl ( g _ { w _ { 1 } } ( x ) , g _ { w _ { 2 } } ( x ) \bigr ) ,$ , where $x \in D _ { 2 }$ .
+
+![](images/553c253f415acce00b77a9a4b2b8c7e63417d1e4b35a2ef71a731e268585a30b.jpg)  
+Figure 1: The blue part is baseline causal effect learning model for the first observational data. After baseline model training, store a subset of feature representations $R _ { 1 }$ into $M _ { 1 }$ by herding algorithm. The green part helps to map $R _ { 1 }$ to transformed feature representations $\tilde { R } _ { 1 }$ compatible with new feature representations space $R _ { 2 }$ . Then the red part is used for continual causal effect estimation based on feature distillation and balanced global feature representation learning for $\tilde { R } _ { 1 }$ and $R _ { 2 }$ .
+
+Feature Representation Transformation. We have previous feature representations $R _ { 1 }$ stored in $M _ { 1 }$ and new feature representations $R _ { 2 }$ extracted from newly available data. $R _ { 1 }$ and $R _ { 2 }$ lie in different feature representation space and they are not compatible with each other because they are learned from different models. In addition, we cannot learn the feature representations of previous data from the new model $g _ { w _ { 2 } }$ , as we no longer have access to previous data. Therefore, to balance the global feature representation space including previous and new representations between treatment and control groups, a feature transformation function is needed from previous feature representations $R _ { 1 }$ to transformed feature representations $\tilde { R } _ { 1 }$ compatible with new feature representations space $R _ { 2 }$ . We define a feature transformation function as $\phi _ { 1 \to 2 } : R _ { 1 } \to \tilde { R } _ { 1 }$ . We also input the feature representations of new data $D _ { 2 }$ learned from old model, i.e., $g _ { w _ { 1 } } ( x )$ , to get the transformed feature representations of new data, i.e., $\phi _ { 1  2 } ( g _ { w _ { 1 } } ( x ) )$ . To keep the transformed space compatible with the new feature representation space, we train the transformation function $\phi _ { 1  2 }$ by making the $\phi _ { 1  2 } ( g _ { w _ { 1 } } ( x ) )$ and $g _ { w _ { 2 } } ( x )$ similar, where $x \ \in \ D _ { 2 }$ . The loss function is defined as $L _ { F T } ( x ) \stackrel { } { = }$ $1 - c o s \bigl ( \phi _ { 1 \to 2 } ( g _ { w _ { 1 } } ( x ) ) , g _ { w _ { 2 } } ( x ) \bigr )$ , which is used to train the function $\phi _ { 1  2 }$ to transform feature representations between different feature spaces. Then, we can attain the transformed old feature representations $\tilde { R } _ { 1 } = \phi _ { 1  2 } ( R _ { 1 } )$ , which is in the same space as $R _ { 2 }$ .
+
+Balancing Global Feature Representation Space. We have obtained a global feature representation space including the transformed representations of stored old data and new representations of new available data. We adopt the same integral probability metrics as baseline model to make sure that the representation distributions are balanced for treatment and control groups in the global feature representation space. In addition, we define a potential outcome function $h _ { \theta _ { 2 } } : ( \tilde { R } _ { 1 } , R _ { 2 } ) \times T $ $Y$ . Let $\hat { y } _ { i } ^ { M } = h _ { \theta _ { 2 } } ( \phi _ { 1  2 } ( r _ { i } ) , t )$ , where $r _ { i } \in M _ { 1 }$ , and $\hat { y } _ { j } ^ { D } = h _ { \theta _ { 2 } } \big ( g _ { w _ { 2 } } ( x _ { j } ) , t \big )$ , where $x _ { j } \in D _ { 2 }$ denote the inferred observed outcomes. We aim to minimize the mean squared error in predicting factual outcomes for global feature representations including transformed old feature representations and new feature representations: $\begin{array} { r } { L _ { G } = \frac { 1 } { \tilde { n } _ { 1 } } \sum _ { i = 1 } ^ { \tilde { n } _ { 1 } } ( \hat { y } _ { i } ^ { M } - y _ { i } ^ { M } ) ^ { 2 } + \frac { 1 } { n _ { 2 } } \sum _ { j = 1 } ^ { n _ { 2 } } ( \hat { y } _ { j } ^ { D } - y _ { j } ^ { D } ) ^ { 2 } } \end{array}$ , where $\tilde { n } _ { 1 }$ is the number of units stored in $M _ { 1 }$ by herding algorithm, $y _ { i } ^ { M } \in M _ { 1 }$ , and $y _ { j } ^ { D } \in D _ { 2 }$ .
+
+In summary, the objective function of our continual causal effect learning model is $L = L _ { G } +$ $\alpha W a s s ( P , Q ) + \lambda L _ { w _ { 2 } } + \beta L _ { F D } + \delta L _ { F T }$ , where $\alpha , \lambda , \beta$ , and $\delta$ denote the hyper-parameters controlling the trade-off among $W a s s ( P , Q ) , L _ { w _ { 2 } } , L _ { F D } , L _ { F T }$ , and $L _ { G }$ in the final objective function.
+
+# 3.2 OVERVIEW OF CERL
+
+In the above sections, we have provided the baseline and continual causal effect learning models. When the continual causal effect learning model for the second data is trained, we can extract the $R _ { 2 } = \{ g _ { w _ { 2 } } ( x ) | x \in D _ { 2 } \}$ and $\tilde { R } _ { 1 } = \{ \stackrel { - } { \phi _ { 1  2 } } ( r ) | r \in M _ { 1 } \}$ . We define a new memory set as $M _ { 2 } =$ $\{ R _ { 2 } , Y _ { 2 } , T _ { 2 } \} \cup \phi _ { 1  2 } ( M _ { 1 } )$ , where $\phi _ { 1  2 } ( M _ { 1 } )$ includes $\bar { \tilde { R } } _ { 1 }$ and the corresponding $\{ Y , T \}$ stored in $M _ { 1 }$ . Similarly, to satisfy the pre-specified memory constraint, $M _ { 2 }$ can be reduced by conducting the herding algorithm to store the same number of feature representations from treatment and control groups. We only store the new memory set $M _ { 2 }$ and new model $g _ { w _ { 2 } }$ , which are used to train the following model and balance the global feature representation space. It is unnecessary to store the original data $D _ { 1 }$ and $D _ { 2 }$ ) any longer.
+
+We follow the same procedure for the subsequently available observational data. When we obtain the new observational data $D _ { d }$ , we can train $h _ { \theta _ { d } } ( g _ { w _ { d } } )$ and $\phi _ { d - 1 \to d } : R _ { d - 1 } \to \tilde { R } _ { d - 1 }$ based on the continual causal effect learning model. Besides, the new memory set is defined as: $M _ { d } =$ $\{ R _ { d } , Y _ { d } , T _ { d } \} \cup \phi _ { d - 1  d } ( M _ { d - 1 } )$ . So far, our model $h _ { \theta _ { d } } ( g _ { w _ { d } } )$ can estimate causal effect for all seen observational data regardless of the data source and it doesn’t require access to previous data. The detailed procedures of our CERL method are summarized in Algorithm 1 in Section B of Appendix.
+
+# 4 EXPERIMENTS
+
+We adapt the traditional benchmarks, i.e., News (Johansson et al., 2016; Schwab et al., 2018) and BlogCatalog (Guo et al., 2020) to continual causal effect estimation. Specifically, we consider three scenarios to represent the different degrees of domain shifts among the incrementally available observational data, including the substantial shift, moderate shift, and no shift. Besides, we generate a series of synthetic datasets and also conduct ablation studies to demonstrate the effectiveness of our model on multiple sequential datasets. The model performance with different numbers of preserved feature representations, and the robustness to hyperparameters are also evaluated.
+
+# 4.1 DATASET DESCRIPTION
+
+We utilize two semi-synthetic benchmarks for the task of continual causal effect estimation, which are based on real-world features, synthesized treatments and outcomes.
+
+News. The News dataset consists of 5000 randomly sampled news articles from the NY Times corpus1. It simulates the opinions of media consumers on news items. The units are different news items represented by word counts $\cdot$ and outcome $\cdot$ is the news item. The intervention $\cdot$ represents the viewing device, desktop $t = 0$ ) or mobile $\cdot$ ). We extend the original dataset specification in Johansson et al. (2016); Schwab et al. (2018) to enable the simulation of incrementally available observational data with different degrees of domain shifts. Assuming consumers prefer to read certain media items on specific viewing devices, we train a topic model on a large set of documents and define $\cdot$ as the topic distribution of news item $x$ . We define one topic distribution of a randomly sampled document as centroid $\cdot$ for mobile and the average topic representation of all document as centroid $\cdot$ for desktop. Therefore, the reader’s opinion of news item $x$ on device $t$ is determined by the similarity between $\cdot$ and $\cdot$ , i.e., $-$ , where $\cdot$ is a scaling factor and $\epsilon \sim N ( 0 , 1 )$ .
+
+Table 1: Performance on two sequential data and $\cdot$ . We present the mean value of $\cdot$ and $\epsilon _ { \mathrm { A T E } }$ on test sets from two datasets. The standard deviations are tiny. Lower is better. Under no domain shift scenario, the three strategies and CERL have the similar performance, because the previous and new data are from the same distribution. Under substantial shift and moderate shift scenarios, CFR-A performs well on previous data, but significantly declines on new dataset; straxtegy CFR-B shows the catastrophic forgetting problem; CERL has a similar performance to strategy CFR-C, while CERL does not require access to previous data. Besides, the larger domain shift leads to worse performance of CFR-A and CFR-B. CERL has remained stable against shift.
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="6">News</td><td colspan="6">BlogCatalog</td></tr><tr><td colspan="2">Previous data</td><td colspan="2"></td><td colspan="2">New data</td><td colspan="2">Previous data</td><td colspan="2"></td><td colspan="2">New data</td></tr><tr><td></td><td>Strategy</td><td>VEPEHE</td><td>EATE</td><td></td><td>VEPEHE</td><td>EATE</td><td></td><td>VePEHE</td><td>EATE</td><td></td><td>√EPEHE</td><td>EATE</td><td></td></tr><tr><td rowspan="4">Substantial shift</td><td>CFR-A</td><td>2.49</td><td>0.80</td><td></td><td>3.62</td><td>1.18</td><td>个</td><td>9.92</td><td>4.25</td><td></td><td>13.65</td><td>6.21</td><td>个</td></tr><tr><td>CFR-B</td><td>3.23</td><td>1.06</td><td>个</td><td>2.71</td><td>0.91</td><td></td><td>14.21</td><td>6.98</td><td>个</td><td>9.77</td><td>4.11</td><td></td></tr><tr><td>CFR-C</td><td>2.51</td><td>0.82</td><td></td><td>2.70</td><td>0.92</td><td></td><td>9.93</td><td>4.24</td><td></td><td>9.77</td><td>4.12</td><td></td></tr><tr><td>CERL</td><td>2.55</td><td>0.84</td><td></td><td>2.71</td><td>0.91</td><td></td><td>9.96</td><td>4.25</td><td></td><td>9.78</td><td>4.12</td><td></td></tr><tr><td rowspan="4">Moderate shift</td><td>CFR-A</td><td>2.58</td><td>0.85</td><td></td><td>3.06</td><td>1.02</td><td>个</td><td>9.89</td><td>4.22</td><td></td><td>11.26</td><td>5.03</td><td>个</td></tr><tr><td>CFR-B</td><td>2.98</td><td>0.99</td><td>个</td><td>2.65</td><td>0.92</td><td></td><td>12.35</td><td>5.67</td><td>个</td><td>9.83</td><td>4.18</td><td></td></tr><tr><td>CFR-C</td><td>2.56</td><td>0.85</td><td></td><td>2.63</td><td>0.90</td><td></td><td>9.88</td><td>4.21</td><td></td><td>9.81</td><td>4.16</td><td></td></tr><tr><td>CERL</td><td>2.59</td><td>0.86</td><td></td><td>2.66</td><td>0.92</td><td></td><td>9.90</td><td>4.24</td><td></td><td>9.82</td><td>4.17</td><td></td></tr><tr><td rowspan="4">No shift</td><td>CFR-A</td><td>2.58</td><td>0.87</td><td></td><td>2.62</td><td>0.88</td><td></td><td>9.86</td><td>4.20</td><td></td><td>9.85</td><td>4.19</td><td></td></tr><tr><td>CFR-B</td><td>2.60</td><td>0.88</td><td></td><td>2.60</td><td>0.87</td><td></td><td>9.85</td><td>4.18</td><td></td><td>9.83</td><td>4.18</td><td></td></tr><tr><td>CFR-C</td><td>2.58</td><td>0.87</td><td></td><td>2.59</td><td>0.87</td><td></td><td>9.84</td><td>4.18 4.19</td><td></td><td>9.83</td><td>4.18</td><td></td></tr><tr><td>CERL</td><td>2.59</td><td>0.87</td><td></td><td>2.60</td><td>0.87</td><td></td><td>9.85</td><td></td><td></td><td>9.83</td><td>4.18</td><td></td></tr></table>
+
+Besides, the intervention $\cdot$ is defined by $\begin{array} { r } { p ( t = 1 | x ) = \frac { e ^ { k \cdot z ( x ) \Psi { z } _ { 1 } ^ { c } } } { e ^ { k \cdot z ( x ) \Psi { z } _ { 0 } ^ { c } } + e ^ { k \cdot z ( x ) \Psi z _ { 1 } ^ { c } } } } \end{array}$ e  1ek·z(x)|zc0 +ek·z(x)|zc1 , where k = 10 indicates an expected selection bias. In the experiments, 50 LDA topics are learned from the training corpus and 3477 bag-of-words features are in the dataset. To generate two sequential datasets with different domain shifts, we combine the news items belonging to LDA topics from 1 to 25 into first dataset and the news items belonging to LDA topics from 26 to 50 into second dataset. There is no overlap of the LDA topics between the first dataset and second dataset, which is considered as substantial domain shift. In addition, the news items belonging to LDA topics from 1 to 35 and items belonging to from 16 to 50 are used to construct the first dataset and second dataset, respectively, which is regarded as moderate domain shift. Finally, randomly sampled items from 50 LDA topics compose the first and second dataset, resulting in no domain shift, because they are from the same distribution. Under each domain shift scenario and each dataset, we randomly sample $6 0 \%$ and $2 0 \%$ of the units as the training set and validation set and let the remaining be the test set.
+
+BlogCatalog. BlogCatalog (Guo et al., 2020) is a blog directory that manages the bloggers and their blogs. In this semi-synthetic dataset, each unit is a blogger and the features are bag-of-words representations of keywords in bloggers’ descriptions collected from real-world source. We adopt the same settings and assumptions to simulate the treatment options and outcomes as we do for the News dataset. 50 LDA topics are learned from the training corpus. 5196 units and 2160 bag-ofwords features are in the dataset. Similar to the generation procedure of News datasets with domain shifts, we create two datasets for each of the three domain shift scenarios. Under each domain shift scenario and each dataset, we randomly sample $\cdot$ and $\cdot$ of the units as the training set and validation set and let the remaining be the test set.
+
+# 4.2 RESULTS AND ANALYSIS
+
+Evaluation Metrics. We adopt two commonly used evaluation metrics. The first one is the error of ATE estimation, which is defined as $\_$ , where ATE is the true value and $\cdot$ is an estimated ATE. The second one is the error of expected precision in estimation of heterogeneous effect (PEHE) Hill (2011), which is defined as $-$ , where $\cdot$ is the true ITE for unit $i$ and $\cdot$ is an estimated ITE for unit $\cdot$ .
+
+We employ three strategies to adapt traditional causal effect estimation models to incrementally available observational data: (A) directly apply the model previously trained based on original data to new observational data; (B) utilize newly available data to fine-tune the previously learned model;
+
+![](images/228d380dba5aa9e0d43148355cc333c6a4220656e625feb519ad9c20b4de2f24.jpg)  
+Figure 2: The types of variables.
+
+![](images/c6aa01613980b5005e9a1185e5db6830054ee3ce10ff0337dacdc135d41be106.jpg)  
+Figure 3: The work flow of task.
+
+Table 2: Performance on two sequential data and $M = 1 0 0 0 0$ . We present the mean value of $\sqrt { \epsilon _ { \mathrm { P E H E } } }$ and $\epsilon _ { \mathrm { A T E } }$ on test sets from two datasets. The standard deviations are tiny. Lower is better.   
+
+<table><tr><td></td><td colspan="2">Previous data</td><td colspan="2">New data</td></tr><tr><td>Strategy</td><td>VPEHE</td><td>EATE</td><td>VPEHE</td><td>EATE</td></tr><tr><td>CFR-A</td><td>1.47</td><td>0.35</td><td>2.51</td><td>0.73</td></tr><tr><td>CFR-B</td><td>1.82</td><td>0.47</td><td>1.63</td><td>0.45</td></tr><tr><td>CFR-C</td><td>1.49</td><td>0.36</td><td>1.62</td><td>0.44</td></tr><tr><td>CERL</td><td>1.49</td><td>0.37</td><td>1.63</td><td>0.44</td></tr><tr><td>CERL (w/o FRT)</td><td>1.71</td><td>0.43</td><td>1.63</td><td>0.44</td></tr><tr><td>CERL (w/o herding)</td><td>1.57</td><td>0.40</td><td>1.63</td><td>0.44</td></tr><tr><td>CERL (w/o cosine norm)</td><td>1.51</td><td>0.38</td><td>1.65</td><td>0.44</td></tr></table>
+
+(C) store all previous data and combine with new data to re-train the model from scratch. Among these three strategies, (C) is expected to be the best performer and get the ideal performance with respect to ATE and PEHE, although it needs to take up the most resources (all the data from previous and new dataset). We implement the three strategies based on the counterfactual regression model (CFR) (Shalit et al., 2017), which is a representative causal effect estimation method.
+
+As shown in Table 1, under no domain shift scenario, the three strategies and our model have the similar performance on the News and BlogCatalog datasets, because the previous and new data are from the same distribution. CFR-A, CFR-B, and CERL need less resources than CFR-C. Under substantial shift and moderate shift scenarios, we find strategy CFR-A performs well on previous data, but significantly declines on new dataset; strategy CFR-B shows the catastrophic forgetting problem where the performance on previous dataset is poor; strategy CFR-C performs well on both previous and new data, but it re-trains the whole model using both previous and new data. However, if there is a memory constraint or a barrier to accessing previous data, the strategy CFR-C cannot be conducted. Our CERL has a similar performance to strategy CFR-C, while CERL does not require access to previous data. Besides, by comparing the performance under substantial and moderate shift scenarios, the larger domain shift leads to worse performance of CFR-A and CFR-B. However, no matter what the domain shift is, the performance of our model CERL is consistent with the ideal strategy CFR-C.
+
+# 4.3 MODEL EVALUATION
+
+Synthetic Dataset. Our synthetic data include confounders, instrumental, adjustment, and irrelevant variables. The interrelations among these variables, treatments, and outcomes are illustrated in Figure 2. We totally simulate five different data sources with five different multivariate normal distributions to represent the incrementally available observational data. In each data source, we randomly draw 10000 samples including treatment units and control units. Therefore, for five datasets, they have different selection bias, magnitude of covariates, covariance matrices for variables, and number of treatment and control units. To ensure a robust estimation of model performance, for each data source, we repeat the simulation procedure 10 times and obtain 10 synthetic datasets. The details of data simulation are provided in Section A of Appendix.
+
+Results. Similar to the experiments for News and BlogCatalog benchmarks, we still utilize two sequential datasets to compare our model with CFR under three strategies on the more complex synthetic data. As shown in Table 2, the result is consistent with the conclusions on News and BlogCatalog. Our model’s performance demonstrates its superiority over CFR-A and CFR-B. CERL is comparable with CFR-C, while it does not need to have access to the raw data from previous dataset. Besides, we also conduct three ablation studies to test the effectiveness of the important components in CERL, i.e., CERL (w/o FRT), CERL (w/o herding), and CERL (w/o cosine norm). CERL (w/o FRT) is the simplified CERL without the feature representation transformation, which is based on traditional continual learning with knowledge distillation and integral probability metrics. In CERL (w/o FRT), we do not store and transform the previous feature representation into new feature space, and only utilize the knowledge distillation to realize the continual learning task and balance the bias between treatment and control groups with each new data. CERL (w/o herding) adopts random subsampling strategy to select samples into memory, instead of herding algorithm. CERL (w/o cosine norm) removes the cosine normalization in the last representation layer. Table 2 shows that the performance becomes poor after removing anyone in the feature representation transformation, herding, or cosine normalization modules compared to the original CERL. More specifically, after removing the feature representation transformation, $\cdot$ and $\cdot$ increase dramatically, which demonstrates that the knowledge distillation always used in continual learning task is not enough for the continual causal effect estimation. Also, using herding to select a representative set of samples from treatment and control distributions is crucial for the feature representation transformation.
+
+CERL Performance Evaluation. As illustrated in Figure 3, the five observational data are incrementally available in sequence, and the model will continue to estimate the causal effect without having access to previous data. We further evaluate the performance of CERL from three perspectives, i.e., the impact of memory constraint, effeteness of cosine normalization, and its robustness to hyper-parameters. As shown in Figure 4 (a) and (b), as the model continually learns a new√ dataset, every time when finishing training one new dataset, we report the $\sqrt { \epsilon _ { \mathrm { P E H E } } }$ and $\epsilon _ { \mathrm { A T E } }$ on test sets composed of previous data and new data. Our model with memory constraints has a similar performance to the ideal situation, where all data are available to train the model from scratch. However, our model can effectively save memory space, e.g., when facing the fifth dataset, our model only stores 1000, 5000, or 10000 feature representations, but the ideal situation needs to store $5 \times 1 0 0 0 0 = 5 0 0 0 0$ observations with all covariates. For the cosine normalization, we perform an ablation study of CERL $( \mathrm { M } { = } 5 0 0 0$ , 5 datasets), where we remove cosine normalization in the representation learning procedure. We find the $\sqrt { \epsilon _ { \mathrm { P E H E } } }$ increases from 1.80 and 1.92 and $\epsilon _ { \mathrm { A T E } }$ from 0.55 to 0.61. Next, we explore the model’s sensitivity to the most important parameter $\alpha$ and $\delta$ , which controls the representation balance and representation transformation. From Fig. 4 (c) and (d), we observe that the performance is stable over a large parameter range. In addition, the parameter $\beta$ for feature representation distillation is set to 1 (Rebuffi et al., 2017; Iscen et al., 2020).
+
+![](images/a46437bc9727544c696429722d8b3def3eda5510599b5a571faa65a0cc2e7e08.jpg)  
+Figure 4: Performance of CERL under different settings.
+
+# 5 CONCLUSION
+
+It is the first time to propose the continual lifelong causal effect inference problem and the corresponding evaluation criteria. As the real world evidence is becoming more prominent, how to integrate and utilize these powerful data for causal effect estimation becomes a new research challenge. To address this challenge, we propose the Continual Causal Effect Representation Learning method for estimating causal effect with observational data, which are incrementally available from non-stationary data distributions. Extensive experiments demonstrate the superiority of our method over baselines for continual causal effect estimation.
+
+# REFERENCES
+
+Ahmed M Alaa and Mihaela van der Schaar. Bayesian inference of individualized treatment effects using multi-task gaussian processes. In Advances in Neural Information Processing Systems, pp. 3424–3432, 2017.
+
+Hugh A Chipman, Edward I George, and Robert E McCulloch. Bart: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1):266–298, 2010.
+
+Zhixuan Chu, Stephen L Rathbun, and Sheng Li. Matching in selective and balanced representation space for treatment effects estimation. arXiv preprint arXiv:2009.06828, 2020.
+
+Prithviraj Dhar, Rajat Vikram Singh, Kuan-Chuan Peng, Ziyan Wu, and Rama Chellappa. Learning without memorizing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5138–5146, 2019.
+
+Robert M French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences, 3(4):128–135, 1999.
+
+Ruocheng Guo, Jundong Li, and Huan Liu. Learning individual causal effects from networked observational data. In Proceedings of the 13th International Conference on Web Search and Data Mining, pp. 232–240, 2020.
+
+Johanna Hardin, Stephan Ramon Garcia, and David Golan. A method for generating realistic correlation matrices. The Annals of Applied Statistics, pp. 1733–1762, 2013.
+
+Jennifer L Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1):217–240, 2011.
+
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
+
+Saihui Hou, Xinyu Pan, Chen Change Loy, Zilei Wang, and Dahua Lin. Learning a unified classifier incrementally via rebalancing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 831–839, 2019.
+
+Guido W Imbens and Donald B Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge University Press, 2015.
+
+Ahmet Iscen, Jeffrey Zhang, Svetlana Lazebnik, and Cordelia Schmid. Memory-efficient incremental learning through feature adaptation. arXiv preprint arXiv:2004.00713, 2020.
+
+Daniel Jacob, Wolfgang Karl Hardle, and Stefan Lessmann. Group average treatment effects for ¨ observational studies. arXiv preprint arXiv:1911.02688, 2019.
+
+Fredrik Johansson, Uri Shalit, and David Sontag. Learning representations for counterfactual inference. In International conference on machine learning, pp. 3020–3029, 2016.
+
+Peter Langfelder, Bin Zhang, and Steve Horvath. Defining clusters from a hierarchical cluster tree: the dynamic tree cut package for r. Bioinformatics, 24(5):719–720, 2008.
+
+Sheng Li and Yun Fu. Matching on balanced nonlinear representations for treatment effects estimation. In Advances in Neural Information Processing Systems, pp. 929–939, 2017.
+
+Zhizhong Li and Derek Hoiem. Learning without forgetting. IEEE transactions on pattern analysis and machine intelligence, 40(12):2935–2947, 2017.
+
+Christos Louizos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling. Causal effect inference with deep latent-variable models. In Advances in Neural Information Processing Systems, pp. 6446–6456, 2017.
+
+Chunjie Luo, Jianfeng Zhan, Xiaohe Xue, Lei Wang, Rui Ren, and Qiang Yang. Cosine normalization: Using cosine similarity instead of dot product in neural networks. In International Conference on Artificial Neural Networks, pp. 382–391. Springer, 2018.
+
+Michael McCloskey and Neal J Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pp. 109–165. Elsevier, 1989.
+
+Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. icarl: Incremental classifier and representation learning. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 2001–2010, 2017.
+
+Peter M Robinson. Root-n-consistent semiparametric regression. Econometrica: Journal of the Econometric Society, pp. 931–954, 1988.
+
+Donald B Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology, 66(5):688, 1974.
+
+Saeed Samet, Ali Miri, and Eric Granger. Incremental learning of privacy-preserving bayesian networks. Applied Soft Computing, 13(8):3657–3667, 2013.
+
+Patrick Schwab, Lorenz Linhardt, and Walter Karlen. Perfect match: A simple method for learning representations for counterfactual inference with neural networks. arXiv preprint arXiv:1810.00656, 2018.
+
+Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: generalization bounds and algorithms. In International Conference on Machine Learning, pp. 3076– 3085. PMLR, 2017.
+
+Jerzy Splawa-Neyman, Dorota M Dabrowska, and TP Speed. On the application of probability theory to agricultural experiments. essay on principles. section 9. Statistical Science, pp. 465– 472, 1990.
+
+Bharath K Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Scholkopf, Gert RG Lanck- ¨ riet, et al. On the empirical estimation of integral probability metrics. Electronic Journal of Statistics, 6:1550–1599, 2012.
+
+Stefan Wager and Susan Athey. Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 113(523):1228–1242, 2018. doi: 10.1080/01621459.2017.1319839. URL https://doi.org/10.1080/01621459. 2017.1319839.
+
+Max Welling. Herding dynamical weights to learn. In Proceedings of the 26th Annual International Conference on Machine Learning, pp. 1121–1128, 2009.
+
+Liuyi Yao, Zhixuan Chu, Sheng Li, Yaliang Li, Jing Gao, and Aidong Zhang. A survey on causal inference. arXiv preprint arXiv:2002.02770, 2020.
+
+Jinsung Yoon, James Jordon, and Mihaela van der Schaar. Ganite: Estimation of individualized treatment effects using generative adversarial nets. In International Conference on Learning Representations, 2018.
+
+Bin Zhang and Steve Horvath. A general framework for weighted gene co-expression network analysis. Statistical applications in genetics and molecular biology, 4(1), 2005.
+
+Junting Zhang, Jie Zhang, Shalini Ghosh, Dawei Li, Serafettin Tasci, Larry Heck, Heming Zhang, and C-C Jay Kuo. Class-incremental learning via deep model consolidation. In The IEEE Winter Conference on Applications of Computer Vision, pp. 1131–1140, 2020.
+
+Hui Zou and Trevor Hastie. Regularization and variable selection via the elastic net. Journal of the royal statistical society: series B (statistical methodology), 67(2):301–320, 2005.
+
+# A SIMULATION PROCEDURE
+
+Our synthetic data include confounders, instrumental, adjustment, and irrelevant variables. The interrelations among these variables, treatments, and outcomes are illustrated in Figure 2. The number of observed variables in the vector $X = ( C ^ { \mathsf { \tau } } , Z ^ { \mathsf { \tau } } , I ^ { \mathsf { \tau } } , A ^ { \mathsf { \tau } } ) ^ { \mathsf { \tau } }$ is set to 100, including 35 confounders in $C$ , 35 adjustment variables in $A$ , 10 instrumental variables in $Z$ , and 20 irrelevant variables in $I$ . The model used to generate the continuous outcome variable $Y$ in this simulation is the partially linear regression model, extending the ideas described in Robinson (1988); Jacob et al. (2019); Chu et al. (2020):
+
+$$
+Y = \tau ( ( C ^ { \mathsf { \tau } } , A ^ { \mathsf { \tau } } ) ^ { \mathsf { \tau } } ) T + g ( ( C ^ { \mathsf { \tau } } , A ^ { \mathsf { \tau } } ) ^ { \mathsf { \tau } } ) + \epsilon ,
+$$
+
+where $\epsilon$ are unobserved covariates, which follow a standard normal distribution $N ( 0 , 1 )$ and $E [ \epsilon | C , A , T ] = 0$ . $T \stackrel { i n d . } { \sim }$ . Bernoulli $( e _ { 0 } ( ( C ^ { \boldsymbol { \mathsf { T } } } , Z ^ { \boldsymbol { \mathsf { T } } } ) ^ { \boldsymbol { \mathsf { T } } } ) ) ,$ ) and $e _ { 0 } { \big ( } ( C ^ { \tau } , Z ^ { \tau } ) ^ { \tau } { \big ) }$ is the propensity score, which represents the treatment selection bias based on their own confounders $C$ and instrumental variables $Z$ . Because we aim to simulate multiple data sources $\{ D _ { d } ; d = 1 , . . . , D \}$ , the vector of all observed covariates $X = ( C ^ { \mathsf { \tau } } , Z ^ { \mathsf { \tau } } , I ^ { \mathsf { \tau } } , A ^ { \mathsf { \tau } } ) ^ { \mathsf { \tau } }$ is sampled from different multivariate normal distribution with mean vector $\mu _ { C } ^ { d } , \mu _ { Z } ^ { d } , \mu _ { I } ^ { d }$ , and $\mu _ { A } ^ { d }$ and different random positive definite covariance matrices $\Sigma ^ { d }$ .
+
+For each data source, except for the different magnitude of mean vector and structure of covariance matrix, the simulation procedure is the same. Let $D$ be the diagonal matrix with the square roots of the diagonal entries of $\Sigma$ on its diagonal, i.e., $D = { \sqrt { d i a g ( \sigma ) } }$ , then the correlation matrix is given as:
+
+$$
+R = D ^ { - 1 } \Sigma D ^ { - 1 } .
+$$
+
+We use algorithm 3 in Hardin et al. (2013) to simulate positive definite correlation matrices consisting of different types of variables. Our correlation matrices are based on the hub correlation structure which has a known correlation between a hub variable and each of the remaining variables (Zhang & Horvath, 2005; Langfelder et al., 2008). Each variable in one type of variables is correlated to the hub-variable with decreasing strength from specified maximum correlation to minimum correlation, and different types of variables are generated independently or with weaker correlation among variable types. Defining the first variable as the hub, for the ith variable $( i = 2 , 3 , . . . , n )$ ), the correlation between it and the hub-variable in one type of variables is given as:
+
+$$
+R _ { i , 1 } = \rho _ { \mathrm { { m a x } } } - \left( \frac { i - 2 } { d - 2 } \right) ^ { \gamma } ( \rho _ { \mathrm { { m a x } } } - \rho _ { \mathrm { { m i n } } } ) ,
+$$
+
+where $\rho _ { \mathrm { m a x } }$ and $\rho _ { \mathrm { m i n } }$ are specified maximum and minimum correlations, and the rate $\gamma$ controls rate at which correlations decay.
+
+After specifying the relationship between the hub variable and the remaining variables in the same type of variables, we use Toeplitz structure to fill out the remainder of the hub correlation matrix and get the hub-Toeplitz correlation matrix $R _ { t y p e }$ for other type of variables. Here, $R$ is the $n \times n$ matrix having the blocks $R _ { Z } , R _ { C } , R _ { A }$ , and $R _ { I }$ along the diagonal and zeros at off-diagonal elements. This yields a correlation matrix with nonzero correlations within the same type and zero correlation among other types. The amount of correlations among types which can be added to the positivedefinite correlation matrix $R$ is determined by its smallest eigenvalue.
+
+The function $\tau ( ( C ^ { \tau } , A ^ { \tau } ) ^ { \tau } )$ describes the true treatment effect as a function of the values of adjustment variables $A$ and confounders $C$ ; namely $\tau ( ( C ^ { \mathsf { \tau } } , A ^ { \mathsf { \tau } } ) ^ { \mathsf { \tau } } ) = ( \sin { ( ( C ^ { \mathsf { \tau } } , A ^ { \mathsf { \tau } } ) ^ { \mathsf { \tau } } \times b _ { \tau } ) } ) ^ { 2 } .$ where $b _ { \tau }$ represents weights for every covariate in the function, which is generated by uniform $( 0 , 1 )$ . The variable treatment effect implies that its strength differs among the units and is therefore conditioned on $C$ and $A$ . The function $g ( ( C ^ { \mathsf { T } } , A ^ { \mathsf { T } } ) ^ { \mathsf { T } } )$ can have an influence on outcome regardless of treatment assignment. It is calculated via a trigonometric function to make the covariates nonlinear, which is defined as $g ( ( C ^ { \mathsf { T } } , A ^ { \mathsf { T } } ) ^ { \mathsf { T } } ) = ( \cos { ( ( C ^ { \mathsf { T } } , A ^ { \mathsf { T } } ) ^ { \mathsf { T } } \times b _ { g } ) } ) ^ { 2 }$ . Here, $b _ { g }$ represents a weight for each covariate in this function, which is generated by uniform $( 0 , 1 )$ . The bias is attributed to unobserved covariates which follow a random normal distribution $N ( 0 , 1 )$ . The treatment assignment $T$ follows the Bernoulli distribution, i.e., $T \stackrel { i n d . } { \sim }$ . Bernoulli $( e _ { 0 } ( ( C ^ { \boldsymbol { \mathsf { T } } } , Z ^ { \boldsymbol { \mathsf { T } } } ) ^ { \boldsymbol { \mathsf { T } } } ) $ ) with probability $\begin{array} { r } { e _ { 0 } \bigl ( ( C ^ { \mathsf { T } } , Z ^ { \mathsf { T } } ) ^ { \mathsf { T } } \bigr ) = \Phi ( \frac { a - \mu ( a ) } { \sigma ( a ) } ) } \end{array}$ , where $e _ { 0 } { \big ( } ( C ^ { \tau } , Z ^ { \tau } ) ^ { \tau } { \big ) }$ represents the propensity score, which is the cumulative distribution function for a standard normal random variable based on confounders $C$ and instrumental variables $Z$ , i.e., $a = \sin \left( ( C ^ { \mathsf { \tau } } , Z ^ { \mathsf { \tau } } ) ^ { \mathsf { \tau } } \times b _ { a } \right)$ , where $b _ { a }$ is generated by uniform $( 0 , 1 )$ .
+
+We totally simulate five different data sources with five different multivariate normal distributions to represent the incrementally available observational data. In each data source, we randomly draw 10000 samples including treatment units and control units. Therefore, for five datasets, they have different selection bias, magnitude of covariates, covariance matrices for variables, and number of treatment and control units. To ensure a robust estimation of model performance, for each data source, we repeat the simulation procedure 10 times and obtain 10 synthetic datasets.
+
+# B ALGORITHM 1
+
+# Algorithm 1 Continual Causal Effect Representation Learning
+
+Data: Given $d$ incrementally available observational data from $D _ { 1 }$ to $D _ { d }$   
+if $\{ x , y , t \} \in D _ { 1 }$ then $^ { \ast \ast \ast }$ Train baseline causal effect model $h _ { \theta _ { 1 } } ( g _ { w _ { 1 } } )$ \*\*\* $w _ { 1 } , \theta _ { 1 } = \mathrm { O P T I M I Z E } ( L _ { Y } + \alpha W a s s ( P , Q ) + \lambda L _ { w _ { 1 } } )$ $R _ { 1 } = \{ g _ { w _ { 1 } } ( x ) | x \in D _ { 1 } \}$ $M _ { 1 } = { \mathrm { H E R D I N G } } \{ R _ { 1 } , Y _ { 1 } , T _ { 1 } \}$   
+else for $\{ x , y , t \} \in D _ { 2 } , . . . , D _ { d }$ do $^ { \ast } { ^ { \ast } } \ast \ast$ Train continual causal effect model $h _ { \theta _ { d } } ( g _ { w _ { d } } )$ \*\*\* $w _ { d } , \theta _ { d } , \phi _ { d - 1  d } = 0 \mathrm { P T I M I Z E } ( L _ { G } + \alpha \bar { W } a s s ( P , Q ) + \lambda L _ { w _ { 2 } } + \beta L _ { F D } + \delta L _ { F T } )$ $\tilde { R } _ { d - 1 } = \phi _ { d - 1 \to d } ( R _ { d - 1 } )$ $R _ { d } = \{ g _ { w _ { d } } ( x ) | x \in D _ { d } \}$ $M _ { d } = \overset { \sim } { \operatorname { H E R D I N G } } \big ( \{ R _ { d } , Y _ { d } , T _ { d } \} \cup \{ \tilde { R } _ { d - 1 } , Y _ { d - 1 } \in M _ { d - 1 } , T _ { d - 1 } \in M _ { d - 1 } \} \big )$ end   
+end

parse/train/IOqr2ZyXHz1/IOqr2ZyXHz1_content_list.json

@@ -0,0 +1,1382 @@
+[
+    {
+        "type": "text",
+        "text": "CONTINUAL LIFELONG CAUSAL EFFECT INFERENCEWITH REAL WORLD EVIDENCE",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The era of real world evidence has witnessed an increasing availability of observational data, which much facilitates the development of causal effect inference. Although significant advances have been made to overcome the challenges in causal effect estimation, such as missing counterfactual outcomes and selection bias, they only focus on source-specific and stationary observational data. In this paper, we investigate a new research problem of causal effect inference from incrementally available observational data, and present three new evaluation criteria accordingly, including extensibility, adaptability, and accessibility. We propose a Continual Causal Effect Representation Learning method for estimating causal effect with observational data, which are incrementally available from non-stationary data distributions. Instead of having access to all seen observational data, our method only stores a limited subset of feature representations learned from previous data. Combining the selective and balanced representation learning, feature representation distillation, and feature transformation, our method achieves the continual causal effect estimation for new data without compromising the estimation capability for original data. Extensive experiments demonstrate the significance of continual causal effect inference and the effectiveness of our method. ",
+        "bbox": [
+            232,
+            263,
+            764,
+            500
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            525,
+            336,
+            540
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Causal effect inference is a critical research topic across many domains, such as statistics, computer science, public policy, and economics. Randomized controlled trials (RCT) are usually considered as the gold-standard for causal effect inference, which randomly assigns participants into a treatment or control group. As the RCT is conducted, the only expected difference between the treatment and control groups is the outcome variable being studied. However, in reality, randomized controlled trials are always time-consuming and expensive, and thus the study cannot involve many subjects, which may be not representative of the real-world population the intervention would eventually target. Nowadays, estimating causal effects from observational data has become an appealing research direction owing to a large amount of available data and low budget requirements, compared with RCT (Yao et al., 2020). Researchers have developed various strategies for causal effect inference with observational data, such as tree-based methods (Chipman et al., 2010; Wager & Athey, 2018), representation learning methods (Johansson et al., 2016; Li & Fu, 2017; Shalit et al., 2017; Chu et al., 2020), adapting Bayesian algorithms (Alaa & van der Schaar, 2017), generative adversarial nets (Yoon et al., 2018), variational autoencoders (Louizos et al., 2017) and so on. ",
+        "bbox": [
+            174,
+            556,
+            825,
+            750
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although significant advances have been made to overcome the challenges in causal effect estimation with observational data, such as missing counterfactual outcomes and selection bias between treatment and control groups, the existing methods only focus on source-specific and stationary observational data. Such learning strategies assume that all observational data are already available during the training phase and from the only one source. This assumption is unsubstantial in practice due to two reasons. The first one is based on the characteristics of observational data, which are incrementally available from non-stationary data distributions. For instance, the number of electronic medical records in one hospital is growing every day, or the electronic medical records for one disease may be from different hospitals or even different countries. This characteristic implies that one cannot have access to all observational data at one time point and from one single source. The second reason is based on the realistic consideration of accessibility. For example, when the new observational are available, if we want to refine the model previously trained by original data, maybe the original training data are no longer accessible due to a variety of reasons, e.g., legacy data may be unrecorded, proprietary, too large to store, or subject to privacy constraint (Zhang et al., 2020). This practical concern of accessibility is ubiquitous in various academic and industrial applications. That’s what it boiled down to: in the era of big data, we face the new challenges in causal inference with observational data: the extensibility for incrementally available observational data, the adaptability for extra domain adaptation problem except for the imbalance between treatment and control groups in one source, and the accessibility for a huge amount of data. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            202
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Existing causal effect inference methods, however, are unable to deal with the aforementioned new challenges, i.e., extensibility, adaptability, and accessibility. Although it is possible to adapt existing causal inference methods to address the new challenges, these adapted methods still have inevitable defects. Three straightforward adaptation strategies are described as follows. (1) If we directly apply the model previously trained based on original data to new observational data, the performance on new task will be very poor due to the domain shift issues among different data sources; (2) If we utilize newly available data to re-train the previously learned model, adapting changes in the data distribution, old knowledge will be completely or partially overwritten by the new one, which can result in severe performance degradation on old tasks. This is the well-known catastrophic forgetting problem (McCloskey & Cohen, 1989; French, 1999); (3) To overcome the catastrophic forgetting problem, we may rely on the storage of old data and combine the old and new data together, and then re-train the model from scratch. However, this strategy is memory inefficient and time-consuming, and it brings practical concerns such as copyright or privacy issues when storing data for a long time (Samet et al., 2013). Our empirical evaluations in Section 4 demonstrate that any of these three strategies in combination with the existing causal effect inference methods is deficient. ",
+        "bbox": [
+            174,
+            208,
+            825,
+            416
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To address the above issues, we propose a Continual Causal Effect Representation Learning method (CERL) for estimating causal effect with incrementally available observational data. Instead of having access to all previous observational data, we only store a limited subset of feature representations learned from previous data. Combining the selective and balanced representation learning, feature representation distillation, and feature transformation, our method preserves the knowledge learned from previous data and update the knowledge by leveraging new data, so that it can achieve the continual causal effect estimation for new data without compromising the estimation capability for previous data. To summarize, our main contributions include: ",
+        "bbox": [
+            174,
+            422,
+            825,
+            535
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Our work is the first to introduce the continual lifelong causal effect inference problem for the incrementally available observational data and three corresponding evaluation criteria, i.e., extensibility, adaptability, and accessibility.   \n• We propose a new framework for continual lifelong causal effect inference based on deep representation learning and continual learning.   \n• Extensive experiments demonstrate the deficiency of existing methods when facing the incrementally available observational data and our model’s outstanding performance. ",
+        "bbox": [
+            217,
+            546,
+            825,
+            654
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND AND PROBLEM STATEMENT",
+        "text_level": 1,
+        "bbox": [
+            174,
+            662,
+            562,
+            679
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Suppose that the observational data contain $\\cdot$ units collected from $\\cdot$ different domains and the $\\cdot$ -th dataset $D _ { d }$ contains the data $-$ collected from $d$ -th domain, which contains $\\cdot$ units. Let $X$ denote all observed variables, $Y$ denote the outcomes in the observational data, and $\\cdot$ is a binary variable. Let $D _ { 1 : d } = \\{ D _ { 1 } , D _ { 2 } , . . . , D _ { d } \\}$ be the set of combination of $\\cdot$ dataset, separately collected from $d$ different domains. For $d$ datasets $\\_$ , they have the common observed variables but due to the fact that they are collected from different domains, they have different distributions with respect to $X$ , $\\cdot$ , and $T$ in each dataset. Each unit in the observational data received one of two treatments. Let $\\cdot$ denote the treatment assignment for unit $\\cdot$ ; $\\cdot$ . For binary treatments, $t _ { i } = 1$ is for the treatment group, and $\\cdot$ for the control group. The outcome for unit $i$ is denoted by $y _ { t } ^ { \\ i }$ when treatment $t$ is applied to unit $i$ ; that is, $\\cdot$ is the potential outcome of unit $\\cdot$ in the treatment group and $\\cdot$ is the potential outcome of unit $i$ in the control group. For observational data, only one of the potential outcomes is observed. The observed outcome is called the factual outcome and the remaining unobserved potential outcomes are called counterfactual outcomes. ",
+        "bbox": [
+            173,
+            694,
+            825,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we follow the potential outcome framework for estimating treatment effects (Rubin, 1974; Splawa-Neyman et al., 1990). The individual treatment effect (ITE) for unit $i$ is the difference between the potential treated and control outcomes, and is defined as $\\mathrm { I T E } _ { i } = y _ { 1 } ^ { i } - y _ { 0 } ^ { i }$ . The average treatment effect (ATE) is the difference between the mean potential treated and control outcomes, which is defined as $\\begin{array} { r } { \\mathrm { A } \\dot { \\mathrm { T E } } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( y _ { 1 } ^ { i } - y _ { 0 } ^ { i } ) } \\end{array}$ . ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            825,
+            147
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The success of the potential outcome framework is based on the following assumptions (Imbens & Rubin, 2015), which ensure that the treatment effect can be identified. Stable Unit Treatment Value Assumption (SUTVA): The potential outcomes for any units do not vary with the treatments assigned to other units, and, for each unit, there are no different forms or versions of each treatment level, which lead to different potential outcomes. Consistency: The potential outcome of treatment $t$ is equal to the observed outcome if the actual treatment received is $t$ . Positivity: For any value of $x$ , treatment assignment is not deterministic, i.e., $P ( T = t | X = x ) > 0$ , for all $t$ and $x$ . Ignorability: Given covariates, treatment assignment is independent to the potential outcomes, i.e., $( y _ { 1 } , y _ { 0 } ) \\perp t | x$ . ",
+        "bbox": [
+            174,
+            152,
+            825,
+            265
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The goal of our work is to develop a novel continual causal inference framework, given new available observational data $D _ { d }$ , to estimate the causal effect for newly available data $D _ { d }$ as well as the previous data $D _ { 1 : ( d - 1 ) }$ without having access to previous training data in $D _ { 1 : ( d - 1 ) }$ . ",
+        "bbox": [
+            174,
+            271,
+            825,
+            314
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 THE PROPOSED FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            334,
+            452,
+            351
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The availability of “real world evidence” is expected to facilitate the development of causal effect inference models for various academic and industrial applications. How to achieve continual learning from incrementally available observational data from non-stationary data domains is a new direction in causal effect inference. Rather than only focusing on handling the selection bias problem, we also need to take into comprehensive consideration three aspects of the model, i.e., the extensibility for incrementally available observational data, the adaptability for various data sources, and the accessibility for a huge amount of data. ",
+        "bbox": [
+            174,
+            366,
+            825,
+            463
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose the Continual Causal Effect Representation Learning method (CERL) for estimating causal effect with incrementally available observational data. Based on selective and balanced representation learning for treatment effect estimation, CERL incorporates feature representation distillation to preserve the knowledge learned from previous observational data. Besides, aiming at adapting the updated model to original and new data without having access to the original data, and solving the selection bias between treatment and control groups, we propose one representation transformation function, which maps partial original feature representations into new feature representation space and makes the global feature representation space balanced with respect to treatment and control groups. Therefore, CERL can achieve the continual causal effect estimation for new data and meanwhile preserve the estimation capability for previous data, without the aid of original data. ",
+        "bbox": [
+            174,
+            469,
+            825,
+            609
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 MODEL ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            627,
+            383,
+            641
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To estimate the incrementally available observational data, the framework of CERL is mainly composed of two components: (1) the baseline causal effect learning model is only for the first available observational data, and thus we don’t need to consider the domain shift issue among different data sources. This component is equivalent to the traditional causal effect estimation problem; (2) the continual causal effect learning model is for the sequentially available observational data, where we need to handle more complex issues, such as knowledge transfer, catastrophic forgetting, global representation balance, and memory constraint. We present the details of each component as follows. ",
+        "bbox": [
+            174,
+            652,
+            825,
+            751
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 THE BASELINE CAUSAL EFFECT LEARNING MODEL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            766,
+            580,
+            780
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We first describe the baseline causal effect learning model for the initial observational dataset and then bring in subsequent datasets. For causal effect estimation in the initial dataset, it can be transformed into the traditional causal effect estimation problem. Motivated by the empirical success of deep representation learning for counterfactual inference (Shalit et al., 2017; Chu et al., 2020), we propose to learn the selective and balanced feature representations for treated and control units, and then infer the potential outcomes based on learned representation space. ",
+        "bbox": [
+            174,
+            790,
+            825,
+            875
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learning Selective and Balanced Representation. Firstly, we adopt a deep feature selection model that enables variable selection in one deep neural network, i.e., $g _ { w _ { 1 } } : X \\to R$ , where $X$ denotes the original covariate space, $R$ denotes the representation space, and $w _ { 1 }$ are the learnable parameters in function $g$ . The elastic net regularization term (Zou & Hastie, 2005) is adopted in our model, i.e., $L _ { w _ { 1 } } = \\| \\dot { w } _ { 1 } \\| _ { 2 } ^ { 2 } + \\| w _ { 1 } \\| _ { 1 }$ . Elastic net throughout the fully connected representation layers assigns larger weights to important features. This strategy can effectively filter out the irrelevant variables and highlight the important variables. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Due to the selection bias between treatment and control groups and among the sequential different data sources, the magnitudes of confounders may be significantly different. To effectively eliminate the imbalance caused by the significant difference in magnitudes between treatment and control groups and among different data sources, we propose to use cosine normalization in the last representation layer. In the multi-layer neural networks, we traditionally use dot products between the output vector of the previous layer and the incoming weight vector, and then input the products to the activation function. The result of dot product is unbounded. Cosine normalization uses cosine similarity instead of simple dot products in neural networks, which can bound the pre-activation between $- 1$ and 1. The result could be even smaller when the dimension is high. As a result, the fined as variance can be controlled within a very narrow range (Luo et al., 2018). Cosine normalization is de- $\\begin{array} { r } { r = \\sigma ( r _ { n o r m } ) = \\sigma \\big ( \\cos ( w , x ) \\big ) = \\sigma ( \\frac { w \\cdot x } { | w | | x | } ) } \\end{array}$ , where $r _ { n o r m }$ is the normalized pre-activation, $w$ is the incoming weight vector, $x$ is the input vector, and $\\sigma$ is nonlinear activation function. ",
+        "bbox": [
+            173,
+            166,
+            825,
+            335
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Motivated by Shalit et al. (2017), we adopt integral probability metrics (IPM) when learning the representation space to balance the treatment and control groups. The IPM measures the divergence between the representation distributions of treatment and control groups, so we want to minimize the IPM to make two distributions more similar. Let $P ( g ( x ) | t = \\bar { 1 } )$ ) and $Q ( g ( x ) | t = 0 )$ ) denote the empirical distributions of the representation vectors for the treatment and control groups, respectively. We adopt the IPM defined in the family of 1-Lipschitz functions, which leads to IPM being the Wasserstein distance (Sriperumbudur et al., 2012; Shalit et al., 2017). In particular, the IPM term with Wasserstein distance is defined as $\\begin{array} { r } { { \\bf W } { \\ a s s } ( P , Q ) = \\operatorname* { i n f } _ { k \\in { \\mathcal K } } \\int _ { g ( x ) } \\| k \\bar { ( g ( x ) ) } - } \\end{array}$ $g ( x ) \\| P ( g ( x ) ) d ( g ( x ) )$ , where $\\gamma$ denotes the hyper-parameter controlling the trade-off between $\\mathrm { W a s s } ( P , Q )$ and other terms in the final objective function. $\\begin{array} { r } { \\mathcal { K } = \\{ k | Q ( k ( g ( x ) ) ) = P ( g ( x ) ) \\} } \\end{array}$ defines the set of push-forward functions that transform the representation distribution of the treatment distribution $P$ to that of the control $Q$ and $g ( x ) \\in \\{ g ( x ) _ { i } \\} _ { i : t _ { i } = 1 }$ . ",
+        "bbox": [
+            173,
+            342,
+            825,
+            513
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Inferring Potential Outcomes. We aim to learn a function $h _ { \\theta _ { 1 } } : R \\times T  Y$ that maps the representation vectors and treatment assignment to the corresponding observed outcomes, and it can be parameterized by deep neural networks. To overcome the risk of losing the influence of $T$ on $R$ ${ \\bf \\xi } , h _ { \\theta _ { 1 } } ( g _ { w _ { 1 } } ( x ) , t )$ is partitioned into two separate tasks for treatment and control groups, respectively. Each unit is only updated in the task corresponding to its observed treatment. Let $\\hat { y } _ { i } =$ $h _ { \\theta _ { 1 } } ( g _ { w _ { 1 } } ( x ) , t )$ denote the inferred observed outcome of unit he mean squared error in predicting factual ou $i$ correspcomes: t . $t _ { i }$ . $\\begin{array} { r } { L _ { Y } = \\frac { 1 } { n _ { 1 } } \\sum _ { i = 1 } ^ { n _ { 1 } } ( \\hat { y } _ { i } - y _ { i } ) ^ { 2 } } \\end{array}$ ",
+        "bbox": [
+            173,
+            520,
+            825,
+            618
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Putting all the above together, the objective function of our baseline causal effect learning model is: $L \\doteq L _ { Y } + \\alpha W a s s ( \\bar { P } , Q ) + \\lambda L _ { w _ { 1 } }$ , where $\\alpha$ and $\\lambda$ denote the hyper-parameters controlling the trade-off among $W a s s ( P , Q )$ , $L _ { w }$ , and $L _ { Y }$ in the objective function. ",
+        "bbox": [
+            176,
+            626,
+            821,
+            669
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.2 THE SUSTAINABILITY OF MODEL LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            713,
+            540,
+            727
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "By far, we have built the baseline model for causal effect estimation with observational data from a single source. To avoid catastrophic forgetting when learning new data, we propose to preserve a subset of lower-dimensional feature representations rather than all original covariates. We also can adjust the number of preserved feature representations according to the memory constraint. ",
+        "bbox": [
+            174,
+            750,
+            823,
+            805
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "After the completion of baseline model training, we store a subset of feature representations $R _ { 1 } = \\{ g _ { w _ { 1 } } ( x ) \\bar { | } x \\in D _ { 1 } \\}$ and the corresponding $\\{ Y , T \\} \\in D _ { 1 }$ as memory $M _ { 1 }$ . The size of stored representation vectors can be reduced to satisfy the pre-specified memory constraint by a herding algorithm (Welling, 2009; Rebuffi et al., 2017). The herding algorithm can create a representative set of samples from distribution and requires fewer samples to achieve a high approximation quality than random subsampling. We run the herding algorithm separately for treatment and control groups to store the same number of feature representations from treatment and control groups. At this point, we only store the memory set $M _ { 1 }$ and model $g _ { w 1 }$ , without the original data $( D _ { 1 } )$ . ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.3 THE CONTINUAL CAUSAL EFFECT LEARNING MODEL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            599,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For now, we have stored memory $M _ { 1 }$ and baseline model. To continually estimate the causal effect for incrementally available observational data, we incorporate feature representation distillation and feature representation transformation to estimate causal effect for all seen data based on balanced global feature representation space. The framework of CERL is shown in Fig. 1. ",
+        "bbox": [
+            174,
+            127,
+            825,
+            184
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Feature Representation Distillation. For next available dataset $D _ { 2 } \\ = \\ \\{ ( x , y , t ) | x \\ \\in \\ X , y \\ \\in$ $Y , t \\in T \\}$ collected from second domain, we adopt the same selective representation learning $g _ { w _ { 2 } } : X \\to R _ { 2 }$ with elastic net regularization $( L _ { w _ { 2 } } )$ on new parameters $w _ { 2 }$ . Because we expect our model can estimate causal effect for both previous and new data, we want the new model to inherit some knowledge from previous model. In continual learning, knowledge distillation (Hinton et al., 2015; Li & Hoiem, 2017) is commonly adopted to alleviate the catastrophic forgetting, where knowledge is transferred from one network to another network by encouraging the outputs of the original and new network to be similar. However, for the continual causal effect estimation problem, we focus more on the feature representations, which are required to be balanced between treatment and control, and among different data domains. Inspired by Hou et al. (2019); Dhar et al. (2019); Iscen et al. (2020), we propose feature representation distillation to encourage the representation vector $\\{ g _ { w _ { 1 } } ( x ) | x \\in D _ { 2 } \\}$ based on baseline model to be similar to the representation vector $\\{ g _ { w _ { 2 } } ( x ) | \\dot { x } \\in D _ { 2 } \\}$ based on new model by Euclidean distance. This feature distillation can help prevent the learned representations from drifting too much in the new feature representation space. Because we apply the cosine normalization to feature representations and $\\| { \\dot { A } } - B \\| ^ { 2 } = ( { \\dot { A } } - B ) ^ { \\tau } ( A - B ) { \\stackrel { \\sim } { = } } \\| A \\| ^ { 2 } + \\| B \\| ^ { 2 } - 2 A ^ { \\tau } B = 2 { \\bigl ( } 1 - c o s ( { \\dot { A , } } B ) { \\bigr ) }$ , the feature representation distillation is defined as $L _ { F D } ( x ) = 1 - c o s \\bigl ( g _ { w _ { 1 } } ( x ) , g _ { w _ { 2 } } ( x ) \\bigr ) ,$ , where $x \\in D _ { 2 }$ . ",
+        "bbox": [
+            173,
+            189,
+            825,
+            433
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/553c253f415acce00b77a9a4b2b8c7e63417d1e4b35a2ef71a731e268585a30b.jpg",
+        "image_caption": [
+            "Figure 1: The blue part is baseline causal effect learning model for the first observational data. After baseline model training, store a subset of feature representations $R _ { 1 }$ into $M _ { 1 }$ by herding algorithm. The green part helps to map $R _ { 1 }$ to transformed feature representations $\\tilde { R } _ { 1 }$ compatible with new feature representations space $R _ { 2 }$ . Then the red part is used for continual causal effect estimation based on feature distillation and balanced global feature representation learning for $\\tilde { R } _ { 1 }$ and $R _ { 2 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            445,
+            723,
+            550
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Feature Representation Transformation. We have previous feature representations $R _ { 1 }$ stored in $M _ { 1 }$ and new feature representations $R _ { 2 }$ extracted from newly available data. $R _ { 1 }$ and $R _ { 2 }$ lie in different feature representation space and they are not compatible with each other because they are learned from different models. In addition, we cannot learn the feature representations of previous data from the new model $g _ { w _ { 2 } }$ , as we no longer have access to previous data. Therefore, to balance the global feature representation space including previous and new representations between treatment and control groups, a feature transformation function is needed from previous feature representations $R _ { 1 }$ to transformed feature representations $\\tilde { R } _ { 1 }$ compatible with new feature representations space $R _ { 2 }$ . We define a feature transformation function as $\\phi _ { 1 \\to 2 } : R _ { 1 } \\to \\tilde { R } _ { 1 }$ . We also input the feature representations of new data $D _ { 2 }$ learned from old model, i.e., $g _ { w _ { 1 } } ( x )$ , to get the transformed feature representations of new data, i.e., $\\phi _ { 1  2 } ( g _ { w _ { 1 } } ( x ) )$ . To keep the transformed space compatible with the new feature representation space, we train the transformation function $\\phi _ { 1  2 }$ by making the $\\phi _ { 1  2 } ( g _ { w _ { 1 } } ( x ) )$ and $g _ { w _ { 2 } } ( x )$ similar, where $x \\ \\in \\ D _ { 2 }$ . The loss function is defined as $L _ { F T } ( x ) \\stackrel { } { = }$ $1 - c o s \\bigl ( \\phi _ { 1 \\to 2 } ( g _ { w _ { 1 } } ( x ) ) , g _ { w _ { 2 } } ( x ) \\bigr )$ , which is used to train the function $\\phi _ { 1  2 }$ to transform feature representations between different feature spaces. Then, we can attain the transformed old feature representations $\\tilde { R } _ { 1 } = \\phi _ { 1  2 } ( R _ { 1 } )$ , which is in the same space as $R _ { 2 }$ . ",
+        "bbox": [
+            174,
+            632,
+            825,
+            861
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Balancing Global Feature Representation Space. We have obtained a global feature representation space including the transformed representations of stored old data and new representations of new available data. We adopt the same integral probability metrics as baseline model to make sure that the representation distributions are balanced for treatment and control groups in the global feature representation space. In addition, we define a potential outcome function $h _ { \\theta _ { 2 } } : ( \\tilde { R } _ { 1 } , R _ { 2 } ) \\times T $ $Y$ . Let $\\hat { y } _ { i } ^ { M } = h _ { \\theta _ { 2 } } ( \\phi _ { 1  2 } ( r _ { i } ) , t )$ , where $r _ { i } \\in M _ { 1 }$ , and $\\hat { y } _ { j } ^ { D } = h _ { \\theta _ { 2 } } \\big ( g _ { w _ { 2 } } ( x _ { j } ) , t \\big )$ , where $x _ { j } \\in D _ { 2 }$ denote the inferred observed outcomes. We aim to minimize the mean squared error in predicting factual outcomes for global feature representations including transformed old feature representations and new feature representations: $\\begin{array} { r } { L _ { G } = \\frac { 1 } { \\tilde { n } _ { 1 } } \\sum _ { i = 1 } ^ { \\tilde { n } _ { 1 } } ( \\hat { y } _ { i } ^ { M } - y _ { i } ^ { M } ) ^ { 2 } + \\frac { 1 } { n _ { 2 } } \\sum _ { j = 1 } ^ { n _ { 2 } } ( \\hat { y } _ { j } ^ { D } - y _ { j } ^ { D } ) ^ { 2 } } \\end{array}$ , where $\\tilde { n } _ { 1 }$ is the number of units stored in $M _ { 1 }$ by herding algorithm, $y _ { i } ^ { M } \\in M _ { 1 }$ , and $y _ { j } ^ { D } \\in D _ { 2 }$ . ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            195
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In summary, the objective function of our continual causal effect learning model is $L = L _ { G } +$ $\\alpha W a s s ( P , Q ) + \\lambda L _ { w _ { 2 } } + \\beta L _ { F D } + \\delta L _ { F T }$ , where $\\alpha , \\lambda , \\beta$ , and $\\delta$ denote the hyper-parameters controlling the trade-off among $W a s s ( P , Q ) , L _ { w _ { 2 } } , L _ { F D } , L _ { F T }$ , and $L _ { G }$ in the final objective function. ",
+        "bbox": [
+            176,
+            200,
+            821,
+            244
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 OVERVIEW OF CERL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            260,
+            362,
+            275
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the above sections, we have provided the baseline and continual causal effect learning models. When the continual causal effect learning model for the second data is trained, we can extract the $R _ { 2 } = \\{ g _ { w _ { 2 } } ( x ) | x \\in D _ { 2 } \\}$ and $\\tilde { R } _ { 1 } = \\{ \\stackrel { - } { \\phi _ { 1  2 } } ( r ) | r \\in M _ { 1 } \\}$ . We define a new memory set as $M _ { 2 } =$ $\\{ R _ { 2 } , Y _ { 2 } , T _ { 2 } \\} \\cup \\phi _ { 1  2 } ( M _ { 1 } )$ , where $\\phi _ { 1  2 } ( M _ { 1 } )$ includes $\\bar { \\tilde { R } } _ { 1 }$ and the corresponding $\\{ Y , T \\}$ stored in $M _ { 1 }$ . Similarly, to satisfy the pre-specified memory constraint, $M _ { 2 }$ can be reduced by conducting the herding algorithm to store the same number of feature representations from treatment and control groups. We only store the new memory set $M _ { 2 }$ and new model $g _ { w _ { 2 } }$ , which are used to train the following model and balance the global feature representation space. It is unnecessary to store the original data $D _ { 1 }$ and $D _ { 2 }$ ) any longer. ",
+        "bbox": [
+            173,
+            285,
+            825,
+            416
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We follow the same procedure for the subsequently available observational data. When we obtain the new observational data $D _ { d }$ , we can train $h _ { \\theta _ { d } } ( g _ { w _ { d } } )$ and $\\phi _ { d - 1 \\to d } : R _ { d - 1 } \\to \\tilde { R } _ { d - 1 }$ based on the continual causal effect learning model. Besides, the new memory set is defined as: $M _ { d } =$ $\\{ R _ { d } , Y _ { d } , T _ { d } \\} \\cup \\phi _ { d - 1  d } ( M _ { d - 1 } )$ . So far, our model $h _ { \\theta _ { d } } ( g _ { w _ { d } } )$ can estimate causal effect for all seen observational data regardless of the data source and it doesn’t require access to previous data. The detailed procedures of our CERL method are summarized in Algorithm 1 in Section B of Appendix. ",
+        "bbox": [
+            174,
+            421,
+            825,
+            508
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            527,
+            326,
+            542
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We adapt the traditional benchmarks, i.e., News (Johansson et al., 2016; Schwab et al., 2018) and BlogCatalog (Guo et al., 2020) to continual causal effect estimation. Specifically, we consider three scenarios to represent the different degrees of domain shifts among the incrementally available observational data, including the substantial shift, moderate shift, and no shift. Besides, we generate a series of synthetic datasets and also conduct ablation studies to demonstrate the effectiveness of our model on multiple sequential datasets. The model performance with different numbers of preserved feature representations, and the robustness to hyperparameters are also evaluated. ",
+        "bbox": [
+            173,
+            559,
+            825,
+            656
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASET DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            672,
+            379,
+            686
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We utilize two semi-synthetic benchmarks for the task of continual causal effect estimation, which are based on real-world features, synthesized treatments and outcomes. ",
+        "bbox": [
+            173,
+            699,
+            823,
+            727
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "News. The News dataset consists of 5000 randomly sampled news articles from the NY Times corpus1. It simulates the opinions of media consumers on news items. The units are different news items represented by word counts $\\cdot$ and outcome $\\cdot$ is the news item. The intervention $\\cdot$ represents the viewing device, desktop $t = 0$ ) or mobile $\\cdot$ ). We extend the original dataset specification in Johansson et al. (2016); Schwab et al. (2018) to enable the simulation of incrementally available observational data with different degrees of domain shifts. Assuming consumers prefer to read certain media items on specific viewing devices, we train a topic model on a large set of documents and define $\\cdot$ as the topic distribution of news item $x$ . We define one topic distribution of a randomly sampled document as centroid $\\cdot$ for mobile and the average topic representation of all document as centroid $\\cdot$ for desktop. Therefore, the reader’s opinion of news item $x$ on device $t$ is determined by the similarity between $\\cdot$ and $\\cdot$ , i.e., $-$ , where $\\cdot$ is a scaling factor and $\\epsilon \\sim N ( 0 , 1 )$ . ",
+        "bbox": [
+            173,
+            733,
+            825,
+            901
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Performance on two sequential data and $\\cdot$ . We present the mean value of $\\cdot$ and $\\epsilon _ { \\mathrm { A T E } }$ on test sets from two datasets. The standard deviations are tiny. Lower is better. Under no domain shift scenario, the three strategies and CERL have the similar performance, because the previous and new data are from the same distribution. Under substantial shift and moderate shift scenarios, CFR-A performs well on previous data, but significantly declines on new dataset; straxtegy CFR-B shows the catastrophic forgetting problem; CERL has a similar performance to strategy CFR-C, while CERL does not require access to previous data. Besides, the larger domain shift leads to worse performance of CFR-A and CFR-B. CERL has remained stable against shift. ",
+        "bbox": [
+            173,
+            104,
+            825,
+            217
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/e2231cf1a6fc70bddd284d6260138997a5c31c275890a16570a0e9e0197c32a6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"6\">News</td><td colspan=\"6\">BlogCatalog</td></tr><tr><td colspan=\"2\">Previous data</td><td colspan=\"2\"></td><td colspan=\"2\">New data</td><td colspan=\"2\">Previous data</td><td colspan=\"2\"></td><td colspan=\"2\">New data</td></tr><tr><td></td><td>Strategy</td><td>VEPEHE</td><td>EATE</td><td></td><td>VEPEHE</td><td>EATE</td><td></td><td>VePEHE</td><td>EATE</td><td></td><td>√EPEHE</td><td>EATE</td><td></td></tr><tr><td rowspan=\"4\">Substantial shift</td><td>CFR-A</td><td>2.49</td><td>0.80</td><td></td><td>3.62</td><td>1.18</td><td>个</td><td>9.92</td><td>4.25</td><td></td><td>13.65</td><td>6.21</td><td>个</td></tr><tr><td>CFR-B</td><td>3.23</td><td>1.06</td><td>个</td><td>2.71</td><td>0.91</td><td></td><td>14.21</td><td>6.98</td><td>个</td><td>9.77</td><td>4.11</td><td></td></tr><tr><td>CFR-C</td><td>2.51</td><td>0.82</td><td></td><td>2.70</td><td>0.92</td><td></td><td>9.93</td><td>4.24</td><td></td><td>9.77</td><td>4.12</td><td></td></tr><tr><td>CERL</td><td>2.55</td><td>0.84</td><td></td><td>2.71</td><td>0.91</td><td></td><td>9.96</td><td>4.25</td><td></td><td>9.78</td><td>4.12</td><td></td></tr><tr><td rowspan=\"4\">Moderate shift</td><td>CFR-A</td><td>2.58</td><td>0.85</td><td></td><td>3.06</td><td>1.02</td><td>个</td><td>9.89</td><td>4.22</td><td></td><td>11.26</td><td>5.03</td><td>个</td></tr><tr><td>CFR-B</td><td>2.98</td><td>0.99</td><td>个</td><td>2.65</td><td>0.92</td><td></td><td>12.35</td><td>5.67</td><td>个</td><td>9.83</td><td>4.18</td><td></td></tr><tr><td>CFR-C</td><td>2.56</td><td>0.85</td><td></td><td>2.63</td><td>0.90</td><td></td><td>9.88</td><td>4.21</td><td></td><td>9.81</td><td>4.16</td><td></td></tr><tr><td>CERL</td><td>2.59</td><td>0.86</td><td></td><td>2.66</td><td>0.92</td><td></td><td>9.90</td><td>4.24</td><td></td><td>9.82</td><td>4.17</td><td></td></tr><tr><td rowspan=\"4\">No shift</td><td>CFR-A</td><td>2.58</td><td>0.87</td><td></td><td>2.62</td><td>0.88</td><td></td><td>9.86</td><td>4.20</td><td></td><td>9.85</td><td>4.19</td><td></td></tr><tr><td>CFR-B</td><td>2.60</td><td>0.88</td><td></td><td>2.60</td><td>0.87</td><td></td><td>9.85</td><td>4.18</td><td></td><td>9.83</td><td>4.18</td><td></td></tr><tr><td>CFR-C</td><td>2.58</td><td>0.87</td><td></td><td>2.59</td><td>0.87</td><td></td><td>9.84</td><td>4.18 4.19</td><td></td><td>9.83</td><td>4.18</td><td></td></tr><tr><td>CERL</td><td>2.59</td><td>0.87</td><td></td><td>2.60</td><td>0.87</td><td></td><td>9.85</td><td></td><td></td><td>9.83</td><td>4.18</td><td></td></tr></table>",
+        "bbox": [
+            176,
+            218,
+            839,
+            435
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Besides, the intervention $\\cdot$ is defined by $\\begin{array} { r } { p ( t = 1 | x ) = \\frac { e ^ { k \\cdot z ( x ) \\Psi { z } _ { 1 } ^ { c } } } { e ^ { k \\cdot z ( x ) \\Psi { z } _ { 0 } ^ { c } } + e ^ { k \\cdot z ( x ) \\Psi z _ { 1 } ^ { c } } } } \\end{array}$ e  1ek·z(x)|zc0 +ek·z(x)|zc1 , where k = 10 indicates an expected selection bias. In the experiments, 50 LDA topics are learned from the training corpus and 3477 bag-of-words features are in the dataset. To generate two sequential datasets with different domain shifts, we combine the news items belonging to LDA topics from 1 to 25 into first dataset and the news items belonging to LDA topics from 26 to 50 into second dataset. There is no overlap of the LDA topics between the first dataset and second dataset, which is considered as substantial domain shift. In addition, the news items belonging to LDA topics from 1 to 35 and items belonging to from 16 to 50 are used to construct the first dataset and second dataset, respectively, which is regarded as moderate domain shift. Finally, randomly sampled items from 50 LDA topics compose the first and second dataset, resulting in no domain shift, because they are from the same distribution. Under each domain shift scenario and each dataset, we randomly sample $6 0 \\%$ and $2 0 \\%$ of the units as the training set and validation set and let the remaining be the test set. ",
+        "bbox": [
+            173,
+            441,
+            826,
+            616
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "BlogCatalog. BlogCatalog (Guo et al., 2020) is a blog directory that manages the bloggers and their blogs. In this semi-synthetic dataset, each unit is a blogger and the features are bag-of-words representations of keywords in bloggers’ descriptions collected from real-world source. We adopt the same settings and assumptions to simulate the treatment options and outcomes as we do for the News dataset. 50 LDA topics are learned from the training corpus. 5196 units and 2160 bag-ofwords features are in the dataset. Similar to the generation procedure of News datasets with domain shifts, we create two datasets for each of the three domain shift scenarios. Under each domain shift scenario and each dataset, we randomly sample $\\cdot$ and $\\cdot$ of the units as the training set and validation set and let the remaining be the test set. ",
+        "bbox": [
+            173,
+            622,
+            825,
+            748
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 RESULTS AND ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            768,
+            388,
+            782
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics. We adopt two commonly used evaluation metrics. The first one is the error of ATE estimation, which is defined as $\\_$ , where ATE is the true value and $\\cdot$ is an estimated ATE. The second one is the error of expected precision in estimation of heterogeneous effect (PEHE) Hill (2011), which is defined as $-$ , where $\\cdot$ is the true ITE for unit $i$ and $\\cdot$ is an estimated ITE for unit $\\cdot$ . ",
+        "bbox": [
+            173,
+            795,
+            825,
+            875
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We employ three strategies to adapt traditional causal effect estimation models to incrementally available observational data: (A) directly apply the model previously trained based on original data to new observational data; (B) utilize newly available data to fine-tune the previously learned model; ",
+        "bbox": [
+            174,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/228d380dba5aa9e0d43148355cc333c6a4220656e625feb519ad9c20b4de2f24.jpg",
+        "image_caption": [
+            "Figure 2: The types of variables. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            97,
+            424,
+            236
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c6aa01613980b5005e9a1185e5db6830054ee3ce10ff0337dacdc135d41be106.jpg",
+        "image_caption": [
+            "Figure 3: The work flow of task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            450,
+            103,
+            823,
+            233
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/c1444ca92be463050f4dd70d22190b602aef58962a997250c46c85c702435590.jpg",
+        "table_caption": [
+            "Table 2: Performance on two sequential data and $M = 1 0 0 0 0$ . We present the mean value of $\\sqrt { \\epsilon _ { \\mathrm { P E H E } } }$ and $\\epsilon _ { \\mathrm { A T E } }$ on test sets from two datasets. The standard deviations are tiny. Lower is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Previous data</td><td colspan=\"2\">New data</td></tr><tr><td>Strategy</td><td>VPEHE</td><td>EATE</td><td>VPEHE</td><td>EATE</td></tr><tr><td>CFR-A</td><td>1.47</td><td>0.35</td><td>2.51</td><td>0.73</td></tr><tr><td>CFR-B</td><td>1.82</td><td>0.47</td><td>1.63</td><td>0.45</td></tr><tr><td>CFR-C</td><td>1.49</td><td>0.36</td><td>1.62</td><td>0.44</td></tr><tr><td>CERL</td><td>1.49</td><td>0.37</td><td>1.63</td><td>0.44</td></tr><tr><td>CERL (w/o FRT)</td><td>1.71</td><td>0.43</td><td>1.63</td><td>0.44</td></tr><tr><td>CERL (w/o herding)</td><td>1.57</td><td>0.40</td><td>1.63</td><td>0.44</td></tr><tr><td>CERL (w/o cosine norm)</td><td>1.51</td><td>0.38</td><td>1.65</td><td>0.44</td></tr></table>",
+        "bbox": [
+            287,
+            310,
+            709,
+            444
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(C) store all previous data and combine with new data to re-train the model from scratch. Among these three strategies, (C) is expected to be the best performer and get the ideal performance with respect to ATE and PEHE, although it needs to take up the most resources (all the data from previous and new dataset). We implement the three strategies based on the counterfactual regression model (CFR) (Shalit et al., 2017), which is a representative causal effect estimation method. ",
+        "bbox": [
+            173,
+            462,
+            825,
+            532
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 1, under no domain shift scenario, the three strategies and our model have the similar performance on the News and BlogCatalog datasets, because the previous and new data are from the same distribution. CFR-A, CFR-B, and CERL need less resources than CFR-C. Under substantial shift and moderate shift scenarios, we find strategy CFR-A performs well on previous data, but significantly declines on new dataset; strategy CFR-B shows the catastrophic forgetting problem where the performance on previous dataset is poor; strategy CFR-C performs well on both previous and new data, but it re-trains the whole model using both previous and new data. However, if there is a memory constraint or a barrier to accessing previous data, the strategy CFR-C cannot be conducted. Our CERL has a similar performance to strategy CFR-C, while CERL does not require access to previous data. Besides, by comparing the performance under substantial and moderate shift scenarios, the larger domain shift leads to worse performance of CFR-A and CFR-B. However, no matter what the domain shift is, the performance of our model CERL is consistent with the ideal strategy CFR-C. ",
+        "bbox": [
+            173,
+            539,
+            825,
+            719
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 MODEL EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            737,
+            362,
+            752
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Synthetic Dataset. Our synthetic data include confounders, instrumental, adjustment, and irrelevant variables. The interrelations among these variables, treatments, and outcomes are illustrated in Figure 2. We totally simulate five different data sources with five different multivariate normal distributions to represent the incrementally available observational data. In each data source, we randomly draw 10000 samples including treatment units and control units. Therefore, for five datasets, they have different selection bias, magnitude of covariates, covariance matrices for variables, and number of treatment and control units. To ensure a robust estimation of model performance, for each data source, we repeat the simulation procedure 10 times and obtain 10 synthetic datasets. The details of data simulation are provided in Section A of Appendix. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            888
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results. Similar to the experiments for News and BlogCatalog benchmarks, we still utilize two sequential datasets to compare our model with CFR under three strategies on the more complex synthetic data. As shown in Table 2, the result is consistent with the conclusions on News and BlogCatalog. Our model’s performance demonstrates its superiority over CFR-A and CFR-B. CERL is comparable with CFR-C, while it does not need to have access to the raw data from previous dataset. Besides, we also conduct three ablation studies to test the effectiveness of the important components in CERL, i.e., CERL (w/o FRT), CERL (w/o herding), and CERL (w/o cosine norm). CERL (w/o FRT) is the simplified CERL without the feature representation transformation, which is based on traditional continual learning with knowledge distillation and integral probability metrics. In CERL (w/o FRT), we do not store and transform the previous feature representation into new feature space, and only utilize the knowledge distillation to realize the continual learning task and balance the bias between treatment and control groups with each new data. CERL (w/o herding) adopts random subsampling strategy to select samples into memory, instead of herding algorithm. CERL (w/o cosine norm) removes the cosine normalization in the last representation layer. Table 2 shows that the performance becomes poor after removing anyone in the feature representation transformation, herding, or cosine normalization modules compared to the original CERL. More specifically, after removing the feature representation transformation, $\\cdot$ and $\\cdot$ increase dramatically, which demonstrates that the knowledge distillation always used in continual learning task is not enough for the continual causal effect estimation. Also, using herding to select a representative set of samples from treatment and control distributions is crucial for the feature representation transformation. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            104,
+            825,
+            353
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "CERL Performance Evaluation. As illustrated in Figure 3, the five observational data are incrementally available in sequence, and the model will continue to estimate the causal effect without having access to previous data. We further evaluate the performance of CERL from three perspectives, i.e., the impact of memory constraint, effeteness of cosine normalization, and its robustness to hyper-parameters. As shown in Figure 4 (a) and (b), as the model continually learns a new√ dataset, every time when finishing training one new dataset, we report the $\\sqrt { \\epsilon _ { \\mathrm { P E H E } } }$ and $\\epsilon _ { \\mathrm { A T E } }$ on test sets composed of previous data and new data. Our model with memory constraints has a similar performance to the ideal situation, where all data are available to train the model from scratch. However, our model can effectively save memory space, e.g., when facing the fifth dataset, our model only stores 1000, 5000, or 10000 feature representations, but the ideal situation needs to store $5 \\times 1 0 0 0 0 = 5 0 0 0 0$ observations with all covariates. For the cosine normalization, we perform an ablation study of CERL $( \\mathrm { M } { = } 5 0 0 0$ , 5 datasets), where we remove cosine normalization in the representation learning procedure. We find the $\\sqrt { \\epsilon _ { \\mathrm { P E H E } } }$ increases from 1.80 and 1.92 and $\\epsilon _ { \\mathrm { A T E } }$ from 0.55 to 0.61. Next, we explore the model’s sensitivity to the most important parameter $\\alpha$ and $\\delta$ , which controls the representation balance and representation transformation. From Fig. 4 (c) and (d), we observe that the performance is stable over a large parameter range. In addition, the parameter $\\beta$ for feature representation distillation is set to 1 (Rebuffi et al., 2017; Iscen et al., 2020). ",
+        "bbox": [
+            173,
+            359,
+            825,
+            597
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/a46437bc9727544c696429722d8b3def3eda5510599b5a571faa65a0cc2e7e08.jpg",
+        "image_caption": [
+            "Figure 4: Performance of CERL under different settings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            613,
+            787,
+            744
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            791,
+            318,
+            806
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is the first time to propose the continual lifelong causal effect inference problem and the corresponding evaluation criteria. As the real world evidence is becoming more prominent, how to integrate and utilize these powerful data for causal effect estimation becomes a new research challenge. To address this challenge, we propose the Continual Causal Effect Representation Learning method for estimating causal effect with observational data, which are incrementally available from non-stationary data distributions. Extensive experiments demonstrate the superiority of our method over baselines for continual causal effect estimation. ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ahmed M Alaa and Mihaela van der Schaar. Bayesian inference of individualized treatment effects using multi-task gaussian processes. In Advances in Neural Information Processing Systems, pp. 3424–3432, 2017. ",
+        "bbox": [
+            173,
+            126,
+            823,
+            167
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hugh A Chipman, Edward I George, and Robert E McCulloch. Bart: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1):266–298, 2010. ",
+        "bbox": [
+            169,
+            178,
+            823,
+            207
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhixuan Chu, Stephen L Rathbun, and Sheng Li. Matching in selective and balanced representation space for treatment effects estimation. arXiv preprint arXiv:2009.06828, 2020. ",
+        "bbox": [
+            171,
+            215,
+            823,
+            244
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Prithviraj Dhar, Rajat Vikram Singh, Kuan-Chuan Peng, Ziyan Wu, and Rama Chellappa. Learning without memorizing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5138–5146, 2019. ",
+        "bbox": [
+            174,
+            253,
+            823,
+            296
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Robert M French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences, 3(4):128–135, 1999. ",
+        "bbox": [
+            171,
+            305,
+            823,
+            334
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ruocheng Guo, Jundong Li, and Huan Liu. Learning individual causal effects from networked observational data. In Proceedings of the 13th International Conference on Web Search and Data Mining, pp. 232–240, 2020. ",
+        "bbox": [
+            173,
+            343,
+            823,
+            387
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Johanna Hardin, Stephan Ramon Garcia, and David Golan. A method for generating realistic correlation matrices. The Annals of Applied Statistics, pp. 1733–1762, 2013. ",
+        "bbox": [
+            171,
+            395,
+            821,
+            426
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jennifer L Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1):217–240, 2011. ",
+        "bbox": [
+            171,
+            434,
+            821,
+            463
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015. ",
+        "bbox": [
+            173,
+            472,
+            821,
+            501
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Saihui Hou, Xinyu Pan, Chen Change Loy, Zilei Wang, and Dahua Lin. Learning a unified classifier incrementally via rebalancing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 831–839, 2019. ",
+        "bbox": [
+            173,
+            510,
+            823,
+            554
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Guido W Imbens and Donald B Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge University Press, 2015. ",
+        "bbox": [
+            173,
+            561,
+            821,
+            592
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ahmet Iscen, Jeffrey Zhang, Svetlana Lazebnik, and Cordelia Schmid. Memory-efficient incremental learning through feature adaptation. arXiv preprint arXiv:2004.00713, 2020. ",
+        "bbox": [
+            173,
+            599,
+            820,
+            630
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Daniel Jacob, Wolfgang Karl Hardle, and Stefan Lessmann. Group average treatment effects for ¨ observational studies. arXiv preprint arXiv:1911.02688, 2019. ",
+        "bbox": [
+            173,
+            638,
+            823,
+            667
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fredrik Johansson, Uri Shalit, and David Sontag. Learning representations for counterfactual inference. In International conference on machine learning, pp. 3020–3029, 2016. ",
+        "bbox": [
+            171,
+            676,
+            821,
+            707
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Peter Langfelder, Bin Zhang, and Steve Horvath. Defining clusters from a hierarchical cluster tree: the dynamic tree cut package for r. Bioinformatics, 24(5):719–720, 2008. ",
+        "bbox": [
+            171,
+            714,
+            821,
+            744
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sheng Li and Yun Fu. Matching on balanced nonlinear representations for treatment effects estimation. In Advances in Neural Information Processing Systems, pp. 929–939, 2017. ",
+        "bbox": [
+            171,
+            752,
+            823,
+            782
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhizhong Li and Derek Hoiem. Learning without forgetting. IEEE transactions on pattern analysis and machine intelligence, 40(12):2935–2947, 2017. ",
+        "bbox": [
+            171,
+            790,
+            823,
+            820
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Christos Louizos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling. Causal effect inference with deep latent-variable models. In Advances in Neural Information Processing Systems, pp. 6446–6456, 2017. ",
+        "bbox": [
+            178,
+            829,
+            823,
+            872
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chunjie Luo, Jianfeng Zhan, Xiaohe Xue, Lei Wang, Rui Ren, and Qiang Yang. Cosine normalization: Using cosine similarity instead of dot product in neural networks. In International Conference on Artificial Neural Networks, pp. 382–391. Springer, 2018. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Michael McCloskey and Neal J Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pp. 109–165. Elsevier, 1989. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. icarl: Incremental classifier and representation learning. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 2001–2010, 2017. ",
+        "bbox": [
+            176,
+            155,
+            821,
+            198
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Peter M Robinson. Root-n-consistent semiparametric regression. Econometrica: Journal of the Econometric Society, pp. 931–954, 1988. ",
+        "bbox": [
+            174,
+            207,
+            823,
+            236
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Donald B Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology, 66(5):688, 1974. ",
+        "bbox": [
+            173,
+            244,
+            821,
+            273
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Saeed Samet, Ali Miri, and Eric Granger. Incremental learning of privacy-preserving bayesian networks. Applied Soft Computing, 13(8):3657–3667, 2013. ",
+        "bbox": [
+            173,
+            281,
+            821,
+            311
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Patrick Schwab, Lorenz Linhardt, and Walter Karlen. Perfect match: A simple method for learning representations for counterfactual inference with neural networks. arXiv preprint arXiv:1810.00656, 2018. ",
+        "bbox": [
+            174,
+            319,
+            826,
+            362
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: generalization bounds and algorithms. In International Conference on Machine Learning, pp. 3076– 3085. PMLR, 2017. ",
+        "bbox": [
+            174,
+            371,
+            825,
+            414
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jerzy Splawa-Neyman, Dorota M Dabrowska, and TP Speed. On the application of probability theory to agricultural experiments. essay on principles. section 9. Statistical Science, pp. 465– 472, 1990. ",
+        "bbox": [
+            174,
+            422,
+            825,
+            465
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Bharath K Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Scholkopf, Gert RG Lanck- ¨ riet, et al. On the empirical estimation of integral probability metrics. Electronic Journal of Statistics, 6:1550–1599, 2012. ",
+        "bbox": [
+            173,
+            473,
+            826,
+            517
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Stefan Wager and Susan Athey. Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 113(523):1228–1242, 2018. doi: 10.1080/01621459.2017.1319839. URL https://doi.org/10.1080/01621459. 2017.1319839. ",
+        "bbox": [
+            173,
+            525,
+            825,
+            582
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Max Welling. Herding dynamical weights to learn. In Proceedings of the 26th Annual International Conference on Machine Learning, pp. 1121–1128, 2009. ",
+        "bbox": [
+            171,
+            590,
+            823,
+            621
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Liuyi Yao, Zhixuan Chu, Sheng Li, Yaliang Li, Jing Gao, and Aidong Zhang. A survey on causal inference. arXiv preprint arXiv:2002.02770, 2020. ",
+        "bbox": [
+            171,
+            628,
+            823,
+            659
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jinsung Yoon, James Jordon, and Mihaela van der Schaar. Ganite: Estimation of individualized treatment effects using generative adversarial nets. In International Conference on Learning Representations, 2018. ",
+        "bbox": [
+            173,
+            666,
+            823,
+            709
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Bin Zhang and Steve Horvath. A general framework for weighted gene co-expression network analysis. Statistical applications in genetics and molecular biology, 4(1), 2005. ",
+        "bbox": [
+            169,
+            718,
+            823,
+            748
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Junting Zhang, Jie Zhang, Shalini Ghosh, Dawei Li, Serafettin Tasci, Larry Heck, Heming Zhang, and C-C Jay Kuo. Class-incremental learning via deep model consolidation. In The IEEE Winter Conference on Applications of Computer Vision, pp. 1131–1140, 2020. ",
+        "bbox": [
+            174,
+            756,
+            823,
+            799
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hui Zou and Trevor Hastie. Regularization and variable selection via the elastic net. Journal of the royal statistical society: series B (statistical methodology), 67(2):301–320, 2005. ",
+        "bbox": [
+            174,
+            808,
+            823,
+            837
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A SIMULATION PROCEDURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            426,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Our synthetic data include confounders, instrumental, adjustment, and irrelevant variables. The interrelations among these variables, treatments, and outcomes are illustrated in Figure 2. The number of observed variables in the vector $X = ( C ^ { \\mathsf { \\tau } } , Z ^ { \\mathsf { \\tau } } , I ^ { \\mathsf { \\tau } } , A ^ { \\mathsf { \\tau } } ) ^ { \\mathsf { \\tau } }$ is set to 100, including 35 confounders in $C$ , 35 adjustment variables in $A$ , 10 instrumental variables in $Z$ , and 20 irrelevant variables in $I$ . The model used to generate the continuous outcome variable $Y$ in this simulation is the partially linear regression model, extending the ideas described in Robinson (1988); Jacob et al. (2019); Chu et al. (2020): ",
+        "bbox": [
+            173,
+            132,
+            825,
+            231
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/276fd3f423809b0e09938a40d83baeb0d31f20ce1bda7bd5fb75dcc4ac83c0f5.jpg",
+        "text": "$$\nY = \\tau ( ( C ^ { \\mathsf { \\tau } } , A ^ { \\mathsf { \\tau } } ) ^ { \\mathsf { \\tau } } ) T + g ( ( C ^ { \\mathsf { \\tau } } , A ^ { \\mathsf { \\tau } } ) ^ { \\mathsf { \\tau } } ) + \\epsilon ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            250,
+            638,
+            268
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ are unobserved covariates, which follow a standard normal distribution $N ( 0 , 1 )$ and $E [ \\epsilon | C , A , T ] = 0$ . $T \\stackrel { i n d . } { \\sim }$ . Bernoulli $( e _ { 0 } ( ( C ^ { \\boldsymbol { \\mathsf { T } } } , Z ^ { \\boldsymbol { \\mathsf { T } } } ) ^ { \\boldsymbol { \\mathsf { T } } } ) ) ,$ ) and $e _ { 0 } { \\big ( } ( C ^ { \\tau } , Z ^ { \\tau } ) ^ { \\tau } { \\big ) }$ is the propensity score, which represents the treatment selection bias based on their own confounders $C$ and instrumental variables $Z$ . Because we aim to simulate multiple data sources $\\{ D _ { d } ; d = 1 , . . . , D \\}$ , the vector of all observed covariates $X = ( C ^ { \\mathsf { \\tau } } , Z ^ { \\mathsf { \\tau } } , I ^ { \\mathsf { \\tau } } , A ^ { \\mathsf { \\tau } } ) ^ { \\mathsf { \\tau } }$ is sampled from different multivariate normal distribution with mean vector $\\mu _ { C } ^ { d } , \\mu _ { Z } ^ { d } , \\mu _ { I } ^ { d }$ , and $\\mu _ { A } ^ { d }$ and different random positive definite covariance matrices $\\Sigma ^ { d }$ . ",
+        "bbox": [
+            174,
+            276,
+            825,
+            382
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For each data source, except for the different magnitude of mean vector and structure of covariance matrix, the simulation procedure is the same. Let $D$ be the diagonal matrix with the square roots of the diagonal entries of $\\Sigma$ on its diagonal, i.e., $D = { \\sqrt { d i a g ( \\sigma ) } }$ , then the correlation matrix is given as: ",
+        "bbox": [
+            173,
+            388,
+            825,
+            445
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d456acf1107d71170eb705ac4183bba58b84088a7fdef7e2d191c2af7435fe9.jpg",
+        "text": "$$\nR = D ^ { - 1 } \\Sigma D ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            444,
+            558,
+            460
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use algorithm 3 in Hardin et al. (2013) to simulate positive definite correlation matrices consisting of different types of variables. Our correlation matrices are based on the hub correlation structure which has a known correlation between a hub variable and each of the remaining variables (Zhang & Horvath, 2005; Langfelder et al., 2008). Each variable in one type of variables is correlated to the hub-variable with decreasing strength from specified maximum correlation to minimum correlation, and different types of variables are generated independently or with weaker correlation among variable types. Defining the first variable as the hub, for the ith variable $( i = 2 , 3 , . . . , n )$ ), the correlation between it and the hub-variable in one type of variables is given as: ",
+        "bbox": [
+            173,
+            472,
+            825,
+            585
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c07f8016b12c49af49c4132992deeda9f886e207b6105a7b53583b6812fe727d.jpg",
+        "text": "$$\nR _ { i , 1 } = \\rho _ { \\mathrm { { m a x } } } - \\left( \\frac { i - 2 } { d - 2 } \\right) ^ { \\gamma } ( \\rho _ { \\mathrm { { m a x } } } - \\rho _ { \\mathrm { { m i n } } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            590,
+            632,
+            627
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\rho _ { \\mathrm { m a x } }$ and $\\rho _ { \\mathrm { m i n } }$ are specified maximum and minimum correlations, and the rate $\\gamma$ controls rate at which correlations decay. ",
+        "bbox": [
+            173,
+            640,
+            823,
+            669
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "After specifying the relationship between the hub variable and the remaining variables in the same type of variables, we use Toeplitz structure to fill out the remainder of the hub correlation matrix and get the hub-Toeplitz correlation matrix $R _ { t y p e }$ for other type of variables. Here, $R$ is the $n \\times n$ matrix having the blocks $R _ { Z } , R _ { C } , R _ { A }$ , and $R _ { I }$ along the diagonal and zeros at off-diagonal elements. This yields a correlation matrix with nonzero correlations within the same type and zero correlation among other types. The amount of correlations among types which can be added to the positivedefinite correlation matrix $R$ is determined by its smallest eigenvalue. ",
+        "bbox": [
+            173,
+            674,
+            825,
+            773
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The function $\\tau ( ( C ^ { \\tau } , A ^ { \\tau } ) ^ { \\tau } )$ describes the true treatment effect as a function of the values of adjustment variables $A$ and confounders $C$ ; namely $\\tau ( ( C ^ { \\mathsf { \\tau } } , A ^ { \\mathsf { \\tau } } ) ^ { \\mathsf { \\tau } } ) = ( \\sin { ( ( C ^ { \\mathsf { \\tau } } , A ^ { \\mathsf { \\tau } } ) ^ { \\mathsf { \\tau } } \\times b _ { \\tau } ) } ) ^ { 2 } .$ where $b _ { \\tau }$ represents weights for every covariate in the function, which is generated by uniform $( 0 , 1 )$ . The variable treatment effect implies that its strength differs among the units and is therefore conditioned on $C$ and $A$ . The function $g ( ( C ^ { \\mathsf { T } } , A ^ { \\mathsf { T } } ) ^ { \\mathsf { T } } )$ can have an influence on outcome regardless of treatment assignment. It is calculated via a trigonometric function to make the covariates nonlinear, which is defined as $g ( ( C ^ { \\mathsf { T } } , A ^ { \\mathsf { T } } ) ^ { \\mathsf { T } } ) = ( \\cos { ( ( C ^ { \\mathsf { T } } , A ^ { \\mathsf { T } } ) ^ { \\mathsf { T } } \\times b _ { g } ) } ) ^ { 2 }$ . Here, $b _ { g }$ represents a weight for each covariate in this function, which is generated by uniform $( 0 , 1 )$ . The bias is attributed to unobserved covariates which follow a random normal distribution $N ( 0 , 1 )$ . The treatment assignment $T$ follows the Bernoulli distribution, i.e., $T \\stackrel { i n d . } { \\sim }$ . Bernoulli $( e _ { 0 } ( ( C ^ { \\boldsymbol { \\mathsf { T } } } , Z ^ { \\boldsymbol { \\mathsf { T } } } ) ^ { \\boldsymbol { \\mathsf { T } } } ) $ ) with probability $\\begin{array} { r } { e _ { 0 } \\bigl ( ( C ^ { \\mathsf { T } } , Z ^ { \\mathsf { T } } ) ^ { \\mathsf { T } } \\bigr ) = \\Phi ( \\frac { a - \\mu ( a ) } { \\sigma ( a ) } ) } \\end{array}$ , where $e _ { 0 } { \\big ( } ( C ^ { \\tau } , Z ^ { \\tau } ) ^ { \\tau } { \\big ) }$ represents the propensity score, which is the cumulative distribution function for a standard normal random variable based on confounders $C$ and instrumental variables $Z$ , i.e., $a = \\sin \\left( ( C ^ { \\mathsf { \\tau } } , Z ^ { \\mathsf { \\tau } } ) ^ { \\mathsf { \\tau } } \\times b _ { a } \\right)$ , where $b _ { a }$ is generated by uniform $( 0 , 1 )$ . ",
+        "bbox": [
+            173,
+            779,
+            825,
+            925
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            101,
+            825,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We totally simulate five different data sources with five different multivariate normal distributions to represent the incrementally available observational data. In each data source, we randomly draw 10000 samples including treatment units and control units. Therefore, for five datasets, they have different selection bias, magnitude of covariates, covariance matrices for variables, and number of treatment and control units. To ensure a robust estimation of model performance, for each data source, we repeat the simulation procedure 10 times and obtain 10 synthetic datasets. ",
+        "bbox": [
+            173,
+            155,
+            825,
+            239
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B ALGORITHM 1 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            258,
+            328,
+            275
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Continual Causal Effect Representation Learning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            297,
+            586,
+            313
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Data: Given $d$ incrementally available observational data from $D _ { 1 }$ to $D _ { d }$   \nif $\\{ x , y , t \\} \\in D _ { 1 }$ then $^ { \\ast \\ast \\ast }$ Train baseline causal effect model $h _ { \\theta _ { 1 } } ( g _ { w _ { 1 } } )$ \\*\\*\\* $w _ { 1 } , \\theta _ { 1 } = \\mathrm { O P T I M I Z E } ( L _ { Y } + \\alpha W a s s ( P , Q ) + \\lambda L _ { w _ { 1 } } )$ $R _ { 1 } = \\{ g _ { w _ { 1 } } ( x ) | x \\in D _ { 1 } \\}$ $M _ { 1 } = { \\mathrm { H E R D I N G } } \\{ R _ { 1 } , Y _ { 1 } , T _ { 1 } \\}$   \nelse for $\\{ x , y , t \\} \\in D _ { 2 } , . . . , D _ { d }$ do $^ { \\ast } { ^ { \\ast } } \\ast \\ast$ Train continual causal effect model $h _ { \\theta _ { d } } ( g _ { w _ { d } } )$ \\*\\*\\* $w _ { d } , \\theta _ { d } , \\phi _ { d - 1  d } = 0 \\mathrm { P T I M I Z E } ( L _ { G } + \\alpha \\bar { W } a s s ( P , Q ) + \\lambda L _ { w _ { 2 } } + \\beta L _ { F D } + \\delta L _ { F T } )$ $\\tilde { R } _ { d - 1 } = \\phi _ { d - 1 \\to d } ( R _ { d - 1 } )$ $R _ { d } = \\{ g _ { w _ { d } } ( x ) | x \\in D _ { d } \\}$ $M _ { d } = \\overset { \\sim } { \\operatorname { H E R D I N G } } \\big ( \\{ R _ { d } , Y _ { d } , T _ { d } \\} \\cup \\{ \\tilde { R } _ { d - 1 } , Y _ { d - 1 } \\in M _ { d - 1 } , T _ { d - 1 } \\in M _ { d - 1 } \\} \\big )$ end   \nend ",
+        "bbox": [
+            171,
+            315,
+            767,
+            534
+        ],
+        "page_idx": 12
+    }
+]

parse/train/SklTQCNtvS/SklTQCNtvS.md

@@ -0,0 +1,414 @@
+# SIGN-OPT: A QUERY-EFFICIENT HARD-LABEL ADVERSARIAL ATTACK
+
+Minhao Cheng1\*, Simranjit Singh1∗, Patrick Chen1, Pin-Yu Chen2, Sijia Liu2, Cho-Jui Hsieh1   
+1Department of Computer Science, UCLA, 2IBM Research   
+{mhcheng, simranjit, patrickchen, chohsieh}@cs.ucla.edu   
+{pin-yu.chen, sijia.liu}@ibm.com
+
+# ABSTRACT
+
+We study the most practical problem setup for evaluating adversarial robustness of a machine learning system with limited access: the hard-label black-box attack setting for generating adversarial examples, where limited model queries are allowed and only the decision is provided to a queried data input. Several algorithms have been proposed for this problem but they typically require huge amount $( > 2 0 , 0 0 0 )$ of queries for attacking one example. Among them, one of the state-of-the-art approaches (Cheng et al., 2019) showed that hard-label attack can be modeled as an optimization problem where the objective function can be evaluated by binary search with additional model queries, thereby a zeroth order optimization algorithm can be applied. In this paper, we adopt the same optimization formulation but propose to directly estimate the sign of gradient at any direction instead of the gradient itself, which enjoys the benefit of single query. Using this single query oracle for retrieving sign of directional derivative, we develop a novel query-efficient Sign-OPT approach for hard-label black-box attack. We provide a convergence analysis of the new algorithm and conduct experiments on several models on MNIST, CIFAR-10 and ImageNet. We find that Sign-OPT attack consistently requires $5 \times$ to $1 0 \times$ fewer queries when compared to the current state-of-the-art approaches, and usually converges to an adversarial example with smaller perturbation.
+
+# 1 INTRODUCTION
+
+It has been shown that neural networks are vulnerable to adversarial examples (Szegedy et al., 2016; Goodfellow et al., 2015; Carlini & Wagner, 2017; Athalye et al., 2018). Given a victim neural network model and a correctly classified example, an adversarial attack aims to compute a small perturbation such that with this perturbation added, the original example will be misclassified. Many adversarial attacks have been proposed in the literature. Most of them consider the white-box setting, where the attacker has full knowledge about the victim model, and thus gradient based optimization can be used for attack. Popular Examples include C&W (Carlini & Wagner, 2017) and PGD (Madry et al., 2017) attacks. On the other hand, some more recent attacks have considered the probability black-box setting where the attacker does not know the victim model’s structure and weights, but can iteratively query the model and get the corresponding probability output. In this setting, although gradient (of output probability to the input layer) is not computable, it can still be estimated using finite differences, and algorithms many attacks are based on this (Chen et al., 2017; Ilyas et al., 2018a; Tu et al., 2019; Jun et al., 2018).
+
+In this paper, we consider the most challenging and practical attack setting – hard-label black-box setting – where the model is hidden to the attacker and the attacker can only make queries and get the corresponding hard-label decisions (e.g., predicted labels) of the model. A commonly used algorithm proposed in this setting, also called Boundary attack (Brendel et al., 2017), is based on random walks on the decision surface, but it does not have any convergence guarantee. More recently, Cheng et al. (2019) showed that finding the minimum adversarial perturbation in the hard-label setting can be reformulated as another optimization problem (we call this Cheng’s formulation in this paper). This new formulation enjoys the benefit of having a smooth boundary in most tasks and the function value is computable using hard-label queries. Therefore, the authors of (Cheng et al., 2019) are able to use standard zeroth order optimization to solve the new formulation. Although their algorithm converges quickly, it still requires large number of queries (e.g., 20,000) for attacking a single image since every function evaluation of Cheng’s formulation has to be computed using binary search requiring tens of queries.
+
+In this paper, we follow the same optimization formulation of (Cheng et al., 2019) which has the advantage of smoothness, but instead of using finite differences to estimate the magnitude of directional derivative, we propose to evaluate its sign using only a single query. With this single-query sign oracle, we design novel algorithms for solving the Cheng’s formulation, and we theoretically prove and empirically demonstrate the significant reduction in the number of queries required for hard-label black box attack.
+
+Our contribution are summarized below:
+
+• Novelty in terms of adversarial attack. We elucidate an efficient approach to compute the sign of directional derivative of Cheng’s formulation using a single query, and based on this technique we develop a novel optimization algorithm called Sign-OPT for hard-label black-box attack. Novelty in terms of optimization. Our method can be viewed as a new zeroth order optimization algorithm that features fast convergence of signSGD. Instead of directly taking the sign of gradient estimation, our algorithm utilizes the scale of random direction. This make existing analysis inappropriate to our case, and we provide a new recipe to prove the convergence of this new optimizer. We conduct comprehensive experiments on several datasets and models. We show that the proposed algorithm consistently reduces the query count by 5–10 times across different models and datasets, suggesting a practical and query-efficient robustness evaluation tool. Furthermore, on most datasets our algorithm can find an adversarial example with smaller distortion compared with previous approaches.
+
+# 2 RELATED WORK
+
+White-box attack Since it was firstly found that neural networks are easy to be fooled by adversarial examples (Goodfellow et al., 2015), a lot of work has been proposed in the white-box attack setting, where the classifier $f$ is completely exposed to the attacker. For neural networks, under this assumption, back-propagation can be conducted on the target model because both network structure and weights are known by the attacker. Algorithms including (Goodfellow et al., 2015; Kurakin et al., 2016; Carlini & Wagner, 2017; Chen et al., 2018; Madry et al., 2017) are then proposed based on gradient computation. Recently, the BPDA attack introduced by Athalye et al. (2018) bypasses some models with obfuscated gradients and is shown to successfully circumvent many defenses. In addition to typical attacks based on small $\ell _ { p }$ norm perturbation, non- $\ell _ { p }$ norm perturbations such as scaling or shifting have also been considered (Zhang et al., 2019).
+
+Black-box attack Recently, black-box setting is drawing rapidly increasing attention. In black-box setting, the attacker can query the model but has no (direct) access to any internal information inside the model. Although there are some works based on transfer attack (Papernot et al., 2017), we consider the query-based attack in the paper. Depending on the model’s feedback for a given query, an attack can be classified as a soft-label or hard-label attack. In the soft-label setting, the model outputs a probability score for each decision. Chen et al. (2017) uses a finite difference in a coordinate-wise manner to approximately estimate the output probability changes and does a coordinate descent to conduct the attack. Ilyas et al. (2018a) uses Neural evolution strategy (NES) to approximately estimate the gradient directly. Later, some variants (Ilyas et al., 2018b; Tu et al., 2019) were proposed to utilize the side information to further speed up the attack procedure. Alzantot et al. (2019) uses a evolutionary algorithm as a black-box optimizer for the soft-label setting. Recently, Al-Dujaili & O’Reilly (2019) proposes SignHunter algorithm based on signSGD (Bernstein et al., 2018) to achieve faster convergence in the soft-label setting. The recent work (Al-Dujaili & O’Reilly, 2019) proposes SignHunter algorithm to achieve a more query-efficent sign estimate when crafting black-box adversarial examples through soft-label information.
+
+In the hard-label case, only the final decision, i.e. the top-1 predicted class, is observed. As a result, the attacker can only make queries to acquire the corresponding hard-label decision instead of the probability outputs. Brendel et al. (2017) first studied this problem and proposed an algorithm based on random walks near the decision boundary. By selecting a random direction and projecting it onto a boundary sphere in each iteration, it aims to generate a high-quality adversarial example. Query-Limited attack (Ilyas et al., 2018a) tries to estimate the output probability scores with model query and turn the hard-label into a soft-label problem. Cheng et al. (2019) instead re-formalizes the hard-label attack into an optimization problem that finds a direction which could produce the shortest distance to decision boundary.
+
+The recent arXiv paper (Chen et al., 2019) applied the zeroth-order sign oracle to improve Boundary attack, and also demonstrated significant improvement. The major differences to our algorithm are that we propose a new zeroth-order gradient descent algorithm, provide its algorithmic convergence guarantees, and aim to improve the query complexity of the attack formulation proposed in (Cheng et al., 2019). For completeness, we also compare with this method in Section A.1. Moreover, (Chen et al., 2019) uses one-point gradient estimate, which is unbiased but may encounter larger variance compared with the gradient estimate in our paper. Thus, we can observe in Section A.1 that although they are slightly faster in the initial stage, Sign-OPT will catch up and eventually lead to a slightly better solution.
+
+# 3 PROPOSED METHOD
+
+We follow the same formulation in (Cheng et al., 2019) and consider the hard-label attack as the problem of finding the direction with shortest distance to the decision boundary. Specifically, for a given example $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ , true label $y _ { 0 }$ and the hard-label black-box function $f : \mathbb { R } ^ { d }  \mathbf { \bar { \{ 1 , \dots , } }  \dot { K } \}$ , the objective function $g : \mathbb { R } ^ { d }  \mathbb { R }$ (for the untargeted attack) can be written as:
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } g ( \pmb { \theta } ) \ \mathrm { w h e r e } \ g ( \pmb { \theta } ) = \arg \operatorname* { m i n } _ { \lambda > 0 } \left( f ( x _ { 0 } + \lambda \frac { \pmb \theta } { \lVert \pmb \theta \rVert } ) \neq y _ { 0 } \right) .
+$$
+
+It has been shown that this objective function is usually smooth and the objective function $g$ can be evaluated by a binary search procedure locally. At each binary search step, we query the function $f ( x _ { 0 } + \lambda \frac { \tilde { \theta } } { | | \theta | | } )$ and determine whether the distance to decision boundary in the direction $\pmb \theta$ is greater or smaller than $\lambda$ based on the hard-label prediction1.
+
+As the objective function is computable, the directional derivative of $g$ can be estimated by finite differences:
+
+$$
+{ \hat { \nabla } } g ( \pmb { \theta } ; \mathbf { u } ) : = \frac { g ( \pmb { \theta } + \epsilon \mathbf { u } ) - g ( \pmb { \theta } ) } { \epsilon } \mathbf { u }
+$$
+
+where $\textbf { \em u }$ is a random Gaussian vector and $\epsilon \mathrm { ~  ~ { ~ \gamma ~ } ~ } > \mathrm { ~  ~ { ~ 0 ~ } ~ }$ is a very small smoothing parameter. This is a standard zeroth order oracle for estimating directional derivative and based on this we can apply many different zeroth order optimization algorithms to minimize $g$
+
+For example, Cheng et al. (2019) used the Random Derivative Free algorithm Nesterov & Spokoiny (2017) to solve problem (1). However, each computation of (2) requires many hard-label queries due to binary search, so Cheng et al. (2019) still requires a huge number of queries despite having fast convergence.
+
+In this work, we introduce an algorithm that hugely improves the query complexity over Cheng et al. (2019). Our algorithm is based on the following key ideas: (i) one does not need very accurate values of directional derivative in order to make the algorithm converge, and (ii) there exists an imperfect but informative estimation of directional derivative of $g$ that can be computed by a single query.
+
+![](images/e2e24a8d0882e69441defcaeee7cf390d1accda886d80d77ebb1c754cfa01bca.jpg)  
+Figure 1: Illustration
+
+# Algorithm 1: Sign-OPT attack
+
+<table><tr><td>Input: Hard-label model f,original image xo,initial 0o ;</td><td></td></tr><tr><td>for t=1,2,...,Tdo Randomly sample u1,...,uQ from a Gaussian or Uniform distribution;</td><td></td></tr><tr><td></td><td></td></tr><tr><td>Computeg ←∑q=1sign(g(0t + εuq)-g(0t)) ·uq;</td><td></td></tr><tr><td>Update0t+1←0t-ng ;</td><td></td></tr><tr><td></td><td>Evaluate g(0t） using the same search algorithm in Cheng et al. (2019) ;</td></tr></table>
+
+# 3.1 A SINGLE QUERY ORACLE
+
+As mentioned before, the previous approach requires com  
+puting $g ( \pmb \theta + \epsilon \pmb u ) - g ( \pmb \theta )$ which consumes a lot of queries. However, based on the definition of $g ( \cdot )$ , we can compute the sign of this value $\mathrm { s i g n } ( g ( \pmb { \theta } + \epsilon \mathbf { \bar { u } } ) - g ( \pmb { \theta } ) )$ using a single query. Considering the untargeted attack case, the sign can be computed by
+
+$$
+\mathrm { s i g n } ( g ( \pmb \theta + \epsilon \mathbf u ) - g ( \pmb \theta ) ) = \left\{ \begin{array} { l l } { + 1 , } & { f ( x _ { 0 } + g ( \pmb \theta ) \frac { ( \pmb \theta + \epsilon \mathbf u ) } { \| \pmb \theta + \epsilon \mathbf u \| } ) = y _ { 0 } , } \\ { - 1 , } & { \mathrm { O t h e r w i s e } . } \end{array} \right.
+$$
+
+This is illustrated in Figure 1. Essentially, for a new direction ${ \pmb \theta } + \epsilon { \bf u }$ , we test whether a point at the original distance $g ( \pmb \theta )$ from $x _ { 0 }$ in this direction lies inside or outside the decision boundary, i.e. if the produced perturbation will result in a wrong prediction by classifier. If the produced perturbation is outside the boundary i.e. $\begin{array} { r } { f ( x _ { 0 } + g ( \pmb { \theta } ) \frac { ( \pmb { \theta } + \epsilon \mathbf { u } ) } { \lVert \pmb { \theta } + \epsilon \mathbf { u } \rVert } ) \neq y _ { 0 } } \end{array}$ , the new direction has a smaller distance to decision boundary, and thus giving a smaller value of $g$ . It indicates that $\mathbf { u }$ is a descent direction to minimize $g$ .
+
+# 3.2 SIGN-OPT ATTACK
+
+By sampling random Gaussian vector $Q$ times, we can estimate the imperfect gradient by
+
+$$
+\hat { \nabla } g ( \pmb \theta ) \approx \hat { \pmb g } : = \sum _ { q = 1 } ^ { Q } \mathrm { s i g n } ( g ( \pmb \theta + \epsilon \mathbf { u } _ { q } ) - g ( \pmb \theta ) ) \mathbf { u } _ { q } ,
+$$
+
+which only requires $Q$ queries. We then use this imperfect gradient estimate to update our search direction $\pmb { \theta }$ as $\pmb { \theta }  \pmb { \theta } - \eta \hat { \mathbf { g } }$ with a step size $\eta$ and use the same search procedure to compute $g ( \pmb \theta )$ up to a certain accuracy. The detailed procedure is shown in Algorithm 1.
+
+We note that Liu et al. (2019) designed a Zattack (not hard-label setting). They use $\begin{array} { r } { \hat { \nabla } g ( \pmb { \theta } ) \approx \hat { \pmb { g } } : = \sum _ { q = 1 } ^ { Q } \mathrm { s i g n } ( g ( \pmb { \theta } + \epsilon \mathbf { u } _ { q } ) - g ( \pmb { \theta } ) \mathbf { u } _ { q } ) } \end{array}$ box and shows that it could achieve a comparable or even better convergence rate than zeroth order stochastic gradient descent by using only sign information of gradient estimation. Although it is possible to combine ZO-SignSGD with our proposed single query oracle for solving hard-label attack, their estimator will take sign of the whole vector and thus ignore the direction of $\mathbf { u } _ { q }$ , which leads to slower convergence in practice (please refer to Section 4.4 and Figure 5(b) for more details).
+
+To the best of our knowledge, no previous analysis can be used to prove convergence of Algorithm 1. In the following, we show that Algorithm 1 can in fact converge and furthermore, with similar convergence rate compared with (Liu et al., 2019) despite using a different gradient estimator.
+
+Assumption 1. Function $g ( \theta )$ is $L$ -smooth with a finite value of L.
+
+Assumption 2. At any iteration step t, the gradient of the function $g$ is upper bounded by $\lVert \nabla g ( \pmb { \theta } _ { t } ) \rVert _ { 2 } \leq \sigma$ .
+
+Theorem 3.1. Suppose that the conditions in the assumptions hold, and the distribution of gradient noise is unimodal and symmetric. Then, Sign-OPT attack with learning rate $\begin{array} { r } { \eta _ { t } = O \big ( \frac { - 1 } { Q \sqrt { d T } } \big ) } \end{array}$ and $\begin{array} { r } { \epsilon = O ( \frac { 1 } { d T } ) } \end{array}$ will give following bound on $\mathbb { E } [ \| \nabla g ( \pmb { \theta } ) \| _ { 2 } ]$ :
+
+$$
+\mathbb { E } [ \| \nabla g ( \pmb { \theta } ) \| _ { 2 } ] = O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { d } { \sqrt { Q } } ) .
+$$
+
+The proof can be found in subsection A.2. The main difference with the original analysis provided by Liu et al. (2019) is that they only only deal with sign of each element, while our analysis also takes the magnitudes of each element of $\pmb { u } _ { q }$ into account.
+
+# 3.3 OTHER GRADIENT ESTIMATIONS
+
+Note that the value $\operatorname { s i g n } ( g ( \pmb \theta + \epsilon \pmb u ) - g ( \pmb \theta ) )$ computed by our single query oracle is actually the sign of the directional derivative:
+
+$$
+\operatorname { s i g n } ( \langle \nabla g ( \theta ) , u \rangle ) = \operatorname { s i g n } ( \operatorname* { l i m } _ { \epsilon \to \infty } \frac { g ( \theta + \epsilon u ) - g ( \theta ) } { \epsilon } ) = \operatorname { s i g n } ( g ( \theta + \epsilon u ) - g ( \theta ) )
+$$
+
+Therefore, we can use this information to estimate the original gradient. The Sign-OPT approach in the previous section uses $\begin{array} { r l } { \sum _ { q } \mathrm { s i g n } ( \langle \nabla g ( \pmb { \theta } ) , \pmb { u } _ { q } \rangle ) \pmb { u } _ { q } } \end{array}$ as an estimation of gradient. Let $y _ { q } : =$ $\operatorname { s i g n } ( \langle \nabla g ( \pmb \theta ) , \pmb u _ { q } \rangle )$ , a more accurate gradient estimation can be cast as the following constraint optimization problem:
+
+Find a vector $_ { z }$ such that $\operatorname { s i g n } ( \langle z , \mathbf { u } _ { q } \rangle ) = y _ { q } \forall q = 1 , \ldots , Q .$
+
+Therefore, this is equivalent to a hard constraint SVM problem where each $\pmb { u } _ { q }$ is a training sample and $y _ { q }$ is the corresponding label. The gradient can then be recovered by solving the following quadratic programming problem:
+
+$$
+\operatorname* { m i n } _ { z } \ z ^ { T } z \ \mathrm { ~ s . t . ~ } \ z ^ { T } { \pmb u } _ { q } \geq y _ { q } , \ \forall q = 1 , \ldots , Q .
+$$
+
+By solving this problem, we can get a good estimation of the gradient. As explained earlier, each $y _ { q }$ can be determined with a single query. Therefore, we propose a variant of Sign-OPT, which is called SVM-OPT attack. The detailed procedure is shown in Algorithm 2. We will present an empirical comparison of our two algorithms in subsection 4.1.
+
+# Algorithm 2: SVM-OPT attack
+
+<table><tr><td>Input: Hard-label model f,original image xo,initial 0o ; fort=1,2,...,Tdo</td></tr><tr><td>Sample u1,...,uq from Gaussian or orthogonal basis ; Solve z defined by (5) ;</td></tr><tr><td>Update0t+1 ←0t-ηz； Evaluate g(0t） using search algorithm in (Cheng et al., 2019) ;</td></tr></table>
+
+# 4 EXPERIMENTAL RESULTS
+
+We evaluate the SIGN-OPT algorithm for attacking black-box models in a hard-label setting on three different standard datasets - MNIST (LeCun et al., 1998), CIFAR-10 (Krizhevsky et al.) and ImageNet-1000 (Deng et al., 2009) and compare it with existing methods. For fair and easy comparison, we use the CNN networks provided by (Carlini & Wagner, 2017), which have also been used by other previous hard-label attacks as well. Specifically, for both MNIST and CIFAR-10, the model consists of nine layers in total - four convolutional layers, two max-pooling layers and two fully-connected layers. Further details about implementation, training and parameters are available on (Carlini & Wagner, 2017). As reported in (Carlini & Wagner, 2017) and (Cheng et al., 2019), we were able to achieve an accuracy of $9 9 . 5 \%$ on MNIST and $8 2 . 5 \%$ on CIFAR-10. We use the pretrained Resnet-50 (He et al., 2016) network provided by torchvision (Marcel & Rodriguez, 2010) for ImageNet-1000, which achieves a Top-1 accuracy of $7 6 . 1 5 \%$ .
+
+In our experiments, we found that Sign-OPT and SVM-OPT perform quite similarly in terms of query efficiency. Hence we compare only Sign-OPT attack with previous approaches and provide a comparison between Sign-OPT and SVM-OPT in subsection 4.1. We compare the following attacks:
+
+• Sign-OPT attack (black box): The approach presented in this paper. • Opt-based attack (black box): The method proposed in Cheng et al. (2019) where they use Randomized Gradient-Free method to optimize the same objective function. We use the implementation provided at https://github.com/LeMinhThong/blackbox-attack.
+
+![](images/6adc2699a9c34120996184bdd226bbc5005dbaed5494821c2793f5d8ecaadeb8.jpg)  
+Figure 2: Example of Sign-OPT targeted attack. $L _ { 2 }$ distortions and queries used are shown above and below the images. First two rows: Example comparison of Sign-OPT attack and OPT attack. Third and fourth rows: Examples of Sign-OPT attack on CIFAR-10 and ImageNet
+
+• Boundary attack (black box): The method proposed in Brendel et al. (2017). This is compared only in $L _ { 2 }$ setting as it is designed for the same. We use the implementation provided in Foolbox (https://github.com/bethgelab/foolbox).
+
+• Guessing Smart Attack (black box): The method proposed in (Brunner et al., 2018). This attack enhances boundary attack by biasing sampling towards three priors. Note that one of the priors assumes access to a similar model as the target model and for a fair comparison we do not incorporate this bias in our experiments. We use the implementation provided at https://github.com/ttbrunner/biased_boundary_attack.
+
+• C&W attack (white box): One of the most popular methods in the white-box setting proposed in Carlini & Wagner (2017). We use C&W $L _ { 2 }$ norm attack as a baseline for the white-box attack performance.
+
+For each attack, we randomly sample 100 examples from validation set and generate adversarial perturbations for them. For untargeted attack, we only consider examples that are correctly predicted by model and for targeted attack, we consider examples that are already not predicted as target label by the model. To compare different methods, we mainly use median distortion as the metric. Median distortion for $x$ queries is the median adversarial perturbation of all examples achieved by a method using less than $x$ queries. Since all the hard-label attack algorithms will start from an adversarial exmample and keep reduce the distortion, if we stop at any time they will always give an adversarial example and medium distortion will be the most suitable metric to compare their performance. Besides, we also show success rate (SR) for $x$ queries for a given threshold (), which is the percentage of number of examples that have achieved an adversarial perturbation below $\epsilon$ with less than $x$ queries. We evaluate success rate on different thresholds which depend on the dataset being used. For comparison of different algorithms in each setting, we chose the same set of examples across all attacks.
+
+Implementation details: To optimize algorithm 1, we estimate the step size $\eta$ using the same line search procedure implemented in Cheng et al. (2019). At the cost of a relatively small number of queries, this provides significant speedup in the optimization. Similar to Cheng et al. (2019), $g ( \theta )$ in last step of algorithm 1 is approximated via binary search. The initial $\theta _ { 0 }$ in algorithm 1 is calculated by evaluating $g ( \theta )$ on 100 random directions and taking the best one. We provide our implementation publicly2.
+
+![](images/7d598643ee79cd36d606570d9be113a92462e1df62bc0750c09bf38866c17ab8.jpg)  
+Figure 3: Median $L _ { 2 }$ distortion vs Queries. First two: Comparison of Sign-OPT and SVM-OPT attack for MNIST and CIFAR-10. Third: Performance of Sign-OPT for different values of $Q$ .
+
+# 4.1 COMPARISON BETWEEN SIGN-OPT AND SVM-OPT
+
+In our experiments, we found that the performance in terms of queries of both these attacks is remarkably similar in all settings (both $L _ { \mathrm { 2 } } / L _ { \infty }$ & Targeted/Untargeted) and datasets. We present a comparison for MNIST and CIFAR-10 ( $L _ { 2 }$ norm-based) for both targeted and untargeted attacks in Figure 3. We see that the median distortion achieved for a given number of queries is quite on part for both Sign-OPT and SVM-OPT.
+
+Number of queries per gradient estimate: In Figure 3, we show the comparison of Sign-OPT attack with different values of $Q$ . Our experiments suggest that $Q$ does not have an impact on the convergence point reached by the algorithm. Although, small values of $Q$ provide a noisy gradient estimate and hence delayed convergence to an adversarial perturbation. Large values of $Q$ , on the other hand, require large amount of time per gradient estimate. After fine tuning on a small set of examples, we found that $Q = 2 0 0$ provides a good balance between the two. Hence, we set the value of $Q = 2 0 0$ for all our experiments in this section.
+
+# 4.2 UNTARGETED ATTACK
+
+In this attack, the objective is to generate an adversary from an original image for which the prediction by model is different from that of original image. Figure 4 provides an elaborate comparison of different attacks for $L _ { 2 }$ case for the three datasets. Sign-OPT attack consistently outperforms the current approaches in terms of queries. Not only is Sign-OPT more efficient in terms of queries, in most cases it converges to a lower distortion than what is possible by other hard-label attacks. Furthermore, we observe Sign-OPT converges to a solution comparable with C&W white-box attack (better on CIFAR-10, worse on MNIST, comparable on ImageNet). This is significant for a hard-label attack algorithm since we are given very limited information.
+
+We highlight some of the comparisons of Boundary attack, OPT-based attack and Sign-OPT attack $L _ { 2 }$ norm-based) in Table 1. Particularly for ImageNet dataset on ResNet-50 model, Sign-OPT attack reaches a median distortion below 3.0 in less than $3 0 k$ queries while other attacks need more than $2 0 0 k$ queries for the same.
+
+# 4.3 TARGETED ATTACK
+
+In targeted attack, the goal is to generate an adversarial perturbation for an image so that the prediction of resulting image is the same as a specified target. For each example, we randomly specify the target label, keeping it consistent across different attacks. We calculate the initial $\theta _ { 0 }$ in algorithm 1 using 100 samples in target label class from training dataset and this $\theta _ { 0 }$ is the same across different attacks. Figure 2 shows some examples of adversarial examples generated by Sign-OPT attack and the Opt-based attack. The first two rows show comparison of Sign-OPT and Opt attack respectively on an example from MNIST dataset. The figures show adversarial examples generated at almost same number of queries for both attacks. Sign-OPT method generates an $L _ { 2 }$ adversarial perturbation of $0 . 9 4 \mathrm { i n } \sim 6 k$ queries for this particular example while Opt-based attack requires $\sim 3 5 k$ for the same. Figure 5 displays a comparison among different attacks in targeted setting. In our experiments, average distortion achieved by white box attack C&W for MNIST dataset is 1.51, for which Sign-OPT requires $\sim 1 2 k$ queries while others need $> 1 2 0 k$ queries. We present a comparison of success rate of different attacks for CIFAR-10 dataset in Figure 6 for both targeted and untargeted cases.
+
+![](images/c77af3f7f932e2c10526900a3c551fd30c7b4a47018903a30f58cc5f856c18ec.jpg)  
+Figure 4: Untargeted attack: Median distortion vs Queries for different datasets.
+
+![](images/7981157c333e149995ee82c4d63afc30d48232891e24b5d3636cd0096328e122.jpg)  
+Figure 5: (a) Targeted Attack: Median distortion vs Queries of different attacks on MNIST and CIFAR-10. (b) Comparing Sign-OPT and ZO-SignSGD with and without single query oracle (SQO).
+
+# 4.4 THE POWER OF SINGLE QUERY ORACLE
+
+In this subsection, we conduct several experiments to prove the effectiveness of our proposed single query oracle in hard-label adversarial attack setting. ZO-SignSGD algorithm (Liu et al., 2019) is proposed for soft-label black box attack and we extend it into hard-label setting. A straightforward way is simply applying ZO-SignSGD to solve the hard-label objective proposed in Cheng et al. (2019), estimate the gradient using binary search as (Cheng et al., 2019) and take its sign. In Figure 5(b), we clearly observe that simply combining ZO-SignSGD and Cheng et al. (2019) is not efficient. With the proposed single query sign oracle, we can also reduce the query count of this method, as demonstrated in Figure 5(b). This verifies the effectiveness of single query oracle, which can universally improve many different optimization methods in the hard-label attack setting. To be noted, there is still improvement on Sign-OPT over ZO-SignSGD with single query oracle because instead of directly taking the sign of gradient estimation, our algorithm utilizes the scale of random direction $u$ as well. In other words, signSGD’s gradient norm is always 1 while our gradient norm takes into account the magnitude of $u$ . Therefore, our signOPT optimization algorithm is fundamentally different (Liu et al., 2019) or any other proposed signSGD varieties. Our method can be viewed as a new zeroth order optimization algorithm that features fast convergence in signSGD.
+
+![](images/0945c58d97bf988454efbbf57040f74f0d4abb417b4d99256220231da9ffe5c0.jpg)  
+Figure 6: Success Rate vs Queries for CIFAR-10 ( $L _ { 2 }$ norm-based attack). First two and last two depict untargeted and targeted attacks respectively. Success rate threshold is at the top of each plot.
+
+Table 1: $L _ { 2 }$ Untargeted attack - Comparison of average $L _ { 2 }$ distortion achieved using a given number of queries for different attacks. SR stands for success rate.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">MNIST</td><td colspan="3">CIFAR10</td><td colspan="3">ImageNet (ResNet-50)</td></tr><tr><td>#Queries</td><td>AvgL2</td><td>SR(e=1.5)</td><td>#Queries</td><td>AvgL2</td><td>SR(∈=0.5)|</td><td>#Queries</td><td>AvgL2</td><td>SR(∈=3.0)</td></tr><tr><td rowspan="2">Boundary attack</td><td>4.000 8.000</td><td>4.24 4.24</td><td>1.0% 1.0%</td><td>4,000 8.000</td><td>3.12 2.84</td><td>2.3% 7.6%</td><td>4,000 30,000</td><td>209.63 17.40</td><td>0% 16.6%</td></tr><tr><td></td><td>2.13</td><td>16.3%</td><td>12.000</td><td>0.78</td><td>29.2%</td><td>160,000</td><td>4.62</td><td>41.6%</td></tr><tr><td rowspan="2">OPT attack</td><td>14,000</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>4,000</td><td>3.65</td><td>3.0%</td><td>4,000</td><td>0.77</td><td>37.0%</td><td>4,000</td><td>83.85</td><td>2.0%</td></tr><tr><td rowspan="2"></td><td>8.000</td><td>2.41</td><td>18.0%</td><td>8.000 12.000</td><td>0.43</td><td>53.0%</td><td>30.000</td><td>16.77</td><td>14.0%</td></tr><tr><td>14,000</td><td>1.76</td><td>36.0%</td><td></td><td>0.33</td><td>61.0%</td><td>160,000</td><td>4.27</td><td>34.0%</td></tr><tr><td rowspan="2">Guessing Smart</td><td>4,000</td><td>1.74</td><td>41.0%</td><td>4,000</td><td>0.29</td><td>75.0%</td><td>4,000</td><td>16.69</td><td>12.0%</td></tr><tr><td>8.000</td><td>1.69</td><td>42.0%</td><td>8.000</td><td>0.25</td><td>80.0%</td><td>30,000</td><td>13.27</td><td>12.0%</td></tr><tr><td rowspan="2">Sign-OPT attack</td><td>14,000</td><td>1.68</td><td>43.0%</td><td>12.000</td><td>0.24</td><td>80.0%</td><td>160,000</td><td>12.88</td><td>12.0%</td></tr><tr><td>4,000</td><td>1.54</td><td>46.0%</td><td>4,000 8.000</td><td>0.26</td><td>73.0%</td><td>4,000</td><td>23.19</td><td>8.0%</td></tr><tr><td rowspan="2">C&amp;W(white-box)</td><td>8.000</td><td>1.18</td><td>84.0% 94.0%</td><td>12.000</td><td>0.16</td><td>90.0%</td><td>30.000</td><td>2.99</td><td>50.0%</td></tr><tr><td>14,000 -</td><td>1.09 0.88</td><td>99.0%</td><td>-</td><td>0.13 0.25</td><td>95.0% 85.0%</td><td>160,000 1</td><td>1.21 1.51</td><td>90.0% 80.0%</td></tr></table>
+
+# 5 CONCLUSION
+
+We developed a new and ultra query-efficient algorithm for adversarial attack in the hard-label black-box setting. Using the same smooth reformulation in Cheng et al. (2019), we design a novel zeroth order oracle that can compute the sign of directional derivative of the attack objective using single query. Equipped with this single-query oracle, we design a new optimization algorithm that can dramatically reduce number of queries compared with Cheng et al. (2019). We prove the convergence of the proposed algorithm and show our new algorithm is overwhelmingly better than current hard-label black-box attacks.
+
+# ACKNOWLEDGEMENT
+
+This work is based upon work supported by the Department of Energy National Energy Technology Laboratory under Award Number DE-OE0000911 and by NSF under IIS1719097.
+
+# REFERENCES
+
+Abdullah Al-Dujaili and Una-May O’Reilly. There are no bit parts for sign bits in black-box attacks. arXiv preprint arXiv:1902.06894, 2019.
+
+Moustafa Alzantot, Yash Sharma, Supriyo Chakraborty, Huan Zhang, Cho-Jui Hsieh, and Mani B Srivastava. Genattack: Practical black-box attacks with gradient-free optimization. In Proceedings of the Genetic and Evolutionary Computation Conference, pp. 1111–1119, 2019.
+
+Anish Athalye, Nicholas Carlini, and David Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. In ICML, 2018.
+
+Jeremy Bernstein, Yu-Xiang Wang, Kamyar Azizzadenesheli, and Animashree Anandkumar. signSGD: Compressed optimisation for non-convex problems. In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 560–569, Stockholmsmssan, Stockholm Sweden, 10– 15 Jul 2018. PMLR. URL http://proceedings.mlr.press/v80/bernstein18a. html.
+
+Wieland Brendel, Jonas Rauber, and Matthias Bethge. Decision-based adversarial attacks: Reliable attacks against black-box machine learning models. arXiv preprint arXiv:1712.04248, 2017.
+
+Thomas Brunner, Frederik Diehl, Michael Truong Le, and Alois Knoll. Guessing smart: Biased sampling for efficient black-box adversarial attacks. arXiv preprint arXiv:1812.09803, 2018.
+
+Nicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In Security and Privacy (SP), 2017 IEEE Symposium on, pp. 39–57. IEEE, 2017.
+
+Jianbo Chen, Michael I. Jordan, and Martin J. Wainwright. Hopskipjumpattack: A query-efficient decision-based attack. arXiv preprint arXiv:1904.02144, 2019.
+
+Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh. Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security, pp. 15–26. ACM, 2017.
+
+Pin-Yu Chen, Yash Sharma, Huan Zhang, Jinfeng Yi, and Cho-Jui Hsieh. Ead: elastic-net attacks to deep neural networks via adversarial examples. In Thirty-second AAAI conference on artificial intelligence, 2018.
+
+Minhao Cheng, Thong Le, Pin-Yu Chen, Huan Zhang, JinFeng Yi, and Cho-Jui Hsieh. Queryefficient hard-label black-box attack: An optimization-based approach. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id= rJlk6iRqKX.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 248–255. IEEE, 2009.
+
+John C Duchi, Peter L Bartlett, and Martin J Wainwright. Randomized smoothing for stochastic optimization. SIAM Journal on Optimization, 22(2):674–701, 2012.
+
+Ian Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. In International Conference on Learning Representations, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Andrew Ilyas, Logan Engstrom, Anish Athalye, and Jessy Lin. Black-box adversarial attacks with limited queries and information. In International Conference on Machine Learning, pp. 2142–2151, 2018a.
+
+Andrew Ilyas, Logan Engstrom, and Aleksander Madry. Prior convictions: Black-box adversarial attacks with bandits and priors. arXiv preprint arXiv:1807.07978, 2018b.
+
+Kwang-Sung Jun, Lihong Li, Yuzhe Ma, and Jerry Zhu. Adversarial attacks on stochastic bandits. In Advances in Neural Information Processing Systems, pp. 3640–3649, 2018.
+
+Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. Cifar-10 (canadian institute for advanced research). URL http://www.cs.toronto.edu/˜kriz/cifar.html.
+
+Alexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial machine learning at scale. arXiv preprint arXiv:1611.01236, 2016.
+
+Yann LeCun, Leon Bottou, Yoshua Bengio, Patrick Haffner, et al. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Sijia Liu, Bhavya Kailkhura, Pin-Yu Chen, Paishun Ting, Shiyu Chang, and Lisa Amini. Zeroth-order stochastic variance reduction for nonconvex optimization. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 3727– 3737. Curran Associates, Inc., 2018. URL http://papers.nips.cc/paper/ 7630-zeroth-order-stochastic-variance-reduction-for-nonconvex-optimizatio pdf.
+
+Sijia Liu, Pin-Yu Chen, Xiangyi Chen, and Mingyi Hong. signSGD via zeroth-order oracle. In International Conference on Learning Representations, 2019. URL https://openreview. net/forum?id ${ . } = { }$ BJe-DsC5Fm.
+
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083, 2017.
+
+Sebastien Marcel and Yann Rodriguez. Torchvision the machine-vision package of torch. In ´ Proceedings of the 18th ACM International Conference on Multimedia, MM ’10, pp. 1485–1488, New York, NY, USA, 2010. ACM. ISBN 978-1-60558-933-6. doi: 10.1145/1873951.1874254. URL http://doi.acm.org/10.1145/1873951.1874254.
+
+Yurii Nesterov and Vladimir Spokoiny. Random gradient-free minimization of convex functions. Foundations of Computational Mathematics, 17(2):527–566, 2017.
+
+Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against machine learning. In Proceedings of the 2017 ACM on Asia Conference on Computer and Communications Security, pp. 506–519. ACM, 2017.
+
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2818–2826, 2016.
+
+Chun-Chen Tu, Paishun Ting, Pin-Yu Chen, Sijia Liu, Huan Zhang, Jinfeng Yi, Cho-Jui Hsieh, and Shin-Ming Cheng. Autozoom: Autoencoder-based zeroth order optimization method for attacking black-box neural networks. AAAI, 2019.
+
+Huan Zhang, Hongge Chen, Zhao Song, Duane Boning, Inderjit S Dhillon, and Cho-Jui Hsieh. The limitations of adversarial training and the blind-spot attack. arXiv preprint arXiv:1901.04684, 2019.
+
+# A APPENDIX
+
+# A.1 COMPARISON WITH HOPSKIPJUMPATTACK
+
+There is a recent paper (Chen et al., 2019) that applied the zeroth-order sign oracle to improve Boundary attack, and also demonstrated significant improvement. The major differences to our algorithm are that we propose a new zeroth-order gradient descent algorithm, provide its algorithmic convergence guarantees, and aim to improve the query complexity of the attack formulation proposed in (Cheng et al., 2019). To be noted, HopSkipJumpAttack only provides the bias and variance analysis (Theorem 2 and 3) without convergence rate analysis.
+
+Also, HopSkipJumpAttack uses one-point gradient estimate compared to the 2-point gradient estimate used by SignOPT. Therefore, although the estimation is unbiased, it has large variance, which achieves successful attack faster but generates a worse adversarial example with larger distortion than ours. For completeness, we also compare with this method (and mention the results) as follows.
+
+Figure 7 shows a comparison of Sign-OPT and HopSkipJumpAttack for CIFAR-10 and MNIST datasets for the case of $L _ { 2 }$ norm based attack. We find in our experiments that performance of both attacks is comparable in terms of queries consumed. In some cases, Sign-OPT converges to a better solution.
+
+![](images/386a464e969663492f95a03331f1486e5e9a48aa6d76c2d2f90266fc1322f088.jpg)  
+Figure 7: Comparison with HopSkipJumpAttack for CIFAR and MNIST: Median distortion vs Queries. (U) represents untargeted attack and (T) represents targeted attack.
+
+# A.2 PROOF
+
+Define following notations:
+
+$$
+\begin{array} { l } { \displaystyle \hat { \nabla } g ( \theta _ { t } ; u _ { q } ) : = \mathrm { s i g n } ( g ( \pmb \theta _ { t } + \epsilon \pmb u _ { q } ) - g ( \pmb \theta _ { t } ) ) \pmb u _ { q } } \\ { \displaystyle \dot { \nabla } g ( \pmb \theta _ { t } ; \pmb u _ { q } ) : = \cfrac { 1 } { \epsilon } ( g ( \pmb \theta _ { t } + \epsilon \pmb u _ { q } ) - g ( \pmb \theta _ { t } ) ) \pmb u _ { q } } \\ { \displaystyle \bar { \nabla } g ( \pmb \theta _ { t } ; \pmb u _ { q } ) : = \mathrm { s i g n } ( \cfrac { 1 } { \epsilon } ( g ( \pmb \theta _ { t } + \epsilon \pmb u _ { q } ) - g ( \pmb \theta _ { t } ) ) \pmb u _ { q } ) } \end{array}
+$$
+
+Thus we could write the corresponding estimate of gradients as follow:
+
+$$
+\begin{array} { l } { \displaystyle \dot { g } _ { t } = \frac { 1 } { Q } \sum _ { q = 1 } ^ { Q } \dot { \mathrm { s i g n } } ( g ( \theta _ { t } + \epsilon { \bf u } _ { q } ) - g ( \theta _ { t } ) ) u _ { q } = \frac { 1 } { Q } \sum _ { q = 1 } ^ { Q } \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) } \\ { \displaystyle \dot { g } _ { t } = \frac { 1 } { Q } \sum _ { q = 1 } ^ { Q } \frac { 1 } { \epsilon } ( g ( \theta _ { t } + \epsilon u _ { q } ) - g ( \theta _ { t } ) ) u _ { q } = \frac { 1 } { Q } \sum _ { q = 1 } ^ { Q } \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) } \\ { \displaystyle \ddot { g } _ { t } = \frac { 1 } { Q } \sum _ { q = 1 } ^ { Q } \dot { \mathrm { s i g n } } ( \frac { 1 } { \epsilon } ( g ( \theta _ { t } + \epsilon u _ { q } ) - g ( \theta _ { t } ) ) u _ { q } ) = \frac { 1 } { Q } \sum _ { q = 1 } ^ { Q } \ddot { \nabla } g ( \theta _ { t } ; u _ { q } ) } \end{array}
+$$
+
+Clearly, we have $\bar { \nabla } g ( \pmb \theta _ { t } ; \pmb u _ { q } ) = \mathrm { s i g n } ( \dot { \nabla } g ( \pmb \theta _ { t } ; \pmb u _ { q } ) )$ and we could relate $\bar { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } )$ and $\hat { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } )$ by writing ${ \hat { \nabla } } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) = G _ { q } \odot { \bar { \nabla } } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } )$ where where $G _ { q } \in \mathbb { R } ^ { d }$ is absolute value of vector $\pmb { u } _ { q }$ (i.e. $G _ { q } = ( | \boldsymbol { u } _ { q , 1 } | , | \boldsymbol { u } _ { q , 2 } | , \cdot \cdot \cdot , | \boldsymbol { u } _ { q , d } | ) ^ { T } )$ .
+
+Note that Zeroth-order gradient estimate $\dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } )$ is a biased approximation to the true gradient of $\mathbf { g }$ . Instead, it becomes unbiased to the gradient of the randomized smoothing function $g _ { \epsilon } ( \pmb { \theta } ) =$ $\mathbb { E } _ { \pmb { u } } \mathbf { \bar { [ { g } ( \pmb { \theta } + \epsilon { u } ) ] } }$ Duchi et al. (2012).
+
+Our analysis is based on the following two assumptions:
+
+Assumption 1 function $\mathbf { g }$ is L-smooth with a finite value of L.
+
+Assumption 2 At any iteration step t, the gradient of the function $\mathbf { g }$ is upper bounded by $\| \nabla g ( \pmb \theta _ { t } ) \| _ { 2 } \le \sigma$ .
+
+To prove the convergence of proposed method, we need the information on variance of the update $\dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } )$ . Here, we introduce a lemma from previous works.
+
+Lemma 1 The variance of Zeroth-Order gradient estimate $\dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } )$ is upper bounded by
+
+$$
+\mathbb { E } \big [ \| \dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) - \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) \| _ { 2 } ^ { 2 } \big ] \leq \frac { 4 ( Q + 1 ) } { Q } \sigma ^ { 2 } + \frac { 2 } { Q } C ( d , \epsilon ) ,
+$$
+
+where $C ( d , \epsilon ) : = 2 d \sigma ^ { 2 } + \epsilon ^ { 2 } L ^ { 2 } d ^ { 2 } / 2$
+
+Proof of Lemma 1 This lemma could be proved by using proposition 2 in Liu et al. (2019) with b $= 1$ and $\mathbf q = \mathbf Q$ . When $\mathbf b = 1$ there is no difference between with/without replacement, and we opt for with replacement case to obtain above bound.
+
+By talking $\mathrm { Q } = 1$ , we know that $\mathbb { E } \big [ \| \dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) - \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) \| _ { 2 } ^ { 2 } \big ]$ is upper bounded. And by Jensen’s inequality, we also know that the
+
+$$
+\begin{array} { r } { \mathbb { E } \big [ | ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) - \nabla g _ { \epsilon } ( \theta _ { t } ) ) \iota | \big ] \leq \sqrt { \mathbb { E } \big [ ( ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) - \nabla g _ { \epsilon } ( \theta _ { t } ) ) ) _ { l } ^ { 2 } \big ] } : = \delta _ { l } , } \end{array}
+$$
+
+where $\delta _ { l }$ denotes the upper bound of $l t h$ coordinate of $\mathbb { E } \big [ | \dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) - \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) | \big ]$ , and $\delta _ { l }$ is finite since $\mathbb { E } \big [ \lVert \dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) - \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) \rVert _ { 2 } ^ { 2 } \big ]$ is upper bounded.
+
+Next, we want to show the $\operatorname { P r o b } [ \operatorname { s i g n } ( ( { \bar { g } } _ { t } ) _ { l } ) \neq \operatorname { s i g n } ( ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ) ]$ by following lemma.
+
+$$
+\begin{array} { r } { | ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } | \mathrm { P r o b } [ \mathrm { s i g n } ( ( \bar { g } _ { t } ) _ { l } ) \neq \mathrm { s i g n } ( ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ) ] \leq \frac { \delta _ { l } } { \sqrt { Q } } } \end{array}
+$$
+
+Proof of Lemma 2 Similar to Bernstein et al. (2018), we first relax $\operatorname { P r o b } [ \operatorname { s i g n } ( ( \dot { \nabla } g ( \pmb \theta _ { t } ; \pmb u _ { q } ) ) _ { l } ) \neq$ $\mathrm { s i g n } ( \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) ) _ { l } ]$ by Markov inequality:
+
+$$
+\begin{array} { r l } & { \mathrm { P r o b } [ \mathrm { s i g n } ( ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) ) _ { l } ) \neq \mathrm { s i g n } ( ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ) ] \leq \mathrm { P r o b } [ | \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) _ { l } ) | \geq | \nabla g _ { \epsilon } ( \theta _ { t } ) _ { l } | ] } \\ & { \quad \quad \quad \quad \quad \leq \frac { \mathbb { E } \left[ | ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) - \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } | \right] } { | \nabla g _ { \epsilon } ( \theta _ { t } ) _ { l } | } } \\ & { \quad \quad \quad \quad \quad \leq \frac { \delta _ { l } } { | \nabla g _ { \epsilon } ( \theta _ { t } ) _ { l } | } , } \end{array}
+$$
+
+where the last inequality comes from eq (6). Recall that $( \dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) ) _ { l } )$ is an unbiased estimation to $( \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) ) _ { l }$ . Under the assumption that the noise distribution is unimodal and symmetric, from Bernstein et al. (2018) Lemma D1, we will have
+
+$$
+\operatorname { P r o b } [ \mathrm { s i g n } ( ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) ) _ { l } ) \neq \mathrm { s i g n } ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ] : = M \leq \left\{ \begin{array} { l l } { \frac { 2 } { 9 } \frac { 1 } { S ^ { 2 } } , } & { \mathrm { S } \geq \frac { 2 } { \sqrt { 3 } } } \\ { \frac { 1 } { 2 } - \frac { S } { 2 \sqrt { 3 } } , } & { o t h e r w i s e } \end{array} \right. < \frac { 1 } { 2 } ,
+$$
+
+where $S : = | \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) _ { l } | / \delta _ { l }$ .
+
+Note that this probability bound applies uniformly to all $q \in Q$ regardless of the magnitude $| ( { \pmb u } _ { q } ) _ { l } |$ . That is,
+
+$$
+\begin{array} { r l r } {  { \mathrm { P r o b } [ \mathrm { s i g n } ( \sum _ { q = 1 } ^ { Q } \vert ( { \boldsymbol u } _ { q } ) _ { l } \vert \mathrm { s i g n } ( ( \dot { \nabla } g ( { \boldsymbol \theta } _ { t } ; { \boldsymbol u } _ { q } ) ) _ { l } ) \neq \mathrm { s i g n } ( \nabla g _ { \epsilon } ( { \boldsymbol \theta } _ { t } ) ) _ { l } ] = } } \\ & { } & { \mathrm { P r o b } [ \mathrm { s i g n } ( ( \sum _ { q = 1 } ^ { Q } \mathrm { s i g n } ( \dot { \nabla } g ( { \boldsymbol \theta } _ { t } ; { \boldsymbol u } _ { q } ) ) _ { l } ) \neq \mathrm { s i g n } ( \nabla g _ { \epsilon } ( { \boldsymbol \theta } _ { t } ) ) _ { l } ] . } \end{array}
+$$
+
+This is true as when all $\begin{array} { r } { u _ { q } ) _ { l } \vert = 1 , \mathrm { P r o b } [ \mathrm { s i g n } ( ( \sum _ { q = 1 } ^ { Q } \mathrm { s i g n } ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) ) _ { l } ) \neq \mathrm { s i g n } ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ] } \end{array}$ is equivalent to majority voting of each estimate q yielding correct sign. This is the same as sum of Q bernoulli trials (i.e. binomial distribution) with error rate $\mathbf { M }$ . And since error probability $\mathbf { M }$ is independent of sampling of $| ( { \pmb u } _ { q } ) _ { l } |$ , calculating $\mathrm { P r o b } [ \mathrm { s i g n } ( \sum _ { q = 1 } ^ { Q } | ( u _ { q } ) _ { l } | \mathrm { s i g n } ( ( \dot { \nabla } g ( \pmb _ { t } ; \pmb { u } _ { q } ) ) _ { l } ) \neq \mathrm { s i g n } ( \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) ) _ { l } ]$ could be thought as taking Q bernoulli experiments and then independently draw a weight from unit length for each of $\mathrm { Q }$ experiment. Since the weight is uniform, we will have expectation of weights on correct counts and incorrect counts are the same and equal to $1 / 2$ . Therefore, the probability of $\mathrm { P r o b } [ \mathrm { s i g n } ( \sum _ { q = 1 } ^ { Q } | ( u _ { q } ) _ { l } | \mathrm { s i g n } ( ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) ) _ { l } ) \neq \mathrm { s i g n } ( \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) ) _ { l } ]$ is still the $\operatorname { s i g n } ( \dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) _ { l } ) = \bar { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) _ { l }$ inomthus $\begin{array} { r } { \frac { 1 } { Q } \sum _ { q = 1 } ^ { Q } \mathrm { s i g n } ( \dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) ) _ { l } = ( \bar { \pmb { g } } _ { t } ) _ { l } } \end{array}$ nota. Let $Z$ n, we will havecounts the number of estimates $\dot { \nabla } g ( \pmb { \theta } _ { t } ; \pmb { u } _ { q } ) _ { l }$ yielding correct sign of $\nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) _ { l }$ . Probability in eq (7) could be written as:
+
+$$
+\operatorname { P r o b } [ \operatorname { s i g n } ( \operatorname { s i g n } ( ( { \bar { g } } _ { t } ) _ { l } ) \neq \operatorname { s i g n } ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ] = P [ Z \leq { \frac { Q } { 2 } } ] .
+$$
+
+Following the derivation of theorem 2b in Bernstein et al. (2018), we could get
+
+$$
+\begin{array} { c } { \displaystyle P [ Z \leq \frac { Q } { 2 } ] \leq \frac { 1 } { \sqrt { Q S } } } \\ { \displaystyle \Rightarrow \vert ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } \vert \mathrm { P r o b } [ \mathrm { s i g n } ( ( \bar { g } _ { t } ) _ { l } ) \neq \mathrm { s i g n } ( ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ) ] \leq \frac { \delta _ { l } } { \sqrt { Q } } } \end{array}
+$$
+
+We also need few more lemmas on properties of function g.
+
+Lemma 3 $g _ { \epsilon } ( \pmb { \theta } _ { 1 } ) - g _ { \epsilon } ( \pmb { \theta } _ { T } ) \leq g _ { \epsilon } ( \pmb { \theta } _ { 1 } ) - g ^ { * } + \epsilon ^ { 2 } L$
+
+Proof of Lemma 3 The proof can be found in Liu et al. (2018) Lemma C.
+
+Lemma 4 $\begin{array} { r } { \mathbb { E } [ \| \nabla g ( \pmb { \theta } ) \| _ { 2 } ] \le \sqrt { 2 } \mathbb { E } [ \| \nabla g _ { \epsilon } ( \pmb { \theta } ) \| _ { 2 } ] + \frac { \epsilon L d } { \sqrt { 2 } } } \end{array}$ , where $g ^ { * } = \operatorname* { m i n } _ { \pmb { \theta } } g ( \pmb { \theta } )$
+
+Proof of Lemma 4 The proof can be found in Liu et al. (2019).
+
+Theorem 1 Suppose that the conditions in the assumptions hold, and the distribution of gradient noise is unimodal and symmetric. Then, Sign-OPT attack with learning rate $\begin{array} { r } { \eta _ { t } = O \big ( \frac { 1 } { Q \sqrt { d T } } \big ) } \end{array}$ ( 1Q√dT ) and $\begin{array} { r } { \epsilon = O ( \frac { 1 } { d T } ) } \end{array}$ will give following bound on $\mathbb { E } [ \| \nabla g ( \pmb { \theta } ) \| _ { 2 } ]$
+
+$$
+\mathbb { E } [ \| \nabla g ( \pmb { \theta } ) \| _ { 2 } ] = O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { d } { \sqrt { Q } } )
+$$
+
+Proof of Theorem 1 From L-smoothness assumption we could have
+
+$$
+\begin{array} { r l } & { g _ { \epsilon } ( \theta _ { t + 1 } ) \leq g _ { \epsilon } ( \theta _ { t } ) + \langle \nabla g _ { \epsilon } ( \theta _ { t } ) , \theta _ { t + 1 } - \theta _ { t } \rangle + \displaystyle \frac { L } { 2 } \| \theta _ { t + 1 } - \theta _ { t } \| _ { 2 } ^ { 2 } } \\ & { \quad \quad \quad = g _ { \epsilon } ( \theta _ { t } ) - \eta _ { k } \langle \nabla g _ { \epsilon } ( \theta _ { t } ) , \hat { g } _ { t } \rangle + \displaystyle \frac { L } { 2 } \eta _ { t } ^ { 2 } \| \hat { g } _ { t } \| _ { 2 } ^ { 2 } } \\ & { \quad \quad \quad = g _ { \epsilon } ( \theta _ { t } ) - \eta _ { t } \odot \bar { G } _ { t } \| \nabla g _ { \epsilon } ( \theta _ { t } ) \| _ { 1 } + \displaystyle \frac { d L } { 2 } \eta _ { t } ^ { 2 } \odot \bar { G } _ { t } ^ { \ 2 } } \\ & { \quad \quad \quad \quad + 2 \eta _ { t } \odot \bar { G } _ { t } \sum _ { l = 1 } ^ { d } | ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } | \mathrm { P r o b } [ \mathrm { s i g n } ( ( \bar { g } _ { t } ) _ { l } ) \neq \mathrm { s i g n } ( ( \nabla g _ { \epsilon } ( \theta _ { t } ) ) _ { l } ) ] , } \end{array}
+$$
+
+where $\hat { G } _ { t }$ is defined as $\begin{array} { r } { ( \bar { G } _ { t } ) _ { l } = \sum _ { q = 1 } ^ { Q } { ( G _ { q } ) _ { l } } \bar { \nabla } g ( \theta _ { t } ; \boldsymbol { u } _ { q } ) _ { l } = \sum _ { q = 1 } ^ { Q } | ( \boldsymbol { u } _ { q } ) _ { l } | \bar { \nabla } g ( \theta _ { t } ; \boldsymbol { u } _ { q } ) _ { l } } \end{array}$ . Continue the inequality,
+
+by eq (8)
+
+$$
+\begin{array} { r l } & { \mathcal { G } _ { t } ( \theta _ { t } ) - \mathcal { H } _ { t } \odot \bar { G } _ { t } | | \nabla _ { \mathcal { A } _ { t } } \{ \theta _ { t } \} | | _ { 1 } + \frac { d L } { 2 } \eta _ { t } ^ { 2 } \odot \bar { G } _ { t } ^ { 2 } } \\ & { + 2 \eta _ { t } \odot \bar { G } _ { t } \sum _ { l = 1 } ^ { 1 } | | \nabla _ { \mathcal { A } _ { t } } \{ \theta _ { t } \} | | \operatorname* { P o b } \{ \bar { \mathrm { s i g n } } ( ( \overline { { g } } _ { t } ) _ { l } ) \} \times \mathrm { s i g n } ( ( \nabla _ { \mathcal { A } _ { t } } \{ \theta _ { t } \} ) _ { l } ) | } \\ & { \le \mathcal { E } _ { t } ( \theta _ { t } ) - \mathcal { H } _ { t } \odot \bar { G } _ { t } | | \nabla _ { \mathcal { A } _ { t } } \{ \theta _ { t } \} | | 1 + \frac { d L } { 2 } \eta _ { t } ^ { 2 } \odot \bar { G } _ { t } ^ { 2 } + 2 \eta _ { t } \odot \bar { G } _ { t } \underset { t = 1 } { \overset { d , d } { \sum } } \frac { \bar { G } _ { t } } { \nabla | \mathcal { Q } } } \\ & { \le \mathcal { E } _ { t } ( \theta _ { t } ) - \eta _ { t } \odot \bar { G } _ { t } | | \nabla _ { \mathcal { A } _ { t } } \{ \theta _ { t } \} | | 1 + \frac { d L } { 2 } \eta _ { t } ^ { 2 } \odot \bar { G } _ { t } ^ { 2 } + 2 \eta _ { t } \odot \bar { G } _ { t } \frac { | | \hat { \mathcal { A } } | } { \sqrt { d } } } \\ & { \le \mathcal { E } _ { t } ( \theta _ { t } ) - \eta _ { t } \odot \bar { G } _ { t } | | \nabla _ { \mathcal { A } _ { t } } \{ \theta _ { t } \} | | 1 + \frac { d L } { 2 } \eta _ { t } ^ { 2 } \odot \bar { G } _ { t } ^ { 2 } + 2 \eta _ { t } \odot \bar { G } _ { t } \frac { | | \hat { \mathcal { A } } | } { \sqrt { d } } } \\ &  = \mathcal { E } _ { t } ( \theta _ { t } ) - \eta _ { t } \odot \bar { G } _ { t } | | \nabla _ { \mathcal { A } _ { t } } \{ \end{array}
+$$
+
+Thus we will have,
+
+$$
+\begin{array} { r l } & { \displaystyle _ { t _ { \epsilon } } ( \theta _ { t + 1 } ) - g _ { \epsilon } ( \theta _ { t } ) \leq - \eta _ { t } \odot \bar { G } _ { t } \| \nabla g _ { \epsilon } ( \theta _ { t } ) \| _ { 1 } + \frac { d L } { 2 } \eta _ { t } ^ { 2 } \odot \bar { G _ { t } } ^ { 2 } + 2 \eta _ { t } \odot \bar { G } _ { t } \frac { \sqrt { d } \sqrt { \mathbb { E } \big [ ( ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) - \nabla g _ { \epsilon } ( \theta _ { t } ) ) ) _ { l } \big ] } } { \sqrt { Q } } , } \\ & { \displaystyle \Rightarrow \eta _ { t } \odot \bar { G } _ { t } \| \nabla g _ { \epsilon } ( \theta _ { t } ) \| _ { 1 } \le g _ { \epsilon } ( \theta _ { t } ) - g _ { \epsilon } ( \theta _ { t + 1 } ) + \frac { d L } { 2 } \eta _ { t } ^ { 2 } \odot \bar { G _ { t } } ^ { 2 } + 2 \eta _ { t } \odot \bar { G } _ { t } \frac { \sqrt { d } \sqrt { \mathbb { E } \big [ ( ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) - \nabla g _ { \epsilon } ( \theta _ { t } ) ) ) _ { l } \big ] } } { \sqrt { Q } } , } \\ & { \displaystyle \Rightarrow \hat { \eta } _ { t } \| \nabla g _ { \epsilon } ( \theta _ { t } ) \| _ { 1 } \le g _ { \epsilon } ( \theta _ { t } ) - g _ { \epsilon } ( \theta _ { t + 1 } ) + \frac { d L } { 2 } \hat { \eta } _ { t } ^ { 2 } + 2 \hat { \eta } _ { t } \sqrt { d } \frac { \sqrt { \mathbb { E } \big [ ( ( \dot { \nabla } g ( \theta _ { t } ; u _ { q } ) - \nabla g _ { \epsilon } ( \theta _ { t } ) ) ) _ { l } ^ { 2 } \big ] } } { \sqrt { Q } } , } \end{array}
+$$
+
+where we define $\hat { \eta _ { t } } : = \eta _ { t } \odot \bar { G } _ { t }$ . Sum up all inequalities for all ts and take expectation on both side, we will have
+
+$$
+\begin{array} { r l } & { \displaystyle \hat { \eta } _ { t } \mathbb { E } [ \| \nabla g _ { \epsilon } ( \theta _ { t } ) \| _ { 1 } ] \le \mathbb { E } [ g _ { \epsilon } ( \theta _ { 1 } ) - g _ { \epsilon } ( \theta _ { T } ) ] + \frac { d L } { 2 } \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } ^ { 2 } + \sum _ { t = 1 } ^ { T } 2 \hat { \eta } _ { t } \sqrt { d } \sqrt { \mathbb { E } \big [ ( ( \hat { \nabla } g ( \theta _ { t } ; u _ { q } ) - \nabla g _ { \epsilon } ( \theta _ { t } ) ) ) \big ] } } \\ & { \displaystyle \le \mathbb { E } \big [ g _ { \epsilon } ( \theta _ { 1 } ) - g _ { \epsilon } ( \theta _ { T } ) \big ] + \frac { d L } { 2 } \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } ^ { 2 } + \sum _ { t = 1 } ^ { T } 2 \hat { \eta } _ { t } \sqrt { d } \sqrt { \frac { 4 ( Q + 1 ) } { Q } \sigma ^ { 2 } + \frac { 2 } { Q } C ( d , \epsilon ) } } \end{array}
+$$
+
+by Lemma 1.
+
+Substitute Lemma 3 into above inequality, we get
+
+$$
+\sum _ { = 1 } ^ { T } \hat { \eta } _ { t } \mathbb { E } [ \| \nabla g _ { \epsilon } ( \theta _ { t } ) \| _ { 1 } ] \le g _ { \epsilon } ( \theta _ { 1 } ) - g ^ { * } + \epsilon ^ { 2 } L + \frac { d L } { 2 } \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } ^ { 2 } + \sum _ { t = 1 } ^ { T } 2 \hat { \eta } _ { t } \sqrt { d } \sqrt { \frac { 4 ( Q + 1 ) } { Q } } \sigma ^ { 2 } + \frac { 2 } { Q } C ( d , \epsilon ) .
+$$
+
+Since $\| \cdot \| _ { 2 } \leq \| \cdot \| _ { 1 }$ and we could divide $\textstyle \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t }$ on both side to get
+
+$$
+\sum _ { i = 1 } ^ { T } \frac { \hat { \eta } _ { t } } { \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } } \mathbb { E } [ \| \nabla g _ { \epsilon } ( \theta _ { t } ) \| _ { 2 } ] \leq \frac { g _ { \epsilon } ( \theta _ { 1 } ) - g ^ { * } + \epsilon ^ { 2 } L } { \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } } + \frac { d L } { 2 } \frac { \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } ^ { 2 } } { \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } } + \sum _ { t = 1 } ^ { T } \frac { 2 \sqrt { d } } { \sqrt { Q } } \sqrt { 4 ( Q + 1 ) \sigma ^ { 2 } + 2 C \| \hat { \eta } _ { t } \| ^ { 2 } } .
+$$
+
+Define a new random variable $\mathbf { R }$ with probability $\begin{array} { r } { P ( R = t ) = \frac { \eta _ { t } } { \sum _ { t = 1 } ^ { T } \eta _ { t } } } \end{array}$ , we will have
+
+$$
+\mathbb { E } [ \| \nabla g _ { \epsilon } ( \pmb { \theta } _ { R } ) \| _ { 2 } ] = \mathbb { E } [ \mathbb { E } _ { R } [ \| \nabla g _ { \epsilon } ( \pmb { \theta } _ { R } ) \| _ { 2 } ] ] = \mathbb { E } \Big [ \sum _ { t = 1 } ^ { T } P ( R = t ) \| \nabla g _ { \epsilon } ( \pmb { \theta } _ { t } ) \| _ { 2 } \Big ] .
+$$
+
+Substitute all the quantities into Lemma 4, we will get
+
+$$
+\mathbb { Z } [ \| \nabla g ( \theta ) \| _ { 2 } ] \le \frac { \sqrt { 2 } \big ( g _ { \epsilon } ( \theta _ { 1 } ) - g ^ { * } + \epsilon ^ { 2 } L \big ) } { \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } } + \frac { d L } { \sqrt { 2 } } \frac { \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } ^ { 2 } } { \sum _ { t = 1 } ^ { T } \hat { \eta } _ { t } } + \frac { \epsilon L d } { \sqrt { 2 } } + \sum _ { t = 1 } ^ { T } \frac { 2 \sqrt { 2 } \sqrt { d } } { \sqrt { Q } } \sqrt { 4 ( Q + 1 ) \sigma ^ { 2 } + 2 \sigma ^ { 2 } } ,
+$$
+
+By choosing $\begin{array} { r } { \epsilon = O ( \frac { 1 } { d T } ) } \end{array}$ and $\begin{array} { r } { \eta _ { t } = O ( \frac { 1 } { Q \sqrt { d T } } ) } \end{array}$ , then the convergence rate as shown in above is $\begin{array} { r } { O ( \frac { d } { T } + \frac { d } { \sqrt { Q } } ) } \end{array}$ .

parse/train/rJeW1yHYwH/rJeW1yHYwH.md

@@ -0,0 +1,465 @@
+# INDUCTIVE REPRESENTATION LEARNING ON TEMPORAL GRAPHS
+
+Da $\mathbf { X } \mathbf { u } ^ { * }$ , Chuanwei Ruan∗, Evren Korpeoglu , Sushant Kumar , Kannan Achan   
+Walmart Labs   
+Sunnyvale, CA 94086, USA   
+{Da.Xu,Chuanwei.Ruan,EKorpeoglu,SKumar4,KAchan}@walmartlabs.com
+
+# ABSTRACT
+
+Inductive representation learning on temporal graphs is an important step toward salable machine learning on real-world dynamic networks. The evolving nature of temporal dynamic graphs requires handling new nodes as well as capturing temporal patterns. The node embeddings, which are now functions of time, should represent both the static node features and the evolving topological structures. Moreover, node and topological features can be temporal as well, whose patterns the node embeddings should also capture. We propose the temporal graph attention (TGAT) layer to efficiently aggregate temporal-topological neighborhood features as well as to learn the time-feature interactions. For TGAT, we use the self-attention mechanism as building block and develop a novel functional time encoding technique based on the classical Bochner’s theorem from harmonic analysis. By stacking TGAT layers, the network recognizes the node embeddings as functions of time and is able to inductively infer embeddings for both new and observed nodes as the graph evolves. The proposed approach handles both node classification and link prediction task, and can be naturally extended to include the temporal edge features. We evaluate our method with transductive and inductive tasks under temporal settings with two benchmark and one industrial dataset. Our TGAT model compares favorably to state-of-the-art baselines as well as the previous temporal graph embedding approaches.
+
+# 1 INTRODUCTION
+
+The technique of learning lower-dimensional vector embeddings on graphs have been widely applied to graph analysis tasks (Perozzi et al., 2014; Tang et al., 2015; Wang et al., 2016) and deployed in industrial systems (Ying et al., 2018; Wang et al., 2018a). Most of the graph representation learning approaches only accept static or non-temporal graphs as input, despite the fact that many graph-structured data are time-dependent. In social network, citation network, question answering forum and user-item interaction system, graphs are created as temporal interactions between nodes. Using the final state as a static portrait of the graph is reasonable in some cases, such as the proteinprotein interaction network, as long as node interactions are timeless in nature. Otherwise, ignoring the temporal information can severely diminish the modelling efforts and even causing questionable inference. For instance, models may mistakenly utilize future information for predicting past interactions during training and testing if the temporal constraints are disregarded. More importantly, the dynamic and evolving nature of many graph-related problems demand an explicitly modelling of the timeliness whenever nodes and edges are added, deleted or changed over time.
+
+Learning representations on temporal graphs is extremely challenging, and it is not until recently that several solutions are proposed (Nguyen et al., 2018; Li et al., 2018; Goyal et al., 2018; Trivedi et al., 2018). We conclude the challenges in three folds. Firstly, to model the temporal dynamics, node embeddings should not be only the projections of topological structures and node features but also functions of the continuous time. Therefore, in addition to the usual vector space, temporal representation learning should be operated in some functional space as well. Secondly, graph topological structures are no longer static since the nodes and edges are evolving over time, which poses temporal constraints on neighborhood aggregation methods. Thirdly, node features and topological structures can exhibit temporal patterns. For example, node interactions that took place long ago may have less impact on the current topological structure and thus the node embeddings. Also, some nodes may possess features that allows them having more regular or recurrent interactions with others. We provide sketched plots for visual illustration in Figure 1.
+
+![](images/d4a2840296d0968e6709f8d305ac0d0542a4acdb461922ad9a73399c06d34c7b.jpg)  
+Figure 1: Visual illustration for several complications from the temporal graphs. (A). The generation process of a temporal graph and its snapshots. It is obvious that the static graphs in the snapshots only reflect partial temporal information. (B). The final state of the temporal graph when projected to the time-independent 2-D plane. Other than the missing temporal information, the multi-edge situation arises as well. (C). When predicting the link between node A and C at time $t _ { 3 }$ , the message-passing paths should be subject to temporal contraints. The solid lines give the appropriate directions, and the dashed lines violates the temporal constraints.
+
+Similar to its non-temporal counterparts, in the real-world applications, models for representation learning on temporal graphs should be able to quickly generate embeddings whenever required, in an inductive fashion. GraphSAGE (Hamilton et al., 2017a) and graph attention network (GAT) (Velickovi ˇ c et al., 2017) are capable of inductively generating embeddings for unseen nodes ´ based on their features, however, they do not consider the temporal factors. Most of the temporal graph embedding methods can only handle transductive tasks, since they require re-training or the computationally-expensive gradient calculations to infer embeddings for unseen nodes or node embeddings for a new timepoint. In this work, we aim at developing an architecture to inductively learn representations for temporal graphs such that the time-aware embeddings (for unseen and observed nodes) can be obtained via a single network forward pass. The key to our approach is the combination of the self-attention mechanism (Vaswani et al., 2017) and a novel functional time encoding technique derived from the Bochner’s theorem from classical harmonic analysis (Loomis, 2013).
+
+The motivation for adapting self-attention to inductive representation learning on temporal graphs is to identify and capture relevant pieces of the temporal neighborhood information. Both graph convolutional network (GCN) (Kipf & Welling, 2016a) and $G A T$ are implicitly or explicitly assigning different weights to neighboring nodes (Velickovi ˇ c et al., 2017) when aggregating node features. The ´ self-attention mechanism was initially designed to recognize the relevant parts of input sequence in natural language processing. As a discrete-event sequence learning method, self-attention outputs a vector representation of the input sequence as a weighted sum of individual entry embeddings. Selfattention enjoys several advantages such as parallelized computation and interpretability (Vaswani et al., 2017). Since it captures sequential information only through the positional encoding, temporal features can not be handled. Therefore, we are motivated to replace positional encoding with some vector representation of time. Since time is a continuous variable, the mapping from the time domain to vector space has to be functional. We gain insights from harmonic analysis and propose a theoretical-grounded functional time encoding approach that is compatible with the self-attention mechanism. The temporal signals are then modelled by the interactions between the functional time encoding and nodes features as well as the graph topological structures.
+
+To evaluate our approach, we consider future link prediction on the observed nodes as transductive learning task, and on the unseen nodes as inductive learning task. We also examine the dynamic node classification task using node embeddings (temporal versus non-temporal) as features to demonstrate the usefulness of our functional time encoding. We carry out extensive ablation studies and sensitivity analysis to show the effectiveness of the proposed functional time encoding and TGAT-layer.
+
+# 2 RELATED WORK
+
+Graph representation learning. Spectral graph embedding models operate on the graph spectral domain by approximating, projecting or expanding the graph Laplacian (Kipf & Welling, 2016a; Henaff et al., 2015; Defferrard et al., 2016). Since their training and inference are conditioned on the specific graph spectrum, they are not directly extendable to temporal graphs. Non-spectral approaches, such as GAT, GraphSAGE and MoNET, (Monti et al., 2017) rely on the localized neighbourhood aggregations and thus are not restricted to the training graph. GraphSAGE and GAT also have the flexibility to handle evolving graphs inductively. To extend classical graph representation learning approaches to the temporal domain, several attempts have been done by cropping the temporal graph into a sequence of graph snapshots (Li et al., 2018; Goyal et al., 2018; Rahman et al., 2018; Xu et al., 2019b), and some others work with temporally persistent node (edges) (Trivedi et al., 2018; Ma et al., 2018). Nguyen et al. (2018) proposes a node embedding method based on temporal random walk and reported state-of-the-art performances. However, their approach only generates embeddings for the final state of temporal graph and can not directly apply to the inductive setting.
+
+Self-attention mechanism. Self-attention mechanisms often have two components: the embedding layer and the attention layer. The embedding layer takes an ordered entity sequence as input. Selfattention uses the positional encoding, i.e. each position $k$ is equipped with a vector $\mathbf { p } _ { k }$ (fixed or learnt) which is shared for all sequences. For the entity sequence $\mathbf { e } = ( e _ { 1 } , \ldots , e _ { l } )$ , the embedding layer takes the sum or concatenation of entity embeddings (or features) $( \mathbf { z } \in \mathbb { R } ^ { d } )$ ) and their positional encodings as input:
+
+$$
+\mathbf { Z _ { e } } = \left[ \mathbf { z } _ { e _ { 1 } } + \mathbf { p } _ { 1 } , \dots , \mathbf { z } _ { e _ { 1 } } + \mathbf { p } _ { l } \right] ^ { \intercal } \in \mathbb { R } ^ { l \times d } , \mathrm { o r } \quad \mathbf { Z _ { e } } = \left[ \mathbf { z } _ { e _ { 1 } } | | \mathbf { p } _ { 1 } , \dots , \mathbf { z } _ { e _ { 1 } } | | \mathbf { p } _ { l } \right] ^ { \intercal } \in \mathbb { R } ^ { l \times ( d + d _ { \mathrm { p s } } ) } .
+$$
+
+where $| |$ denotes concatenation operation and $d _ { \mathrm { p o s } }$ is the dimension for positional encoding. Selfattention layers can be constructed using the scaled dot-product attention, which is defined as:
+
+$$
+\operatorname { A t t n } \ ( \mathbf { Q } , \mathbf { K } , \mathbf { V } ) = \operatorname { s o f t m a x } \Big ( \frac { \mathbf { Q } \mathbf { K } ^ { \intercal } } { \sqrt { d } } \Big ) \mathbf { V } ,
+$$
+
+where $\mathbf { Q }$ denotes the ’queries’, $\mathbf { K }$ the ’keys’ and $\mathbf { V }$ the ’values’. In Vaswani et al. (2017), they are treated as projections of the output $\mathbf { Z } _ { \mathbf { e } } \colon \mathbf { Q } = \mathbf { Z } _ { \mathbf { e } } \mathbf { W } _ { Q } , \quad \mathbf { K } = \mathbf { Z } _ { \mathbf { e } } \mathbf { W } _ { K } , \quad \mathbf { V } = \mathbf { Z } _ { \mathbf { e } } \mathbf { W } _ { V } .$ , where $\mathbf { W } _ { Q }$ , ${ \bf W } _ { K }$ and $\mathbf { W } _ { V }$ are the projection matrices. Since each row of $\mathbf { Q }$ , $\mathbf { K }$ and $\mathbf { V }$ represents an entity, the dot-product attention takes a weighted sum of the entity ’values’ in $\mathbf { V }$ where the weights are given by the interactions of entity ’query-key’ pairs. The hidden representation for the entity sequence under the dot-product attention is then given by $h _ { \mathbf { e } } = \mathrm { A t t n } ( \mathbf { Q } , \mathbf { K } , \mathbf { V } )$ .
+
+# 3 TEMPORAL GRAPH ATTENTION NETWORK ARCHITECTURE
+
+We first derive the mapping from time domain to the continuous differentiable functional domain as the functional time encoding such that resulting formulation is compatible with self-attention mechanism as well as the backpropagation-based optimization frameworks. The same idea was explored in a concurrent work ( $\mathrm { { X u } }$ et al., 2019a). We then present the temporal graph attention layer and show how it can be naturally extended to incorporate the edge features.
+
+# 3.1 FUNCTIONAL TIME ENCODING
+
+Recall that our starting point is to obtain a continuous functional mapping $\Phi : T  \mathbb { R } ^ { d _ { T } }$ from time domain to the $d _ { T }$ -dimensional vector space to replace the positional encoding in (1). Without loss of generality, we assume that the time domain can be represented by the interval starting from origin: $\hat { T } = [ 0 , \hat { t } _ { \mathrm { m a x } } ]$ , where $t _ { \mathrm { m a x } }$ is determined by the observed data. For the inner-product selfattention in (2), often the ’key’ and ’query’ matrices $( \mathbf { K } , \mathbf { Q } )$ are given by identity or linear projection of $\mathbf { Z _ { e } }$ defined in (1), leading to terms that only involve inner-products between positional (time) encodings. Consider two time points $t _ { 1 } , t _ { 2 }$ and inner product between their functional encodings $\left. \Phi ( t _ { 1 } ) , \mathbf { \bar { \Phi } } ( t _ { 2 } ) \right.$ . Usually, the relative timespan, rather than the absolute value of time, reveals critical temporal information. Therefore, we are more interested in learning patterns related to the timespan of $| t _ { 2 } - t _ { 1 } |$ , which should be ideally expressed by $\left. \Phi ( t _ { 1 } ) , \Phi ( t _ { 2 } ) \right.$ to be compatible with self-attention.
+
+Formally, we define the temporal kernel $K : T \times T  \mathbb { R }$ with $\mathcal { K } ( t _ { 1 } , t _ { 2 } ) : = \left. { \Phi ( t _ { 1 } ) , \Phi ( t _ { 2 } ) } \right.$ and $\mathcal { K } ( t _ { 1 } , t _ { 2 } ) = \psi ( t _ { 1 } - t _ { 2 } ) , \forall t _ { 1 } , t _ { 2 } \in T$ for some $\psi : [ - t _ { \mathrm { m a x } } , t _ { \mathrm { m a x } } ]  \mathbb { R }$ . The temporal kernel is then translation-invariant, since $\mathcal { K } ( t _ { 1 } + c , t _ { 2 } + c ) = \psi ( t _ { 1 } - t _ { 2 } ) = \mathcal { K } ( t _ { 1 } , t _ { 2 } )$ for any constant $c$ . Generally speaking, functional learning is extremely complicated since it operates on infinite-dimensional spaces, but now we have transformed the problem into learning the temporal kernel $\kappa$ expressed by $\Phi$ . Nonetheless, we still need to figure out an explicit parameterization for $\Phi$ in order to conduct efficient gradient-based optimization. Classical harmonic analysis theory, i.e. the Bochner’s theorem, motivates our final solution. We point out that the temporal kernel $\kappa$ is positive-semidefinite (PSD) and continuous, since it is defined via Gram matrix and the mapping $\Phi$ is continuous. Therefore, the kernel $\kappa$ defined above satisfy the assumptions of the Bochner’s theorem, which we state below.
+
+Theorem 1 (Bochner’s Theorem). A continuous, translation-invariant kernel ${ \mathcal { K } } ( \mathbf { x } , \mathbf { y } ) = \psi ( \mathbf { x } - \mathbf { y } )$ on $\mathbb { R } ^ { d }$ is positive definite if and only if there exists a non-negative measure on $\mathbb { R }$ such that $\psi$ is the Fourier transform of the measure.
+
+Consequently, when scaled properly, our temporal kernel $\kappa$ have the alternate expression:
+
+$$
+{ \cal K } ( t _ { 1 } , t _ { 2 } ) = \psi ( t _ { 1 } , t _ { 2 } ) = \int _ { \mathbb { R } } e ^ { i \omega ( t _ { 1 } - t _ { 2 } ) } p ( \omega ) d \omega = \mathbb { E } _ { \omega } [ \xi _ { \omega } ( t _ { 1 } ) \xi _ { \omega } ( t _ { 2 } ) ^ { * } ] ,
+$$
+
+where $\xi _ { \omega } ( t ) = e ^ { i \omega t }$ . Since the kernel $\kappa$ and the probability measure $p ( \omega )$ are real, we extract the real part of (3) and obtain:
+
+$$
+\begin{array} { r } { K ( t _ { 1 } , t _ { 2 } ) = \mathbb { E } _ { \omega } \big [ \cos ( \omega ( t _ { 1 } - t _ { 2 } ) ) \big ] = \mathbb { E } _ { \omega } \big [ \cos ( \omega t _ { 1 } ) \cos ( \omega t _ { 2 } ) + \sin ( \omega t _ { 1 } ) \sin ( \omega t _ { 2 } ) \big ] . } \end{array}
+$$
+
+The above formulation suggests approximating the expectation by the Monte Carlo integral 008), i.e. $\begin{array} { r } { \mathcal { K } ( t _ { 1 } , t _ { 2 } ) \approx \frac { 1 } { d } \sum _ { i = 1 } ^ { d } \cos ( \omega _ { i } t _ { 1 } ) \cos ( \omega _ { i } t _ { 2 } ) + \sin ( \omega _ { i } t _ { 1 } ) \sin ( \omega _ { i } t _ { 2 } ) } \end{array}$ , with $\omega _ { 1 } , \ldots , \omega _ { d } \stackrel { \mathrm { i . i . d } } { \sim } p ( \omega )$ $\mathbb { R } ^ { d }$
+
+$$
+t \mapsto \Phi _ { d } ( t ) : = { \sqrt { \frac { 1 } { d } } } { \big [ } \cos ( \omega _ { 1 } t ) , \sin ( \omega _ { 1 } t ) , \ldots , \cos ( \omega _ { d } t ) , \sin ( \omega _ { d } t ) { \big ] } ,
+$$
+
+and it is easy to show that $\left. \Phi _ { d } ( t _ { 1 } ) , \Phi _ { d } ( t _ { 2 } ) \right. \approx K ( t _ { 1 } , t _ { 2 } )$ . As a matter of fact, we prove the stochastic uniform convergence of $\left. \Phi _ { d } ( t _ { 1 } ) , \Phi _ { d } ( t _ { 2 } ) \right.$ to the underlying $\displaystyle \kappa ( t _ { 1 } , t _ { 2 } )$ and shows that it takes only a reasonable amount of samples to achieve proper estimation, which is stated in Claim 1.
+
+Claim 1. Let $p ( \omega )$ be the corresponding probability measure stated in Bochner’s Theorem for kernel function $\kappa$ . Suppose the feature map $\Phi$ is constructed as described above using samples $\{ \omega _ { i } \} _ { i = 1 } ^ { d }$ , then we only need $\begin{array} { r } { d = \Omega \big ( \frac { 1 } { \epsilon ^ { 2 } } \log \frac { \sigma _ { p } ^ { 2 } t _ { \mathrm { m a x } } } { \epsilon } \big ) } \end{array}$ samples to have
+
+$$
+\operatorname* { s u p } _ { t _ { 1 } , t _ { 2 } \in T } \left| \Phi _ { d } ( t _ { 1 } ) ^ { ' } \Phi _ { d } ( t _ { 2 } ) - K ( t _ { 1 } , t _ { 2 } ) \right| < \epsilon w i t h a n y p r o b a b i l i t y f o r \forall \epsilon > 0 ,
+$$
+
+where $\sigma _ { p } ^ { 2 }$ is the second momentum with respect to $p ( \omega )$
+
+The proof is provided in supplement material.
+
+By applying Bochner’s theorem, we convert the problem of kernel learning to distribution learning, i.e. estimating the $p ( \omega )$ in Theorem 1. A straightforward solution is to apply the reparameterization trick by using auxiliary random variables with a known marginal distribution as in variational autoencoders (Kingma & Welling, 2013). However, the reparameterization trick is often limited to certain distributions such as the ’local-scale’ family, which may not be rich enough for our purpose. For instance, when $p ( \omega )$ is multimodal it is difficult to reconstruct the underlying distribution via direct reparameterizations. An alternate approach is to use the inverse cumulative distribution function (CDF) transformation. Rezende & Mohamed (2015) propose using parameterized normalizing flow, i.e. a sequence of invertible transformation functions, to approximate arbitrarily complicated CDF and efficiently sample from it. Dinh et al. (2016) further considers stacking bijective transformations, known as affine coupling layer, to achieve more effective CDF estimation. The above methods learns the inverse CDF function $\overline { { F _ { \theta } ^ { - 1 } } } ( . )$ parameterized by flow-based networks and draw samples from the corresponding distribution. On the other hand, if we consider an non-parameterized approach for estimating distribution, then learning $F ^ { - 1 } ( . )$ and obtain $d$ samples from it is equivalent to directly optimizing the $\{ \omega _ { 1 } , \ldots , \omega _ { d } \}$ in (4) as free model parameters. In practice, we find these two approaches to have highly comparable performances (see supplement material). Therefore we focus on the non-parametric approach, since it is more parameter-efficient and has faster training speed (as no sampling during training is required).
+
+The above functional time encoding is fully compatible with self-attention, thus they can replace the positional encodings in (1) and their parameters are jointly optimized as part of the whole model.
+
+![](images/ccb26592e0ccb09c7b0b3eb051b883ab02ac64ccf31d0dc34257e1c4ff9a5d23.jpg)  
+Figure 2: The architecture of the $l ^ { t h }$ TGAT layer with $k = 3$ attention heads for node $v _ { 0 }$ at time $t$ .
+
+# 3.2 TEMPORAL GRAPH ATTENTION LAYER
+
+We use $v _ { i }$ and $\mathbf { x } _ { i } \in \mathbb { R } ^ { d _ { 0 } }$ to denote node $i$ and its raw node features. The proposed TGAT architecture depends solely on the temporal graph attention layer (TGAT layer). In analogy to GraphSAGE and GAT, the TGAT layer can be thought of as a local aggregation operator that takes the temporal neighborhood with their hidden representations (or features) as well as timestamps as input, and the output is the time-aware representation for target node at any time point $t$ . We denote the hidden representation output for node $i$ at time $t$ from the ${ { l } ^ { t h } }$ layer as $\tilde { \mathbf { h } } _ { i } ^ { ( l ) } ( t )$ .
+
+Similar to $G A T$ , we perform the masked self-attention to take account of the structural information (Velickovi ˇ c et al., 2017). For node ´ $v _ { 0 }$ at time $t$ , we consider its neighborhood $\mathcal { N } ( v _ { 0 } ; t ) = \{ v _ { 1 } , . . . , v _ { N } \}$ such that the interaction between $v _ { 0 }$ and $v _ { i } \ \in \mathcal { N } ( v _ { 0 } ; t )$ , which takes place at time $t _ { i }$ , is prior to $t ^ { 1 }$ . The input of TGAT layer is the neighborhood information ${ \textbf { Z } } =$ $\{ \tilde { \mathbf { h } } _ { 1 } ^ { ( l - 1 ) } ( t _ { 1 } ) , \ldots , \tilde { \mathbf { h } } _ { N } ^ { ( l - 1 ) } ( t _ { N } ) \}$ and the target node information with some time point  layer, the inputs are just raw node features. The laye $( \tilde { \mathbf { h } } _ { 0 } ^ { ( l - 1 ) } ( t ) , t )$ $l = 1$ time-aware representation of target node $v _ { 0 }$ at time $t$ , denoted by $\tilde { \mathbf { h } } _ { 0 } ^ { ( l ) } ( t )$ , as its output. Due to the translation-invariant assumption for the temporal kernel, we can alternatively use $\{ t - t _ { 1 } , \ldots , t - t _ { N } \}$ as interaction times, since $| t _ { i } - t _ { j } | = { \big | } ( t - t _ { i } ) - ( t - t _ { j } ) { \big | }$ and we only care for the timespan.
+
+In line with original self-attention mechanism, we first obtain the entity-temporal feature matrix as
+
+$$
+\mathbf { Z } ( t ) = \left[ \tilde { \mathbf { h } } _ { 0 } ^ { ( l - 1 ) } ( t ) | | \Phi _ { d _ { T } } ( 0 ) , \tilde { \mathbf { h } } _ { 1 } ^ { ( l - 1 ) } ( t _ { 1 } ) | | \Phi _ { d _ { T } } ( t - t _ { 1 } ) , . . . , \tilde { \mathbf { h } } _ { N } ^ { ( l - 1 ) } ( t _ { N } ) | | \Phi _ { d _ { T } } ( t - t _ { N } ) \right] ^ { \top } ( \mathrm { o r ~ u s e ~ s u m } )
+$$
+
+and forward it to three different linear projections to obtain the ’query’, ’key’ and ’value’:
+
+$$
+\mathbf { q } ( t ) = \left[ \mathbf { Z } ( t ) \right] _ { 0 } \mathbf { W } _ { Q } , \mathbf { K } ( t ) = \left[ \mathbf { Z } ( t ) \right] _ { 1 : N } \mathbf { W } _ { K } , \mathbf { V } ( t ) = \left[ \mathbf { Z } ( t ) \right] _ { 1 : N } \mathbf { W } _ { V } ,
+$$
+
+where $\mathbf { W } _ { Q } , \mathbf { W } _ { K } , \mathbf { W } _ { V } \ \in \ \mathbb { R } ^ { ( d + d _ { T } ) \times d _ { h } }$ are the weight matrices that are employed to capture the interactions between time encoding and node features. For notation simplicity, in the following discussion we treat the dependence of the intermediate outputs on target time $t$ as implicit. The attention weights $\{ \alpha _ { i } \} _ { i = 1 } ^ { N }$ of the softmax function output in (2) is given by: $\alpha _ { i } ~ =$ $\begin{array} { r l } { \exp \big ( \mathbf { q } ^ { \mathsf { T } } \mathbf { K } _ { i } \big ) / \Big ( \sum _ { q } \exp \big ( \mathbf { q } ^ { \mathsf { T } } \mathbf { K } _ { q } \big ) \Big ) } & { { } } \end{array}$ . The attention weight $\alpha _ { i }$ reveals how node $i$ attends to the features of node $v _ { 0 }$ within the topological structure defined as $\mathcal { N } ( v _ { 0 } ; t )$ after accounting for their interaction time with $v _ { 0 }$ . The self-attention therefore captures the temporal interactions with both node features and topological features and defines a local temporal aggregation operator on graph. The hidden representation for any node $v _ { i } \in \mathcal { N } ( v _ { 0 } ; t )$ is given by: $\alpha _ { i } \mathbf { V } _ { i }$ . The mechanism can be effectively shared across all nodes for any time point. We then take the row-wise sum from the above dot-product self-attention output as the hidden neighborhood representations, i.e.
+
+$$
+\mathbf { h } ( t ) = \mathrm { A t t n } \big ( \mathbf { q } ( t ) , \mathbf { K } ( t ) , \mathbf { V } ( t ) \big ) \in \mathbb { R } ^ { d _ { h } } .
+$$
+
+To combine neighbourhood representation with the target node features, we adopt the same practice from GraphSAGE and concatenate the neighbourhood representation with the target node’s feature vector $\mathbf { z } _ { 0 }$ . We then pass it to a feed-forward neural network to capture non-linear interactions between the features as in (Vaswani et al., 2017):
+
+$$
+\begin{array} { r l } & { \widetilde { \mathbf { h } } _ { 0 } ^ { ( l ) } ( t ) = \mathrm { F F N } \Big ( \mathbf { h } ( t ) | | \mathbf { x } _ { 0 } \Big ) \equiv \mathrm { R e L U } \Big ( [ \mathbf { h } ( t ) | | \mathbf { x } _ { 0 } ] \mathbf { W } _ { 0 } ^ { ( l ) } + \mathbf { b } _ { 0 } ^ { ( l ) } \Big ) \mathbf { W } _ { 1 } ^ { ( l ) } + \mathbf { b } _ { 1 } ^ { ( l ) } , } \\ & { \mathbf { W } _ { 0 } ^ { ( l ) } \in \mathbb { R } ^ { ( d _ { h } + d _ { 0 } ) \times d _ { f } } , \mathbf { W } _ { 1 } ^ { ( l ) } \in \mathbb { R } ^ { d _ { f } \times d } , \mathbf { b } _ { 0 } ^ { ( l ) } \in \mathbb { R } ^ { d _ { f } } , \mathbf { b } _ { 1 } ^ { ( l ) } \in \mathbb { R } ^ { d } , } \end{array}
+$$
+
+where $\tilde { \mathbf { h } } _ { 0 } ^ { ( l ) } ( t ) \in \mathbb { R } ^ { d }$ is the final output representing the time-aware node embedding at time $t$ for the target node. Therefore, the TGAT layer can be implemented for node classification task using the semi-supervised learning framework proposed in Kipf & Welling (2016a) as well as the link prediction task with the encoder-decoder framework summarized by Hamilton et al. (2017b).
+
+Velickovi ˇ c et al. (2017) suggests that using ´ multi-head attention improves performances and stabilizes training for GAT. For generalization purposes, we also show that the proposed TGAT layer can be easily extended to the multi-head setting. Consider the dot-product self-attention outputs from a total of $k$ different heads, i.e. $\mathbf { h } ^ { ( i ) } \equiv \mathrm { A t t n } ^ { ( i ) } \left( \mathbf { q } ( t ) , \mathbf { K } ( t ) , \mathbf { V } ( t ) \right)$ , $i = 1 , \ldots , k$ . We first concatenate the $k$ neighborhood representations into a combined vector and then carry out the same procedure:
+
+$$
+\tilde { \mathbf { h } } _ { 0 } ^ { ( l ) } ( t ) = \mathrm { F F N } \Big ( \mathbf { h } ^ { ( 1 ) } ( t ) | | \dots | | \mathbf { h } ^ { ( k ) } ( t ) | | \mathbf { x } _ { 0 } \Big ) .
+$$
+
+Just like GraphSAGE, a single TGAT layer aggregates the localized one-hop neighborhood, and by stacking $L$ TGAT layers the aggregation extends to $L$ hops. Similar to $G A T ,$ , out approach does not restrict the size of neighborhood. We provide a graphical illustration of our TGAT layer in Figure 2.
+
+# 3.3 EXTENSION TO INCORPORATE EDGE FEATURES
+
+We show that the TGAT layer can be naturally extended to handle edge features in a messagepassing fashion. Simonovsky & Komodakis (2017) and Wang et al. (2018b) modify classical spectral-based graph convolutional networks to incorporate edge features. Battaglia et al. (2018) propose general graph neural network frameworks where edges features can be processed. For temporal graphs, we consider the general setting where each dynamic edge is associated with a feature vector, i.e. the interaction between $v _ { i }$ and $v _ { j }$ at time $t$ induces the feature vector $\mathbf { x } _ { i , j } ( t )$ . To propagate edge features during the TGAT aggregation, we simply extend the $\mathbf { Z } ( t )$ in (6) to:
+
+$$
+\mathbf { Z } ( t ) = \left[ \ldots , \tilde { \mathbf { h } } _ { i } ^ { ( l - 1 ) } ( t _ { i } ) | | \mathbf { x } _ { 0 , i } ( t _ { i } ) | | \Phi _ { d _ { T } } ( t - t _ { i } ) , \ldots \right] \mathrm { ~ ( o r ~ u s e ~ s u m m a t i o n ) , }
+$$
+
+such that the edge information is propagated to the target node’s hidden representation, and then passed on to the next layer (if exists). The remaining structures stay the same as in Section 3.2.
+
+# 3.4 TEMPORAL SUB-GRAPH BATCHING
+
+Stacking $L$ TGAT layers is equivalent to aggregate over the $L$ -hop neighborhood. For each $L$ -hop sub-graph that is constructed during the batch-wise training, all message passing directions must be aligned with the observed chronological orders. Unlike the non-temporal setting where each edge appears only once, in temporal graphs two node can have multiple interactions at different time points. Whether or not to allow loops that involve the target node should be judged case-bycase. Sampling from neighborhood, or known as neighborhood dropout, may speed up and stabilize model training. For temporal graphs, neighborhood dropout can be carried uniformly or weighted by the inverse timespan such that more recent interactions has higher probability of being sampled.
+
+# 3.5 COMPARISONS TO RELATED WORK
+
+The functional time encoding technique and TGAT layer introduced in Section 3.1 and 3.2 solves several critical challenges, and the TGAT network intrinsically connects to several prior methods.
+
+• Instead of cropping temporal graphs into a sequence of snapshots or constructing timeconstraint random walks, which inspired most of the current temporal graph embedding methods, we directly learn the functional representation of time. The proposed approach is motivated by and thus fully compatible with the well-established self-attention mechanism. Also, to the best of our knowledge, no previous work has discussed the temporal-feature interactions for temporal graphs, which is also considered in our approach.
+
+• The TGAT layer is computationally efficient compared with RNN-based models, since the masked self-attention operation is parallelizable, as suggested by Vaswani et al. (2017). The per-batch time complexity of the TGAT layer with $k$ heads and $l$ layers can be expressed as $\bar { O } \big ( ( k \tilde { N } ) ^ { l } \big )$ where $\bar { \tilde { N } }$ is the average neighborhood size, which is comparable to GAT. When using multi-head attention, the computation for each head can be parallelized as well.
+
+• The inference with TGAT is entirely inductive. With an explicit functional expression $\tilde { h } ( t )$ for each node, the time-aware node embeddings can be easily inferred for any timestamp via a single network forward pass. Similarity, whenever the graph is updated, the embeddings for both unseen and observed nodes can be quickly inferred in an inductive fashion similar to that of GraphSAGE, and the computations can be parallelized across all nodes.
+
+• GraphSAGE with mean pooling (Hamilton et al., 2017a) can be interpreted as a special case of the proposed method, where the temporal neighborhood is aggregated with equal attention coefficients. GAT is like the time-agnostic version of our approach but with a different formulation for self-attention, as they refer to the work of Bahdanau et al. (2014). We discuss the differences in detail in the Appendix. It is also straightforward to show our connections with the menory networks (Sukhbaatar et al., 2015) by thinking of the temporal neighborhoods as memory. The techniques developed in our work may also help adapting GAT and GraphSAGE to temporal settings as we show in our experiments.
+
+# 4 EXPERIMENT AND RESULTS
+
+We test the performance of the proposed method against a variety of strong baselines (adapted for temporal settings when possible) and competing approaches, for both the inductive and transductive tasks on two benchmark and one large-scale industrial dataset.
+
+# 4.1 DATASETS
+
+Real-world temporal graphs consist of time-sensitive node interactions, evolving node labels as well as new nodes and edges. We choose the following datasets which contain all scenarios.
+
+Reddit dataset.2 We use the data from active users and their posts under subreddits, leading to a temporal graph with 11,000 nodes, $\sim 7 0 0 { , } 0 0 0$ temporal edges and dynamic labels indicating whether a user is banned from posting. The user posts are transformed into edge feature vectors.
+
+Wikipedia dataset.3 We use the data from top edited pages and active users, yielding a temporal graph $\sim 9 { , } 3 0 0$ nodes and around 160,000 temporal edges. Dynamic labels indicate if users are temporarily banned from editing. The user edits are also treated as edge features.
+
+Industrial dataset. We choose 70,000 popular products and 100,000 active customers as nodes from the online grocery shopping website4 and use the customer-product purchase as temporal edges ${ \sim } 2$ million). The customers are tagged with labels indicating if they have a recent interest in dietary products. Product features are given by the pre-trained product embeddings (Xu et al., 2020).
+
+We do the chronological train-validation-test split with $7 0 \% - 1 5 \% - 1 5 \%$ according to node interaction timestamps. The dataset and preprocessing details are provided in the supplement material.
+
+# 4.2 TRANSDUCTIVE AND INDUCTIVE LEARNING TASKS
+
+Since the majority of temporal information is reflected via the timely interactions among nodes, we choose to use a more revealing link prediction setup for training. Node classification is then treated as the downstream task using the obtained time-aware node embeddings as input.
+
+Table 1: Transductive learning task results for predicting future edges of nodes that have been observed in training data. All results are converted to percentage by multiplying by 100, and the standard deviations computed over ten runs (in parenthesis). The best and second-best results in each column are highlighted in bold font and underlined. GraphSAGE is short for GraphSAGE-LSTM.   
+
+<table><tr><td rowspan="2">Dataset Metric</td><td colspan="2">Reddit</td><td colspan="2">Wikipedia</td><td colspan="2">Industrial</td></tr><tr><td>Accuracy</td><td>AP</td><td>Accuracy</td><td>AP</td><td> Accuracy</td><td>AP</td></tr><tr><td>GAE</td><td>74.31 (0.5)</td><td>93.23 (0.3)</td><td>72.85 (0.7)</td><td>91.44 (0.1)</td><td>68.92 (0.3)</td><td>81.15 (0.2)</td></tr><tr><td>VAGE</td><td>74.19 (0.4)</td><td>92.92 (0.2)</td><td>78.01 (0.3)</td><td>91.34 (0.3)</td><td>67.81 (0.4)</td><td>80.87 (0.3)</td></tr><tr><td>DeepWalk</td><td>71.43 (0.6)</td><td>83.10 (0.5)</td><td>76.67 (0.5)</td><td>90.71 (0.6)</td><td>65.87 (0.3)</td><td>80.93 (0.2)</td></tr><tr><td>Node2vec</td><td>72.53 (0.4)</td><td>84.58 (0.5)</td><td>78.09 (0.4)</td><td>91.48 (0.3)</td><td>66.64 (0.3)</td><td>81.39 (0.3)</td></tr><tr><td>CTDNE</td><td>73.76 (0.5)</td><td>91.41 (0.3)</td><td>79.42 (0.4)</td><td>92.17 (0.5)</td><td>67.81 (0.3)</td><td>80.95 (0.5)</td></tr><tr><td>GAT</td><td>92.14 (0.2)</td><td>97.33 (0.2)</td><td>87.34 (0.3)</td><td>94.73 (0.2)</td><td>69.58 (0.4)</td><td>81.51 (0.2)</td></tr><tr><td>GAT+T</td><td>92.47 (0.2)</td><td>97.62 (0.2)</td><td>87.57 (0.2)</td><td>95.14 (0.4)</td><td>70.15 (0.3)</td><td>82.66 (0.4)</td></tr><tr><td>GraphSAGE</td><td>92.31(0.2)</td><td>97.65 (0.2)</td><td>85.93 (0.3)</td><td>93.56 (0.3)</td><td>70.19 (0.2)</td><td>83.27 (0.3)</td></tr><tr><td>GraphSAGE+T</td><td>92.58 (0.2)</td><td>97.89 (0.3)</td><td>86.31 (0.3)</td><td>93.72 (0.3)</td><td>71.84 (0.3)</td><td>84.95(0.)</td></tr><tr><td>Const-TGAT</td><td>91.39 (0.2)</td><td>97.86 (0.2)</td><td>86.03 (0.4)</td><td>93.50 (0.3)</td><td>68.52 (0.2)</td><td>81.91 (0.3)</td></tr><tr><td>TGAT</td><td>92.92 (0.3)</td><td>98.12 (0.2)</td><td>88.14 (0.2)</td><td>95.34 (0.1)</td><td>73.28 (0.2)</td><td>86.32 (0.1)</td></tr></table>
+
+Table 2: Inductive learning task results for predicting future edges of unseen nodes.   
+
+<table><tr><td rowspan="2">Dataset Metric</td><td colspan="2">Reddit</td><td colspan="2">Wikipedia</td><td colspan="2">Industrial</td></tr><tr><td> Accuracy</td><td>AP</td><td>Accuracy</td><td>AP</td><td>Accuracy</td><td>AP</td></tr><tr><td>GAT</td><td>89.86 (0.2)</td><td>95.37 (0.3)</td><td>82.36 (0.3)</td><td>91.27 (0.4)</td><td>68.28 (0.2)</td><td>79.93 (0.3)</td></tr><tr><td>GAT+T</td><td>90.44 (0.3)</td><td>96.31 (0.3)</td><td>84.82 (0.3)</td><td>93.57 (0.3)</td><td>69.51 (0.3)</td><td>81.68 (0.3)</td></tr><tr><td>GraphSAGE</td><td>89.43 (0.1)</td><td>96.27 (0.2)</td><td>82.43 (0.3)</td><td>91.09 (0.3)</td><td>67.49 (0.2)</td><td>80.54 (0.3)</td></tr><tr><td>GraphSAGE+T</td><td>90.07 (0.2)</td><td>95.83 (0.2)</td><td>84.03 (0.4)</td><td>92.37 (0.5)</td><td>69.66 (0.3)</td><td>82.74 (0.3)</td></tr><tr><td>Const-TGAT</td><td>88.28 (0.3)</td><td>94.12 (0.2)</td><td>83.60 (0.4)</td><td>91.93 (0.3)</td><td>65.87 (0.3)</td><td>77.03 (0.4)</td></tr><tr><td>TGAT</td><td>90.73 (0.2)</td><td>96.62 (0.3)</td><td>85.35 (0.2)</td><td>93.99 (0.3)</td><td>72.08 (0.3)</td><td>84.99 (0.2)</td></tr></table>
+
+Transductive task examines embeddings of the nodes that have been observed in training, via the future link prediction task and the node classification. To avoid violating temporal constraints, we predict the links that strictly take place posterior to all observations in the training data.
+
+Inductive task examines the inductive learning capability using the inferred representations of unseen nodes, by predicting the future links between unseen nodes and classify them based on their inferred embedding dynamically. We point out that it suffices to only consider the future sub-graph for unseen nodes since they are equivalent to new graphs under the non-temporal setting.
+
+As for the evaluation metrics, in the link prediction tasks, we first sample an equal amount of negative node pairs to the positive links and then compute the average precision $( A P )$ and classification accuracy. In the downstream node classification tasks, due to the label imbalance in the datasets, we employ the area under the ROC curve (AUC).
+
+# 4.3 BASELINES
+
+Transductive task: for link prediction of observed nodes, we choose the compare our approach with the state-of-the-art graph embedding methods: GAE and VGAE (Kipf & Welling, 2016b). For complete comparisons, we also include the skip-gram-based node2vec (Grover & Leskovec, 2016) as well as the spectral-based DeepWalk model (Perozzi et al., 2014), using the same inner-product decoder as GAE for link prediction. The CDTNE model based on the temporal random walk has been reported with superior performance on transductive learning tasks (Nguyen et al., 2018), so we include CDTNE as the representative for temporal graph embedding approaches.
+
+Inductive task: few approaches are capable of managing inductive learning on graphs even in the non-temporal setting. As a consequence, we choose GraphSAGE and GAT as baselines after adapting them to the temporal setting. In particular, we equip them with the same temporal sub-graph batching describe in Section 3.4 to maximize their usage on temporal information. Also, we implement the extended version for the baselines to include edge features in the same way as ours (in Section 3.3). We experiment on different aggregation functions for GraphSAGE, i.e. Graph
+
+SAGE-mean, GraphSAGE-pool and GraphSAGE-LSTM. In accordance with the original work of Hamilton et al. (2017a), GraphSAGE-LSTM gives the best validation performance among the three approaches, which is reasonable under temporal setting since LSTM aggregation takes account of the sequential information. Therefore we report the results of GraphSAGE-LSTM.
+
+In addition to the above baselines, we implement a version of TGAT with all temporal attention weights set to equal value (Const-TGAT). Finally, to show that the superiority of our approach owes to both the time encoding and the network architecture, we experiment with the enhanced GAT and GraphSAGE-mean by concatenating the proposed time encoding to the original features during temporal aggregations ( $G A T { + } T$ and $G r a p h S A G E { + } T$ ).
+
+<table><tr><td>Dataset</td><td>Reddit</td><td>Wikipedia</td><td>Industrial</td></tr><tr><td>GAE VGAE</td><td>58.39 (0.5)</td><td>74.85 (0.6)</td><td>76.59 (0.3)</td></tr><tr><td></td><td>57.98 (0.6)</td><td>73.67 (0.8)</td><td>75.38 (0.4)</td></tr><tr><td>CTDNE</td><td>59.43 (0.6)</td><td>75.89 (0.5)</td><td>78.36 (0.5)</td></tr><tr><td>GAT</td><td>64.52 (0.5)</td><td>82.34 (0.8)</td><td>87.43 (0.4)</td></tr><tr><td>GAT+T</td><td>64.76 (0.6)</td><td>82.95 5(0.7)</td><td>88.24 (0.5)</td></tr><tr><td>GraphSAGE</td><td>61.24 (0.6)</td><td>82.42 (0.7)</td><td>88.28 (0.3)</td></tr><tr><td>GraphSAGE+T</td><td>62.31 (0.7)</td><td>82.87 (0.6)</td><td>89.81 (0.3)</td></tr><tr><td>Const-TGAT</td><td>60.97 (0.5)</td><td>75.18 (0.7)</td><td>82.59 (0.6)</td></tr><tr><td>TGAT</td><td>65.56 (0.7)</td><td>83.69 (0.7)</td><td>92.31 (0.3)</td></tr></table>
+
+![](images/2c825379ae24193e814c3707276ecd6ce1fecedcfb29efd3f0285259783e4f13.jpg)  
+Figure 3: Results of node classification task in the ablation study.
+
+Table 3: Dynamic node classification task results, where the reported metric is the AUC.
+
+# 4.4 EXPERIMENT SETUP
+
+We use the time-sensitive link prediction loss function for training the $l$ -layer TGAT network:
+
+$$
+\ell = \sum _ { ( v _ { i } , v _ { j } , t _ { i j } ) \in \mathcal E } - \log \Big ( \sigma \big ( - \tilde { \mathbf { h } } _ { i } ^ { l } ( t _ { i j } ) ^ { \top } \tilde { \mathbf { h } } _ { j } ^ { l } ( t _ { i j } ) \big ) \Big ) - Q . \mathbb E _ { v _ { q } \sim P _ { n } ( v ) } \log \Big ( \sigma \big ( \tilde { \mathbf { h } } _ { i } ^ { l } ( t _ { i j } ) ^ { \top } \tilde { \mathbf { h } } _ { q } ^ { l } ( t _ { i j } ) \big ) \Big ) ,
+$$
+
+where the summation is over the observed edges on $v _ { i }$ and $v _ { j }$ that interact at time $t _ { i j }$ , and $\sigma ( . )$ is the sigmoid function, $Q$ is the number of negative samples and $P _ { n } ( v )$ is the negative sampling distribution over the node space. As for tuning hyper-parameters, we fix the node embedding dimension and the time encoding dimension to be the original feature dimension for simplicity, and then select the number of TGAT layers from $\{ 1 , 2 , 3 \}$ , the number of attention heads from $\{ 1 , \dot { 2 } , 3 , 4 , 5 \}$ , according to the link prediction $A P$ score in the validation dataset. Although our method does not put restriction on the neighborhood size during aggregations, to speed up training, specially when using the multi-hop aggregations, we use neighborhood dropout (selected among $p = \{ 0 . 1 , 0 . 3 , 0 . 5 \} )$ with the uniform sampling. During training, we use 0.0001 as learning rate for Reddit and Wikipedia dataset and 0.001 for the industrial dataset, with Glorot initialization and the Adam SGD optimizer. We do not experiment on applying regularization since our approach is parameter-efficient and only requires $\Omega \big ( ( \dot { d } + d _ { T } ) d _ { h } + \dot { ( d _ { h } + d _ { 0 } ) d _ { f } } + d _ { f } d \big )$ parameters for each attention head, which is independent of the graph and neighborhood size. Using two TGAT layers and two attention heads with dropout rate as 0.1 give the best validation performance. For inference, we inductively compute the embeddings for both the unseen and observed nodes at each time point that the graph evolves, or when the node labels are updated. We then use these embeddings as features for the future link prediction and dynamic node classifications with multilayer perceptron.
+
+We further conduct ablation study to demonstrate the effectiveness of the proposed functional time encoding approach. We experiment on abandoning time encoding or replacing it with the original positional encoding (both fixed and learnt). We also compare the uniform neighborhood dropout to sampling with inverse timespan (where the recent edges are more likely to be sampled), which is provided in supplement material along with other implementation details and setups for baselines.
+
+# 4.5 RESULTS
+
+The results in Table 1 and Table 2 demonstrates the state-of-the-art performances of our approach on both transductive and inductive learning tasks. In the inductive learning task, our TGAT network significantly improves upon the the upgraded GraphSAGE-LSTM and GAT in accuracy and average precision by at least $5 \%$ for both metrics, and in the transductive learning task TGAT consistently outperforms all baselines across datasets. While $G A T { + } T$ and GraphSAGE+T slightly outperform or tie with GAT and GraphSAGE-LSTM, they are nevertheless outperformed by our approach. On one hand, the results suggest that the time encoding have potential to extend non-temporal graph representation learning methods to temporal settings. On the other, we note that the time encoding still works the best with our network architecture which is designed for temporal graphs. Overall, the results demonstrate the superiority of our approach in learning representations on temporal graphs over prior models. We also see the benefits from assigning temporal attention weights to neighboring nodes, where GAT significantly outperforms the Const-TGAT in all three tasks. The dynamic node classification outcome (in Table 3) further suggests the usefulness of our time-aware node embeddings for downstream tasks as they surpass all the baselines. The ablation study results of Figure 3 successfully reveals the effectiveness of the proposed functional time encoding approach in capturing temporal signals as it outperforms the positional encoding counterparts.
+
+# 4.6 ATTENTION ANALYSIS
+
+To shed some insights into the temporal signals captured by the proposed TGAT, we analyze the pattern of the attention weights $\{ \bar { \alpha _ { i j } } ( t ) \}$ as functions of both time $t$ and node pairs $( i , j )$ in the inference stage. Firstly, we analyze how the attention weights change with respect to the timespans of previous interactions, by plotting the attention weights $\{ \breve { \alpha } _ { j q } ( t _ { i j } ) | \breve { q } \in \mathcal { N } ( v _ { j } ; \dot { t } _ { i j } ) \} \cup \{ \alpha _ { i k } ( t _ { i j } ) | k \in$ $\mathcal { N } ( v _ { i } ; t _ { i j } ) \big \}$ against the timespans $\{ t _ { i j } - t _ { j q } \} \cup \{ t _ { i j } - t _ { i k } \}$ when predicting the link for $( v _ { i } , v _ { j } , t _ { i j } ) \in \mathcal { E }$ (Figure 4a). This gives us an empirical estimation on the $\alpha ( \Delta t )$ , where a smaller $\Delta t$ means a more recent interaction. Secondly, we analyze how the topological structures affect the attention weights as time elapses. Specifically, we focus on the topological structure of the recurring neighbours, by finding out what attention weights the model put on the neighbouring nodes with different number of reoccurrences. Since the functional forms of all $\{ \alpha _ { i j } ( . ) \}$ are fixed after training, we are able to feed in different target time $t$ and then record their value on neighbouring nodes with different number of occurrences (Figure 4b). From Figure 4a we observe that TGAT captures the pattern of having less attention on more distant interactions in all three datasets. In Figure 4b, it is obvious that when predicting a more future interaction, TGAT will consider neighbouring nodes who have a higher number of occurrences of more importance. The patterns of the attention weights are meaningful, since the more recent and repeated actions often have larger influence on users’ future interests.
+
+![](images/a34091e2ba83cd5ddd835e31e60c68387967347a6b04ef7ab0aaab44d9b47248.jpg)  
+Figure 4: Attention weight analysis. We apply the Loess smoothing method for visualization.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+We introduce a novel time-aware graph attention network for inductive representation learning on temporal graphs. We adapt the self-attention mechanism to handle the continuous time by proposing a theoretically-grounded functional time encoding. Theoretical and experimental analysis demonstrate the effectiveness of our approach for capturing temporal-feature signals in terms of both node and topological features on temporal graphs. Self-attention mechanism often provides useful model interpretations (Vaswani et al., 2017), which is an important direction of our future work. Developing tools to visualize the evolving graph dynamics and temporal representations efficiently is another important direction for both research and application. Also, the functional time encoding technique has huge potential for adapting other deep learning methods to the temporal graph domain.
+
+# REFERENCES
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.
+
+Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al. Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv:1806.01261, 2018.
+
+Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on ¨ graphs with fast localized spectral filtering. In Advances in neural information processing systems, pp. 3844–3852, 2016.
+
+Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. arXiv preprint arXiv:1605.08803, 2016.
+
+Matthias Fey and Jan E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019.
+
+Palash Goyal, Nitin Kamra, Xinran He, and Yan Liu. Dyngem: Deep embedding method for dynamic graphs. arXiv preprint arXiv:1805.11273, 2018.
+
+Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 855–864. ACM, 2016.
+
+Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems, pp. 1024–1034, 2017a.
+
+William L Hamilton, Rex Ying, and Jure Leskovec. Representation learning on graphs: Methods and applications. arXiv preprint arXiv:1709.05584, 2017b.
+
+Mikael Henaff, Joan Bruna, and Yann LeCun. Deep convolutional networks on graph-structured data. arXiv preprint arXiv:1506.05163, 2015.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
+
+Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016a.
+
+Thomas N Kipf and Max Welling. Variational graph auto-encoders. arXiv preprint arXiv:1611.07308, 2016b.
+
+Taisong Li, Jiawei Zhang, S Yu Philip, Yan Zhang, and Yonghong Yan. Deep dynamic network embedding for link prediction. IEEE Access, 6:29219–29230, 2018.
+
+Lynn H Loomis. Introduction to abstract harmonic analysis. Courier Corporation, 2013.
+
+Yao Ma, Ziyi Guo, Eric Zhao Zhaochun Ren, and Dawei Yin Jiliang Tang. Streaming graph neural networks. arXiv preprint arXiv:1810.10627, 2018.
+
+Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M Bronstein. Geometric deep learning on graphs and manifolds using mixture model cnns. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5115– 5124, 2017.
+
+Giang Hoang Nguyen, John Boaz Lee, Ryan A Rossi, Nesreen K Ahmed, Eunyee Koh, and Sungchul Kim. Continuous-time dynamic network embeddings. In Companion Proceedings of the The Web Conference 2018, pp. 969–976. International World Wide Web Conferences Steering Committee, 2018.
+
+James W Pennebaker, Martha E Francis, and Roger J Booth. Linguistic inquiry and word count: Liwc 2001. Mahway: Lawrence Erlbaum Associates, 71(2001):2001, 2001.
+
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 701–710. ACM, 2014.
+
+Ali Rahimi and Benjamin Recht. Random features for large-scale kernel machines. In Advances in neural information processing systems, pp. 1177–1184, 2008.
+
+Mahmudur Rahman, Tanay Kumar Saha, Mohammad Al Hasan, Kevin S Xu, and Chandan K Reddy. Dylink2vec: Effective feature representation for link prediction in dynamic networks. arXiv preprint arXiv:1804.05755, 2018.
+
+Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015.
+
+Martin Simonovsky and Nikos Komodakis. Dynamic edge-conditioned filters in convolutional neural networks on graphs. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 3693–3702, 2017.
+
+Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015.
+
+Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large-scale information network embedding. In Proceedings of the 24th international conference on world wide web, pp. 1067–1077. International World Wide Web Conferences Steering Committee, 2015.
+
+Rakshit Trivedi, Mehrdad Farajtabar, Prasenjeet Biswal, and Hongyuan Zha. Representation learning over dynamic graphs. arXiv preprint arXiv:1803.04051, 2018.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017.
+
+Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua ´ Bengio. Graph attention networks. arXiv preprint arXiv:1710.10903, 2017.
+
+Daixin Wang, Peng Cui, and Wenwu Zhu. Structural deep network embedding. In Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 1225–1234. ACM, 2016.
+
+Jizhe Wang, Pipei Huang, Huan Zhao, Zhibo Zhang, Binqiang Zhao, and Dik Lun Lee. Billion-scale commodity embedding for e-commerce recommendation in alibaba. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 839–848. ACM, 2018a.
+
+Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E Sarma, Michael M Bronstein, and Justin M Solomon. Dynamic graph cnn for learning on point clouds. arXiv preprint arXiv:1801.07829, 2018b.
+
+Da Xu, Chuanwei Ruan, Evren Korpeoglu, Sushant Kumar, and Kannan Achan. Self-attention with functional time representation learning. In Advances in Neural Information Processing Systems, pp. 15889–15899, 2019a.
+
+Da Xu, Chuanwei Ruan, Kamiya Motwani, Evren Korpeoglu, Sushant Kumar, and Kannan Achan. Generative graph convolutional network for growing graphs. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3167–3171. IEEE, 2019b.
+
+Da Xu, Chuanwei Ruan, Evren Korpeoglu, Sushant Kumar, and Kannan Achan. Product knowledge graph embedding for e-commerce. In Proceedings of the 13th International Conference on Web Search and Data Mining, pp. 672–680, 2020.
+
+Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L Hamilton, and Jure Leskovec. Graph convolutional neural networks for web-scale recommender systems. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 974– 983. ACM, 2018.
+
+# A APPENDIX
+
+# A.1 PROOF FOR CLAIM 1
+
+Proof. The proof is also shown in our concurrent work $\mathrm { X u }$ et al. (2019a). We also provide it here for completeness. To prove the results in Claim 1, we alternatively show that under the same condition,
+
+$$
+\operatorname* { P r } \bigl ( \operatorname* { s u p } _ { t _ { 1 } , t _ { 2 } \in T } \bigl | \Phi _ { d } ^ { B } ( t _ { 1 } ) ^ { ' } \Phi _ { d } ^ { B } ( t _ { 2 } ) - K ( t _ { 1 } , t _ { 2 } ) \bigr | \geq \epsilon \bigr ) \leq 4 \sigma _ { p } \sqrt { \frac { t _ { \operatorname* { m a x } } } { \epsilon } } e x p \bigl ( \frac { - d \epsilon ^ { 2 } } { 3 2 } \bigr ) .
+$$
+
+Define the score $S ( t _ { 1 } , t _ { 2 } ) ~ = ~ \Phi _ { d } ^ { B } ( t _ { 1 } ) ^ { ' } \Phi _ { d } ^ { B } ( t _ { 2 } )$ . The goal is to derive a uniform upper bound for $s ( t _ { 1 } , t _ { 2 } ) \ - \ \mathcal { K } ( t _ { 1 } , t _ { 2 } )$ . By assumption $S ( t _ { 1 } , t _ { 2 } )$ is an unbiased estimator for $\displaystyle \kappa ( t _ { 1 } , t _ { 2 } )$ , i.e. $E [ S ( t _ { 1 } , t _ { 2 } ) ] = \mathcal { K } ( t _ { 1 } , t _ { 2 } )$ . Due to the translation-invariant property of $S$ and $\kappa$ , we let $\Delta ( t ) \equiv$ $s ( t _ { 1 } , t _ { 2 } ) - \mathcal { K } ( t _ { 1 } , t _ { 2 } )$ , where $t \equiv t _ { 1 } - t _ { 2 }$ for all $t _ { 1 } , t _ { 2 } \in [ 0 , \bar { t } _ { \mathrm { m a x } } ]$ . Also we define $s ( t _ { 1 } - t _ { 2 } ) : =$ $S ( t _ { 1 } , t _ { 2 } )$ . Therefore $t \in [ - t _ { \operatorname* { m a x } } , t _ { \operatorname* { m a x } } ]$ , and we use $t \in \tilde { T }$ as the shorthand notation. The LHS in (1) now becomes $\mathrm { P r } \big ( \operatorname* { s u p } _ { t \in \tilde { T } } | \Delta ( t ) | \geq \epsilon \big )$ .
+
+t  i $\tilde { T } \subseteq \cup _ { i = 0 } ^ { N - 1 } T _ { i }$ wi of . $\begin{array} { r } { T _ { i } = [ - t _ { \mathrm { m a x } } + \frac { 2 i t _ { \mathrm { m a x } } } { N } , - t _ { \mathrm { m a x } } + \frac { 2 ( i + 1 ) t _ { \mathrm { m a x } } } { N } ] } \end{array}$ 2(i+1)tmax ] for i = 1, . . . , N . So $\cup _ { i = 0 } ^ { N - 1 } T _ { i }$ $\tilde { T }$ $\begin{array} { r } { t _ { i } = - t _ { \operatorname* { m a x } } + \frac { ( 2 i + 1 ) t _ { \operatorname* { m a x } } } { N } } \end{array}$ (2i+1)tmaxN , then for any t ∈ Ti, i = 1, . . . , N we have
+
+$$
+\begin{array} { r l } { | \Delta ( t ) | = | \Delta ( t ) - \Delta ( t _ { i } ) + \Delta ( t _ { i } ) | } & { } \\ { \leq | \Delta ( t ) - \Delta ( t _ { i } ) | + | \Delta ( t _ { i } ) | } & { } \\ { \leq L _ { \Delta } | t - t _ { i } | + | \Delta ( t _ { i } ) | } & { } \\ { \leq L _ { \Delta } \frac { 2 t _ { \operatorname* { m a x } } } { N } + | \Delta ( t _ { i } ) | , } \end{array}
+$$
+
+where $L _ { \Delta } = \operatorname* { m a x } _ { t \in \tilde { T } } \| \nabla \Delta ( t ) \|$ (since $\Delta$ is differentiable) with the maximum achieved at $t ^ { * }$ . So we may bound the two events separately.
+
+For $| \Delta ( t _ { i } ) |$ we simply notice that trigeometric functions are bounded between $[ - 1 , 1 ]$ , and therefore $- 1 \leq \Phi _ { d } ^ { B } ( t _ { 1 } ) ^ { \prime } \Phi _ { d } ^ { B } ( t _ { 2 } ) \leq 1$ . The Hoeffding’s inequality for bounded random variables immediately gives us:
+
+$$
+\operatorname* { P r } \bigl ( | \Delta ( t _ { i } ) | > \frac { \epsilon } { 2 } \bigr ) \leq 2 e x p ( - \frac { d \epsilon ^ { 2 } } { 1 6 } ) .
+$$
+
+So applying the Hoeffding-type union bound to the finite cover gives
+
+$$
+\mathrm { P r } \big ( \cup _ { i = 0 } ^ { N - 1 } | \Delta ( t _ { i } ) | \geq \frac { \epsilon } { 2 } \big ) \leq 2 N \exp ( - \frac { d \epsilon ^ { 2 } } { 1 6 } )
+$$
+
+For the other event we first apply Markov inequality and obtain:
+
+$$
+\operatorname* { P r } \bigl ( L _ { \Delta } \frac { 2 t _ { \mathrm { m a x } } } { N } \geq \frac { \epsilon } { 2 } \bigr ) = \operatorname* { P r } \bigl ( L _ { \Delta } \geq \frac { \epsilon N } { 4 t _ { \mathrm { m a x } } } \bigr ) \leq \frac { 4 t _ { \mathrm { m a x } } E [ L _ { \Delta } ^ { 2 } ] } { \epsilon N } .
+$$
+
+Also, since $E [ s ( t _ { 1 } - t _ { 2 } ) ] = \psi ( t _ { 1 } - t _ { 2 } )$ , we have
+
+$$
+\begin{array} { r } { E [ L _ { \Delta } ^ { 2 } ] = E \| \nabla s ( t ^ { * } ) - \nabla \psi ( t ^ { * } ) \| ^ { 2 } = E \| \nabla s ( t ^ { * } ) \| ^ { 2 } - E \| \nabla \psi ( t ^ { * } ) \| ^ { 2 } \leq E \| \nabla s ( t ^ { * } ) \| ^ { 2 } = \sigma _ { p } ^ { 2 } , } \end{array}
+$$
+
+where $\sigma _ { p } ^ { 2 }$ is the second momentum with respect to $p ( \omega )$ .
+
+Combining (11), (12) and (11) gives us:
+
+$$
+\operatorname* { P r } \bigl ( \operatorname* { s u p } _ { t \in \tilde { T } } | \Delta ( t ) | \geq \epsilon \bigr ) \leq 2 N \exp ( - \frac { d \epsilon ^ { 2 } } { 1 6 } ) + \frac { 4 t _ { \mathrm { m a x } } \sigma _ { p } ^ { 2 } } { \epsilon N } .
+$$
+
+It is straightforward to examine that the RHS of (14) is a convex function of $N$ and is minimized by $\begin{array} { r } { N ^ { * } = \sigma _ { p } \sqrt { \frac { 2 t _ { \mathrm { m a x } } } { \epsilon } } e x p ( \frac { d \epsilon ^ { 2 } } { 3 2 } ) } \end{array}$ . Plug $N ^ { * }$ back to (14) and we obtain (9). We then solve for $d$ according to (9) and obtain the results in Claim 1.
+
+In this part, we provide detailed comparisons between the attention mechanism employed by our proposed TGAT and the GAT proposed by Velickovi ˇ c et al. (2017). Other than the obvious fact that ´ GAT does not handle temporal information, the main difference lies in the formulation of attention weights. While GAT depends on the attention mechanism proposed by Bahdanau et al. (2014), our architecture refers to the self-attention mechanism of Vaswani et al. (2017). Firstly, the attention mechanism used by GAT does not involve the notions of ’query’, ’key’ and ’value’ nor the dotproduct formulation introduced in (2). As a consequence, the attention weight between node $v _ { i }$ and its neighbor $v _ { j }$ is computed via
+
+$$
+\alpha _ { i j } = \frac { \mathrm { { e x p } \Big ( L e a k y R e L U \big ( \mathbf { a } ^ { \top } [ \mathbf { W } \mathbf { h } _ { i } | | \mathbf { W } \mathbf { h } _ { j } ] \big ) \Big ) } } { { \sum _ { k \in { \mathcal { N } ( v _ { i } ) } } \mathrm { { e x p } \Big ( L e a k y R e L U \big ( \mathbf { a } ^ { \top } [ \mathbf { W } \mathbf { h } _ { i } | | \mathbf { W } \mathbf { h } _ { k } ] \big ) \Big ) } } } ,
+$$
+
+where $\mathbf { a }$ is a weight vector, $\mathbf { W }$ is a weight matrix, $\mathcal { N } ( v _ { i } )$ is the neighorhood set for node $v _ { i }$ and $\mathbf { h } _ { i }$ is the hidden representation of node $v _ { i }$ . It is then obvious that their computation of $\alpha _ { i j }$ is very different from our approach. In TGAT, after expanding the expressions in Section 3, the attention weight is computed by:
+
+$$
+\alpha _ { i j } ( t ) = \frac { \exp \Big ( \big ( | \tilde { \mathbf { h } } _ { i } ( t _ { i } ) | | \Phi _ { d _ { T } } ( t - t _ { i } ) | \mathbf { W } _ { Q } \big ) ^ { \mathsf { T } } \big ( | \tilde { \mathbf { h } } _ { j } ( t _ { j } ) | | \Phi _ { d _ { T } } ( t - t _ { j } ) | \mathbf { W } _ { K } \big ) \Big ) } { \sum _ { k \in \mathcal { N } ( v _ { i } ; t ) } \exp \Big ( \big ( [ \tilde { \mathbf { h } } _ { i } ( t _ { i } ) | | \Phi _ { d _ { T } } ( t - t _ { i } ) ] \mathbf { W } _ { Q } \big ) ^ { \mathsf { T } } \big ( [ \tilde { \mathbf { h } } _ { k } ( t _ { k } ) | | \Phi _ { d _ { T } } ( t - t _ { k } ) ] \mathbf { W } _ { K } \big ) \Big ) } .
+$$
+
+Intuitively speaking, the attention mechanism of $G A T$ relies on the parameter vector $\mathbf { a }$ and the LeakyReLU(.) to capture the hidden factor interactions between entities in the sequence, while we use the linear transformation followed by the dot-product to capture pair-wise interactions of the hidden factors between entities and the time embeddings. The dot-product formulation is important for our approach. From the theoretical perspective, the time encoding functional form is derived according to the notion of temporal kernel $\kappa$ and its inner-product decomposition (Section 3). As for the practical performances, we see from Table 1, 2 and 3 that even after we equip GAT with the same time encoding, the performance is still inferior to our TGAT.
+
+# A.3 DETAILS ON DATASETS AND PREPROCESSING
+
+Reddit dataset: this benchmark dataset contains users interacting with subreddits by posting under the subreddits. The timestamps tell us when the user makes the posts. The dataset uses the posts made in a one-month span, and selects the most active users and subreddits as nodes, giving a total of 11,000 nodes and around 700,000 temporal edges. The user posts have textual features that are transformed into a 172-dimensional vector representing under the linguistic inquiry and word count (LIWC) categories (Pennebaker et al., 2001). The dynamic binary labels indicate if a user is banned from posting under a subreddit. Since node features are not provided in the original dataset, we use the all-zero vector instead.
+
+Wikipedia dataset: the dataset also collects one-month of interactions induced by users’ editing the Wikipedia pages. The the top edited pages and active users are considered, leading to ${ \sim } 9 { , } 3 0 0$ nodes and around 160,000 temporal edges. Similar to the Reddit dataset, we also have the ground-truth dynamic labels on whether a user is banned from editing a Wikipedia page. User edits consist of the textual features and are also converted into 172-dimensional LIWC feature vectors. Node features are also not provided, so we also use the all-zero vector as well.
+
+Industrial dataset: we obtain the large-scale customer-product interaction graph from the online grocery shopping platform grocery.walmart.com. We select $\sim 7 0 { , } 0 0 0$ most popular products and 100,000 active customers as nodes and use the customer-product purchase interactions over a one-month period as temporal edges ${ \sim } 2$ million). Each purchase interaction is timestamped, which we use to construct the temporal graph. The customers are labelled with business tags, indicating if they are interested in dietary products according to their most recent purchase records. Each product node possesses contextual features containing their name, brand, categories and short description. The previous LIWC categories no longer apply since the product contextual features are not natural sentences. We use product embedding approach (Xu et al., 2020) to embed each product’s contextual
+
+<table><tr><td></td><td>Reddit</td><td>Wikipedia</td><td>Industrial</td></tr><tr><td># Nodes</td><td>11,000</td><td>9,227</td><td>170,243</td></tr><tr><td># Edges</td><td>672,447</td><td>157,474</td><td>2,135,762</td></tr><tr><td>#Feature dimension</td><td>172</td><td>172</td><td>100</td></tr><tr><td># Feature type</td><td>LIWC category vector</td><td>LIWC category</td><td>document embeddings</td></tr><tr><td># Timespan</td><td>30 days</td><td>vector 30 days</td><td>30 days</td></tr><tr><td>% Training nodes</td><td>90%</td><td>90%</td><td>90%</td></tr><tr><td>% Unseen nodes</td><td>10%</td><td>10%</td><td>10%</td></tr><tr><td>% Training edges</td><td>~67%</td><td>~65%</td><td>~64%</td></tr><tr><td>% Future edges between observed nodes</td><td>~27%</td><td>~28%</td><td>~29%</td></tr><tr><td>% Future edges between unseen nodes</td><td>~6%</td><td>~7%</td><td>~7%</td></tr><tr><td># Nodes with dynamic labels</td><td>366</td><td>217</td><td>5,236</td></tr><tr><td>Label type</td><td>binary</td><td>binary</td><td>binary</td></tr><tr><td></td><td>banned from</td><td>banned from</td><td>interested in</td></tr><tr><td>Positive label meaning</td><td>posting</td><td>editting</td><td>dietary products</td></tr></table>
+
+Table 4: Data statistics for the three datasets. Since we sample a proportion of unseen nodes, the percentage of the edge statistics reported here are approximations.
+
+features into a 100-dimensional vector space as preprocessing. The user nodes and edges do not possess features.
+
+We then split the temporal graphs chronologically into $7 0 \% - 1 5 \% - 1 5 \%$ for training, validation and testing according to the time epochs of edges, as illustrated in Figure 5 with the Reddit dataset. Since all three datasets have a relatively stationary edge count distribution over time, using the 70 and 85 percentile time points to split the dataset results in approximately $7 0 \% - 1 5 \% - 1 5 \%$ of total edges, as suggested by Figure 5.
+
+![](images/fd302ebc40c99b608adfc802a13d17d876dac4a25204dafee97b64f3700eca75.jpg)  
+Figure 5: The distribution of temporal edge count for the Reddit dataset, and the illustration on the train-validation-test splitting.
+
+To ensure that an appropriate amount of future edges among the unseen nodes will show up during validation and testing, for each dataset, we randomly sample $10 \%$ of nodes, mask them during training and treat them as unseen nodes by only considering their interactions in validation and testing period. This manipulation is necessary since the new nodes that show up during validation and testing period may not have much interaction among themselves. The statistics for the three datasets are summarized in Table 4.
+
+# Preprocessing.
+
+For the Node2vec and DeepWalk baselines who only take static graphs as input, the graph is constructed using all edges in training data regardless of temporal information. For DeepWalk, we treat the recurrent edges as appearing only once, so the graph is unweighted. Although our approach handles both directed and undirected graphs, for the sake of training stability of the baselines, we treat the graphs as undirected. For Node2vec, we use the count of recurrent edges as their weights and construct the weighted graph. For all three datasets, the obtained graphs in both cases are undirected and do not have isolated nodes. Since we choose from active users and popular items, the graphs are all connected.
+
+For the graph convolutional network baselines, i.e. GAE and VGAE, we construct the same undirected weighted graph as for Node2vec. Since GAE and VGAE do not take edge features as input, we use the posts/edits as user node features. For each user in Reddit and Wikipedia dataset, we take the average of their post/edit feature vectors as the node feature. For the industrial dataset where user features are not available, we use the all-zero feature vector instead.
+
+As for the downstream dynamic node classification task, we use the same training, validation and testing dataset as above. Since we aim at predicting the dynamic node labels, for Reddit and Wikipedia dataset we predict if the user node is banned and for the industrial dataset we predict the customers’ business labels, at different time points. Due to the label imbalance, in each of the batch when training for the node label classifier, we conduct stratified sampling such that the label distributions are similar across batches.
+
+# A.4 EXPERIMENT SETUP FOR BASELINES
+
+For all baselines, we set the node embedding dimension to $d = 1 0 0$ to keep in accordance with our approach.
+
+# Transductive baselines.
+
+Since Node2vec and DeepWalk do not provide room for task-specific manipulation or hacking, we do not modify their default loss function and input format. For both approaches, we select the number of walks among $\{ 6 0 , 8 0 , 1 0 0 \}$ and the walk-length among $\{ 2 0 , 3 0 , 4 0 \}$ according to the validation $A P$ . Setting number of walks $scriptstyle = 8 0$ and walk-length $\scriptstyle 1 = 3 0$ give slightly better validation performance compared to others for both approaches. Notice that both Node2vec and DeepWalk use the sigmoid function with embedding inner-products as the decoder to predict neighborhood probabilities. So when predicting whether $v _ { i }$ and $v _ { j }$ will interact in the future, we use $\bar { \sigma ( - \mathbf { z } _ { i } ^ { \intercal } \mathbf { z } _ { j } ) }$ as the score, where $\mathbf { z } _ { i }$ and $\mathbf { z } _ { j }$ are the node embeddings. Notice that Node2vec has the extra hyper-parameter $p$ and $q$ which controls the likelihood of immediately revisiting a node in the walk and interpolation between breadth-first strategy and depth-first strategy. After selecting the optimal number of walks and walklength under $p = 1$ and $q = 1$ , we further tune the different values of $p$ in $\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 . 0 \}$ while fixing $q = 1$ . According to validation, $p = 0 . 6$ and 0.8 give comparable optimal performance.
+
+For the $G A E$ and VGAE baselines, we experiment on using one, two and three graph convolutional layers as the encoder (Kipf & Welling, 2016a) and use the ReLU(.) as the activation function. By referencing the official implementation, we also set the dimension of hidden layers to 200. Similar to previous findings, using two layers gives significant performances to using only one layer. Adding the third layer, on the other hand, shows almost identical results for both models. Therefore the results reported are based on two-layer GCN as the encoder. For GAE, we use the standard inner-product decoder as our approach and optimize over the reconstruction loss, and for VGAE, we restrict the Gaussian latent factor space (Kipf & Welling, 2016b). Since we have eliminated the temporal information when constructing the input, we find that the optimal hyper-parameters selected according to the tuning have similar patterns as in the previous non-temporal settings.
+
+For the temporal network embedding model CTDNE, the walk length for the temporal random walk is also selected among $\left\{ 6 0 , 8 0 , 1 0 0 \right\}$ , where setting walk length to 80 gives slightly better validation outcome. The original paper considers several temporal edge selection (sampling) methods (uniform, linear and exponential) and finds uniform sampling with best performances (Nguyen et al., 2018). Since our setting is similar to theirs, we adopt the uniform sampling approach.
+
+# Inductive baselines.
+
+For the GraphSAGE and GAT baselines, as mentioned before, we train the models in an identical way as our approach with the temporal subgraph batching, despite several slight differences. Firstly, the aggregation layers in GraphSAGE usually considers a fixed neighborhood size via sampling, whereas our approach can take an arbitrary neighborhood as input. Therefore, we only consider the most recent $d _ { \mathrm { s a m p l e } }$ edges during each aggregation for all layers, and we find $d _ { \mathrm { s a m p l e } } = 2 0$ gives the best performance among $\{ 1 0 , 1 \bar { 5 } , 2 0 , 2 5 \}$ . Secondly, GAT implements a uniform neighborhood dropout. We also experiment with the inverse timespan sampling for neighborhood dropout, and find that it gives slightly better performances but at the cost of computational efficiency, especially for large graphs. We consider aggregating over one, two and three-hop neighborhood for both GAT and GraphSAGE. When working with three hops, we only experiment on GraphSAGE with the mean pooling aggregation. In general, using two hops gives comparable performance to using three hops. Notice that computations with three-hop are costly, since the number of edges during aggregation increase exponentially to the number of hops. Thus we stick to using two hops for GraphSAGE, $G A T$ and our approach. It is worth mentioning that when implementing GraphSAGE-LSTM, the input neighborhood sequences of LSTM are also ordered by their interaction time.
+
+# Node classification with baselines.
+
+The dynamic node classification with GraphSAGE and GAT can be conducted similarity to our approach, where we inductively compute the most up-to-date node embeddings and then input them as features to an MLP classifier. For the transductive baselines, it is not reasonable to predict the dynamic node labels with only the fixed node embeddings. Instead, we combine the node embedding with the other node embedding it is interacting with when the label changes, e.g. combine the user embedding with the Wikipedia page embedding that the user attempts on editing when the system bans the user. To combine the pair of node embeddings, we experimented on summation, concatenation and bi-linear transformation. Under summation and concatenation, the combined embeddings are then used as input to an MLP classifier, where the bi-linear transformation directly outputs scores for classification. The validation outcomes suggest that using concatenation with MLP yields the best performance.
+
+# A.5 IMPLEMENTATION DETAILS
+
+Training. We implement Node2vec using the official C code5 on a 16-core Linux server with 500 Gb memory. DeepWalk is implemented with the official python code6. We refer to the PyTorch geometric library for implementing the GAE and VGAE baselines (Fey & Lenssen, 2019). To accommodate the temporal setting and incorporate edges features, we develop off-the-shelf implementation for GraphSAGE and $G A T$ in PyTorch by referencing their original implementations7 8. We also implement our model using PyTorch. All the deep learning models are trained on a machine with one Tesla V100 GPU. We use the Glorot initialization and the Adam SGD optimizer for all models, and apply the early-stopping strategy during training where we terminate the training process if the validation $A P$ score does not improve for 10 epochs.
+
+Downstream node classification. As we discussed before, we use the three-layer MLP as classifier and the (combined) node embeddings as input features from all the experimented approaches, for all three datasets. The MLP is trained with the Glorot initialization and the Adam SGD optimizer in PyTorch as well. The $\ell _ { 2 }$ regularization parameter $\lambda$ is selected in $\{ 0 . 0 0 1 , 0 . 0 1 , 0 . 0 5 , \bar { 0 . 1 } , 0 . 2 \}$ case-by-case during training. The early-stopping strategy is also employed.
+
+# A.6 SENSITIVITY ANALYSIS AND EXTRA ABLATION STUDY
+
+Firstly, we focus on the output node embedding dimension as well as the functional time encoding dimension in this sensitivity analysis. The reported results are averaged over five runs. We experiment on $d \in \{ 6 0 , 8 0 , 1 0 0 , \dot { 1 } 2 0 , 1 \dot { 4 } 0 \}$ and $d _ { T } \in \{ 6 0 , 8 0 , 1 0 0 , 1 2 0 , 1 4 \bar { 0 } \}$ , and the results are reported in Figure 7a and 7c. The remaining model setups reported in Section 4.4 are untouched when varying $d$ or $d _ { T }$ . We observe slightly better outcome when increasing either $d$ or $d _ { T }$ on the industrial dataset. The patterns on Reddit and Wikipedia dataset are almost identical.
+
+Secondly, we compare between the two methods of learning functional encoding, i.e. using flowbased model or using the non-parametric method introduced in Section 3.1. We experiment on two
+
+![](images/2905e5d65cf3724afab5ac971814bb545895b66092f226ce4d914464e30fab28.jpg)  
+(a) Comparison between uniform and inverse timespan weighted sampling on the link prediction task
+
+![](images/57a3318159b69c4a05402389ae0fc1db680bcac7f14edb8bbf40e17350d2348c.jpg)
+
+(b) Comparison between three different ways of learning the functional time encoding, on link prediction task.
+
+![](images/43088b3a68bf07a5e311e925bc4e3e7ddd72b2e3ea118f2295b01bfafaac82f1.jpg)  
+Figure 6: Extra ablation study.
+
+(a) Sensitivity analysis on node embeddings dimension.
+
+![](images/e74ea6d23aa3310e5da6c1ab68aa2d62a55f2261dabc073f31f15916982d7790.jpg)  
+(b) Sensitivity analysis on time embeddings dimension.
+
+![](images/385115506444b730386723f8ec7a45588a6875a4312f03ffc0c5d0513307ed10.jpg)  
+Figure 7: Sensitivity analysis on the Industrial dataset.
+
+(c) Sensitivity analysis on number of attention heads and layers (hops) with $d = 1 0 0$ and $d _ { T } = 1 0 0$ .
+
+flow-based state-of-the-art CDF learning method: normalizing flow (Rezende & Mohamed, 2015) and RealNVP (Dinh et al., 2016). We use the default model setups and hyper-parameters in their reference implementations9 10. We provide the results in Figure 6b. As we mentioned before, using flow-based models leads to highly comparable outcomes as the non-parametric approach, but they require longer training time since they implement sampling during each training batch. However, it is possible that carefully-tuned flow-based models can lead to nontrivial improvements, which we leave to the future work.
+
+Finally, we provide sensitivity analysis on the number of attention heads and layers for TGAT. Recall that by stacking two layers in TGAT we are aggregating information from the two-hop neighbourhood. For both accuracy and $A P$ , using three-head attention and two-layers gives the best outcome. In general, the results are relatively stable to the number of heads, and stacking two layers leads to significant improvements compared with using only a single layer.
+
+The ablation study for comparing between uniform neighborhood dropout and sampling with inverse timespan is given in Figure 6a. The two experiments are carried out under the same setting which we reported in Section 4.4. We see that using the inverse timespan sampling gives slightly worse performances. This is within expectation since uniform sampling has advantage in capturing the recurrent patterns, which can be important for predicting user actions. On the other hand, the results also suggest the effectiveness of the proposed time encoding for capturing such temporal patterns. Moreover, we point out that using the inverse timespan sampling slows down training, particularly for large graphs where a weighted sampling is conducted within a large number of nodes for each training batch construction. Nonetheless, inverse timespan sampling can help capturing the more recent interactions which may be more useful for certain tasks. Therefore, we suggest to choose the neighborhood dropout method according to the specific use cases.