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+# A Probabilistic Representation for Deep Learning: Delving into The Information Bottleneck Principle
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 The Information Bottleneck (IB) principle has recently attracted great attention to   
+2 explaining Deep Neural Networks (DNNs), and the key is to accurately estimate the   
+3 mutual information between a hidden layer and dataset. However, some unsettled   
+4 limitations weaken the validity of the IB explanation for DNNs. To address these   
+5 limitations and fully explain deep learning in an information theoretic fashion, we   
+6 propose a probabilistic representation for deep learning that allows the framework   
+7 to estimate the mutual information, more accurately than existing non-parametric   
+8 models, and also quantify how the components of a hidden layer affect the mutual   
+9 information. Leveraging the probabilistic representation, we take into account the   
+10 back-propagation training and derive two novel Markov chains to characterize the   
+11 information flow in DNNs. We show that different hidden layers achieve different   
+12 IB trade-offs depending on the architecture and the position of the layers in DNNs,   
+13 whereas a DNN satisfies the IB principle no matter the architecture of the DNN.
+
+# 14 1 Introduction
+
+15 Deep learning [18] has already achieved great success in numerous applications. Deep Neural   
+16 Networks (DNNs), however, are still commonly viewed as ‘black boxes’ [27]. Considerable efforts   
+17 have been devoted to explaining the internal mechanism of DNNs from various perspectives, such as   
+18 mathematics [5, 12], statistics [14, 20, 23], computer vision [37, 21], etc. Recently, the Information   
+19 Bottleneck (IB) principle has attracted attention in opening the ‘black boxes’ of DNNs [30, 33].   
+20 Given a joint distribution $P ( X , Y )$ , the IB principle posits a random variable $T = f ( X )$ obeying the   
+21 Markov chain $Y  X  T$ and optimizes $T$ by the IB Lagrangian [32, 31]
+
+$$
+\operatorname* { m i n } _ { P ( T \mid X ) } I ( X ; T ) - \beta I ( Y ; T ) ,
+$$
+
+22 where $f ( \cdot )$ is an arbitrary function, $I ( \cdot ; \cdot )$ denotes mutual information, and the Lagrange multiplier   
+23 $\beta > 0$ controls the IB trade-off between compressing the input $X$ and preserving the information   
+24 of the label $Y$ . In the seminal work [30], Tishby et al. manifest the IB trade-off in every layer of   
+25 $\mathrm { D N N s } = \{ \pmb { x } ; t _ { 1 } ; \cdot \cdot \cdot ; t _ { I } ; \hat { \pmb { y } } \}$ via studying $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ , where $T _ { i }$ is the random variable of   
+26 the ith hidden layer $\mathbf { \Delta } \mathbf { t } _ { i }$ . Especially, the authors ascribe DNN generalization to the compression [29].   
+27 In the context of deterministic DNNs, recent works reveal some limitations of the IB principle for   
+28 explaining DNNs. Amjad et al. argue that the IB principle becomes an ill-posed optimization problem   
+29 due to $I ( { \bar { X } } ; T _ { i } ) = \infty$ [1], and Kolchinsky et al. demonstrate that not every layer of DNNs satisfies a   
+30 strict IB trade-off, i.e., different layers only differ in $I ( X ; T _ { i } )$ but $I ( Y ; T _ { i } )$ keeps consistent in all   
+31 layers [15]. In addition, Saxe et al. experimentally show that the compression does not occur in   
+32 DNNs with non-saturating activation functions, e.g., the popular ReLU function [28], and Goldfeld   
+33 et al. doubt the causality between the generalization of DNNs and the compression [10, 7]. These   
+34 unsettled limitations greatly weakens the validity of the IB explanations for DNNs.   
+35 The key to examining the IB principle in DNNs is the accurate estimation of the mutual information.   
+36 However, regarding DNNs as deterministic models hinders us from specifying the random variable   
+37 $T _ { i }$ and the distribution $P ( T _ { i } )$ , thus it is difficult to accurately estimate $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ . More   
+38 specifically, in the absence of a clear definition of $T _ { i }$ , simply assuming the activations of $\mathbf { \Delta } _ { t _ { i } }$ as the i.i.d.   
+39 samples of $T _ { i }$ induces $T _ { i }$ being a continuous random variable and $I ( X ; T _ { i } ) = \infty$ in deterministic   
+40 DNNs (see Appendix C in [28]). The complicated architecture of DNNs makes it challenging to   
+41 specify $P ( T _ { i } )$ . Therefore, most previous works have to indirectly estimate $P ( T _ { i } )$ via non-parametric   
+42 models [35], such as the empirical distribution [30], Kernel Density Estimation (KDE) [28], and   
+43 Gaussian convolution [10]. However, we experimentally confirm that classical non-parametric models   
+44 derives poor mutual information estimation [24, 22] in DNNs, and one reason is because activations   
+45 do not satisfy the i.i.d. prerequisite of non-parametric models (see Appendix G). In summary, the   
+46 limitations mainly stem from the lack of an explicit probabilistic representation for deep learning.   
+47 The IB principle only formulates the information flow in $\mathbf { D N N s } = \{ \pmb { x } , \pmb { t } _ { 1 } , \cdot \cdot \cdot , \pmb { t } _ { I } , \hat { \pmb { y } } \}$ after training,   
+48 and the corresponding Markov chain (see Fig. 1 in [30])
+
+$$
+Y  X  T _ { 1 } \cdots  T _ { I }  { \hat { Y } }
+$$
+
+49 indicates that the information of $Y$ transfers to $T _ { i }$ in the forward direction and $T _ { i }$ receives the   
+50 information of $Y$ only via $X$ . However, training DNNs by the back-propagation [25] implies that the   
+51 information of $Y$ transfers to $T _ { i }$ in the backward direction during training and retains information   
+52 in $T _ { i }$ after training. Notably, Zhang et al. show that a DNN can fit labels well even using Gaussian   
+53 noise as input to train the DNN [38], which implies that $T _ { i }$ can directly receive the information of $Y$ .   
+54 Hence, the IB principle does not comprehensively characterize the information flow in DNNs.   
+55 To address the above limitations and comprehensively explain DNNs in an information theoretic   
+56 fashion, we introduce the probability space $\left( \Omega _ { T _ { i } } , \mathcal { F } , P _ { T _ { i } } \right)$ [6] for the ith hidden layer $\mathbf { \Delta } _ { t _ { i } }$ in DNNs.   
+57 Compared to previous works, the probability space $\left( \Omega _ { T _ { i } } , \mathcal { F } , P _ { T _ { i } } \right)$ enables us to: (i) accurately estimate   
+58 $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ via specifying $T _ { i }$ and $P ( T _ { i } )$ , and (ii) quantify the effect of the architecture of   
+59 $\mathbf { \Delta } _ { t _ { i } }$ and the back-propagation on $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ via explicitly modeling all the ingredients of $\mathbf { \Delta } _ { t _ { i } }$ ,   
+60 such as the activation function and the weights in a probabilistic way. To the best of our knowledge,   
+61 this is the first time the probability space of a hidden layer in DNNs is as defined.
+
+Leveraging 62 $\left( \Omega _ { T _ { i } } , \mathcal { F } , P _ { T _ { i } } \right)$ , we derive information theoretic explanations for DNNs as follows:
+
+• Two Markov chains1 characterize the information flow in $\mathrm { D N N s } = \{ \pmb { x } , t _ { 1 } , \cdot \cdot \cdot , t _ { I } , \hat { \pmb { y } } \}$
+
+$$
+\begin{array} { c } { { \bar { X } \right. T _ { 1 } \right. \cdots \right. T _ { I } \right. \hat { Y } } } \\ { { \nonumber } } \\ { { T _ { 1 } \left. \cdot \cdot \cdot \left. T _ { I } \left. \hat { Y } \left. Y . } } \end{array}
+$$
+
+• Different hidden layers manifest different IB trade-offs depending on the architecture and the position of hidden layers in DNNs.
+
+• A DNN satisfies the IB principle no matter the architecture of the DNN.
+
+67 Preliminaries. $P ( X , Y ) = P ( X ) P ( Y | X )$ is an unknown joint distribution between $X$ and $Y$ . A   
+68 dataset $\pmb { \mathcal { D } } = \{ ( \pmb { x } ^ { j } , \pmb { y } ^ { j } ) | \pmb { x } ^ { j } \in \mathbb { R } ^ { M } , \pmb { y } ^ { j } \in \mathbb { Z } \} _ { j = 1 } ^ { J }$ consists of $J$ i.i.d. samples generated from $P ( X , Y )$   
+69 with finite $L$ labels, i.e., $y ^ { j } \in \{ 1 , \cdots , L \}$ . In the context of supervised learning, we focus on   
+70 feedfworad fully connected $\mathrm { D N N s } = \{ \pmb { x } , t _ { 1 } , \cdot \cdot \cdot , t _ { I } , \hat { \pmb { y } } \}$ , i.e., Multi-Layer Perceptions (MLPs) [8]   
+71 for the image classification task. Without loss of generality, we use the $\mathrm { M L P } = \{ { \pmb x } , t _ { 1 } , t _ { 2 } , { \hat { \pmb y } } \}$ with   
+72 the cross-entropy loss $\ell _ { \mathrm { C E } }$ for most theoretical derivations. In addition, $H ( \cdot )$ denotes entropy.
+
+In the MLP, 73 where 74 $\begin{array} { r } { \langle \omega _ { n } ^ { ( 1 ) } , \pmb { x } \rangle = \sum _ { m = 1 } ^ { M } \omega _ { m n } ^ { ( 1 ) } \cdot \pmb { x } _ { m } + b _ { 1 n } } \end{array}$ $\mathbf { \delta t } _ { 1 }$ and $\mathbf { { t } } _ { 2 }$ have $N$ and $K$ neurons, respectively, and is the $n$ 1  1n  th dot-product given the weight $\pmb { t } _ { 1 } = \{ t _ { 1 n } = \sigma _ { 1 } [ \langle \pmb { \omega } _ { n } ^ { ( 1 ) } , \pmb { x } \rangle ] \} _ { n = 1 } ^ { N }$ $\omega _ { m n } ^ { ( 1 ) }$ n  n=1 and the bias 75 $b _ { 1 n }$ , and $\sigma _ { 1 } ( \cdot )$ denotes an activation function, e.g., ReLU. Similarly, $t _ { 2 } = \{ t _ { 2 k } = \sigma _ { 2 } [ \langle \omega _ { k } ^ { ( 2 ) } , t _ { 1 } \rangle ] \} _ { k = 1 } ^ { K }$ where 76 $\begin{array} { r } { \langle \omega _ { k } ^ { ( 2 ) } , t _ { 1 } \rangle = \sum _ { n = 1 } ^ { N } \omega _ { n k } ^ { ( 2 ) } \cdot t _ { 1 n } + b _ { 2 k } } \end{array}$ . The output layer $\hat { \pmb { y } }$ is softmax with $L$ nodes
+
+$$
+\hat { \pmb { y } } = \{ \hat { y } _ { l } = \frac { 1 } { Z _ { Y } } \mathrm { e x p } [ \langle { \pmb \omega } _ { l } ^ { ( 3 ) } , { \pmb t } _ { 2 } \rangle ] = \frac { 1 } { Z _ { Y } } \mathrm { e x p } [ g _ { l } ( { \pmb t } _ { 2 } ( { \pmb t } _ { 1 } ( { \pmb x } ) ) ) ] \} _ { l = 1 } ^ { L } ,
+$$
+
+$\begin{array} { r } { \langle \omega _ { l } ^ { ( 3 ) } , t _ { 2 } \rangle = \sum _ { k = 1 } ^ { K } \omega _ { k l } ^ { ( 3 ) } \cdot t _ { 2 k } + b _ { y l } } \end{array}$ $\begin{array} { r } { Z _ { Y } = \sum _ { l = 1 } ^ { L } \exp [ \langle \omega _ { l } ^ { ( 3 ) } , t _ { 2 } \rangle ] } \end{array}$
+
+1In which the virtual random variable $\bar { X }$ has all the information of $X$ except $Y$ , namely $H ( { \bar { X } } ) = H ( X | Y )$
+
+![](images/6b11a040923f0c2af4a1cb0fdb895b0bb60f125cb5f7a1fcf74e4d971d618c08.jpg)  
+Figure 1: Given a $4 \times 4$ input $_ z$ , a fully connected layer $\pmb { t }$ is equivalent to a convolution layer with $4 \times 4$ convolution kernels. The definition of convolution (Chapter 9.1 in [11]) implies that the $4 \times 4$ weights $\omega _ { 1 }$ and $\omega _ { 2 }$ define two global features, and the two activations $t _ { 1 } , t _ { 2 }$ indicate the cross-correlation between $\omega _ { 1 } , \omega _ { 2 }$ and $_ z$ , respectively. $P _ { T | , Z } ( \omega _ { 1 } | z )$ and $P _ { T | Z } ( \omega _ { 2 } | z )$ measure the probability of $\omega _ { 1 }$ and $\omega _ { 2 }$ being recognized as the feature with the largest cross-correlation to $_ z$ , respectively.
+
+# 78 2 A probabilistic representation for deep learning
+
+9 To accurately estimate $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ , in this section, we specify the probability space [6] for   
+80 a fully connected layer and derive the probabilistic explanations of the entire MLP.   
+81 It is known that a convolution kernel (namely the weights of convolution) defines a local feature,   
+82 and a convolution operation derives a feature map to measure the cross-correlation between the   
+83 local feature and input in a receptive field (Chapter 9.1 in [11]). Notably, a fully connected layer   
+84 is equivalent to a convolution layer with the kernel size having the same dimension as input. Thus   
+85 the weights of a neuron can be viewed as a global feature, and a fully connected layer with multiple   
+86 neurons derives activations to measure the cross-correlation between the multiple global features and   
+87 the input. The cross-correlation explanation for a fully connected layer is visualized in Figure 1.
+
+Assuming that88 is the input of 89 , $\pmb { t }$ $N$ neurons s the dot- $\{ t _ { n } = \sigma [ \langle \omega _ { n } , z \rangle ] \} _ { n = 1 } ^ { N }$ , ere , a $z \in \mathbb { R } ^ { M }$ $\pmb { t }$ $\begin{array} { r } { \langle \omega _ { n } , z \rangle = \sum _ { m = 1 } ^ { M } \omega _ { m n } \cdot z _ { m } + b _ { n } } \end{array}$ $_ { z }$ $\omega _ { n }$ $\sigma ( \cdot )$ 90 an activation function. Based othe cross-correlations between $_ z$ the crosand the $N$ orrelation explanation, the behavior of possible features defined by the the we $\pmb { t }$ is thts $\{ \omega _ { n } \} _ { n = 1 } ^ { N }$ 92 In the context of pattern recognition [34], we define a virtual random process or ‘experiment’ as $\pmb { t }$ 93 recognizing one of the patterns/features with the largest cross-correlation to $_ { z }$ from the $N$ possible 94 features. The experiment characterizes the behavior of $\pmb { t }$ (i.e., before recognizing the features with 95 the largest cross-correlation, $\pmb { t }$ must measure the cross-correlations between $_ { z }$ and all the $N$ possible 96 features) while meets the requirement of the ‘experiment’ definition (i.e., only one outcome will 97 occur on each trial of the experiment [6]). The probability space $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ is defined as follows:
+
+98 Definition 1. $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ consists of three components: the sample space $\Omega _ { T }$ has $N$ possible   
+99 outcomes (features) $\{ \omega _ { n } = \{ \omega _ { m n } \} _ { m = 1 } ^ { M } \} _ { n = 1 } ^ { N }$ defined by the weights2 of the $N$ neurons; the event   
+100 space is the $\sigma$ -algebra; and the probability measure $P _ { T }$ is a Gibbs distribution [19] to quantify the   
+101 probability of $\omega _ { n }$ being recognized as the feature with the largest cross-correlation to $_ z$ .
+
+102 Taking into account the randomness of $_ z$ , the conditional distribution $P _ { T | Z }$ is formulated as
+
+$$
+P _ { T | Z } ( \omega _ { n } | z ) = \frac { 1 } { Z _ { T } } \mathrm { e x p } ( t _ { n } ) = \frac { 1 } { Z _ { T } } \mathrm { e x p } [ \sigma ( \langle \omega _ { n } , z \rangle ) ] ,
+$$
+
+where 103 $Z$ is the random variable of $_ z$ and $\begin{array} { r } { Z _ { T } = \sum _ { n = 1 } ^ { N } \exp ( f _ { n } ) } \end{array}$ is the partition function.
+
+$\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ clearly explains all the ingredients of $\pmb { t }$ in a probabilistic fashion. The $n$ th neuron defines a global feature by the weights ${ \pmb w } _ { n }$ and the activation $t _ { n } = \sigma ( \langle \omega _ { n } , z \rangle )$ measures the crosscorrelation between ${ \pmb w } _ { n }$ and $_ { z }$ . The Gibbs distribution $P _ { T | Z }$ indicates that if ${ \pmb w } _ { n }$ has the higher activation, i.e., the larger cross-correlation to $_ z$ , it has the larger probability being recognized as the feature with largest cross-correlation to $_ { z }$ . For instance, if $\dot { z } \in \mathbb { R } ^ { 1 6 }$ and $\pmb { t }$ includes $N = 2$ neurons, then $\Omega _ { T } = \bar { \{ \omega _ { 1 } , \omega _ { 2 } \} }$ defines two possible outcomes (features), where $\omega _ { n } = \{ \omega _ { m n } \} _ { m = 1 } ^ { 1 6 }$ $\mathcal { F } = \{ \emptyset , \{ \omega _ { 1 } \} , \{ \omega _ { 2 } \} , \{ \omega _ { 1 } , \omega _ { 2 } \} \}$ means that neither, one, or both of the features are recognized by $\pmb { t }$ given $_ z$ , respectively. $\dot { P _ { T | Z } } ( \omega _ { 1 } | z )$ and $P _ { T | Z } ( \omega _ { 2 } | z )$ are the probability of $\omega _ { 1 }$ and $\omega _ { 2 }$ being recognized as the feature with the largest cross-correlation to $_ z$ , respectively.
+
+113 $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ explains the representation ability of deep learning. Compared to Restricted Boltzmann   
+114 Machines (RBMs) [26] simply using binary units to indicate features being recognized or not given   
+115 input, the Gibbs distribution3 $P _ { T | Z } \bar { ( \omega _ { n } | z ) }$ measures the probability of $\omega _ { n }$ being recognized with   
+116 the largest cross-correlation to $_ { z }$ , i.e., it characterizes the relation between features and input more   
+117 accurately. Moreover, Equation 5 shows that $t _ { n } = \sigma ( \langle \omega _ { n } , z \rangle )$ is the negative energy function [19] of   
+118 the Gibbs distribution, thus $P _ { T | Z } ( \omega _ { n } | z )$ can be derived as long as $\sigma ( \langle \omega _ { n } , z \rangle )$ are known because   
+119 the energy function is the sufficient statistics [2] of the Gibbs distribution. That enables subsequent   
+120 hidden layers to generate high-level features of input via directly processing the activations $\{ t _ { n } \} _ { n = 1 } ^ { N }$   
+121 thus deep learning can form a hierarchical structure to represent much complex features.   
+122 $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ answers a fundamental question: which component of a hidden layer contains the   
+123 information of the layer? Since $\omega _ { n }$ defines $\Omega _ { T }$ , the weights contain all the information of a layer. In   
+124 particular, since the activation $t _ { n } = \sigma ( \langle \omega _ { n } , z \rangle )$ is a function of $\omega _ { n }$ , the data processing inequality   
+125 [4] indicates that the information of $t _ { n }$ is no more than the information of $\omega _ { n }$ . Simulations in Section   
+126 4.2 demonstrate that if activations do not correctly characterize the cross-correlation between weights   
+127 and input, activations contain less information than weights do.
+
+Based on $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ , we define the random variable $T$ as follows:
+
+29 Definition 2. Given the fully connected layer $\pmb { t }$ , we define the random variable $T : \Omega _ { T } \to E _ { T }$ as
+
+$$
+T ( \omega _ { n } ) \triangleq n ,
+$$
+
+130 where the measurable space $E _ { T } = \{ 1 , \cdots , N \}$ .
+
+31 Since $\Omega _ { T }$ is composed of finite $N$ possible outcomes, $T$ is a discrete random variable. Notably, the   
+132 one-to-one correspondence between $\omega _ { n }$ and $n$ indicates
+
+$$
+P _ { T | Z } ( \omega _ { n } | z ) = P _ { T | Z } ( n | z ) .
+$$
+
+133 If not considering the back-propagation training, the weights (namely $\Omega _ { T _ { i } }$ ) of each layer are fixed.   
+134 Thus $T _ { i + 1 }$ entirely depends on $T _ { i }$ and the $\mathrm { M L P } = \{ { \pmb x } ; t _ { 1 } ; t _ { 2 } ; { \hat { \pmb y } } \}$ forms a Markov chain
+
+$$
+X  T _ { 1 }  T _ { 2 }  { \hat { Y } } .
+$$
+
+135 Based on the corresponding joint distribution $P ( \hat { Y } , T _ { 2 } , T _ { 1 } | X ) = P ( T _ { 1 } | X ) P ( T _ { 2 } | T _ { 1 } ) P ( \hat { Y } | T _ { 2 } )$ and   
+136 Definition 2, we derive a probabilistic explanation for the entire MLP, which is summarized in   
+137 Theorem 1. The detailed derivation is presented in Appendix $\mathbf { B }$ .
+
+138 Theorem 1. The $\mathrm { M L P } = \{ { \pmb x } ; { \pmb t } _ { 1 } ; { \pmb t } _ { 2 } ; { \hat { \pmb y } } \}$ formulates a conditional Gibbs distribution
+
+$$
+P _ { \hat { Y } | X } ( l | x ) = \sum _ { k = 1 } ^ { K } \sum _ { n = 1 } ^ { N } P ( \hat { Y } = l , T _ { 2 } = k , T _ { 1 } = n | X = x ) = \frac { 1 } { Z _ { \mathrm { M L P } } ( x ) } \mathrm { e x p } [ g _ { l } ( t _ { 2 } ( t _ { 1 } ( x ) ) ) ] ,
+$$
+
+where 139 $\begin{array} { r } { Z _ { \mathrm { M L P } } ( \boldsymbol { x } ) = \sum _ { l = 1 } ^ { L } \sum _ { k = 1 } ^ { K } \sum _ { n = 1 } ^ { N } P _ { \hat { Y } , T _ { 2 } , T _ { 1 } | \boldsymbol { X } } ( l , k , n | \boldsymbol { x } ) } \end{array}$ is the partition function.
+
+140 Since $P _ { \hat { Y } | X } ( l | \pmb { x } )$ exactly equals the output $\hat { y } _ { l }$ of the MLP, namely Equation (4), we conclude that   
+141 the entire architecture of the MLP forms a family of Gibbs distribution $P _ { \hat { Y } | X } ( l | \mathbf { \boldsymbol { x } } )$ . In general, the   
+142 back-propagation updates a weight $\omega$ based on the gradient of $\ell _ { \mathrm { C E } }$ with respect to $\omega$ ,
+
+$$
+\omega ( s + 1 ) = \omega ( s ) - \alpha \cdot \frac { \partial \ell _ { \mathrm { C E } } } { \partial \omega ( s ) } = \omega ( s ) - \alpha \cdot \frac { \partial \mathrm { K L } [ P ( Y | X ) | | P ( \hat { Y } | X ) ] } { \partial \omega ( s ) } ,
+$$
+
+143 where $s$ is the index of training iteration, $\alpha$ is the training rate, and $\mathrm { K L } [ \cdot | | \cdot ]$ is the KL-divergence.
+
+144 Figure 2 summarizes the probabilistic explanation for deep learning based on the MLP. In general,   
+145 a single learning iteration, an epoch, consists of two phases: training and inference (after training).   
+146 During inference, the MLP bridges $X$ and $\hat { Y }$ via multiple intermediate features $\Omega _ { T _ { 1 } }$ , $\Omega _ { T _ { 2 } }$ , and $\Omega _ { \hat { Y } }$   
+147 defined by weights, and formulates the statistical relation between $\hat { Y }$ and $X$ as a family of conditional   
+148 Gibbs distribution $P ( { \hat { Y } } | X )$ . During training, the back-propagation updates weights to learn optimal   
+149 intermediate features for searching an optimal $P ( { \hat { Y } } | X )$ to accurately approximate $P ( { Y \vert } X )$ .
+
+![](images/4783b0e35de39034e8ce539d14ecdda10d599541289305cdd2ef3b9b1d3f69c0.jpg)  
+Figure 2: The visualization of the probabilistic explanation for deep learning based on the MLP.
+
+# 150 3 The information theoretic explanations for deep learning
+
+51 To address the limitations of existing IB explanations, this section proposes some novel information   
+52 theoretic explanations for DNNs based on the proposed probabilistic representation.
+
+153 Proposition 1. The mutual information between a fully connected layer and dataset is finite.
+
+$$
+I ( X ; T ) < \infty .
+$$
+
+154 Proof: Definition 2 shows $E _ { T } = \{ 1 , \cdots N \}$ . Thus $T$ is a discrete random variable and $H ( T ) < \infty$ ,   
+155 thereby $I ( X ; T ) \leq H ( T ) < \infty$ .   
+156 Proposition 1 circumvents the infinite mutual information problem. In the absence of a clear definition   
+157 $T : \Omega _ { T } \to E _ { T }$ , most previous works [28, 3, 1] simply viewing the activation $t _ { n }$ as the sample of $T$ ,   
+158 namely $t _ { n } \in E _ { T } = \mathbb { R }$ , implies $T$ being continuous and gives rise to the infinite mutual information   
+159 problem in deterministic DNNs. However, $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ indicates that $t _ { n }$ actually is a variable   
+160 measuring the cross-correlation between ${ \pmb w } _ { n }$ and $_ { z }$ rather than the sample of $T$ , namely $t _ { n } \notin E _ { T }$ .
+
+161 Theorem 2. The information of $Y$ flows into the MLP in the backward direction during training
+
+$$
+T _ { 1 } \gets T _ { 2 } \gets \hat { Y } \gets Y .
+$$
+
+162 Proof: First, since $\Omega _ { T }$ is defined by $\omega$ in $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ and Equation (10) shows that $\omega ( s + 1 )$ is   
+163 determined by all the previous gradients $\{ \frac { \partial \ell _ { \mathrm { C E } } } { \partial \omega ( s ) } \} _ { s = 1 } ^ { S }$ Es) }Ss=1 , and ω(0) is randomly initialized and α is a   
+164 constant, we can derive that $\Omega _ { T }$ is determined by ∂\`CE∂ω . Second, based on the back-propagation, the   
+165 relation between gradients in two adjacent layers in the $\mathrm { M L P } = \{ { \pmb x } ; t _ { 1 } ; t _ { 2 } ; { \hat { \pmb y } } \}$ is formulated as
+
+$$
+\begin{array} { r l } & { \qquad \displaystyle \frac { \partial \ell _ { \mathrm { C E } } ^ { \star } } { \partial \omega _ { k l } ^ { ( 3 ) } } = [ P _ { \hat { Y } | X } ( l | x ) - P _ { Y | X } ( l | x ) ] \cdot t _ { 2 k } , } \\ { \displaystyle \partial \ell _ { \mathrm { C E } } ^ { ( \zeta ) } = \sum _ { l = 1 } ^ { L } \displaystyle \frac { \partial \ell _ { \mathrm { C E } } ^ { \star } } { \partial \omega _ { k l } ^ { ( 3 ) } } \cdot \omega _ { k l } ^ { ( 3 ) } \cdot \displaystyle \frac { \sigma _ { 2 } ^ { \prime } \big ( \langle \omega _ { k } ^ { ( 2 ) } , t _ { 1 } \rangle \big ) } { f _ { 2 k } } \cdot t _ { 1 n } , ~ \displaystyle \frac { \partial \ell _ { \mathrm { C E } } ^ { ( \zeta ) } } { \partial \omega _ { m n } ^ { ( 1 ) } } = \sum _ { k = 1 } ^ { K } \displaystyle \frac { \partial \ell _ { \mathrm { C E } } ^ { ( \zeta ) } } { \partial \omega _ { n k } ^ { ( 2 ) } } \cdot \omega _ { n k } ^ { ( 2 ) } \cdot \displaystyle \frac { \sigma _ { 1 } ^ { \prime } \big ( \langle \omega _ { n } ^ { ( 1 ) } , x \rangle \big ) } { t _ { 1 n } } \cdot x _ { m } . } \end{array}
+$$
+
+166 Equation 13 shows that $\frac { \partial \ell _ { \mathrm { C E } } } { \partial \omega ^ { ( 3 ) } }$ is a function of $P _ { Y \mid X } ( l \mid x )$ and $\frac { \partial \ell _ { \mathrm { C E } } } { \partial \omega ^ { ( i ) } }$ is a function of $\frac { \partial \ell _ { \mathrm { C E } } } { \partial \omega ^ { ( i + 1 ) } }$ , where   
+167 $\boldsymbol { \omega } ^ { ( 3 ) }$ denotes the weight of $\hat { \pmb { y } }$ . The two points above enable us to derive that $\Omega _ { T _ { i } }$ is a function of $\Omega _ { T _ { i + 1 } }$   
+168 and $\Omega _ { \hat { Y } }$ is a function of $P ( { Y \vert } X )$ . Based on Definition 2, we can further derive that $T _ { i }$ is a function   
+169 of $T _ { i + 1 }$ and $\hat { Y }$ is a function of $Y$ , i.e., $T _ { 1 } \gets T _ { 2 } \gets \hat { Y } \gets Y$ . (See the detailed proof in Appendix C).   
+170 Theorem 2 is consistent with the prevailing explanation for deep learning. LeCunn et al. show that   
+171 deep learning exploits the hierarchical property of signals [18], i.e., the layers farther from output   
+172 learn lower-level features, such as edges, whereas the layers closer to output assemble lower-level   
+173 features into the higher-level features corresponding to labels (see Figure 2 in [37]). Notably, since   
+174 lower-level features commonly exist in signals with different labels (e.g., lower-level features, such   
+175 as the edges of the vehicle frame and the circular contour of wheels, exist in both the car and the   
+176 truck classes in the CIFAR-10 dataset [16] in Figure 2), lower-level features do not contain much   
+177 information of labels. Therefore, the layers farther from output do not have much information of   
+178 labels, which is consistent with the Markov chain $T _ { 1 } \gets T _ { 2 } \gets \hat { Y } \gets Y$ .   
+179 Since all the information of $Y$ stems from $X$ (i.e., $H ( Y ) = I ( X ; Y )$ proven in Appendix D),   
+180 Theorem 2 implies that partial information of $X$ flows into the MLP in the backward direction during   
+181 training. Equation (2) shows the information of $X$ flowing into the MLP in the forward direction   
+182 during inference. Overall, the information of $X$ flows in the backward and forward directions during   
+183 training and inference, respectively. As a result, the Markov chain, Equation (2), proposed by recent   
+184 works could not fully characterize the information flow of $X$ in the MLP in each epoch. In other   
+185 words, $I ( X ; T _ { i } )$ is not necessarily greater than $I ( X ; T _ { i + 1 } )$ in the MLP in each epoch.   
+186 Equation (2) shows that $T _ { i }$ receives the information of $Y$ via $X$ during inference. Theorem 2 shows   
+187 that $T _ { i }$ also directly receives information of $Y$ during training, because the back-propagation updates   
+188 weights (i.e., $\Omega _ { T _ { i } }$ ) based on the label $Y$ . Thus Equation (2) cannot fully characterize the information   
+189 flow of $Y$ in the MLP in each epoch, when we take into account the back-propagation training.   
+190 To fully characterize the information flow in the MLP in each epoch, we introduce Corollary 1.   
+191 Corollary 1. The information flow in the MLP can be characterized by two Markov chains as
+
+$$
+\begin{array} { c } { { \bar { X } \right. T _ { 1 } \right. T _ { 2 } \right. \hat { Y } } } \\ { { T _ { 1 } \left. T _ { 2 } \left. \hat { Y } \left. Y . } } \end{array}
+$$
+
+The virtual random variable 192 $\bar { X }$ contains all the information of $X$ except $Y$ , i.e., $H ( { \bar { X } } ) = H ( X | Y )$
+
+193 Proof of the first Markov chain: Since $\bar { X }$ does not have any information of $Y$ , it can only flow into   
+194 the MLP in the forward direction during inference. Again since $\bar { X }$ does not have any information of   
+195 $Y$ , the information flow of $Y$ during training will not affect the information flow of $\bar { X }$ . Therefore,   
+196 ${ \bar { X } }  T _ { 1 }  T _ { 2 }  { \hat { Y } }$ characterizes the information flow of $\bar { X }$ in both training and inference phases.   
+197 Proof of the second Markov chain: Since the weights are fixed after training, the sample space and   
+198 the distribution of hidden layers are fixed after training. Therefore, the information of $Y$ transferred   
+199 into hidden layers during training will retain there after training (i.e., during inference). In addition,   
+200 Definition 1 indicates that a fully connected layer $t = \{ t _ { n } = \bar { \sigma } ( \langle \omega _ { n } , z \rangle ) \bar  \} _ { n = 1 } ^ { N }$ measures the cross  
+201 correlation between $\omega _ { n } ^ { ( 1 ) }$ ) and z during inference, thus {ω(1)n }Nn=1 can be viewed as a representation   
+202 of $Z$ . As a result, even though has all the information of $Y$ , the information of that can learn   
+203 from $Z$ is determined by how much information of $Y$ the representation $\{ \omega _ { n } ^ { ( 1 ) } \} _ { n = 1 } ^ { N }$ has. Overall, the   
+204 information flow of $Y$ during inference will be the same as that during training. Based on Theorem 2,   
+205 we conclude that $T _ { 1 } \gets T _ { 2 } \gets \hat { Y } \gets Y$ characterizes the information flow of $Y$ in the MLP in both   
+206 training and inference phases. Detailed derivations and explanations are presented in Appendix E.   
+07 To quantify how much information of $X$ and $Y$ is learned by the MLP, we introduce Corollary 2.   
+08 Corollary 2. The mutual information between dataset and the entire MLP can be expressed as
+
+$$
+\begin{array} { r l } & { I ( X ; T _ { \mathrm { M L P } } ) = I ( \bar { X } ; T _ { 1 } ) + I ( Y ; \hat { Y } ) } \\ & { I ( Y ; T _ { \mathrm { M L P } } ) = I ( Y ; \hat { Y } ) } \end{array}
+$$
+
+09 where $T _ { \mathrm { M L P } }$ denotes a random variable corresponding to the entire architecture of the MLP.
+
+Proof: Since 210 $H ( Y ) = I ( X ; Y )$ (Appendix D), $H ( X ) = H ( { \bar { X } } ) + I ( X ; Y ) = H ( { \bar { X } } ) + H ( Y )$ .   
+211 Hence, Corollary 2 can be derived by Corollary 1 and the chain rule. The proof is in Appendix $\mathrm { F }$ .
+
+# 4 Simulations
+
+In this section, we propose a mutual information estimator based on $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ and demonstrate the probabilistic representation and information theoretic explanations for deep learning on a synthetic dataset with known entropy. Additional experiments on benchmark datasets are in Appendix $_ \mathrm { H }$ .
+
+# 4.1 Setup
+
+Mutual information estimator. Based on the definition of mutual information, we have
+
+$$
+I ( X ; T _ { i } ) = H ( T _ { i } ) - H ( T _ { i } | X ) .
+$$
+
+218 Previous works simply estimate $I ( X ; T _ { i } ) = H ( T _ { i } )$ , because $T _ { i }$ is assumed to be entirely dependent   
+219 on $X$ in the Markov chain, Equation (2), thereby $H ( T _ { i } | X ) = 0$ . However, Corollary 1 shows that $T _ { i }$   
+220 depends on both $X$ and $Y$ if taking into account the training phase, thereby $H ( T _ { i } | \dot { X } ) \neq 0$ .
+
+![](images/d35890b35d8c63aacf56c438ea3467ac26a9d308e54ecd338e8f4f95ff3f52f6.jpg)  
+Figure 3: (A) the deterministic image $\hat { \textbf { \textit { x } } }$ . Image0 is generated by adding ${ \mathcal { N } } ( \mu , \sigma ^ { 2 } )$ without rotation, Image1 is generated by rotating $\hat { \textbf { \textit { x } } }$ along the secondary diagonal direction and adding $\mathcal { N } ( \mu , \sigma ^ { 2 } )$ , Image2 and Image are generated by rotating $\hat { \textbf { \textit { x } } }$ along the vertical and horizontal directions, respectively, and adding ${ \mathcal { N } } ( \mu , \sigma ^ { 2 } )$ .
+
+Table 1: The number of neurons(nodes) and the activation function in the layers of the MLPs
+
+$$
+\begin{array} { r } { \begin{array} { r l r l r l r } & { } & { x } & { t _ { 1 } } & { t _ { 2 } } & { \hat { y } } & { \sigma ( \cdot ) } \\ & { \mathrm { M L P 1 } } & { 1 0 2 4 ( 3 2 \times 3 2 ) } & { 8 } & { 6 } & { 2 } & { \mathrm { R e L U } ( z ) = \operatorname* { m a x } ( 0 , z ) } \\ & { \mathrm { M L P 2 } } & { 1 0 2 4 ( 3 2 \times 3 2 ) } & { 8 } & { 6 } & { 2 } & { \mathrm { T a n h } ( z ) = ( e ^ { z } - e ^ { - z } ) / ( e ^ { z } + e ^ { - z } ) } \\ & { \mathrm { M L P 3 } } & { 1 0 2 4 ( 3 2 \times 3 2 ) } & { 2 } & { 6 } & { 2 } & & { \mathrm { R e L U } } \end{array} } \end{array}
+$$
+
+To accurately estimate 221 $I ( X ; T _ { i } )$ , we need to specify $P ( T _ { i } | X )$ and $P ( T _ { i } )$ . Based on $\left( \Omega _ { T _ { i } } , \mathcal { F } , P _ { T _ { i } } \right)$ , we formulate 222 $P _ { T _ { i } | X } ( n | \mathbf { x } ^ { j } )$ of the three fully connected layers in the MLP as
+
+$$
+\begin{array} { r l } & { P _ { T _ { 1 } | X } ( n | x ^ { j } ) = \frac { 1 } { Z _ { F _ { 1 } } } \mathrm { e x p } [ \sigma _ { 1 } ( \langle \omega _ { n } ^ { ( 1 ) } , x ^ { j } \rangle ) ] , ~ P _ { T _ { 2 } | X } ( k | x ^ { j } ) = \frac { 1 } { Z _ { F _ { 2 } } } \mathrm { e x p } [ \sigma _ { 2 } ( \langle \omega _ { k } ^ { ( 2 ) } , t _ { 1 } ( x ^ { j } ) \rangle ) ] , } \\ & { ~ P _ { T _ { Y } | X } ( l | x ^ { j } ) = \frac { 1 } { Z _ { F _ { Y } } } \mathrm { e x p } [ \langle \omega _ { l } ^ { ( 3 ) } , t _ { 2 } ( t _ { 1 } ( x ^ { j } ) ) \rangle ] . } \end{array}
+$$
+
+To derive the marginal distribution 223 $P ( T _ { i } )$ , we sum the joint distribution $P ( T _ { i } , X )$ over $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ ,
+
+$$
+\begin{array} { r } { ^ { \circ } ( T _ { i } = n ) = \sum _ { \mathbf { x } \in \mathcal { X } } P _ { X } ( \mathbf { x } ) P _ { T _ { i } | X } ( n | \mathbf { x } ) \approx \sum _ { \mathbf { x } ^ { \prime } \in \mathcal { D } } P _ { X } ( \mathbf { x } ^ { j } ) P _ { T _ { i } | X } ( n | \mathbf { x } ^ { j } ) = \frac { 1 } { J } \sum _ { \mathbf { x } ^ { j } \in \mathcal { D } } P _ { T _ { i } | X } ( n | \mathbf { x } ^ { j } ) , } \end{array}
+$$
+
+where 224 $P _ { X } ( \pmb { x } ^ { j } )$ is estimated by the empirical distribution $1 / J$ given $\mathcal { D }$ . Finally, we can derive $I ( X ; T _ { i } )$ 225 by Equation 16, 17, and 18. Similarly, based on the definition of mutual information, we have
+
+$$
+I ( Y ; T _ { i } ) = H ( T _ { i } ) - H ( T _ { i } | Y ) .
+$$
+
+To estimate 226 $H ( T _ { i } | Y )$ , we reformulate $P ( T _ { i } | Y )$ as
+
+$$
+\begin{array} { r } { P _ { T _ { i } | Y } ( n | l ) = \sum _ { \pmb { x } \in \mathcal { X } } P _ { T _ { i } | X } ( n | \pmb { x } ) P _ { X | Y } ( \pmb { x } | l ) \approx \frac { 1 } { N ( l ) } \sum _ { \pmb { x } ^ { j } \in \mathcal { D } , y ^ { j } = l } P _ { T _ { i } | X } ( n | \pmb { x } ^ { j } ) , } \end{array}
+$$
+
+where 27 $P _ { X | Y } ( \pmb { x } ^ { j } | l )$ is estimated by the empirical distribution $1 / N ( l )$ and $N ( l )$ denotes the number of 28 samples with the label $l$ in $\mathcal { D }$ . Finally, we can derive $I ( Y ; T _ { i } )$ by Equation 18, 19, and 20.
+
+Synthetic dataset. The dataset consists of 512 gray-scale $3 2 \times 3 2$ images, which are evenly generated by rotating a deterministic image $\hat { \pmb x }$ in four different orientations and adding Gaussian noise with expectation $\boldsymbol { \mu } = \mathbb { E } ( \hat { \boldsymbol { x } } )$ and variance $\sigma ^ { 2 } = 1$ , namely $\pmb { x } = r ( \hat { \pmb { x } } ) + \mathcal { N } ( \mu , \sigma ^ { 2 } )$ , where $r ( \cdot )$ denotes the rotation method shown in Figure 3. The reason for adding Gaussian noise is to avoid DNNs directly memorizing the deterministic image. In addition, the binary labels [1,0] and [0,1] evenly divide the synthetic dataset into two classes. As a result, the synthetic dataset has (approximately) 2 bits information and the labels have 1 bit information. Compared to popular benchmark dataset with unknown features and entropy, e.g., MNIST [17] and Fashion-MNIST [36], the features and the entropy of the synthetic dataset are clear and known, which enables us to examine the probabilistic representation and the mutual information estimator.
+
+Neural Networks. We train three MLPs, namely MLP1, MLP2 and MLP3, on the synthetic dataset by a variant of Stochastic Gradient Descent (SGD) method, namely Adam [13], over 1000 epochs with the learning rate $\alpha = 0 . 0 3$ . Table 1 summarizes the architecture of the three MLPs.
+
+# 4.2 Validating the probability space and the mutual information estimator
+
+We demonstrate the sample space $\Omega _ { T }$ by visualizing the weights4 of the eight neurons in $\mathbf { t } _ { 1 }$ , i.e., $\omega _ { n } ^ { ( 1 ) } = \{ \omega _ { m n } ^ { ( 1 ) } \} _ { m = 1 } ^ { 1 0 2 4 }$ , in 5 different epochs (i.e., 0,1,4,128,1000) in Figure 4 (Left). As training continues, we observe that $\omega _ { n } ^ { ( 1 ) }$ quickly learns all the spatial features of the synthetic dataset. For instance, ${ \boldsymbol \omega } _ { 2 } ^ { ( 1 ) }$ has low magnitude at top-left positions and high magnitude at bottom-right positions, 2which correctly characterizes the spatial feature of Image0. Similarly, ${ \boldsymbol \omega } _ { 3 } ^ { ( 1 ) }$ , ${ \omega } _ { 4 } ^ { ( 1 ) }$ , and ${ \omega } _ { 5 } ^ { ( 1 ) }$ correctly characterize the spatial feature of Image1, Image2, and Image3 in Figure 3, respectively.
+
+![](images/9b9266acd8487034d9e99b73d4c9145c4cb7e1632f21f4a5511e1eb1aeb9e872.jpg)  
+Figure 4: (Left) The eight features $\{ \omega _ { n } ^ { ( 1 ) } \} _ { n = 1 } ^ { 8 }$ learned by the weights of the eight neurons in 5 different epochs (i.e., 0,1,4,128,1000), where $\omega _ { n } ^ { ( 1 ) } = \{ \omega _ { m n } ^ { ( 1 ) } \} _ { m = 1 } ^ { 1 0 2 4 }$ are reshaped into $3 2 \times 3 2$ to show the spatial structure. (Right) in the MLP1, MLP2, and MLP3 during 1000 epochs.
+
+Table 2: The Gibbs probability $P _ { F _ { 1 } | X } ( \omega _ { n } ^ { ( 1 ) } | \mathrm { I m a g e 0 } )$ in MLP1 and MLP2 in the 1000 epoch   
+
+<table><tr><td></td><td></td><td>1</td><td></td><td>1</td><td>w</td><td>(1</td><td>（1）</td><td></td></tr><tr><td>（，）</td><td>-63.6</td><td>208.8</td><td>-181.6</td><td>45.1</td><td>-55.6</td><td>157.5</td><td>-210.0</td><td>-30.1</td></tr><tr><td>fReLU (x) in</td><td>0.0</td><td>208.8</td><td>0.0</td><td>45.1</td><td>0.0</td><td>157.5</td><td>0.0</td><td>0.0</td></tr><tr><td>expReLu(）</td><td>1.0</td><td>4.79e+90</td><td>1.0</td><td>3.86e+19</td><td>1.0</td><td>2.51e+68</td><td>1.0</td><td>1.0</td></tr><tr><td>PU</td><td>0.0</td><td>1.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>fTanh(（x） Jin</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>-1.0</td></tr><tr><td>exp(）</td><td>0.36</td><td>2.71</td><td>0.36</td><td>2.71</td><td>0.36</td><td>2.71</td><td>0.36</td><td>0.36</td></tr><tr><td>P</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.037</td></tr></table>
+
+$f _ { 1 n } ^ { \mathrm { T a n h } } ( \pmb { x } ) = \sigma ^ { \mathrm { T a n h } } ( \langle \pmb { \omega } _ { n } ^ { ( 1 ) } , \pmb { x } \rangle )$ and $f _ { 1 n } ^ { \mathrm { R e L U } } ( \pmb { x } ) = \sigma ^ { \mathrm { R e L U } } ( \langle \pmb { \omega } _ { n } ^ { ( 1 ) } , \pmb { x } \rangle )$ are the activations given the same $\overline { { \langle { \omega _ { n } ^ { ( 1 ) } , \pmb x } \rangle } }$
+
+249 We demonstrate that feature with the larg $P ( T _ { 1 } | X )$ correctly mecorrelation to $_ { \textbf { \em x } }$ res the probability of in Table 2. For insta $\{ \omega _ { n } ^ { ( 1 ) } \} _ { n = 1 } ^ { 8 }$ ${ \boldsymbol \omega } _ { 2 } ^ { ( 1 ) }$ being recognized theorrectly characterizes   
+251 the feature of Image0 and has the largest cross-correlation $\langle \omega _ { 2 } ^ { ( 1 ) } , \pmb { x } \rangle = 1 9 0 . 8$ , thus it has the largest   
+252 probability $P _ { T _ { 1 } | X } ^ { \mathrm { R e L U } } ( \omega _ { 2 } ^ { ( 1 ) } | \mathrm { I m a g e 0 } ) = 1 . 0$ being recognized as the feature with largest cross-correlation   
+253 to Image0. In contrast, since ${ \omega } _ { 7 } ^ { ( 1 ) }$ incorrectly characterizes the feature of Image0 and has the lowest   
+254 cross-correlation $\langle \omega _ { 7 } ^ { ( 1 ) } , \pmb { x } \rangle = - 2 1 0 . 0$ , so it has the lowest probability $P _ { T _ { 1 } | X } ^ { \mathrm { R e L U } } ( \omega _ { 7 } ^ { ( 1 ) } | \mathrm { I m a g e 0 } ) = 0 . 0$   
+255 being recognized as the feature with largest cross-correlation to Image0.
+
+6 We observe that an activation function (abbr. ACT) plays an important role in the distribution. Specifically, ReLU, a non-saturating (unbounded) ACT [9], preserves the positive cross-correlations while resets all the negative ones as zero. $P _ { T _ { 1 } | X } ^ { \mathrm { R e L U } } ( \omega _ { 2 } ^ { ( 1 ) } | \mathrm { I m a g e 0 } ) = 1 . 0$ shows that ReLU derives the correct probability of ${ \boldsymbol \omega } _ { 2 } ^ { ( 1 ) }$ being recognized as the feature with largest cross-correlation. In contrast, though ${ \boldsymbol \omega } _ { 2 } ^ { ( 1 ) }$ has stronger cross-correlation to Image0 than ${ \omega } _ { 4 } ^ { ( 1 ) }$ , i.e., $\langle \omega _ { 2 } ^ { ( 1 ) } , \pmb { x } \rangle > \langle \pmb { \omega } _ { 4 } ^ { ( 1 ) } , \pmb { x } \rangle$ , Tanh, a saturating (bounded) ACT, derives $f _ { 1 2 } ^ { \mathrm { T a n h } } ( { \pmb x } ) = f _ { 1 4 } ^ { \mathrm { T a n h } } ( { \pmb x } ) = 1 . 0$ , and makes ${ \omega } _ { 4 } ^ { ( 1 ) }$ to incorrectly have the same probability 0.272 to ${ \boldsymbol \omega } _ { 2 } ^ { ( 1 ) }$ being recognized as the feature with the largest cross-correlation 3 to Image0, i.e., Tanh hinders $\mathbf { t } _ { 1 }$ from correctly recognizing the features of input. The simulations for validating the probability space based on other synthetic images are presented in Appendix G.
+
+265 To validate the mutual information estimator, we follow recent works [30, 28] to train the three   
+266 MLPs with 50 different random initialization and study the average mutual information. Figure 4   
+267 (Right) shows that $I ( X ; T _ { 1 } )$ quickly increases to 1.81 and keeps stable in the MLP1, i.e., $\mathbf { t } _ { 1 }$ learns   
+268 most information of the dataset as $H ( X ) = 2 . 0$ . Notably, the result is consistent with the variation   
+269 of the weights in Figure 4 (Left), which shows that the weights correctly characterize the features   
+270 of the dataset and keeps stable after the fourth epoch. As a comparison, we observe that $I ( X ; T _ { 1 } )$   
+271 keeps stable at 0.44 in the MLP2, which confirms the statement that Tanh hinders $\mathbf { \delta t } _ { 1 }$ from correctly   
+272 recognizing the features of input. In addition, Figure 4 (Right) shows that $I ( X ; T _ { 1 } ) \approx 0 . 7 9$ in MLP3   
+273 is smaller than $I ( X ; T _ { 1 } ) \approx \bar { 1 } . 8 1$ in MLP1, which is consistent with Definition 1, i.e., a layer with   
+274 fewer neurons would represent fewer possible features, thus it contains less information.   
+275 In summary, we demonstrate the probability space $\left( \Omega _ { T } , \mathcal { F } , P _ { T } \right)$ and show that if an ACT cannot   
+276 preserve the cross-correlation between weights(features) and input, it would distort the distribution   
+277 of a layer, thereby affecting the mutual information between the layer and data/labels. In addition,   
+278 we show that the proposed mutual information estimator outperforms the existing non-parametric   
+279 models, e.g., empirical distribution [30] and KDE [28], based on the synthetic dataset. Especially,   
+280 activations do not satisfy the i.i.d. prerequisite of non-parametric models is an important reason for   
+281 non-parametric models deriving inaccurate mutual information in DNNs. Due to limited space, the   
+282 experimental comparison and study of non-parametric models are presented in Appendix G.
+
+![](images/3c88b2ed333f4d3d7b5765f256c0655f9d9c4b4ba232bb3dfdb758309fa8fad6.jpg)  
+Figure 5: All the $\mathbf { X }$ -axis index training epochs. In each column, the first three figures show $I ( X ; T _ { i } )$ , $I ( \bar { X } ; T _ { i } )$ , and $I ( Y ; T _ { i } )$ respectively. The forth figure shows $I ( X ; T _ { \mathrm { M L P } } )$ and $I ( Y ; T _ { \mathrm { M L P } } )$ in a MLP. The pink line denotes $H ( Y ) = 1 . 0$ and the orange line denotes $H ( X ) = 2 . 0$ .
+
+# 283 4.3 Validating the information theoretic explanations for DNNs
+
+In Figure 5, we observe $I ( X ; T _ { i } ) \leq I ( X ; { \hat { Y } } )$ in MLP2 and MLP3, which confirms that the Markov chain proposed by previous works, Equation (2), cannot fully explain the information flow in MLPs, if taking into account the back-propagation training. As a comparison, the second and third row show $I ( \bar { X } ; T _ { 1 } ) \ge I ( \bar { X } ; T _ { 2 } ) \ge I ( \bar { X } ; \hat { Y } )$ and $I ( Y ; T _ { 1 } ) \leq I ( Y ; T _ { 2 } ) \geq I ( Y ; \hat { Y } )$ in all the three MLPs, which validates that Corollary 1, i.e., Equation (14) characterizes the information flow in MLPs.
+
+Figure 5 demonstrates that different hidden layers achieve different IB trade-offs depending on the architecture and the position of the layers in MLPs. In terms of architecture, $I ( Y ; T _ { 1 } ) > 0 . 8$ and $I ( \bar { X } ; T _ { 1 } ) > 0 . 7 5$ in MLP1 indicate that $\mathbf { \delta t } _ { 1 }$ , with ReLU, achieves a good prediction without much compression, whereas $I ( Y ; T _ { 1 } ) < 0 . 5$ and $I ( \bar { X } ; T _ { 1 } ) < 0 . 1$ in MLP2 show that $\mathbf { \delta t } _ { 1 }$ , with Tanh, achieves a different IB trade-off. In addition, $I ( Y ; T _ { 1 } ) \approx 0 . 4 5$ and $I ( \bar { X } ; T _ { 1 } ) \approx 0 . 2 5$ in MLP3 show the effect of neuron numbers on the IB trade-off. In terms of position, $I ( Y ; { \hat { Y } } ) = 1$ and $I ( { \bar { X } } ; { \hat { Y } } ) = 0$ in MLP1 means that $\hat { \pmb { y } }$ has a different IB trade-off to $\mathbf { \delta t } _ { 1 }$ in MLP1.
+
+We demonstrate that a MLP satisfies the IB principle no matter what the architecture of the MLP is. Figure 5 visualizes $I ( X ; T _ { \mathrm { M L P } } )$ and $I ( Y ; { \bar { T } } _ { \mathrm { M L P } } )$ based on Corollary 2. It shows that all of three MLPs satisfy the IB principle, namely $I ( X ; T _ { \mathrm { M L P } } ) < H ( X ) = 2$ and $I ( Y ; T _ { \mathrm { M L P } } ) = H ( Y ) = 1$ though they have different architectures. Importantly, in contrast to previous work [28] claiming that the compression not exists in DNNs with non-saturating ACT, such as ReLU, Figure 5 clearly shows that the compression exists in all the MLPs, no matter the activation function of MLPs.
+
+We further demonstrate the information theoretic explanations for DNNs on the benchmark MNIST and Fashion-MNIST datasets. The experiments are presented in Appendix $_ \mathrm { H }$ .
+
+# 04 5 Conclusion and future work
+
+In this work, we (1) specify the probability space for a hidden layer for (2) accurately estimating the mutual information and (3) clearly explaining how the components of the layer affect the mutual information. We take into account the back-propagation training and derive two novel Markov chains to characterize the information flow in DNNs. Furthermore, we demonstrate that a DNN satisfies the IB principle no matter the architecture of the DNN. In contrast, different hidden layers show different IB trade-offs depending on the architecture and the position of the layers in DNNs. A potential direction is to study the generalization of DNNs based on the probabilistic representation.
+
+References   
+[1] Rana Ali Amjad and Bernhard Claus Geiger. Learning representations for neural network-based classification using the information bottleneck principle. IEEE transactions on pattern analysis and machine intelligence, 2019.   
+[2] George Casella and Roger L Berger. Statistical inference. Cengage Learning, 2021.   
+[3] Ivan Chelombiev, Conor Houghton, and Cian O’Donnell. Adaptive estimators show information compression in deep neural networks. In International Conference on Learning Representations, 2019.   
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+[11] Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio. Deep learning, volume 1. MIT press Cambridge, 2016.   
+[12] Patrick Kidger and Terry Lyons. Universal approximation with deep narrow networks. In Conference on Learning Theory, pages 2306–2327. PMLR, 2020.   
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+[15] Artemy Kolchinsky, Brendan D Tracey, and Steven Van Kuyk. Caveats for information bottleneck in deterministic scenarios. arXiv preprint arXiv:1808.07593, 2018.   
+[16] Alex Krizhevsky. Learning multiple layers of features from tiny images. Technique Report, 2009.   
+[17] Y. LeCun, BE. Boser, and JS. Denker. Handwritten digit recognition with a back-propagation network. In NeurIPS, pages 396–494, 1990.   
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+[19] Yann LeCun, Sumit Chopra, Raia Hadsell, Marc’Aurelio Ranzato, and Fu Jie Huang. A tutorial on energy-based learning. MIT Press, 2006.   
+[20] Jaehoon Lee, Yasaman Bahri, Roman Novak, Samuel S. Schoenholz, Jeffrey Pennington, and Jascha Sohl-Dickstein. Deep neural networks as gaussian processes. In ICLR, 2018.   
+[21] Aravindh Mahendran and Andrea Vedaldi. Understanding deep image representations by inverting them. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5188–5196, 2015.   
+[22] David McAllester and Karl Stratos. Formal limitations on the measurement of mutual information. In International Conference on Artificial Intelligence and Statistics, pages 875–884. PMLR, 2020.   
+[23] Julian D Olden and Donald A Jackson. Illuminating the “black box”: a randomization approach for understanding variable contributions in artificial neural networks. Ecological modelling, 154(1-2):135–150, 2002.   
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+[25] David E. Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams. Learning representations by backpropagating errors. Nature, 323:533–536, October 1986.   
+[26] Ruslan Salakhutdinov and Geoffrey Hinton. Deep boltzmann machines. In Artificial intelligence and statistics, pages 448–455. PMLR, 2009.   
+[27] Wojciech Samek, Thomas Wiegand, and Klaus-Robert Müller. Explainable artificial intelligence: Understanding, visualizing and interpreting deep learning models. arXiv preprint arXiv:1708.08296, 2017.   
+[28] Andrew Saxe, Yamini Bansal, Joel Dapello, Madhu Advani, Artemy Kolchinsky, Brendan Tracey, and David Cox. On the information bottleneck theory of deep learning. In International Conference on Representation Learning, 2018.   
+[29] Ohad Shamir, Sivan Sabato, and Naftali Tishby. Learning and generalization with the information bottleneck. Theoretical Computer Science, 411(29-30):2696–2711, 2010.   
+[30] Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via information. arXiv preprint arXiv:1703.00810, 2017.   
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+[38] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] see Section 5   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] see the URL in Appendix G   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] see Section 4.1, Appendix G, and Appendix H   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] see Section 4   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] see Appendix G
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] see Appendix H   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
\ No newline at end of file
diff --git a/parse/train/6ugK-RQhIP5/6ugK-RQhIP5_content_list.json b/parse/train/6ugK-RQhIP5/6ugK-RQhIP5_content_list.json
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index 0000000000000000000000000000000000000000..28b807004079eb6641a02f9619260de026066eb0
--- /dev/null
+++ b/parse/train/6ugK-RQhIP5/6ugK-RQhIP5_content_list.json
@@ -0,0 +1,1454 @@
+[
+    {
+        "type": "text",
+        "text": "A Probabilistic Representation for Deep Learning: Delving into The Information Bottleneck Principle ",
+        "text_level": 1,
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
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+    },
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+        "text": "1 The Information Bottleneck (IB) principle has recently attracted great attention to   \n2 explaining Deep Neural Networks (DNNs), and the key is to accurately estimate the   \n3 mutual information between a hidden layer and dataset. However, some unsettled   \n4 limitations weaken the validity of the IB explanation for DNNs. To address these   \n5 limitations and fully explain deep learning in an information theoretic fashion, we   \n6 propose a probabilistic representation for deep learning that allows the framework   \n7 to estimate the mutual information, more accurately than existing non-parametric   \n8 models, and also quantify how the components of a hidden layer affect the mutual   \n9 information. Leveraging the probabilistic representation, we take into account the   \n10 back-propagation training and derive two novel Markov chains to characterize the   \n11 information flow in DNNs. We show that different hidden layers achieve different   \n12 IB trade-offs depending on the architecture and the position of the layers in DNNs,   \n13 whereas a DNN satisfies the IB principle no matter the architecture of the DNN. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "14 1 Introduction ",
+        "text_level": 1,
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+    },
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+        "text": "15 Deep learning [18] has already achieved great success in numerous applications. Deep Neural   \n16 Networks (DNNs), however, are still commonly viewed as ‘black boxes’ [27]. Considerable efforts   \n17 have been devoted to explaining the internal mechanism of DNNs from various perspectives, such as   \n18 mathematics [5, 12], statistics [14, 20, 23], computer vision [37, 21], etc. Recently, the Information   \n19 Bottleneck (IB) principle has attracted attention in opening the ‘black boxes’ of DNNs [30, 33].   \n20 Given a joint distribution $P ( X , Y )$ , the IB principle posits a random variable $T = f ( X )$ obeying the   \n21 Markov chain $Y  X  T$ and optimizes $T$ by the IB Lagrangian [32, 31] ",
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+    {
+        "type": "equation",
+        "img_path": "images/11aa05fc87c0e926984011995c27a67adcdc75d29a8c9a8582819583cb7571ba.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { P ( T \\mid X ) } I ( X ; T ) - \\beta I ( Y ; T ) ,\n$$",
+        "text_format": "latex",
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+        "text": "22 where $f ( \\cdot )$ is an arbitrary function, $I ( \\cdot ; \\cdot )$ denotes mutual information, and the Lagrange multiplier   \n23 $\\beta > 0$ controls the IB trade-off between compressing the input $X$ and preserving the information   \n24 of the label $Y$ . In the seminal work [30], Tishby et al. manifest the IB trade-off in every layer of   \n25 $\\mathrm { D N N s } = \\{ \\pmb { x } ; t _ { 1 } ; \\cdot \\cdot \\cdot ; t _ { I } ; \\hat { \\pmb { y } } \\}$ via studying $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ , where $T _ { i }$ is the random variable of   \n26 the ith hidden layer $\\mathbf { \\Delta } \\mathbf { t } _ { i }$ . Especially, the authors ascribe DNN generalization to the compression [29].   \n27 In the context of deterministic DNNs, recent works reveal some limitations of the IB principle for   \n28 explaining DNNs. Amjad et al. argue that the IB principle becomes an ill-posed optimization problem   \n29 due to $I ( { \\bar { X } } ; T _ { i } ) = \\infty$ [1], and Kolchinsky et al. demonstrate that not every layer of DNNs satisfies a   \n30 strict IB trade-off, i.e., different layers only differ in $I ( X ; T _ { i } )$ but $I ( Y ; T _ { i } )$ keeps consistent in all   \n31 layers [15]. In addition, Saxe et al. experimentally show that the compression does not occur in   \n32 DNNs with non-saturating activation functions, e.g., the popular ReLU function [28], and Goldfeld   \n33 et al. doubt the causality between the generalization of DNNs and the compression [10, 7]. These   \n34 unsettled limitations greatly weakens the validity of the IB explanations for DNNs.   \n35 The key to examining the IB principle in DNNs is the accurate estimation of the mutual information.   \n36 However, regarding DNNs as deterministic models hinders us from specifying the random variable   \n37 $T _ { i }$ and the distribution $P ( T _ { i } )$ , thus it is difficult to accurately estimate $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ . More   \n38 specifically, in the absence of a clear definition of $T _ { i }$ , simply assuming the activations of $\\mathbf { \\Delta } _ { t _ { i } }$ as the i.i.d.   \n39 samples of $T _ { i }$ induces $T _ { i }$ being a continuous random variable and $I ( X ; T _ { i } ) = \\infty$ in deterministic   \n40 DNNs (see Appendix C in [28]). The complicated architecture of DNNs makes it challenging to   \n41 specify $P ( T _ { i } )$ . Therefore, most previous works have to indirectly estimate $P ( T _ { i } )$ via non-parametric   \n42 models [35], such as the empirical distribution [30], Kernel Density Estimation (KDE) [28], and   \n43 Gaussian convolution [10]. However, we experimentally confirm that classical non-parametric models   \n44 derives poor mutual information estimation [24, 22] in DNNs, and one reason is because activations   \n45 do not satisfy the i.i.d. prerequisite of non-parametric models (see Appendix G). In summary, the   \n46 limitations mainly stem from the lack of an explicit probabilistic representation for deep learning.   \n47 The IB principle only formulates the information flow in $\\mathbf { D N N s } = \\{ \\pmb { x } , \\pmb { t } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { t } _ { I } , \\hat { \\pmb { y } } \\}$ after training,   \n48 and the corresponding Markov chain (see Fig. 1 in [30]) ",
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+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ead84a6b0118b658e9a1bf913fc61d857e2125f8496d78d794591ac5e0c3e24c.jpg",
+        "text": "$$\nY  X  T _ { 1 } \\cdots  T _ { I }  { \\hat { Y } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "text": "49 indicates that the information of $Y$ transfers to $T _ { i }$ in the forward direction and $T _ { i }$ receives the   \n50 information of $Y$ only via $X$ . However, training DNNs by the back-propagation [25] implies that the   \n51 information of $Y$ transfers to $T _ { i }$ in the backward direction during training and retains information   \n52 in $T _ { i }$ after training. Notably, Zhang et al. show that a DNN can fit labels well even using Gaussian   \n53 noise as input to train the DNN [38], which implies that $T _ { i }$ can directly receive the information of $Y$ .   \n54 Hence, the IB principle does not comprehensively characterize the information flow in DNNs.   \n55 To address the above limitations and comprehensively explain DNNs in an information theoretic   \n56 fashion, we introduce the probability space $\\left( \\Omega _ { T _ { i } } , \\mathcal { F } , P _ { T _ { i } } \\right)$ [6] for the ith hidden layer $\\mathbf { \\Delta } _ { t _ { i } }$ in DNNs.   \n57 Compared to previous works, the probability space $\\left( \\Omega _ { T _ { i } } , \\mathcal { F } , P _ { T _ { i } } \\right)$ enables us to: (i) accurately estimate   \n58 $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ via specifying $T _ { i }$ and $P ( T _ { i } )$ , and (ii) quantify the effect of the architecture of   \n59 $\\mathbf { \\Delta } _ { t _ { i } }$ and the back-propagation on $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ via explicitly modeling all the ingredients of $\\mathbf { \\Delta } _ { t _ { i } }$ ,   \n60 such as the activation function and the weights in a probabilistic way. To the best of our knowledge,   \n61 this is the first time the probability space of a hidden layer in DNNs is as defined. ",
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+        "text": "Leveraging 62 $\\left( \\Omega _ { T _ { i } } , \\mathcal { F } , P _ { T _ { i } } \\right)$ , we derive information theoretic explanations for DNNs as follows: ",
+        "bbox": [
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+        "text": "• Two Markov chains1 characterize the information flow in $\\mathrm { D N N s } = \\{ \\pmb { x } , t _ { 1 } , \\cdot \\cdot \\cdot , t _ { I } , \\hat { \\pmb { y } } \\}$ ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f3a81709a1c4d97473189579df11bdad3aee210f893f908a8a45e57b7327c1ab.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\bar { X } \\right. T _ { 1 } \\right. \\cdots \\right. T _ { I } \\right. \\hat { Y } } } \\\\ { { \\nonumber } } \\\\ { { T _ { 1 } \\left. \\cdot \\cdot \\cdot \\left. T _ { I } \\left. \\hat { Y } \\left. Y . } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "• Different hidden layers manifest different IB trade-offs depending on the architecture and the position of hidden layers in DNNs. ",
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+    {
+        "type": "text",
+        "text": "• A DNN satisfies the IB principle no matter the architecture of the DNN. ",
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+        "text": "67 Preliminaries. $P ( X , Y ) = P ( X ) P ( Y | X )$ is an unknown joint distribution between $X$ and $Y$ . A   \n68 dataset $\\pmb { \\mathcal { D } } = \\{ ( \\pmb { x } ^ { j } , \\pmb { y } ^ { j } ) | \\pmb { x } ^ { j } \\in \\mathbb { R } ^ { M } , \\pmb { y } ^ { j } \\in \\mathbb { Z } \\} _ { j = 1 } ^ { J }$ consists of $J$ i.i.d. samples generated from $P ( X , Y )$   \n69 with finite $L$ labels, i.e., $y ^ { j } \\in \\{ 1 , \\cdots , L \\}$ . In the context of supervised learning, we focus on   \n70 feedfworad fully connected $\\mathrm { D N N s } = \\{ \\pmb { x } , t _ { 1 } , \\cdot \\cdot \\cdot , t _ { I } , \\hat { \\pmb { y } } \\}$ , i.e., Multi-Layer Perceptions (MLPs) [8]   \n71 for the image classification task. Without loss of generality, we use the $\\mathrm { M L P } = \\{ { \\pmb x } , t _ { 1 } , t _ { 2 } , { \\hat { \\pmb y } } \\}$ with   \n72 the cross-entropy loss $\\ell _ { \\mathrm { C E } }$ for most theoretical derivations. In addition, $H ( \\cdot )$ denotes entropy. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the MLP, 73 where 74 $\\begin{array} { r } { \\langle \\omega _ { n } ^ { ( 1 ) } , \\pmb { x } \\rangle = \\sum _ { m = 1 } ^ { M } \\omega _ { m n } ^ { ( 1 ) } \\cdot \\pmb { x } _ { m } + b _ { 1 n } } \\end{array}$ $\\mathbf { \\delta t } _ { 1 }$ and $\\mathbf { { t } } _ { 2 }$ have $N$ and $K$ neurons, respectively, and is the $n$ 1  1n  th dot-product given the weight $\\pmb { t } _ { 1 } = \\{ t _ { 1 n } = \\sigma _ { 1 } [ \\langle \\pmb { \\omega } _ { n } ^ { ( 1 ) } , \\pmb { x } \\rangle ] \\} _ { n = 1 } ^ { N }$ $\\omega _ { m n } ^ { ( 1 ) }$ n  n=1 and the bias 75 $b _ { 1 n }$ , and $\\sigma _ { 1 } ( \\cdot )$ denotes an activation function, e.g., ReLU. Similarly, $t _ { 2 } = \\{ t _ { 2 k } = \\sigma _ { 2 } [ \\langle \\omega _ { k } ^ { ( 2 ) } , t _ { 1 } \\rangle ] \\} _ { k = 1 } ^ { K }$ where 76 $\\begin{array} { r } { \\langle \\omega _ { k } ^ { ( 2 ) } , t _ { 1 } \\rangle = \\sum _ { n = 1 } ^ { N } \\omega _ { n k } ^ { ( 2 ) } \\cdot t _ { 1 n } + b _ { 2 k } } \\end{array}$ . The output layer $\\hat { \\pmb { y } }$ is softmax with $L$ nodes ",
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+        "img_path": "images/4f2456371ba4872c2a70ce1d8314e74e6343ea929ed8c9110bb4228bd42583ce.jpg",
+        "text": "$$\n\\hat { \\pmb { y } } = \\{ \\hat { y } _ { l } = \\frac { 1 } { Z _ { Y } } \\mathrm { e x p } [ \\langle { \\pmb \\omega } _ { l } ^ { ( 3 ) } , { \\pmb t } _ { 2 } \\rangle ] = \\frac { 1 } { Z _ { Y } } \\mathrm { e x p } [ g _ { l } ( { \\pmb t } _ { 2 } ( { \\pmb t } _ { 1 } ( { \\pmb x } ) ) ) ] \\} _ { l = 1 } ^ { L } ,\n$$",
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+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { \\langle \\omega _ { l } ^ { ( 3 ) } , t _ { 2 } \\rangle = \\sum _ { k = 1 } ^ { K } \\omega _ { k l } ^ { ( 3 ) } \\cdot t _ { 2 k } + b _ { y l } } \\end{array}$ $\\begin{array} { r } { Z _ { Y } = \\sum _ { l = 1 } ^ { L } \\exp [ \\langle \\omega _ { l } ^ { ( 3 ) } , t _ { 2 } \\rangle ] } \\end{array}$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1In which the virtual random variable $\\bar { X }$ has all the information of $X$ except $Y$ , namely $H ( { \\bar { X } } ) = H ( X | Y )$ ",
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+            "Figure 1: Given a $4 \\times 4$ input $_ z$ , a fully connected layer $\\pmb { t }$ is equivalent to a convolution layer with $4 \\times 4$ convolution kernels. The definition of convolution (Chapter 9.1 in [11]) implies that the $4 \\times 4$ weights $\\omega _ { 1 }$ and $\\omega _ { 2 }$ define two global features, and the two activations $t _ { 1 } , t _ { 2 }$ indicate the cross-correlation between $\\omega _ { 1 } , \\omega _ { 2 }$ and $_ z$ , respectively. $P _ { T | , Z } ( \\omega _ { 1 } | z )$ and $P _ { T | Z } ( \\omega _ { 2 } | z )$ measure the probability of $\\omega _ { 1 }$ and $\\omega _ { 2 }$ being recognized as the feature with the largest cross-correlation to $_ z$ , respectively. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "78 2 A probabilistic representation for deep learning ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "9 To accurately estimate $I ( X ; T _ { i } )$ and $I ( Y ; T _ { i } )$ , in this section, we specify the probability space [6] for   \n80 a fully connected layer and derive the probabilistic explanations of the entire MLP.   \n81 It is known that a convolution kernel (namely the weights of convolution) defines a local feature,   \n82 and a convolution operation derives a feature map to measure the cross-correlation between the   \n83 local feature and input in a receptive field (Chapter 9.1 in [11]). Notably, a fully connected layer   \n84 is equivalent to a convolution layer with the kernel size having the same dimension as input. Thus   \n85 the weights of a neuron can be viewed as a global feature, and a fully connected layer with multiple   \n86 neurons derives activations to measure the cross-correlation between the multiple global features and   \n87 the input. The cross-correlation explanation for a fully connected layer is visualized in Figure 1. ",
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+    {
+        "type": "text",
+        "text": "Assuming that88 is the input of 89 , $\\pmb { t }$ $N$ neurons s the dot- $\\{ t _ { n } = \\sigma [ \\langle \\omega _ { n } , z \\rangle ] \\} _ { n = 1 } ^ { N }$ , ere , a $z \\in \\mathbb { R } ^ { M }$ $\\pmb { t }$ $\\begin{array} { r } { \\langle \\omega _ { n } , z \\rangle = \\sum _ { m = 1 } ^ { M } \\omega _ { m n } \\cdot z _ { m } + b _ { n } } \\end{array}$ $_ { z }$ $\\omega _ { n }$ $\\sigma ( \\cdot )$ 90 an activation function. Based othe cross-correlations between $_ z$ the crosand the $N$ orrelation explanation, the behavior of possible features defined by the the we $\\pmb { t }$ is thts $\\{ \\omega _ { n } \\} _ { n = 1 } ^ { N }$ 92 In the context of pattern recognition [34], we define a virtual random process or ‘experiment’ as $\\pmb { t }$ 93 recognizing one of the patterns/features with the largest cross-correlation to $_ { z }$ from the $N$ possible 94 features. The experiment characterizes the behavior of $\\pmb { t }$ (i.e., before recognizing the features with 95 the largest cross-correlation, $\\pmb { t }$ must measure the cross-correlations between $_ { z }$ and all the $N$ possible 96 features) while meets the requirement of the ‘experiment’ definition (i.e., only one outcome will 97 occur on each trial of the experiment [6]). The probability space $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ is defined as follows: ",
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+    {
+        "type": "text",
+        "text": "98 Definition 1. $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ consists of three components: the sample space $\\Omega _ { T }$ has $N$ possible   \n99 outcomes (features) $\\{ \\omega _ { n } = \\{ \\omega _ { m n } \\} _ { m = 1 } ^ { M } \\} _ { n = 1 } ^ { N }$ defined by the weights2 of the $N$ neurons; the event   \n100 space is the $\\sigma$ -algebra; and the probability measure $P _ { T }$ is a Gibbs distribution [19] to quantify the   \n101 probability of $\\omega _ { n }$ being recognized as the feature with the largest cross-correlation to $_ z$ . ",
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+    {
+        "type": "text",
+        "text": "102 Taking into account the randomness of $_ z$ , the conditional distribution $P _ { T | Z }$ is formulated as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3f1f4ef89e2f58eaa03cac182729e8910f8d7536c6a81a74316daeadbb1e3acf.jpg",
+        "text": "$$\nP _ { T | Z } ( \\omega _ { n } | z ) = \\frac { 1 } { Z _ { T } } \\mathrm { e x p } ( t _ { n } ) = \\frac { 1 } { Z _ { T } } \\mathrm { e x p } [ \\sigma ( \\langle \\omega _ { n } , z \\rangle ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where 103 $Z$ is the random variable of $_ z$ and $\\begin{array} { r } { Z _ { T } = \\sum _ { n = 1 } ^ { N } \\exp ( f _ { n } ) } \\end{array}$ is the partition function. ",
+        "bbox": [
+            153,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ clearly explains all the ingredients of $\\pmb { t }$ in a probabilistic fashion. The $n$ th neuron defines a global feature by the weights ${ \\pmb w } _ { n }$ and the activation $t _ { n } = \\sigma ( \\langle \\omega _ { n } , z \\rangle )$ measures the crosscorrelation between ${ \\pmb w } _ { n }$ and $_ { z }$ . The Gibbs distribution $P _ { T | Z }$ indicates that if ${ \\pmb w } _ { n }$ has the higher activation, i.e., the larger cross-correlation to $_ z$ , it has the larger probability being recognized as the feature with largest cross-correlation to $_ { z }$ . For instance, if $\\dot { z } \\in \\mathbb { R } ^ { 1 6 }$ and $\\pmb { t }$ includes $N = 2$ neurons, then $\\Omega _ { T } = \\bar { \\{ \\omega _ { 1 } , \\omega _ { 2 } \\} }$ defines two possible outcomes (features), where $\\omega _ { n } = \\{ \\omega _ { m n } \\} _ { m = 1 } ^ { 1 6 }$ $\\mathcal { F } = \\{ \\emptyset , \\{ \\omega _ { 1 } \\} , \\{ \\omega _ { 2 } \\} , \\{ \\omega _ { 1 } , \\omega _ { 2 } \\} \\}$ means that neither, one, or both of the features are recognized by $\\pmb { t }$ given $_ z$ , respectively. $\\dot { P _ { T | Z } } ( \\omega _ { 1 } | z )$ and $P _ { T | Z } ( \\omega _ { 2 } | z )$ are the probability of $\\omega _ { 1 }$ and $\\omega _ { 2 }$ being recognized as the feature with the largest cross-correlation to $_ z$ , respectively. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "113 $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ explains the representation ability of deep learning. Compared to Restricted Boltzmann   \n114 Machines (RBMs) [26] simply using binary units to indicate features being recognized or not given   \n115 input, the Gibbs distribution3 $P _ { T | Z } \\bar { ( \\omega _ { n } | z ) }$ measures the probability of $\\omega _ { n }$ being recognized with   \n116 the largest cross-correlation to $_ { z }$ , i.e., it characterizes the relation between features and input more   \n117 accurately. Moreover, Equation 5 shows that $t _ { n } = \\sigma ( \\langle \\omega _ { n } , z \\rangle )$ is the negative energy function [19] of   \n118 the Gibbs distribution, thus $P _ { T | Z } ( \\omega _ { n } | z )$ can be derived as long as $\\sigma ( \\langle \\omega _ { n } , z \\rangle )$ are known because   \n119 the energy function is the sufficient statistics [2] of the Gibbs distribution. That enables subsequent   \n120 hidden layers to generate high-level features of input via directly processing the activations $\\{ t _ { n } \\} _ { n = 1 } ^ { N }$   \n121 thus deep learning can form a hierarchical structure to represent much complex features.   \n122 $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ answers a fundamental question: which component of a hidden layer contains the   \n123 information of the layer? Since $\\omega _ { n }$ defines $\\Omega _ { T }$ , the weights contain all the information of a layer. In   \n124 particular, since the activation $t _ { n } = \\sigma ( \\langle \\omega _ { n } , z \\rangle )$ is a function of $\\omega _ { n }$ , the data processing inequality   \n125 [4] indicates that the information of $t _ { n }$ is no more than the information of $\\omega _ { n }$ . Simulations in Section   \n126 4.2 demonstrate that if activations do not correctly characterize the cross-correlation between weights   \n127 and input, activations contain less information than weights do. ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Based on $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ , we define the random variable $T$ as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "29 Definition 2. Given the fully connected layer $\\pmb { t }$ , we define the random variable $T : \\Omega _ { T } \\to E _ { T }$ as ",
+        "bbox": [
+            155,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ad0b415d9e5e503212ddc52194e3581a8ae4ff8450544cd1937f2441cf866c5.jpg",
+        "text": "$$\nT ( \\omega _ { n } ) \\triangleq n ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "130 where the measurable space $E _ { T } = \\{ 1 , \\cdots , N \\}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "31 Since $\\Omega _ { T }$ is composed of finite $N$ possible outcomes, $T$ is a discrete random variable. Notably, the   \n132 one-to-one correspondence between $\\omega _ { n }$ and $n$ indicates ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2247299639c415f9334a06869ea782e214240aa9a4c9b348e157d823dfc6218b.jpg",
+        "text": "$$\nP _ { T | Z } ( \\omega _ { n } | z ) = P _ { T | Z } ( n | z ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "133 If not considering the back-propagation training, the weights (namely $\\Omega _ { T _ { i } }$ ) of each layer are fixed.   \n134 Thus $T _ { i + 1 }$ entirely depends on $T _ { i }$ and the $\\mathrm { M L P } = \\{ { \\pmb x } ; t _ { 1 } ; t _ { 2 } ; { \\hat { \\pmb y } } \\}$ forms a Markov chain ",
+        "bbox": [
+            140,
+            457,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4da2d70d9dacbaa625b7984616acefca401c5937c8e66798e413744240256425.jpg",
+        "text": "$$\nX  T _ { 1 }  T _ { 2 }  { \\hat { Y } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "135 Based on the corresponding joint distribution $P ( \\hat { Y } , T _ { 2 } , T _ { 1 } | X ) = P ( T _ { 1 } | X ) P ( T _ { 2 } | T _ { 1 } ) P ( \\hat { Y } | T _ { 2 } )$ and   \n136 Definition 2, we derive a probabilistic explanation for the entire MLP, which is summarized in   \n137 Theorem 1. The detailed derivation is presented in Appendix $\\mathbf { B }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "138 Theorem 1. The $\\mathrm { M L P } = \\{ { \\pmb x } ; { \\pmb t } _ { 1 } ; { \\pmb t } _ { 2 } ; { \\hat { \\pmb y } } \\}$ formulates a conditional Gibbs distribution ",
+        "bbox": [
+            135,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1a6333cf5aa7f26f3f32459a0ba19f9ef6407814128bf70d9f5ddb5d6c1237a2.jpg",
+        "text": "$$\nP _ { \\hat { Y } | X } ( l | x ) = \\sum _ { k = 1 } ^ { K } \\sum _ { n = 1 } ^ { N } P ( \\hat { Y } = l , T _ { 2 } = k , T _ { 1 } = n | X = x ) = \\frac { 1 } { Z _ { \\mathrm { M L P } } ( x ) } \\mathrm { e x p } [ g _ { l } ( t _ { 2 } ( t _ { 1 } ( x ) ) ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where 139 $\\begin{array} { r } { Z _ { \\mathrm { M L P } } ( \\boldsymbol { x } ) = \\sum _ { l = 1 } ^ { L } \\sum _ { k = 1 } ^ { K } \\sum _ { n = 1 } ^ { N } P _ { \\hat { Y } , T _ { 2 } , T _ { 1 } | \\boldsymbol { X } } ( l , k , n | \\boldsymbol { x } ) } \\end{array}$ is the partition function. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "140 Since $P _ { \\hat { Y } | X } ( l | \\pmb { x } )$ exactly equals the output $\\hat { y } _ { l }$ of the MLP, namely Equation (4), we conclude that   \n141 the entire architecture of the MLP forms a family of Gibbs distribution $P _ { \\hat { Y } | X } ( l | \\mathbf { \\boldsymbol { x } } )$ . In general, the   \n142 back-propagation updates a weight $\\omega$ based on the gradient of $\\ell _ { \\mathrm { C E } }$ with respect to $\\omega$ , ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb511190153234912764c973280cbdfc4de5c1615372f7986068825cf4016c92.jpg",
+        "text": "$$\n\\omega ( s + 1 ) = \\omega ( s ) - \\alpha \\cdot \\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ( s ) } = \\omega ( s ) - \\alpha \\cdot \\frac { \\partial \\mathrm { K L } [ P ( Y | X ) | | P ( \\hat { Y } | X ) ] } { \\partial \\omega ( s ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "143 where $s$ is the index of training iteration, $\\alpha$ is the training rate, and $\\mathrm { K L } [ \\cdot | | \\cdot ]$ is the KL-divergence. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "144 Figure 2 summarizes the probabilistic explanation for deep learning based on the MLP. In general,   \n145 a single learning iteration, an epoch, consists of two phases: training and inference (after training).   \n146 During inference, the MLP bridges $X$ and $\\hat { Y }$ via multiple intermediate features $\\Omega _ { T _ { 1 } }$ , $\\Omega _ { T _ { 2 } }$ , and $\\Omega _ { \\hat { Y } }$   \n147 defined by weights, and formulates the statistical relation between $\\hat { Y }$ and $X$ as a family of conditional   \n148 Gibbs distribution $P ( { \\hat { Y } } | X )$ . During training, the back-propagation updates weights to learn optimal   \n149 intermediate features for searching an optimal $P ( { \\hat { Y } } | X )$ to accurately approximate $P ( { Y \\vert } X )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4783b0e35de39034e8ce539d14ecdda10d599541289305cdd2ef3b9b1d3f69c0.jpg",
+        "image_caption": [
+            "Figure 2: The visualization of the probabilistic explanation for deep learning based on the MLP. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "150 3 The information theoretic explanations for deep learning ",
+        "text_level": 1,
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "51 To address the limitations of existing IB explanations, this section proposes some novel information   \n52 theoretic explanations for DNNs based on the proposed probabilistic representation. ",
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+    {
+        "type": "text",
+        "text": "153 Proposition 1. The mutual information between a fully connected layer and dataset is finite. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c2322c37754743cf88c1e39576148863252d3766f16220cff196fcfea7814eff.jpg",
+        "text": "$$\nI ( X ; T ) < \\infty .\n$$",
+        "text_format": "latex",
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+        "text": "154 Proof: Definition 2 shows $E _ { T } = \\{ 1 , \\cdots N \\}$ . Thus $T$ is a discrete random variable and $H ( T ) < \\infty$ ,   \n155 thereby $I ( X ; T ) \\leq H ( T ) < \\infty$ .   \n156 Proposition 1 circumvents the infinite mutual information problem. In the absence of a clear definition   \n157 $T : \\Omega _ { T } \\to E _ { T }$ , most previous works [28, 3, 1] simply viewing the activation $t _ { n }$ as the sample of $T$ ,   \n158 namely $t _ { n } \\in E _ { T } = \\mathbb { R }$ , implies $T$ being continuous and gives rise to the infinite mutual information   \n159 problem in deterministic DNNs. However, $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ indicates that $t _ { n }$ actually is a variable   \n160 measuring the cross-correlation between ${ \\pmb w } _ { n }$ and $_ { z }$ rather than the sample of $T$ , namely $t _ { n } \\notin E _ { T }$ . ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "161 Theorem 2. The information of $Y$ flows into the MLP in the backward direction during training ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e0bff2c4bdcdef0b2f4947e8e3d58ede655d10bd1f77754eb744a4829bcc63f.jpg",
+        "text": "$$\nT _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "162 Proof: First, since $\\Omega _ { T }$ is defined by $\\omega$ in $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ and Equation (10) shows that $\\omega ( s + 1 )$ is   \n163 determined by all the previous gradients $\\{ \\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ( s ) } \\} _ { s = 1 } ^ { S }$ Es) }Ss=1 , and ω(0) is randomly initialized and α is a   \n164 constant, we can derive that $\\Omega _ { T }$ is determined by ∂\\`CE∂ω . Second, based on the back-propagation, the   \n165 relation between gradients in two adjacent layers in the $\\mathrm { M L P } = \\{ { \\pmb x } ; t _ { 1 } ; t _ { 2 } ; { \\hat { \\pmb y } } \\}$ is formulated as ",
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+        "text": "$$\n\\begin{array} { r l } & { \\qquad \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { \\star } } { \\partial \\omega _ { k l } ^ { ( 3 ) } } = [ P _ { \\hat { Y } | X } ( l | x ) - P _ { Y | X } ( l | x ) ] \\cdot t _ { 2 k } , } \\\\ { \\displaystyle \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } = \\sum _ { l = 1 } ^ { L } \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { \\star } } { \\partial \\omega _ { k l } ^ { ( 3 ) } } \\cdot \\omega _ { k l } ^ { ( 3 ) } \\cdot \\displaystyle \\frac { \\sigma _ { 2 } ^ { \\prime } \\big ( \\langle \\omega _ { k } ^ { ( 2 ) } , t _ { 1 } \\rangle \\big ) } { f _ { 2 k } } \\cdot t _ { 1 n } , ~ \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } } { \\partial \\omega _ { m n } ^ { ( 1 ) } } = \\sum _ { k = 1 } ^ { K } \\displaystyle \\frac { \\partial \\ell _ { \\mathrm { C E } } ^ { ( \\zeta ) } } { \\partial \\omega _ { n k } ^ { ( 2 ) } } \\cdot \\omega _ { n k } ^ { ( 2 ) } \\cdot \\displaystyle \\frac { \\sigma _ { 1 } ^ { \\prime } \\big ( \\langle \\omega _ { n } ^ { ( 1 ) } , x \\rangle \\big ) } { t _ { 1 n } } \\cdot x _ { m } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            608,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "166 Equation 13 shows that $\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( 3 ) } }$ is a function of $P _ { Y \\mid X } ( l \\mid x )$ and $\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( i ) } }$ is a function of $\\frac { \\partial \\ell _ { \\mathrm { C E } } } { \\partial \\omega ^ { ( i + 1 ) } }$ , where   \n167 $\\boldsymbol { \\omega } ^ { ( 3 ) }$ denotes the weight of $\\hat { \\pmb { y } }$ . The two points above enable us to derive that $\\Omega _ { T _ { i } }$ is a function of $\\Omega _ { T _ { i + 1 } }$   \n168 and $\\Omega _ { \\hat { Y } }$ is a function of $P ( { Y \\vert } X )$ . Based on Definition 2, we can further derive that $T _ { i }$ is a function   \n169 of $T _ { i + 1 }$ and $\\hat { Y }$ is a function of $Y$ , i.e., $T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y$ . (See the detailed proof in Appendix C).   \n170 Theorem 2 is consistent with the prevailing explanation for deep learning. LeCunn et al. show that   \n171 deep learning exploits the hierarchical property of signals [18], i.e., the layers farther from output   \n172 learn lower-level features, such as edges, whereas the layers closer to output assemble lower-level   \n173 features into the higher-level features corresponding to labels (see Figure 2 in [37]). Notably, since   \n174 lower-level features commonly exist in signals with different labels (e.g., lower-level features, such   \n175 as the edges of the vehicle frame and the circular contour of wheels, exist in both the car and the   \n176 truck classes in the CIFAR-10 dataset [16] in Figure 2), lower-level features do not contain much   \n177 information of labels. Therefore, the layers farther from output do not have much information of   \n178 labels, which is consistent with the Markov chain $T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y$ .   \n179 Since all the information of $Y$ stems from $X$ (i.e., $H ( Y ) = I ( X ; Y )$ proven in Appendix D),   \n180 Theorem 2 implies that partial information of $X$ flows into the MLP in the backward direction during   \n181 training. Equation (2) shows the information of $X$ flowing into the MLP in the forward direction   \n182 during inference. Overall, the information of $X$ flows in the backward and forward directions during   \n183 training and inference, respectively. As a result, the Markov chain, Equation (2), proposed by recent   \n184 works could not fully characterize the information flow of $X$ in the MLP in each epoch. In other   \n185 words, $I ( X ; T _ { i } )$ is not necessarily greater than $I ( X ; T _ { i + 1 } )$ in the MLP in each epoch.   \n186 Equation (2) shows that $T _ { i }$ receives the information of $Y$ via $X$ during inference. Theorem 2 shows   \n187 that $T _ { i }$ also directly receives information of $Y$ during training, because the back-propagation updates   \n188 weights (i.e., $\\Omega _ { T _ { i } }$ ) based on the label $Y$ . Thus Equation (2) cannot fully characterize the information   \n189 flow of $Y$ in the MLP in each epoch, when we take into account the back-propagation training.   \n190 To fully characterize the information flow in the MLP in each epoch, we introduce Corollary 1.   \n191 Corollary 1. The information flow in the MLP can be characterized by two Markov chains as ",
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+        "bbox": [
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+            256,
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+    {
+        "type": "equation",
+        "img_path": "images/5009ee51df98562fce5a0f511b8df83d1af573616fc90fd96cb436e7cdb2eab5.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\bar { X } \\right. T _ { 1 } \\right. T _ { 2 } \\right. \\hat { Y } } } \\\\ { { T _ { 1 } \\left. T _ { 2 } \\left. \\hat { Y } \\left. Y . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            287,
+            588,
+            329
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The virtual random variable 192 $\\bar { X }$ contains all the information of $X$ except $Y$ , i.e., $H ( { \\bar { X } } ) = H ( X | Y )$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "193 Proof of the first Markov chain: Since $\\bar { X }$ does not have any information of $Y$ , it can only flow into   \n194 the MLP in the forward direction during inference. Again since $\\bar { X }$ does not have any information of   \n195 $Y$ , the information flow of $Y$ during training will not affect the information flow of $\\bar { X }$ . Therefore,   \n196 ${ \\bar { X } }  T _ { 1 }  T _ { 2 }  { \\hat { Y } }$ characterizes the information flow of $\\bar { X }$ in both training and inference phases.   \n197 Proof of the second Markov chain: Since the weights are fixed after training, the sample space and   \n198 the distribution of hidden layers are fixed after training. Therefore, the information of $Y$ transferred   \n199 into hidden layers during training will retain there after training (i.e., during inference). In addition,   \n200 Definition 1 indicates that a fully connected layer $t = \\{ t _ { n } = \\bar { \\sigma } ( \\langle \\omega _ { n } , z \\rangle ) \\bar  \\} _ { n = 1 } ^ { N }$ measures the cross  \n201 correlation between $\\omega _ { n } ^ { ( 1 ) }$ ) and z during inference, thus {ω(1)n }Nn=1 can be viewed as a representation   \n202 of $Z$ . As a result, even though has all the information of $Y$ , the information of that can learn   \n203 from $Z$ is determined by how much information of $Y$ the representation $\\{ \\omega _ { n } ^ { ( 1 ) } \\} _ { n = 1 } ^ { N }$ has. Overall, the   \n204 information flow of $Y$ during inference will be the same as that during training. Based on Theorem 2,   \n205 we conclude that $T _ { 1 } \\gets T _ { 2 } \\gets \\hat { Y } \\gets Y$ characterizes the information flow of $Y$ in the MLP in both   \n206 training and inference phases. Detailed derivations and explanations are presented in Appendix E.   \n07 To quantify how much information of $X$ and $Y$ is learned by the MLP, we introduce Corollary 2.   \n08 Corollary 2. The mutual information between dataset and the entire MLP can be expressed as ",
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+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d5b8ff95f0bad5682c7c714c8b67c980309ad74b0febef5e44e741a02bb75cf3.jpg",
+        "text": "$$\n\\begin{array} { r l } & { I ( X ; T _ { \\mathrm { M L P } } ) = I ( \\bar { X } ; T _ { 1 } ) + I ( Y ; \\hat { Y } ) } \\\\ & { I ( Y ; T _ { \\mathrm { M L P } } ) = I ( Y ; \\hat { Y } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "09 where $T _ { \\mathrm { M L P } }$ denotes a random variable corresponding to the entire architecture of the MLP. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof: Since 210 $H ( Y ) = I ( X ; Y )$ (Appendix D), $H ( X ) = H ( { \\bar { X } } ) + I ( X ; Y ) = H ( { \\bar { X } } ) + H ( Y )$ .   \n211 Hence, Corollary 2 can be derived by Corollary 1 and the chain rule. The proof is in Appendix $\\mathrm { F }$ . ",
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+    {
+        "type": "text",
+        "text": "4 Simulations ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we propose a mutual information estimator based on $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ and demonstrate the probabilistic representation and information theoretic explanations for deep learning on a synthetic dataset with known entropy. Additional experiments on benchmark datasets are in Appendix $_ \\mathrm { H }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 Setup",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Mutual information estimator. Based on the definition of mutual information, we have ",
+        "bbox": [
+            168,
+            830,
+            764,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/970136fd3bee2b75a4bac9104899d20bdee6043ba65170ace25fc78acd861873.jpg",
+        "text": "$$\nI ( X ; T _ { i } ) = H ( T _ { i } ) - H ( T _ { i } | X ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "218 Previous works simply estimate $I ( X ; T _ { i } ) = H ( T _ { i } )$ , because $T _ { i }$ is assumed to be entirely dependent   \n219 on $X$ in the Markov chain, Equation (2), thereby $H ( T _ { i } | X ) = 0$ . However, Corollary 1 shows that $T _ { i }$   \n220 depends on both $X$ and $Y$ if taking into account the training phase, thereby $H ( T _ { i } | \\dot { X } ) \\neq 0$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/d35890b35d8c63aacf56c438ea3467ac26a9d308e54ecd338e8f4f95ff3f52f6.jpg",
+        "image_caption": [
+            "Figure 3: (A) the deterministic image $\\hat { \\textbf { \\textit { x } } }$ . Image0 is generated by adding ${ \\mathcal { N } } ( \\mu , \\sigma ^ { 2 } )$ without rotation, Image1 is generated by rotating $\\hat { \\textbf { \\textit { x } } }$ along the secondary diagonal direction and adding $\\mathcal { N } ( \\mu , \\sigma ^ { 2 } )$ , Image2 and Image are generated by rotating $\\hat { \\textbf { \\textit { x } } }$ along the vertical and horizontal directions, respectively, and adding ${ \\mathcal { N } } ( \\mu , \\sigma ^ { 2 } )$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 1: The number of neurons(nodes) and the activation function in the layers of the MLPs "
+        ],
+        "table_footnote": [],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/78ed5d663a137bb6729dedb1c3caef8157323f41452164a292289bd5861434b1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { r l r l r l r } & { } & { x } & { t _ { 1 } } & { t _ { 2 } } & { \\hat { y } } & { \\sigma ( \\cdot ) } \\\\ & { \\mathrm { M L P 1 } } & { 1 0 2 4 ( 3 2 \\times 3 2 ) } & { 8 } & { 6 } & { 2 } & { \\mathrm { R e L U } ( z ) = \\operatorname* { m a x } ( 0 , z ) } \\\\ & { \\mathrm { M L P 2 } } & { 1 0 2 4 ( 3 2 \\times 3 2 ) } & { 8 } & { 6 } & { 2 } & { \\mathrm { T a n h } ( z ) = ( e ^ { z } - e ^ { - z } ) / ( e ^ { z } + e ^ { - z } ) } \\\\ & { \\mathrm { M L P 3 } } & { 1 0 2 4 ( 3 2 \\times 3 2 ) } & { 2 } & { 6 } & { 2 } & & { \\mathrm { R e L U } } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            196,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To accurately estimate 221 $I ( X ; T _ { i } )$ , we need to specify $P ( T _ { i } | X )$ and $P ( T _ { i } )$ . Based on $\\left( \\Omega _ { T _ { i } } , \\mathcal { F } , P _ { T _ { i } } \\right)$ , we formulate 222 $P _ { T _ { i } | X } ( n | \\mathbf { x } ^ { j } )$ of the three fully connected layers in the MLP as ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2b324456286d1acddea3632abf02add7e95062d7bd704f47c187d30112325571.jpg",
+        "text": "$$\n\\begin{array} { r l } & { P _ { T _ { 1 } | X } ( n | x ^ { j } ) = \\frac { 1 } { Z _ { F _ { 1 } } } \\mathrm { e x p } [ \\sigma _ { 1 } ( \\langle \\omega _ { n } ^ { ( 1 ) } , x ^ { j } \\rangle ) ] , ~ P _ { T _ { 2 } | X } ( k | x ^ { j } ) = \\frac { 1 } { Z _ { F _ { 2 } } } \\mathrm { e x p } [ \\sigma _ { 2 } ( \\langle \\omega _ { k } ^ { ( 2 ) } , t _ { 1 } ( x ^ { j } ) \\rangle ) ] , } \\\\ & { ~ P _ { T _ { Y } | X } ( l | x ^ { j } ) = \\frac { 1 } { Z _ { F _ { Y } } } \\mathrm { e x p } [ \\langle \\omega _ { l } ^ { ( 3 ) } , t _ { 2 } ( t _ { 1 } ( x ^ { j } ) ) \\rangle ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To derive the marginal distribution 223 $P ( T _ { i } )$ , we sum the joint distribution $P ( T _ { i } , X )$ over $\\mathbf { \\boldsymbol { x } } \\in \\mathcal { X }$ , ",
+        "bbox": [
+            140,
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/338336e0e6b08caad18fa6446b1c2ca4822cf1d36932b61a865fda6bba463246.jpg",
+        "text": "$$\n\\begin{array} { r } { ^ { \\circ } ( T _ { i } = n ) = \\sum _ { \\mathbf { x } \\in \\mathcal { X } } P _ { X } ( \\mathbf { x } ) P _ { T _ { i } | X } ( n | \\mathbf { x } ) \\approx \\sum _ { \\mathbf { x } ^ { \\prime } \\in \\mathcal { D } } P _ { X } ( \\mathbf { x } ^ { j } ) P _ { T _ { i } | X } ( n | \\mathbf { x } ^ { j } ) = \\frac { 1 } { J } \\sum _ { \\mathbf { x } ^ { j } \\in \\mathcal { D } } P _ { T _ { i } | X } ( n | \\mathbf { x } ^ { j } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            368,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where 224 $P _ { X } ( \\pmb { x } ^ { j } )$ is estimated by the empirical distribution $1 / J$ given $\\mathcal { D }$ . Finally, we can derive $I ( X ; T _ { i } )$ 225 by Equation 16, 17, and 18. Similarly, based on the definition of mutual information, we have ",
+        "bbox": [
+            137,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08591322dff3a2a6bdf86ca7a5aa904e6540cd4ff726c9491295f0ad798bfdeb.jpg",
+        "text": "$$\nI ( Y ; T _ { i } ) = H ( T _ { i } ) - H ( T _ { i } | Y ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            444,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To estimate 226 $H ( T _ { i } | Y )$ , we reformulate $P ( T _ { i } | Y )$ as ",
+        "bbox": [
+            140,
+            465,
+            503,
+            482
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be6ebf67fab350423a7426074d6cb414acb27a634668fffcee744eeafe6a6947.jpg",
+        "text": "$$\n\\begin{array} { r } { P _ { T _ { i } | Y } ( n | l ) = \\sum _ { \\pmb { x } \\in \\mathcal { X } } P _ { T _ { i } | X } ( n | \\pmb { x } ) P _ { X | Y } ( \\pmb { x } | l ) \\approx \\frac { 1 } { N ( l ) } \\sum _ { \\pmb { x } ^ { j } \\in \\mathcal { D } , y ^ { j } = l } P _ { T _ { i } | X } ( n | \\pmb { x } ^ { j } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where 27 $P _ { X | Y } ( \\pmb { x } ^ { j } | l )$ is estimated by the empirical distribution $1 / N ( l )$ and $N ( l )$ denotes the number of 28 samples with the label $l$ in $\\mathcal { D }$ . Finally, we can derive $I ( Y ; T _ { i } )$ by Equation 18, 19, and 20. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Synthetic dataset. The dataset consists of 512 gray-scale $3 2 \\times 3 2$ images, which are evenly generated by rotating a deterministic image $\\hat { \\pmb x }$ in four different orientations and adding Gaussian noise with expectation $\\boldsymbol { \\mu } = \\mathbb { E } ( \\hat { \\boldsymbol { x } } )$ and variance $\\sigma ^ { 2 } = 1$ , namely $\\pmb { x } = r ( \\hat { \\pmb { x } } ) + \\mathcal { N } ( \\mu , \\sigma ^ { 2 } )$ , where $r ( \\cdot )$ denotes the rotation method shown in Figure 3. The reason for adding Gaussian noise is to avoid DNNs directly memorizing the deterministic image. In addition, the binary labels [1,0] and [0,1] evenly divide the synthetic dataset into two classes. As a result, the synthetic dataset has (approximately) 2 bits information and the labels have 1 bit information. Compared to popular benchmark dataset with unknown features and entropy, e.g., MNIST [17] and Fashion-MNIST [36], the features and the entropy of the synthetic dataset are clear and known, which enables us to examine the probabilistic representation and the mutual information estimator. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Neural Networks. We train three MLPs, namely MLP1, MLP2 and MLP3, on the synthetic dataset by a variant of Stochastic Gradient Descent (SGD) method, namely Adam [13], over 1000 epochs with the learning rate $\\alpha = 0 . 0 3$ . Table 1 summarizes the architecture of the three MLPs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Validating the probability space and the mutual information estimator ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We demonstrate the sample space $\\Omega _ { T }$ by visualizing the weights4 of the eight neurons in $\\mathbf { t } _ { 1 }$ , i.e., $\\omega _ { n } ^ { ( 1 ) } = \\{ \\omega _ { m n } ^ { ( 1 ) } \\} _ { m = 1 } ^ { 1 0 2 4 }$ , in 5 different epochs (i.e., 0,1,4,128,1000) in Figure 4 (Left). As training continues, we observe that $\\omega _ { n } ^ { ( 1 ) }$ quickly learns all the spatial features of the synthetic dataset. For instance, ${ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }$ has low magnitude at top-left positions and high magnitude at bottom-right positions, 2which correctly characterizes the spatial feature of Image0. Similarly, ${ \\boldsymbol \\omega } _ { 3 } ^ { ( 1 ) }$ , ${ \\omega } _ { 4 } ^ { ( 1 ) }$ , and ${ \\omega } _ { 5 } ^ { ( 1 ) }$ correctly characterize the spatial feature of Image1, Image2, and Image3 in Figure 3, respectively. ",
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+        "image_caption": [
+            "Figure 4: (Left) The eight features $\\{ \\omega _ { n } ^ { ( 1 ) } \\} _ { n = 1 } ^ { 8 }$ learned by the weights of the eight neurons in 5 different epochs (i.e., 0,1,4,128,1000), where $\\omega _ { n } ^ { ( 1 ) } = \\{ \\omega _ { m n } ^ { ( 1 ) } \\} _ { m = 1 } ^ { 1 0 2 4 }$ are reshaped into $3 2 \\times 3 2$ to show the spatial structure. (Right) in the MLP1, MLP2, and MLP3 during 1000 epochs. "
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+            "Table 2: The Gibbs probability $P _ { F _ { 1 } | X } ( \\omega _ { n } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } )$ in MLP1 and MLP2 in the 1000 epoch "
+        ],
+        "table_footnote": [
+            "$f _ { 1 n } ^ { \\mathrm { T a n h } } ( \\pmb { x } ) = \\sigma ^ { \\mathrm { T a n h } } ( \\langle \\pmb { \\omega } _ { n } ^ { ( 1 ) } , \\pmb { x } \\rangle )$ and $f _ { 1 n } ^ { \\mathrm { R e L U } } ( \\pmb { x } ) = \\sigma ^ { \\mathrm { R e L U } } ( \\langle \\pmb { \\omega } _ { n } ^ { ( 1 ) } , \\pmb { x } \\rangle )$ are the activations given the same $\\overline { { \\langle { \\omega _ { n } ^ { ( 1 ) } , \\pmb x } \\rangle } }$ "
+        ],
+        "table_body": "<table><tr><td></td><td></td><td>1</td><td></td><td>1</td><td>w</td><td>(1</td><td>（1）</td><td></td></tr><tr><td>（，）</td><td>-63.6</td><td>208.8</td><td>-181.6</td><td>45.1</td><td>-55.6</td><td>157.5</td><td>-210.0</td><td>-30.1</td></tr><tr><td>fReLU (x) in</td><td>0.0</td><td>208.8</td><td>0.0</td><td>45.1</td><td>0.0</td><td>157.5</td><td>0.0</td><td>0.0</td></tr><tr><td>expReLu(）</td><td>1.0</td><td>4.79e+90</td><td>1.0</td><td>3.86e+19</td><td>1.0</td><td>2.51e+68</td><td>1.0</td><td>1.0</td></tr><tr><td>PU</td><td>0.0</td><td>1.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>fTanh(（x） Jin</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>-1.0</td></tr><tr><td>exp(）</td><td>0.36</td><td>2.71</td><td>0.36</td><td>2.71</td><td>0.36</td><td>2.71</td><td>0.36</td><td>0.36</td></tr><tr><td>P</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.037</td></tr></table>",
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+        "type": "text",
+        "text": "249 We demonstrate that feature with the larg $P ( T _ { 1 } | X )$ correctly mecorrelation to $_ { \\textbf { \\em x } }$ res the probability of in Table 2. For insta $\\{ \\omega _ { n } ^ { ( 1 ) } \\} _ { n = 1 } ^ { 8 }$ ${ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }$ being recognized theorrectly characterizes   \n251 the feature of Image0 and has the largest cross-correlation $\\langle \\omega _ { 2 } ^ { ( 1 ) } , \\pmb { x } \\rangle = 1 9 0 . 8$ , thus it has the largest   \n252 probability $P _ { T _ { 1 } | X } ^ { \\mathrm { R e L U } } ( \\omega _ { 2 } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } ) = 1 . 0$ being recognized as the feature with largest cross-correlation   \n253 to Image0. In contrast, since ${ \\omega } _ { 7 } ^ { ( 1 ) }$ incorrectly characterizes the feature of Image0 and has the lowest   \n254 cross-correlation $\\langle \\omega _ { 7 } ^ { ( 1 ) } , \\pmb { x } \\rangle = - 2 1 0 . 0$ , so it has the lowest probability $P _ { T _ { 1 } | X } ^ { \\mathrm { R e L U } } ( \\omega _ { 7 } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } ) = 0 . 0$   \n255 being recognized as the feature with largest cross-correlation to Image0. ",
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+        "type": "text",
+        "text": "6 We observe that an activation function (abbr. ACT) plays an important role in the distribution. Specifically, ReLU, a non-saturating (unbounded) ACT [9], preserves the positive cross-correlations while resets all the negative ones as zero. $P _ { T _ { 1 } | X } ^ { \\mathrm { R e L U } } ( \\omega _ { 2 } ^ { ( 1 ) } | \\mathrm { I m a g e 0 } ) = 1 . 0$ shows that ReLU derives the correct probability of ${ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }$ being recognized as the feature with largest cross-correlation. In contrast, though ${ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }$ has stronger cross-correlation to Image0 than ${ \\omega } _ { 4 } ^ { ( 1 ) }$ , i.e., $\\langle \\omega _ { 2 } ^ { ( 1 ) } , \\pmb { x } \\rangle > \\langle \\pmb { \\omega } _ { 4 } ^ { ( 1 ) } , \\pmb { x } \\rangle$ , Tanh, a saturating (bounded) ACT, derives $f _ { 1 2 } ^ { \\mathrm { T a n h } } ( { \\pmb x } ) = f _ { 1 4 } ^ { \\mathrm { T a n h } } ( { \\pmb x } ) = 1 . 0$ , and makes ${ \\omega } _ { 4 } ^ { ( 1 ) }$ to incorrectly have the same probability 0.272 to ${ \\boldsymbol \\omega } _ { 2 } ^ { ( 1 ) }$ being recognized as the feature with the largest cross-correlation 3 to Image0, i.e., Tanh hinders $\\mathbf { t } _ { 1 }$ from correctly recognizing the features of input. The simulations for validating the probability space based on other synthetic images are presented in Appendix G. ",
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+        "type": "text",
+        "text": "265 To validate the mutual information estimator, we follow recent works [30, 28] to train the three   \n266 MLPs with 50 different random initialization and study the average mutual information. Figure 4   \n267 (Right) shows that $I ( X ; T _ { 1 } )$ quickly increases to 1.81 and keeps stable in the MLP1, i.e., $\\mathbf { t } _ { 1 }$ learns   \n268 most information of the dataset as $H ( X ) = 2 . 0$ . Notably, the result is consistent with the variation   \n269 of the weights in Figure 4 (Left), which shows that the weights correctly characterize the features   \n270 of the dataset and keeps stable after the fourth epoch. As a comparison, we observe that $I ( X ; T _ { 1 } )$   \n271 keeps stable at 0.44 in the MLP2, which confirms the statement that Tanh hinders $\\mathbf { \\delta t } _ { 1 }$ from correctly   \n272 recognizing the features of input. In addition, Figure 4 (Right) shows that $I ( X ; T _ { 1 } ) \\approx 0 . 7 9$ in MLP3   \n273 is smaller than $I ( X ; T _ { 1 } ) \\approx \\bar { 1 } . 8 1$ in MLP1, which is consistent with Definition 1, i.e., a layer with   \n274 fewer neurons would represent fewer possible features, thus it contains less information.   \n275 In summary, we demonstrate the probability space $\\left( \\Omega _ { T } , \\mathcal { F } , P _ { T } \\right)$ and show that if an ACT cannot   \n276 preserve the cross-correlation between weights(features) and input, it would distort the distribution   \n277 of a layer, thereby affecting the mutual information between the layer and data/labels. In addition,   \n278 we show that the proposed mutual information estimator outperforms the existing non-parametric   \n279 models, e.g., empirical distribution [30] and KDE [28], based on the synthetic dataset. Especially,   \n280 activations do not satisfy the i.i.d. prerequisite of non-parametric models is an important reason for   \n281 non-parametric models deriving inaccurate mutual information in DNNs. Due to limited space, the   \n282 experimental comparison and study of non-parametric models are presented in Appendix G. ",
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+    {
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+        "img_path": "images/3c88b2ed333f4d3d7b5765f256c0655f9d9c4b4ba232bb3dfdb758309fa8fad6.jpg",
+        "image_caption": [
+            "Figure 5: All the $\\mathbf { X }$ -axis index training epochs. In each column, the first three figures show $I ( X ; T _ { i } )$ , $I ( \\bar { X } ; T _ { i } )$ , and $I ( Y ; T _ { i } )$ respectively. The forth figure shows $I ( X ; T _ { \\mathrm { M L P } } )$ and $I ( Y ; T _ { \\mathrm { M L P } } )$ in a MLP. The pink line denotes $H ( Y ) = 1 . 0$ and the orange line denotes $H ( X ) = 2 . 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "283 4.3 Validating the information theoretic explanations for DNNs ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            435,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we observe $I ( X ; T _ { i } ) \\leq I ( X ; { \\hat { Y } } )$ in MLP2 and MLP3, which confirms that the Markov chain proposed by previous works, Equation (2), cannot fully explain the information flow in MLPs, if taking into account the back-propagation training. As a comparison, the second and third row show $I ( \\bar { X } ; T _ { 1 } ) \\ge I ( \\bar { X } ; T _ { 2 } ) \\ge I ( \\bar { X } ; \\hat { Y } )$ and $I ( Y ; T _ { 1 } ) \\leq I ( Y ; T _ { 2 } ) \\geq I ( Y ; \\hat { Y } )$ in all the three MLPs, which validates that Corollary 1, i.e., Equation (14) characterizes the information flow in MLPs. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 demonstrates that different hidden layers achieve different IB trade-offs depending on the architecture and the position of the layers in MLPs. In terms of architecture, $I ( Y ; T _ { 1 } ) > 0 . 8$ and $I ( \\bar { X } ; T _ { 1 } ) > 0 . 7 5$ in MLP1 indicate that $\\mathbf { \\delta t } _ { 1 }$ , with ReLU, achieves a good prediction without much compression, whereas $I ( Y ; T _ { 1 } ) < 0 . 5$ and $I ( \\bar { X } ; T _ { 1 } ) < 0 . 1$ in MLP2 show that $\\mathbf { \\delta t } _ { 1 }$ , with Tanh, achieves a different IB trade-off. In addition, $I ( Y ; T _ { 1 } ) \\approx 0 . 4 5$ and $I ( \\bar { X } ; T _ { 1 } ) \\approx 0 . 2 5$ in MLP3 show the effect of neuron numbers on the IB trade-off. In terms of position, $I ( Y ; { \\hat { Y } } ) = 1$ and $I ( { \\bar { X } } ; { \\hat { Y } } ) = 0$ in MLP1 means that $\\hat { \\pmb { y } }$ has a different IB trade-off to $\\mathbf { \\delta t } _ { 1 }$ in MLP1. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate that a MLP satisfies the IB principle no matter what the architecture of the MLP is. Figure 5 visualizes $I ( X ; T _ { \\mathrm { M L P } } )$ and $I ( Y ; { \\bar { T } } _ { \\mathrm { M L P } } )$ based on Corollary 2. It shows that all of three MLPs satisfy the IB principle, namely $I ( X ; T _ { \\mathrm { M L P } } ) < H ( X ) = 2$ and $I ( Y ; T _ { \\mathrm { M L P } } ) = H ( Y ) = 1$ though they have different architectures. Importantly, in contrast to previous work [28] claiming that the compression not exists in DNNs with non-saturating ACT, such as ReLU, Figure 5 clearly shows that the compression exists in all the MLPs, no matter the activation function of MLPs. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We further demonstrate the information theoretic explanations for DNNs on the benchmark MNIST and Fashion-MNIST datasets. The experiments are presented in Appendix $_ \\mathrm { H }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "04 5 Conclusion and future work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "In this work, we (1) specify the probability space for a hidden layer for (2) accurately estimating the mutual information and (3) clearly explaining how the components of the layer affect the mutual information. We take into account the back-propagation training and derive two novel Markov chains to characterize the information flow in DNNs. Furthermore, we demonstrate that a DNN satisfies the IB principle no matter the architecture of the DNN. In contrast, different hidden layers show different IB trade-offs depending on the architecture and the position of the layers in DNNs. A potential direction is to study the generalization of DNNs based on the probabilistic representation. ",
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+    {
+        "type": "text",
+        "text": "References   \n[1] Rana Ali Amjad and Bernhard Claus Geiger. Learning representations for neural network-based classification using the information bottleneck principle. IEEE transactions on pattern analysis and machine intelligence, 2019.   \n[2] George Casella and Roger L Berger. Statistical inference. Cengage Learning, 2021.   \n[3] Ivan Chelombiev, Conor Houghton, and Cian O’Donnell. Adaptive estimators show information compression in deep neural networks. In International Conference on Learning Representations, 2019.   \n[4] Thomas Cover and Joy Thomas. Elements of Information Theory. Wiley-Interscience, Hoboken, New Jersy, 2006.   \n[5] Balázs Csanád Csáji et al. Approximation with artificial neural networks. Faculty of Sciences, Etvs Lornd University, Hungary, 24(48):7, 2001.   \n[6] Rick Durrett. Probability: theory and examples, volume 49. Cambridge university press, 2019.   \n[7] Marylou Gabrié, Andre Manoel, Clément Luneau, Nicolas Macris, Florent Krzakala, Lenka Zdeborová, et al. Entropy and mutual information in models of deep neural networks. In Advances in Neural Information Processing Systems, pages 1821–1831, 2018.   \n[8] Matt W Gardner and SR Dorling. Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences. Atmospheric environment, 32(14-15):2627–2636, 1998.   \n[9] Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pages 249–256, 2010.   \n[10] Ziv Goldfeld, Ewout Van Den Berg, Kristjan Greenewald, Igor Melnyk, Nam Nguyen, Brian Kingsbury, and Yury Polyanskiy. Estimating information flow in deep neural networks. In Proceedings of the 36th International Conference on Machine Learning, volume 97, pages 2299–2308, 2019.   \n[11] Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio. Deep learning, volume 1. MIT press Cambridge, 2016.   \n[12] Patrick Kidger and Terry Lyons. Universal approximation with deep narrow networks. In Conference on Learning Theory, pages 2306–2327. PMLR, 2020.   \n[13] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \n[14] Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. In International Conference on Machine Learning, pages 1885–1894. PMLR, 2017.   \n[15] Artemy Kolchinsky, Brendan D Tracey, and Steven Van Kuyk. Caveats for information bottleneck in deterministic scenarios. arXiv preprint arXiv:1808.07593, 2018.   \n[16] Alex Krizhevsky. Learning multiple layers of features from tiny images. Technique Report, 2009.   \n[17] Y. LeCun, BE. Boser, and JS. Denker. Handwritten digit recognition with a back-propagation network. In NeurIPS, pages 396–494, 1990.   \n[18] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. nature, 521(7553):436–444, 2015.   \n[19] Yann LeCun, Sumit Chopra, Raia Hadsell, Marc’Aurelio Ranzato, and Fu Jie Huang. A tutorial on energy-based learning. MIT Press, 2006.   \n[20] Jaehoon Lee, Yasaman Bahri, Roman Novak, Samuel S. Schoenholz, Jeffrey Pennington, and Jascha Sohl-Dickstein. Deep neural networks as gaussian processes. In ICLR, 2018.   \n[21] Aravindh Mahendran and Andrea Vedaldi. Understanding deep image representations by inverting them. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5188–5196, 2015.   \n[22] David McAllester and Karl Stratos. Formal limitations on the measurement of mutual information. In International Conference on Artificial Intelligence and Statistics, pages 875–884. PMLR, 2020.   \n[23] Julian D Olden and Donald A Jackson. Illuminating the “black box”: a randomization approach for understanding variable contributions in artificial neural networks. Ecological modelling, 154(1-2):135–150, 2002.   \n[24] Liam Paninski. Estimation of entropy and mutual information. Neural computation, 15(6):1191–1253, 2003.   \n[25] David E. Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams. Learning representations by backpropagating errors. Nature, 323:533–536, October 1986.   \n[26] Ruslan Salakhutdinov and Geoffrey Hinton. Deep boltzmann machines. In Artificial intelligence and statistics, pages 448–455. PMLR, 2009.   \n[27] Wojciech Samek, Thomas Wiegand, and Klaus-Robert Müller. Explainable artificial intelligence: Understanding, visualizing and interpreting deep learning models. arXiv preprint arXiv:1708.08296, 2017.   \n[28] Andrew Saxe, Yamini Bansal, Joel Dapello, Madhu Advani, Artemy Kolchinsky, Brendan Tracey, and David Cox. On the information bottleneck theory of deep learning. In International Conference on Representation Learning, 2018.   \n[29] Ohad Shamir, Sivan Sabato, and Naftali Tishby. Learning and generalization with the information bottleneck. Theoretical Computer Science, 411(29-30):2696–2711, 2010.   \n[30] Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via information. arXiv preprint arXiv:1703.00810, 2017.   \n[31] Noam Slonim. The information bottleneck: Theory and applications. PhD thesis, Citeseer, 2002.   \n[32] Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method. arXiv preprint physics/0004057, 2000.   \n[33] Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In 2015 IEEE Information Theory Workshop (ITW), pages 1–5. IEEE, 2015.   \n[34] Vladimir N Vapnik. An overview of statistical learning theory. IEEE transactions on neural networks, 10(5):988–999, 1999.   \n[35] Larry Wasserman. All of nonparametric statistics. Springer Science & Business Media, 2006.   \n[36] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.   \n[37] Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pages 818–833. Springer, 2014.   \n[38] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] see Section 5   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] see the URL in Appendix G   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] see Section 4.1, Appendix G, and Appendix H   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] see Section 4   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] see Appendix G ",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": ". Table 1 summarizes the architecture of the three MLPs.",
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+                                    "score": 0.93,
+                                    "content": "\\omega _ { n } ^ { ( 1 ) } = \\{ \\omega _ { m n } ^ { ( 1 ) } \\} _ { m = 1 } ^ { 1 0 2 4 }",
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", in 5 different epochs (i.e., 0,1,4,128,1000) in Figure 4 (Left). As training",
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+                        },
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "quickly learns all the spatial features of the synthetic dataset. For",
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+                                    "score": 0.91,
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+                                    "score": 1.0,
+                                    "content": "has low magnitude at top-left positions and high magnitude at bottom-right positions,",
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+                                    "score": 1.0,
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+                                    "score": 0.78,
+                                    "content": "{ \\boldsymbol \\omega } _ { 3 } ^ { ( 1 ) }",
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+                                    "score": 0.71,
+                                    "content": "{ \\omega } _ { 4 } ^ { ( 1 ) }",
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+                                    "score": 0.91,
+                                    "content": "{ \\omega } _ { 5 } ^ { ( 1 ) }",
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+                                {
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+                                    "score": 1.0,
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "characterize the spatial feature of Image1, Image2, and Image3 in Figure 3, respectively.",
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+                                    "content": "divide the synthetic dataset into two classes. As a result, the synthetic dataset has (approximately)",
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+                                    "score": 1.0,
+                                    "content": "with unknown features and entropy, e.g., MNIST [17] and Fashion-MNIST [36], the features and the",
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+                        },
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+                                    "content": "by a variant of Stochastic Gradient Descent (SGD) method, namely Adam [13], over 1000 epochs",
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+                                    "type": "inline_equation"
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+                                    "content": ". Table 1 summarizes the architecture of the three MLPs.",
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+                                    "score": 0.93,
+                                    "content": "f _ { 1 2 } ^ { \\mathrm { T a n h } } ( { \\pmb x } ) = f _ { 1 4 } ^ { \\mathrm { T a n h } } ( { \\pmb x } ) = 1 . 0",
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+                                    "score": 1.0,
+                                    "content": "validating the probability space based on other synthetic images are presented in Appendix G.",
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+                                    "content": "keeps stable at 0.44 in the MLP2, which confirms the statement that Tanh hinders",
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+                                    "score": 1.0,
+                                    "content": "recognizing the features of input. In addition, Figure 4 (Right) shows that",
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+                                    "content": "is smaller than",
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+                                    "content": "I ( X ; T _ { 1 } ) \\approx \\bar { 1 } . 8 1",
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+                                    "score": 1.0,
+                                    "content": "in MLP1, which is consistent with Definition 1, i.e., a layer with",
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+                                    "content": "fewer neurons would represent fewer possible features, thus it contains less information.",
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+                                    "score": 1.0,
+                                    "content": "of a layer, thereby affecting the mutual information between the layer and data/labels. In addition,",
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+                                    "content": "we show that the proposed mutual information estimator outperforms the existing non-parametric",
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+                                    "score": 1.0,
+                                    "content": "models, e.g., empirical distribution [30] and KDE [28], based on the synthetic dataset. Especially,",
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+                                    "content": "activations do not satisfy the i.i.d. prerequisite of non-parametric models is an important reason for",
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+                                            "content": "to show the spatial structure. (Right)",
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+                                            "score": 1.0,
+                                            "content": "in the MLP1, MLP2, and MLP3 during 1000 epochs.",
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+                                            "score": 0.87,
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+                                            "html": "<table><tr><td></td><td></td><td>1</td><td></td><td>1</td><td>w</td><td>(1</td><td>（1）</td><td></td></tr><tr><td>（，）</td><td>-63.6</td><td>208.8</td><td>-181.6</td><td>45.1</td><td>-55.6</td><td>157.5</td><td>-210.0</td><td>-30.1</td></tr><tr><td>fReLU (x) in</td><td>0.0</td><td>208.8</td><td>0.0</td><td>45.1</td><td>0.0</td><td>157.5</td><td>0.0</td><td>0.0</td></tr><tr><td>expReLu(）</td><td>1.0</td><td>4.79e+90</td><td>1.0</td><td>3.86e+19</td><td>1.0</td><td>2.51e+68</td><td>1.0</td><td>1.0</td></tr><tr><td>PU</td><td>0.0</td><td>1.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>fTanh(（x） Jin</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>1.0</td><td>-1.0</td><td>-1.0</td></tr><tr><td>exp(）</td><td>0.36</td><td>2.71</td><td>0.36</td><td>2.71</td><td>0.36</td><td>2.71</td><td>0.36</td><td>0.36</td></tr><tr><td>P</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.272</td><td>0.037</td><td>0.037</td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "04 5 Conclusion and future work",
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+                                    "content": "to characterize the information flow in DNNs. Furthermore, we demonstrate that a DNN satisfies the",
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+                                    "content": "IB trade-offs depending on the architecture and the position of the layers in DNNs. A potential",
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+                                    "type": "text"
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+                        },
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+                                    "score": 1.0,
+                                    "content": "[1] Rana Ali Amjad and Bernhard Claus Geiger. Learning representations for neural network-based classi-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 1
+                        },
+                        {
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+}
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index 0000000000000000000000000000000000000000..47fa28561a88ff2804bf15a7b7ac4917d0cb4be1
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diff --git a/parse/train/AJTAcS7SZzf/AJTAcS7SZzf.md b/parse/train/AJTAcS7SZzf/AJTAcS7SZzf.md
new file mode 100644
index 0000000000000000000000000000000000000000..754dcb20e77172341a9ca0293574f879860ef847
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@@ -0,0 +1,264 @@
+# AUTOSAMPLING: SEARCH FOR EFFECTIVE DATA SAMPLING SCHEDULES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Data sampling acts as a pivotal role in training deep learning models. However, an effective sampling schedule is difficult to learn due to its inherent high-dimension as a hyper-parameter. In this paper, we propose the AutoSampling method to automatically learn sampling schedules for model training, which consists of the multi-exploitation step aiming for optimal local sampling schedules and the exploration step for the ideal sampling distribution. More specifically, we achieve sampling schedule search with shortened exploitation cycle to provide enough supervision. In addition, we periodically estimate the sampling distribution from the learned sampling schedules and perturb it to search in the distribution space. The combination of two searches allows us to learn a robust sampling schedule. We apply our AutoSampling method to a variety of image classification tasks illustrating the effectiveness of the proposed method.
+
+# 1 INTRODUCTION
+
+Data sampling policies can greatly influence the performance of model training in computer vision tasks, and therefore finding robust sampling policies can be important. Handcrafted rules, e.g. data resampling, reweighting, and importance sampling, promote better model performance by adjusting the training data frequency and order (Estabrooks et al., 2004; Weiss et al., 2007; Bengio et al., 2009; Johnson & Guestrin, 2018; Katharopoulos & Fleuret, 2018; Shrivastava et al., 2016; Jesson et al., 2017). Handcrafted rules heavily rely on the assumption over the dataset and cannot adapt well to datasets with their own characteristics. To handle this issue, learning-based methods (Li et al., 2019; Jiang et al., 2017; Fan et al., 2017) were designed to automatically reweight or select training data utilizing meta-learning techniques or a policy network.
+
+However existing learning-based sampling methods still rely on human priors as proxies to optimize sampling policies, which may fail in practice. Such priors often include assumptions on policy network design for data selection (Fan et al., 2017), or dataset conditions like noisiness (Li et al., 2019; Loshchilov & Hutter, 2015) or imbalance (Wang et al., 2019). These approaches take images features, losses, importance or their representations as inputs and use the policy network or other learning approaches with small amount of parameters for estimating the sampling probability. However, for example, images with similar visual features can be redundant in training, but their losses or features fed into the policy network are more likely to be close, causing the same probability to be sampled for redundant samples if we rely on aforementioned priors. Therefore, we propose to directly optimize the sampling schedule itself so that no prior knowledge is required for the dataset. Specifically, the sampling schedule refers to order by which data are selected for the entire training course. In this way, we only rely on data themselves to determine the optimal sampling schedule without any prior.
+
+Directly optimizing a sampling schedule is challenging due to its inherent high dimension. For example, for the ImageNet classification dataset (Deng et al., 2009) with around one million samples, the dimension of parameters would be in the same order. While popular approaches such as deep reinforcement learning (Cubuk et al., 2018; Zhang et al., 2020), Bayesian optimization (Snoek et al., 2015), population-based training (Jaderberg et al., 2017) or simple random search (Bergstra & Bengio, 2012) have already been utilized to tune low-dimensional hyper-parameters like augmentation schedules, their applications in directly finding good sampling schedules remain unexploited. For instance, the dimension of a data augmentation policy is generally only in dozens, and it needs thousands of training runs (Cubuk et al., 2018) to sample enough rewards to find an optimal augmentation policy because high-quality rewards require many epochs of training to obtain. As such, optimizing a sampling schedule may require orders of magnitude more rewards than data augmentation to gather and hence training runs, which result in prohibitively slow convergence.
+
+To overcome the aforementioned challenge, we propose a data sampling policy search framework, named AutoSampling, to sufficiently learn an optimal sampling schedule in a population-based training fashion (Jaderberg et al., 2017). Unlike previous methods, which focus on collecting longterm rewards and updating hyper-parameters or agents offline, our AutoSampling method collects rewards online with a shortened collection cycle but without priors. Specifically, the AutoSampling collects rewards within several training iterations, tens or hundred times shorter than that in existing works (Ho et al., 2019; Cubuk et al., 2018). In this manner, we provide the search process with much more frequent feedback to ensure sufficient optimization of the sampling schedule. Each time when a few training iterations pass, we collect the reward from the previous several iterations, accumulate them and later update the sampling distribution using the rewards. Then we perturb the sampling distribution to search in distribution space, and use it to generate new mini-batches for later iterations, which are recorded into the output sampling schedule. As illustrated in Sec. 4.1, shortened collection cycles with less interference also can better reflect the training value of each data.
+
+Our contributions are as follows:
+
+• To our best knowledge, we are the first to propose to directly learn a robust sampling schedule from the data themselves without any human prior or condition on the dataset. We propose the AutoSampling method to handle the optimization difficulty due to the high dimension of sampling schedules, and efficiently learn a robust sampling schedule through shortened reward collection cycle and online update of the sampling schedule.
+
+Comprehensive experiments on CIFAR-10/100 and ImageNet datasets (Krizhevsky, 2009; Deng et al., 2009) with different networks show that the Autosampling can increase the top-1 accuracy by up to $2 . 8 5 \%$ on CIFAR-10, $2 . 1 9 \%$ on CIFAR-100, and $2 . 8 3 \%$ on ImageNet.
+
+# 2 BACKGROUND
+
+# 2.1 RELATED WORK
+
+Data sampling is of great significance to deep learning, and has been extensively studied. Approaches with human-designed rules take pre-defined heuristic rules to modify the frequency and order by which training data is presented. In particular, one intuitive method is to resample or reweight data according to their frequencies, difficulties or importance in training (Estabrooks et al., 2004; Weiss et al., 2007; Drummond et al., 2003; Bengio et al., 2009; Lin et al., 2017; Shrivastava et al., 2016; Loshchilov & Hutter, 2015; Wang et al., 2019; Johnson & Guestrin, 2018; Katharopoulos & Fleuret, 2018; Byrd & Lipton, 2018; Jesson et al., 2017). These methods have been widely used in imbalanced training or hard mining problems. However, they are often restricted to certain tasks and datasets based on which they are proposed, and their ability to generalize to a broader range of tasks with different data distribution may be limited. In another word, these methods often implicitly assume certain conditions on the dataset, such as cleanness or imbalance. In addition, learning-based methods have been proposed for finding suitable sampling schemes automatically. Methods using meta-learning or reinforcement learning are also utilized to automatically select or reweight data during training (Li et al., 2019; Jiang et al., 2017; Ren et al., 2018; Fan et al., 2017), but they are only tested on small-scale or noisy datasets. Whether or not they can generalize over tasks of other datasets still remain untested. In this work, we directly study the data sampling without any prior, and we also investigate its wide generalization ability across different datasets such as CIFAR-10, CIFAR-100 and ImageNet using many typical networks.
+
+As for hyper-parameter tuning, popular approaches such as deep reinforcement learning (Cubuk et al., 2018; Zhang et al., 2020), Bayesian optimization (Snoek et al., 2015) or simply random search (Bergstra & Bengio, 2012) have already been utilized to tune low-dimensional hyper-parameters and proven to be effective. Nevertheless, they have not been adopted to find good sampling schedule due to its inherent high dimensiona. Some recent works tackle the challenge of optimizing highdimensional hyper-parameter. MacKay et al. (2019) uses structured best-response functions and Jonathan Lorraine (2019) achieve this goal through the combinations of the implicit function theorem and efficient inverse Hessian approximations. However, they have not been tested on the task of optimizing sampling schedules, which is the major focus of our work in this paper.
+
+![](images/660fc98379b8fda0cbceb221c021ef383238ffbbe012ec81eb21c03d04754de5.jpg)  
+Figure 1: Overview of AutoSampling illustrated through one multi-exploitation-and-exploration cycle. a) The multi-exploitation step, illustrated by the left half, is the process of learning optimal sampling schedule locally. The same color of model for each worker indicates that the same model weight is cloned into it. Also for simplicity, in this figure we adopt the exploitation interval of length 1. b) The exploration step, shown by the right half, is to search in the sampling distribution space. Specifically, we estimate the sampling distribution from the schedules collected in the multi-exploitation step and perturb it to generate new sampling schedules for all workers.
+
+# 2.2 POPULATION BASED TRAINING
+
+Hyper-parameter tuning task can be framed as a bi-level optimization problem with the following objective function,
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { h \in \mathcal { H } } \mathcal { L } ( \theta ^ { * } , h ) } \\ { \theta ^ { * } = \mathop { \mathrm { a r g } \operatorname* { m a x } } \mathop { \mathrm { e v a l } } ( \theta , h ) } \\ { \theta \in \Theta } \end{array}
+$$
+
+where $\theta$ represents the model weight and $h = ( h _ { 1 } , h _ { 2 } , \cdot \cdot \cdot , h _ { T } )$ is the hyper-parameter schedule for $T$ training intervals. Population based training (PBT) (Jaderberg et al., 2017) solves the bilevel optimization problem by training a population $\mathcal { P }$ of child models in parallel with different hyper-parameter schedules initialized:
+
+$$
+\mathcal { P } = \{ ( \theta _ { i } , h _ { i } , t ) \} _ { i = 1 } ^ { N _ { p } }
+$$
+
+where $\theta _ { i } , h _ { i }$ respectively represents the child model weight, the corresponding hyper-parameter schedule for the training interval $t$ on worker $i$ , and $N _ { p }$ is the number of workers. PBT proceeds in intervals, which usually consists of several epochs of training. During the interval, the population of models are trained in parallel to finish the lower-level optimization of weights $\theta _ { i }$ .
+
+Between intervals, an exploit-and-explore procedure is adopted to conduct the upper-level optimization of the hyper-parameter schedule. In particular for interval $t$ , to exploit we evaluate child models on a held-out validation dataset:
+
+$$
+h _ { t } ^ { * } , \theta _ { t } ^ { * } = \operatorname * { a r g m a x } _ { p _ { i } = ( \theta _ { i } , h _ { i } , t ) \in \mathcal { P } } \mathrm { e v a l } ( \theta _ { i } , h _ { i } )
+$$
+
+$$
+\theta ^ { * }  \theta _ { i } , i = 1 , \cdots , N _ { p }
+$$
+
+We record the best performing hyper-parameter setting $h _ { t } ^ { * }$ and broadcast the top-performing model $\theta _ { t } ^ { * }$ to all workers. To explore, we initialize new hyper-parameter schedules for interval $t + 1$ with different random seeds on all workers, which can be viewed as a search in the hyper-parameter space. The next exploit-and-explore cycle will then be continued. In the end, the top-performing hyper-parameter schedule $h ^ { * } = ( h _ { 1 } ^ { * } , h _ { 2 } ^ { * } , \cdot \cdot \cdot , h _ { T } ^ { * } )$ can be obtained.
+
+PBT is applied to tune low-dimenisal hyper-parameters such as data augmentation schedules (Ho et al., 2019; Jaderberg et al., 2017). However, it cannot be directly used for finding sampling strategies due to the high dimension. Unlike PBT, our AutoSampling adopts a multi-exploitation-andexploration structure, leading to much shorter reward collection cycles that contribute to much more and effective rewards for sufficient optimization within a practical computational budget.
+
+# 3 AUTOSAMPLING WITH SEARCHING
+
+The overview of our AutoSampling is illustrated in Fig.1. AutoSampling alternately runs multiexploitation step and exploration step. In the exploration step, we 1) update the sampling distribution using the rewards collected from the multi-exploitation step (the sampling distribution is uniform distribution initially); 2) perturb the updated sampling distribution for child models so that different child models have different sampling distributions; 3) use the corresponding perturbed sampling distribution for each child model to sample mini-batches of training data. In the multi-exploitation step, we 1) train multiple child models using the mini-batches sampled from the exploration step; 2) collect short-term rewards from the child models. AutoSampling finishes with a recorded topperforming sampling schedule, which can be transferred to other models.
+
+Algorithm 1: The Multi-Exploitation Step   
+
+<table><tr><td>number of workers Np, number of i=1 exploitation intervals T,exploitation interval length Ns Initialize H*←(</td></tr><tr><td>for t=1toTdo</td></tr><tr><td>for j = 1 to Ns do for(0i,ht,i,t) ∈Pdo</td></tr><tr><td>0i←L(θi,ht,i) &gt; update the weight of child model i end for</td></tr><tr><td>h*,0* = arg maxp eval(0i,hi)</td></tr><tr><td>H*←H*+h D update the sampling for child model i for i = 1 to Np do</td></tr><tr><td>θ← &gt; clone the optimal weight</td></tr><tr><td>end for end for</td></tr></table>
+
+# 3.1 MULTI-EXPLOITATION BY SEARCHING IN THE DATA SPACE
+
+In the multi-exploitation step, we aim to search locally in the data space by collecting short-term rewards and sub-schedules. Specifically, we wish to learn a sampling schedule for $T$ exploitation intervals. In each interval, there are a population $\mathcal { P }$ of $N _ { p }$ child models. Denote $\mathbf { h } _ { t , i }$ as the training data sub-schedule in the $t ^ { t h }$ interval for the $i ^ { t h }$ child model. When all of the $T$ exploitation intervals for the $i ^ { t h }$ child model are considered, we have $\mathbf { H } _ { i } = \{ \mathbf { h } _ { t , i } | t = 1 , \dots , T \} = \{ x _ { 1 } , \cdot \cdot \cdot , x _ { N } \}$ , where $N$ is the number of training data for the multi-exploitation step. Each interval consists of $N _ { s }$ training iterations that is also equivalent to $N _ { s }$ training mini-batches, where $N _ { s }$ is the length of the interval. AutoSampling is expected to produce a sequence of training samples, denoted by $\mathbf { H } ^ { * }$ , so that a given model is optimally trained. The population $\left\{ \mathbf { H } _ { i } \right\}$ forms the local search space, from which we aim to search for an optimal sampling schedule $\mathbf { H } ^ { * }$ .
+
+Given the population $\mathcal { P }$ , we train them in parallel on $N _ { p }$ workers. Once an interval of data $\mathbf { h } _ { t , i }$ containing $N _ { s }$ training batches have been used for training, we evaluate all child models and use the top evaluation performance as the reward. According to the reward, we record the top-performing weight and sub-schedule for the current interval $t$ , in particular,
+
+$$
+h _ { t } ^ { * } , \theta _ { t } ^ { * } = \operatorname * { a r g m a x } _ { p _ { i } = ( \theta _ { i } , h _ { i } , t ) \in \mathcal { P } } \operatorname { e v a l } ( \theta _ { i } , h _ { t , i } )
+$$
+
+On the other hand, we update all child model weights of $\mathcal { P }$ by cloning into them with the topperforming weight $\theta _ { t } ^ { * }$ so we can continue searching based on the more promising child. We will continue the exploit steps through the whole training process, and output the recorded optimal sampling schedule $\mathbf { H } ^ { * } = \{ h _ { 1 } ^ { * } , h _ { 2 } ^ { * } , \cdot \cdot \cdot , h _ { T } ^ { * } \}$ . By using exploitation interval of mini-batches rather than epochs or even entire training runs adopted by earlier methods, AutoSampling may yield a better and more robust sampling schedule. It should be pointed out that even though in AutoSampling rewards are collected within a much shorter interval, they remain effective. As we directly optimize the sampling schedule, we are concerned with only the data themselves. The short-term rewards reflect the training value of data from the exploitation interval they are collected. But for global hyperparameters such as augmentation schedules, short-term rewards may lead to inferior performance as these hyper-parameters are concerned with the overall training outcome. We describe the multiexploitation with details in Alg.1.
+
+Algorithm 2: Search based AutoSampling   
+
+<table><tr><td>Input: Training dataset D, population size Np Initialize H* ←O,P(D) ←uniform(D) and initialize child models 01,. ,0Np while not end of training do for i= 1 to Np do</td></tr><tr><td>Sample hi from Mixture(log(P(D) + β),Nu × uniform(D)) end for</td></tr><tr><td>InitializeP={(0,hi,t）}1</td></tr><tr><td>H*,P← Alg.1 Estimate P(D) according to Equation (5)</td></tr><tr><td>Update P(D) according to Equation (6)</td></tr><tr><td>H*←H*+H* end while</td></tr><tr><td>Return H*,P(D)</td></tr><tr><td></td></tr></table>
+
+# 3.2 EXPLORATION BY SEARCHING IN SAMPLING DISTRIBUTION SPACE
+
+In the exploration, we search in sampling distribution space by updating and perturbing the sampling distribution. We first estimate the underlying sampling distribution $P ( D )$ from the top sampling schedule $h ^ { * }$ produced in the multi-exploitation, that is, for $x \in D$ ,
+
+$$
+P ( x ) = \frac { c o u n t ( x \in \mathbf { H } ^ { * } ) } { \sum _ { x \in D } c o u n t ( x \in \mathbf { H } ^ { * } ) }
+$$
+
+where $c o u n t ( x \in \mathbf { H } ^ { * } )$ denotes the number of $x$ ’s appearances in $\mathbf { H } ^ { * }$ . We further perturb the $P ( D )$ and generate the sampling schedules on each worker for the later multi-exploitation. We introduce perturbations into the generated schedules by simply sampling from the multinomial distribution $P ( D )$ using different random seeds. However, in our experiments, we observe that the distribution produced by $P ( D )$ tends to be extremely skewed and a majority of the data actually have zero frequencies. Such skewness causes highly imbalanced training mini-batches, and therefore destabilizes subsequent model training.
+
+Distribution Smoothing To tackle the above issue, we first smooth $P ( D )$ through the logarithmic function, and then apply a probability mixture with uniform distributions. In particular for the dataset $D$ ,
+
+$$
+P ^ { \prime } ( D ) = M i x t u r e ( \log ( P ( D ) + \beta ) , N _ { u } \times \mathrm { u n i f o r m } ( D ) )
+$$
+
+where $\beta \geq 1$ is the smoothing factor and $N _ { u } \times \mathrm { u n i f o r m } ( D )$ denotes $N _ { u }$ uniform multinomial distributions on the dataset $D$ . The smoothing through the log function can greatly reduce the skewness, however, $\log ( P ( D ) + \beta )$ may still contain zero probabilities for some training data, resulting in unstable training. Therefore, we further smooth it through a probability mixture with $N _ { u }$ uniform distribution uniform $( D )$ to ensure presence of all data. This is equivalent to combining $N _ { u }$ epochs of training data to the training batches sampled from $P ( D )$ , and shuffling the union. Once we have new diverse sampling schedules for the population, we proceed to the next multi-exploitation step.
+
+We continue this alternation between multi-exploitation and exploration steps until the end of training. Note that to generate sampling schedule for the first multi-exploitation run, we initialize $P ( D )$ to be an uniform multinomial distribution. In the end, we output a sequence of optimal sampling schedules $H ^ { * } = ( \mathbf { H } _ { 1 } ^ { * } , \cdots , \mathbf { H } _ { n } ^ { * } )$ for $n$ alternations. The entire process is illustrated in details in Alg.2.
+
+# 4 EXPERIMENTS
+
+In this section, we present comprehensive experiments on various datasets to illustrate the performance of AutoSampling, and also demonstrate the process of progressively learning better sampling distribution.
+
+# 4.1 ABLATION STUDY
+
+For this part, we gradually build up and test components of AutoSampling on CIFAR-100, and then examine their performances on CIFAR-10 and ImageNet datasets. The training implementation details and computational complexity can be found in Appendix A.1.
+
+Table 1: Performance on CIFAR-100 using different configurations of AutoSampling and baselines. Worker is the number of workers used and Interval is the exploitation interval in terms of batches.   
+
+<table><tr><td>NETWORK</td><td>WORKER</td><td>INTERVAL</td><td>EXPLORATION TYPE</td><td>TOP1(%)</td></tr><tr><td>RESNET18 (ZHANG ET AL., 2019)</td><td>-</td><td>-</td><td></td><td>78.34±0.05</td></tr><tr><td>RESNET18</td><td>1</td><td>-</td><td>UNIFORM</td><td>78.46±0.035</td></tr><tr><td>RESNET18</td><td>20</td><td>80 BATCHES</td><td>RANDOM</td><td>78.76±0.003</td></tr><tr><td>RESNET18</td><td>20</td><td>20 BATCHES</td><td>RANDOM</td><td>78.99±0.003</td></tr><tr><td>RESNET18</td><td>80</td><td>20 BATCHES</td><td>RANDOM</td><td>79.09±0.017</td></tr><tr><td>RESNET18</td><td>20</td><td>20 BATCHES</td><td>MIXTURE</td><td>79.44±0.020</td></tr><tr><td>RESNET50 (JINET AL.,2019)</td><td>-</td><td>-</td><td>-</td><td>79.34</td></tr><tr><td>RESNET50</td><td>1</td><td></td><td>UNIFORM</td><td>79.70±0.023</td></tr><tr><td>RESNET50</td><td>20</td><td>80 BATCHES</td><td>RANDOM</td><td>80.55±0.129</td></tr><tr><td>RESNET50</td><td>20</td><td>20 BATCHES</td><td>RANDOM</td><td>81.05±0.064</td></tr><tr><td>RESNET50</td><td>80</td><td>20 BATCHES</td><td>RANDOM</td><td>81.19±0.072</td></tr><tr><td>RESNET50</td><td>20</td><td>20 BATCHES</td><td>MIXTURE</td><td>81.53±0.088</td></tr><tr><td>DENSENET121</td><td>1</td><td>-</td><td>UNIFORM</td><td>80.13±0.028</td></tr><tr><td>DENSENET121</td><td>20</td><td>80 BATCHES</td><td>RANDOM</td><td>80.62±0.694</td></tr><tr><td>DENSENET121</td><td>20</td><td>20 BATCHES</td><td>RANDOM</td><td>81.11±0.127</td></tr><tr><td>DENSENET121</td><td>80</td><td>20 BATCHES</td><td>RANDOM</td><td>81.08±0.021</td></tr><tr><td>DENSENET121</td><td>20</td><td>20 BATCHES</td><td>MIXTURE</td><td>80.97±0.006</td></tr></table>
+
+Table 2: Experiments on CIFAR-10.   
+
+<table><tr><td>NETWORK</td><td>EXPLORATION TYPE</td><td>ToP1(%)</td></tr><tr><td>RESNET18</td><td>UNIFORM</td><td>93.01±0.009</td></tr><tr><td>RESNET18</td><td>RANDOM</td><td>95.86±0.003</td></tr><tr><td>RESNET18</td><td>MIXTURE</td><td>95.80±0.018</td></tr><tr><td>RESNET50</td><td>UNIFORM</td><td>93.60±0.004</td></tr><tr><td>RESNET50</td><td>RANDOM</td><td>96.10±0.002</td></tr><tr><td>RESNET50</td><td>MIXTURE</td><td>96.09±0.070</td></tr></table>
+
+Table 3: Experiments on ImageNet.   
+
+<table><tr><td>NETWORK</td><td>EXPLORATION TYPE</td><td>TOP1(%)</td></tr><tr><td>RESNET18</td><td>UNIFORM</td><td>70.38</td></tr><tr><td>RESNET18</td><td>RANDOM</td><td>72.07</td></tr><tr><td>RESNET18</td><td>MIXTURE</td><td>72.91</td></tr><tr><td>RESNET34</td><td>UNIFORM</td><td>74.09</td></tr><tr><td>RESNET34</td><td>RANDOM</td><td>76.11</td></tr><tr><td>RESNET34</td><td>MIXTURE</td><td>76.92</td></tr></table>
+
+Adding Workers To look into the influence of the worker numbers, we conduct experiments using worker numbers of 1, 20, 80 respectively with the same setting ( $N _ { s } = 2 0$ with random exploration). With the worker number of 1, the experiment is simply the normal model training using stochastic gradient descent. To show the competitiveness of our baselines, we also include state-of-the-art results on CIFAR-100 with ResNet-18 and ResNet-50 (Zhang et al., 2019; Jin et al., 2019). We notice significant performance gain using the worker number of 20 for ResNet-18, ResNet-50 and DenseNet-121 (He et al., 2015; Huang et al., 2017), as illustrated in Table 1. However, we note that increasing worker number from 20 to 80 only brings marginal performance gains across various model structures, as shown in Table 1. Therefore, we set the worker number to be 20 for the rest of the experiments.
+
+Shortening Exploitation Intervals To study the effects of the shortened exploitation interval, we run experiments using different exploitation intervals of 20 and 80 batches(iterations) respectively. As shown in Table 1, models with the shorter exploitation interval of 20 batches(iterations) perform better than the one with the longer exploitation interval across all three network structures, conforming to our assumptions that the reward collected reflects value of each data used in the exploitation interval. This result adheres to our intuition that shorter exploitation interval can encourage the sampler to accumulate more rewards to learn better sampling schedules. For the rest of this section we keep the exploitation interval of 20.
+
+Adding Exploration Type We further add mixture as the exploration type to see the effects of learning the underlying sampling distribution, and completing the proposed method. As shown in Table 1, with ResNet-18 and ResNet-50 we push performance higher with the mixture exploration, and outperform the baseline method by about 1 and 1.8 percentage on CIFAR-100 respectively. However, we found that it is not true in the case of DenseNet-121 and this case may be attributed to the bigger capacity of DenseNet-121.
+
+Generalization Over Datasets In addition, we experiment on other datasets. We report the results on CIFAR10 in Table 2 and the results of ResNet-18, ResNet-34 on ImageNet in Table 3. For CIFAR-10, we notice that the mixture and random exploration methods are comparable while both outperforming the uniform baseline, and we believe it is due to the simplicity of the dataset. In the more challenging
+
+![](images/fe3c88399ad6162c77b6025bd2e83af8d156bc7688326885cf936bf3a3ec6d4f.jpg)  
+Figure 2: The comparison between histograms estimated from the sampling schedules of Epoch 80, 160 and 240 from CIFAR-100 with ResNet-18. We divide the 50000 training images into 500 segments of 100 images, and calculate the histograms of total data counts of all segments. We reorder the $x$ -axis based on the ranking of data counts for epoch 240 for easier comparison.
+
+Table 4: Static vs dynamic sampling schedule on CIFAR-100 $( \% )$   
+
+<table><tr><td rowspan="2">NETWORK</td><td colspan="3">SAMPLING TYPE</td></tr><tr><td>UNIFORM</td><td>STATIC</td><td>DYNAMIC</td></tr><tr><td>RESNET18</td><td>78.46±0.035</td><td>78.80±0.007</td><td>79.44±0.020</td></tr><tr><td>RESNET50</td><td>79.70±0.023</td><td>80.21±0.014</td><td>81.53±0.088</td></tr></table>
+
+ImageNet, the mixture exploration outperforms the random exploration by a clear margin. We also compare our AutoSampling with some recent non-uniform sampling methods on CIFAR-100, which can be found in Appendix A.2.
+
+# 4.2 STATIC VS DYNAMIC SCHEDULES
+
+We aim to see if the final sampling distribution estimated by our AutoSampling is sufficient to produce robust sampling schedules. In another word, we wish to know training with the AutoSampling is either a process of learning a robust sampling distribution, or a process of dynamically adjusting the sampling schedule for optimal training. To this end, we conduct training using different sampling schedules. First, we calculate the sampling distribution estimated throughout the learning steps of AutoSampling, and use it to generate the sampling schedule of a full training process, which we denote as STATIC. Moreover, we denote the sampling schedule learned using AutoSampling as DYNAMIC, since AutoSampling dynamically adjust the sampling schedule alongside the training process. Finally, we denote the baseline method as UNIFORM, which uses the sampling schedule generated from uniform distribution.
+
+We report results on CIFAR-100 with ResNet-18 and ResNet-50 in Table 4. Model trained with STATIC sampling schedules exceeds the baseline UNIFORM significantly, indicating the superiority of the learned sampling distribution over the uniform distribution. It shows the ability of AutoSampling to learn good sampling distribution. Nonetheless, note that models trained with DYNAMIC sampling schedules outperform models trained with STATIC, by a margin bigger than the one between STATIC and UNIFORM. This result shows the fact that despite the AutoSampling’s capability of learning good sampling distribution, its flexibility during training matters even more. Moreover, this phenomenon also indicates that models at different stages of learning process may require different sampling distributions to achieve optimal training. One single sampling distribution, even gradually estimated using AutoSampling, seems incapable of covering the needs from different learning stages. We plot the histograms of data counts in training estimated from schedules of different learning stages with ResNet-18 on CIFAR-100 in Fig.2, showing the great differences between optimized sampling distributions from different epochs.
+
+# 4.3 ANALYZING SAMPLING SCHEDULES LEARNED BY AUTOSAMPLING
+
+To further investigate the sampling schedule learned by AutoSampling, we review the images at the tail and head part of the sampling spectrum. In particular, given a sampling schedule learned we rank all images based on their appearances in training. Training images at the top and bottom of the order are extracted, corresponding to high and low probabilities of being sampled respectively. In Fig.3, we show 4 classes of exemplary images. The images of low probability tend to have clearer imagery features enabling easy recognition, while the images of high probability tend to be more obscure, indicating that the sampling schedule may show hard samples mining effects. However, as shown in A.3 and Fig. 4, the loss values and probabilities of being sampled seem to be not highly correlated, which indicates more potential of AutoSampling beyond visually hard example mining. In addition, we notice the images of low probability also contain low quality images. For instance, in Fig.3 the leftmost image of CAMAL class contains only legs. This shows that AutoSampling may potentially rule out problematic training data for better training.
+
+![](images/809ba11d034a648f9392e8b6bf5244948a22bc2430e9efcce77a6fcafe43227f.jpg)  
+Figure 3: Example images on the head and tail of the sampling spectrum. The images on the left are the ones with low sampling probability, while the images on the right more likely to be sampled. We obtain these images using AutoSampling with the ResNet-18 model on CIFAR-100.
+
+Table 5: Transfer of sampling distributions learned by three model structures to ResNet-50 on CIFAR-100 $( \% )$ . UNIFORM denotes the baseline result using uniform sampling distribution.   
+
+<table><tr><td rowspan="2">NETWORK</td><td colspan="4">SAMPLING SCHEDULE SOURCE</td></tr><tr><td>UNIFORM</td><td>RESNET18</td><td>RESNET50</td><td>DENSENET121</td></tr><tr><td>RESNET50</td><td>79.70±0.023</td><td>80.27±0.014</td><td>80.21±0.014</td><td>80.47±0.194</td></tr></table>
+
+Furthermore, we examine the transfer ability of sampling distributions learned by AutoSampling to other network structures. Specifically, we run training on ResNet-50 (He et al., 2015) using STATIC sampling schedule generated by three distributions learned by AutoSampling on 3 different models. As shown in Table 5, using sampling schedules learned by AutoSampling from other models, we demonstrate similar improvements over the UNIFORM baseline. This result, combined with the above observations on images of different sampling probability, indicates that there may exist a common optimal sampling schedule determined by the intrinsic property of the data rather than the model being optimized. Our AutoSampling is an effort to gradually converge to such an optimal schedule.
+
+# 4.4 DISCUSSIONS
+
+The experimental results and observations from Section 4.2 and 4.3 shed light on the possible existence of an optimal sampling schedule, which relies only on the intrinsic property of the data and the learning stage of the model, regardless of the specific model structure or any prior knowledge. The learned sampling schedule may provide enough rewards in the searching process, leading to sufficient convergence compared to other related works. Once obtained, the optimal sampling schedule may also be generalized over other model structures for robust training. Although AutoSampling requires relatively large amount of computing resources to find a robust sampler, we want to point out that the efficiency of our method can be improved through better training techniques. Moreover, the possibility of an optimal sampling schedule relying solely on the data themselves may indicate more efficient sampling policy search algorithms, if one can quickly and effectively determine data value based on its property.
+
+# 5 CONCLUSIONS
+
+In this paper, we introduce a new search based AutoSampling scheme to overcome the issue of insufficient rewards for optimizing high-dimensional sampling hyper-parameter by utilizing a shorter period of reward collection. We use a shortened exploitation interval to search in the local data space and provide sufficient rewards. For the exploration step, we estimate sampling distribution from the searched sampling schedule and perturb it to search in the distribution space. We test our method and it consistently outperforms the baseline methods across different benchmarks.
+
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+
+# A APPENDIX
+
+# A.1 IMPLEMENTATION DETAILS
+
+Experiments on CIFAR We use the same training configuration for both CIFAR-100 and CIFAR-10 datasets, which both consist of 50000 training images. In particular, for model training we use the base learning rate of 0.1 and a step decay learning rate schedule where the learning rate is divided by 10 after each 60 epochs. We run the experiments for 240 epochs. In addition, we set the training batch size to be 128 per worker, and each worker is for one Nvidia V100 GPU card.
+
+We run the explore step for each $N _ { u } + 1$ epochs with $N _ { u } = 3$ , but note that we take the first explore step after the initial 20 epochs to better accumulate enough rewards. The experiments require 4800 epochs of training for 20 workers, and roughly 14 hours of training time.
+
+Experiments on ImageNet For ImageNet which consists of 1.28 million training images, we adopted the base learning rate of 0.2 and a cosine decay learning rate schedule. We run the experiments with 100 epochs of training. For each worker we utilize eight Nvidia V100 GPU cards and a total batch size of 512. Eight workers are used for all ImageNet experiments, and the rest of the setting adheres to that of CIFAR experiments. In addition, we utilize FP16 computation to achieve faster training, which has almost no drop in accuracy in practice. The experiments require 800 epochs of training for 8 workers, and roughly 4 days of training time.
+
+# A.2 COMPARISON WITH EXISTING SAMPLING METHODS
+
+To better illustrate the effectiveness of our AutoSampling method, we conduct experiments in comparison with recent non-uniform sampling methods DLIS (Johnson & Guestrin, 2018) and RAIS (Katharopoulos & Fleuret, 2018). DLIS (Johnson & Guestrin, 2018) achieves faster convergence by selecting data reducing gradient norm variance, while RAIS (Katharopoulos & Fleuret, 2018) does so through approximating the ideal sampling distribution using robust optimization. The comparison is recorded in Table 6.
+
+First, we run AutoSampling using Wide Resnet-28-2 (Zagoruyko & Komodakis, 2016) on CIFAR-100 with the training setting aligned roughly to (Katharopoulos & Fleuret, 2018). AutoSampling achievs improvement of roughly 3 percentage points $( 7 3 . 3 7 \pm 1 . 0 9 \%  7 6 . 2 4 \pm 1 . 0 2 \% )$ , while Katharopoulos & Fleuret shows improvement of 2 percentage points $( 6 6 . 0 \%  6 8 . 0 \%$ ). Second, we report the comparison between AutoSampling and RAIS on CIFAR-100. Johnson & Guestrin shows no improvement $7 6 . 4 \%  7 6 . 4 \%$ ) on accuracy and 0.027 $( 0 . 9 8 9  0 . 9 6 2$ ) decrease in validation loss, while our method shows improvement of 0.008 $7 8 . 6 \%  7 9 . 4 \% )$ on accuracy and 0.014 $0 . 8 8 6 $ 0.872 ) decrease in validation loss. As such, our method demonstrates significant improvements over existing non-uniform sampling methods.
+
+Table 6: Comparisons between AutoSampling and existing sampling methods on CIFAR-100   
+
+<table><tr><td>Methods</td><td>Network</td><td>Baseline (%)</td><td>With method (%)</td><td>Improvement (%)</td></tr><tr><td>DLIS</td><td>WRN-28-2</td><td>66.0</td><td>68.0</td><td>2.0</td></tr><tr><td>AutoSampling (ours)</td><td>WRN-28-2</td><td>73.37±1.09</td><td>76.24±1.02</td><td>2.87</td></tr><tr><td>RAIS</td><td>ResNet18</td><td>76.4</td><td>76.4</td><td>0.0</td></tr><tr><td>AutoSampling (ours)</td><td>ResNet18</td><td>78.46±0.035</td><td>79.44±0.020</td><td>0.98</td></tr></table>
+
+To further interpret the learned sampling schedules, we compare the sampling frequency of each training image and its loss values in different epochs during training of CIFAR-100 with ResNet-18. We draw the comparison for randomly selected 500 training images in Fig. 4 for epoch 80, 160, and 240. As shown in the figure, across different learning stages, the correlation between loss values and sampling frequencies of training data is not obvious. The high chance of being sampled by AutoSampling does not necessarily lead to high loss values, which demonstrates that AutoSampling is not merely over-sampling difficult samples as pointed by the loss. The resulting sampling schedule learned by AutoSampling would be significantly different from the one guided by loss. Moreover as the training progresses the loss values of data are reduced, which is expected.
+
+![](images/0db2448746ee565759d1c566b2bb9f36f75f6049646b65c3c95dda54f6e6be15.jpg)  
+Epoch 80   
+Epoch 160   
+Epoch 240   
+Figure 4: The comparison between the sampling frequency of each training image and its loss values of Epoch 80, 160 and 240 from CIFAR-100 with ResNet-18. We randomly selected 500 training images, and calculate their sampling frequency and loss values. The $\mathbf { X }$ -axis is the indexes of 500 training images, while the left y-axis denotes loss values and the right y-axis denotes the sampling frequency. The blue line represents the sampling frequencies and the red lines represents the loss values of all 500 images. As we can see from the figure, the two lines are not obviously correlated.
\ No newline at end of file
diff --git a/parse/train/AJTAcS7SZzf/AJTAcS7SZzf_content_list.json b/parse/train/AJTAcS7SZzf/AJTAcS7SZzf_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..9b6f105efa82dc83fe705931545a792864b5463c
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+++ b/parse/train/AJTAcS7SZzf/AJTAcS7SZzf_content_list.json
@@ -0,0 +1,1392 @@
+[
+    {
+        "type": "text",
+        "text": "AUTOSAMPLING: SEARCH FOR EFFECTIVE DATA SAMPLING SCHEDULES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            101,
+            826,
+            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,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Data sampling acts as a pivotal role in training deep learning models. However, an effective sampling schedule is difficult to learn due to its inherent high-dimension as a hyper-parameter. In this paper, we propose the AutoSampling method to automatically learn sampling schedules for model training, which consists of the multi-exploitation step aiming for optimal local sampling schedules and the exploration step for the ideal sampling distribution. More specifically, we achieve sampling schedule search with shortened exploitation cycle to provide enough supervision. In addition, we periodically estimate the sampling distribution from the learned sampling schedules and perturb it to search in the distribution space. The combination of two searches allows us to learn a robust sampling schedule. We apply our AutoSampling method to a variety of image classification tasks illustrating the effectiveness of the proposed method. ",
+        "bbox": [
+            233,
+            266,
+            766,
+            434
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            336,
+            477
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Data sampling policies can greatly influence the performance of model training in computer vision tasks, and therefore finding robust sampling policies can be important. Handcrafted rules, e.g. data resampling, reweighting, and importance sampling, promote better model performance by adjusting the training data frequency and order (Estabrooks et al., 2004; Weiss et al., 2007; Bengio et al., 2009; Johnson & Guestrin, 2018; Katharopoulos & Fleuret, 2018; Shrivastava et al., 2016; Jesson et al., 2017). Handcrafted rules heavily rely on the assumption over the dataset and cannot adapt well to datasets with their own characteristics. To handle this issue, learning-based methods (Li et al., 2019; Jiang et al., 2017; Fan et al., 2017) were designed to automatically reweight or select training data utilizing meta-learning techniques or a policy network. ",
+        "bbox": [
+            174,
+            492,
+            825,
+            618
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However existing learning-based sampling methods still rely on human priors as proxies to optimize sampling policies, which may fail in practice. Such priors often include assumptions on policy network design for data selection (Fan et al., 2017), or dataset conditions like noisiness (Li et al., 2019; Loshchilov & Hutter, 2015) or imbalance (Wang et al., 2019). These approaches take images features, losses, importance or their representations as inputs and use the policy network or other learning approaches with small amount of parameters for estimating the sampling probability. However, for example, images with similar visual features can be redundant in training, but their losses or features fed into the policy network are more likely to be close, causing the same probability to be sampled for redundant samples if we rely on aforementioned priors. Therefore, we propose to directly optimize the sampling schedule itself so that no prior knowledge is required for the dataset. Specifically, the sampling schedule refers to order by which data are selected for the entire training course. In this way, we only rely on data themselves to determine the optimal sampling schedule without any prior. ",
+        "bbox": [
+            174,
+            625,
+            825,
+            791
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Directly optimizing a sampling schedule is challenging due to its inherent high dimension. For example, for the ImageNet classification dataset (Deng et al., 2009) with around one million samples, the dimension of parameters would be in the same order. While popular approaches such as deep reinforcement learning (Cubuk et al., 2018; Zhang et al., 2020), Bayesian optimization (Snoek et al., 2015), population-based training (Jaderberg et al., 2017) or simple random search (Bergstra & Bengio, 2012) have already been utilized to tune low-dimensional hyper-parameters like augmentation schedules, their applications in directly finding good sampling schedules remain unexploited. For instance, the dimension of a data augmentation policy is generally only in dozens, and it needs thousands of training runs (Cubuk et al., 2018) to sample enough rewards to find an optimal augmentation policy because high-quality rewards require many epochs of training to obtain. As such, optimizing a sampling schedule may require orders of magnitude more rewards than data augmentation to gather and hence training runs, which result in prohibitively slow convergence. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To overcome the aforementioned challenge, we propose a data sampling policy search framework, named AutoSampling, to sufficiently learn an optimal sampling schedule in a population-based training fashion (Jaderberg et al., 2017). Unlike previous methods, which focus on collecting longterm rewards and updating hyper-parameters or agents offline, our AutoSampling method collects rewards online with a shortened collection cycle but without priors. Specifically, the AutoSampling collects rewards within several training iterations, tens or hundred times shorter than that in existing works (Ho et al., 2019; Cubuk et al., 2018). In this manner, we provide the search process with much more frequent feedback to ensure sufficient optimization of the sampling schedule. Each time when a few training iterations pass, we collect the reward from the previous several iterations, accumulate them and later update the sampling distribution using the rewards. Then we perturb the sampling distribution to search in distribution space, and use it to generate new mini-batches for later iterations, which are recorded into the output sampling schedule. As illustrated in Sec. 4.1, shortened collection cycles with less interference also can better reflect the training value of each data. ",
+        "bbox": [
+            174,
+            154,
+            825,
+            333
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are as follows: ",
+        "bbox": [
+            174,
+            340,
+            390,
+            354
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• To our best knowledge, we are the first to propose to directly learn a robust sampling schedule from the data themselves without any human prior or condition on the dataset. We propose the AutoSampling method to handle the optimization difficulty due to the high dimension of sampling schedules, and efficiently learn a robust sampling schedule through shortened reward collection cycle and online update of the sampling schedule. ",
+        "bbox": [
+            215,
+            357,
+            825,
+            431
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Comprehensive experiments on CIFAR-10/100 and ImageNet datasets (Krizhevsky, 2009; Deng et al., 2009) with different networks show that the Autosampling can increase the top-1 accuracy by up to $2 . 8 5 \\%$ on CIFAR-10, $2 . 1 9 \\%$ on CIFAR-100, and $2 . 8 3 \\%$ on ImageNet. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+            494,
+            326,
+            511
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            526,
+            331,
+            540
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data sampling is of great significance to deep learning, and has been extensively studied. Approaches with human-designed rules take pre-defined heuristic rules to modify the frequency and order by which training data is presented. In particular, one intuitive method is to resample or reweight data according to their frequencies, difficulties or importance in training (Estabrooks et al., 2004; Weiss et al., 2007; Drummond et al., 2003; Bengio et al., 2009; Lin et al., 2017; Shrivastava et al., 2016; Loshchilov & Hutter, 2015; Wang et al., 2019; Johnson & Guestrin, 2018; Katharopoulos & Fleuret, 2018; Byrd & Lipton, 2018; Jesson et al., 2017). These methods have been widely used in imbalanced training or hard mining problems. However, they are often restricted to certain tasks and datasets based on which they are proposed, and their ability to generalize to a broader range of tasks with different data distribution may be limited. In another word, these methods often implicitly assume certain conditions on the dataset, such as cleanness or imbalance. In addition, learning-based methods have been proposed for finding suitable sampling schemes automatically. Methods using meta-learning or reinforcement learning are also utilized to automatically select or reweight data during training (Li et al., 2019; Jiang et al., 2017; Ren et al., 2018; Fan et al., 2017), but they are only tested on small-scale or noisy datasets. Whether or not they can generalize over tasks of other datasets still remain untested. In this work, we directly study the data sampling without any prior, and we also investigate its wide generalization ability across different datasets such as CIFAR-10, CIFAR-100 and ImageNet using many typical networks. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As for hyper-parameter tuning, popular approaches such as deep reinforcement learning (Cubuk et al., 2018; Zhang et al., 2020), Bayesian optimization (Snoek et al., 2015) or simply random search (Bergstra & Bengio, 2012) have already been utilized to tune low-dimensional hyper-parameters and proven to be effective. Nevertheless, they have not been adopted to find good sampling schedule due to its inherent high dimensiona. Some recent works tackle the challenge of optimizing highdimensional hyper-parameter. MacKay et al. (2019) uses structured best-response functions and Jonathan Lorraine (2019) achieve this goal through the combinations of the implicit function theorem and efficient inverse Hessian approximations. However, they have not been tested on the task of optimizing sampling schedules, which is the major focus of our work in this paper. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/660fc98379b8fda0cbceb221c021ef383238ffbbe012ec81eb21c03d04754de5.jpg",
+        "image_caption": [
+            "Figure 1: Overview of AutoSampling illustrated through one multi-exploitation-and-exploration cycle. a) The multi-exploitation step, illustrated by the left half, is the process of learning optimal sampling schedule locally. The same color of model for each worker indicates that the same model weight is cloned into it. Also for simplicity, in this figure we adopt the exploitation interval of length 1. b) The exploration step, shown by the right half, is to search in the sampling distribution space. Specifically, we estimate the sampling distribution from the schedules collected in the multi-exploitation step and perturb it to generate new sampling schedules for all workers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 POPULATION BASED TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hyper-parameter tuning task can be framed as a bi-level optimization problem with the following objective function, ",
+        "bbox": [
+            178,
+            390,
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+            417
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/50edf36fee92ccdc04a98d255f0f5994bb66257138be32898ebd79bba6a18ed5.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { m i n } _ { h \\in \\mathcal { H } } \\mathcal { L } ( \\theta ^ { * } , h ) } \\\\ { \\theta ^ { * } = \\mathop { \\mathrm { a r g } \\operatorname* { m a x } } \\mathop { \\mathrm { e v a l } } ( \\theta , h ) } \\\\ { \\theta \\in \\Theta } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta$ represents the model weight and $h = ( h _ { 1 } , h _ { 2 } , \\cdot \\cdot \\cdot , h _ { T } )$ is the hyper-parameter schedule for $T$ training intervals. Population based training (PBT) (Jaderberg et al., 2017) solves the bilevel optimization problem by training a population $\\mathcal { P }$ of child models in parallel with different hyper-parameter schedules initialized: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7096fe11231e6a2be42df01848117ab311a35d42f62b06ff8854fc0422c80e95.jpg",
+        "text": "$$\n\\mathcal { P } = \\{ ( \\theta _ { i } , h _ { i } , t ) \\} _ { i = 1 } ^ { N _ { p } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            529,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta _ { i } , h _ { i }$ respectively represents the child model weight, the corresponding hyper-parameter schedule for the training interval $t$ on worker $i$ , and $N _ { p }$ is the number of workers. PBT proceeds in intervals, which usually consists of several epochs of training. During the interval, the population of models are trained in parallel to finish the lower-level optimization of weights $\\theta _ { i }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Between intervals, an exploit-and-explore procedure is adopted to conduct the upper-level optimization of the hyper-parameter schedule. In particular for interval $t$ , to exploit we evaluate child models on a held-out validation dataset: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/baa50128f023546003052d0fc995e30e4a472a7721e1ff2ab8d73a3afd472070.jpg",
+        "text": "$$\nh _ { t } ^ { * } , \\theta _ { t } ^ { * } = \\operatorname * { a r g m a x } _ { p _ { i } = ( \\theta _ { i } , h _ { i } , t ) \\in \\mathcal { P } } \\mathrm { e v a l } ( \\theta _ { i } , h _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/285997ae87dff049fced817c2d5502b0325ef83bdb353bbd10c152ba2664f0bb.jpg",
+        "text": "$$\n\\theta ^ { * }  \\theta _ { i } , i = 1 , \\cdots , N _ { p }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We record the best performing hyper-parameter setting $h _ { t } ^ { * }$ and broadcast the top-performing model $\\theta _ { t } ^ { * }$ to all workers. To explore, we initialize new hyper-parameter schedules for interval $t + 1$ with different random seeds on all workers, which can be viewed as a search in the hyper-parameter space. The next exploit-and-explore cycle will then be continued. In the end, the top-performing hyper-parameter schedule $h ^ { * } = ( h _ { 1 } ^ { * } , h _ { 2 } ^ { * } , \\cdot \\cdot \\cdot , h _ { T } ^ { * } )$ can be obtained. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "PBT is applied to tune low-dimenisal hyper-parameters such as data augmentation schedules (Ho et al., 2019; Jaderberg et al., 2017). However, it cannot be directly used for finding sampling strategies due to the high dimension. Unlike PBT, our AutoSampling adopts a multi-exploitation-andexploration structure, leading to much shorter reward collection cycles that contribute to much more and effective rewards for sufficient optimization within a practical computational budget. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 AUTOSAMPLING WITH SEARCHING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            875,
+            501,
+            891
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The overview of our AutoSampling is illustrated in Fig.1. AutoSampling alternately runs multiexploitation step and exploration step. In the exploration step, we 1) update the sampling distribution using the rewards collected from the multi-exploitation step (the sampling distribution is uniform distribution initially); 2) perturb the updated sampling distribution for child models so that different child models have different sampling distributions; 3) use the corresponding perturbed sampling distribution for each child model to sample mini-batches of training data. In the multi-exploitation step, we 1) train multiple child models using the mini-batches sampled from the exploration step; 2) collect short-term rewards from the child models. AutoSampling finishes with a recorded topperforming sampling schedule, which can be transferred to other models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/c398aa5dd9369bdf8545b46a6faa4ae22e1f4fd350352be9dcdab52e3d14db76.jpg",
+        "table_caption": [
+            "Algorithm 1: The Multi-Exploitation Step "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>number of workers Np, number of i=1 exploitation intervals T,exploitation interval length Ns Initialize H*←(</td></tr><tr><td>for t=1toTdo</td></tr><tr><td>for j = 1 to Ns do for(0i,ht,i,t) ∈Pdo</td></tr><tr><td>0i←L(θi,ht,i) &gt; update the weight of child model i end for</td></tr><tr><td>h*,0* = arg maxp eval(0i,hi)</td></tr><tr><td>H*←H*+h D update the sampling for child model i for i = 1 to Np do</td></tr><tr><td>θ← &gt; clone the optimal weight</td></tr><tr><td>end for end for</td></tr></table>",
+        "bbox": [
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+            354
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 MULTI-EXPLOITATION BY SEARCHING IN THE DATA SPACE ",
+        "text_level": 1,
+        "bbox": [
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+            496,
+            627,
+            510
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the multi-exploitation step, we aim to search locally in the data space by collecting short-term rewards and sub-schedules. Specifically, we wish to learn a sampling schedule for $T$ exploitation intervals. In each interval, there are a population $\\mathcal { P }$ of $N _ { p }$ child models. Denote $\\mathbf { h } _ { t , i }$ as the training data sub-schedule in the $t ^ { t h }$ interval for the $i ^ { t h }$ child model. When all of the $T$ exploitation intervals for the $i ^ { t h }$ child model are considered, we have $\\mathbf { H } _ { i } = \\{ \\mathbf { h } _ { t , i } | t = 1 , \\dots , T \\} = \\{ x _ { 1 } , \\cdot \\cdot \\cdot , x _ { N } \\}$ , where $N$ is the number of training data for the multi-exploitation step. Each interval consists of $N _ { s }$ training iterations that is also equivalent to $N _ { s }$ training mini-batches, where $N _ { s }$ is the length of the interval. AutoSampling is expected to produce a sequence of training samples, denoted by $\\mathbf { H } ^ { * }$ , so that a given model is optimally trained. The population $\\left\\{ \\mathbf { H } _ { i } \\right\\}$ forms the local search space, from which we aim to search for an optimal sampling schedule $\\mathbf { H } ^ { * }$ . ",
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+    {
+        "type": "text",
+        "text": "Given the population $\\mathcal { P }$ , we train them in parallel on $N _ { p }$ workers. Once an interval of data $\\mathbf { h } _ { t , i }$ containing $N _ { s }$ training batches have been used for training, we evaluate all child models and use the top evaluation performance as the reward. According to the reward, we record the top-performing weight and sub-schedule for the current interval $t$ , in particular, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f64294f660ab593e563d3de4c389d356c19a7dfad042ff1964dac94c6f6ffb8a.jpg",
+        "text": "$$\nh _ { t } ^ { * } , \\theta _ { t } ^ { * } = \\operatorname * { a r g m a x } _ { p _ { i } = ( \\theta _ { i } , h _ { i } , t ) \\in \\mathcal { P } } \\operatorname { e v a l } ( \\theta _ { i } , h _ { t , i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, we update all child model weights of $\\mathcal { P }$ by cloning into them with the topperforming weight $\\theta _ { t } ^ { * }$ so we can continue searching based on the more promising child. We will continue the exploit steps through the whole training process, and output the recorded optimal sampling schedule $\\mathbf { H } ^ { * } = \\{ h _ { 1 } ^ { * } , h _ { 2 } ^ { * } , \\cdot \\cdot \\cdot , h _ { T } ^ { * } \\}$ . By using exploitation interval of mini-batches rather than epochs or even entire training runs adopted by earlier methods, AutoSampling may yield a better and more robust sampling schedule. It should be pointed out that even though in AutoSampling rewards are collected within a much shorter interval, they remain effective. As we directly optimize the sampling schedule, we are concerned with only the data themselves. The short-term rewards reflect the training value of data from the exploitation interval they are collected. But for global hyperparameters such as augmentation schedules, short-term rewards may lead to inferior performance as these hyper-parameters are concerned with the overall training outcome. We describe the multiexploitation with details in Alg.1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/49080fd4c9f0e1affd16af372d8ec1bfb96efc8d3abe8dcd110eeba4448af75a.jpg",
+        "table_caption": [
+            "Algorithm 2: Search based AutoSampling "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Input: Training dataset D, population size Np Initialize H* ←O,P(D) ←uniform(D) and initialize child models 01,. ,0Np while not end of training do for i= 1 to Np do</td></tr><tr><td>Sample hi from Mixture(log(P(D) + β),Nu × uniform(D)) end for</td></tr><tr><td>InitializeP={(0,hi,t）}1</td></tr><tr><td>H*,P← Alg.1 Estimate P(D) according to Equation (5)</td></tr><tr><td>Update P(D) according to Equation (6)</td></tr><tr><td>H*←H*+H* end while</td></tr><tr><td>Return H*,P(D)</td></tr><tr><td></td></tr></table>",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3.2 EXPLORATION BY SEARCHING IN SAMPLING DISTRIBUTION SPACE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the exploration, we search in sampling distribution space by updating and perturbing the sampling distribution. We first estimate the underlying sampling distribution $P ( D )$ from the top sampling schedule $h ^ { * }$ produced in the multi-exploitation, that is, for $x \\in D$ , ",
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+    {
+        "type": "equation",
+        "img_path": "images/0ca1c4ff7f6dd3d3d53b67fbcd7d0052180af5739a2157501e32a679ffbe4c2e.jpg",
+        "text": "$$\nP ( x ) = \\frac { c o u n t ( x \\in \\mathbf { H } ^ { * } ) } { \\sum _ { x \\in D } c o u n t ( x \\in \\mathbf { H } ^ { * } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $c o u n t ( x \\in \\mathbf { H } ^ { * } )$ denotes the number of $x$ ’s appearances in $\\mathbf { H } ^ { * }$ . We further perturb the $P ( D )$ and generate the sampling schedules on each worker for the later multi-exploitation. We introduce perturbations into the generated schedules by simply sampling from the multinomial distribution $P ( D )$ using different random seeds. However, in our experiments, we observe that the distribution produced by $P ( D )$ tends to be extremely skewed and a majority of the data actually have zero frequencies. Such skewness causes highly imbalanced training mini-batches, and therefore destabilizes subsequent model training. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Distribution Smoothing To tackle the above issue, we first smooth $P ( D )$ through the logarithmic function, and then apply a probability mixture with uniform distributions. In particular for the dataset $D$ , ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08bd2ef6d0bd4396daae5bab7e35fda77a516ac458ce6f41c8419762a02f6d25.jpg",
+        "text": "$$\nP ^ { \\prime } ( D ) = M i x t u r e ( \\log ( P ( D ) + \\beta ) , N _ { u } \\times \\mathrm { u n i f o r m } ( D ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta \\geq 1$ is the smoothing factor and $N _ { u } \\times \\mathrm { u n i f o r m } ( D )$ denotes $N _ { u }$ uniform multinomial distributions on the dataset $D$ . The smoothing through the log function can greatly reduce the skewness, however, $\\log ( P ( D ) + \\beta )$ may still contain zero probabilities for some training data, resulting in unstable training. Therefore, we further smooth it through a probability mixture with $N _ { u }$ uniform distribution uniform $( D )$ to ensure presence of all data. This is equivalent to combining $N _ { u }$ epochs of training data to the training batches sampled from $P ( D )$ , and shuffling the union. Once we have new diverse sampling schedules for the population, we proceed to the next multi-exploitation step. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We continue this alternation between multi-exploitation and exploration steps until the end of training. Note that to generate sampling schedule for the first multi-exploitation run, we initialize $P ( D )$ to be an uniform multinomial distribution. In the end, we output a sequence of optimal sampling schedules $H ^ { * } = ( \\mathbf { H } _ { 1 } ^ { * } , \\cdots , \\mathbf { H } _ { n } ^ { * } )$ for $n$ alternations. The entire process is illustrated in details in Alg.2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we present comprehensive experiments on various datasets to illustrate the performance of AutoSampling, and also demonstrate the process of progressively learning better sampling distribution. ",
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+    {
+        "type": "text",
+        "text": "4.1 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For this part, we gradually build up and test components of AutoSampling on CIFAR-100, and then examine their performances on CIFAR-10 and ImageNet datasets. The training implementation details and computational complexity can be found in Appendix A.1. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/be621a325539e739104018c4e948dbda4dcc4adf2469fe3d683dfd4df6568e67.jpg",
+        "table_caption": [
+            "Table 1: Performance on CIFAR-100 using different configurations of AutoSampling and baselines. Worker is the number of workers used and Interval is the exploitation interval in terms of batches. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>NETWORK</td><td>WORKER</td><td>INTERVAL</td><td>EXPLORATION TYPE</td><td>TOP1(%)</td></tr><tr><td>RESNET18 (ZHANG ET AL., 2019)</td><td>-</td><td>-</td><td></td><td>78.34±0.05</td></tr><tr><td>RESNET18</td><td>1</td><td>-</td><td>UNIFORM</td><td>78.46±0.035</td></tr><tr><td>RESNET18</td><td>20</td><td>80 BATCHES</td><td>RANDOM</td><td>78.76±0.003</td></tr><tr><td>RESNET18</td><td>20</td><td>20 BATCHES</td><td>RANDOM</td><td>78.99±0.003</td></tr><tr><td>RESNET18</td><td>80</td><td>20 BATCHES</td><td>RANDOM</td><td>79.09±0.017</td></tr><tr><td>RESNET18</td><td>20</td><td>20 BATCHES</td><td>MIXTURE</td><td>79.44±0.020</td></tr><tr><td>RESNET50 (JINET AL.,2019)</td><td>-</td><td>-</td><td>-</td><td>79.34</td></tr><tr><td>RESNET50</td><td>1</td><td></td><td>UNIFORM</td><td>79.70±0.023</td></tr><tr><td>RESNET50</td><td>20</td><td>80 BATCHES</td><td>RANDOM</td><td>80.55±0.129</td></tr><tr><td>RESNET50</td><td>20</td><td>20 BATCHES</td><td>RANDOM</td><td>81.05±0.064</td></tr><tr><td>RESNET50</td><td>80</td><td>20 BATCHES</td><td>RANDOM</td><td>81.19±0.072</td></tr><tr><td>RESNET50</td><td>20</td><td>20 BATCHES</td><td>MIXTURE</td><td>81.53±0.088</td></tr><tr><td>DENSENET121</td><td>1</td><td>-</td><td>UNIFORM</td><td>80.13±0.028</td></tr><tr><td>DENSENET121</td><td>20</td><td>80 BATCHES</td><td>RANDOM</td><td>80.62±0.694</td></tr><tr><td>DENSENET121</td><td>20</td><td>20 BATCHES</td><td>RANDOM</td><td>81.11±0.127</td></tr><tr><td>DENSENET121</td><td>80</td><td>20 BATCHES</td><td>RANDOM</td><td>81.08±0.021</td></tr><tr><td>DENSENET121</td><td>20</td><td>20 BATCHES</td><td>MIXTURE</td><td>80.97±0.006</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/029c09aff4dbeafb5217d5430a43331bf4a8597adc1625f639f9b0825f3afe3a.jpg",
+        "table_caption": [
+            "Table 2: Experiments on CIFAR-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>NETWORK</td><td>EXPLORATION TYPE</td><td>ToP1(%)</td></tr><tr><td>RESNET18</td><td>UNIFORM</td><td>93.01±0.009</td></tr><tr><td>RESNET18</td><td>RANDOM</td><td>95.86±0.003</td></tr><tr><td>RESNET18</td><td>MIXTURE</td><td>95.80±0.018</td></tr><tr><td>RESNET50</td><td>UNIFORM</td><td>93.60±0.004</td></tr><tr><td>RESNET50</td><td>RANDOM</td><td>96.10±0.002</td></tr><tr><td>RESNET50</td><td>MIXTURE</td><td>96.09±0.070</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/bc1ee2af3122bea19c36cb086ddd7fabb00704975fe563d7759b847e4cff591a.jpg",
+        "table_caption": [
+            "Table 3: Experiments on ImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>NETWORK</td><td>EXPLORATION TYPE</td><td>TOP1(%)</td></tr><tr><td>RESNET18</td><td>UNIFORM</td><td>70.38</td></tr><tr><td>RESNET18</td><td>RANDOM</td><td>72.07</td></tr><tr><td>RESNET18</td><td>MIXTURE</td><td>72.91</td></tr><tr><td>RESNET34</td><td>UNIFORM</td><td>74.09</td></tr><tr><td>RESNET34</td><td>RANDOM</td><td>76.11</td></tr><tr><td>RESNET34</td><td>MIXTURE</td><td>76.92</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Adding Workers To look into the influence of the worker numbers, we conduct experiments using worker numbers of 1, 20, 80 respectively with the same setting ( $N _ { s } = 2 0$ with random exploration). With the worker number of 1, the experiment is simply the normal model training using stochastic gradient descent. To show the competitiveness of our baselines, we also include state-of-the-art results on CIFAR-100 with ResNet-18 and ResNet-50 (Zhang et al., 2019; Jin et al., 2019). We notice significant performance gain using the worker number of 20 for ResNet-18, ResNet-50 and DenseNet-121 (He et al., 2015; Huang et al., 2017), as illustrated in Table 1. However, we note that increasing worker number from 20 to 80 only brings marginal performance gains across various model structures, as shown in Table 1. Therefore, we set the worker number to be 20 for the rest of the experiments. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Shortening Exploitation Intervals To study the effects of the shortened exploitation interval, we run experiments using different exploitation intervals of 20 and 80 batches(iterations) respectively. As shown in Table 1, models with the shorter exploitation interval of 20 batches(iterations) perform better than the one with the longer exploitation interval across all three network structures, conforming to our assumptions that the reward collected reflects value of each data used in the exploitation interval. This result adheres to our intuition that shorter exploitation interval can encourage the sampler to accumulate more rewards to learn better sampling schedules. For the rest of this section we keep the exploitation interval of 20. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Adding Exploration Type We further add mixture as the exploration type to see the effects of learning the underlying sampling distribution, and completing the proposed method. As shown in Table 1, with ResNet-18 and ResNet-50 we push performance higher with the mixture exploration, and outperform the baseline method by about 1 and 1.8 percentage on CIFAR-100 respectively. However, we found that it is not true in the case of DenseNet-121 and this case may be attributed to the bigger capacity of DenseNet-121. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Generalization Over Datasets In addition, we experiment on other datasets. We report the results on CIFAR10 in Table 2 and the results of ResNet-18, ResNet-34 on ImageNet in Table 3. For CIFAR-10, we notice that the mixture and random exploration methods are comparable while both outperforming the uniform baseline, and we believe it is due to the simplicity of the dataset. In the more challenging ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/fe3c88399ad6162c77b6025bd2e83af8d156bc7688326885cf936bf3a3ec6d4f.jpg",
+        "image_caption": [
+            "Figure 2: The comparison between histograms estimated from the sampling schedules of Epoch 80, 160 and 240 from CIFAR-100 with ResNet-18. We divide the 50000 training images into 500 segments of 100 images, and calculate the histograms of total data counts of all segments. We reorder the $x$ -axis based on the ranking of data counts for epoch 240 for easier comparison. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            118,
+            828,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/35fe25bd3314aff6534d669d89ffbd077293701a8dc7ccb3d1f4e5ce28da6652.jpg",
+        "table_caption": [
+            "Table 4: Static vs dynamic sampling schedule on CIFAR-100 $( \\% )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">NETWORK</td><td colspan=\"3\">SAMPLING TYPE</td></tr><tr><td>UNIFORM</td><td>STATIC</td><td>DYNAMIC</td></tr><tr><td>RESNET18</td><td>78.46±0.035</td><td>78.80±0.007</td><td>79.44±0.020</td></tr><tr><td>RESNET50</td><td>79.70±0.023</td><td>80.21±0.014</td><td>81.53±0.088</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ImageNet, the mixture exploration outperforms the random exploration by a clear margin. We also compare our AutoSampling with some recent non-uniform sampling methods on CIFAR-100, which can be found in Appendix A.2. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 STATIC VS DYNAMIC SCHEDULES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We aim to see if the final sampling distribution estimated by our AutoSampling is sufficient to produce robust sampling schedules. In another word, we wish to know training with the AutoSampling is either a process of learning a robust sampling distribution, or a process of dynamically adjusting the sampling schedule for optimal training. To this end, we conduct training using different sampling schedules. First, we calculate the sampling distribution estimated throughout the learning steps of AutoSampling, and use it to generate the sampling schedule of a full training process, which we denote as STATIC. Moreover, we denote the sampling schedule learned using AutoSampling as DYNAMIC, since AutoSampling dynamically adjust the sampling schedule alongside the training process. Finally, we denote the baseline method as UNIFORM, which uses the sampling schedule generated from uniform distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We report results on CIFAR-100 with ResNet-18 and ResNet-50 in Table 4. Model trained with STATIC sampling schedules exceeds the baseline UNIFORM significantly, indicating the superiority of the learned sampling distribution over the uniform distribution. It shows the ability of AutoSampling to learn good sampling distribution. Nonetheless, note that models trained with DYNAMIC sampling schedules outperform models trained with STATIC, by a margin bigger than the one between STATIC and UNIFORM. This result shows the fact that despite the AutoSampling’s capability of learning good sampling distribution, its flexibility during training matters even more. Moreover, this phenomenon also indicates that models at different stages of learning process may require different sampling distributions to achieve optimal training. One single sampling distribution, even gradually estimated using AutoSampling, seems incapable of covering the needs from different learning stages. We plot the histograms of data counts in training estimated from schedules of different learning stages with ResNet-18 on CIFAR-100 in Fig.2, showing the great differences between optimized sampling distributions from different epochs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 ANALYZING SAMPLING SCHEDULES LEARNED BY AUTOSAMPLING ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To further investigate the sampling schedule learned by AutoSampling, we review the images at the tail and head part of the sampling spectrum. In particular, given a sampling schedule learned we rank all images based on their appearances in training. Training images at the top and bottom of the order are extracted, corresponding to high and low probabilities of being sampled respectively. In Fig.3, we show 4 classes of exemplary images. The images of low probability tend to have clearer imagery features enabling easy recognition, while the images of high probability tend to be more obscure, indicating that the sampling schedule may show hard samples mining effects. However, as shown in A.3 and Fig. 4, the loss values and probabilities of being sampled seem to be not highly correlated, which indicates more potential of AutoSampling beyond visually hard example mining. In addition, we notice the images of low probability also contain low quality images. For instance, in Fig.3 the leftmost image of CAMAL class contains only legs. This shows that AutoSampling may potentially rule out problematic training data for better training. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/809ba11d034a648f9392e8b6bf5244948a22bc2430e9efcce77a6fcafe43227f.jpg",
+        "image_caption": [
+            "Figure 3: Example images on the head and tail of the sampling spectrum. The images on the left are the ones with low sampling probability, while the images on the right more likely to be sampled. We obtain these images using AutoSampling with the ResNet-18 model on CIFAR-100. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            305,
+            101,
+            692,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/43678316f0182e70f8861746bf17b58bce51dfda83e45ffa5193b90232b0949a.jpg",
+        "table_caption": [
+            "Table 5: Transfer of sampling distributions learned by three model structures to ResNet-50 on CIFAR-100 $( \\% )$ . UNIFORM denotes the baseline result using uniform sampling distribution. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">NETWORK</td><td colspan=\"4\">SAMPLING SCHEDULE SOURCE</td></tr><tr><td>UNIFORM</td><td>RESNET18</td><td>RESNET50</td><td>DENSENET121</td></tr><tr><td>RESNET50</td><td>79.70±0.023</td><td>80.27±0.014</td><td>80.21±0.014</td><td>80.47±0.194</td></tr></table>",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Furthermore, we examine the transfer ability of sampling distributions learned by AutoSampling to other network structures. Specifically, we run training on ResNet-50 (He et al., 2015) using STATIC sampling schedule generated by three distributions learned by AutoSampling on 3 different models. As shown in Table 5, using sampling schedules learned by AutoSampling from other models, we demonstrate similar improvements over the UNIFORM baseline. This result, combined with the above observations on images of different sampling probability, indicates that there may exist a common optimal sampling schedule determined by the intrinsic property of the data rather than the model being optimized. Our AutoSampling is an effort to gradually converge to such an optimal schedule. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 DISCUSSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The experimental results and observations from Section 4.2 and 4.3 shed light on the possible existence of an optimal sampling schedule, which relies only on the intrinsic property of the data and the learning stage of the model, regardless of the specific model structure or any prior knowledge. The learned sampling schedule may provide enough rewards in the searching process, leading to sufficient convergence compared to other related works. Once obtained, the optimal sampling schedule may also be generalized over other model structures for robust training. Although AutoSampling requires relatively large amount of computing resources to find a robust sampler, we want to point out that the efficiency of our method can be improved through better training techniques. Moreover, the possibility of an optimal sampling schedule relying solely on the data themselves may indicate more efficient sampling policy search algorithms, if one can quickly and effectively determine data value based on its property. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduce a new search based AutoSampling scheme to overcome the issue of insufficient rewards for optimizing high-dimensional sampling hyper-parameter by utilizing a shorter period of reward collection. We use a shortened exploitation interval to search in the local data space and provide sufficient rewards. For the exploration step, we estimate sampling distribution from the searched sampling schedule and perturb it to search in the distribution space. We test our method and it consistently outperforms the baseline methods across different benchmarks. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
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+        "text": "Xinyu Zhang, Qiang Wang, Jian Zhang, and Zhao Zhong. Adversarial autoaugment. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum? id $=$ ByxdUySKvS. ",
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+            102,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            410,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Experiments on CIFAR We use the same training configuration for both CIFAR-100 and CIFAR-10 datasets, which both consist of 50000 training images. In particular, for model training we use the base learning rate of 0.1 and a step decay learning rate schedule where the learning rate is divided by 10 after each 60 epochs. We run the experiments for 240 epochs. In addition, we set the training batch size to be 128 per worker, and each worker is for one Nvidia V100 GPU card. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We run the explore step for each $N _ { u } + 1$ epochs with $N _ { u } = 3$ , but note that we take the first explore step after the initial 20 epochs to better accumulate enough rewards. The experiments require 4800 epochs of training for 20 workers, and roughly 14 hours of training time. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Experiments on ImageNet For ImageNet which consists of 1.28 million training images, we adopted the base learning rate of 0.2 and a cosine decay learning rate schedule. We run the experiments with 100 epochs of training. For each worker we utilize eight Nvidia V100 GPU cards and a total batch size of 512. Eight workers are used for all ImageNet experiments, and the rest of the setting adheres to that of CIFAR experiments. In addition, we utilize FP16 computation to achieve faster training, which has almost no drop in accuracy in practice. The experiments require 800 epochs of training for 8 workers, and roughly 4 days of training time. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 COMPARISON WITH EXISTING SAMPLING METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            402,
+            573,
+            415
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To better illustrate the effectiveness of our AutoSampling method, we conduct experiments in comparison with recent non-uniform sampling methods DLIS (Johnson & Guestrin, 2018) and RAIS (Katharopoulos & Fleuret, 2018). DLIS (Johnson & Guestrin, 2018) achieves faster convergence by selecting data reducing gradient norm variance, while RAIS (Katharopoulos & Fleuret, 2018) does so through approximating the ideal sampling distribution using robust optimization. The comparison is recorded in Table 6. ",
+        "bbox": [
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+            428,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "First, we run AutoSampling using Wide Resnet-28-2 (Zagoruyko & Komodakis, 2016) on CIFAR-100 with the training setting aligned roughly to (Katharopoulos & Fleuret, 2018). AutoSampling achievs improvement of roughly 3 percentage points $( 7 3 . 3 7 \\pm 1 . 0 9 \\%  7 6 . 2 4 \\pm 1 . 0 2 \\% )$ , while Katharopoulos & Fleuret shows improvement of 2 percentage points $( 6 6 . 0 \\%  6 8 . 0 \\%$ ). Second, we report the comparison between AutoSampling and RAIS on CIFAR-100. Johnson & Guestrin shows no improvement $7 6 . 4 \\%  7 6 . 4 \\%$ ) on accuracy and 0.027 $( 0 . 9 8 9  0 . 9 6 2$ ) decrease in validation loss, while our method shows improvement of 0.008 $7 8 . 6 \\%  7 9 . 4 \\% )$ on accuracy and 0.014 $0 . 8 8 6 $ 0.872 ) decrease in validation loss. As such, our method demonstrates significant improvements over existing non-uniform sampling methods. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/64c0fdbdfc9f5b47027578bde15966cd04b78d2a2380cd960919a10fef747e4b.jpg",
+        "table_caption": [
+            "Table 6: Comparisons between AutoSampling and existing sampling methods on CIFAR-100 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Network</td><td>Baseline (%)</td><td>With method (%)</td><td>Improvement (%)</td></tr><tr><td>DLIS</td><td>WRN-28-2</td><td>66.0</td><td>68.0</td><td>2.0</td></tr><tr><td>AutoSampling (ours)</td><td>WRN-28-2</td><td>73.37±1.09</td><td>76.24±1.02</td><td>2.87</td></tr><tr><td>RAIS</td><td>ResNet18</td><td>76.4</td><td>76.4</td><td>0.0</td></tr><tr><td>AutoSampling (ours)</td><td>ResNet18</td><td>78.46±0.035</td><td>79.44±0.020</td><td>0.98</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To further interpret the learned sampling schedules, we compare the sampling frequency of each training image and its loss values in different epochs during training of CIFAR-100 with ResNet-18. We draw the comparison for randomly selected 500 training images in Fig. 4 for epoch 80, 160, and 240. As shown in the figure, across different learning stages, the correlation between loss values and sampling frequencies of training data is not obvious. The high chance of being sampled by AutoSampling does not necessarily lead to high loss values, which demonstrates that AutoSampling is not merely over-sampling difficult samples as pointed by the loss. The resulting sampling schedule learned by AutoSampling would be significantly different from the one guided by loss. Moreover as the training progresses the loss values of data are reduced, which is expected. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0db2448746ee565759d1c566b2bb9f36f75f6049646b65c3c95dda54f6e6be15.jpg",
+        "image_caption": [
+            "Epoch 80 ",
+            "Epoch 160 ",
+            "Epoch 240 ",
+            "Figure 4: The comparison between the sampling frequency of each training image and its loss values of Epoch 80, 160 and 240 from CIFAR-100 with ResNet-18. We randomly selected 500 training images, and calculate their sampling frequency and loss values. The $\\mathbf { X }$ -axis is the indexes of 500 training images, while the left y-axis denotes loss values and the right y-axis denotes the sampling frequency. The blue line represents the sampling frequencies and the red lines represents the loss values of all 500 images. As we can see from the figure, the two lines are not obviously correlated. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+                                    "content": "collects rewards within several training iterations, tens or hundred times shorter than that in existing",
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+                                    "score": 1.0,
+                                    "content": "works (Ho et al., 2019; Cubuk et al., 2018). In this manner, we provide the search process with much",
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+                                    "score": 1.0,
+                                    "content": "more frequent feedback to ensure sufficient optimization of the sampling schedule. Each time when a",
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+                                    "score": 1.0,
+                                    "content": "few training iterations pass, we collect the reward from the previous several iterations, accumulate",
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+                                    "score": 1.0,
+                                    "content": "them and later update the sampling distribution using the rewards. Then we perturb the sampling",
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+                            "index": 12
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+                                    "score": 1.0,
+                                    "content": "distribution to search in distribution space, and use it to generate new mini-batches for later iterations,",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "which are recorded into the output sampling schedule. As illustrated in Sec. 4.1, shortened collection",
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+                            ],
+                            "index": 14
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+                        {
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+                                    "content": "cycles with less interference also can better reflect the training value of each data.",
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+                                    "score": 0.91,
+                                    "content": "\\mathbf { H } _ { i } = \\{ \\mathbf { h } _ { t , i } | t = 1 , \\dots , T \\} = \\{ x _ { 1 } , \\cdot \\cdot \\cdot , x _ { N } \\}",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "is the number of training data for the multi-exploitation step. Each interval consists of",
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+                                    "score": 1.0,
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+                                {
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+                                    "score": 1.0,
+                                    "content": "training mini-batches, where",
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+                                    "score": 1.0,
+                                    "content": "is the length of the interval.",
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+                                    "score": 1.0,
+                                    "content": "AutoSampling is expected to produce a sequence of training samples, denoted by",
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", so that a given",
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+                                    "score": 1.0,
+                                    "content": "model is optimally trained. The population",
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+                                    "score": 0.92,
+                                    "content": "\\left\\{ \\mathbf { H } _ { i } \\right\\}",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "forms the local search space, from which we aim to",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "training batches have been used for training, we evaluate all child models and use the",
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+                                    "score": 1.0,
+                                    "content": "top evaluation performance as the reward. According to the reward, we record the top-performing",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": ", in particular,",
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+                                    "score": 0.94,
+                                    "content": "h _ { t } ^ { * } , \\theta _ { t } ^ { * } = \\operatorname * { a r g m a x } _ { p _ { i } = ( \\theta _ { i } , h _ { i } , t ) \\in \\mathcal { P } } \\operatorname { e v a l } ( \\theta _ { i } , h _ { t , i } )",
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+                                    "content": "On the other hand, we update all child model weights of",
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+                                    "score": 1.0,
+                                    "content": "so we can continue searching based on the more promising child. We will",
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+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "continue the exploit steps through the whole training process, and output the recorded optimal",
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+                                    "type": "text"
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+                                    "score": 0.91,
+                                    "content": "\\mathbf { H } ^ { * } = \\{ h _ { 1 } ^ { * } , h _ { 2 } ^ { * } , \\cdot \\cdot \\cdot , h _ { T } ^ { * } \\}",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ". By using exploitation interval of mini-batches rather",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "than epochs or even entire training runs adopted by earlier methods, AutoSampling may yield a better",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and more robust sampling schedule. It should be pointed out that even though in AutoSampling",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "rewards are collected within a much shorter interval, they remain effective. As we directly optimize",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the sampling schedule, we are concerned with only the data themselves. The short-term rewards",
+                                    "type": "text"
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+                            ],
+                            "index": 34
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "reflect the training value of data from the exploitation interval they are collected. But for global hyper-",
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+                            ],
+                            "index": 35
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "parameters such as augmentation schedules, short-term rewards may lead to inferior performance",
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+                            "index": 36
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "as these hyper-parameters are concerned with the overall training outcome. We describe the multi-",
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+                                    "score": 1.0,
+                                    "content": "exploitation with details in Alg.1.",
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+                                    "content": "h _ { t } ^ { * } , \\theta _ { t } ^ { * } = \\operatorname * { a r g m a x } _ { p _ { i } = ( \\theta _ { i } , h _ { i } , t ) \\in \\mathcal { P } } \\operatorname { e v a l } ( \\theta _ { i } , h _ { t , i } )",
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+                                    "content": "so we can continue searching based on the more promising child. We will",
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+                                    "content": ". By using exploitation interval of mini-batches rather",
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+                                    "content": "than epochs or even entire training runs adopted by earlier methods, AutoSampling may yield a better",
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+                                    "content": "and more robust sampling schedule. It should be pointed out that even though in AutoSampling",
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+                                    "content": "rewards are collected within a much shorter interval, they remain effective. As we directly optimize",
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+                                            "html": "<table><tr><td>Input: Training dataset D, population size Np Initialize H* ←O,P(D) ←uniform(D) and initialize child models 01,. ,0Np while not end of training do for i= 1 to Np do</td></tr><tr><td>Sample hi from Mixture(log(P(D) + β),Nu × uniform(D)) end for</td></tr><tr><td>InitializeP={(0,hi,t）}1</td></tr><tr><td>H*,P← Alg.1 Estimate P(D) according to Equation (5)</td></tr><tr><td>Update P(D) according to Equation (6)</td></tr><tr><td>H*←H*+H* end while</td></tr><tr><td>Return H*,P(D)</td></tr><tr><td></td></tr></table>",
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+                                    "content": "P ( x ) = \\frac { c o u n t ( x \\in \\mathbf { H } ^ { * } ) } { \\sum _ { x \\in D } c o u n t ( x \\in \\mathbf { H } ^ { * } ) }",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "using different random seeds. However, in our experiments, we observe that the",
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+                                    "score": 1.0,
+                                    "content": "distribution produced by",
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+                                    "score": 0.92,
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+                                    "score": 1.0,
+                                    "content": "tends to be extremely skewed and a majority of the data actually",
+                                    "type": "text"
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+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "have zero frequencies. Such skewness causes highly imbalanced training mini-batches, and therefore",
+                                    "type": "text"
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+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "destabilizes subsequent model training.",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Distribution Smoothing To tackle the above issue, we first smooth",
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+                                    "score": 1.0,
+                                    "content": "function, and then apply a probability mixture with uniform distributions. In particular for the dataset",
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+                                    "score": 0.8,
+                                    "content": "D",
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+                                    "content": ",",
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+                                    "content": "P ^ { \\prime } ( D ) = M i x t u r e ( \\log ( P ( D ) + \\beta ) , N _ { u } \\times \\mathrm { u n i f o r m } ( D ) )",
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+                                    "score": 1.0,
+                                    "content": ". The smoothing through the log function can greatly reduce the",
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+                                    "score": 1.0,
+                                    "content": "skewness, however,",
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+                                    "score": 0.92,
+                                    "content": "\\log ( P ( D ) + \\beta )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "may still contain zero probabilities for some training data,",
+                                    "type": "text"
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+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "resulting in unstable training. Therefore, we further smooth it through a probability mixture with",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.87,
+                                    "content": "N _ { u }",
+                                    "type": "inline_equation"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "uniform distribution uniform",
+                                    "type": "text"
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+                                    "score": 0.73,
+                                    "content": "( D )",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "to ensure presence of all data. This is equivalent to combining",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.85,
+                                    "content": "N _ { u }",
+                                    "type": "inline_equation"
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+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "epochs of training data to the training batches sampled from",
+                                    "type": "text"
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+                                    "score": 0.93,
+                                    "content": "P ( D )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", and shuffling the union. Once we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "have new diverse sampling schedules for the population, we proceed to the next multi-exploitation",
+                                    "type": "text"
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+                            "index": 26
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "step.",
+                                    "type": "text"
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+                    ],
+                    "index": 23.5
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+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We continue this alternation between multi-exploitation and exploration steps until the end of training.",
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+                                    "content": "Note that to generate sampling schedule for the first multi-exploitation run, we initialize",
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+                                    "score": 0.91,
+                                    "content": "H ^ { * } = ( \\mathbf { H } _ { 1 } ^ { * } , \\cdots , \\mathbf { H } _ { n } ^ { * } )",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "alternations. The entire process is illustrated in details in Alg.2.",
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+                                    "content": "4 EXPERIMENTS",
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+                                    "content": "In this section, we present comprehensive experiments on various datasets to illustrate the perfor-",
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+                                    "score": 1.0,
+                                    "content": "mance of AutoSampling, and also demonstrate the process of progressively learning better sampling",
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+                                    "content": "distribution.",
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+                    "type": "title",
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+                                    "content": "4.1 ABLATION STUDY",
+                                    "type": "text"
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+                    ],
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+                    "type": "text",
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+                                    "content": "For this part, we gradually build up and test components of AutoSampling on CIFAR-100, and then",
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+                                    "content": "examine their performances on CIFAR-10 and ImageNet datasets. The training implementation",
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+                                    "score": 1.0,
+                                    "content": "details and computational complexity can be found in Appendix A.1.",
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+                                    "score": 1.0,
+                                    "content": "using different random seeds. However, in our experiments, we observe that the",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": ". The smoothing through the log function can greatly reduce the",
+                                    "type": "text"
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+                                    "score": 0.92,
+                                    "content": "\\log ( P ( D ) + \\beta )",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "may still contain zero probabilities for some training data,",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "resulting in unstable training. Therefore, we further smooth it through a probability mixture with",
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+                                    "content": "uniform distribution uniform",
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+                                    "content": "( D )",
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+                                    "score": 1.0,
+                                    "content": "to ensure presence of all data. This is equivalent to combining",
+                                    "type": "text"
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+                                    "content": "N _ { u }",
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+                            "index": 24
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "epochs of training data to the training batches sampled from",
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+                                    "score": 0.93,
+                                    "content": "P ( D )",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", and shuffling the union. Once we",
+                                    "type": "text"
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+                            "index": 25
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "have new diverse sampling schedules for the population, we proceed to the next multi-exploitation",
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+                                    "score": 1.0,
+                                    "content": "step.",
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+                                    "content": "We continue this alternation between multi-exploitation and exploration steps until the end of training.",
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+                                    "content": "Note that to generate sampling schedule for the first multi-exploitation run, we initialize",
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+                                    "content": "H ^ { * } = ( \\mathbf { H } _ { 1 } ^ { * } , \\cdots , \\mathbf { H } _ { n } ^ { * } )",
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+                                    "score": 1.0,
+                                    "content": "alternations. The entire process is illustrated in details in Alg.2.",
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+                                    "content": "4 EXPERIMENTS",
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+                                            "content": "Table 3: Experiments on ImageNet.",
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+                                            "html": "<table><tr><td>NETWORK</td><td>EXPLORATION TYPE</td><td>TOP1(%)</td></tr><tr><td>RESNET18</td><td>UNIFORM</td><td>70.38</td></tr><tr><td>RESNET18</td><td>RANDOM</td><td>72.07</td></tr><tr><td>RESNET18</td><td>MIXTURE</td><td>72.91</td></tr><tr><td>RESNET34</td><td>UNIFORM</td><td>74.09</td></tr><tr><td>RESNET34</td><td>RANDOM</td><td>76.11</td></tr><tr><td>RESNET34</td><td>MIXTURE</td><td>76.92</td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "Adding Workers To look into the influence of the worker numbers, we conduct experiments using",
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+                                    "content": "worker numbers of 1, 20, 80 respectively with the same setting (",
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+                                    "score": 1.0,
+                                    "content": "with random exploration).",
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+                                    "content": "With the worker number of 1, the experiment is simply the normal model training using stochastic",
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+                                    "score": 1.0,
+                                    "content": "gradient descent. To show the competitiveness of our baselines, we also include state-of-the-art",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "results on CIFAR-100 with ResNet-18 and ResNet-50 (Zhang et al., 2019; Jin et al., 2019). We",
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+                            "index": 23
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+                                    "score": 1.0,
+                                    "content": "notice significant performance gain using the worker number of 20 for ResNet-18, ResNet-50 and",
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+                            "index": 24
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+                                    "score": 1.0,
+                                    "content": "DenseNet-121 (He et al., 2015; Huang et al., 2017), as illustrated in Table 1. However, we note that",
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+                            "index": 25
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+                                    "score": 1.0,
+                                    "content": "increasing worker number from 20 to 80 only brings marginal performance gains across various",
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+                                    "score": 1.0,
+                                    "content": "model structures, as shown in Table 1. Therefore, we set the worker number to be 20 for the rest of",
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+                            "index": 27
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+                                    "score": 1.0,
+                                    "content": "the experiments.",
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+                                    "score": 1.0,
+                                    "content": "Shortening Exploitation Intervals To study the effects of the shortened exploitation interval, we",
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+                            "index": 29
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+                                    "score": 1.0,
+                                    "content": "shown in Table 1, models with the shorter exploitation interval of 20 batches(iterations) perform better",
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+                                    "score": 1.0,
+                                    "content": "than the one with the longer exploitation interval across all three network structures, conforming to",
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+                                    "score": 1.0,
+                                    "content": "our assumptions that the reward collected reflects value of each data used in the exploitation interval.",
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+                                    "score": 1.0,
+                                    "content": "This result adheres to our intuition that shorter exploitation interval can encourage the sampler to",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "accumulate more rewards to learn better sampling schedules. For the rest of this section we keep the",
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+                                    "score": 1.0,
+                                    "content": "exploitation interval of 20.",
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+                                    "score": 1.0,
+                                    "content": "Adding Exploration Type We further add mixture as the exploration type to see the effects of",
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+                            "index": 37
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+                                    "score": 1.0,
+                                    "content": "learning the underlying sampling distribution, and completing the proposed method. As shown in",
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+                                    "score": 1.0,
+                                    "content": "Table 1, with ResNet-18 and ResNet-50 we push performance higher with the mixture exploration,",
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+                                    "score": 1.0,
+                                    "content": "and outperform the baseline method by about 1 and 1.8 percentage on CIFAR-100 respectively.",
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+                            "index": 40
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "However, we found that it is not true in the case of DenseNet-121 and this case may be attributed to",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "the bigger capacity of DenseNet-121.",
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+                                    "score": 1.0,
+                                    "content": "Generalization Over Datasets In addition, we experiment on other datasets. We report the results on",
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+                                    "score": 1.0,
+                                    "content": "CIFAR10 in Table 2 and the results of ResNet-18, ResNet-34 on ImageNet in Table 3. For CIFAR-10,",
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+                                    "score": 1.0,
+                                    "content": "we notice that the mixture and random exploration methods are comparable while both outperforming",
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+                                    "score": 1.0,
+                                    "content": "the uniform baseline, and we believe it is due to the simplicity of the dataset. In the more challenging",
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+                                            "content": "Table 3: Experiments on ImageNet.",
+                                            "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Adding Workers To look into the influence of the worker numbers, we conduct experiments using",
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+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "worker numbers of 1, 20, 80 respectively with the same setting (",
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+                                    "score": 0.91,
+                                    "content": "N _ { s } = 2 0",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "with random exploration).",
+                                    "type": "text"
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+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "With the worker number of 1, the experiment is simply the normal model training using stochastic",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "gradient descent. To show the competitiveness of our baselines, we also include state-of-the-art",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "results on CIFAR-100 with ResNet-18 and ResNet-50 (Zhang et al., 2019; Jin et al., 2019). We",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "notice significant performance gain using the worker number of 20 for ResNet-18, ResNet-50 and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "DenseNet-121 (He et al., 2015; Huang et al., 2017), as illustrated in Table 1. However, we note that",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "increasing worker number from 20 to 80 only brings marginal performance gains across various",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
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+                                506
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "model structures, as shown in Table 1. Therefore, we set the worker number to be 20 for the rest of",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the experiments.",
+                                    "type": "text"
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+                            "index": 28
+                        }
+                    ],
+                    "index": 23.5,
+                    "bbox_fs": [
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+                    "type": "text",
+                    "bbox": [
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+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Shortening Exploitation Intervals To study the effects of the shortened exploitation interval, we",
+                                    "type": "text"
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+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "run experiments using different exploitation intervals of 20 and 80 batches(iterations) respectively. As",
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+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "shown in Table 1, models with the shorter exploitation interval of 20 batches(iterations) perform better",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "than the one with the longer exploitation interval across all three network structures, conforming to",
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+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "our assumptions that the reward collected reflects value of each data used in the exploitation interval.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "This result adheres to our intuition that shorter exploitation interval can encourage the sampler to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "accumulate more rewards to learn better sampling schedules. For the rest of this section we keep the",
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+                                    "content": "Table 1, with ResNet-18 and ResNet-50 we push performance higher with the mixture exploration,",
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+                                    "score": 1.0,
+                                    "content": "and outperform the baseline method by about 1 and 1.8 percentage on CIFAR-100 respectively.",
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+                                    "score": 1.0,
+                                    "content": "However, we found that it is not true in the case of DenseNet-121 and this case may be attributed to",
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+                                    "score": 1.0,
+                                    "content": "the bigger capacity of DenseNet-121.",
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+                                    "content": "Generalization Over Datasets In addition, we experiment on other datasets. We report the results on",
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+                                    "content": "CIFAR10 in Table 2 and the results of ResNet-18, ResNet-34 on ImageNet in Table 3. For CIFAR-10,",
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+                                    "score": 1.0,
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+                                            "html": "<table><tr><td rowspan=\"2\">NETWORK</td><td colspan=\"3\">SAMPLING TYPE</td></tr><tr><td>UNIFORM</td><td>STATIC</td><td>DYNAMIC</td></tr><tr><td>RESNET18</td><td>78.46±0.035</td><td>78.80±0.007</td><td>79.44±0.020</td></tr><tr><td>RESNET50</td><td>79.70±0.023</td><td>80.21±0.014</td><td>81.53±0.088</td></tr></table>",
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+                                    "content": "compare our AutoSampling with some recent non-uniform sampling methods on CIFAR-100, which",
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+                                    "content": "can be found in Appendix A.2.",
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+                                    "content": "either a process of learning a robust sampling distribution, or a process of dynamically adjusting the",
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+                                    "score": 1.0,
+                                    "content": "sampling schedule for optimal training. To this end, we conduct training using different sampling",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "schedules. First, we calculate the sampling distribution estimated throughout the learning steps of",
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+                                    "score": 1.0,
+                                    "content": "AutoSampling, and use it to generate the sampling schedule of a full training process, which we",
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+                                    "score": 1.0,
+                                    "content": "denote as STATIC. Moreover, we denote the sampling schedule learned using AutoSampling as",
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+                        },
+                        {
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+                                    "content": "DYNAMIC, since AutoSampling dynamically adjust the sampling schedule alongside the training",
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+                                    "score": 1.0,
+                                    "content": "process. Finally, we denote the baseline method as UNIFORM, which uses the sampling schedule",
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+                                    "content": "We report results on CIFAR-100 with ResNet-18 and ResNet-50 in Table 4. Model trained with",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "to learn good sampling distribution. Nonetheless, note that models trained with DYNAMIC sampling",
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+                                    "score": 1.0,
+                                    "content": "and UNIFORM. This result shows the fact that despite the AutoSampling’s capability of learning good",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "sampling distribution, its flexibility during training matters even more. Moreover, this phenomenon",
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+                            "index": 32
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+                        {
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+                                    "content": "also indicates that models at different stages of learning process may require different sampling",
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+                                    "score": 1.0,
+                                    "content": "using AutoSampling, seems incapable of covering the needs from different learning stages. We",
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+                        {
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+                                    "content": "plot the histograms of data counts in training estimated from schedules of different learning stages",
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+                                    "score": 1.0,
+                                    "content": "with ResNet-18 on CIFAR-100 in Fig.2, showing the great differences between optimized sampling",
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+                            ],
+                            "index": 37
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+                                    "score": 1.0,
+                                    "content": "distributions from different epochs.",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "are extracted, corresponding to high and low probabilities of being sampled respectively. In Fig.3, we",
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+                        {
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+                                            "content": "-axis based on the ranking of data counts for epoch 240 for easier comparison.",
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "We aim to see if the final sampling distribution estimated by our AutoSampling is sufficient to produce",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                                414
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        401,
+                                        506,
+                                        414
+                                    ],
+                                    "score": 1.0,
+                                    "content": "robust sampling schedules. In another word, we wish to know training with the AutoSampling is",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
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+                                425
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        506,
+                                        425
+                                    ],
+                                    "score": 1.0,
+                                    "content": "either a process of learning a robust sampling distribution, or a process of dynamically adjusting the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        422,
+                                        506,
+                                        437
+                                    ],
+                                    "score": 1.0,
+                                    "content": "sampling schedule for optimal training. To this end, we conduct training using different sampling",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                                446
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        433,
+                                        506,
+                                        446
+                                    ],
+                                    "score": 1.0,
+                                    "content": "schedules. First, we calculate the sampling distribution estimated throughout the learning steps of",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                                457
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        444,
+                                        506,
+                                        457
+                                    ],
+                                    "score": 1.0,
+                                    "content": "AutoSampling, and use it to generate the sampling schedule of a full training process, which we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                                454,
+                                506,
+                                469
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        454,
+                                        506,
+                                        469
+                                    ],
+                                    "score": 1.0,
+                                    "content": "denote as STATIC. Moreover, we denote the sampling schedule learned using AutoSampling as",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                480
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        466,
+                                        505,
+                                        480
+                                    ],
+                                    "score": 1.0,
+                                    "content": "DYNAMIC, since AutoSampling dynamically adjust the sampling schedule alongside the training",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                                491
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        506,
+                                        491
+                                    ],
+                                    "score": 1.0,
+                                    "content": "process. Finally, we denote the baseline method as UNIFORM, which uses the sampling schedule",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
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+                                489,
+                                254,
+                                500
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        489,
+                                        254,
+                                        500
+                                    ],
+                                    "score": 1.0,
+                                    "content": "generated from uniform distribution.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        }
+                    ],
+                    "index": 20.5,
+                    "bbox_fs": [
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+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We report results on CIFAR-100 with ResNet-18 and ResNet-50 in Table 4. Model trained with",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "STATIC sampling schedules exceeds the baseline UNIFORM significantly, indicating the superiority of",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "the learned sampling distribution over the uniform distribution. It shows the ability of AutoSampling",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "to learn good sampling distribution. Nonetheless, note that models trained with DYNAMIC sampling",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "schedules outperform models trained with STATIC, by a margin bigger than the one between STATIC",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        573
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+                                    "score": 1.0,
+                                    "content": "and UNIFORM. This result shows the fact that despite the AutoSampling’s capability of learning good",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        584
+                                    ],
+                                    "score": 1.0,
+                                    "content": "sampling distribution, its flexibility during training matters even more. Moreover, this phenomenon",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "also indicates that models at different stages of learning process may require different sampling",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "distributions to achieve optimal training. One single sampling distribution, even gradually estimated",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        617
+                                    ],
+                                    "score": 1.0,
+                                    "content": "using AutoSampling, seems incapable of covering the needs from different learning stages. We",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
+                        },
+                        {
+                            "bbox": [
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+                                629
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        614,
+                                        506,
+                                        629
+                                    ],
+                                    "score": 1.0,
+                                    "content": "plot the histograms of data counts in training estimated from schedules of different learning stages",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        624,
+                                        505,
+                                        640
+                                    ],
+                                    "score": 1.0,
+                                    "content": "with ResNet-18 on CIFAR-100 in Fig.2, showing the great differences between optimized sampling",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        650
+                                    ],
+                                    "score": 1.0,
+                                    "content": "distributions from different epochs.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        }
+                    ],
+                    "index": 32,
+                    "bbox_fs": [
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+                },
+                {
+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "4.3 ANALYZING SAMPLING SCHEDULES LEARNED BY AUTOSAMPLING",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "To further investigate the sampling schedule learned by AutoSampling, we review the images at the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "tail and head part of the sampling spectrum. In particular, given a sampling schedule learned we rank",
+                                    "type": "text"
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+                            ],
+                            "index": 41
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "all images based on their appearances in training. Training images at the top and bottom of the order",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "are extracted, corresponding to high and low probabilities of being sampled respectively. In Fig.3, we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "show 4 classes of exemplary images. The images of low probability tend to have clearer imagery",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 44
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "features enabling easy recognition, while the images of high probability tend to be more obscure,",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 17
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+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "indicating that the sampling schedule may show hard samples mining effects. However, as shown in",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
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+                            "index": 18
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+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "A.3 and Fig. 4, the loss values and probabilities of being sampled seem to be not highly correlated,",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "which indicates more potential of AutoSampling beyond visually hard example mining. In addition,",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
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+                            "index": 20
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "we notice the images of low probability also contain low quality images. For instance, in Fig.3 the",
+                                    "type": "text",
+                                    "cross_page": true
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+                            "index": 21
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+                        {
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+                                    "score": 1.0,
+                                    "content": "leftmost image of CAMAL class contains only legs. This shows that AutoSampling may potentially",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
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+                            "index": 22
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+                        {
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+                                    "score": 1.0,
+                                    "content": "rule out problematic training data for better training.",
+                                    "type": "text",
+                                    "cross_page": true
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+                                            "content": "( \\% )",
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+                                            "score": 1.0,
+                                            "content": ". UNIFORM denotes the baseline result using uniform sampling distribution.",
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+                                            "html": "<table><tr><td rowspan=\"2\">NETWORK</td><td colspan=\"4\">SAMPLING SCHEDULE SOURCE</td></tr><tr><td>UNIFORM</td><td>RESNET18</td><td>RESNET50</td><td>DENSENET121</td></tr><tr><td>RESNET50</td><td>79.70±0.023</td><td>80.27±0.014</td><td>80.21±0.014</td><td>80.47±0.194</td></tr></table>",
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+                                    "content": "features enabling easy recognition, while the images of high probability tend to be more obscure,",
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+                                    "content": "indicating that the sampling schedule may show hard samples mining effects. However, as shown in",
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+                                    "content": "A.3 and Fig. 4, the loss values and probabilities of being sampled seem to be not highly correlated,",
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+                                    "content": "which indicates more potential of AutoSampling beyond visually hard example mining. In addition,",
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+                                    "content": "we notice the images of low probability also contain low quality images. For instance, in Fig.3 the",
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+                                    "content": "leftmost image of CAMAL class contains only legs. This shows that AutoSampling may potentially",
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+                                    "content": "demonstrate similar improvements over the UNIFORM baseline. This result, combined with the above",
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+                                    "content": "observations on images of different sampling probability, indicates that there may exist a common",
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+                                    "content": "being optimized. Our AutoSampling is an effort to gradually converge to such an optimal schedule.",
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+                                    "content": "The experimental results and observations from Section 4.2 and 4.3 shed light on the possible",
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+                                    "content": "existence of an optimal sampling schedule, which relies only on the intrinsic property of the data and",
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+                                    "score": 1.0,
+                                    "content": "the learning stage of the model, regardless of the specific model structure or any prior knowledge. The",
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+                                    "score": 1.0,
+                                    "content": "convergence compared to other related works. Once obtained, the optimal sampling schedule may",
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+                                    "content": "also be generalized over other model structures for robust training. Although AutoSampling requires",
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+                                    "score": 1.0,
+                                    "content": "based on its property.",
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+                                    "content": "5 CONCLUSIONS",
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+                                            "score": 1.0,
+                                            "content": ". UNIFORM denotes the baseline result using uniform sampling distribution.",
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+                                    "content": "demonstrate similar improvements over the UNIFORM baseline. This result, combined with the above",
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+                                    "content": "being optimized. Our AutoSampling is an effort to gradually converge to such an optimal schedule.",
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+                                    "score": 1.0,
+                                    "content": "efficient sampling policy search algorithms, if one can quickly and effectively determine data value",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "based on its property.",
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+                                    "content": "5 CONCLUSIONS",
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+                                    "score": 1.0,
+                                    "content": "Gary M Weiss, Kate McCarthy, and Bibi Zabar. Cost-sensitive learning vs. sampling: Which is best",
+                                    "type": "text"
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+                        },
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+                                    "content": "for handling unbalanced classes with unequal error costs? Dmin, 7(35-41):24, 2007.",
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+                                    "content": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. CoRR, abs/1605.07146, 2016.",
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+                                    "score": 1.0,
+                                    "content": "Michael R. Zhang, James Lucas, Geoffrey E. Hinton, and Jimmy Ba. Lookahead optimizer: k steps",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
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+                                    "content": "forward, 1 step back. CoRR, abs/1907.08610, 2019. URL http://arxiv.org/abs/1907.",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "ByxdUySKvS.",
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+                    "bbox_fs": [
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+            "preproc_blocks": [
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+                                    "score": 1.0,
+                                    "content": "A APPENDIX",
+                                    "type": "text"
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+                            "index": 0
+                        }
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+                    "index": 0
+                },
+                {
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+                                    "content": "A.1 IMPLEMENTATION DETAILS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 1
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "score": 1.0,
+                                    "content": "Experiments on CIFAR We use the same training configuration for both CIFAR-100 and CIFAR-10",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                        151
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+                                    "score": 1.0,
+                                    "content": "datasets, which both consist of 50000 training images. In particular, for model training we use the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                        162
+                                    ],
+                                    "score": 1.0,
+                                    "content": "base learning rate of 0.1 and a step decay learning rate schedule where the learning rate is divided",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        105,
+                                        159,
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+                                        174
+                                    ],
+                                    "score": 1.0,
+                                    "content": "by 10 after each 60 epochs. We run the experiments for 240 epochs. In addition, we set the training",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                        443,
+                                        183
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+                                    "score": 1.0,
+                                    "content": "batch size to be 128 per worker, and each worker is for one Nvidia V100 GPU card.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "index": 4
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+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "We run the explore step for each",
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+                                    "content": "N _ { u } + 1",
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+                                    "content": "N _ { u } = 3",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", but note that we take the first explore",
+                                    "type": "text"
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+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "step after the initial 20 epochs to better accumulate enough rewards. The experiments require 4800",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                                210,
+                                399,
+                                223
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "epochs of training for 20 workers, and roughly 14 hours of training time.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Experiments on ImageNet For ImageNet which consists of 1.28 million training images, we adopted",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                        250
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+                                    "score": 1.0,
+                                    "content": "the base learning rate of 0.2 and a cosine decay learning rate schedule. We run the experiments with",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                        261
+                                    ],
+                                    "score": 1.0,
+                                    "content": "100 epochs of training. For each worker we utilize eight Nvidia V100 GPU cards and a total batch",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "size of 512. Eight workers are used for all ImageNet experiments, and the rest of the setting adheres",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                                284
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        284
+                                    ],
+                                    "score": 1.0,
+                                    "content": "to that of CIFAR experiments. In addition, we utilize FP16 computation to achieve faster training,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                                295
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        295
+                                    ],
+                                    "score": 1.0,
+                                    "content": "which has almost no drop in accuracy in practice. The experiments require 800 epochs of training for",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        292,
+                                        297,
+                                        306
+                                    ],
+                                    "score": 1.0,
+                                    "content": "8 workers, and roughly 4 days of training time.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "type": "title",
+                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "A.2 COMPARISON WITH EXISTING SAMPLING METHODS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "To better illustrate the effectiveness of our AutoSampling method, we conduct experiments in",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        362
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+                                    "score": 1.0,
+                                    "content": "comparison with recent non-uniform sampling methods DLIS (Johnson & Guestrin, 2018) and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                                506,
+                                374
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        361,
+                                        506,
+                                        374
+                                    ],
+                                    "score": 1.0,
+                                    "content": "RAIS (Katharopoulos & Fleuret, 2018). DLIS (Johnson & Guestrin, 2018) achieves faster conver-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                371,
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+                                386
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        371,
+                                        507,
+                                        386
+                                    ],
+                                    "score": 1.0,
+                                    "content": "gence by selecting data reducing gradient norm variance, while RAIS (Katharopoulos & Fleuret,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                                396
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        382,
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+                                        396
+                                    ],
+                                    "score": 1.0,
+                                    "content": "2018) does so through approximating the ideal sampling distribution using robust optimization. The",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                394,
+                                249,
+                                406
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        394,
+                                        249,
+                                        406
+                                    ],
+                                    "score": 1.0,
+                                    "content": "comparison is recorded in Table 6.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        }
+                    ],
+                    "index": 20.5
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                        505,
+                        510
+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                423
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "First, we run AutoSampling using Wide Resnet-28-2 (Zagoruyko & Komodakis, 2016) on CIFAR-100",
+                                    "type": "text"
+                                }
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+                                    "content": "with the training setting aligned roughly to (Katharopoulos & Fleuret, 2018). AutoSampling achievs",
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+                                    "content": "( 7 3 . 3 7 \\pm 1 . 0 9 \\%  7 6 . 2 4 \\pm 1 . 0 2 \\% )",
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+                                    "score": 1.0,
+                                    "content": ", while Katharopoulos",
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+                                    "score": 1.0,
+                                    "content": "& Fleuret shows improvement of 2 percentage points",
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+                                    "score": 1.0,
+                                    "content": "). Second, we report the",
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+                                    "score": 1.0,
+                                    "content": "comparison between AutoSampling and RAIS on CIFAR-100. Johnson & Guestrin shows no",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "0.872 ) decrease in validation loss. As such, our method demonstrates significant improvements over",
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+                                    "score": 1.0,
+                                    "content": "existing non-uniform sampling methods.",
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+                                            "score": 1.0,
+                                            "content": "Table 6: Comparisons between AutoSampling and existing sampling methods on CIFAR-100",
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+                                            "html": "<table><tr><td>Methods</td><td>Network</td><td>Baseline (%)</td><td>With method (%)</td><td>Improvement (%)</td></tr><tr><td>DLIS</td><td>WRN-28-2</td><td>66.0</td><td>68.0</td><td>2.0</td></tr><tr><td>AutoSampling (ours)</td><td>WRN-28-2</td><td>73.37±1.09</td><td>76.24±1.02</td><td>2.87</td></tr><tr><td>RAIS</td><td>ResNet18</td><td>76.4</td><td>76.4</td><td>0.0</td></tr><tr><td>AutoSampling (ours)</td><td>ResNet18</td><td>78.46±0.035</td><td>79.44±0.020</td><td>0.98</td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "To further interpret the learned sampling schedules, we compare the sampling frequency of each",
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+                                    "score": 1.0,
+                                    "content": "We draw the comparison for randomly selected 500 training images in Fig. 4 for epoch 80, 160, and",
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+                                    "score": 1.0,
+                                    "content": "240. As shown in the figure, across different learning stages, the correlation between loss values",
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+                                    "score": 1.0,
+                                    "content": "and sampling frequencies of training data is not obvious. The high chance of being sampled by",
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+                                    "score": 1.0,
+                                    "content": "AutoSampling does not necessarily lead to high loss values, which demonstrates that AutoSampling",
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+                                    "content": "Experiments on CIFAR We use the same training configuration for both CIFAR-100 and CIFAR-10",
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+                                    "score": 1.0,
+                                    "content": "datasets, which both consist of 50000 training images. In particular, for model training we use the",
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+                                    "score": 1.0,
+                                    "content": "base learning rate of 0.1 and a step decay learning rate schedule where the learning rate is divided",
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+                                    "score": 1.0,
+                                    "content": "by 10 after each 60 epochs. We run the experiments for 240 epochs. In addition, we set the training",
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+                                    "content": "batch size to be 128 per worker, and each worker is for one Nvidia V100 GPU card.",
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+                                    "content": ", but note that we take the first explore",
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+                                    "score": 1.0,
+                                    "content": "step after the initial 20 epochs to better accumulate enough rewards. The experiments require 4800",
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+                                    "score": 1.0,
+                                    "content": "epochs of training for 20 workers, and roughly 14 hours of training time.",
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+                                    "content": "size of 512. Eight workers are used for all ImageNet experiments, and the rest of the setting adheres",
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+                                    "score": 1.0,
+                                    "content": "to that of CIFAR experiments. In addition, we utilize FP16 computation to achieve faster training,",
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+                                    "score": 1.0,
+                                    "content": "which has almost no drop in accuracy in practice. The experiments require 800 epochs of training for",
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+                                    "score": 1.0,
+                                    "content": "To better illustrate the effectiveness of our AutoSampling method, we conduct experiments in",
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+                                    "content": "comparison with recent non-uniform sampling methods DLIS (Johnson & Guestrin, 2018) and",
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+                                    "score": 1.0,
+                                    "content": "2018) does so through approximating the ideal sampling distribution using robust optimization. The",
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+                                    "score": 1.0,
+                                    "content": "comparison is recorded in Table 6.",
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+                                    "score": 1.0,
+                                    "content": "First, we run AutoSampling using Wide Resnet-28-2 (Zagoruyko & Komodakis, 2016) on CIFAR-100",
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+                                    "score": 1.0,
+                                    "content": "with the training setting aligned roughly to (Katharopoulos & Fleuret, 2018). AutoSampling achievs",
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+                                    "score": 1.0,
+                                    "content": "improvement of roughly 3 percentage points",
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+                                    "score": 0.89,
+                                    "content": "( 7 3 . 3 7 \\pm 1 . 0 9 \\%  7 6 . 2 4 \\pm 1 . 0 2 \\% )",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", while Katharopoulos",
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+                                    "score": 1.0,
+                                    "content": "& Fleuret shows improvement of 2 percentage points",
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+                                    "content": "( 6 6 . 0 \\%  6 8 . 0 \\%",
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+                                    "score": 1.0,
+                                    "content": "). Second, we report the",
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+                                    "score": 1.0,
+                                    "content": "comparison between AutoSampling and RAIS on CIFAR-100. Johnson & Guestrin shows no",
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+                                    "score": 0.92,
+                                    "content": "7 6 . 4 \\%  7 6 . 4 \\%",
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+                                    "score": 1.0,
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+                                    "score": 0.87,
+                                    "content": "( 0 . 9 8 9  0 . 9 6 2",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 0.88,
+                                    "content": "7 8 . 6 \\%  7 9 . 4 \\% )",
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+                                    "score": 1.0,
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+                                    "score": 0.79,
+                                    "content": "0 . 8 8 6 ",
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+                                    "score": 1.0,
+                                    "content": "0.872 ) decrease in validation loss. As such, our method demonstrates significant improvements over",
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+                                    "score": 1.0,
+                                    "content": "existing non-uniform sampling methods.",
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+                                            "content": "Table 6: Comparisons between AutoSampling and existing sampling methods on CIFAR-100",
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+                                            "html": "<table><tr><td>Methods</td><td>Network</td><td>Baseline (%)</td><td>With method (%)</td><td>Improvement (%)</td></tr><tr><td>DLIS</td><td>WRN-28-2</td><td>66.0</td><td>68.0</td><td>2.0</td></tr><tr><td>AutoSampling (ours)</td><td>WRN-28-2</td><td>73.37±1.09</td><td>76.24±1.02</td><td>2.87</td></tr><tr><td>RAIS</td><td>ResNet18</td><td>76.4</td><td>76.4</td><td>0.0</td></tr><tr><td>AutoSampling (ours)</td><td>ResNet18</td><td>78.46±0.035</td><td>79.44±0.020</td><td>0.98</td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "To further interpret the learned sampling schedules, we compare the sampling frequency of each",
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+                                    "content": "training image and its loss values in different epochs during training of CIFAR-100 with ResNet-18.",
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+                                    "score": 1.0,
+                                    "content": "We draw the comparison for randomly selected 500 training images in Fig. 4 for epoch 80, 160, and",
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+                                    "content": "240. As shown in the figure, across different learning stages, the correlation between loss values",
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+                                    "score": 1.0,
+                                    "content": "and sampling frequencies of training data is not obvious. The high chance of being sampled by",
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diff --git a/parse/train/AJTAcS7SZzf/AJTAcS7SZzf_model.json b/parse/train/AJTAcS7SZzf/AJTAcS7SZzf_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..f6a2a9ec5e5e8942b46ae0869822a1d767e8eaa0
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index 0000000000000000000000000000000000000000..aedd177a21f78f9a55f1239188d6618ca1bc5bb8
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@@ -0,0 +1,190 @@
+# PROTOTYPICAL NETWORKS FOR FEW-SHOT LEARNING
+
+Jake Snell1∗, Kevin Swersky2& Richard S. Zemel1   
+1University of Toronto   
+2Twitter
+
+# ABSTRACT
+
+A recent approach to few-shot classification called matching networks has demonstrated the benefits of coupling metric learning with a training procedure that mimics test. This approach relies on an attention scheme that forms a distribution over all points in the support set, scaling poorly with its size. We propose a more streamlined approach, prototypical networks, that learns a metric space in which few-shot classification can be performed by computing Euclidean distances to prototype representations of each class, rather than individual points. Our method is competitive with state-of-the-art few-shot classification approaches while being much simpler and more scalable with the size of the support set. We empirically demonstrate the performance of our approach on the Omniglot and miniImageNet datasets. We further demonstrate that a similar idea can be used for zero-shot learning, where each class is described by a set of attributes, and achieve state-ofthe-art results on the Caltech UCSD bird dataset.
+
+# 1 INTRODUCTION
+
+One-shot classification (Miller et al., 2000; Lake et al., 2011; Koch, 2015) (and more generally, fewshot classification) is a problem in which a classifier must be adapted to accommodate new classes not seen in training, given only a single $( n )$ example(s) of these classes. A classical approach, such as retraining the model on the new data, would severely overfit. While the problem is quite difficult, it has been demonstrated that people have the ability to successfully perform one-shot classification (Lake et al., 2011). Nonparametric models such as nearest neighbors are useful in one-shot classification because they naturally adapt to new data, however this comes at the cost of storing the entire set of examples per class, the “support set”.
+
+To overcome this, much progress has been made recently in applying metric learning (Goldberger et al., 2004; Kulis, 2012; Bellet et al., 2013) to one-shot tasks. Most recently, (Vinyals et al., 2016) proposed a metric learning approach that they call matching networks. This approach uses an attention mechanism over a learned embedding of the support set in order to predict class labels for the points to be classified, a.k.a the “query set”. It optionally allows the embeddings to be conditioned on other points in the support set (“full context embeddings”) or for the embeddings to be fine-tuned at test time. A particularly interesting feature of the matching networks model is that it utilizes sampled mini-batches called “episodes” during training, where each episode is designed to mimic the one-shot task. This makes the training problem more faithful to the test environment. Matching networks however optionally utilize additional components such as an attention-based LSTM to change the embedding based on the support set. This complexity makes implementation more difficult in addition to the aforementioned poor scaling characteristics due to computing attention over the entire support set.
+
+In this paper, we propose a few-shot learning classifier based on a relatively simple idea: there exists an embedding, where points belonging to a class cluster around a single prototype. This inductive bias is a useful one to combat overfitting for one-shot tasks. Our approach also comes with the benefit that it is very simple to implement, and computationally fast. In order to do this, we learn a non-linear mapping of the input into an embedding space using a neural network, and take the class prototype to be the mean of the support set in the embedding space. Classification is then performed by simply finding the nearest prototype to the embedded query point. We find that this approach yields competitive results with matching networks and other one-shot learning approaches, despite being much simpler.
+
+A related problem is known as zero-shot learning, where instead of being given a small number of examples of a new class at test-time, each class comes with a set of meta-information, often attributes, that give a high level description of that class. The idea then is to learn a mapping from input examples to the high-level attributes of their member class. We adapt the idea of prototypical networks to this setting by learning a secondary embedding of the attribute vector such that the image embeddings and attribute embeddings lie within the same space. In this case, we use the attribute embedding as the class prototype, rather than the class mean.
+
+# 2 RELATED WORK
+
+Neighborhood Components Analysis (NCA) (Goldberger et al., 2004) learns a Mahalanobis distance to maximize K-nearest-neighbour’s (KNN) leave-one-out accuracy in the transformed space. A distribution over the neighbors of each data point is computed according to a softmax over the corresponding Mahalanobis distances. This distribution is marginalized to form a distribution over class assignments and the projection matrix is updated via gradient descent to maximize the probability of the true class. (Salakhutdinov & Hinton, 2007) extend NCA by using a neural network to perform the transformation. Our approach is similar in that we optimize a softmax based on distances in the transformed space. Ours differs because it is a softmax over classes, rather than points, computed from Euclidean distances to each class’s prototype representation. This is more appropriate for few-shot learning for two reasons: (a) the number of support points can vary by class, and (b) each class has a succinct representation independent of the number of data points, and this representation can optionally be updated in an online manner.
+
+Our approach is similar to the nearest class mean approach of (Mensink et al., 2013) from the metric learning literature, where each class is represented by the mean of its examples, and classification is performed by finding the prototype that is closest to the query point. Their approach was developed to rapidly incorporate new classes into a classifier without retraining, however it relies on a linear embedding and is designed to handle the case where the novel classes come with many examples. In our approach, we utilize neural networks to learn a non-linear embedding of the features and we couple this with episodic training in order to handle the one-shot scenario. Mensink et al. do attempt to extend their approach to perform non-linear classification, but they do this by allowing classes to have multiple prototypes. They find these prototypes in a pre-processing step by using $\mathbf { k }$ -means on the input space, and then perform a multi-modal variant of their linear embedding. By contrast, we learn a non-linear embedding in an end-to-end manner with no such pre-processing, producing a non-linear classifier that still only requires one prototype per class.
+
+In matching networks (Vinyals et al., 2016) they propose a meta-learning strategy in which training mimics test by stochastically creating one-shot “episodes”. We adopt the same strategy when training our models. They, like us, use neural networks to non-linearly transform data points into a space that is more amenable to classification. However, matching networks make predictions by computing attention weights over each point in the support set. This becomes computationally expensive as the size of the support set grows. Our approach, on the other hand, first summarizes each class in the support set by a prototype and then computes a distribution over classes. Ours thus has flexibility in the way the prototypes are computed and can handle additional support points gracefully by updating prototypes online.
+
+The neural statistician (Edwards & Storkey, 2016) extends the variational autoencoder (Kingma & Welling, 2013) to learn generative models of datasets rather than individual points. One component of the neural statistician is the “statistic network” which summarizes a set of data points into a statistic vector. It does this by encoding each point within a dataset, taking a sample mean, and applying a post-processing network to obtain an approximate posterior over the statistic vector. Edwards & Storkey test their model for one-shot classification on the Omniglot dataset (Lake et al., 2011) by considering each character to be a separate dataset and making predictions based on the class whose approximate posterior over the statistic vector had minimal KL-divergence from the test point. Like the neural statistician, we also produce a summary statistic for each class. However, ours is a discriminative model which is more appropriate because our primary task, one-shot learning, is also discriminative. Discriminative training has the added benefit of lending our model more flexibility in both the way we compute summary statistics and use them to make predictions at test time.
+
+There are many other approaches to one-shot learning that employ very different techniques from ours. Koch uses siamese networks to predict the probability that two images belong to the same class. Lake et al. devise a hierarchical Bayesian generative model of how a handwritten character is created in order to perform one-shot learning on the Omniglot dataset. Santoro et al. propose memory augmented neural networks (MANN) that reference an external memory in a similar fashion to neural Turing machines (Graves et al., 2014). This allows them to store support examples in an external memory and reference them later when making classification decisions. They also introduce a form of episodic training, similar to that in matching networks.
+
+# 3 PROTOTYPICAL NETWORKS
+
+At prediction time we are given a support set of $N$ labeled examples: ${ \cal { S } } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N } = { \cal { S } } ^ { 1 } \cup . . . \cup$ $S ^ { K }$ where $S ^ { k } = \{ ( x , y ) \in S | y = k \}$ . Our method computes a class representation $c _ { k }$ , or prototype, of each class through an embedding function $f _ { \theta } ( x )$ parameterized by learnable parameters $\theta$ :
+
+$$
+c _ { k } = \frac { 1 } { | S ^ { k } | } \sum _ { ( x , y ) \in S ^ { k } } f _ { \theta } ( x )
+$$
+
+Given a test point $\tilde { x }$ , prototypical networks forms a distribution over classes based on a softmax over the Euclidean distances between its embedding and the prototypes:
+
+$$
+p ( y = k | \tilde { x } ) = \frac { \exp ( - \| f _ { \theta } ( \tilde { x } ) - c _ { k } \| ^ { 2 } ) } { \sum _ { k ^ { \prime } } \exp ( - \| f _ { \theta } ( \tilde { x } ) - c _ { k } ^ { \prime } \| ^ { 2 } ) }
+$$
+
+Learning proceeds by maximizing the log-probability of the true class $\tilde { y }$ :
+
+$$
+\operatorname* { m a x } _ { \theta } \log p ( \tilde { y } | \tilde { x } )
+$$
+
+We train in an episodic manner similar to Vinyals et al. (2016) by randomly selecting a subset of classes from the training set, then choosing a subset of examples within each class to act as the support set and the remainder to serve as test points.
+
+# 3.1 PROTOTYPE NORMALIZATION
+
+In episodic training, the support set is randomly chosen from among the training points. In datasets with high variability this can lead to a large variance in the class prototypes, $c$ , between episodes. In order to reduce this variability, we found that it can sometimes be beneficial to normalize the prototypes, $\begin{array} { r } { \bar { c } _ { k } = \frac { c _ { k } } { \| c _ { k } \| } } \end{array}$ and use $\bar { c } _ { k }$ in place of $c _ { k }$ in Equation (2). This ensures that the prototypes always lie on the unit sphere, although the query points are still allowed to be embedded off of the unit sphere. Normalization has two benefits: the reduction in variance helps to greatly speed up training, while the restriction of the prototypes to the unit sphere confers additional regularization.
+
+# 3.2 PREDICTING THE WEIGHTS OF A LINEAR CLASSIFIER
+
+A simple analysis is useful in gaining insight into the nature of the learned classifier (a similar analysis appears in Mensink et al. (2013)). When we use Euclidean distance to measure the distance between a query point and the class prototypes, then the loss function in (2) is equivalent to a linear classifier with a particular parameterization. To see this, we expand the term within the exponent:
+
+$$
+\begin{array} { c } { { - \| f _ { \theta } ( \tilde { x } ) - c _ { k } \| ^ { 2 } = - ( f _ { \theta } ( \tilde { x } ) - c _ { k } ) ^ { \top } ( f _ { \theta } ( \tilde { x } ) - c _ { k } ) } } \\ { { = - f _ { \theta } ( \tilde { x } ) ^ { \top } f _ { \theta } ( \tilde { x } ) + 2 c _ { k } ^ { \top } f _ { \theta } ( \tilde { x } ) ^ { \top } - c _ { k } ^ { \top } c _ { k } } } \end{array}
+$$
+
+The first term in Equation (4) is constant with respect to the class $k$ , so it does not affect the softmax probabilities. We can write the remaining terms as a linear classifier as follows:
+
+$$
+\begin{array} { r l } & { 2 c _ { k } ^ { \top } f _ { \theta } ( \tilde { x } ) - c _ { k } ^ { \top } c _ { k } = w _ { k } ^ { \top } f _ { \theta } ( \tilde { x } ) + b _ { k } } \\ & { \qquad w _ { k } = 2 c _ { k } } \\ & { \qquad b _ { k } = - c _ { k } ^ { \top } c _ { k } } \end{array}
+$$
+
+We can view this through the lens of meta-learning, where the model is predicting the weights and biases of a linear classifier using a simple function of the mean of the embedded support set. By contrast, the predictive function in matching networks is a generalization of a nearest neighbor classifier, rather than a linear classifier.
+
+When using prototype normalization, the biases $b _ { k }$ will all be 1, and the class weights $w _ { k }$ will be restricted to have a norm of 2. In this case, using Euclidean distance becomes proportional to cosine distance.
+
+A natural question is whether it makes sense to use multiple prototypes per class instead of just one. If each support point were to be considered a prototype, then this would be analogous to doing nearest neighbor classification in the embedding space, which would be computationally expensive. On the other hand, if the number of prototypes per class is fixed, then this would require a partitioning scheme. This has been proposed in Mensink et al. (2013) and Rippel et al. (2016), however both methods require a separate partitioning phase that is decoupled from the weight updates, while our approach is simple to learn with ordinary gradient methods. Finally, the equivalence to a linear classifier suggests that this may be sufficient, as all of the required non-linearity can be learned within the embedding function. Indeed, this is the approach that state-of-the-art neural network classification systems currently use, e.g., (Krizhevsky et al., 2012).
+
+# 3.3 DESIGN CHOICES
+
+There are still a number of design choices that need to be made with this model in order to achieve optimal performance. One such choice is in deciding how many classes we would like the classifier to operate over during each training episode. For example, at test time we might be evaluating on 5-way classification, but at training time we could train each episode with 20-way classification. We found in general that training on a larger number of classes per episode improves performance, even if the number of classes we need to decide between at test-time is fewer.
+
+Another choice involves the possible decoupling of the $n$ in $n$ -shot between training and testing. We could train on 1-shot, but test on 5-shot or vice-versa. We found that it is typically better to match the shot at training and testing; that is, when it comes to the shot, to match the training procedure to the test procedure. We demonstrate this empirically in the Experiments section below.
+
+Finally, we need to specify whether to use prototype normalization. We found that normalization generally acts as a regularizer and speeds up training.
+
+# 4 EXPERIMENTS
+
+For few-shot learning, we performed experiments on Omniglot (Lake et al., 2011), the miniImagenet version of ILSVRC-2012 (Russakovsky et al., 2015) proposed by (Vinyals et al., 2016). For zeroshot learning, we perform experiments on the 2011 version of the Caltech UCSD bird dataset (CUB200 2011) (Welinder et al., 2010).
+
+# 4.1 OMNIGLOT
+
+Omniglot (Lake et al., 2011) is a dataset of 1623 handwritten characters collected from 50 alphabets. There are 20 examples associated with each character, where each example was drawn by a different human subject. We follow the procedure of (Vinyals et al., 2016) by augmenting the characters with rotations in multiples of 90 degrees and using 1200 characters for training and the remainder for evaluation. Our embedding architecture mirrors that of Matching Nets and is composed of four blocks of a 64 filter $3 \times 3$ convolution, batch normalization (Ioffe & Szegedy, 2015), a ReLU nonlinearity and a $2 \times 2$ max-pooling, resulting in a 64-dimensional output space. The results of our model trained to perform Omniglot classification are shown in Table 1.
+
+We trained prototypical networks using episodes designed for 1-shot learning, i.e., the support sets during training consist of a single input example, and we train using 20-way classification. Our results are as good or better than those reported in matching networks, and to our knowledge represent the state-of-the-art on this dataset using these splits.
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">5-way</td><td colspan="2">20-way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Pixels</td><td>41.7%</td><td>63.2%</td><td>26.7%</td><td>42.6%</td></tr><tr><td>Baseline Classifier</td><td>80.0%</td><td>95.0%</td><td>69.5%</td><td>89.1%</td></tr><tr><td>Neural Statistician (Edwards &amp; Storkey,2016)*</td><td>1</td><td>=</td><td>88%</td><td>95%</td></tr><tr><td>Matching Nets (non-FCE, no fine-tune)</td><td>98.1%</td><td>98.9%</td><td>93.8%</td><td>98.5%</td></tr><tr><td>Prototypical Nets (1-shot)</td><td>98.1%</td><td>99.5%</td><td>94.2%</td><td>98.6%</td></tr></table>
+
+Table 1: Omniglot few-shot classification accuracy. \*Note that the Neural Statistician used nonstandard class splits.
+
+# 4.2 miniIMAGENET
+
+The miniImageNet dataset (Vinyals et al., 2016) is derived from the larger ImageNet dataset (Deng et al., 2009). It consists of 60,000 color images of size $8 4 \times 8 4$ divided into 100 classes with 600 examples each. It is designed for testing one-shot learning algorithms, where 80 classes are chosen for training, and 20 for testing.
+
+Classification results for miniImageNet are shown in Table 2. The embedding architecture we used for miniImagenet is the same as our experiments for Omniglot, though here it results in a 1600- dimensional output space due to the increased size of the images. We trained two versions of prototypical networks, one with episodes containing a single support examples per class (denoted by 1-shot) and one with five support examples per class (denoted by 5-shot). All episodes contained 20 randomly sampled classes, as 20-way classification represents a more difficult task than 5-way. We evaluated both models on 1-shot and 5-shot for 5-way and 20-way classification at test and find that each model performs best on the number of support examples it was trained for.
+
+Table 2: miniImageNet classification accuracy   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">5-way</td><td colspan="2">20-way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Pixels</td><td>23.0%</td><td>26.6%</td><td>6.7%</td><td>7.8%</td></tr><tr><td>Baseline Classifier</td><td>36.6%</td><td>46.0%</td><td>1</td><td>-</td></tr><tr><td>Matching Nets (non-FCE, no fine-tune)</td><td>41.2%</td><td>56.2%</td><td>=</td><td>=</td></tr><tr><td>Prototypical Nets (1-shot)</td><td>40.6%</td><td>55.8%</td><td>16.5%</td><td>27.5%</td></tr><tr><td>Prototypical Nets (5-shot)</td><td>37.0%</td><td>57.0%</td><td>14.2 %</td><td>29.4%</td></tr></table>
+
+# 4.3 CUB ZERO-SHOT CLASSIFICATION
+
+In order to assess the suitability of our approach for zero-shot learning, we also run experiments on the Caltech-UCSD Birds (CUB) 200-2011 dataset (Welinder et al., 2010). In the zero-shot setting, the goal is to classify query images in the absence of any support examples. Instead, class metadata (such as attributes or a textual description) is provided for each of the test classes. We adapt our few-shot approach to the zero-shot setting by learning to jointly embed images and metadata in a shared space. The embedded metadata serve as class prototypes and classification is performed by embedding the query image and selecting the class whose prototype is nearest in the Euclidean space.
+
+Table 3: CUB-200 zero-shot classification accuracy for methods utilizing attribute vectors as class metadata.   
+
+<table><tr><td>Method</td><td>Image Features</td><td>Top-1 Acc (50-way)</td></tr><tr><td>ALE (Akata et al., 2013)</td><td>Fisher Vectors</td><td>26.9%</td></tr><tr><td>SJE (Akata et al., 2015)</td><td>AlexNet</td><td>40.3%</td></tr><tr><td>Sample-Clustering (Liao et al., 2016)</td><td>AlexNet</td><td>44.3%</td></tr><tr><td>SJE (Akata et al., 2015)</td><td>GoogLeNet</td><td>50.1%</td></tr><tr><td>DS-SJE (Reed et al., 2016)</td><td>GoogLeNet</td><td>50.4%</td></tr><tr><td>DA-SJE (Reed et al., 2016)</td><td>GoogLeNet</td><td>50.9%</td></tr><tr><td>Prototypical Networks</td><td>GoogLeNet</td><td>54.6%</td></tr></table>
+
+The CUB dataset contains 11,788 images of 200 bird species. We closely follow the procedure of Reed et al. (2016) in preparing the data. We use their splits to divide the classes into disjoint sets of 100 training, 50 validation, and 50 test. For images we use 1,024-dimensional features extracted by applying GoogLeNet (Szegedy et al., 2015) to middle, upper left, upper right, lower left, and lower right crops of the original and horizontally-flipped image1. At test time we use only the middle crop of the original image. For class metadata we use the 312-dimensional continuous attribute vectors provided with the CUB dataset. These attributes encode various characteristics of the bird species such as their color, shape, and feather patterns.
+
+We learned a simple linear mapping on top of both the 1,024-dimensional image features and the 312-dimensional attribute vectors to produce a 1,024-dimensional output space. We apply prototype normalization to the embedded attributes so that the class prototypes are always of unit length. This serves as a form of regularization to help our embedding functions generalize better. The model parameters were optimized according to our objective via SGD with Adam (Kingma & Ba, 2014) at learning rate of $\mathrm { \dot { 1 } 0 ^ { - 4 } }$ and weight decay of $1 \dot { 0 } ^ { - 5 }$ . Early stopping on validation loss was used to determine the optimal number of epochs for retraining on the training $^ +$ validation set.
+
+Table 3 shows that of methods utilizing attributes as class metadata, we achieve state-of-the-art results by a large margin. Our approach is much simpler than that of other recent approaches (Liao et al., 2016) which train an SVM on a learned feature space obtained by fine-tuning AlexNet (Krizhevsky et al., 2012). These zero-shot classification results demonstrate that our approach is general enough to be applied even when the data points (images) are from a different domain relative to the classes (attributes).
+
+Figure 1 shows a t-SNE (Maaten & Hinton, 2008) visualization of attribute embeddings learned using prototypical networks for zero-shot classification. We can see that the embeddings group the bird species by characteristics such as their color and shape.
+
+# 5 CONCLUSION
+
+We have proposed a simple method called prototypical networks for few-shot learning based on the idea that we can represent each class by the mean of its examples in a representation space learned by a neural network. We train these networks to specifically perform well in the few-shot setting by using episodic training. Prototypical networks are simple to implement, and computationally efficient. We showed that this approach is equivalent to predicting the weights of a linear classifier, where the weights and biases are a function of the prototypes. Prototypical networks achieve stateof-the-art results on the Omniglot dataset, and competitive results on the miniImagenet dataset. We further showed how this approach can be adapted to the zero-shot setting by taking an embedding of an attribute vector for each class to be the prototype. This approach achieves state-of-the-art results on zero-shot classification of the Caltech UCSD birds dataset.
+
+![](images/95c4cfdf50699d30e0c17e3540ba1470c5429eec19f7ed0d7dc2862c30033750.jpg)  
+Figure 1: A t-SNE visualization of the attribute embeddings learned by a prototypical network on the CUB dataset. Each image is an arbitrarily chosen example from the corresponding test class. The learned space successfully clusters unseen bird species by characteristics such as color, shape, and pattern.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Sachin Ravi and Hugo Larochelle for help in setting up the Omniglot and miniImage data. We would also like to thank Renjie Liao for assistance with the CUB-200 zero-shot procedure and Oriol Vinyals for confirming details regarding the Omniglot and miniImagenet splits and matching nets architectures.
+
+# REFERENCES
+
+Zeynep Akata, Florent Perronnin, Zaid Harchaoui, and Cordelia Schmid. Label-embedding for attribute-based classification. In Computer Vision and Pattern Recognition, pp. 819–826, 2013.
+
+Zeynep Akata, Scott Reed, Daniel Walter, Honglak Lee, and Bernt Schiele. Evaluation of output embeddings for fine-grained image classification. In Computer Vision and Pattern Recognition, pp. 2927–2936, 2015.
+
+Aurelien Bellet, Amaury Habrard, and Marc Sebban. A survey on metric learning for feature vectors ´ and structured data. arXiv preprint arXiv:1306.6709, 2013.
+
+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.
+
+Harrison Edwards and Amos Storkey. Towards a neural statistician. arXiv preprint arXiv:1606.02185, 2016.
+
+Jacob Goldberger, Geoffrey E. Hinton, Sam T. Roweis, and Ruslan Salakhutdinov. Neighbourhood components analysis. In Advances in Neural Information Processing Systems, pp. 513–520, 2004.
+
+Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
+
+Gregory Koch. Siamese neural networks for one-shot image recognition. Master’s thesis, University of Toronto, 2015.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+
+Brian Kulis. Metric learning: A survey. Foundations and Trends in Machine Learning, 5(4):287– 364, 2012.
+
+Brenden M. Lake, Ruslan Salakhutdinov, Jason Gross, and Joshua B. Tenenbaum. One shot learning of simple visual concepts. In CogSci, 2011.
+
+Renjie Liao, Alexander Schwing, Richard Zemel, and Raquel Urtasun. Learning deep parsimonious representations. Advances in Neural Information Processing Systems, 2016.
+
+Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(Nov):2579–2605, 2008.
+
+Thomas Mensink, Jakob Verbeek, Florent Perronnin, and Gabriela Csurka. Distance-based image classification: Generalizing to new classes at near-zero cost. IEEE transactions on pattern analysis and machine intelligence, 35(11):2624–2637, 2013.
+
+Erik G Miller, Nicholas E Matsakis, and Paul A Viola. Learning from one example through shared densities on transforms. In CVPR, volume 1, pp. 464–471. IEEE, 2000.
+
+Scott Reed, Zeynep Akata, Bernt Schiele, and Honglak Lee. Learning deep representations of finegrained visual descriptions. arXiv preprint arXiv:1605.05395, 2016.
+
+Oren Rippel, Manohar Paluri, Piotr Dollar, and Lubomir Bourdev. Metric learning with adaptive density discrimination. International Conference on Learning Representations, 2016.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li FeiFei. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.
+
+Ruslan Salakhutdinov and Geoffrey E. Hinton. Learning a nonlinear embedding by preserving class neighbourhood structure. In AISTATS, pp. 412–419, 2007.
+
+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.
+
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–9, 2015.
+
+Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Koray Kavukcuoglu, and Daan Wierstra. Matching networks for one shot learning. arXiv preprint arXiv:1606.04080, 2016.
+
+P. Welinder, S. Branson, T. Mita, C. Wah, F. Schroff, S. Belongie, and P. Perona. Caltech-UCSD Birds 200. Technical Report CNS-TR-2010-001, California Institute of Technology, 2010.
\ No newline at end of file
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+++ b/parse/train/B1-Hhnslg/B1-Hhnslg_content_list.json
@@ -0,0 +1,1025 @@
+[
+    {
+        "type": "text",
+        "text": "PROTOTYPICAL NETWORKS FOR FEW-SHOT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            562,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jake Snell1∗, Kevin Swersky2& Richard S. Zemel1   \n1University of Toronto   \n2Twitter ",
+        "bbox": [
+            184,
+            169,
+            532,
+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            251,
+            544,
+            265
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A recent approach to few-shot classification called matching networks has demonstrated the benefits of coupling metric learning with a training procedure that mimics test. This approach relies on an attention scheme that forms a distribution over all points in the support set, scaling poorly with its size. We propose a more streamlined approach, prototypical networks, that learns a metric space in which few-shot classification can be performed by computing Euclidean distances to prototype representations of each class, rather than individual points. Our method is competitive with state-of-the-art few-shot classification approaches while being much simpler and more scalable with the size of the support set. We empirically demonstrate the performance of our approach on the Omniglot and miniImageNet datasets. We further demonstrate that a similar idea can be used for zero-shot learning, where each class is described by a set of attributes, and achieve state-ofthe-art results on the Caltech UCSD bird dataset. ",
+        "bbox": [
+            233,
+            282,
+            764,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            491,
+            336,
+            507
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One-shot classification (Miller et al., 2000; Lake et al., 2011; Koch, 2015) (and more generally, fewshot classification) is a problem in which a classifier must be adapted to accommodate new classes not seen in training, given only a single $( n )$ example(s) of these classes. A classical approach, such as retraining the model on the new data, would severely overfit. While the problem is quite difficult, it has been demonstrated that people have the ability to successfully perform one-shot classification (Lake et al., 2011). Nonparametric models such as nearest neighbors are useful in one-shot classification because they naturally adapt to new data, however this comes at the cost of storing the entire set of examples per class, the “support set”. ",
+        "bbox": [
+            174,
+            523,
+            825,
+            635
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To overcome this, much progress has been made recently in applying metric learning (Goldberger et al., 2004; Kulis, 2012; Bellet et al., 2013) to one-shot tasks. Most recently, (Vinyals et al., 2016) proposed a metric learning approach that they call matching networks. This approach uses an attention mechanism over a learned embedding of the support set in order to predict class labels for the points to be classified, a.k.a the “query set”. It optionally allows the embeddings to be conditioned on other points in the support set (“full context embeddings”) or for the embeddings to be fine-tuned at test time. A particularly interesting feature of the matching networks model is that it utilizes sampled mini-batches called “episodes” during training, where each episode is designed to mimic the one-shot task. This makes the training problem more faithful to the test environment. Matching networks however optionally utilize additional components such as an attention-based LSTM to change the embedding based on the support set. This complexity makes implementation more difficult in addition to the aforementioned poor scaling characteristics due to computing attention over the entire support set. ",
+        "bbox": [
+            174,
+            642,
+            825,
+            821
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a few-shot learning classifier based on a relatively simple idea: there exists an embedding, where points belonging to a class cluster around a single prototype. This inductive bias is a useful one to combat overfitting for one-shot tasks. Our approach also comes with the benefit that it is very simple to implement, and computationally fast. In order to do this, we learn a non-linear mapping of the input into an embedding space using a neural network, and take the class prototype to be the mean of the support set in the embedding space. Classification is then performed by simply finding the nearest prototype to the embedded query point. We find that this approach yields competitive results with matching networks and other one-shot learning approaches, despite being much simpler. ",
+        "bbox": [
+            174,
+            829,
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+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A related problem is known as zero-shot learning, where instead of being given a small number of examples of a new class at test-time, each class comes with a set of meta-information, often attributes, that give a high level description of that class. The idea then is to learn a mapping from input examples to the high-level attributes of their member class. We adapt the idea of prototypical networks to this setting by learning a secondary embedding of the attribute vector such that the image embeddings and attribute embeddings lie within the same space. In this case, we use the attribute embedding as the class prototype, rather than the class mean. ",
+        "bbox": [
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+            166,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            286,
+            344,
+            303
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Neighborhood Components Analysis (NCA) (Goldberger et al., 2004) learns a Mahalanobis distance to maximize K-nearest-neighbour’s (KNN) leave-one-out accuracy in the transformed space. A distribution over the neighbors of each data point is computed according to a softmax over the corresponding Mahalanobis distances. This distribution is marginalized to form a distribution over class assignments and the projection matrix is updated via gradient descent to maximize the probability of the true class. (Salakhutdinov & Hinton, 2007) extend NCA by using a neural network to perform the transformation. Our approach is similar in that we optimize a softmax based on distances in the transformed space. Ours differs because it is a softmax over classes, rather than points, computed from Euclidean distances to each class’s prototype representation. This is more appropriate for few-shot learning for two reasons: (a) the number of support points can vary by class, and (b) each class has a succinct representation independent of the number of data points, and this representation can optionally be updated in an online manner. ",
+        "bbox": [
+            174,
+            319,
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+            486
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our approach is similar to the nearest class mean approach of (Mensink et al., 2013) from the metric learning literature, where each class is represented by the mean of its examples, and classification is performed by finding the prototype that is closest to the query point. Their approach was developed to rapidly incorporate new classes into a classifier without retraining, however it relies on a linear embedding and is designed to handle the case where the novel classes come with many examples. In our approach, we utilize neural networks to learn a non-linear embedding of the features and we couple this with episodic training in order to handle the one-shot scenario. Mensink et al. do attempt to extend their approach to perform non-linear classification, but they do this by allowing classes to have multiple prototypes. They find these prototypes in a pre-processing step by using $\\mathbf { k }$ -means on the input space, and then perform a multi-modal variant of their linear embedding. By contrast, we learn a non-linear embedding in an end-to-end manner with no such pre-processing, producing a non-linear classifier that still only requires one prototype per class. ",
+        "bbox": [
+            174,
+            492,
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+            660
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In matching networks (Vinyals et al., 2016) they propose a meta-learning strategy in which training mimics test by stochastically creating one-shot “episodes”. We adopt the same strategy when training our models. They, like us, use neural networks to non-linearly transform data points into a space that is more amenable to classification. However, matching networks make predictions by computing attention weights over each point in the support set. This becomes computationally expensive as the size of the support set grows. Our approach, on the other hand, first summarizes each class in the support set by a prototype and then computes a distribution over classes. Ours thus has flexibility in the way the prototypes are computed and can handle additional support points gracefully by updating prototypes online. ",
+        "bbox": [
+            174,
+            666,
+            825,
+            791
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The neural statistician (Edwards & Storkey, 2016) extends the variational autoencoder (Kingma & Welling, 2013) to learn generative models of datasets rather than individual points. One component of the neural statistician is the “statistic network” which summarizes a set of data points into a statistic vector. It does this by encoding each point within a dataset, taking a sample mean, and applying a post-processing network to obtain an approximate posterior over the statistic vector. Edwards & Storkey test their model for one-shot classification on the Omniglot dataset (Lake et al., 2011) by considering each character to be a separate dataset and making predictions based on the class whose approximate posterior over the statistic vector had minimal KL-divergence from the test point. Like the neural statistician, we also produce a summary statistic for each class. However, ours is a discriminative model which is more appropriate because our primary task, one-shot learning, is also discriminative. Discriminative training has the added benefit of lending our model more flexibility in both the way we compute summary statistics and use them to make predictions at test time. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are many other approaches to one-shot learning that employ very different techniques from ours. Koch uses siamese networks to predict the probability that two images belong to the same class. Lake et al. devise a hierarchical Bayesian generative model of how a handwritten character is created in order to perform one-shot learning on the Omniglot dataset. Santoro et al. propose memory augmented neural networks (MANN) that reference an external memory in a similar fashion to neural Turing machines (Graves et al., 2014). This allows them to store support examples in an external memory and reference them later when making classification decisions. They also introduce a form of episodic training, similar to that in matching networks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PROTOTYPICAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            301,
+            436,
+            318
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "At prediction time we are given a support set of $N$ labeled examples: ${ \\cal { S } } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N } = { \\cal { S } } ^ { 1 } \\cup . . . \\cup$ $S ^ { K }$ where $S ^ { k } = \\{ ( x , y ) \\in S | y = k \\}$ . Our method computes a class representation $c _ { k }$ , or prototype, of each class through an embedding function $f _ { \\theta } ( x )$ parameterized by learnable parameters $\\theta$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f0349705dc19dbfad2e965f8c8e272dfb1972ffd2e3b94a7649e3e9b104a8fb.jpg",
+        "text": "$$\nc _ { k } = \\frac { 1 } { | S ^ { k } | } \\sum _ { ( x , y ) \\in S ^ { k } } f _ { \\theta } ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a test point $\\tilde { x }$ , prototypical networks forms a distribution over classes based on a softmax over the Euclidean distances between its embedding and the prototypes: ",
+        "bbox": [
+            171,
+            454,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3433a34d0b9a10e30344785e57ac15e68ed3b27633300fda98101cebee43d5ce.jpg",
+        "text": "$$\np ( y = k | \\tilde { x } ) = \\frac { \\exp ( - \\| f _ { \\theta } ( \\tilde { x } ) - c _ { k } \\| ^ { 2 } ) } { \\sum _ { k ^ { \\prime } } \\exp ( - \\| f _ { \\theta } ( \\tilde { x } ) - c _ { k } ^ { \\prime } \\| ^ { 2 } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            503,
+            643,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learning proceeds by maximizing the log-probability of the true class $\\tilde { y }$ : ",
+        "bbox": [
+            173,
+            553,
+            648,
+            569
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9102b9c83693e5c9103982c39703de79543273c7fc1d8fff1aaa2d11c34ac327.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\log p ( \\tilde { y } | \\tilde { x } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            592,
+            552,
+            614
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We train in an episodic manner similar to Vinyals et al. (2016) by randomly selecting a subset of classes from the training set, then choosing a subset of examples within each class to act as the support set and the remainder to serve as test points. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 PROTOTYPE NORMALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            693,
+            419,
+            707
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In episodic training, the support set is randomly chosen from among the training points. In datasets with high variability this can lead to a large variance in the class prototypes, $c$ , between episodes. In order to reduce this variability, we found that it can sometimes be beneficial to normalize the prototypes, $\\begin{array} { r } { \\bar { c } _ { k } = \\frac { c _ { k } } { \\| c _ { k } \\| } } \\end{array}$ and use $\\bar { c } _ { k }$ in place of $c _ { k }$ in Equation (2). This ensures that the prototypes always lie on the unit sphere, although the query points are still allowed to be embedded off of the unit sphere. Normalization has two benefits: the reduction in variance helps to greatly speed up training, while the restriction of the prototypes to the unit sphere confers additional regularization. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 PREDICTING THE WEIGHTS OF A LINEAR CLASSIFIER ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            839,
+            586,
+            854
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A simple analysis is useful in gaining insight into the nature of the learned classifier (a similar analysis appears in Mensink et al. (2013)). When we use Euclidean distance to measure the distance between a query point and the class prototypes, then the loss function in (2) is equivalent to a linear classifier with a particular parameterization. To see this, we expand the term within the exponent: ",
+        "bbox": [
+            174,
+            867,
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+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67cc9b2469264385d156fb8d007b36858223108d3e11942aa63975ff27ce3f53.jpg",
+        "text": "$$\n\\begin{array} { c } { { - \\| f _ { \\theta } ( \\tilde { x } ) - c _ { k } \\| ^ { 2 } = - ( f _ { \\theta } ( \\tilde { x } ) - c _ { k } ) ^ { \\top } ( f _ { \\theta } ( \\tilde { x } ) - c _ { k } ) } } \\\\ { { = - f _ { \\theta } ( \\tilde { x } ) ^ { \\top } f _ { \\theta } ( \\tilde { x } ) + 2 c _ { k } ^ { \\top } f _ { \\theta } ( \\tilde { x } ) ^ { \\top } - c _ { k } ^ { \\top } c _ { k } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            127,
+            691,
+            170
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The first term in Equation (4) is constant with respect to the class $k$ , so it does not affect the softmax probabilities. We can write the remaining terms as a linear classifier as follows: ",
+        "bbox": [
+            173,
+            186,
+            823,
+            215
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b8b560f29316f959d9ba74f38dca66758dee26669499275953935e6acf599188.jpg",
+        "text": "$$\n\\begin{array} { r l } & { 2 c _ { k } ^ { \\top } f _ { \\theta } ( \\tilde { x } ) - c _ { k } ^ { \\top } c _ { k } = w _ { k } ^ { \\top } f _ { \\theta } ( \\tilde { x } ) + b _ { k } } \\\\ & { \\qquad w _ { k } = 2 c _ { k } } \\\\ & { \\qquad b _ { k } = - c _ { k } ^ { \\top } c _ { k } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            244,
+            619,
+            306
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can view this through the lens of meta-learning, where the model is predicting the weights and biases of a linear classifier using a simple function of the mean of the embedded support set. By contrast, the predictive function in matching networks is a generalization of a nearest neighbor classifier, rather than a linear classifier. ",
+        "bbox": [
+            174,
+            323,
+            825,
+            378
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When using prototype normalization, the biases $b _ { k }$ will all be 1, and the class weights $w _ { k }$ will be restricted to have a norm of 2. In this case, using Euclidean distance becomes proportional to cosine distance. ",
+        "bbox": [
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+            428
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A natural question is whether it makes sense to use multiple prototypes per class instead of just one. If each support point were to be considered a prototype, then this would be analogous to doing nearest neighbor classification in the embedding space, which would be computationally expensive. On the other hand, if the number of prototypes per class is fixed, then this would require a partitioning scheme. This has been proposed in Mensink et al. (2013) and Rippel et al. (2016), however both methods require a separate partitioning phase that is decoupled from the weight updates, while our approach is simple to learn with ordinary gradient methods. Finally, the equivalence to a linear classifier suggests that this may be sufficient, as all of the required non-linearity can be learned within the embedding function. Indeed, this is the approach that state-of-the-art neural network classification systems currently use, e.g., (Krizhevsky et al., 2012). ",
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+    {
+        "type": "text",
+        "text": "3.3 DESIGN CHOICES ",
+        "text_level": 1,
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+        "text": "There are still a number of design choices that need to be made with this model in order to achieve optimal performance. One such choice is in deciding how many classes we would like the classifier to operate over during each training episode. For example, at test time we might be evaluating on 5-way classification, but at training time we could train each episode with 20-way classification. We found in general that training on a larger number of classes per episode improves performance, even if the number of classes we need to decide between at test-time is fewer. ",
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+        "text": "Another choice involves the possible decoupling of the $n$ in $n$ -shot between training and testing. We could train on 1-shot, but test on 5-shot or vice-versa. We found that it is typically better to match the shot at training and testing; that is, when it comes to the shot, to match the training procedure to the test procedure. We demonstrate this empirically in the Experiments section below. ",
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+        "text": "Finally, we need to specify whether to use prototype normalization. We found that normalization generally acts as a regularizer and speeds up training. ",
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+        "text": "4 EXPERIMENTS ",
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+        "text": "For few-shot learning, we performed experiments on Omniglot (Lake et al., 2011), the miniImagenet version of ILSVRC-2012 (Russakovsky et al., 2015) proposed by (Vinyals et al., 2016). For zeroshot learning, we perform experiments on the 2011 version of the Caltech UCSD bird dataset (CUB200 2011) (Welinder et al., 2010). ",
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+    {
+        "type": "text",
+        "text": "4.1 OMNIGLOT ",
+        "text_level": 1,
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+        "text": "Omniglot (Lake et al., 2011) is a dataset of 1623 handwritten characters collected from 50 alphabets. There are 20 examples associated with each character, where each example was drawn by a different human subject. We follow the procedure of (Vinyals et al., 2016) by augmenting the characters with rotations in multiples of 90 degrees and using 1200 characters for training and the remainder for evaluation. Our embedding architecture mirrors that of Matching Nets and is composed of four blocks of a 64 filter $3 \\times 3$ convolution, batch normalization (Ioffe & Szegedy, 2015), a ReLU nonlinearity and a $2 \\times 2$ max-pooling, resulting in a 64-dimensional output space. The results of our model trained to perform Omniglot classification are shown in Table 1. ",
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+        "text": "We trained prototypical networks using episodes designed for 1-shot learning, i.e., the support sets during training consist of a single input example, and we train using 20-way classification. Our results are as good or better than those reported in matching networks, and to our knowledge represent the state-of-the-art on this dataset using these splits. ",
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+        "type": "table",
+        "img_path": "images/e87618dcdea34ecbe1a131bbc55445ddf0fc6d961cfb678d5714f6aa10a477c8.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Omniglot few-shot classification accuracy. \\*Note that the Neural Statistician used nonstandard class splits. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">5-way</td><td colspan=\"2\">20-way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Pixels</td><td>41.7%</td><td>63.2%</td><td>26.7%</td><td>42.6%</td></tr><tr><td>Baseline Classifier</td><td>80.0%</td><td>95.0%</td><td>69.5%</td><td>89.1%</td></tr><tr><td>Neural Statistician (Edwards &amp; Storkey,2016)*</td><td>1</td><td>=</td><td>88%</td><td>95%</td></tr><tr><td>Matching Nets (non-FCE, no fine-tune)</td><td>98.1%</td><td>98.9%</td><td>93.8%</td><td>98.5%</td></tr><tr><td>Prototypical Nets (1-shot)</td><td>98.1%</td><td>99.5%</td><td>94.2%</td><td>98.6%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 miniIMAGENET ",
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+        "type": "text",
+        "text": "The miniImageNet dataset (Vinyals et al., 2016) is derived from the larger ImageNet dataset (Deng et al., 2009). It consists of 60,000 color images of size $8 4 \\times 8 4$ divided into 100 classes with 600 examples each. It is designed for testing one-shot learning algorithms, where 80 classes are chosen for training, and 20 for testing. ",
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+        "type": "text",
+        "text": "Classification results for miniImageNet are shown in Table 2. The embedding architecture we used for miniImagenet is the same as our experiments for Omniglot, though here it results in a 1600- dimensional output space due to the increased size of the images. We trained two versions of prototypical networks, one with episodes containing a single support examples per class (denoted by 1-shot) and one with five support examples per class (denoted by 5-shot). All episodes contained 20 randomly sampled classes, as 20-way classification represents a more difficult task than 5-way. We evaluated both models on 1-shot and 5-shot for 5-way and 20-way classification at test and find that each model performs best on the number of support examples it was trained for. ",
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+    {
+        "type": "table",
+        "img_path": "images/7b1c4dacb743ca4589ece530454a8c473974847ff7402ffa8db204f61b7883b1.jpg",
+        "table_caption": [
+            "Table 2: miniImageNet classification accuracy "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">5-way</td><td colspan=\"2\">20-way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Pixels</td><td>23.0%</td><td>26.6%</td><td>6.7%</td><td>7.8%</td></tr><tr><td>Baseline Classifier</td><td>36.6%</td><td>46.0%</td><td>1</td><td>-</td></tr><tr><td>Matching Nets (non-FCE, no fine-tune)</td><td>41.2%</td><td>56.2%</td><td>=</td><td>=</td></tr><tr><td>Prototypical Nets (1-shot)</td><td>40.6%</td><td>55.8%</td><td>16.5%</td><td>27.5%</td></tr><tr><td>Prototypical Nets (5-shot)</td><td>37.0%</td><td>57.0%</td><td>14.2 %</td><td>29.4%</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 CUB ZERO-SHOT CLASSIFICATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to assess the suitability of our approach for zero-shot learning, we also run experiments on the Caltech-UCSD Birds (CUB) 200-2011 dataset (Welinder et al., 2010). In the zero-shot setting, the goal is to classify query images in the absence of any support examples. Instead, class metadata (such as attributes or a textual description) is provided for each of the test classes. We adapt our few-shot approach to the zero-shot setting by learning to jointly embed images and metadata in a shared space. The embedded metadata serve as class prototypes and classification is performed by embedding the query image and selecting the class whose prototype is nearest in the Euclidean space. ",
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+    {
+        "type": "table",
+        "img_path": "images/840c2dbcaf9b93cae92d4352825c984b25baf47c7303c7dfadca703c9e2d03e9.jpg",
+        "table_caption": [
+            "Table 3: CUB-200 zero-shot classification accuracy for methods utilizing attribute vectors as class metadata. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Image Features</td><td>Top-1 Acc (50-way)</td></tr><tr><td>ALE (Akata et al., 2013)</td><td>Fisher Vectors</td><td>26.9%</td></tr><tr><td>SJE (Akata et al., 2015)</td><td>AlexNet</td><td>40.3%</td></tr><tr><td>Sample-Clustering (Liao et al., 2016)</td><td>AlexNet</td><td>44.3%</td></tr><tr><td>SJE (Akata et al., 2015)</td><td>GoogLeNet</td><td>50.1%</td></tr><tr><td>DS-SJE (Reed et al., 2016)</td><td>GoogLeNet</td><td>50.4%</td></tr><tr><td>DA-SJE (Reed et al., 2016)</td><td>GoogLeNet</td><td>50.9%</td></tr><tr><td>Prototypical Networks</td><td>GoogLeNet</td><td>54.6%</td></tr></table>",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The CUB dataset contains 11,788 images of 200 bird species. We closely follow the procedure of Reed et al. (2016) in preparing the data. We use their splits to divide the classes into disjoint sets of 100 training, 50 validation, and 50 test. For images we use 1,024-dimensional features extracted by applying GoogLeNet (Szegedy et al., 2015) to middle, upper left, upper right, lower left, and lower right crops of the original and horizontally-flipped image1. At test time we use only the middle crop of the original image. For class metadata we use the 312-dimensional continuous attribute vectors provided with the CUB dataset. These attributes encode various characteristics of the bird species such as their color, shape, and feather patterns. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We learned a simple linear mapping on top of both the 1,024-dimensional image features and the 312-dimensional attribute vectors to produce a 1,024-dimensional output space. We apply prototype normalization to the embedded attributes so that the class prototypes are always of unit length. This serves as a form of regularization to help our embedding functions generalize better. The model parameters were optimized according to our objective via SGD with Adam (Kingma & Ba, 2014) at learning rate of $\\mathrm { \\dot { 1 } 0 ^ { - 4 } }$ and weight decay of $1 \\dot { 0 } ^ { - 5 }$ . Early stopping on validation loss was used to determine the optimal number of epochs for retraining on the training $^ +$ validation set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 3 shows that of methods utilizing attributes as class metadata, we achieve state-of-the-art results by a large margin. Our approach is much simpler than that of other recent approaches (Liao et al., 2016) which train an SVM on a learned feature space obtained by fine-tuning AlexNet (Krizhevsky et al., 2012). These zero-shot classification results demonstrate that our approach is general enough to be applied even when the data points (images) are from a different domain relative to the classes (attributes). ",
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+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows a t-SNE (Maaten & Hinton, 2008) visualization of attribute embeddings learned using prototypical networks for zero-shot classification. We can see that the embeddings group the bird species by characteristics such as their color and shape. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have proposed a simple method called prototypical networks for few-shot learning based on the idea that we can represent each class by the mean of its examples in a representation space learned by a neural network. We train these networks to specifically perform well in the few-shot setting by using episodic training. Prototypical networks are simple to implement, and computationally efficient. We showed that this approach is equivalent to predicting the weights of a linear classifier, where the weights and biases are a function of the prototypes. Prototypical networks achieve stateof-the-art results on the Omniglot dataset, and competitive results on the miniImagenet dataset. We further showed how this approach can be adapted to the zero-shot setting by taking an embedding of an attribute vector for each class to be the prototype. This approach achieves state-of-the-art results on zero-shot classification of the Caltech UCSD birds dataset. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/95c4cfdf50699d30e0c17e3540ba1470c5429eec19f7ed0d7dc2862c30033750.jpg",
+        "image_caption": [
+            "Figure 1: A t-SNE visualization of the attribute embeddings learned by a prototypical network on the CUB dataset. Each image is an arbitrarily chosen example from the corresponding test class. The learned space successfully clusters unseen bird species by characteristics such as color, shape, and pattern. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Sachin Ravi and Hugo Larochelle for help in setting up the Omniglot and miniImage data. We would also like to thank Renjie Liao for assistance with the CUB-200 zero-shot procedure and Oriol Vinyals for confirming details regarding the Omniglot and miniImagenet splits and matching nets architectures. ",
+        "bbox": [
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+    {
+        "type": "text",
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Ruslan Salakhutdinov and Geoffrey E. Hinton. Learning a nonlinear embedding by preserving class neighbourhood structure. In AISTATS, pp. 412–419, 2007. ",
+        "bbox": [
+            171,
+            843,
+            821,
+            873
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "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. ",
+        "bbox": [
+            176,
+            881,
+            821,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–9, 2015. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Koray Kavukcuoglu, and Daan Wierstra. Matching networks for one shot learning. arXiv preprint arXiv:1606.04080, 2016. ",
+        "bbox": [
+            173,
+            155,
+            821,
+            184
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "P. Welinder, S. Branson, T. Mita, C. Wah, F. Schroff, S. Belongie, and P. Perona. Caltech-UCSD Birds 200. Technical Report CNS-TR-2010-001, California Institute of Technology, 2010. ",
+        "bbox": [
+            174,
+            193,
+            823,
+            222
+        ],
+        "page_idx": 8
+    }
+]
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diff --git a/parse/train/B1-Hhnslg/B1-Hhnslg_middle.json b/parse/train/B1-Hhnslg/B1-Hhnslg_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..d98b0e4e7eb3c2880912e1710cbcb6981f53e792
--- /dev/null
+++ b/parse/train/B1-Hhnslg/B1-Hhnslg_middle.json
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+                    ],
+                    "index": 23.5
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+                                    "score": 1.0,
+                                    "content": "networks to this setting by learning a secondary embedding of the attribute vector such that the",
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+                                    "content": "2 RELATED WORK",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "classifier with a particular parameterization. To see this, we expand the term within the exponent:",
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+                                    "score": 0.92,
+                                    "content": "\\begin{array} { c } { { - \\| f _ { \\theta } ( \\tilde { x } ) - c _ { k } \\| ^ { 2 } = - ( f _ { \\theta } ( \\tilde { x } ) - c _ { k } ) ^ { \\top } ( f _ { \\theta } ( \\tilde { x } ) - c _ { k } ) } } \\\\ { { = - f _ { \\theta } ( \\tilde { x } ) ^ { \\top } f _ { \\theta } ( \\tilde { x } ) + 2 c _ { k } ^ { \\top } f _ { \\theta } ( \\tilde { x } ) ^ { \\top } - c _ { k } ^ { \\top } c _ { k } } } \\end{array}",
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+                                    "content": "The first term in Equation (4) is constant with respect to the class",
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+                                    "score": 1.0,
+                                    "content": ", so it does not affect the softmax",
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+                                    "score": 1.0,
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+                    "type": "text",
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+                                    "content": "will be",
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+                            "index": 10
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+                                    "score": 1.0,
+                                    "content": "restricted to have a norm of 2. In this case, using Euclidean distance becomes proportional to cosine",
+                                    "type": "text"
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+                            ],
+                            "index": 11
+                        },
+                        {
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+                                    "content": "distance.",
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+                    ],
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+                },
+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "A natural question is whether it makes sense to use multiple prototypes per class instead of just",
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "one. If each support point were to be considered a prototype, then this would be analogous to doing",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "nearest neighbor classification in the embedding space, which would be computationally expensive.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "On the other hand, if the number of prototypes per class is fixed, then this would require a parti-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "tioning scheme. This has been proposed in Mensink et al. (2013) and Rippel et al. (2016), however",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "both methods require a separate partitioning phase that is decoupled from the weight updates, while",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "our approach is simple to learn with ordinary gradient methods. Finally, the equivalence to a lin-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "ear classifier suggests that this may be sufficient, as all of the required non-linearity can be learned",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "within the embedding function. Indeed, this is the approach that state-of-the-art neural network",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "classification systems currently use, e.g., (Krizhevsky et al., 2012).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        }
+                    ],
+                    "index": 17.5
+                },
+                {
+                    "type": "title",
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+                                    "content": "3.3 DESIGN CHOICES",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "type": "text",
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+                                    "content": "There are still a number of design choices that need to be made with this model in order to achieve",
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+                                    "score": 1.0,
+                                    "content": "optimal performance. One such choice is in deciding how many classes we would like the classifier",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "to operate over during each training episode. For example, at test time we might be evaluating on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "5-way classification, but at training time we could train each episode with 20-way classification. We",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "found in general that training on a larger number of classes per episode improves performance, even",
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "if the number of classes we need to decide between at test-time is fewer.",
+                                    "type": "text"
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+                        }
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+                                    "content": "Another choice involves the possible decoupling of the",
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+                                    "score": 1.0,
+                                    "content": "-shot between training and testing. We",
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "could train on 1-shot, but test on 5-shot or vice-versa. We found that it is typically better to match",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
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+                                            "html": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">5-way</td><td colspan=\"2\">20-way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Pixels</td><td>41.7%</td><td>63.2%</td><td>26.7%</td><td>42.6%</td></tr><tr><td>Baseline Classifier</td><td>80.0%</td><td>95.0%</td><td>69.5%</td><td>89.1%</td></tr><tr><td>Neural Statistician (Edwards &amp; Storkey,2016)*</td><td>1</td><td>=</td><td>88%</td><td>95%</td></tr><tr><td>Matching Nets (non-FCE, no fine-tune)</td><td>98.1%</td><td>98.9%</td><td>93.8%</td><td>98.5%</td></tr><tr><td>Prototypical Nets (1-shot)</td><td>98.1%</td><td>99.5%</td><td>94.2%</td><td>98.6%</td></tr></table>",
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+                                    "content": "We would like to thank Sachin Ravi and Hugo Larochelle for help in setting up the Omniglot and",
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+                                    "content": "miniImage data. We would also like to thank Renjie Liao for assistance with the CUB-200 zero-shot",
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+                                    "content": "procedure and Oriol Vinyals for confirming details regarding the Omniglot and miniImagenet splits",
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+                                    "score": 1.0,
+                                    "content": "and matching nets architectures.",
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+                                    "content": "REFERENCES",
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+                                    "score": 1.0,
+                                    "content": "attribute-based classification. In Computer Vision and Pattern Recognition, pp. 819–826, 2013.",
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+                                    "bbox": [
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+                                    "score": 1.0,
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+                                    "type": "text"
+                                }
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+                                    "score": 1.0,
+                                    "content": "pp. 2927–2936, 2015.",
+                                    "type": "text"
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+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and structured data. arXiv preprint arXiv:1306.6709, 2013.",
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+                                }
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+                        }
+                    ],
+                    "index": 18.5
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diff --git a/parse/train/HkEI22jeg/HkEI22jeg.md b/parse/train/HkEI22jeg/HkEI22jeg.md
new file mode 100644
index 0000000000000000000000000000000000000000..5932f6de624c8db3c750385a587977388e956cdc
--- /dev/null
+++ b/parse/train/HkEI22jeg/HkEI22jeg.md
@@ -0,0 +1,200 @@
+# MULTILAYER RECURRENT NETWORK MODELS OF PRIMATE RETINAL GANGLION CELL RESPONSES
+
+Eleanor Batty, Josh Merel ∗ Doctoral Program in Neurobiology & Behavior, Columbia University erb2180@columbia.edu,jsmerel@gmail.com
+
+Nora Brackbill \* Department of Physics, Stanford University nbrack@stanford.edu
+
+Alexander Heitman   
+Neurosciences Graduate Program, University of California, San Diego   
+alexkenheitman@gmail.com   
+Alexander Sher & Alan Litke   
+Santa Cruz Institute for Particle Physics, University of California, Santa Cruz   
+sashake3@ucsc.edu, Alan.Litke@cern.ch   
+E.J. Chichilnisky   
+Department of Neurosurgery and Hansen Experimental Physics Laboratory, Stanford University   
+ej@stanford.edu
+
+# Liam Paninski
+
+Departments of Statistics and Neuroscience, Columbia University liam@stat.columbia.edu
+
+# ABSTRACT
+
+Developing accurate predictive models of sensory neurons is vital to understanding sensory processing and brain computations. The current standard approach to modeling neurons is to start with simple models and to incrementally add interpretable features. An alternative approach is to start with a more complex model that captures responses accurately, and then probe the fitted model structure to understand the neural computations. Here, we show that a multitask recurrent neural network (RNN) framework provides the flexibility necessary to model complex computations of neurons that cannot be captured by previous methods. Specifically, multilayer recurrent neural networks that share features across neurons outperform generalized linear models (GLMs) in predicting the spiking responses of parasol ganglion cells in the primate retina to natural images. The networks achieve good predictive performance given surprisingly small amounts of experimental training data. Additionally, we present a novel GLM-RNN hybrid model with separate spatial and temporal processing components which provides insights into the aspects of retinal processing better captured by the recurrent neural networks.
+
+# 1 INTRODUCTION
+
+Our understanding of sensory processing in the brain is most straightforwardly reflected in our ability to model the process by which stimuli presented at the sensory periphery are transformed into the spiking activity of populations of neurons. For decades, researchers have interrogated stimulus-response neural properties using simplified targeted stimuli, such as bars, spots, or gratings. While these types of stimuli uncovered many interesting aspects of visual computation, they have several limitations (Barlow & Levick, 1965). These stimuli may not fully drive important components of neural response, and modeling efforts have often assumed a quasi-linear mapping from stimulus to firing rate. Subsequent efforts to characterize cells relied on white noise stimulation and building models through reverse correlation (de Boer R & Kuyper, 1968; Marmarelis & Naka, 1972; Chichilnisky, 2001). A standard model used to relate white noise to spiking responses is the linear-nonlinear-Poisson (LN) or generalized linear model (GLM) which consists of a spatiotemporal linear filtering of the stimulus followed by a nonlinearity and probabilistic spike generation (Chichilnisky, 2001; Simoncelli et al., 2004; Schwartz et al., 2006). Although this family of models have advanced our understanding, they do not optimally capture neural responses, especially to natural scenes which can lead to more complex responses than white noise stimuli (David et al., 2004). Even in the retina, early in the visual processing stream, these commonly-used models capture retinal ganglion cell (RGC) responses to natural stimuli less accurately than to white noise (Heitman et al., 2016).
+
+Recently, deep neural networks have been used to dramatically improve performance on a diverse array of machine learning tasks (Krizhevsky et al., 2012; LeCun et al., 2015). Furthermore, these networks bear a loose resemblance to real neural networks, and provide a sufficiently rich model class that can still be roughly constrained to match the biological architecture (Kriegeskorte, 2015). Most previous research at this intersection of neuroscience and artificial neural networks has focused on training networks on a certain task, such as object recognition, and then comparing the computations performed in different layers of the artificial network to those performed by real neurons (Yamins et al., 2014). Here we take a different approach: we fit multilayer models directly to the spiking responses of neurons, an approach that has not been explored in detail (but see (McIntosh et al., 2016) for some recent independent parallel developments).
+
+# 2 APPROACH
+
+We fit a range of models, detailed below, to spiking responses of primate RGCs. Our baseline comparisons are the GLM architectures that have been widely used to construct previous neural models (Pillow et al., 2008), though here we focus on individual neuronal responses (we leave modeling of correlations between neurons for future work). We focused on RNNs as a flexible framework in which to model more complex temporal and spatial nonlinearities. We also explored a number of network architectures involving features or weights shared across observed neurons. Given the complexity of the network architectures, we reasoned that sharing statistical strength across neurons by learning a shared feature space might improve predictive performance. This is conceptually a form of multitask learning - we are using a shared representation to achieve better generalization (Baxter, 2000). Motivated by previous research showing significant differences in the processing properties of the two cell types examined, ON and OFF parasol retinal ganglion cells, we fit separate models for each of these cell types (Chichilnisky & Kalmar, 2002).
+
+# 3 METHODS
+
+# 3.1 DATA COLLECTION
+
+We fit spiking responses of OFF and ON parasol retinal ganglion cells to natural scenes. Recordings were performed on isolated retina using a large-scale multi-electrode recording system (Litke et al., 2004; Frechette et al., 2005; Field et al., 2007). A standard spike sorting algorithm was used to identify spikes from different cells from the voltage signals on each electrode during visual stimulation (Litke et al., 2004). We focus on two separate experiments (the same experimental procedure in two separate retinas) here; analyses of other datasets yielded similar results. Models were fit separately for the two experiments due to animal to animal variability in cell properties, such as receptive field size and firing rate. Almost all spike sorted cells were used for training (exp $1 = 1 1 8$ OFF cells, 66 ON cells; exp $2 = 1 4 2$ OFF cells, 103 ON cells): two cells were removed due to data quality issues (see sec 3.3). Performance metrics in this paper are reported for the same subset of cells used in a previous study (Heitman et al., 2016). These cells passed passed a manual screen for spike sorting accuracy, demonstrated stable light responses, and met a convergence criteria in prior linear-nonlinear modeling (exp $1 = 1 0$ OFF cells, $1 8 \mathrm { \ : O N }$ cells; exp $2 = 6 5$ OFF cells, $1 4 \mathrm { { \ O N } }$ cells). The naturalistic movie stimulus consisted of images from the Van Hateren database shown for one second each, with spatial jitter based on eye movements during fixation by awake macaque monkeys (Z.M. Hafed and R.J. Krauzlis, personal communication), (van Hateren & van der Schaaf, 1998). An example stimulus can be found at https://youtu.be/sG_18Uz_6OE. 59 distinct natural scenes movies of length one minute (the training data) were interleaved with 59 repetitions of a 30 second movie (the test data). Interleaving ensured that the test movie repetitions spanned the same period of time as the training data and therefore experienced the same range of experimental conditions (in case of neural response drifts over time). The first 4 movies shown (2 training movies and 2 repetitions of the test movie) were excluded to avoid initial transients. Test metrics are reported for the last 29 seconds of the 30 second test movie for the same reason. For further details on the experimental set-up, data preprocessing, and visual stimulation, see Heitman et al. (2016).
+
+![](images/ba7501c108dec1599568c9756264f2e7f9540f733be3724d116380a263a47fb7.jpg)  
+Figure 1: Example model architectures. (A) Shared LN model. The past few frames of the stimulus images are presented as inputs which are spatiotemporally filtered and passed through a nonlinearity to produce a firing rate, which drives a Poisson spiking process. (B) Two-layer RNN. The current frame of the stimulus feeds into a sequence of RNN layers (history dependence is implicit in the hidden unit activations) and a Poisson GLM draws weighted inputs from the activations of the hidden units of the last RNN layer and outputs predicted spike trains. Thus the last RNN layer represents a shared feature pool that all the RGCs can draw from.
+
+# 3.2 MODEL TRAINING
+
+All models were implemented in Theano and trained on a combination of CPUs and GPUs (Theano Development Team, 2016). Training was performed using the Adam optimizer on the mean squared error (MSE) between predicted firing rate and true spikes (Kingma & Ba, 2014). We also experimented with optimizing a Poisson likelihood; this led to qualitatively similar results but occasionally less stable fits, so we focus on the MSE results here. All recurrent dynamics and temporal filters operated on time bins of $8 . 3 3 \mathrm { m s }$ (the frame rate of the movie). Spike history terms and performance metrics were calculated for $0 . 8 3 3 ~ \mathrm { m s }$ bins. We used the same split of training and validation data for both experiments: 104 thirty-second movies as training data and 10 thirty-second movies as a held-out validation set.
+
+During training, the performance on the held-out validation set is checked after every pass through the training data. After each iteration through the training data, if the model exhibits significantly better validation performance than our previous best, we reset the minimum number of iterations to be twice the current iteration number. If we make it through those iterations without another significant improvement, we stop. We train for a maximum of 150 epochs, where we define one epoch as one pass through all the training data. The model with the best validation performance is saved and used to assess test performance. All models with shared parameters were trained on a combined MSE over
+
+all neurons and the parameters picked were those which minimized validation MSE for all neurons.   
+For individual LNs/GLMs/RNNs, the validation MSE was minimized for each neuron separately.
+
+# 3.3 RECEPTIVE FIELD CENTER ESTIMATION
+
+In all models used in this paper, we estimate the receptive field (RF) center of each neuron in order to identify the appropriate portion of the image to use as input. We calculate a $2 5 0 ~ \mathrm { m s }$ long spike triggered average (STA) using reverse correlation of the neuron’s spikes with a white noise stimulus. We reduce the noise in this STA by using a rank 1 approximation (singular value decomposition followed by reconstruction using the primary temporal and spatial components). We then smooth each frame of the STA via convolution with a Gaussian spatial filter. The center location is defined as the pixel location that has the maximum absolute magnitude over time. The center locations were visually assessed to check accuracy of the algorithm. Rare cases where the algorithm failed to identify the correct center indicated neurons that responded to very little of the image as their receptive field was more than half-way displaced out of the image. These two neurons (two Exp 1 ON cells) were removed from further analysis. If the receptive field center is close to the edge of the image, the image patch is padded with the average training stimulus value.
+
+# 3.4 PERFORMANCE EVALUATION
+
+To quantitatively evaluate the accuracy of model spike predictions, we used the fraction of explainable variance, which has been described in previous literature (Heitman et al., 2016). Average firing rates over time are obtained after generating spikes from the model in $0 . 8 3 3 \mathrm { m s }$ bins and smoothing with a Gaussian temporal filter $\mathrm { \ S D = 1 0 m s } ,$ ). The fraction of variance is computed as
+
+$$
+F ( r , r _ { s } ) = 1 - \frac { \sum _ { t } ( r ( t ) - r _ { s } ( t ) ) ^ { 2 } } { \sum _ { t } ( r ( t ) - \mu ) ^ { 2 } }
+$$
+
+where $r ( t )$ is the smoothed recorded firing rate, $r _ { s } ( t )$ is the smoothed predicted firing rate, and $\mu$ is the average recorded rate. Finally, to account for the reproducibility of responses over repeated trials, we normalize by the fraction of variance captured by using the average firing rate on the odd $( r _ { o } )$ trials of the repeated test movie to predict responses on the even $( r _ { e } )$ trials:
+
+$$
+F V = \frac { F ( r , r _ { s } ) } { F ( r _ { e } , r _ { o } ) } .
+$$
+
+# 4 MODEL ANALYSIS
+
+# 4.1 NETWORK ARCHITECTURES
+
+Individual LNs and GLMs: The linear-nonlinear model (LN) consists of a spatiotemporal filtering of the $3 1 \mathrm { x } 3 1 \mathrm { x } 3 0$ movie patch $X _ { t }$ , width by height by time) surrounding the estimated center of the neuron’s receptive field plus a bias term $( b )$ , followed by a sigmoid nonlinearity $( f )$ , and Poisson spike generation to produce the responses $r _ { t }$ . The generalized linear model (GLM), given by
+
+$$
+r _ { t } \sim P o i s s \left[ f \bigg ( \vec { w } _ { s } ^ { T } ( X _ { t } \vec { w } _ { t } ) + b + \sum _ { i } { h _ { i } r _ { t - i } } \bigg ) \right] ,
+$$
+
+has the same architecture with the addition of a post-spike history filter $h$ before the nonlinearity $f$ (Pillow et al., 2008). We used a rank 1 approximation of the full spatiotemporal filter (higher rank models did not significantly improve fits on a subset of examined neurons), resulting in a vectorized 31x31 spatial filter $( \vec { w } _ { s } )$ and a 30 bin temporal filter $( \vec { w } _ { t } )$ which spans $2 5 0 \mathrm { m s }$ (Heitman et al., 2016). The post-spike history filter consists of a weighted sum of a basis of 20 raised cosines spanning approximately $1 0 0 \mathrm { m s }$ (Pillow et al., 2008). The models with spike history were fit by initializing with the model fit without spike history. The filter either operates on the recorded spikes (training and validation) or the spikes generated by the model (testing). The nonlinearity is the logistic sigmoid: $f = L / ( 1 + \exp ( \bar { - } x ) )$ , which has been shown to improve fitting over an exponential nonlinearity for modeling RGC responses to natural scenes (Heitman et al., 2016).
+
+![](images/07780a793573a414ea2f391472bf6f2c726ecc672ededb70f868499ffd486ed6.jpg)  
+Figure 2: Model performance. (A) Mean $\pm$ std. err. of the fraction of explainable variance for criteria-passing subset of OFF and ON cells for various model architectures. (B) Scatter plots show individual neural performance from LN and RNN model; each dot corresponds to one cell. Negative FV values are shown on relevant axis as $\operatorname { F V } { = } 0$ (C) Hybrid model performance, quantified by the ratio between the multitask LN to multitask hybrid performance gap and the multitask LN to multitask RNN performance gap (one high outlier not pictured for both Exp 1 ON and Exp 2 ON )
+
+Shared LN: In this model, the architecture is similar to the individual LNs but all cells of a given type (OFF or ON) share the same temporal and spatial filters (Figure 1A; note that the spatial filters are displaced to the RF center of each individual RGC). All other parameters are individually tuned for each observed neuron. There is an additional gain term that weights the output of the filtering individually for each observed neuron.
+
+Two-layer RNN, 50 units: In this architecture, there are two recurrent neural network (RNN) layers between the image patch and Poisson neural unit:
+
+$$
+\begin{array} { r l } & { \vec { h } _ { j , t } ^ { ( 1 ) } = \operatorname* { m a x } ( 0 , { U } _ { 1 } \vec { s } _ { j , t } + { V } _ { 1 } \vec { h } _ { j , t - 1 } ^ { ( 1 ) } + \vec { c } ) } \\ & { \vec { h } _ { j , t } ^ { ( 2 ) } = \operatorname* { m a x } ( 0 , { U } _ { 2 } \vec { h } _ { j , t } ^ { ( 1 ) } + { V } _ { 2 } \vec { h } _ { j , t - 1 } ^ { ( 2 ) } + \vec { d } ) } \\ & { { r } _ { j , t } \sim P o i s s \left[ f ( \vec { w } _ { j } ^ { T } \vec { h } _ { j , t } ^ { ( 2 ) } + b _ { j } ) \right] . } \end{array}
+$$
+
+The activity of the 50 units in the first RNN layer at time $t$ is given by $\vec { h } _ { j , t } ^ { ( 1 ) }$ in Eqn. 4. These units are rectified linear, and receive input from the vectorized 31x31 image patch surrounding the center of neuron $j$ ’s receptive field, $\vec { s } _ { j , t }$ , with weights $U _ { 1 }$ , along with input from the other units in the layer with weights $V _ { 1 }$ and a bias $\vec { c }$ . The output of the first RNN is then fed into a second RNN with similar architecture. The firing rate for each observed neuron in the final layer is then given by Eqn. 6, and is a weighted sum of the recurrent units plus a bias $b _ { j }$ , followed by a softplus nonlinearity $f = \log ( 1 + \exp ( - x ) )$ . Note that all parameters are shared across neurons except for the weights to the final layer and the final bias terms ( $\overrightarrow { w } _ { j }$ and $b _ { j }$ ).
+
+GLM-RNN Hybrid: The GLM-RNN hybrid model consists of a spatial filter followed by a two-layer RNN. The architecture resembles that of the full two-layer RNN with 50 units, except the input to the first layer is a scalar (post multiplication with the spatial filter) at each time step instead of the full image patch; thus the RNN in this model is responsible for shaping the temporal properties of the output, but does not affect spatial processing after the first linear spatial filtering stage. All weights are shared across neurons except for weights to the final layer $( \vec { w } _ { j } )$ and the final bias terms $( b _ { j } )$
+
+![](images/91e264c19d450d0ab71845440ff4e8c2c4a4c17d7cc2e0d9c27f007c9d1915e0.jpg)  
+Figure 3: (A,B) Rasters showing spiking responses for 57 trials (each row corresponds to a single trial) for an OFF and ON cell from experiment 1 for 10 seconds of a novel natural scenes movie. Example cells chosen had near average difference between LN and RNN performance. Red ticks denote time at which one natural image was replaced by another. (C,D) Average predicted spikes over trials smoothed with Gaussian $\mathrm { S D } = 1 0 \mathrm { m s }$ ) for same 10 seconds of the novel natural scenes movie show qualitative differences among models. Dotted vertical lines align with red ticks in (A,B).
+
+$$
+\begin{array} { l } { { y _ { j , t } = \vec { w } _ { s } ^ { T } \vec { s } _ { j , t } } } \\ { { \vec { h } _ { j , t } ^ { ( 1 ) } = m a x ( 0 , \vec { u _ { 1 } } y _ { j , t } + V _ { 1 } \vec { h } _ { j , t - 1 } ^ { ( 1 ) } + \vec { c } ) } } \\ { { \vec { h } _ { j , t } ^ { ( 2 ) } = m a x ( 0 , U _ { 2 } h _ { j , t } ^ { ( 1 ) } + V _ { 2 } \vec { h } _ { j , t - 1 } ^ { ( 2 ) } + \vec { d } ) } } \\ { { r _ { j , t } \sim P o i s s \left[ f ( \vec { w } _ { j } ^ { T } \vec { h } _ { t } ^ { ( 2 ) } + b _ { j } ) \right] . } } \end{array}
+$$
+
+# 4.2 MODEL PERFORMANCE
+
+RNNs of varying architectures consistently outperformed LNs and GLMs in predicting neural spiking responses to a novel natural scene movie for both OFF and ON parasol retinal ganglion cells in both experiments (Figure 2). A shared two-layer recurrent network consistently captures around $8 0 \%$ of the explainable variance across experiments and cell types. Other recurrent architectures (1-3 layer RNNs and a 2 layer LSTM) led to similar levels of performance (Supplementary Figure 6). The increase in performance according to the fraction of explainable variance metric was not an average effect: almost all neurons were significantly better predicted by the RNN (Figure 2B). A 2 layer RNN model with additional trained spike history filters outperformed GLMs and LNs according to a normalized log likelihood metric (Supplementary Figure 7).
+
+Inspection of the mean predicted firing rate traces for LNs and RNNs in Figure 3 reveals that the recurrent network seems to be capturing the timing of firing more precisely. The LN often predicts a general increase in firing rate at the correct times, but the RNN captures the sudden increase in firing rate followed by decay which often occurs when the image changes. On the other hand, the LN models sometimes predict modest increases or decreases in firing rate that the recurrent nets miss.
+
+Understanding why the recurrent models improve performance is a challenging task due to the black-box nature of deep networks. The first layer filters $U _ { 1 }$ , from image patches to recurrent units) have an interpretable structure resembling traditional receptive fields expected in the retina (Supplementary Figure 8). However, the computations performed by the recurrent units are difficult to tease apart, because the weights are less interpretable. Thus, instead of attempting a mechanistic explanation of the internals of the RNN, we focused on what additional captured information resulted in the improved RNN performance.
+
+![](images/e211a5f688b38b0a2e9fd82f71903fe4167ac4ec30a222c8edc9289a38ac0304.jpg)  
+Figure 4: Model predictive performance on held-out data as a function of the amount of training data. Error bars show SEM over 3 iterations of the mean FV over all neurons
+
+One possibility is that capturing nonlinear effects in parts of the image far from the receptive field center improved predictions (McIlwain, 1964; Passaglia et al., 2009). We restricted the size of the image patch surrounding each receptive field center from 31x31 to 15x15 (Supplementary Figure 9). Shared RNNs trained on the smaller image patch size did as well, or better, than those trained on the larger patch across almost all combinations of cell type and experiment. (We see a similar small improvement when training the LN models on the small patch.) Thus we concluded that long-range nonlinear spatial interactions do not contribute to the increased performance produced by the RNNs.
+
+We also investigated whether nonlinear spatial interactions or nonlinear temporal processing primarily contributed to better predictions. To accomplish this, we constructed a GLM-RNN hybrid, described previously, in which a single spatial filter precedes a two-layer RNN - effectively allowing only temporal nonlinearities to be captured. This model improved prediction over the LNs and GLMs but did not reach full RNN performance. The amount by which this model closed the gap differed for different experiments and cell types. We quantified this by computing the difference between multitask RNN and multitask LN performance for each neuron and the difference between multitask hybrid and multitask LN performance. We divide the latter by the former (on a cell-by-cell basis) to obtain the ratios summarized in Figure 2C. The hybrid model closed greater than half of the gap on average between multitask LN and RNN performance, indicating that the richer temporal dynamics of the RNN model account for a large part of the difference between RNN and LN performance, though spatial nonlinearities play a role too.
+
+# 5 MODEST TRAINING DATA LENGTH SUFFICES FOR GOOD PERFORMANCE
+
+Deep networks can be complex and often require large amounts of data to adequately train: convolutional neural networks used for object recognition are trained on over a million images (Krizhevsky et al., 2012). Standard neuroscience experiments yield limited data sets, so it is crucial to assess whether we have enough data to adequately fit our network architectures. We trained the RNN on varying amounts of data, and ran several different iterations of the network to explore variation over random initializations and randomly chosen training sets. These results are shown for both ON and OFF cells in Figure 4. Surprisingly small amounts of training data resulted in good predictive abilities. For larger amounts of training data, different iterations resulted in very similar mean fraction of variance values, indicating fairly robust fitting in these models. See Supplementary Figure 10 for further details.
+
+# 6 BENEFITS OF MULTITASK FRAMEWORK
+
+We investigated whether the multitask framework with shared parameters across neurons actually helps to improve predictive performance with reasonable amounts of experimental data. First, we quantified the benefits of parameter-sharing in the simple LN model. This is a highly constrained framework: every cell has the same spatial and temporal filter. The shared LN does not improve performance for most neurons (Figure 5A).
+
+![](images/44514c8539a1d7b10219249491b9c0775fb9630004f28c50bde26eb5d5022f87.jpg)  
+Figure 5: Shared vs individual fits for LN model (A) and RNN model (B). $1 0 \mathrm { O F F }$ and $1 0 \mathrm { { O N } }$ cells from each experiment are pictured (Light blue $=$ exp 1, dark blue $=$ exp 2). Negative FV values are pictured as $\mathrm { F V } = 0$ .
+
+We expected the multitask framework to be more helpful applied to the RNN model because in this case we are sharing features but not all parameters across neurons. Indeed, the multitask RNN consistently outperformed RNNs trained individually on single neurons (Figure 5B); individuallytrained RNNs also had much more variable losses than did the multitask-trained RNNs. In a realistic experimental setting with limited data, the multitask framework is a useful way to leverage all of the data collected for all neurons.
+
+# 7 CONCLUSION
+
+Using retinal neurons responding to natural scenes as an example, we showed that: using deep networks to model neural spiking responses can significantly improve prediction over current state-ofthe-art models; sharing information across neurons in a multi-task framework leads to better and more stable predictions; and these models work well even given relatively small amounts of experimental data. We believe that the multitask RNN framework presented here will enable new, richer models of complex nonlinear spiking computations in other brain areas.
+
+While one could argue that we have merely exchanged the black box of the brain for another black box, just having a more predictive model is an important tool for research: these predictive models of the primate retina can be used in retinal prosthetics research, to probe decoding, and as a first stage of processing in the modeling of higher visual areas. Additionally, the recurrent network is more accessible and available for experimentation and quantitative analysis. For example, the trained neural network models may guide choices for more accurate simpler models by identifying key computational features that are important to include. Training smaller models on the denoised compression of spiking data (the predicted firing rate) may help them to learn features they otherwise would not (Ba & Caruana, 2014). The deep network approach allows one to determine types of information important to the neuron without having to build an exact mechanistic model of how such information is incorporated, as demonstrated by our finding that both spatial and temporal nonlinearities are not fully captured by the standard pseudo-linear models. We hope in future work to gain a more thorough and quantitative understanding of the dynamics captured by the recurrent networks and to extend this approach to higher sensory areas.
+
+# ACKNOWLEDGMENTS
+
+Funding for this research was provided by the National Science Foundation Graduate Research Fellowship Program under grant No. DGE-114747 (NB), Grant Number No. DGE-16-44869 (EB), the National Science Foundation IGERT Training Grant No. 0801700 (NB), the National Institutes of Health Grant EY017992 (EJC), NSF CRCNS IIS-1430239 (LP, EJC) and Google Faculty Research awards (LP, EJC); in addition, this work was supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/ Interior Business Center (DoI/IBC) contract number D16PC00003 (LP). The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/IBC, or the U.S. Government.
+
+# REFERENCES
+
+Jimmy Ba and Rich Caruana. Do deep nets really need to be deep? Advances in Neural Information Processing Systems, 2014.
+
+HB Barlow and William R Levick. The mechanism of directionally selective units in rabbit’s retina. The Journal of Physiology, 178(3):477, 1965.
+
+Jonathan Baxter. A model of inductive bias learning. Journal of Artificial Intelligence Research, 12: 149–198, 2000.
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+E. J. Chichilnisky and Rachel S. Kalmar. Functional asymmetries in on and off ganglion cells of primate retina. The Journal of Neuroscience, 22(7):2737–2747, 2002.
+
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+
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+
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+
+Alexander Heitman, Nora Brackbill, Martin Greschner, Alexander Sher, Alan M. Litke, and E.J. Chichilnisky. Testing pseudo-linear models of responses to natural scenes in primate retina. bioRxiv, 2016.
+
+Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural Computation, 9(8): 1735–1780, 1997.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint, arXiv:11412.6980, 2014.
+
+Nikolaus Kriegeskorte. Deep neural networks: A new framework for modeling biological vision and brain information processing. Annual Review of Vision Science, 1:417–446, 2015.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, pp. 1097–1105, 2012.
+
+Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521:436–444, 2015.
+
+A. M. Litke, N. Bezayiff, E. J. Chichilnisky, W. Cunningham, W. Dabrowski, A. A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. S. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher. What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity. IEEE Transactions on Nuclear Science, 51(4):1434–1440, 2004.   
+P.Z. Marmarelis and K. Naka. White-noise analysis of a neuron chain: an application of the wiener theory. Science, 175:1276–1278, 1972.   
+James T. McIlwain. Receptive fields of optic tract axons and lateral geniculate cells: peripheral extent and barbiturate sensitivity. Journal of Neurophysiology, 27(6):1154–1173, 1964.   
+Lane McIntosh, Niru Maheswaranathan, Aran Nayebi, Surya Ganguli, and Stephen Baccus. Deep convolutional neural network models of the retinal response to natural scenes. In Cosyne Abstracts, Salt Lake City USA, 2016.   
+CL. Passaglia, Freeman DK., and Troy JB. Effects of remote stimulation of the modulated activity of cat retina. Journal of Neuroscience, 29, 2009.   
+Jonathan W Pillow, Jonathon Shlens, Liam Paninski, Alexander Sher, Alan M Litke, E.J. Chichilnisky, and Eero P Simoncelli. Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207):995–999, 2008.   
+Odelia Schwartz, Jonathan W. Pillow, Nicole C. Rust, and Eero P. Simoncelli. Spike-triggered neural characterization. Journal of Vision, 6(4):13, 2006.   
+Eero Simoncelli, Jonathan W. Pillow, Liam Paninski, and Odelia Schwartz. Characterization of neural responses with stochastic stimuli, pp. 327–338. MIT Press, 2004.   
+Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016.   
+J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings. Biological sciences / The Royal Society, 265(1394):359–366, March 1998.   
+Daniel LK Yamins, Ha Hong, Charles F Cadieu, Ethan A Solomon, Darren Seibert, and James J DiCarlo. Performance-optimized hierarchical models predict neural responses in higher visual cortex. Proceedings of the National Academy of Sciences, 111(23):8619–8624, 2014.
+
+![](images/226db03924c89b79841754fc9f99db1fdc8843a39e6f69a1059af274fee81068.jpg)  
+Figure 6: Multiple different types of RNN architectures lead to similar levels of performance (Light blue $=$ exp 1, dark blue $=$ exp 2). We compare (from left to right): 1) 2 layer RNN with 50 units/layer, 2) 1 layer RNN with 100 units, 3) 3 layer RNN with 33 units/layer, 4) LSTM architecture as detailed in Hochreiter & Schmidhuber (1997)
+
+![](images/df943982a2c00d13c3bc0090bd0d43b27a51a7b2c341f716ea7baba6b7c36c2f.jpg)  
+Figure 7: RNNs with added spike history filters outperform GLMs and LNs according to a normalized log-likelihood metric. RNNs without spike history filters underperformed GLMs in some cases: loglikelihood metrics are calculated using spike history filters generated by actual spikes so these filters can improve log-likelihood without improving the fraction of explainable variance. The parameters of a multitask 2 layer RNN trained with a sigmoid nonlinearity were held fixed while the parameters of the last layer, including spike history filters, were trained. The normalized LL term was computed as detailed in Heitman et al. (2016) except the ideal model was trained using MSE.
+
+![](images/b5911fe677ee340946ba579a905a2ea21cfdb2527f01aefe6eaded4c39e19f6c.jpg)  
+Figure 8: Filters from image to Layer 1 RNN units in 2 layer RNN. Interpretable structures, including OFF and ON centers and surrounds, are visible
+
+![](images/ef91ff4fc37a3b5f73069bebe85e55356bd4a6d39b7a3f8449d4bb814a908b54.jpg)  
+Figure 9: The 31x31 shared LN spatial filter is pictured. The white rectangle indicates the boundary of the smaller $1 5 \mathrm { x } 1 5$ patch
+
+![](images/cb88b74e8c57e3a309d2889f9782467bfb349a13595fcbc5e628b171d74b15a6.jpg)  
+Figure 10: Comparison of two different initializations of a 2 layer RNN trained on Exp 1 OFF cells. Performance differs slightly for individual neurons but the resulting average FV values are very similar (0.819 vs 0.828)
\ No newline at end of file
diff --git a/parse/train/HkEI22jeg/HkEI22jeg_content_list.json b/parse/train/HkEI22jeg/HkEI22jeg_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..b500863a7b9d679992aa52cd5dda954b17bcbb68
--- /dev/null
+++ b/parse/train/HkEI22jeg/HkEI22jeg_content_list.json
@@ -0,0 +1,1038 @@
+[
+    {
+        "type": "text",
+        "text": "MULTILAYER RECURRENT NETWORK MODELS OF PRIMATE RETINAL GANGLION CELL RESPONSES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    {
+        "type": "text",
+        "text": "Eleanor Batty, Josh Merel ∗ Doctoral Program in Neurobiology & Behavior, Columbia University erb2180@columbia.edu,jsmerel@gmail.com ",
+        "bbox": [
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+            170,
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+    },
+    {
+        "type": "text",
+        "text": "Nora Brackbill \\* Department of Physics, Stanford University nbrack@stanford.edu ",
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+            183,
+            233,
+            470,
+            275
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Alexander Heitman   \nNeurosciences Graduate Program, University of California, San Diego   \nalexkenheitman@gmail.com   \nAlexander Sher & Alan Litke   \nSanta Cruz Institute for Particle Physics, University of California, Santa Cruz   \nsashake3@ucsc.edu, Alan.Litke@cern.ch   \nE.J. Chichilnisky   \nDepartment of Neurosurgery and Hansen Experimental Physics Laboratory, Stanford University   \nej@stanford.edu ",
+        "bbox": [
+            183,
+            296,
+            645,
+            338
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Liam Paninski ",
+        "text_level": 1,
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+            289,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Departments of Statistics and Neuroscience, Columbia University liam@stat.columbia.edu ",
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+            184,
+            501,
+            612,
+            527
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            454,
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+            544,
+            580
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Developing accurate predictive models of sensory neurons is vital to understanding sensory processing and brain computations. The current standard approach to modeling neurons is to start with simple models and to incrementally add interpretable features. An alternative approach is to start with a more complex model that captures responses accurately, and then probe the fitted model structure to understand the neural computations. Here, we show that a multitask recurrent neural network (RNN) framework provides the flexibility necessary to model complex computations of neurons that cannot be captured by previous methods. Specifically, multilayer recurrent neural networks that share features across neurons outperform generalized linear models (GLMs) in predicting the spiking responses of parasol ganglion cells in the primate retina to natural images. The networks achieve good predictive performance given surprisingly small amounts of experimental training data. Additionally, we present a novel GLM-RNN hybrid model with separate spatial and temporal processing components which provides insights into the aspects of retinal processing better captured by the recurrent neural networks. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our understanding of sensory processing in the brain is most straightforwardly reflected in our ability to model the process by which stimuli presented at the sensory periphery are transformed into the spiking activity of populations of neurons. For decades, researchers have interrogated stimulus-response neural properties using simplified targeted stimuli, such as bars, spots, or gratings. While these types of stimuli uncovered many interesting aspects of visual computation, they have several limitations (Barlow & Levick, 1965). These stimuli may not fully drive important components of neural response, and modeling efforts have often assumed a quasi-linear mapping from stimulus to firing rate. Subsequent efforts to characterize cells relied on white noise stimulation and building models through reverse correlation (de Boer R & Kuyper, 1968; Marmarelis & Naka, 1972; Chichilnisky, 2001). A standard model used to relate white noise to spiking responses is the linear-nonlinear-Poisson (LN) or generalized linear model (GLM) which consists of a spatiotemporal linear filtering of the stimulus followed by a nonlinearity and probabilistic spike generation (Chichilnisky, 2001; Simoncelli et al., 2004; Schwartz et al., 2006). Although this family of models have advanced our understanding, they do not optimally capture neural responses, especially to natural scenes which can lead to more complex responses than white noise stimuli (David et al., 2004). Even in the retina, early in the visual processing stream, these commonly-used models capture retinal ganglion cell (RGC) responses to natural stimuli less accurately than to white noise (Heitman et al., 2016). ",
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+        "text": "",
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+        "text": "Recently, deep neural networks have been used to dramatically improve performance on a diverse array of machine learning tasks (Krizhevsky et al., 2012; LeCun et al., 2015). Furthermore, these networks bear a loose resemblance to real neural networks, and provide a sufficiently rich model class that can still be roughly constrained to match the biological architecture (Kriegeskorte, 2015). Most previous research at this intersection of neuroscience and artificial neural networks has focused on training networks on a certain task, such as object recognition, and then comparing the computations performed in different layers of the artificial network to those performed by real neurons (Yamins et al., 2014). Here we take a different approach: we fit multilayer models directly to the spiking responses of neurons, an approach that has not been explored in detail (but see (McIntosh et al., 2016) for some recent independent parallel developments). ",
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+    {
+        "type": "text",
+        "text": "2 APPROACH",
+        "text_level": 1,
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+        "text": "We fit a range of models, detailed below, to spiking responses of primate RGCs. Our baseline comparisons are the GLM architectures that have been widely used to construct previous neural models (Pillow et al., 2008), though here we focus on individual neuronal responses (we leave modeling of correlations between neurons for future work). We focused on RNNs as a flexible framework in which to model more complex temporal and spatial nonlinearities. We also explored a number of network architectures involving features or weights shared across observed neurons. Given the complexity of the network architectures, we reasoned that sharing statistical strength across neurons by learning a shared feature space might improve predictive performance. This is conceptually a form of multitask learning - we are using a shared representation to achieve better generalization (Baxter, 2000). Motivated by previous research showing significant differences in the processing properties of the two cell types examined, ON and OFF parasol retinal ganglion cells, we fit separate models for each of these cell types (Chichilnisky & Kalmar, 2002). ",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 DATA COLLECTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We fit spiking responses of OFF and ON parasol retinal ganglion cells to natural scenes. Recordings were performed on isolated retina using a large-scale multi-electrode recording system (Litke et al., 2004; Frechette et al., 2005; Field et al., 2007). A standard spike sorting algorithm was used to identify spikes from different cells from the voltage signals on each electrode during visual stimulation (Litke et al., 2004). We focus on two separate experiments (the same experimental procedure in two separate retinas) here; analyses of other datasets yielded similar results. Models were fit separately for the two experiments due to animal to animal variability in cell properties, such as receptive field size and firing rate. Almost all spike sorted cells were used for training (exp $1 = 1 1 8$ OFF cells, 66 ON cells; exp $2 = 1 4 2$ OFF cells, 103 ON cells): two cells were removed due to data quality issues (see sec 3.3). Performance metrics in this paper are reported for the same subset of cells used in a previous study (Heitman et al., 2016). These cells passed passed a manual screen for spike sorting accuracy, demonstrated stable light responses, and met a convergence criteria in prior linear-nonlinear modeling (exp $1 = 1 0$ OFF cells, $1 8 \\mathrm { \\ : O N }$ cells; exp $2 = 6 5$ OFF cells, $1 4 \\mathrm { { \\ O N } }$ cells). The naturalistic movie stimulus consisted of images from the Van Hateren database shown for one second each, with spatial jitter based on eye movements during fixation by awake macaque monkeys (Z.M. Hafed and R.J. Krauzlis, personal communication), (van Hateren & van der Schaaf, 1998). An example stimulus can be found at https://youtu.be/sG_18Uz_6OE. 59 distinct natural scenes movies of length one minute (the training data) were interleaved with 59 repetitions of a 30 second movie (the test data). Interleaving ensured that the test movie repetitions spanned the same period of time as the training data and therefore experienced the same range of experimental conditions (in case of neural response drifts over time). The first 4 movies shown (2 training movies and 2 repetitions of the test movie) were excluded to avoid initial transients. Test metrics are reported for the last 29 seconds of the 30 second test movie for the same reason. For further details on the experimental set-up, data preprocessing, and visual stimulation, see Heitman et al. (2016). ",
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+        "img_path": "images/ba7501c108dec1599568c9756264f2e7f9540f733be3724d116380a263a47fb7.jpg",
+        "image_caption": [
+            "Figure 1: Example model architectures. (A) Shared LN model. The past few frames of the stimulus images are presented as inputs which are spatiotemporally filtered and passed through a nonlinearity to produce a firing rate, which drives a Poisson spiking process. (B) Two-layer RNN. The current frame of the stimulus feeds into a sequence of RNN layers (history dependence is implicit in the hidden unit activations) and a Poisson GLM draws weighted inputs from the activations of the hidden units of the last RNN layer and outputs predicted spike trains. Thus the last RNN layer represents a shared feature pool that all the RGCs can draw from. "
+        ],
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+    {
+        "type": "text",
+        "text": "3.2 MODEL TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "All models were implemented in Theano and trained on a combination of CPUs and GPUs (Theano Development Team, 2016). Training was performed using the Adam optimizer on the mean squared error (MSE) between predicted firing rate and true spikes (Kingma & Ba, 2014). We also experimented with optimizing a Poisson likelihood; this led to qualitatively similar results but occasionally less stable fits, so we focus on the MSE results here. All recurrent dynamics and temporal filters operated on time bins of $8 . 3 3 \\mathrm { m s }$ (the frame rate of the movie). Spike history terms and performance metrics were calculated for $0 . 8 3 3 ~ \\mathrm { m s }$ bins. We used the same split of training and validation data for both experiments: 104 thirty-second movies as training data and 10 thirty-second movies as a held-out validation set. ",
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+        "type": "text",
+        "text": "During training, the performance on the held-out validation set is checked after every pass through the training data. After each iteration through the training data, if the model exhibits significantly better validation performance than our previous best, we reset the minimum number of iterations to be twice the current iteration number. If we make it through those iterations without another significant improvement, we stop. We train for a maximum of 150 epochs, where we define one epoch as one pass through all the training data. The model with the best validation performance is saved and used to assess test performance. All models with shared parameters were trained on a combined MSE over ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "all neurons and the parameters picked were those which minimized validation MSE for all neurons.   \nFor individual LNs/GLMs/RNNs, the validation MSE was minimized for each neuron separately. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3.3 RECEPTIVE FIELD CENTER ESTIMATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In all models used in this paper, we estimate the receptive field (RF) center of each neuron in order to identify the appropriate portion of the image to use as input. We calculate a $2 5 0 ~ \\mathrm { m s }$ long spike triggered average (STA) using reverse correlation of the neuron’s spikes with a white noise stimulus. We reduce the noise in this STA by using a rank 1 approximation (singular value decomposition followed by reconstruction using the primary temporal and spatial components). We then smooth each frame of the STA via convolution with a Gaussian spatial filter. The center location is defined as the pixel location that has the maximum absolute magnitude over time. The center locations were visually assessed to check accuracy of the algorithm. Rare cases where the algorithm failed to identify the correct center indicated neurons that responded to very little of the image as their receptive field was more than half-way displaced out of the image. These two neurons (two Exp 1 ON cells) were removed from further analysis. If the receptive field center is close to the edge of the image, the image patch is padded with the average training stimulus value. ",
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+    {
+        "type": "text",
+        "text": "3.4 PERFORMANCE EVALUATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To quantitatively evaluate the accuracy of model spike predictions, we used the fraction of explainable variance, which has been described in previous literature (Heitman et al., 2016). Average firing rates over time are obtained after generating spikes from the model in $0 . 8 3 3 \\mathrm { m s }$ bins and smoothing with a Gaussian temporal filter $\\mathrm { \\ S D = 1 0 m s } ,$ ). The fraction of variance is computed as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/524b5fcddaa7cd08ebada43237605099b784cede3500a121d1dc4bc2c9bd55a5.jpg",
+        "text": "$$\nF ( r , r _ { s } ) = 1 - \\frac { \\sum _ { t } ( r ( t ) - r _ { s } ( t ) ) ^ { 2 } } { \\sum _ { t } ( r ( t ) - \\mu ) ^ { 2 } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $r ( t )$ is the smoothed recorded firing rate, $r _ { s } ( t )$ is the smoothed predicted firing rate, and $\\mu$ is the average recorded rate. Finally, to account for the reproducibility of responses over repeated trials, we normalize by the fraction of variance captured by using the average firing rate on the odd $( r _ { o } )$ trials of the repeated test movie to predict responses on the even $( r _ { e } )$ trials: ",
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+        "text": "$$\nF V = \\frac { F ( r , r _ { s } ) } { F ( r _ { e } , r _ { o } ) } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4 MODEL ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 NETWORK ARCHITECTURES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Individual LNs and GLMs: The linear-nonlinear model (LN) consists of a spatiotemporal filtering of the $3 1 \\mathrm { x } 3 1 \\mathrm { x } 3 0$ movie patch $X _ { t }$ , width by height by time) surrounding the estimated center of the neuron’s receptive field plus a bias term $( b )$ , followed by a sigmoid nonlinearity $( f )$ , and Poisson spike generation to produce the responses $r _ { t }$ . The generalized linear model (GLM), given by ",
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+    {
+        "type": "equation",
+        "img_path": "images/c95f1c10bb77a267737d76ad093c01fefc534bbf51438dfa1e2afb44bda5079d.jpg",
+        "text": "$$\nr _ { t } \\sim P o i s s \\left[ f \\bigg ( \\vec { w } _ { s } ^ { T } ( X _ { t } \\vec { w } _ { t } ) + b + \\sum _ { i } { h _ { i } r _ { t - i } } \\bigg ) \\right] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "has the same architecture with the addition of a post-spike history filter $h$ before the nonlinearity $f$ (Pillow et al., 2008). We used a rank 1 approximation of the full spatiotemporal filter (higher rank models did not significantly improve fits on a subset of examined neurons), resulting in a vectorized 31x31 spatial filter $( \\vec { w } _ { s } )$ and a 30 bin temporal filter $( \\vec { w } _ { t } )$ which spans $2 5 0 \\mathrm { m s }$ (Heitman et al., 2016). The post-spike history filter consists of a weighted sum of a basis of 20 raised cosines spanning approximately $1 0 0 \\mathrm { m s }$ (Pillow et al., 2008). The models with spike history were fit by initializing with the model fit without spike history. The filter either operates on the recorded spikes (training and validation) or the spikes generated by the model (testing). The nonlinearity is the logistic sigmoid: $f = L / ( 1 + \\exp ( \\bar { - } x ) )$ , which has been shown to improve fitting over an exponential nonlinearity for modeling RGC responses to natural scenes (Heitman et al., 2016). ",
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+    {
+        "type": "image",
+        "img_path": "images/07780a793573a414ea2f391472bf6f2c726ecc672ededb70f868499ffd486ed6.jpg",
+        "image_caption": [
+            "Figure 2: Model performance. (A) Mean $\\pm$ std. err. of the fraction of explainable variance for criteria-passing subset of OFF and ON cells for various model architectures. (B) Scatter plots show individual neural performance from LN and RNN model; each dot corresponds to one cell. Negative FV values are shown on relevant axis as $\\operatorname { F V } { = } 0$ (C) Hybrid model performance, quantified by the ratio between the multitask LN to multitask hybrid performance gap and the multitask LN to multitask RNN performance gap (one high outlier not pictured for both Exp 1 ON and Exp 2 ON ) "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Shared LN: In this model, the architecture is similar to the individual LNs but all cells of a given type (OFF or ON) share the same temporal and spatial filters (Figure 1A; note that the spatial filters are displaced to the RF center of each individual RGC). All other parameters are individually tuned for each observed neuron. There is an additional gain term that weights the output of the filtering individually for each observed neuron. ",
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+        "type": "text",
+        "text": "Two-layer RNN, 50 units: In this architecture, there are two recurrent neural network (RNN) layers between the image patch and Poisson neural unit: ",
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+        "img_path": "images/a1e76811d4c804e1044b18466984080c4d1ad18aa7835f48d5efffd5b5254f56.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\vec { h } _ { j , t } ^ { ( 1 ) } = \\operatorname* { m a x } ( 0 , { U } _ { 1 } \\vec { s } _ { j , t } + { V } _ { 1 } \\vec { h } _ { j , t - 1 } ^ { ( 1 ) } + \\vec { c } ) } \\\\ & { \\vec { h } _ { j , t } ^ { ( 2 ) } = \\operatorname* { m a x } ( 0 , { U } _ { 2 } \\vec { h } _ { j , t } ^ { ( 1 ) } + { V } _ { 2 } \\vec { h } _ { j , t - 1 } ^ { ( 2 ) } + \\vec { d } ) } \\\\ & { { r } _ { j , t } \\sim P o i s s \\left[ f ( \\vec { w } _ { j } ^ { T } \\vec { h } _ { j , t } ^ { ( 2 ) } + b _ { j } ) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The activity of the 50 units in the first RNN layer at time $t$ is given by $\\vec { h } _ { j , t } ^ { ( 1 ) }$ in Eqn. 4. These units are rectified linear, and receive input from the vectorized 31x31 image patch surrounding the center of neuron $j$ ’s receptive field, $\\vec { s } _ { j , t }$ , with weights $U _ { 1 }$ , along with input from the other units in the layer with weights $V _ { 1 }$ and a bias $\\vec { c }$ . The output of the first RNN is then fed into a second RNN with similar architecture. The firing rate for each observed neuron in the final layer is then given by Eqn. 6, and is a weighted sum of the recurrent units plus a bias $b _ { j }$ , followed by a softplus nonlinearity $f = \\log ( 1 + \\exp ( - x ) )$ . Note that all parameters are shared across neurons except for the weights to the final layer and the final bias terms ( $\\overrightarrow { w } _ { j }$ and $b _ { j }$ ). ",
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+    {
+        "type": "text",
+        "text": "GLM-RNN Hybrid: The GLM-RNN hybrid model consists of a spatial filter followed by a two-layer RNN. The architecture resembles that of the full two-layer RNN with 50 units, except the input to the first layer is a scalar (post multiplication with the spatial filter) at each time step instead of the full image patch; thus the RNN in this model is responsible for shaping the temporal properties of the output, but does not affect spatial processing after the first linear spatial filtering stage. All weights are shared across neurons except for weights to the final layer $( \\vec { w } _ { j } )$ and the final bias terms $( b _ { j } )$ ",
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+        "img_path": "images/91e264c19d450d0ab71845440ff4e8c2c4a4c17d7cc2e0d9c27f007c9d1915e0.jpg",
+        "image_caption": [
+            "Figure 3: (A,B) Rasters showing spiking responses for 57 trials (each row corresponds to a single trial) for an OFF and ON cell from experiment 1 for 10 seconds of a novel natural scenes movie. Example cells chosen had near average difference between LN and RNN performance. Red ticks denote time at which one natural image was replaced by another. (C,D) Average predicted spikes over trials smoothed with Gaussian $\\mathrm { S D } = 1 0 \\mathrm { m s }$ ) for same 10 seconds of the novel natural scenes movie show qualitative differences among models. Dotted vertical lines align with red ticks in (A,B). "
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+        "img_path": "images/08ef2f8f36d5226381b9aa1a594eaa8fff1c35357a015b6f59963d7e90306765.jpg",
+        "text": "$$\n\\begin{array} { l } { { y _ { j , t } = \\vec { w } _ { s } ^ { T } \\vec { s } _ { j , t } } } \\\\ { { \\vec { h } _ { j , t } ^ { ( 1 ) } = m a x ( 0 , \\vec { u _ { 1 } } y _ { j , t } + V _ { 1 } \\vec { h } _ { j , t - 1 } ^ { ( 1 ) } + \\vec { c } ) } } \\\\ { { \\vec { h } _ { j , t } ^ { ( 2 ) } = m a x ( 0 , U _ { 2 } h _ { j , t } ^ { ( 1 ) } + V _ { 2 } \\vec { h } _ { j , t - 1 } ^ { ( 2 ) } + \\vec { d } ) } } \\\\ { { r _ { j , t } \\sim P o i s s \\left[ f ( \\vec { w } _ { j } ^ { T } \\vec { h } _ { t } ^ { ( 2 ) } + b _ { j } ) \\right] . } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4.2 MODEL PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "RNNs of varying architectures consistently outperformed LNs and GLMs in predicting neural spiking responses to a novel natural scene movie for both OFF and ON parasol retinal ganglion cells in both experiments (Figure 2). A shared two-layer recurrent network consistently captures around $8 0 \\%$ of the explainable variance across experiments and cell types. Other recurrent architectures (1-3 layer RNNs and a 2 layer LSTM) led to similar levels of performance (Supplementary Figure 6). The increase in performance according to the fraction of explainable variance metric was not an average effect: almost all neurons were significantly better predicted by the RNN (Figure 2B). A 2 layer RNN model with additional trained spike history filters outperformed GLMs and LNs according to a normalized log likelihood metric (Supplementary Figure 7). ",
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+    {
+        "type": "text",
+        "text": "Inspection of the mean predicted firing rate traces for LNs and RNNs in Figure 3 reveals that the recurrent network seems to be capturing the timing of firing more precisely. The LN often predicts a general increase in firing rate at the correct times, but the RNN captures the sudden increase in firing rate followed by decay which often occurs when the image changes. On the other hand, the LN models sometimes predict modest increases or decreases in firing rate that the recurrent nets miss. ",
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+        "type": "text",
+        "text": "Understanding why the recurrent models improve performance is a challenging task due to the black-box nature of deep networks. The first layer filters $U _ { 1 }$ , from image patches to recurrent units) have an interpretable structure resembling traditional receptive fields expected in the retina (Supplementary Figure 8). However, the computations performed by the recurrent units are difficult to tease apart, because the weights are less interpretable. Thus, instead of attempting a mechanistic explanation of the internals of the RNN, we focused on what additional captured information resulted in the improved RNN performance. ",
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+        "img_path": "images/e211a5f688b38b0a2e9fd82f71903fe4167ac4ec30a222c8edc9289a38ac0304.jpg",
+        "image_caption": [
+            "Figure 4: Model predictive performance on held-out data as a function of the amount of training data. Error bars show SEM over 3 iterations of the mean FV over all neurons "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            106,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "One possibility is that capturing nonlinear effects in parts of the image far from the receptive field center improved predictions (McIlwain, 1964; Passaglia et al., 2009). We restricted the size of the image patch surrounding each receptive field center from 31x31 to 15x15 (Supplementary Figure 9). Shared RNNs trained on the smaller image patch size did as well, or better, than those trained on the larger patch across almost all combinations of cell type and experiment. (We see a similar small improvement when training the LN models on the small patch.) Thus we concluded that long-range nonlinear spatial interactions do not contribute to the increased performance produced by the RNNs. ",
+        "bbox": [
+            174,
+            369,
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+            467
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also investigated whether nonlinear spatial interactions or nonlinear temporal processing primarily contributed to better predictions. To accomplish this, we constructed a GLM-RNN hybrid, described previously, in which a single spatial filter precedes a two-layer RNN - effectively allowing only temporal nonlinearities to be captured. This model improved prediction over the LNs and GLMs but did not reach full RNN performance. The amount by which this model closed the gap differed for different experiments and cell types. We quantified this by computing the difference between multitask RNN and multitask LN performance for each neuron and the difference between multitask hybrid and multitask LN performance. We divide the latter by the former (on a cell-by-cell basis) to obtain the ratios summarized in Figure 2C. The hybrid model closed greater than half of the gap on average between multitask LN and RNN performance, indicating that the richer temporal dynamics of the RNN model account for a large part of the difference between RNN and LN performance, though spatial nonlinearities play a role too. ",
+        "bbox": [
+            174,
+            474,
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+            640
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 MODEST TRAINING DATA LENGTH SUFFICES FOR GOOD PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            661,
+            792,
+            676
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Deep networks can be complex and often require large amounts of data to adequately train: convolutional neural networks used for object recognition are trained on over a million images (Krizhevsky et al., 2012). Standard neuroscience experiments yield limited data sets, so it is crucial to assess whether we have enough data to adequately fit our network architectures. We trained the RNN on varying amounts of data, and ran several different iterations of the network to explore variation over random initializations and randomly chosen training sets. These results are shown for both ON and OFF cells in Figure 4. Surprisingly small amounts of training data resulted in good predictive abilities. For larger amounts of training data, different iterations resulted in very similar mean fraction of variance values, indicating fairly robust fitting in these models. See Supplementary Figure 10 for further details. ",
+        "bbox": [
+            174,
+            691,
+            826,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 BENEFITS OF MULTITASK FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            852,
+            531,
+            867
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We investigated whether the multitask framework with shared parameters across neurons actually helps to improve predictive performance with reasonable amounts of experimental data. First, we quantified the benefits of parameter-sharing in the simple LN model. This is a highly constrained framework: every cell has the same spatial and temporal filter. The shared LN does not improve performance for most neurons (Figure 5A). ",
+        "bbox": [
+            176,
+            882,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/44514c8539a1d7b10219249491b9c0775fb9630004f28c50bde26eb5d5022f87.jpg",
+        "image_caption": [
+            "Figure 5: Shared vs individual fits for LN model (A) and RNN model (B). $1 0 \\mathrm { O F F }$ and $1 0 \\mathrm { { O N } }$ cells from each experiment are pictured (Light blue $=$ exp 1, dark blue $=$ exp 2). Negative FV values are pictured as $\\mathrm { F V } = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            109,
+            647,
+            325
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            431,
+            823,
+            460
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We expected the multitask framework to be more helpful applied to the RNN model because in this case we are sharing features but not all parameters across neurons. Indeed, the multitask RNN consistently outperformed RNNs trained individually on single neurons (Figure 5B); individuallytrained RNNs also had much more variable losses than did the multitask-trained RNNs. In a realistic experimental setting with limited data, the multitask framework is a useful way to leverage all of the data collected for all neurons. ",
+        "bbox": [
+            174,
+            467,
+            825,
+            551
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            593,
+            318,
+            609
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Using retinal neurons responding to natural scenes as an example, we showed that: using deep networks to model neural spiking responses can significantly improve prediction over current state-ofthe-art models; sharing information across neurons in a multi-task framework leads to better and more stable predictions; and these models work well even given relatively small amounts of experimental data. We believe that the multitask RNN framework presented here will enable new, richer models of complex nonlinear spiking computations in other brain areas. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            722
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "While one could argue that we have merely exchanged the black box of the brain for another black box, just having a more predictive model is an important tool for research: these predictive models of the primate retina can be used in retinal prosthetics research, to probe decoding, and as a first stage of processing in the modeling of higher visual areas. Additionally, the recurrent network is more accessible and available for experimentation and quantitative analysis. For example, the trained neural network models may guide choices for more accurate simpler models by identifying key computational features that are important to include. Training smaller models on the denoised compression of spiking data (the predicted firing rate) may help them to learn features they otherwise would not (Ba & Caruana, 2014). The deep network approach allows one to determine types of information important to the neuron without having to build an exact mechanistic model of how such information is incorporated, as demonstrated by our finding that both spatial and temporal nonlinearities are not fully captured by the standard pseudo-linear models. We hope in future work to gain a more thorough and quantitative understanding of the dynamics captured by the recurrent networks and to extend this approach to higher sensory areas. ",
+        "bbox": [
+            174,
+            729,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
+            326,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Funding for this research was provided by the National Science Foundation Graduate Research Fellowship Program under grant No. DGE-114747 (NB), Grant Number No. DGE-16-44869 (EB), the National Science Foundation IGERT Training Grant No. 0801700 (NB), the National Institutes of Health Grant EY017992 (EJC), NSF CRCNS IIS-1430239 (LP, EJC) and Google Faculty Research awards (LP, EJC); in addition, this work was supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/ Interior Business Center (DoI/IBC) contract number D16PC00003 (LP). The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/IBC, or the U.S. Government. ",
+        "bbox": [
+            174,
+            127,
+            825,
+            280
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            301,
+            285,
+            316
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Jimmy Ba and Rich Caruana. Do deep nets really need to be deep? Advances in Neural Information Processing Systems, 2014. ",
+        "bbox": [
+            176,
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+    {
+        "type": "text",
+        "text": "HB Barlow and William R Levick. The mechanism of directionally selective units in rabbit’s retina. The Journal of Physiology, 178(3):477, 1965. ",
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+        "text": "Stephen V. David, William E. Vinje, and Jack L. Gallant. Natural stimulus statistics alter the receptive field structure of v1 neurons. The Journal of Neuroscience, 24(31):6991–7006, 2004. ",
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+        "text": "de Boer R and P Kuyper. Triggered correlation. IEEE Transactions on Biomedical Engineering, 15: 169–179, 1968. ",
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+    {
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+        "text": "Greg D. Field, Alexander Sher, Jeffrey L. Gauthier, Martin Greschner, Jonathon Shlens, Alan M. Litke, and E. J. Chichilnisky. Spatial properties and functional organization of small bistratified ganglion cells in primate retina. The Journal of Neuroscience, 27(48):13261–13272, 2007. ",
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+    {
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+        "text": "E. S. Frechette, A. Sher, M. I. Grivich, D. Petrusca, A. M. Litke, and E. J. Chichilnisky. Fidelity of the ensemble code for visual motion in primate retina. Journal of Neurophysiology, 94(1):119–135, 2005. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Alexander Heitman, Nora Brackbill, Martin Greschner, Alexander Sher, Alan M. Litke, and E.J. Chichilnisky. Testing pseudo-linear models of responses to natural scenes in primate retina. bioRxiv, 2016. ",
+        "bbox": [
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+        "text": "Nikolaus Kriegeskorte. Deep neural networks: A new framework for modeling biological vision and brain information processing. Annual Review of Vision Science, 1:417–446, 2015. ",
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+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, pp. 1097–1105, 2012. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521:436–444, 2015. ",
+        "bbox": [
+            173,
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+    },
+    {
+        "type": "text",
+        "text": "A. M. Litke, N. Bezayiff, E. J. Chichilnisky, W. Cunningham, W. Dabrowski, A. A. Grillo, M. Grivich, P. Grybos, P. Hottowy, S. Kachiguine, R. S. Kalmar, K. Mathieson, D. Petrusca, M. Rahman, and A. Sher. What does the eye tell the brain?: Development of a system for the large-scale recording of retinal output activity. IEEE Transactions on Nuclear Science, 51(4):1434–1440, 2004.   \nP.Z. Marmarelis and K. Naka. White-noise analysis of a neuron chain: an application of the wiener theory. Science, 175:1276–1278, 1972.   \nJames T. McIlwain. Receptive fields of optic tract axons and lateral geniculate cells: peripheral extent and barbiturate sensitivity. Journal of Neurophysiology, 27(6):1154–1173, 1964.   \nLane McIntosh, Niru Maheswaranathan, Aran Nayebi, Surya Ganguli, and Stephen Baccus. Deep convolutional neural network models of the retinal response to natural scenes. In Cosyne Abstracts, Salt Lake City USA, 2016.   \nCL. Passaglia, Freeman DK., and Troy JB. Effects of remote stimulation of the modulated activity of cat retina. Journal of Neuroscience, 29, 2009.   \nJonathan W Pillow, Jonathon Shlens, Liam Paninski, Alexander Sher, Alan M Litke, E.J. Chichilnisky, and Eero P Simoncelli. Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207):995–999, 2008.   \nOdelia Schwartz, Jonathan W. Pillow, Nicole C. Rust, and Eero P. Simoncelli. Spike-triggered neural characterization. Journal of Vision, 6(4):13, 2006.   \nEero Simoncelli, Jonathan W. Pillow, Liam Paninski, and Odelia Schwartz. Characterization of neural responses with stochastic stimuli, pp. 327–338. MIT Press, 2004.   \nTheano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016.   \nJ. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings. Biological sciences / The Royal Society, 265(1394):359–366, March 1998.   \nDaniel LK Yamins, Ha Hong, Charles F Cadieu, Ethan A Solomon, Darren Seibert, and James J DiCarlo. Performance-optimized hierarchical models predict neural responses in higher visual cortex. Proceedings of the National Academy of Sciences, 111(23):8619–8624, 2014. ",
+        "bbox": [
+            169,
+            98,
+            828,
+            599
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/226db03924c89b79841754fc9f99db1fdc8843a39e6f69a1059af274fee81068.jpg",
+        "image_caption": [
+            "Figure 6: Multiple different types of RNN architectures lead to similar levels of performance (Light blue $=$ exp 1, dark blue $=$ exp 2). We compare (from left to right): 1) 2 layer RNN with 50 units/layer, 2) 1 layer RNN with 100 units, 3) 3 layer RNN with 33 units/layer, 4) LSTM architecture as detailed in Hochreiter & Schmidhuber (1997) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            359,
+            140,
+            638,
+            411
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/df943982a2c00d13c3bc0090bd0d43b27a51a7b2c341f716ea7baba6b7c36c2f.jpg",
+        "image_caption": [
+            "Figure 7: RNNs with added spike history filters outperform GLMs and LNs according to a normalized log-likelihood metric. RNNs without spike history filters underperformed GLMs in some cases: loglikelihood metrics are calculated using spike history filters generated by actual spikes so these filters can improve log-likelihood without improving the fraction of explainable variance. The parameters of a multitask 2 layer RNN trained with a sigmoid nonlinearity were held fixed while the parameters of the last layer, including spike history filters, were trained. The normalized LL term was computed as detailed in Heitman et al. (2016) except the ideal model was trained using MSE. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            320,
+            503,
+            678,
+            780
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/b5911fe677ee340946ba579a905a2ea21cfdb2527f01aefe6eaded4c39e19f6c.jpg",
+        "image_caption": [
+            "Figure 8: Filters from image to Layer 1 RNN units in 2 layer RNN. Interpretable structures, including OFF and ON centers and surrounds, are visible "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            108,
+            825,
+            324
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef91ff4fc37a3b5f73069bebe85e55356bd4a6d39b7a3f8449d4bb814a908b54.jpg",
+        "image_caption": [
+            "Figure 9: The 31x31 shared LN spatial filter is pictured. The white rectangle indicates the boundary of the smaller $1 5 \\mathrm { x } 1 5$ patch "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            351,
+            390,
+            656,
+            594
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb88b74e8c57e3a309d2889f9782467bfb349a13595fcbc5e628b171d74b15a6.jpg",
+        "image_caption": [
+            "Figure 10: Comparison of two different initializations of a 2 layer RNN trained on Exp 1 OFF cells. Performance differs slightly for individual neurons but the resulting average FV values are very similar (0.819 vs 0.828) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            305,
+            657,
+            669,
+            861
+        ],
+        "page_idx": 11
+    }
+]
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diff --git a/parse/train/HkEI22jeg/HkEI22jeg_middle.json b/parse/train/HkEI22jeg/HkEI22jeg_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..58b0b21bad03a91bf4a331c3c054f8e0e6cc39ce
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+                                    "score": 1.0,
+                                    "content": "be found at https://youtu.be/sG_18Uz_6OE. 59 distinct natural scenes movies of length",
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+                                            "content": "Figure 1: Example model architectures. (A) Shared LN model. The past few frames of the stimulus",
+                                            "type": "text"
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+                                            "score": 1.0,
+                                            "content": "units of the last RNN layer and outputs predicted spike trains. Thus the last RNN layer represents a",
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+                                            "score": 1.0,
+                                            "content": "shared feature pool that all the RGCs can draw from.",
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+                            "index": 6
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+                                    "content": "Krauzlis, personal communication), (van Hateren & van der Schaaf, 1998). An example stimulus can",
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+                            "index": 12
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+                        {
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+                                    "score": 1.0,
+                                    "content": "be found at https://youtu.be/sG_18Uz_6OE. 59 distinct natural scenes movies of length",
+                                    "type": "text"
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+                            ],
+                            "index": 13
+                        },
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+                                    "score": 1.0,
+                                    "content": "one minute (the training data) were interleaved with 59 repetitions of a 30 second movie (the test",
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+                                    "score": 1.0,
+                                    "content": "data). Interleaving ensured that the test movie repetitions spanned the same period of time as the",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "response drifts over time). The first 4 movies shown (2 training movies and 2 repetitions of the test",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "movie) were excluded to avoid initial transients. Test metrics are reported for the last 29 seconds of",
+                                    "type": "text"
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+                            ],
+                            "index": 18
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the 30 second test movie for the same reason. For further details on the experimental set-up, data",
+                                    "type": "text"
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+                            ],
+                            "index": 19
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "preprocessing, and visual stimulation, see Heitman et al. (2016).",
+                                    "type": "text"
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+                            ],
+                            "index": 20
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+                    ],
+                    "index": 15
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+                                    "content": "3.2 MODEL TRAINING",
+                                    "type": "text"
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+                            "index": 21
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+                                    "content": "All models were implemented in Theano and trained on a combination of CPUs and GPUs (Theano",
+                                    "type": "text"
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+                            ],
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+                        {
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+                        {
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+                                    "score": 1.0,
+                                    "content": "error (MSE) between predicted firing rate and true spikes (Kingma & Ba, 2014). We also experimented",
+                                    "type": "text"
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+                            ],
+                            "index": 24
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "with optimizing a Poisson likelihood; this led to qualitatively similar results but occasionally less",
+                                    "type": "text"
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+                            ],
+                            "index": 25
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+                        {
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+                                    "score": 1.0,
+                                    "content": "stable fits, so we focus on the MSE results here. All recurrent dynamics and temporal filters operated",
+                                    "type": "text"
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+                            ],
+                            "index": 26
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "on time bins of",
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+                                    "score": 0.4,
+                                    "content": "8 . 3 3 \\mathrm { m s }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(the frame rate of the movie). Spike history terms and performance metrics",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "were calculated for",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.26,
+                                    "content": "0 . 8 3 3 ~ \\mathrm { m s }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "bins. We used the same split of training and validation data for both",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "experiments: 104 thirty-second movies as training data and 10 thirty-second movies as a held-out",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
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+                                    "content": "validation set.",
+                                    "type": "text"
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+                    ],
+                    "index": 26
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+                {
+                    "type": "text",
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+                                    "content": "the training data. After each iteration through the training data, if the model exhibits significantly",
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+                                    "score": 1.0,
+                                    "content": "better validation performance than our previous best, we reset the minimum number of iterations to be",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "twice the current iteration number. If we make it through those iterations without another significant",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "to identify the appropriate portion of the image to use as input. We calculate a",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.37,
+                                    "content": "2 5 0 ~ \\mathrm { m s }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "long spike",
+                                    "type": "text"
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+                            ],
+                            "index": 4
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+                        {
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+                                    "score": 1.0,
+                                    "content": "triggered average (STA) using reverse correlation of the neuron’s spikes with a white noise stimulus.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
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+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We reduce the noise in this STA by using a rank 1 approximation (singular value decomposition",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "followed by reconstruction using the primary temporal and spatial components). We then smooth",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "each frame of the STA via convolution with a Gaussian spatial filter. The center location is defined as",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the pixel location that has the maximum absolute magnitude over time. The center locations were",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "visually assessed to check accuracy of the algorithm. Rare cases where the algorithm failed to identify",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the correct center indicated neurons that responded to very little of the image as their receptive field",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "was more than half-way displaced out of the image. These two neurons (two Exp 1 ON cells) were",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "removed from further analysis. If the receptive field center is close to the edge of the image, the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
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+                        {
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+                                    "score": 1.0,
+                                    "content": "image patch is padded with the average training stimulus value.",
+                                    "type": "text"
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+                            "index": 14
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+                                    "content": "3.4 PERFORMANCE EVALUATION",
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+                                    "score": 1.0,
+                                    "content": "To quantitatively evaluate the accuracy of model spike predictions, we used the fraction of explainable",
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+                                    "type": "text"
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+                                    "score": 0.72,
+                                    "content": "\\mathrm { \\ S D = 1 0 m s } ,",
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+                                    "score": 1.0,
+                                    "content": "). The fraction of variance is computed as",
+                                    "type": "text"
+                                }
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+                            "index": 19
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+                                    "score": 0.95,
+                                    "content": "F ( r , r _ { s } ) = 1 - \\frac { \\sum _ { t } ( r ( t ) - r _ { s } ( t ) ) ^ { 2 } } { \\sum _ { t } ( r ( t ) - \\mu ) ^ { 2 } }",
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+                                    "score": 1.0,
+                                    "content": "is",
+                                    "type": "text"
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+                            "index": 22
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+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the average recorded rate. Finally, to account for the reproducibility of responses over repeated trials,",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "we normalize by the fraction of variance captured by using the average firing rate on the odd",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.82,
+                                    "content": "( r _ { o } )",
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "trials of the repeated test movie to predict responses on the even",
+                                    "type": "text"
+                                },
+                                {
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+                                    "content": "( r _ { e } )",
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+                                    "score": 1.0,
+                                    "content": "trials:",
+                                    "type": "text"
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+                                    "type": "interline_equation",
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+                                    "type": "text"
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": ", width by height by time) surrounding the estimated center of the",
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+                                    "content": "neuron’s receptive field plus a bias term",
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+                                    "score": 1.0,
+                                    "content": ", followed by a sigmoid nonlinearity",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": ", and Poisson",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "removed from further analysis. If the receptive field center is close to the edge of the image, the",
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+                                    "content": "the larger patch across almost all combinations of cell type and experiment. (We see a similar small",
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+                                    "content": "improvement when training the LN models on the small patch.) Thus we concluded that long-range",
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+                                    "type": "text"
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+                                    "content": "obtain the ratios summarized in Figure 2C. The hybrid model closed greater than half of the gap on",
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+                                    "content": "5 MODEST TRAINING DATA LENGTH SUFFICES FOR GOOD PERFORMANCE",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "tional neural networks used for object recognition are trained on over a million images (Krizhevsky",
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+                                    "content": "et al., 2012). Standard neuroscience experiments yield limited data sets, so it is crucial to assess",
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+                                    "content": "OFF cells in Figure 4. Surprisingly small amounts of training data resulted in good predictive abilities.",
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+                                    "score": 1.0,
+                                    "content": "variance values, indicating fairly robust fitting in these models. See Supplementary Figure 10 for",
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+                                    "content": "5 MODEST TRAINING DATA LENGTH SUFFICES FOR GOOD PERFORMANCE",
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+                                    "score": 1.0,
+                                    "content": "tional neural networks used for object recognition are trained on over a million images (Krizhevsky",
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+                                    "content": "OFF cells in Figure 4. Surprisingly small amounts of training data resulted in good predictive abilities.",
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+                                    "content": "variance values, indicating fairly robust fitting in these models. See Supplementary Figure 10 for",
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diff --git a/parse/train/HkEI22jeg/HkEI22jeg_model.json b/parse/train/HkEI22jeg/HkEI22jeg_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..4a275936df60669f32edbfd6e56db285bb2cd4cd
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diff --git a/parse/train/HyMTkQZAb/HyMTkQZAb.md b/parse/train/HyMTkQZAb/HyMTkQZAb.md
new file mode 100644
index 0000000000000000000000000000000000000000..a11f12b5ee22ecea5fbe6af51c617816827c698d
--- /dev/null
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@@ -0,0 +1,1063 @@
+# KRONECKER-FACTORED CURVATURE APPROXIMATIONS FOR RECURRENT NEURAL NETWORKS
+
+James Martens   
+DeepMind   
+jamesmartens@google.com
+
+# Jimmy Ba
+
+Department of Computer Science University of Toronto Toronto, Canada jimmy@psi.toronto.edu
+
+Matthew Johnson Google Brain mattjj@google.com
+
+# ABSTRACT
+
+Kronecker-factor Approximate Curvature (Martens & Grosse, 2015) (K-FAC) is a 2nd-order optimization method which has been shown to give state-of-the-art performance on large-scale neural network optimization tasks (Ba et al., 2017). It is based on an approximation to the Fisher information matrix (FIM) that makes assumptions about the particular structure of the network and the way it is parameterized. The original K-FAC method was applicable only to fully-connected networks, although it has been recently extended by Grosse & Martens (2016) to handle convolutional networks as well. In this work we extend the method to handle RNNs by introducing a novel approximation to the FIM for RNNs. This approximation works by modelling the statistical structure between the gradient contributions at different time-steps using a chain-structured linear Gaussian graphical model, summing the various cross-moments, and computing the inverse in closed form. We demonstrate in experiments that our method significantly outperforms general purpose state-of-the-art optimizers like SGD with momentum and Adam on several challenging RNN training tasks.
+
+# 1 INTRODUCTION
+
+As neural networks have become ubiquitous in both research and applications the need to efficiently train has never been greater. The main workhorses for neural net optimization are stochastic gradient descent (SGD) with momentum and various 2nd-order optimizers that use diagonal curvature-matrix approximations, such as RMSprop (Tieleman & Hinton, 2012) and Adam (Ba & Kingma, 2015). While the latter are typically easier to tune and work better out of the box, they unfortunately only offer marginal performance improvements over well-tuned SGD on most problems.
+
+Because modern neural networks have many millions of parameters it is computationally too expensive to compute and invert an entire curvature matrix and so approximations are required. While early work on non-diagonal curvature matrix approximations such as TONGA (Le Roux et al., 2008) and the Hessian-free (HF) approach (Martens, 2010; Martens & Sutskever, 2011; 2012; Desjardins et al., 2013; Sainath et al., 2013) demonstrated the potential of such methods, they never achieved wide adoption due to issues of scalability (to large models in the case of the former, and large datasets in the case of the latter).
+
+Motivated in part by these older results and by the more recent success of centering and normalization methods (e.g. Schraudolph, 1998; Vatanen et al., 2013; Ioffe & Szegedy, 2015) a new family of methods has emerged that are based on non-diagonal curvature matrix approximations the rely on the special structure of neural networks. Such methods, which include Kronecker-factored approximated curvature (K-FAC) (Martens & Grosse, 2015), Natural Neural Nets (Desjardins et al., 2015), Practical Riemannian Neural Networks (Marceau-Caron & Ollivier, 2016), and others (Povey et al., 2015), have achieved state-of-the-art optimization performance on various challenging neural network training tasks and benchmarks.
+
+While the original K-FAC method is applicable only to standard feed-forward networks with fully connected layers, it has recently been extended to handle convolutional networks (Grosse & Martens, 2016) through the introduction of the “Kronecker Factors for Convolution” (KFC) approximation. Ba et al. (2017) later developed a distributed asynchronous version which proposed additional approximations to handle very large hidden layers.
+
+In this work we develop a new family of curvature matrix approximations for recurrent neural networks (RNNs) within the same design space. As in the original K-FAC approximation and the KFC approximation, we focus on the Fisher information matrix (a popular choice of curvature matrix), and show how it can be approximated in different ways through the adoption of various approximating assumptions on the statistics of the network’s gradients. Our main novel technical contribution is an approximation which uses a chain-structured linear Gaussian graphical model to describe the statistical relationship between gradient contributions coming from different time-steps. Somewhat remarkably, it is possible to sum the required cross-moments to obtain a Fisher approximations which has enough special algebraic structure that it can still be efficiently inverted. In experiments we demonstrate the usefulness of our approximations on several challenging RNN training tasks.
+
+# 2 NOTATION AND BACKGROUND
+
+# 2.1 NETWORK, LOSS, AND OBJECTIVE FUNCTION
+
+We denote by $f ( x , \theta )$ the neural network function associated evaluated on input $x$ , where $\theta$ are the parameters. We will assume a loss function of the form $L ( y , z ) = - \log r ( y | \bar { z } )$ , where $r$ is the density function associated with a predictive distribution $R$ . The loss associated with a single training case is then given by $L ( y , f ( x , \bar { \theta } ) ) \equiv - \log r ( y | f ( x , \theta ) )$ . Throughout the rest of this document we will use the following special notation for derivatives of the single-case loss w.r.t. some arbitrary variable $Z$ (possibly matrix-valued):
+
+$$
+\mathcal { D } Z = \frac { \mathrm { d } L ( y , f ( x , \theta ) ) } { \mathrm { d } Z }
+$$
+
+The objective function which we wish to minimize is the expected loss $h ( \theta ) = \mathbb { E } _ { Q } [ L ( y , f ( x , \theta ) ) ]$ over the training distribution $Q$ on $x$ and $y$ .
+
+2.2 THE FISHER, THE NATURAL GRADIENT, AND 2ND-ORDER OPTIMIZATION
+
+The Fisher information matrix (aka “the Fisher”) associated with the model’s predictive distribution $P _ { y | x } ( \theta )$ is given by
+
+$$
+F = \mathbb { E } \left[ { \mathcal { D } } \theta { \mathcal { D } } \theta ^ { \top } \right] = \operatorname { c o v } ( { \mathcal { D } } \theta , { \mathcal { D } } \theta ) .
+$$
+
+Note that here, and for the remainder of this paper, $y$ is taken to be distributed according to the model’s predictive conditional distribution $P _ { y | x } ( \theta )$ , so that $\mathbb { E } [ \mathcal { D } Z ] = 0$ for any variable $Z$ that is conditionally independent of $y$ given the value of $f ( x , \theta )$ (this includes $\mathcal { D } \theta$ ). All expectations and covariances are defined accordingly. This is done because the expectation that defines the Fisher information matrix uses $P _ { y | x } ( \theta )$ . If we were to instead use the training distribution $Q _ { y | x }$ on $y$ , we would essentially be computing to the “empirical Fisher” (or approximations thereof), which as argued by Martens (2014) is a less appropriate choice for a curvature matrix than the true Fisher.
+
+The natural gradient is defined as $F ^ { - 1 } \nabla h$ , and is the update direction used in natural gradient descent. As argued by Amari (1998), natural gradient descent has the two key advantages: it is invariant to the parameterization of the model, and has “Fisher efficient” convergence1. However, as shown by Martens (2014) these two facts have several important caveats. First, the parameterization invariance only holds approximately in practice when non-infinitesimal step-sizes are used. Second, Fisher efficiency is actually a weak property possessed by simpler methods like SGD with
+
+Polyak/parameter averaging (Polyak & Juditsky, 1992), and even then will only be achieved when the method converges to a global minimizer and the model is capable of perfectly capturing the true distribution of $y$ given $x$ .
+
+An alternative explanation for the empirical success of the natural gradient method is that it is a 2ndorder method, whose update minimizes the following local quadratic approximation to the objective $h ( \theta + \delta )$ :
+
+$$
+\frac { 1 } { 2 } \delta ^ { \top } F \delta + \nabla h ( \theta ) ^ { \top } \delta + h ( \theta ) .
+$$
+
+This is similar to the 2nd-order Taylor series approximation of $h ( \theta + \delta )$ , but with the Fisher substituted in for the Hessian. This substitution can be justified by the observation that the Fisher is a kind of PSD approximation to the Hessian (Pascanu & Bengio, 2014; Martens, 2014). And as argued by Martens (2014), while stochastic 2nd-order methods like natural gradient descent cannot beat the asymptotically optimal Fisher efficient convergence achieved by SGD with Polyak averaging, they can enjoy better pre-asymptotic convergence rates. Moreover, insofar as gradient noise can be mitigated through the use of large mini-batches – so that stochastic optimization starts to resemble deterministic optimization – the theoretical advantages of 2nd-order methods become further pronounced, which agrees with the empirical observation that the use of large-minibatches speeds up 2nd-methods much more than 1st-order methods (Martens & Grosse, 2015; Ba et al., 2017).
+
+In addition to providing an arguably better theoretical argument for the success of natural gradient methods, their interpretation as 2nd-order methods also justifies the common practice of computing the natural gradient as $( F + \lambda I ) ^ { - 1 } \nabla h$ instead of $F ^ { - 1 } \dot { \nabla } h$ . In particular, this practice can be viewed as a type of “update damping/regularization”, where one encourages $\delta$ to lie within some region around $\delta = 0$ where eqn. 1 remains a trustworthy approximation (e.g. Nocedal & Wright, 2006; Martens & Sutskever, 2012).
+
+# 2.3 KRONECKER-FACTORED APPROXIMATE CURVATURE (K-FAC)
+
+Because modern neural network have millions (or even billions) of parameters it is computationally too expensive to compute and invert the Fisher. To address this problem, the K-FAC method of Martens $\&$ Grosse (2015) uses a block-diagonal approximation of the Fisher (where the blocks correspond to entire layers/weight matrices), and where the blocks are further approximated as Kronecker products between much smaller matrices. The details of this approximation are given in the brief derivation below.
+
+Let $W$ be a weight matrix in the network which computes the mapping
+
+$$
+s = W a ,
+$$
+
+where $a$ and $s$ are vector-valued inputs and outputs respectively and denote
+
+$$
+g = { \mathcal { D } } s .
+$$
+
+As in the original K-FAC paper we will assume that $a$ includes a homogeneous coordinate with value 1 so that the bias vector may be folded into the matrix $W$ .
+
+Here and throughout the rest of this document, $F$ will refer to the block of the Fisher corresponding to this particular weight-matrix $W$ .
+
+The Kronecker product of matrices $B$ and $C$ , denoted by $B \otimes C$ for matrices $B \in \mathbb { R } ^ { m \times n }$ and $C$ of arbitrary dimensions, is a block matrix defined by
+
+$$
+B \otimes C \equiv \left[ \begin{array} { c c c } { { [ B ] _ { 1 , 1 } C } } & { { \cdot \cdot \cdot } } & { { [ B ] _ { 1 , n } C } } \\ { { \vdots } } & { { \cdot \cdot } } & { { \vdots } } \\ { { [ B ] _ { m , 1 } C } } & { { \cdot \cdot \cdot } } & { { [ B ] _ { m , n } C } } \end{array} \right]
+$$
+
+Note that the Kronecker product has many convenient properties that we will make use of in this paper. (See Van Loan (2000) for a good discussion of the Kronecker product and its properties.)
+
+A simple application of the chain rule gives $\mathcal { D } W = g a ^ { \top }$ . If we approximate $g$ and $a$ as statistically independent, we can write $F$ as
+
+$$
+\begin{array} { r c l } { F } & { = } & { \mathbb { E } [ \mathrm { v e c } ( \mathcal { D } W ) \mathrm { v e c } ( \mathcal { D } W ) ^ { \top } ] = \mathbb { E } [ \mathrm { v e c } ( g a ^ { \top } ) \mathrm { v e c } ( g a ^ { \top } ) ^ { \top } ] = \mathbb { E } [ ( a \otimes g ) ( a \otimes g ) ^ { \top } ] } \\ & { = } & { \mathbb { E } [ ( a a ^ { \top } ) \otimes ( g g ^ { \top } ) ] = \mathbb { E } [ a a ^ { \top } ] \otimes \mathbb { E } [ g g ^ { \top } ] = A \otimes G , } \end{array}
+$$
+
+where we have defined
+
+$$
+A = \mathbb { E } [ a a ^ { \top } ] \quad { \mathrm { a n d } } \quad G = \mathbb { E } [ g g ^ { \top } ] .
+$$
+
+The matrices $A$ and $G$ can be estimated using simple Monte Carlo methods, and averaged over lots of data by taking an exponentially decaying average across mini-batches.
+
+This is the basic Kronecker factored approximation (Heskes, 2000; Martens & Grosse, 2015; Povey et al., 2015), which is related to the approximation made in the Natural Neural Nets approach (Desjardins et al., 2015). It is shown by Martens $\&$ Grosse (2015) that the approximation is equivalent to neglecting the higher-order cumulants of the as and $g \mathrm { s }$ , or equivalently, assuming that they are Gaussian distributed. To see why this approximation is useful, we observe that inversion and multiplication of a vector by $F$ amounts to inverting the factor matrices $A$ and $G$ and performing matrix-matrix multiplications with them, due to the following two basic identities:
+
+$$
+( B \otimes C ) ^ { - 1 } = B ^ { - 1 } \otimes C ^ { - 1 } \quad \mathrm { a n d } \quad ( B \otimes C ) \operatorname { v e c } ( X ) = \operatorname { v e c } ( C X B ^ { \top } ) .
+$$
+
+The required inversion and matrix multiplication operations are usually computational feasible because the factor matrices have dimensions equal to the size of the layers, which is typically just a few thousand. And when they are not, additional approximations can be applied, such as approximate/iterative inversion (Povey et al., 2015), or additional Kronecker-factorization applied to either $A$ or $G$ (Ba et al., 2017). Moreover, the computation of the inverses can be amortized across iterations of the optimizer at the cost of introducing some staleness into the estimates.
+
+# 3 APPROXIMATING $F$ FOR RNNS
+
+The basic Kronecker-factored approximation to the Fisher block $F$ described in the previous section assumed that the weight matrix $W$ was used to compute a single mapping of the form $s = W a$ . When $W$ is used to compute multiple such mappings, as is often the case for RNNs, or a mapping of a different flavor, as is the case for convolutional networks (CNNs), the approximation is not applicable, strictly speaking.
+
+Grosse & Martens (2016) recently showed that by making additional approximating assumptions, the basic Kronecker-factored approximation can be extended to convolutional layers. This new approximation, called “KFC”, is derived by assuming that gradient contributions coming from different spatial locations are uncorrelated, and that their intra and inter-location statistics are spatially homogeneous, in the sense that they look the same from all reference locations. These assumptions are referred to “spatially uncorrelated derivatives” and “spatial homogeneity,” respectively.
+
+In this section we give the main technical contribution of this paper, which is a family of Kroneckerbased approximations of $F$ that can be applied to RNNs. To build this we will apply various combinations of the approximating assumptions used to derive the original K-FAC and KFC approaches, along with several new ones, including an approximation which works by modelling the statistical structure between the gradient contributions from time-steps using a chain-structured linear Gaussian graphical model.
+
+# 3.1 PRELIMINARIES
+
+Let $W$ be some weight matrix which is used at $\tau$ different time-steps (or positions) to compute the mapping
+
+$$
+s _ { t } = W a _ { t } ,
+$$
+
+where $t$ indexes the time-step. $\tau$ is allowed to vary between different training cases.
+
+Defining $g _ { t } = \mathcal { D } s _ { t }$ , the gradient of the single-case loss with respect to $W$ can be written as
+
+$$
+\mathcal { D } W = \sum _ { t = 1 } ^ { \mathcal { T } } g _ { t } a _ { t } ^ { \top } = \sum _ { t = 1 } ^ { \mathcal { T } } \mathcal { D } _ { t } W ,
+$$
+
+where $\mathcal { D } _ { t } W = g _ { t } a _ { t } ^ { \top }$ denotes the contribution to the gradient from time-step $t$ . When it is more convenient to work with the vector-representations of the matrix-valued variables $\mathcal { D } _ { t } W$ we will use the notation
+
+$$
+w _ { t } = \operatorname { v e c } ( \mathscr { D } _ { t } W ) ,
+$$
+
+so that vec(DW ) = PTt= .
+
+Let $F _ { T }$ denote the conditional Fisher of $\mathcal { D } W$ for a particular value of $\tau$ . We have
+
+$$
+F _ { T } = \mathbb { E } [ \mathrm { v e c } ( \mathcal { D } W ) \mathrm { v e c } ( \mathcal { D } W ) ^ { \top } | T ] = \mathbb { E } \left[ \left( \sum _ { t = 1 } ^ { T } w _ { t } \right) \left( \sum _ { t = 1 } ^ { T } w _ { t } \right) ^ { \top } \Bigg | T \right] = \sum _ { t = 1 } ^ { T } \sum _ { s = 1 } ^ { T } \mathbb { E } \left[ w _ { t } w _ { s } ^ { \top } | T \right] .
+$$
+
+Observe that $F$ can be computed from $F _ { T }$ via $F = \mathbb { E } _ { T } [ F _ { T } ]$ .
+
+To proceed with our goal of obtaining a tractable approximation to $F$ we will make several approximating assumptions, as discussed in the next section.
+
+# 3.2 BASIC INITIAL APPROXIMATIONS
+
+# 3.2.1 INDEPENDENCE OF $\tau$
+
+One simplifying approximation we will make immediately is that $\tau$ is independent of the $w _ { t }$ ’s, so that $\mathbb { E } [ \tilde { w _ { t } } w _ { s } ^ { \top } | \breve { \mathcal { T } } ] = \mathbb { E } [ w _ { t } w _ { s } ^ { \top } ]$ . In this case eqn. 3 can be written as
+
+$$
+F _ { \mathcal { T } } = { \sum _ { t = 1 } ^ { \mathcal { T } } } \sum _ { s = 1 } ^ { \mathcal { T } } \mathbb { E } [ w _ { t } w _ { s } ^ { \top } ] = { \sum _ { t = 1 } ^ { \mathcal { T } } } \sum _ { s = 1 } ^ { \mathcal { T } } V _ { t , s } ,
+$$
+
+where we have defined $V _ { t , s } = \mathbb { E } [ w _ { t } w _ { s } ^ { \top } ]$ .
+
+Independence of $\tau$ and the $w _ { t }$ ’s is a reasonable approximation assumption to make because 1) for many datasets $\tau$ is constant (which formally implies independence), and 2) even when $\tau$ varies substantially, shorter sequences will typically have similar statistical properties to longer ones (e.g. short paragraphs of text versus longer paragraphs).
+
+# 3.2.2 TEMPORAL HOMOGENEITY
+
+Another convenient and natural approximating assumption we will make is that the $w _ { t }$ ’s are temporally homogeneous, which is to say that the statistical relationship between any $w _ { t }$ and $w _ { s }$ depends only on their distance in time $( d \ : = \ : t \ : - \ : s )$ . This is analogous to the “spatial homogeneity” assumption of KFC. Under this assumption the following single-subscript notation is well-defined: $V _ { t - s } = \mathbb { E } [ w _ { t } w _ { s } ^ { \top } ]$ . We note that $V _ { - d } \doteq V _ { d } ^ { \top }$ .
+
+Applying this notation to eqn. 4 we have
+
+$$
+F _ { \mathcal { T } } = \sum _ { t = 1 } ^ { \mathcal { T } } \sum _ { s = 1 } ^ { \mathcal { T } } V _ { t , s } = \sum _ { d = - \mathcal { T } } ^ { \mathcal { T } } ( \mathcal { T } - | d | ) V _ { d } = \sum _ { d = 0 } ^ { \mathcal { T } } ( \mathcal { T } - d ) V _ { d } + \sum _ { d = 0 } ^ { \mathcal { T } } ( \mathcal { T } - d ) V _ { d } ^ { \top } - \mathcal { T } I ,
+$$
+
+where we have used the fact that there are $\tau - | d |$ ways to write $d$ as $t - s$ for $t , s \in \{ 1 , 2 , \ldots , T \}$
+
+Temporal homogeneity is a pretty mild approximation, and is analogous to the frequently used “steady-state assumption” from dynamical systems. Essentially, it is the assumption that the Markov chain defined by the system “mixes” and reaches its equilibrium distribution. If the system has any randomness, and its external inputs reach steady-state, the steady-state assumption is quite accurate for states sufficiently far from the beginning of the sequence (which will be most of them).
+
+# 3.2.3 INDEPENDENCE BETWEEN THE $a _ { t }$ ’S AND THE $g _ { t }$ ’ S
+
+If we have that $a _ { t }$ and $g _ { s }$ are pair-wise independent for each $t$ and $s$ , which is the obvious generalization of the basic approximation used to derive the K-FAC approach, then following a similar derivation to the one from Section 2.3 we have
+
+$$
+V _ { t , s } = \mathbb { E } [ ( a _ { t } a _ { s } ^ { \top } ) \otimes ( g _ { t } g _ { s } ^ { \top } ) ] = \mathbb { E } [ ( a _ { t } a _ { s } ^ { \top } ) \otimes ( { \mathcal { D } } s _ { t } { \mathcal { D } } s _ { s } ^ { \top } ) ] = A _ { t , s } \otimes G _ { t , s } ,
+$$
+
+where we have defined
+
+$$
+A _ { t , s } = \mathbb { E } [ a _ { t } a _ { s } ^ { \top } ] \quad \mathrm { a n d } \quad G _ { t , s } = \mathbb { E } [ g _ { t } g _ { s } ^ { \top } ] .
+$$
+
+Extending our temporal homogeneity assumption from the $w _ { t }$ ’s to the $a _ { t }$ ’s and $g _ { t }$ ’s (which is natural to do since $w _ { t } = \dot { \mathrm { v e c } } ( g _ { t } a _ { t } ^ { \top } ) )$ ), the following notation becomes well-defined:
+
+$$
+A _ { t - s } = A _ { t , s } \quad \mathrm { a n d } \quad G _ { t - s } = G _ { t , s } ,
+$$
+
+which allows us to write
+
+$$
+V _ { d } = A _ { d } \otimes G _ { d } .
+$$
+
+3.3 AN INITIAL ATTEMPT TO OBTAIN A TRACTABLE FISHER APPROXIMATION
+
+Given the approximating assumptions made in the previous subsections we have
+
+$$
+F _ { \mathcal { T } } = \sum _ { d = - \mathcal { T } } ^ { \mathcal { T } } ( \mathcal { T } - | d | ) V _ { d } = \sum _ { d = - \mathcal { T } } ^ { \mathcal { T } } ( \mathcal { T } - | d | ) ( A _ { d } \otimes G _ { d } ) .
+$$
+
+Assuming for the moment that all of the training sequences have the same length, so that $F = F _ { \mathcal { T } _ { 0 } }$ for some $\mathcal { T } _ { 0 }$ , we have that $F$ will be the sum of $2 \mathcal { T } _ { 0 } + 1$ Kronecker products.
+
+Without assuming any additional structure, such as a relationship between the various $A _ { d }$ ’s or $G _ { d }$ ’s, there doesn’t appear to be any efficient way to invert such a sum. One can use the elementary identity $( B \otimes C ) ^ { - 1 } = \bar { B } ^ { - 1 } \otimes C ^ { - 1 }$ to invert a single Kronecker product, and there exists decomposition-based methods to efficiently invert sums of two Kronecker products (see Martens & Grosse (2015)), however there is no known efficient algorithm for inverting sums of three or more Kronecker products. Thus is appears that we must make additional approximating assumptions in order to proceed.
+
+# 3.4 ASSUMING INDEPENDENCE OF THE $w _ { t }$ ’S ACROSS TIME
+
+If we assume that the contributions to the gradient (the $w _ { t }$ ’s) are independent across time, or at least uncorrelated, this means that $V _ { d } = 0$ for $d \neq 0$ . This is analogous to the “spatially uncorrelated derivatives” assumption of KFC.
+
+In this case eqn. 5 simplifies to
+
+$$
+F _ { \mathcal { T } } = \sum _ { d = - \mathcal { T } } ^ { \mathcal { T } } ( \mathcal { T } - | d | ) V _ { d } = ( \mathcal { T } - 0 ) V _ { 0 } = \mathcal { T } V _ { 0 } ,
+$$
+
+so that
+
+$$
+F = \mathbb { E } _ { \mathcal { T } } [ F _ { \mathcal { T } } ] = \mathbb { E } _ { \mathcal { T } } [ \mathcal { T } V _ { 0 } ] = \mathbb { E } _ { \mathcal { T } } [ \mathcal { T } ] V _ { 0 } .
+$$
+
+Using the identities in eqn. 2, and the symmetry of $A _ { 0 }$ and $G _ { 0 }$ , we can thus efficiently multiply $F ^ { - 1 }$ by a vector $z = \operatorname { v e c } ( Z )$ using the formula
+
+$$
+F ^ { - 1 } z = { \frac { 1 } { \mathbb { E } _ { T } [ T ] } } \operatorname { v e c } ( G _ { 0 } ^ { - 1 } Z A _ { 0 } ^ { - 1 } ) .
+$$
+
+This is, up to normalization by $\mathbb { E } \tau [ \mathcal { T } ]$ , identical to the inverse multiplication formula used in the original K-FAC approximation for fully-connected layers.
+
+We note that $\begin{array} { r } { \mathbb { E } _ { \mathcal { T } } [ \mathcal { T } ] = \sum _ { i } \omega _ { i } \mathcal { T } _ { i } } \end{array}$ , where $\mathcal { T } _ { i }$ are the different values of $\tau$ , and $\omega _ { i } \geqslant 0$ are normalized weights (with $\textstyle \sum _ { i } \omega _ { i } = 1 $ ) that measure their proportions in the training set.
+
+# 3.5 MODELING THE RELATIONSHIPS BETWEEN THE $w _ { t }$ ’S USING AN LGGM
+
+As we saw in Section 3.3, the approximation assumptions made in Section 3.2 (independence of $\tau$ , temporal homogeneity, and independence between the $a _ { t }$ ’s and the $g _ { t }$ ’s), aren’t sufficient to yield a tractable formula for $\scriptstyle { \dot { F } } ^ { - 1 }$ . And while additionally assuming independence across time of the $w _ { t }$ ’s is sufficient (as shown in Section 3.4), it seems like an overly severe approximation to make.
+
+In this section we consider a less severe approximation which we will show still produces a tractable $F ^ { - 1 }$ . In particular, we will assume that the statistical relationship of the $w _ { t }$ ’s is described by a simple linear Gaussian graphical model (LGGM) with a compact parameterization (whose size is independent of $\tau$ ). Such an approach to computing a tractable Fisher approximations was first explored by Grosse & Salakhutdinov (2015) for RBMs, although our use of it here is substantially different, and requires additional mathematical machinery.
+
+The model we will use is a fairly natural one. It is a linear Gaussian graphical model with a onedimensional chain structure corresponding to time. The graphical structure of our model is given by the following picture:
+
+![](images/9b5321b39344150c6ed0d2d609369eeef29164dc9baac7830ff39342112dba98.jpg)
+
+Variables in the model evolve forward in time according to the following equation:
+
+$$
+w _ { t } = \Psi w _ { t - 1 } + \epsilon _ { t }
+$$
+
+where $\Psi$ is a square matrix and $\epsilon _ { t }$ are i.i.d. from $\mathcal { N } ( 0 , \Sigma )$ for some positive definite matrix $\Sigma$ (which is the conditional covariance of $w _ { t }$ given $w _ { t - 1 }$ ).
+
+Due to the well-known equivalence between directed and undirected Gaussian graphical models for tree-structured graphs like this one, the decision of whether to make the edges directed or undirected, and whether to have them point forwards or backwards in time, are irrelevant from a modeling perspective (and thus to the Fisher approximation we eventually compute). We will use a directed representation purely for mathematical convenience.
+
+We will assume that our model extends infinitely in both directions, with indices in the range $( - \infty , \infty )$ , so that the $w _ { t }$ ’s are all in their stationary distribution (with respect to time). For this to yield a well-defined model we require that $\Psi$ has spectral radius $< 1$ .
+
+The intuition behind this model structure is clear. The correlations between gradient contributions (the $w _ { t }$ ’s) at two different time-steps should be reasonably well explained by the gradient contributions made at time-steps between them. In other words, they should be approximately Markovian.
+
+We know that the gradient computations are generated by a process, Back-prop Through Time (BPTT), where information flows only between consecutive time-steps (forwards through time during the “forward pass”, and backwards during the “backwards pass”). This process involves temporal quantities which are external to the $w _ { t }$ ’s, such as the inputs $x$ and activations for other layers, which essentially act as “hidden variables”. The evolution of these external quantities may be described by their own separate temporal dynamics (e.g. the unknown process which generates the true $x$ ’s), and thus the $w _ { t }$ ’s won’t be Markovian in general. But insofar as the $w _ { t }$ ’s (or equivalently the $a _ { t }$ ’s and $g _ { t }$ ’s) encode the relevant information contained in these external variables, they should be approximately Markovian. (If they contained all of the information they would be exactly Markovian.)
+
+A similar approximation across consecutive layers was made in the “block-tridiagonal” version of the original K-FAC approach. It was shown by Martens & Grosse (2015) that this approximation was a pretty reasonable one. The linear-Gaussian assumption meanwhile is a more severe one to make, but it seems necessary for there to be any hope that the required expectations remain tractable.
+
+# 3.5.1 INITIAL COMPUTATIONS
+
+Define the following “transformed” versions of $F _ { T }$ and $\Psi$ :
+
+$$
+\hat { F } _ { \mathcal { T } } = V _ { 0 } ^ { 1 / 2 } F _ { \mathcal { T } } V _ { 0 } ^ { 1 / 2 } \quad \mathrm { a n d } \quad \hat { \Psi } = \hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } \Psi V _ { 0 } ^ { 1 / 2 } .
+$$
+
+As shown in Section A.1 of the appendix we have
+
+$$
+\begin{array} { r l } & { \hat { F } _ { \mathcal { T } } = { \displaystyle \sum _ { d = 0 } ^ { \mathcal { T } } } ( \mathcal { T } - d ) \hat { \Psi } ^ { d } + \left( { \displaystyle \sum _ { d = 0 } ^ { \mathcal { T } } } ( \mathcal { T } - d ) \hat { \Psi } ^ { d } \right) ^ { \top } - \mathcal { T } I } \\ & { ~ = \zeta \tau ( \hat { \Psi } ) + \zeta \tau ( \hat { \Psi } ^ { \top } ) - \mathcal { T } I } \end{array}
+$$
+
+where
+
+$$
+\zeta \tau ( x ) = \frac { \mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } .
+$$
+
+(Note that rational functions can be evaluated with matrix arguments in this way, as discussed in Section A.1.)
+
+Our goal is to compute $\hat { F } ^ { - 1 }$ , from which we can recover $F ^ { - 1 }$ via the simple relation $F ^ { - 1 } =$ $ { V _ { 0 } } ^ { - 1 / 2 }  { \hat { F } } ^ { - 1 }  { V _ { 0 } } ^ { - 1 / 2 }$
+
+Unfortunately it doesn’t appear to be possible to simplify this formula sufficiently enough to allow for the efficient computation of $\hat { F } ^ { - 1 } = \mathbb { E } _ { T } [ \hat { F } _ { T } ] ^ { - 1 }$ when $\hat { \Psi }$ is a Kronecker product (which it will be when $V _ { 0 }$ and $V _ { 1 }$ are). The difficulty is due to both the appearance of $\hat { \Psi }$ and its transpose (which are not codiagonalizable/commutative in general), and various higher powers of $\hat { \Psi }$ .
+
+To proceed from this point and obtain a formula which can be efficiently evaluated when $\hat { \Psi }$ is a Kronecker product, we will make one of two simplifying assumptions/approximations, which we call “Option $1 ^ { \circ }$ and “Option $2 ^ { \circ }$ respectively. These are explained in the next two subsections.
+
+# 3.5.2 OPTION 1: $V _ { 1 }$ IS SYMMETRIC
+
+If $V _ { 1 }$ (the cross-moment over time) is symmetric, this implies that $\hat { \Psi } = \hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 }$ is also symmetric. Thus by eqn. 7 we have
+
+$$
+\hat { F } _ { T } = \zeta _ { T } ( \hat { \Psi } ) + \zeta _ { T } ( \hat { \Psi } ) - \mathcal { T } I = \eta _ { T } ( \hat { \Psi } ) ,
+$$
+
+where
+
+$$
+\eta _ { T } ( x ) = 2 \zeta \tau ( x ) - \mathcal { T } = \frac { 2 ( \mathcal { T } ( 1 - x ) - x ( 1 - x ^ { T } ) ) } { ( 1 - x ) ^ { 2 } } - \mathcal { T } = \frac { \mathcal { T } ( 1 - x ^ { 2 } ) - 2 x ( 1 - x ^ { T } ) } { ( 1 - x ) ^ { 2 } } .
+$$
+
+Let $U \mathrm { d i a g } ( \hat { \psi } ) U ^ { \top } = \hat { \Psi }$ be the eigen-decomposition of $\hat { \Psi }$ . By the above expression for $\hat { F } _ { T }$ we have that $\hat { F } _ { T } = U \mathrm { d i a g } ( \eta _ { T } ( \hat { \psi } ) ) U ^ { \top }$ , where $f ( b )$ denotes the component-wise evaluation of a function $f$ for each component of the vector $b$ , i.e. $[ { \dot { f } } ( b ) ] _ { i } = f ( [ b ] _ { i } )$ . We thus have
+
+$$
+\hat { F } = \mathbb { E } _ { \mathcal { T } } [ \hat { F } _ { \mathcal { T } } ] = \mathbb { E } _ { \mathcal { T } } [ U \operatorname { d i a g } ( \eta _ { \mathcal { T } } ( \hat { \psi } ) ) U ^ { \top } ] = U \operatorname { d i a g } ( \mathbb { E } _ { \mathcal { T } } [ \eta _ { \mathcal { T } } ( \hat { \psi } ) ] ) U ^ { \top } .
+$$
+
+Inverting both sides of this yields
+
+$$
+\hat { F } ^ { - 1 } = U \mathrm { d i a g } ( \gamma ( \hat { \psi } ) ) U ^ { \top }
+$$
+
+where we have defined $\gamma ( x ) = 1 / \mathbb { E } _ { T } [ \eta _ { T } ( x ) ]$
+
+This expression can be efficiently evaluated when $\hat { \Psi }$ is a Kronecker product since the eigendecomposition of a Kronecker product can be easily obtained from the eigendecomposition of the factors. Evaluation of $\gamma ( \hat { \psi } )$ is done component-wise (i.e. $[ \gamma ( \hat { \psi } ) ] _ { i } = \gamma ( [ \hat { \psi } ] _ { i } ) )$ and is thus easy to perform. See Section 3.5.5 for further details.
+
+$V _ { 1 }$ is symmetric if and only if $\hat { \Psi }$ is symmetric. And as shown in the proof of Proposition 1 (see Appendix A.1) $\hat { \Psi }$ has the interpretation of being the transition matrix of an LGGM which describes the evolution of “whitened” versions of the wt’s (given by wˆt = V −1/20 w . Linear dynamical systems with symmetric transition matrices arise frequently in machine learning and related areas (Huang et al., 2016; Hazan et al., 2017), particularly because of the algorithmic techniques they enable. Intuitively, a symmetric transition matrix allows allows one to model exponential decay of different basis components of the signal over time, but not rotations between these components (which are required to model sinusoidal/oscillating signals).
+
+Note that the observed/measured $V _ { 1 }$ may or may not be exactly symmetric up to numerical precision, even if it well approximated as symmetric. For these calculations to make sense it must be exactly symmetric, and so even if it turns out to be approximately symmetric one should ensure that it is exactly so by using the symmetrized version $( \dot { V _ { 1 } } + V _ { 1 } ^ { \top } ) / 2$ .
+
+# 3.5.3 OPTION 2: COMPUTING THE LIMITING VALUE INSTEAD
+
+If $V _ { 1 }$ is not well approximated as symmetric, another option is to approximate
+
+$$
+\hat { F } = \mathbb { E } _ { T } [ \hat { F } _ { T } ^ { ( \infty ) } ]
+$$
+
+(instead of $\hat { F } = \mathbb { E } _ { T } [ \hat { F } _ { T } ] )$ , where we define
+
+$$
+{ \hat { F } } _ { \mathcal { T } } ^ { ( \infty ) } \equiv \operatorname* { l i m } _ { { \mathcal { T } } ^ { \prime } \to \infty } { \frac { \mathcal { T } } { { \mathcal { T } } ^ { \prime } } } { \hat { F } } _ { { \mathcal { T } } ^ { \prime } } .
+$$
+
+This is essentially equivalent to the assumption that the training sequences are all infinitely long, which may be a reasonable one to make in practice. We re-scale by the factor $\frac { \mathcal { T } } { \mathcal { T } ^ { \prime } }$ to achieve the proper scaling characteristics of $\hat { F } _ { T }$ , and to ensure that the limit actually exists.
+
+As shown in Section A.2 of the appendix this yields the following remarkably simple expression for $\hat { F } ^ { - 1 }$ :
+
+$$
+\hat { F } ^ { - 1 } = \frac { 1 } { \mathbb { E } _ { T } [ \mathcal { T } ] } ( I - \hat { \Psi } ) ( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 } ( I - \hat { \Psi } ^ { \top } ) .
+$$
+
+Despite the fact that it includes both $\hat { \Psi }$ and $\hat { \Psi } ^ { \top }$ , this formula can be efficiently evaluated when $\hat { \Psi }$ is a Kronecker product due to the existance of decomposition-based techniques for inverting matrices of the form $A \otimes B + C \otimes D$ . See Section 3.5.5 for further details.
+
+This approximation can break down if some of the linear components of $\hat { w } _ { t }$ have temporal autocorrelations close to 1 (i.e. $[ \hat { \psi } ] _ { i } \approx 1$ for some $i$ ) and $\tau$ is relatively small. In such a case we will have that $[ \hat { \psi } ] _ { i } ^ { \mathcal { T } }$ is large for some $i$ (despite being raised to the $\tau$ -th power) so that ${ \hat { F } } _ { T } ^ { ( \infty ) }$ may essentially “overcount” the amount of temporal correlation that contributes to the sum.
+
+This can be made more concrete by noting that the approximation is essentially equivalent to taking $\begin{array} { r } { \zeta _ { \mathcal { T } } ( x ) \approx \operatorname* { l i m } _ { \mathcal { T } ^ { \prime }  \infty } \frac { \mathcal { T } } { \mathcal { T } ^ { \prime } } \zeta _ { \mathcal { T } ^ { \prime } } ( x ) \equiv \kappa ( x ) } \end{array}$ for each $x = [ \hat { \psi } ] _ { i }$ . We can express the error of this as
+
+$$
+| \kappa ( x ) - \zeta \tau ( x ) | = \left| { \frac { \mathcal { T } } { 1 - x } } - { \frac { \mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } } \right| = \left| { \frac { x ( 1 - x ^ { \mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } } \right| .
+$$
+
+It is easy to see how this expression, when evaluated at $x = [ \hat { \psi } ] _ { i }$ , might be large when $[ \hat { \psi } ] _ { i }$ is close to 1, and $\tau$ is relatively small.
+
+# 3.5.4 ESTIMATING $\hat { \Psi }$
+
+The formulae for $\hat { F } ^ { - 1 }$ from the previous sections depend on the quantity $\hat { \Psi } = \hat { V } _ { 1 } = { V _ { 0 } ^ { - 1 / 2 } \Psi { V _ { 0 } ^ { 1 / 2 } } }$ and so it remains to compute $\Psi$ . We observe that
+
+$$
+V _ { 1 } = V _ { 1 , 0 } = \mathbb { E } [ w _ { 1 } w _ { 0 } ^ { \top } ] = \mathbb { E } [ ( \Psi w _ { 0 } + \epsilon _ { 1 } ) w _ { 0 } ^ { \top } ] = \Psi \mathbb { E } [ w _ { 0 } w _ { 0 } ^ { \top } ] + \mathbb { E } [ \epsilon _ { 1 } w _ { 0 } ^ { \top } ] = \Psi V _ { 0 } + 0 = \Psi V _ { 0 } .
+$$
+
+Right-multiplying both sides by $V _ { 0 }$ yields $\Psi = { \cal V } _ { 1 } { \cal V } _ { 0 } ^ { - 1 }$ . Thus, given estimates of $V _ { 0 }$ and $V _ { 1 }$ , we may compute an estimate of $\hat { \Psi }$ as
+
+$$
+\begin{array} { r } { \hat { \Psi } = V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 } . } \end{array}
+$$
+
+In practice we estimate $V _ { 0 }$ and $V _ { 1 }$ by forming estimates of their Kronecker factors and taking the product. The factors themselves are estimated using exponentially decayed averages over mini-batch estimates. And the mini-batch estimates are in turn computed by averaging over cases and summing across time-steps, before divide by the expected number of time-steps.
+
+For example, for $A _ { 0 }$ and $A _ { 1 }$ these the mini-batch estimates are averages of 1ET [ ] PTt=1 ata>t and 1ET [ ] PT −1t=1 at+1a>t , respectively. Note that as long as V0 is computed as the 2nd-order moment of some empirical data, and $V _ { 1 }$ computed as the 2nd-order moment between that same data and a temporally shifted version, the spectral radius of $\hat { \Psi } = { V _ { 0 } ^ { - 1 / 2 } } V _ { 1 } V _ { 0 } ^ { - 1 / 2 }$ (and similarly $\Psi = V _ { 1 } V _ { 0 } ^ { - 1 } \dag ,$ ) will indeed be less than or equal to , as we prove in Section B.2 of the appendix. This bound on the spectral radius is a necessary condition for our infinite chain-structured Gaussian graphical model to be well-defined, and for our calculations to make sense.
+
+The sufficient condition that the spectral radius is actually less than 1 will most often be satisfied too, except in the unlikely event that some eigen component remains perfectly constant across time. But even if this somehow happens, the inclusion within the given $V _ { 0 }$ of some damping/regularization term such as $\lambda I$ will naturally deal with this problem.
+
+3.5.5 EFFICIENT IMPLEMENTATION ASSUMING $V _ { 0 }$ AND $V _ { 1 }$ ARE KRONECKER-FACTORED
+
+It remains to show that the approximations developed in Section 3.5 can be combined with the Kronecker-factored approximations for $V _ { 0 }$ and $V _ { 1 }$ from Section 3.2.3 to yield an efficient algorithm for computing $F ^ { - 1 } z$ for an arbitrary vector $z = \mathrm { v e c } ( Z )$ . This is a straightforward although very long computation which we leave to Section C of the appendix.
+
+Full pseudo-code for the resulting algorithms is given in Section C.3. As they only involve symmetric eigen-decomposition and matrix-matrix products with matrices the size of $A _ { 0 }$ and $G _ { 0 }$ they are only several times more expensive to compute than eqn. 6. This extra overhead will often be negligible since the gradient computation via BPTT, whose costs scales with the sequence length $\tau$ , tends to dominate all the other costs.
+
+# 4 EXPERIMENTS
+
+To demonstrate the benefit of our novel curvature matrix approximations for RNNs, we empirically evaluated them within the standard “distributed K-FAC” framework (Ba et al., 2017) on two different RNN training tasks.
+
+The 2nd-order statistics (i.e. the Kronecker factors $A _ { 0 }$ , $A _ { 1 }$ , $G _ { 0 }$ , and $G _ { 1 }$ ) are accumulated through an exponential moving average during training. When computing our approximate inverse Fisher, factored Tikhonov damping (Martens & Grosse, 2015) was applied to $V _ { 0 } = G _ { 0 } \otimes A _ { 0 }$ .
+
+We used a single machine with 16 CPU cores and a Nvidia K40 GPU for all the experiments. The additional computations required to get the approximate Fisher inverse from these statistics (i.e. the “pre-processing steps” described in Section C.3) are performed asynchronously on the CPUs, while the GPU is used for the usual forward evaluation and back-propagation to compute the gradient. Updates are computed using the most recently computed values of these (which are allowed to be stale), so there is minimal per-iteration computational overhead compared SGD.
+
+We adopted the step-size selection technique described in Section 5 of Ba et al. (2017), as we found it let us use larger learning rates without compromising the stability of the optimization. The hyperparameters of our approach, which include the max learning rate and trust-region size for the aforementioned step-size selection procedure, as well as the momentum, damping constants, and the decay-rate for the second-order statistics, as well as the hyper-parameters of the baseline methods, were tuned using a grid search.
+
+Word-level language model: We start by applying our method to a two-layer RNN based on the well-studied Long Short-Term Memory (LSTM) architecture (Hochreiter & Schmidhuber, 1997) for a word-level language modeling task on the Penn-TreeBank (PTB) dataset (Marcus et al., 1993) following the experimental setup in Zaremba et al. (2014). The gradients are computed using a fixed sequence length truncated back-propagation scheme in which the initial states of the recurrent hidden units are inherited from the final state of the preceding sequence. The truncation length used in the experiments is 35 timesteps. The learning rate is given by a carefully tuned decaying schedule (whose base value we tune along with the other hyperparamters).
+
+In our experiments we simply substitute their optimizer with our modified distributed K-FAC optimizer that uses our proposed RNN Fisher approximations. We performed experiments on two different sizes of the same architecture, which use two-layer 650 and 1024 LSTM units respectively.
+
+LSTMs have 4 groups of internal units: input gates, output gates, forget gates, and update candidates. We treat the 4 weight matrices that compute the pre-activations to each of these as distinct for the purposes of defining Fisher blocks (whereas many LSTM implementations treat them as one big matrix). This results in smaller Kronecker factors that are cheaper to compute and invert.
+
+![](images/ad3c90338cf7847de32bfd1df5c5edabca76a0d435c7d9a88c3e87b2b1f8ae9b.jpg)  
+Figure 1: Optimization performance of our method compared to the baselines in perplexity-perword on length-35 word sequences from Penn-TreeBank. All the methods used a mini-batch size of 200. K-FAC indep. uses the update in eqn. 6, K-FAC option1 uses eqn. 8, and K-FAC option2 uses eqn. 9. (left) Training perplexity v.s. the number of updates. Dashed lines denote the training curves for RNNs with 1024 LSTM units and solid lines denote the training curves for RNNs with 650 LSTM units. (right) Training perplexity v.s. the wall-clock time.
+
+# 4.1 LANGUAGE MODELING WITH LONG SHORT-TERM MEMORY UNITS
+
+Because the typical vocabulary size used for PTB is 10,000, the Fisher blocks for the input embedding layer and output layer (computing the logits to the softmax) each contain a 10,000 by 10,000 sizes Kronecker factor, which is too large to be inverted with any reasonable frequency. Given that the input vector uses a one-hot encoding it is easy to see its associated factor is actually diagonal, and so we can store and invert it as such. Meanwhile the large factor associated with the output isn’t diagonal, but we nonetheless approximate it as such for the sake of efficiency.
+
+In our experiments we found that each parameter update of our method required about $80 \%$ more wall-clock time than an SGD update (using mini-batch size of 200) although the updates made more much progress.
+
+In Figure 1, we plot the training progress as a function of the number of parameter updates. While Adam outperforms SGD in the first few epochs, SGD obtains a lower loss at the end of training. We found the recent layer-normalization technique (Ba et al., 2016) helps speed up Adam considerably, but it hurts the SGD performance. Such an observation is consistent with previous findings. In comparison, our proposed method still significantly outperform both the Adam and the SGD baselines even with the help of layer-normalization.
+
+While optimization performance, not generalization performance, is the focus of this paper, we have included validation performance data in the appendix for the sake of completeness. (See Figure 4 in Appendix D.) Not surprisingly, we found that the 2nd-order methods, including our approach and diagonal ones like Adam, tended to overfit more than SGD on these tasks.
+
+The tendency for SGD w/ early-stopping to self-regularize is well-documented, and there are many compelling theories about why this happens (e.g. Duvenaud et al., 2016; Hardt et al., 2015). It is also well-known that 2nd-order methods, including K-FAC and diagonal methods like Adam/RMSprop, dont self-regularize nearly as much (e.g. Wilson et al., 2017; Keskar & Socher, 2017). We feel that this problem can likely be addressed through the careful application of additional explicit regularization (e.g. increased weight decay, drop-out, etc) and/or model modifications, but that exploring this is outside of the scope of this paper.
+
+Character-level model: To further investigate the optimization performance of our proposed Fisher approximation, we use a small two layer LSTM with 128 units to model the character sequences on the Penn-TreeBank (PTB) dataset (Marcus et al., 1993). We employ the same data partition in Mikolov et al. (2012). We plotted the bits-per-character vs the number of parameter updates and the wall-clock times in Figure 2. The K-FAC updates were roughly twice as time-consuming to compute as the Adam updates in our implementation. Despite this, our results demonstrate that K-FAC has a significant advantage over the Adam baseline in terms of wall-clock time.
+
+![](images/195985476ed79c9ed7a96907def60b68725ef3c2df313969a5ef4933221affdf.jpg)  
+Figure 2: Optimization performance in bit-per-character on length-100 character sequences from Penn-TreeBank. batchsize indicates the mini-batch size used to train the baseline methods (our method always used a mini-batch size of 200). K-FAC indep. uses the update in eqn. 6, $K – F A C$ option1 uses eqn. 8, and K-FAC option2 uses eqn. 9. (left) Training perplexity v.s. the number of updates. (right) Training perplexity v.s. the wall-clock time.
+
+![](images/2dcddfdcf44fcf48896aeb8b94df1bf2448c115c90055b93b77f229570a76702.jpg)  
+Figure 3: Optimization performance for differentiable Neural Computers (DNC) on a repeated copy task. $K – F A C$ indep. uses the update in eqn. 6, K-FAC option1 uses eqn. 8, and $K -$ -FAC option2 uses eqn. 9. (left) Training cross entropy loss v.s. the number of updates. (right) Training cross entropy loss v.s. the wall-clock time.
+
+# 4.2 LEARNING DIFFERENTIABLE NEURAL COMPUTERS
+
+To further investigate the potential benefits of using our approach over existing methods, we applied it to the Differentiable Neural Computer (DNC) model (Graves et al., 2016) for learning simple algorithmic programs. Recently, there have been several attempts (Weston et al., 2014; Graves et al., 2016) to extend the existing RNN models to incorporate more long-term memory storage devices in order to help solve problems beyond simple sequence prediction tasks. Although these extended RNNs could potentially be more powerful than simple LSTMs, they often require thousands of parameter updates to learn simple copy tasks (Graves et al., 2016). Both the complexity of these models and the difficulty of the learning tasks have posed a significant challenge to commonly used optimization methods.
+
+The DNC model is designed to solve structured algorithmic tasks by using an LSTM to control an external read-write memory. We applied the Fisher-based precondition to compute the updates for both the weights in the LSTM controller and the read-write weight matrices used to interface with the memory. We trained the model on a simple repeated copy task in which the DNC needs to recreate a series of two random binary sequences after they are presented as inputs. The total length of the sequence is fixed to 22 time-steps. From Figure 3, we see that our method significantly outperforms the Adam baseline in terms of update count, although only provides a modest improvement in wallclock time.
+
+This gap is explained by the fact that the iterations were significantly more time-consuming to compute relative to the gradient computations than they were in previous two experiments on language models. This is likely due to a different trade-off in terms of the gradient computation vs the overheads specific to our method owing to smallness of the model and dataset. With more careful engineering to reduce the communication costs, and/or a larger model and dataset, we would expect to see a bigger improvement in wall-clock time.
+
+# 5 CONCLUSION
+
+We have presented a new family of approximations to the Fisher information matrix of recurrent neural networks (RNNs), extending previous work on Kronecker-factored approximations. With this contribution, recurrent networks can now finally be trained with the K-FAC optimization method. We have demonstrated that our new approximations substantially reduce the required number of iterations for convergence vs standard baseline optimizers on several realistic tasks. And we have also shown that in a modern distributed training setup this results in a substantial savings in wallclock time as well.
+
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+
+# A SUPPLEMENTARY COMPUTATIONS
+
+A.1 PROOFS FOR SECTION 3.5.1
+
+# Proposition 1 Given
+
+$$
+\hat { F } _ { \mathcal { T } } = V _ { 0 } ^ { 1 / 2 } F _ { \mathcal { T } } V _ { 0 } ^ { 1 / 2 } \quad a n d \quad \hat { \Psi } = \hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } \Psi V _ { 0 } ^ { 1 / 2 } .
+$$
+
+we have
+
+where
+
+$$
+\begin{array} { c } { { \hat { F } _ { \mathcal { T } } = \zeta \tau ( \hat { \Psi } ) + \zeta \tau ( \hat { \Psi } ^ { \top } ) - \mathcal { T } I } } \\ { { { } } } \\ { { \zeta \tau ( x ) = \displaystyle \frac { \mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } . } } \end{array}
+$$
+
+# Proof
+
+For $d > 0$ we have that $V _ { d } = \Psi V _ { d - 1 }$ , which can be seen as follows:
+
+$$
+\begin{array} { r c l } { { V _ { d } } } & { { = } } & { { V _ { d , 0 } = \mathbb { E } [ w _ { d } w _ { 0 } ^ { \top } ] = \mathbb { E } [ \Psi w _ { d - 1 } + \epsilon _ { d } w _ { 0 } ^ { \top } ] } } \\ { { } } & { { = } } & { { \Psi \mathbb { E } [ w _ { d - 1 } w _ { 0 } ^ { \top } ] + \mathbb { E } [ \epsilon _ { d } w _ { 0 } ^ { \top } ] } } \\ { { } } & { { = } } & { { \Psi V _ { d - 1 } + 0 = \Psi V _ { d - 1 } . } } \end{array}
+$$
+
+Applying $V _ { d } = \Psi V _ { d - 1 }$ recursively yields
+
+$$
+V _ { d } = \Psi ^ { d } V _ { 0 } \quad \mathrm { f o r } \quad d \geqslant 0 .
+$$
+
+And using $V _ { - d } = V _ { d } ^ { \top }$ it also follows that
+
+$$
+V _ { d } = V _ { 0 } ( \Psi ^ { d } ) ^ { \top } \quad \mathrm { f o r } \quad d \leqslant 0 .
+$$
+
+Setting $d = 1$ and multiplying both sides by $V _ { 0 }$ (which is assumed to be invertible) one can also derive the following simple formula for $\Psi$ :
+
+$$
+\Psi = { \cal V } _ { 1 } { \cal V } _ { 0 } ^ { - 1 } .
+$$
+
+To proceed from here we define a “transformed” version of the original chain-structured linearGaussian graphical model whose variables are $\hat { w } _ { t } = V _ { 0 } ^ { - 1 / 2 } w _ { t }$ . (Here we assume that $V _ { 0 }$ is invertible – it is symmetric by definition.) All quantities related to the original model have their analogues in the transformed model, which we indicate with the hat symbol ˆ·.
+
+In the transformed model the 2nd-order moments of the $\hat { w } _ { t }$ ’s are given by
+
+$$
+\hat { V } _ { d } = \mathbb { E } \left[ \left( V _ { 0 } ^ { - 1 / 2 } w _ { d } \right) \left( V _ { 0 } ^ { - 1 / 2 } w _ { 0 } \right) ^ { \top } \right] = V _ { 0 } ^ { - 1 / 2 } \mathbb { E } \left[ w _ { d } w _ { 0 } ^ { \top } \right] V _ { 0 } ^ { - 1 / 2 } = V _ { 0 } ^ { - 1 / 2 } V _ { d } V _ { 0 } ^ { - 1 / 2 } .
+$$
+
+We observe that $\hat { V } _ { 0 } = I$
+
+Analogously to the original model, the transformed version obeys
+
+$$
+\hat { w } _ { t } = \hat { \Psi } \hat { w } _ { t - 1 } + \hat { \epsilon } _ { t } ,
+$$
+
+with $\hat { \epsilon } _ { t } = V _ { 0 } ^ { - 1 / 2 } \epsilon _ { t }$ and $\hat { \Psi } = \hat { V } _ { 1 } \hat { V } _ { 0 } ^ { - 1 } = \hat { V } _ { 1 }$ (using $\hat { V } _ { 0 } = I$ ). This can be seen by noting that
+
+$$
+\begin{array} { r c l } { \hat { w } _ { t } } & { = } & { V _ { 0 } ^ { - 1 / 2 } w _ { t } } \\ & { = } & { V _ { 0 } ^ { - 1 / 2 } ( \Psi w _ { t - 1 } + \epsilon _ { t } ) } \\ & { = } & { V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 } w _ { t - 1 } + V _ { 0 } ^ { - 1 / 2 } \epsilon _ { t } } \\ & { = } & { ( V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 } ) ( V _ { 0 } ^ { - 1 / 2 } w _ { t - 1 } ) + \hat { \epsilon } _ { t } } \\ & { = } & { \hat { V } _ { 1 } \hat { w } _ { t - 1 } + \hat { \epsilon } _ { t } . } \end{array}
+$$
+
+It also remains true that the spectral radius of $\hat { \Psi }$ is less than 1, which can be seen in at least one of two ways: by noticing that the transformed model is well-defined in the infinite limit if and only if
+
+the original one is, or that $\hat { \Psi } = \hat { V } _ { 1 } = { V _ { 0 } ^ { - 1 / 2 } \Psi { V _ { 0 } ^ { 1 / 2 } } }$ is a similar matrix to $\Psi$ (in the technical sense) and hence has the same eigenvalues.
+
+As the transformed model is isomorphic to the original one, all of the previously derived relationships which held for it also hold here, simply by replacing each quantity with its transformed version (denoted by the hat symbol ˆ·).
+
+Given these relations (included the transformed analogue of equation 5) we can express $\hat { F } _ { T }$ as
+
+$$
+\hat { F } _ { \mathcal { T } } = \sum _ { d = 0 } ^ { \mathcal { T } } ( \mathcal { T } - d ) \hat { \Psi } ^ { d } + \left( \sum _ { d = 0 } ^ { \mathcal { T } } ( \mathcal { T } - d ) \hat { \Psi } ^ { d } \right) ^ { \top } - \mathcal { T } I .
+$$
+
+It is a well-known fact that one can evaluate rational functions, and functions that are the limiting values of sequences of rational functions, with matrix arguments. This is done by replacing scalar multiplication with matrix multiplication, division with matrix inversion, and scalar constants with scalar multiples of the identity matrix, etc. Note that because sums of powers and inverses of matrices are co-diagonalizable/commutative when the matrices themselves are, there is no issue of ambiguity caused by mixing commutative and non-commutative algebra in this way.
+
+Moreover, the value of some such function $f ( x )$ , given a matrix argument $B$ , is
+
+$$
+f ( B ) = V \mathrm { d i a g } ( f ( b ) ) V ^ { - 1 } ,
+$$
+
+where $V \mathrm { d i a g } ( b ) V ^ { - 1 } = B$ is eigendecomposition of $\mathbf { B }$ , and where $f ( b )$ denotes the componentwise evaluation of $f$ for each component of the vector $b$ , i.e. $[ f ( b ) ] _ { i } = f ( [ b ] _ { i } )$ . Note that if $[ { \bar { f } } ( b ) ] _ { i }$ is undefined from some $i$ , either because of a division by zero, or because the limit which defines $f ( x )$ doesn’t converge for $x = [ b ] _ { i }$ , then $f ( B )$ doesn’t exist for that particular $B$ (and otherwise it does).
+
+We observe that our above expression for $\widehat { F } _ { T }$ can be rewritten as
+
+$$
+\hat { F } _ { \mathcal { T } } = \zeta _ { \mathcal { T } } ( \hat { \Psi } ) + \zeta _ { \mathcal { T } } ( \hat { \Psi } ^ { \top } ) - \mathcal { T } I ,
+$$
+
+where $\textstyle \zeta _ { T } ( x ) = \sum _ { d = 0 } ^ { T } ( T - d ) x ^ { d }$ . By Proposition 3 in Appendix B.1, we have for $x \neq 1$ that
+
+$$
+\zeta \tau ( x ) = \frac { \mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } .
+$$
+
+Let $U \mathrm { d i a g } ( \hat { \psi } ) U ^ { - 1 } = \hat { \Psi }$ be the eigendecomposition of $\hat { \Psi }$ . Because $\hat { \Psi }$ has a spectral radius less than 1, we have $| [ \hat { \psi } ] _ { i } | < 1$ for each $i$ (so that in particular $[ \hat { \psi } ] _ { i } \neq 1 \rangle$ ), and thus we can evaluate $\zeta \tau ( { \hat { \Psi } } )$ and $\zeta \tau ( \hat { \Psi } ^ { \top } )$ according to the above formula for $\zeta \tau ( x )$ .
+
+# A.2 PROOFS FOR SECTION 3.5.3
+
+Proposition 2 Suppose we approximate $\hat { F } = \mathbb { E } _ { \mathcal { T } } [ \hat { F } _ { \mathcal { T } } ^ { ( \infty ) } ] ,$ , where we have defined
+
+$$
+{ \hat { F } } _ { \mathcal { T } } ^ { ( \infty ) } \equiv \operatorname* { l i m } _ { { \mathcal { T } } ^ { \prime } \to \infty } { \frac { \mathcal { T } } { { \mathcal { T } } ^ { \prime } } } { \hat { F } } _ { { \mathcal { T } } ^ { \prime } } .
+$$
+
+Then we have
+
+$$
+\hat { F } ^ { - 1 } = \frac { 1 } { \mathbb { E } _ { T } [ \mathcal { T } ] } ( I - \hat { \Psi } ) ( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 } ( I - \hat { \Psi } ^ { \top } ) .
+$$
+
+# Proof
+
+From eqn. 7 we have that
+
+$$
+\begin{array} { l c l } { { \hat { F } _ { 7 } ^ { ( \infty ) } } } & { { = } } & { { \displaystyle \operatorname* { l i m } _ { T ^ { \prime } \to \infty } \frac { T } { T ^ { \prime } } ( \zeta _ { T ^ { \prime } } ( \hat { \Psi } ) + \zeta _ { T ^ { \prime } } ( \hat { \Psi } ^ { \top } ) - { T ^ { \prime } } I ) } } \\ { { } } & { { = } } & { { \displaystyle \operatorname* { l i m } _ { T ^ { \prime } \to \infty } \frac { T } { T ^ { \prime } } \zeta _ { T ^ { \prime } } ( \hat { \Psi } ) + \operatorname* { l i m } _ { T ^ { \prime } \to \infty } \frac { T } { T ^ { \prime } } \zeta _ { T ^ { \prime } } ( \hat { \Psi } ^ { \top } ) - \operatorname* { l i m } _ { T ^ { \prime } \to \infty } \frac { T } { T ^ { \prime } } { T ^ { \prime } } I . } } \end{array}
+$$
+
+To evaluate this we first term note that
+
+$$
+\operatorname * { l i m } _ { T ^ { \prime } \to \infty } \frac { T } { T ^ { \prime } } \mathcal { T } ^ { \prime } I = \mathcal { T } I \quad \mathrm { a n d } \quad \operatorname * { l i m } _ { T ^ { \prime } \to \infty } \frac { T } { T ^ { \prime } } \zeta _ { T ^ { \prime } } ( A ) = \kappa ( A ) ,
+$$
+
+where we have defined
+
+$$
+\kappa ( x ) = \operatorname* { l i m } _ { \mathcal { T } ^ { \prime } \to \infty } \frac { \mathcal { T } } { \mathcal { T } ^ { \prime } } \zeta \tau ^ { \prime } ( x ) .
+$$
+
+For $| x | < 1$ we have that $\begin{array} { r } { \operatorname* { l i m } _ { T ^ { \prime } \to \infty } x ^ { T ^ { \prime } } = 0 } \end{array}$ , from which it follows that
+
+$$
+\begin{array} { r l } { \kappa ( x ) } & { = \underset { T ^ { \prime }  \infty } { \mathrm { \operatorname* { l i m } } } \frac { T } { T ^ { \prime } } \Bigg ( \frac { T ^ { \prime } ( 1 - x ) - x ( 1 - x ^ { T ^ { \prime } } ) } { ( 1 - x ) ^ { 2 } } \Bigg ) } \\ & { = \underset { T ^ { \prime }  \infty } { \mathrm { \operatorname* { l i m } } } \frac { T } { T ^ { \prime } } \frac { T ^ { \prime } } { 1 - x } - \underset { T ^ { \prime }  \infty } { \mathrm { \operatorname* { l i m } } } \frac { T } { T ^ { \prime } } \frac { x ( 1 - x ^ { T ^ { \prime } } ) } { ( 1 - x ) ^ { 2 } } } \\ & { = \underset { T ^ { \prime }  \infty } { \mathrm { \operatorname* { l i m } } } \frac { T } { 1 - x } - \underset { T ^ { \prime }  \infty } { \mathrm { \operatorname* { l i m } } } \frac { T } { T ^ { \prime } } \frac { \sin } { r  \infty } \frac { x ( 1 - x ^ { T ^ { \prime } } ) } { ( 1 - x ) ^ { 2 } } } \\ & { = \underset { 1 - x } { \mathrm { \operatorname* { l i m } } } - \underset { T ^ { \prime }  \infty } { \mathrm { \operatorname* { l i m } } } \frac { T } { T ^ { \prime } } \frac { x ( 1 - 0 ) } { ( 1 - x ) ^ { 2 } } } \\ & { = \underset { 1 - x } { \mathrm { \operatorname* { l i m } } } . } \end{array}
+$$
+
+Let $U \mathrm { d i a g } ( \hat { \psi } ) U ^ { - 1 } = \hat { \Psi }$ be the eigendecomposition of $\hat { \Psi }$ . Using the fact that $| [ \hat { \psi } ] _ { i } | < 1$ (as established in Section A.1) we can use the above expression to evaluate $\kappa ( x )$ at both $x = { \hat { \Psi } }$ and $x = \hat { \Psi } ^ { \top }$ , which yields
+
+$$
+\begin{array} { r } { \hat { F } _ { \mathcal { T } } ^ { ( \infty ) } = \kappa ( \hat { \Psi } ) + \kappa ( \hat { \Psi } ^ { \top } ) = \mathcal { T } ( ( I - \hat { \Psi } ) ^ { - 1 } + ( I - \hat { \Psi } ^ { \top } ) ^ { - 1 } - I ) . } \end{array}
+$$
+
+Pre-multiplying both sides by $I - { \hat { \Psi } } ^ { \top }$ , and post-multiplying both sides by $I - { \hat { \Psi } }$ , we have
+
+$$
+\begin{array} { r c l } { ( I - \hat { \Psi } ^ { \top } ) \hat { F } _ { \mathcal { T } } ^ { ( \infty ) } ( I - \hat { \Psi } ) } & { = } & { \mathcal { T } ( ( I - \hat { \Psi } ^ { \top } ) + ( I - \hat { \Psi } ) - ( I - \hat { \Psi } ^ { \top } ) ( I - \hat { \Psi } ) ) } \\ & { = } & { \mathcal { T } ( I - \hat { \Psi } ^ { \top } + I - \hat { \Psi } - I + \hat { \Psi } ^ { \top } + \hat { \Psi } - \hat { \Psi } ^ { \top } \hat { \Psi } ) } \\ & { = } & { \mathcal { T } ( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) . } \end{array}
+$$
+
+Then applying the reverse operation gives
+
+$$
+\widehat F _ { \tau } ^ { ( \infty ) } = \mathcal T ( I - \hat { \Psi } ^ { \top } ) ^ { - 1 } ( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ( I - \hat { \Psi } ) ^ { - 1 } .
+$$
+
+Taking the expectation over $\tau$ gives
+
+$$
+\hat { F } = \mathbb { E } _ { \mathcal { T } } [ \hat { F } _ { \mathcal { T } } ^ { ( \infty ) } ] = \mathbb { E } _ { \mathcal { T } } [ \mathcal { T } ] ( I - \hat { \Psi } ^ { \top } ) ^ { - 1 } ( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ( I - \hat { \Psi } ) ^ { - 1 } .
+$$
+
+Finally, inverting both sides yields
+
+$$
+\hat { F } ^ { - 1 } = \frac { 1 } { \mathbb { E } _ { T } [ \mathcal { T } ] } ( I - \hat { \Psi } ) ( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 } ( I - \hat { \Psi } ^ { \top } ) .
+$$
+
+# B ADDITIONAL TECHNICAL PROOFS
+
+# B.1
+
+Proposition 3 Suppose $x \in \mathbb { C }$ , $x \neq 0$ , and $\tau$ is a non-negative integer. We have
+
+$$
+\sum _ { i = 0 } ^ { T } ( \mathcal T - i ) x ^ { i } = \frac { \mathcal T ( 1 - x ) - x ( 1 - x ^ { \mathcal T } ) } { ( 1 - x ) ^ { 2 } } .
+$$
+
+Proof Observe that
+
+$$
+x { \frac { \mathrm { d } \sum _ { i = 0 } ^ { \mathcal { T } } x ^ { i } } { \mathrm { d } x } } = x \sum _ { i = 0 } ^ { \mathcal { T } } { \frac { \mathrm { d } x ^ { i } } { \mathrm { d } x } } = x \sum _ { i = 0 } ^ { \mathcal { T } } i x ^ { i - 1 } = \sum _ { i = 0 } ^ { \mathcal { T } } i x ^ { i } .
+$$
+
+Another way to express this is to use the geometric series formula $\textstyle \sum _ { i = 0 } ^ { T } x ^ { i } = { \frac { 1 - x ^ { T + 1 } } { 1 - x } }$ (which holds
+
+$$
+: \frac { \mathrm { d } \sum _ { i = 0 } ^ { T } x ^ { i } } { \mathrm { ~ d } x } = x \frac { \mathrm { d } \left( \frac { 1 - x ^ { T + 1 } } { 1 - x } \right) } { \mathrm { ~ d } x } = x \left( - \frac { ( 1 + T ) x ^ { T } } { 1 - x } + \frac { 1 - x ^ { T + 1 } } { ( 1 - x ) ^ { 2 } } \right) = \frac { x ( 1 - x ^ { T + 1 } - ( 1 + T ) ( 1 - x ) ) } { ( 1 - x ) ^ { 2 } } .
+$$
+
+Thus we have
+
+$$
+\sum _ { i = 0 } ^ { T } i x ^ { i } = \frac { x ( 1 - x ^ { T + 1 } - ( 1 + \mathcal { T } ) ( 1 - x ) x ^ { \mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } .
+$$
+
+And so
+
+$$
+\begin{array} { r l } { \frac { T } { - \alpha } ( T - \epsilon ) x ^ { 2 } } & { = \begin{array} { r l } { \underline { \gamma } } & { \underline { \gamma } } \\ & { - \underline { \gamma } } & { \underline { \gamma } } \end{array} ; } \\ { = \begin{array} { r l } & { \underline { \gamma } ( 1 - x ^ { * } ) } \\ & { - \underline { \gamma } } & { \underline { \gamma } ( 1 - x ^ { * } ) } \\ & { \underline { \gamma } } & { \underline { \gamma } } \end{array} . } \end{array}
+$$
+
+where we have again used the geometric series formula $\textstyle \sum _ { i = 0 } ^ { T } x ^ { i } = { \frac { 1 - x ^ { T + 1 } } { 1 - x } }$ on the second line.
+
+# B.2 SPECTRAL BOUND FOR ESTIMATE OF $\hat { \Psi }$
+
+In what follows all quantities are computed using their defining formulae, starting from the estimated values of $A _ { 0 } , A _ { 1 } , G _ { 0 }$ , and $G _ { 1 }$ .
+
+First we observe that since $\hat { \Psi } = \hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } \Psi V _ { 0 } ^ { 1 / 2 }$ is similar to $\Psi$ (in the technical sense of the word), they share the same eigenvalues. Thus it suffices bound to the spectral radius of $\Psi = { \cal V } _ { 1 } { \cal V } _ { 0 } ^ { - 1 }$ .
+
+Next we observe that $V _ { 0 } = A _ { 0 } \otimes G _ { 0 }$ and $V _ { 1 } = A _ { 1 } \otimes G _ { 1 }$ , so that
+
+$$
+V _ { 1 } V _ { 0 } ^ { - 1 } = ( A _ { 1 } \otimes G _ { 1 } ) ( A _ { 0 } \otimes G _ { 0 } ) ^ { - 1 } = ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \otimes ( G _ { 1 } G _ { 0 } ^ { - 1 } ) .
+$$
+
+Because the eigendecomposition of a Kronecker product is the Kronecker product of the decompositions of the factors we have that $\rho ( V _ { 1 } V _ { 0 } ^ { - 1 } ) \stackrel { { } = } { = } \rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \rho ( G _ { 1 } G _ { 0 } ^ { - 1 } )$ , where $\rho ( X )$ denotes the spectral radius of a matrix $X$ .
+
+Thus it suffices to show that $\rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \leq 1$ and $\rho ( G _ { 1 } G _ { 0 } ^ { - 1 } ) \leq 1$ for $A _ { i }$ and $G _ { i }$ as computed by the estimation scheme outlined in Section 3.5.4. Recall that this is the exponentially decayed average of mini-batch averages of estimators of the form $\begin{array} { r } { \frac { 1 } { \mathbb { E } \tau [ \mathcal { T } ] } \sum _ { t = 1 } ^ { \mathcal { T } } a _ { t } a _ { t } ^ { \top } } \end{array}$ and $\begin{array} { r } { \frac { 1 } { \mathbb { E } \tau [ \mathcal { T } ] } \dot { \sum _ { t = 1 } ^ { \mathcal { T } - 1 } { a _ { t + 1 } } } { a _ { t } ^ { \top } } } \end{array}$ .
+
+In the remainder of this section we will show that $\rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \ \leq \ 1$ . The argument to show that $\rho ( G _ { 1 } G _ { 0 } ^ { - 1 } ) \leq 1$ is identical.
+
+Define
+
+$$
+M _ { 0 } = [ 0 \quad a _ { 1 } \quad a _ { 2 } \quad \cdots \quad a _ { T } ]
+$$
+
+and
+
+$$
+M _ { 1 } = [ a _ { 1 } \quad a _ { 2 } \quad \cdot \cdot \cdot \quad a _ { \mathcal { T } } \quad 0 ]
+$$
+
+$\begin{array} { r } { M _ { 0 } M _ { 0 } ^ { \top } = M _ { 1 } M _ { 1 } ^ { \top } = \sum _ { t = 1 } ^ { \mathcal T } a _ { t } a _ { t } ^ { \top } \mathrm { ~ a n d ~ } M _ { 1 } M _ { 0 } ^ { \top } = \sum _ { t = 1 } ^ { \mathcal T - 1 } a _ { t + 1 } a _ { t } ^ { \top } . } \end{array}$
+
+Provided that the exponentially decayed averages for $A _ { 0 }$ and $A _ { 1 }$ are computed in the same way and use the same normalizers (i.e. $1 / ( m \mathbb { E } _ { \mathcal { T } } [ \mathcal { T } ] )  $ ) we thus have that the matrix
+
+$$
+\begin{array} { r } { \mathcal { A } = \left[ \begin{array} { l l } { A _ { 0 } } & { A _ { 1 } } \\ { A _ { 1 } ^ { \top } } & { A _ { 0 } } \end{array} \right] } \end{array}
+$$
+
+is a positively-weighted linear combination of terms of the form
+
+$$
+\left[ { \cal M } _ { 1 } \right] \left[ { \cal M } _ { 1 } \right] ^ { \top } \succeq 0
+$$
+
+where the various $a _ { t }$ ’s are computed on different data using current and previous model parameters.   
+It thus follows that $A \succeq 0$ . The inequality $\rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \leq 1$ now follows from the following lemma.
+
+Lemma 1 Consider a real, symmetric, positive semi-definite block matrix
+
+$$
+{ \left[ \begin{array} { l l } { B } & { C } \\ { C ^ { \mathsf { T } } } & { B } \end{array} \right] } \succeq 0 .
+$$
+
+If $B$ is invertible then we have $\rho ( C B ^ { - 1 } ) \le 1$ , and further if the block matrix is positive definite we have $\rho ( C B ^ { - 1 } ) < 1$ .
+
+Proof Because similarity transformations preserve eigenvalues, the statement is equivalent to $\rho ( B ^ { - \frac { 1 } { 2 } } C B ^ { - \frac { 1 } { 2 } } ) \le 1$ . Define $X \triangleq B ^ { - \frac { 1 } { 2 } } C \bar { B } ^ { - \frac { 1 } { 2 } }$ . Because any induced matrix norm is an upperbound on the spectral radius, it suffices to show $\| X \| _ { 2 } = \sigma _ { \operatorname* { m a x } } ( X ) \leq 1$ , where $\sigma _ { \mathrm { m a x } } ( X )$ denotes the largest singular value of $X$ .
+
+By taking Schur complements of the block matrix (11) we have
+
+$$
+\begin{array} { c } { { 0 { \preceq } B - C B ^ { - 1 } C ^ { \mathsf { T } } = B ^ { \frac { 1 } { 2 } } ( I - ( B ^ { - \frac { 1 } { 2 } } C B ^ { - \frac { 1 } { 2 } } ) ( B ^ { - \frac { 1 } { 2 } } C B ^ { - \frac { 1 } { 2 } } ) ) B ^ { \frac { 1 } { 2 } } } } \\ { { { } } } \\ { { { } = B ^ { \frac { 1 } { 2 } } ( I - X X ^ { \mathsf { T } } ) B ^ { \frac { 1 } { 2 } } . } } \end{array}
+$$
+
+Note that the Schur complement is PSD because the original block matrix itself is (e.g. Zhang, 2006).
+
+Using the fact that $Z \mapsto B ^ { { \frac { 1 } { 2 } } } Z B ^ { { \frac { 1 } { 2 } } }$ maps positive semidefinite matrices to positive semidefinite matrices, this implies
+
+$$
+0 \preceq I - X X ^ { \mathsf { T } } \Rightarrow X X ^ { \mathsf { T } } \preceq I \Rightarrow \| X \| _ { 2 } ^ { 2 } \leq 1 \Rightarrow \rho ( X ) \leq 1 .
+$$
+
+When the block matrix is positive definite, the inequalities become strict.
+
+# C EFFICIENT IMPLEMENTATION ASSUMING $V _ { 0 }$ AND $V _ { 1 }$ ARE KRONECKER-FACTORED
+
+The ultimate goal of our calculations is to efficiently compute the matrix-vector product of some arbitrary vector $z$ (which will often be the gradient $\mathrm { v e c } ( { \mathcal { D } } W ) ,$ ) with our inverse Fisher approximation $F ^ { - 1 }$ . That is, we wish to compute $F ^ { - 1 } z$ .
+
+It will be convenient to assume that $z$ is given as a matrix $Z$ (with the same dimensions as $\mathcal { D } W$ ) so that $z = \operatorname { v e c } ( Z )$ .
+
+Multiplying the vector $z$ by $F ^ { - 1 } = V _ { 0 } ^ { - 1 / 2 } \hat { F } ^ { - 1 } V _ { 0 } ^ { - 1 / 2 }$ amounts to first multiplying by $V _ { 0 } ^ { - 1 / 2 }$ , then by $\hat { F } ^ { - 1 }$ , and then by $V _ { 0 } ^ { - 1 / 2 }$ again.
+
+We will suppose that we are given $A _ { 0 } , A _ { 1 } , G _ { 0 }$ , and $G _ { 1 }$ such that
+
+$$
+V _ { 0 } = A _ { 0 } \otimes G _ { 0 } \quad { \mathrm { a n d } } \quad V _ { 1 } = A _ { 1 } \otimes G _ { 1 } .
+$$
+
+We then have
+
+$$
+V _ { 0 } ^ { - 1 / 2 } = A _ { 0 } ^ { - 1 / 2 } \otimes G _ { 0 } ^ { - 1 / 2 }
+$$
+
+and thus to multiply by $V _ { 0 } ^ { - 1 / 2 }$ we can use the identity $( C \otimes B ) \operatorname { v e c } ( X ) = \operatorname { v e c } ( B X C ^ { \top } )$ , giving
+
+$$
+V _ { 0 } ^ { - 1 / 2 } z = \mathrm { v e c } ( G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 } ) .
+$$
+
+In matrix form this is simply $G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 }$ . Note that $A _ { 0 } ^ { - 1 / 2 }$ and $G _ { 0 } ^ { - 1 / 2 }$ can be computed using the eigendecompositions of $A _ { 0 }$ and $G _ { 0 }$ , for example.
+
+The procedure to efficiently multiply a vector $z$ by $\hat { F } ^ { - 1 }$ is more involved and depends on which approximation “option” we are using. However one immediate useful insight we can make before specializing to Option 1 or Option 2 is that $\hat { \Psi }$ can be written as a Kronecker product as follows:
+
+$$
+\begin{array} { l c l } { { \hat { \Psi } } } & { { = } } & { { V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 } } } \\ { { } } & { { = } } & { { ( A _ { 0 } \otimes G _ { 0 } ) ^ { - 1 / 2 } ( A _ { 1 } \otimes G _ { 1 } ) ( A _ { 0 } \otimes G _ { 0 } ) ^ { - 1 / 2 } } } \\ { { } } & { { = } } & { { ( A _ { 0 } ^ { - 1 / 2 } \otimes G _ { 0 } ^ { - 1 / 2 } ) ( A _ { 1 } \otimes G _ { 1 } ) ( A _ { 0 } ^ { - 1 / 2 } \otimes G _ { 0 } ^ { - 1 / 2 } ) } } \\ { { } } & { { = } } & { { ( A _ { 0 } ^ { - 1 / 2 } A _ { 1 } A _ { 0 } ^ { - 1 / 2 } ) \otimes ( G _ { 0 } ^ { - 1 / 2 } G _ { 1 } G _ { 0 } ^ { - 1 / 2 } ) } } \\ { { } } & { { = } } & { { \hat { \Psi } _ { A } \otimes \hat { \Psi } _ { G } . } } \end{array}
+$$
+
+where we have defined $\hat { \Psi } _ { A } = A _ { 0 } ^ { - 1 / 2 } A _ { 1 } A _ { 0 } ^ { - 1 / 2 }$ and $\hat { \Psi } _ { G } = G _ { 0 } ^ { - 1 / 2 } G _ { 1 } G _ { 0 } ^ { - 1 / 2 }$
+
+# C.1 OPTION 1
+
+For Option 1 we assume that $V _ { 1 }$ , and hence $\hat { \Psi } = \hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 }$ , is symmetric. We note that $V _ { 1 } = A _ { 1 } \otimes G _ { 1 }$ will be symmetric if and only if both $A _ { 1 }$ and $G _ { 1 }$ are.
+
+Our task is to compute the matrix-vector product
+
+$$
+\hat { F } ^ { - 1 } z = U \mathrm { d i a g } ( \gamma ( \hat { \psi } ) ) U ^ { \top } z
+$$
+
+where $U \mathrm { d i a g } ( \hat { \psi } ) U ^ { \top }$ is the eigendecomposition of $\hat { \Psi }$ , and $\gamma ( x )$ is defined as in Section 3.5.3, eqn. 8.   
+To do this we first multiply the vector by $U ^ { \top }$ , then by $\mathrm { d i a g } ( \gamma ( \hat { \psi } ) )$ , and then finally by $U$ .
+
+The eigendecomposition of $\hat { \Psi }$ can be computed efficiently using its Kronecker product structure. In particular, we compute the eigendecompositions of each factor as
+
+$$
+U _ { A } \mathrm { d i a g } ( \hat { \psi } _ { A } ) U _ { A } ^ { \top } = \hat { \Psi } _ { A } \quad \mathrm { a n d } \quad U _ { G } \mathrm { d i a g } ( \hat { \psi } _ { G } ) U _ { G } ^ { \top } = \hat { \Psi } _ { G } ,
+$$
+
+from which we can write the eigendecomposition $U \mathrm { d i a g } ( \hat { \psi } ) U ^ { - 1 }$ of $\hat { \Psi }$ as
+
+$$
+\begin{array} { l c l } { { \hat { \Psi } } } & { { = } } & { { \hat { \Psi } _ { A } \otimes \hat { \Psi } _ { G } } } \\ { { } } & { { = } } & { { ( U _ { A } \mathrm { d i a g } ( \hat { \psi } _ { A } ) U _ { A } ^ { \top } ) \otimes ( U _ { G } \mathrm { d i a g } ( \hat { \psi } _ { G } ) U _ { G } ^ { \top } ) } } \\ { { } } & { { = } } & { { ( U _ { A } \otimes U _ { G } ) ( \mathrm { d i a g } ( \hat { \psi } _ { A } ) \otimes \mathrm { d i a g } ( \hat { \psi } _ { G } ) ) ( U _ { A } ^ { \top } \otimes U _ { G } ^ { \top } ) } } \\ { { } } & { { = } } & { { ( U _ { A } \otimes U _ { G } ) \mathrm { d i a g } ( \mathrm { v e c } ( \hat { \psi } _ { G } \hat { \psi } _ { A } ^ { \top } ) ) ( U _ { A } ^ { \top } \otimes U _ { G } ^ { \top } ) . } } \end{array}
+$$
+
+In other words, we have $U = U _ { A } \otimes U _ { G }$ and $\hat { \psi } = \mathrm { v e c } ( \hat { \psi } _ { G } \hat { \psi } _ { A } ^ { \top } )$ .
+
+To multiply $z$ by $U ^ { \top } = U _ { A } ^ { \top } \otimes U _ { G } ^ { \top }$ we use the identity $( C \otimes B ) \operatorname { v e c } ( X ) = \operatorname { v e c } ( B X C ^ { \top } )$ which gives
+
+$$
+U ^ { \top } z = ( U _ { A } ^ { \top } \otimes U _ { G } ^ { \top } ) \operatorname { v e c } ( Z ) = \operatorname { v e c } ( U _ { G } ^ { \top } Z U _ { A } ) .
+$$
+
+Similarly, the multiplication of $z$ by $U = U _ { A } \otimes U _ { G }$ can be computed as $\mathrm { v e c } ( U _ { G } Z U _ { A } ^ { \top } )$ .
+
+Finally, multiplying $z$ by $\mathrm { d i a g } ( \gamma ( \hat { \psi } ) ) = \mathrm { d i a g } ( \gamma ( \mathrm { v e c } ( \hat { \psi } _ { G } \hat { \psi } _ { A } ^ { \top } ) ) )$ corresponds to entry-wise multiplication of $Z$ by a matrix $Y$ , where $\mathrm { v e c } ( Y ) ~ = ~ \gamma ( \mathrm { v e c } ( \hat { \psi } _ { G } \hat { \psi } _ { A } ^ { \top } ) )$ , or in other words $[ Y ] _ { i , j } ~ =$ $\gamma ( [ \hat { \psi } _ { G } \hat { \psi } _ { A } ^ { \top } ] _ { i , j } ) = \gamma ( [ \hat { \psi } _ { G } ] _ { i } [ \hat { \psi } _ { A } ] _ { j } )$ . Thus we have
+
+$$
+\mathrm { d i a g } ( \gamma ( \hat { \psi } ) ) z = \mathrm { d i a g } ( \mathrm { v e c } ( Y ) ) \mathrm { v e c } ( Z ) = \mathrm { v e c } ( Z \odot Y ) ,
+$$
+
+where $\odot$ denotes entry-wise multiplication of matrices.
+
+In summary we have that $\hat { F } ^ { - 1 } z$ can be computed in matrix form as
+
+$$
+U _ { G } ( ( U _ { G } ^ { \top } Z U _ { A } ) \odot Y ) U _ { A } ^ { \top }
+$$
+
+for $Y$ s.t. $[ Y ] _ { i , j } = \gamma ( [ \hat { \psi } _ { G } ] _ { i } [ \hat { \psi } _ { A } ] _ { j } )$ .
+
+We note that computing $\gamma ( [ \hat { \psi } _ { G } ] _ { i } [ \hat { \psi } _ { A } ] _ { j } )$ is trivial since it is just a scalar evaluation of the rational function $\gamma ( x )$ .
+
+# C.2 OPTION 2
+
+For Option 2 we must compute the matrix-vector product
+
+$$
+\hat { F } ^ { - 1 } z = \frac { 1 } { \sum _ { i } \omega _ { i } \mathcal { T } _ { i } } ( I - \hat { \Psi } ) ( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 } ( I - \hat { \Psi } ^ { \top } ) v .
+$$
+
+To do this we will first multiply by $I - { \hat { \Psi } } ^ { \top }$ , then by $( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 }$ , and then by $I - { \hat { \Psi } }$ , before finally dividing the result by $\sum _ { i } { \omega _ { i } { \bar { T } } _ { i } }$ .
+
+To compute the matrix-vector product $( I - { \hat { \Psi } } ^ { \top } ) z$ we use the identity $( C \otimes B ) \operatorname { v e c } ( X ) =$ $\mathrm { v e c } ( B X C ^ { \top } )$ while noting that $\hat { \Psi } ^ { \top } = ( \hat { \Psi } _ { A } \otimes \hat { \Psi } _ { G } ) ^ { \top } = \hat { \Psi } _ { A } ^ { \top } \otimes \hat { \Psi } _ { G } ^ { \top }$ . This gives
+
+$$
+\begin{array} { r c l } { { ( I - \hat { \Psi } ^ { \top } ) \mathrm { v e c } ( Z ) } } & { { = } } & { { \mathrm { v e c } ( Z ) - ( \hat { \Psi } _ { A } ^ { \top } \otimes \hat { \Psi } _ { G } ^ { \top } ) \mathrm { v e c } ( Z ) } } \\ { { } } & { { = } } & { { \mathrm { v e c } ( Z ) - \mathrm { v e c } ( \hat { \Psi } _ { G } ^ { \top } Z \hat { \Psi } _ { A } ) . } } \end{array}
+$$
+
+The matrix form of this is simply $Z - \hat { \Psi } _ { G } ^ { \top } Z \hat { \Psi } _ { A }$ .
+
+We may similarly compute $( I - { \hat { \Psi } } ) z$ in matrix form as $Z - \hat { \Psi } _ { G } Z \hat { \Psi } _ { A } ^ { \top }$ .
+
+The harder task is to compute $( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 } z$ , which is what we tackle next.
+
+We first observe that
+
+$$
+\begin{array} { l l l l } { { I - \hat { \Psi } ^ { \top } \hat { \Psi } } } & { { = } } & { { I \otimes I - ( \hat { \Psi } _ { A } ^ { \top } \otimes \hat { \Psi } _ { G } ^ { \top } ) ( \hat { \Psi } _ { A } \otimes \hat { \Psi } _ { G } ) } } \\ { { } } & { { = } } & { { I \otimes I - ( \hat { \Psi } _ { A } ^ { \top } \hat { \Psi } _ { A } ) \otimes ( \hat { \Psi } _ { G } ^ { \top } \hat { \Psi } _ { G } ) } } \\ { { } } & { { \equiv } } & { { I \otimes I - M _ { A } \otimes M _ { G } . } } \end{array}
+$$
+
+Given the eigendecompositions $E _ { A } \mathrm { d i a g } ( m _ { A } ) E _ { A } ^ { \top } = M _ { A }$ and $E _ { G } \mathrm { d i a g } ( m _ { G } ) E _ { G } ^ { \top } = M _ { G }$ we can thus compute the larger eigendecomposition as
+
+$$
+\begin{array} { l l l } { { I \otimes I - M _ { A } \otimes M _ { G } } } & { { = } } & { { ( E _ { A } \otimes E _ { G } ) ( I \otimes I - \mathrm { d i a g } ( m _ { A } ) \otimes \mathrm { d i a g } ( m _ { G } ) ) ( E _ { A } ^ { \top } \otimes E _ { G } ^ { \top } ) } } \\ { { } } & { { = } } & { { ( E _ { A } \otimes E _ { G } ) \mathrm { d i a g } ( \mathbb { I } \otimes \mathbb { I } - m _ { A } \otimes m _ { G } ) ( E _ { A } ^ { \top } \otimes E _ { G } ^ { \top } ) } } \\ { { } } & { { = } } & { { ( E _ { A } \otimes E _ { G } ) \mathrm { d i a g } ( \mathrm { v e c } ( \mathbb { I } \mathbb { I } ^ { \top } - m _ { G } m _ { A } ^ { \top } ) ) ( E _ { A } ^ { \top } \otimes E _ { G } ^ { \top } ) , } } \end{array}
+$$
+
+where l is the vector of ones.
+
+Using the eigendecomposition the inverse can then be easily computed as
+
+$$
+( I \otimes I - M _ { A } \otimes M _ { G } ) ^ { - 1 } = ( E _ { A } \otimes E _ { G } ) \mathrm { d i a g } ( \mathrm { v e c } ( \mathbb { 1 } \mathbb { I } ^ { \top } - m _ { G } m _ { A } ^ { \top } ) ) ^ { - 1 } ( E _ { A } ^ { \top } \otimes E _ { G } ^ { \top } ) .
+$$
+
+Thus $( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 } z$ may be computed by first multiplying by $( E _ { A } ^ { \top } \otimes E _ { G } ^ { \top } )$ , then by $\mathrm { d i a g ( v e c ( I I ^ { \top } - }$ $m _ { G } m _ { A } ^ { \top } ) ) ^ { - 1 }$ (which in matrix form corresponds to element-wise division by $\mathbb { I } \mathbb { I } ^ { \top } - m _ { G } m _ { A } ^ { \top } )$ , and then by $E _ { A } \otimes E _ { G }$ . The matrix form of this is
+
+$$
+E _ { G } ( ( E _ { G } ^ { \top } Z E _ { A } ) \oslash ( \mathbb { I I } ^ { \top } - m _ { G } m _ { A } ^ { \top } ) ) E _ { A } ^ { \top } ,
+$$
+
+where $B \oslash C$ denotes element-wise division of the matrix $B$ by the matrix $C$ .
+
+C.3 PSEUDO-CODE FOR THE COMPUTATION OF $F ^ { - 1 } z$
+
+Given $A _ { 0 }$ , $A _ { 1 }$ , $G _ { 0 }$ , and $G _ { 1 }$ such that $V _ { 0 } = A _ { 0 } \otimes G _ { 0 }$ and $V _ { 1 } = A _ { 1 } \otimes G _ { 1 }$ , the procedure to compute $F ^ { - 1 } z$ for an arbitrary vector $z = \operatorname { v e c } ( Z )$ is as follows.
+
+Pre-processing for Option 1 only:
+
+• Ensure that $A _ { 1 }$ and $G _ { 1 }$ are exactly symmetric, and if not, symmertrize them via:
+
+$$
+A _ { 1 }  { \frac { A _ { 1 } + A _ { 1 } ^ { \top } } { 2 } } \quad { \mathrm { a n d } } \quad G _ { 1 }  { \frac { G _ { 1 } + G _ { 1 } ^ { \top } } { 2 } }
+$$
+
+Pre-processing steps (both options):
+
+• Compute the matrix square roots of the factors of $V _ { 0 }$ (e.g. using eigendecompositions):
+
+$$
+A _ { 0 } ^ { - 1 / 2 } \quad \mathrm { a n d } \quad G _ { 0 } ^ { - 1 / 2 }
+$$
+
+• Compute the factors of the transformed transition matrix $\hat { \Psi }$ :
+
+$$
+\hat { \Psi } _ { A } = A _ { 0 } ^ { - 1 / 2 } A _ { 1 } A _ { 0 } ^ { - 1 / 2 } \quad \mathrm { a n d } \quad \hat { \Psi } _ { G } = G _ { 0 } ^ { - 1 / 2 } G _ { 1 } G _ { 0 } ^ { - 1 / 2 }
+$$
+
+Pre-processing for Option 1 only:
+
+• Compute the eigendecompositions of the factors of $\hat { \Psi }$
+
+$$
+U _ { A } \mathrm { d i a g } ( \hat { \psi } _ { A } ) U _ { A } ^ { \top } = \hat { \Psi } _ { A } \quad \mathrm { a n d } \quad U _ { G } \mathrm { d i a g } ( \hat { \psi } _ { G } ) U _ { G } ^ { \top } = \hat { \Psi } _ { G }
+$$
+
+• Sum across the temporal correlations and then invert by performing the element-wise computation in eigenspace:
+
+$$
+[ Y ] _ { i , j } = \gamma ( [ \hat { \psi } _ { G } ] _ { i } [ \hat { \psi } _ { A } ] _ { j } ) \quad \mathrm { w h e r e } \quad \gamma ( x ) = \frac { ( 1 - x ) ^ { 2 } } { \sum _ { i } \omega _ { i } ( { \mathcal T } _ { i } ( 1 - x ^ { 2 } ) - 2 x ( 1 - x ^ { { \mathcal T } _ { i } } ) ) }
+$$
+
+Pre-processing for Option 2 only:
+
+• Compute the eigendecompositions of the factors of $\hat { \Psi } ^ { \top } \hat { \Psi }$ (or equivalently the SVD’s of $\hat { \Psi } _ { A }$ and $\hat { \Psi } _ { G }$ ):
+
+$$
+E _ { A } \mathrm { d i a g } ( m _ { A } ) E _ { A } ^ { \top } = \hat { \Psi } _ { A } ^ { \top } \hat { \Psi } _ { A } \quad \mathrm { a n d } \quad E _ { G } \mathrm { d i a g } ( m _ { G } ) E _ { G } ^ { \top } = \hat { \Psi } _ { G } ^ { \top } \hat { \Psi } _ { G }
+$$
+
+$z$ -dependent calculations for Option 1:
+
+• Multiply the input vector $z = \operatorname { v e c } ( Z )$ by $V _ { 0 } ^ { - 1 / 2 }$
+
+$$
+Z _ { 0 } = G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 }
+$$
+
+• Multiply by $\hat { F } ^ { - 1 }$ using its eigendecomposition:
+
+$$
+Z _ { 1 } = U _ { G } ( ( U _ { G } ^ { \top } Z _ { 0 } U _ { A } ) \odot Y ) U _ { A } ^ { \top }
+$$
+
+where $B \odot C$ denotes the element-wise product between matrices $B$ and $C$ .
+
+• Multiply the result by $V _ { 0 } ^ { - 1 / 2 }$ and express as a vector:
+
+$$
+Z _ { 2 } = G _ { 0 } ^ { - 1 / 2 } Z _ { 1 } A _ { 0 } ^ { - 1 / 2 } \quad \mathrm { a n d } \quad F ^ { - 1 } z = \mathrm { v e c } ( Z _ { 2 } )
+$$
+
+$z$ -dependent calculations for Option 2:
+
+• Multiply the input vector $z = \operatorname { v e c } ( Z )$ by $V _ { 0 } ^ { - 1 / 2 }$ :
+
+$$
+Z _ { 0 } = G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 }
+$$
+
+• Multiply the result by $I - { \hat { \Psi } } ^ { \top }$ :
+
+$$
+Z _ { 1 } = Z _ { 0 } - \hat { \Psi } _ { G } ^ { \top } Z _ { 0 } \hat { \Psi } _ { A }
+$$
+
+• Multiply the previous result by $( I - \hat { \Psi } ^ { \top } \hat { \Psi } ) ^ { - 1 }$ using its eigendecomposition:
+
+$$
+\begin{array} { r } { Z _ { 2 } = E _ { G } ( ( E _ { G } ^ { \top } Z _ { 1 } E _ { A } ) \oslash ( \mathbb { I } \mathbb { I } ^ { \top } - m _ { G } m _ { A } ^ { \top } ) ) E _ { A } ^ { \top } } \end{array}
+$$
+
+where $X \oslash Y$ denotes the element-wise division of $X$ by $Y$ .
+
+• Multiply the result by $I - { \hat { \Psi } }$
+
+$$
+Z _ { 3 } = Z _ { 2 } - \hat { \Psi } _ { G } Z _ { 2 } \hat { \Psi } _ { A } ^ { \top }
+$$
+
+• Normalize the result by $\begin{array} { r } { \mathbb { E } \tau [ \mathcal { T } ] = \sum _ { i } \omega _ { i } \mathcal { T } _ { i } } \end{array}$ (for $\mathcal { T } _ { i }$ and $\omega _ { i }$ as defined at the bottom of Section 3.4):
+
+$$
+Z _ { 4 } = \frac { 1 } { \sum _ { i } \omega _ { i } \mathcal { T } _ { i } } Z _ { 3 }
+$$
+
+• Multiply the result by $V _ { 0 } ^ { - 1 / 2 }$ and express as a vector:
+
+$$
+Z _ { 5 } = G _ { 0 } ^ { - 1 / 2 } Z _ { 4 } A _ { 0 } ^ { - 1 / 2 } \quad \mathrm { a n d } \quad F ^ { - 1 } z = \mathrm { v e c } ( Z _ { 5 } )
+$$
+
+![](images/16150c71374fcf05e47bd613efbbf69dd41fdf0750837d9fd974275f9db499e0.jpg)  
+Figure 4: Generalization performance of our method compared to the baselines in perplexity-perword on length-35 word sequences from Penn-TreeBank. All the methods used a mini-batch size of 200. K-FAC indep. uses the update in eqn. 6, K-FAC option1 uses eqn. 8, and K-FAC option2 uses eqn. 9. (left) Validation perplexity v.s. the number of updates. The dashed lines correspond to the experiments that used RNNs with 1024 LSTM units, and the solid lines correspond to experiments that used RNNs with 650 LSTM units. (right) Validation perplexity v.s. the wall-clock time.
+
+D TEST PERFORMANCE ON PENN-TREEBANK
\ No newline at end of file
diff --git a/parse/train/HyMTkQZAb/HyMTkQZAb_content_list.json b/parse/train/HyMTkQZAb/HyMTkQZAb_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..b387e3aabc76ff3cdc09ee21b18a4de3f406d89c
--- /dev/null
+++ b/parse/train/HyMTkQZAb/HyMTkQZAb_content_list.json
@@ -0,0 +1,4951 @@
+[
+    {
+        "type": "text",
+        "text": "KRONECKER-FACTORED CURVATURE APPROXIMATIONS FOR RECURRENT NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "James Martens   \nDeepMind   \njamesmartens@google.com ",
+        "bbox": [
+            184,
+            170,
+            413,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jimmy Ba ",
+        "text_level": 1,
+        "bbox": [
+            593,
+            171,
+            666,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science University of Toronto Toronto, Canada jimmy@psi.toronto.edu ",
+        "bbox": [
+            593,
+            185,
+            813,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Matthew Johnson Google Brain mattjj@google.com ",
+        "bbox": [
+            183,
+            261,
+            352,
+            304
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            340,
+            544,
+            354
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kronecker-factor Approximate Curvature (Martens & Grosse, 2015) (K-FAC) is a 2nd-order optimization method which has been shown to give state-of-the-art performance on large-scale neural network optimization tasks (Ba et al., 2017). It is based on an approximation to the Fisher information matrix (FIM) that makes assumptions about the particular structure of the network and the way it is parameterized. The original K-FAC method was applicable only to fully-connected networks, although it has been recently extended by Grosse & Martens (2016) to handle convolutional networks as well. In this work we extend the method to handle RNNs by introducing a novel approximation to the FIM for RNNs. This approximation works by modelling the statistical structure between the gradient contributions at different time-steps using a chain-structured linear Gaussian graphical model, summing the various cross-moments, and computing the inverse in closed form. We demonstrate in experiments that our method significantly outperforms general purpose state-of-the-art optimizers like SGD with momentum and Adam on several challenging RNN training tasks. ",
+        "bbox": [
+            233,
+            373,
+            764,
+            580
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            612,
+            334,
+            627
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As neural networks have become ubiquitous in both research and applications the need to efficiently train has never been greater. The main workhorses for neural net optimization are stochastic gradient descent (SGD) with momentum and various 2nd-order optimizers that use diagonal curvature-matrix approximations, such as RMSprop (Tieleman & Hinton, 2012) and Adam (Ba & Kingma, 2015). While the latter are typically easier to tune and work better out of the box, they unfortunately only offer marginal performance improvements over well-tuned SGD on most problems. ",
+        "bbox": [
+            174,
+            645,
+            825,
+            728
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Because modern neural networks have many millions of parameters it is computationally too expensive to compute and invert an entire curvature matrix and so approximations are required. While early work on non-diagonal curvature matrix approximations such as TONGA (Le Roux et al., 2008) and the Hessian-free (HF) approach (Martens, 2010; Martens & Sutskever, 2011; 2012; Desjardins et al., 2013; Sainath et al., 2013) demonstrated the potential of such methods, they never achieved wide adoption due to issues of scalability (to large models in the case of the former, and large datasets in the case of the latter). ",
+        "bbox": [
+            174,
+            736,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Motivated in part by these older results and by the more recent success of centering and normalization methods (e.g. Schraudolph, 1998; Vatanen et al., 2013; Ioffe & Szegedy, 2015) a new family of methods has emerged that are based on non-diagonal curvature matrix approximations the rely on the special structure of neural networks. Such methods, which include Kronecker-factored approximated curvature (K-FAC) (Martens & Grosse, 2015), Natural Neural Nets (Desjardins et al., 2015), Practical Riemannian Neural Networks (Marceau-Caron & Ollivier, 2016), and others (Povey et al., 2015), have achieved state-of-the-art optimization performance on various challenging neural network training tasks and benchmarks. ",
+        "bbox": [
+            174,
+            840,
+            823,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While the original K-FAC method is applicable only to standard feed-forward networks with fully connected layers, it has recently been extended to handle convolutional networks (Grosse & Martens, 2016) through the introduction of the “Kronecker Factors for Convolution” (KFC) approximation. Ba et al. (2017) later developed a distributed asynchronous version which proposed additional approximations to handle very large hidden layers. ",
+        "bbox": [
+            174,
+            138,
+            823,
+            208
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work we develop a new family of curvature matrix approximations for recurrent neural networks (RNNs) within the same design space. As in the original K-FAC approximation and the KFC approximation, we focus on the Fisher information matrix (a popular choice of curvature matrix), and show how it can be approximated in different ways through the adoption of various approximating assumptions on the statistics of the network’s gradients. Our main novel technical contribution is an approximation which uses a chain-structured linear Gaussian graphical model to describe the statistical relationship between gradient contributions coming from different time-steps. Somewhat remarkably, it is possible to sum the required cross-moments to obtain a Fisher approximations which has enough special algebraic structure that it can still be efficiently inverted. In experiments we demonstrate the usefulness of our approximations on several challenging RNN training tasks. ",
+        "bbox": [
+            174,
+            215,
+            825,
+            354
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 NOTATION AND BACKGROUND",
+        "text_level": 1,
+        "bbox": [
+            176,
+            375,
+            457,
+            390
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 NETWORK, LOSS, AND OBJECTIVE FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            406,
+            529,
+            420
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We denote by $f ( x , \\theta )$ the neural network function associated evaluated on input $x$ , where $\\theta$ are the parameters. We will assume a loss function of the form $L ( y , z ) = - \\log r ( y | \\bar { z } )$ , where $r$ is the density function associated with a predictive distribution $R$ . The loss associated with a single training case is then given by $L ( y , f ( x , \\bar { \\theta } ) ) \\equiv - \\log r ( y | f ( x , \\theta ) )$ . Throughout the rest of this document we will use the following special notation for derivatives of the single-case loss w.r.t. some arbitrary variable $Z$ (possibly matrix-valued): ",
+        "bbox": [
+            174,
+            431,
+            825,
+            516
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f597276de84998e469c9ba6fe64c3b57cc0b4d6a86da66f5d2e3b8bfd30c6352.jpg",
+        "text": "$$\n\\mathcal { D } Z = \\frac { \\mathrm { d } L ( y , f ( x , \\theta ) ) } { \\mathrm { d } Z }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            423,
+            521,
+            573,
+            553
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The objective function which we wish to minimize is the expected loss $h ( \\theta ) = \\mathbb { E } _ { Q } [ L ( y , f ( x , \\theta ) ) ]$ over the training distribution $Q$ on $x$ and $y$ . ",
+        "bbox": [
+            174,
+            559,
+            823,
+            587
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 THE FISHER, THE NATURAL GRADIENT, AND 2ND-ORDER OPTIMIZATION ",
+        "bbox": [
+            173,
+            603,
+            718,
+            618
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The Fisher information matrix (aka “the Fisher”) associated with the model’s predictive distribution $P _ { y | x } ( \\theta )$ is given by ",
+        "bbox": [
+            171,
+            628,
+            823,
+            659
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/31eef2f85f338c7365df403d2f499800adf170e2fcf3907b4d3c811fc0563c8f.jpg",
+        "text": "$$\nF = \\mathbb { E } \\left[ { \\mathcal { D } } \\theta { \\mathcal { D } } \\theta ^ { \\top } \\right] = \\operatorname { c o v } ( { \\mathcal { D } } \\theta , { \\mathcal { D } } \\theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            665,
+            614,
+            686
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Note that here, and for the remainder of this paper, $y$ is taken to be distributed according to the model’s predictive conditional distribution $P _ { y | x } ( \\theta )$ , so that $\\mathbb { E } [ \\mathcal { D } Z ] = 0$ for any variable $Z$ that is conditionally independent of $y$ given the value of $f ( x , \\theta )$ (this includes $\\mathcal { D } \\theta$ ). All expectations and covariances are defined accordingly. This is done because the expectation that defines the Fisher information matrix uses $P _ { y | x } ( \\theta )$ . If we were to instead use the training distribution $Q _ { y | x }$ on $y$ , we would essentially be computing to the “empirical Fisher” (or approximations thereof), which as argued by Martens (2014) is a less appropriate choice for a curvature matrix than the true Fisher. ",
+        "bbox": [
+            173,
+            696,
+            825,
+            797
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The natural gradient is defined as $F ^ { - 1 } \\nabla h$ , and is the update direction used in natural gradient descent. As argued by Amari (1998), natural gradient descent has the two key advantages: it is invariant to the parameterization of the model, and has “Fisher efficient” convergence1. However, as shown by Martens (2014) these two facts have several important caveats. First, the parameterization invariance only holds approximately in practice when non-infinitesimal step-sizes are used. Second, Fisher efficiency is actually a weak property possessed by simpler methods like SGD with ",
+        "bbox": [
+            174,
+            803,
+            825,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Polyak/parameter averaging (Polyak & Juditsky, 1992), and even then will only be achieved when the method converges to a global minimizer and the model is capable of perfectly capturing the true distribution of $y$ given $x$ . ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An alternative explanation for the empirical success of the natural gradient method is that it is a 2ndorder method, whose update minimizes the following local quadratic approximation to the objective $h ( \\theta + \\delta )$ : ",
+        "bbox": [
+            174,
+            152,
+            823,
+            194
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3248f13a2a6cf907ef2f4c5a31b0dd51ccb4c94ac256ba25545783ae57157862.jpg",
+        "text": "$$\n\\frac { 1 } { 2 } \\delta ^ { \\top } F \\delta + \\nabla h ( \\theta ) ^ { \\top } \\delta + h ( \\theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            196,
+            596,
+            227
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This is similar to the 2nd-order Taylor series approximation of $h ( \\theta + \\delta )$ , but with the Fisher substituted in for the Hessian. This substitution can be justified by the observation that the Fisher is a kind of PSD approximation to the Hessian (Pascanu & Bengio, 2014; Martens, 2014). And as argued by Martens (2014), while stochastic 2nd-order methods like natural gradient descent cannot beat the asymptotically optimal Fisher efficient convergence achieved by SGD with Polyak averaging, they can enjoy better pre-asymptotic convergence rates. Moreover, insofar as gradient noise can be mitigated through the use of large mini-batches – so that stochastic optimization starts to resemble deterministic optimization – the theoretical advantages of 2nd-order methods become further pronounced, which agrees with the empirical observation that the use of large-minibatches speeds up 2nd-methods much more than 1st-order methods (Martens & Grosse, 2015; Ba et al., 2017). ",
+        "bbox": [
+            173,
+            228,
+            825,
+            368
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition to providing an arguably better theoretical argument for the success of natural gradient methods, their interpretation as 2nd-order methods also justifies the common practice of computing the natural gradient as $( F + \\lambda I ) ^ { - 1 } \\nabla h$ instead of $F ^ { - 1 } \\dot { \\nabla } h$ . In particular, this practice can be viewed as a type of “update damping/regularization”, where one encourages $\\delta$ to lie within some region around $\\delta = 0$ where eqn. 1 remains a trustworthy approximation (e.g. Nocedal & Wright, 2006; Martens & Sutskever, 2012). ",
+        "bbox": [
+            173,
+            375,
+            825,
+            459
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 KRONECKER-FACTORED APPROXIMATE CURVATURE (K-FAC) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            474,
+            640,
+            489
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Because modern neural network have millions (or even billions) of parameters it is computationally too expensive to compute and invert the Fisher. To address this problem, the K-FAC method of Martens $\\&$ Grosse (2015) uses a block-diagonal approximation of the Fisher (where the blocks correspond to entire layers/weight matrices), and where the blocks are further approximated as Kronecker products between much smaller matrices. The details of this approximation are given in the brief derivation below. ",
+        "bbox": [
+            173,
+            500,
+            825,
+            584
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $W$ be a weight matrix in the network which computes the mapping ",
+        "bbox": [
+            174,
+            590,
+            640,
+            606
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9eb90c829006e5687b6e02f0e611845e86ab3f4c97360e9ddedf386c43cd42cb.jpg",
+        "text": "$$\ns = W a ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            467,
+            608,
+            531,
+            625
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $a$ and $s$ are vector-valued inputs and outputs respectively and denote ",
+        "bbox": [
+            173,
+            627,
+            666,
+            641
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e05cf512c12f2fdf8b6679d84e776786a0d63fc009112f0a6c7ee475e25b2d1d.jpg",
+        "text": "$$\ng = { \\mathcal { D } } s .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            468,
+            643,
+            527,
+            660
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As in the original K-FAC paper we will assume that $a$ includes a homogeneous coordinate with value 1 so that the bias vector may be folded into the matrix $W$ . ",
+        "bbox": [
+            171,
+            662,
+            823,
+            690
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here and throughout the rest of this document, $F$ will refer to the block of the Fisher corresponding to this particular weight-matrix $W$ . ",
+        "bbox": [
+            174,
+            696,
+            823,
+            726
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Kronecker product of matrices $B$ and $C$ , denoted by $B \\otimes C$ for matrices $B \\in \\mathbb { R } ^ { m \\times n }$ and $C$ of arbitrary dimensions, is a block matrix defined by ",
+        "bbox": [
+            174,
+            732,
+            821,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06981a2b7b433c0d78ed61ac6feb61619117678e077c567a6d14d15ea9ed5af1.jpg",
+        "text": "$$\nB \\otimes C \\equiv \\left[ \\begin{array} { c c c } { { [ B ] _ { 1 , 1 } C } } & { { \\cdot \\cdot \\cdot } } & { { [ B ] _ { 1 , n } C } } \\\\ { { \\vdots } } & { { \\cdot \\cdot } } & { { \\vdots } } \\\\ { { [ B ] _ { m , 1 } C } } & { { \\cdot \\cdot \\cdot } } & { { [ B ] _ { m , n } C } } \\end{array} \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            762,
+            630,
+            820
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that the Kronecker product has many convenient properties that we will make use of in this paper. (See Van Loan (2000) for a good discussion of the Kronecker product and its properties.) ",
+        "bbox": [
+            173,
+            821,
+            823,
+            851
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A simple application of the chain rule gives $\\mathcal { D } W = g a ^ { \\top }$ . If we approximate $g$ and $a$ as statistically independent, we can write $F$ as ",
+        "bbox": [
+            173,
+            854,
+            823,
+            885
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bd5bb7d730b2665d40f3d8042d135b0ea5f1b2587641b9672a283163f747b004.jpg",
+        "text": "$$\n\\begin{array} { r c l } { F } & { = } & { \\mathbb { E } [ \\mathrm { v e c } ( \\mathcal { D } W ) \\mathrm { v e c } ( \\mathcal { D } W ) ^ { \\top } ] = \\mathbb { E } [ \\mathrm { v e c } ( g a ^ { \\top } ) \\mathrm { v e c } ( g a ^ { \\top } ) ^ { \\top } ] = \\mathbb { E } [ ( a \\otimes g ) ( a \\otimes g ) ^ { \\top } ] } \\\\ & { = } & { \\mathbb { E } [ ( a a ^ { \\top } ) \\otimes ( g g ^ { \\top } ) ] = \\mathbb { E } [ a a ^ { \\top } ] \\otimes \\mathbb { E } [ g g ^ { \\top } ] = A \\otimes G , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            886,
+            782,
+            928
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where we have defined ",
+        "bbox": [
+            174,
+            104,
+            326,
+            117
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/786396a8bab72b213c9664843550004f1e3b828346e32f8e8cf5b99063244ff1.jpg",
+        "text": "$$\nA = \\mathbb { E } [ a a ^ { \\top } ] \\quad { \\mathrm { a n d } } \\quad G = \\mathbb { E } [ g g ^ { \\top } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            114,
+            614,
+            133
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The matrices $A$ and $G$ can be estimated using simple Monte Carlo methods, and averaged over lots of data by taking an exponentially decaying average across mini-batches. ",
+        "bbox": [
+            174,
+            136,
+            825,
+            164
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This is the basic Kronecker factored approximation (Heskes, 2000; Martens & Grosse, 2015; Povey et al., 2015), which is related to the approximation made in the Natural Neural Nets approach (Desjardins et al., 2015). It is shown by Martens $\\&$ Grosse (2015) that the approximation is equivalent to neglecting the higher-order cumulants of the as and $g \\mathrm { s }$ , or equivalently, assuming that they are Gaussian distributed. To see why this approximation is useful, we observe that inversion and multiplication of a vector by $F$ amounts to inverting the factor matrices $A$ and $G$ and performing matrix-matrix multiplications with them, due to the following two basic identities: ",
+        "bbox": [
+            173,
+            170,
+            825,
+            268
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f060ad2aff638da77736f3b05f727e9a1350fb6acce0663102682d2a310bfce.jpg",
+        "text": "$$\n( B \\otimes C ) ^ { - 1 } = B ^ { - 1 } \\otimes C ^ { - 1 } \\quad \\mathrm { a n d } \\quad ( B \\otimes C ) \\operatorname { v e c } ( X ) = \\operatorname { v e c } ( C X B ^ { \\top } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            272,
+            732,
+            291
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The required inversion and matrix multiplication operations are usually computational feasible because the factor matrices have dimensions equal to the size of the layers, which is typically just a few thousand. And when they are not, additional approximations can be applied, such as approximate/iterative inversion (Povey et al., 2015), or additional Kronecker-factorization applied to either $A$ or $G$ (Ba et al., 2017). Moreover, the computation of the inverses can be amortized across iterations of the optimizer at the cost of introducing some staleness into the estimates. ",
+        "bbox": [
+            174,
+            303,
+            825,
+            388
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 APPROXIMATING $F$ FOR RNNS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            407,
+            465,
+            424
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The basic Kronecker-factored approximation to the Fisher block $F$ described in the previous section assumed that the weight matrix $W$ was used to compute a single mapping of the form $s = W a$ . When $W$ is used to compute multiple such mappings, as is often the case for RNNs, or a mapping of a different flavor, as is the case for convolutional networks (CNNs), the approximation is not applicable, strictly speaking. ",
+        "bbox": [
+            173,
+            438,
+            825,
+            508
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Grosse & Martens (2016) recently showed that by making additional approximating assumptions, the basic Kronecker-factored approximation can be extended to convolutional layers. This new approximation, called “KFC”, is derived by assuming that gradient contributions coming from different spatial locations are uncorrelated, and that their intra and inter-location statistics are spatially homogeneous, in the sense that they look the same from all reference locations. These assumptions are referred to “spatially uncorrelated derivatives” and “spatial homogeneity,” respectively. ",
+        "bbox": [
+            173,
+            515,
+            825,
+            599
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section we give the main technical contribution of this paper, which is a family of Kroneckerbased approximations of $F$ that can be applied to RNNs. To build this we will apply various combinations of the approximating assumptions used to derive the original K-FAC and KFC approaches, along with several new ones, including an approximation which works by modelling the statistical structure between the gradient contributions from time-steps using a chain-structured linear Gaussian graphical model. ",
+        "bbox": [
+            174,
+            606,
+            825,
+            689
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            705,
+            326,
+            720
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $W$ be some weight matrix which is used at $\\tau$ different time-steps (or positions) to compute the mapping ",
+        "bbox": [
+            171,
+            732,
+            823,
+            760
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5dd193cb6f678abe00fa232a30e2d7a80c52a2218275a53f2bd86a99a959ef3e.jpg",
+        "text": "$$\ns _ { t } = W a _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            460,
+            760,
+            535,
+            775
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $t$ indexes the time-step. $\\tau$ is allowed to vary between different training cases. ",
+        "bbox": [
+            176,
+            777,
+            725,
+            792
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Defining $g _ { t } = \\mathcal { D } s _ { t }$ , the gradient of the single-case loss with respect to $W$ can be written as ",
+        "bbox": [
+            169,
+            797,
+            767,
+            814
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ebb253835774b0d533a93c24579e8e89c7efedceda05765f7b4eb0964821acdd.jpg",
+        "text": "$$\n\\mathcal { D } W = \\sum _ { t = 1 } ^ { \\mathcal { T } } g _ { t } a _ { t } ^ { \\top } = \\sum _ { t = 1 } ^ { \\mathcal { T } } \\mathcal { D } _ { t } W ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            818,
+            599,
+            862
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { D } _ { t } W = g _ { t } a _ { t } ^ { \\top }$ denotes the contribution to the gradient from time-step $t$ . When it is more convenient to work with the vector-representations of the matrix-valued variables $\\mathcal { D } _ { t } W$ we will use the notation ",
+        "bbox": [
+            174,
+            867,
+            826,
+            909
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bd479b43862562a48d9e126026fd70606160d7b704c3bc85fde37607577ed9b4.jpg",
+        "text": "$$\nw _ { t } = \\operatorname { v e c } ( \\mathscr { D } _ { t } W ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            439,
+            907,
+            555,
+            925
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "so that vec(DW ) = PTt= . ",
+        "bbox": [
+            173,
+            101,
+            374,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let $F _ { T }$ denote the conditional Fisher of $\\mathcal { D } W$ for a particular value of $\\tau$ . We have ",
+        "bbox": [
+            173,
+            125,
+            709,
+            140
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/497bc8c3483ad7970a4ab86dcec7b774502070e3c71668210bea53f2e9a96f83.jpg",
+        "text": "$$\nF _ { T } = \\mathbb { E } [ \\mathrm { v e c } ( \\mathcal { D } W ) \\mathrm { v e c } ( \\mathcal { D } W ) ^ { \\top } | T ] = \\mathbb { E } \\left[ \\left( \\sum _ { t = 1 } ^ { T } w _ { t } \\right) \\left( \\sum _ { t = 1 } ^ { T } w _ { t } \\right) ^ { \\top } \\Bigg | T \\right] = \\sum _ { t = 1 } ^ { T } \\sum _ { s = 1 } ^ { T } \\mathbb { E } \\left[ w _ { t } w _ { s } ^ { \\top } | T \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            146,
+            818,
+            198
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Observe that $F$ can be computed from $F _ { T }$ via $F = \\mathbb { E } _ { T } [ F _ { T } ]$ . ",
+        "bbox": [
+            173,
+            207,
+            573,
+            223
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To proceed with our goal of obtaining a tractable approximation to $F$ we will make several approximating assumptions, as discussed in the next section. ",
+        "bbox": [
+            174,
+            229,
+            823,
+            258
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 BASIC INITIAL APPROXIMATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            276,
+            444,
+            290
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 INDEPENDENCE OF $\\tau$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            301,
+            379,
+            316
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One simplifying approximation we will make immediately is that $\\tau$ is independent of the $w _ { t }$ ’s, so that $\\mathbb { E } [ \\tilde { w _ { t } } w _ { s } ^ { \\top } | \\breve { \\mathcal { T } } ] = \\mathbb { E } [ w _ { t } w _ { s } ^ { \\top } ]$ . In this case eqn. 3 can be written as ",
+        "bbox": [
+            173,
+            327,
+            823,
+            356
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc036dedc4001fc8db65f7c0c0ddf9d09c16e790bc7c1f74a721bd45b0d69431.jpg",
+        "text": "$$\nF _ { \\mathcal { T } } = { \\sum _ { t = 1 } ^ { \\mathcal { T } } } \\sum _ { s = 1 } ^ { \\mathcal { T } } \\mathbb { E } [ w _ { t } w _ { s } ^ { \\top } ] = { \\sum _ { t = 1 } ^ { \\mathcal { T } } } \\sum _ { s = 1 } ^ { \\mathcal { T } } V _ { t , s } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            363,
+            632,
+            407
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we have defined $V _ { t , s } = \\mathbb { E } [ w _ { t } w _ { s } ^ { \\top } ]$ . ",
+        "bbox": [
+            176,
+            415,
+            442,
+            431
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Independence of $\\tau$ and the $w _ { t }$ ’s is a reasonable approximation assumption to make because 1) for many datasets $\\tau$ is constant (which formally implies independence), and 2) even when $\\tau$ varies substantially, shorter sequences will typically have similar statistical properties to longer ones (e.g. short paragraphs of text versus longer paragraphs). ",
+        "bbox": [
+            174,
+            436,
+            825,
+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.2 TEMPORAL HOMOGENEITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            510,
+            416,
+            525
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Another convenient and natural approximating assumption we will make is that the $w _ { t }$ ’s are temporally homogeneous, which is to say that the statistical relationship between any $w _ { t }$ and $w _ { s }$ depends only on their distance in time $( d \\ : = \\ : t \\ : - \\ : s )$ . This is analogous to the “spatial homogeneity” assumption of KFC. Under this assumption the following single-subscript notation is well-defined: $V _ { t - s } = \\mathbb { E } [ w _ { t } w _ { s } ^ { \\top } ]$ . We note that $V _ { - d } \\doteq V _ { d } ^ { \\top }$ . ",
+        "bbox": [
+            173,
+            534,
+            825,
+            606
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Applying this notation to eqn. 4 we have ",
+        "bbox": [
+            176,
+            611,
+            441,
+            626
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/44504a0154ad7fd681406ed421e2c5eb5e33c9c9485eb7d4f03da6fff0da7aba.jpg",
+        "text": "$$\nF _ { \\mathcal { T } } = \\sum _ { t = 1 } ^ { \\mathcal { T } } \\sum _ { s = 1 } ^ { \\mathcal { T } } V _ { t , s } = \\sum _ { d = - \\mathcal { T } } ^ { \\mathcal { T } } ( \\mathcal { T } - | d | ) V _ { d } = \\sum _ { d = 0 } ^ { \\mathcal { T } } ( \\mathcal { T } - d ) V _ { d } + \\sum _ { d = 0 } ^ { \\mathcal { T } } ( \\mathcal { T } - d ) V _ { d } ^ { \\top } - \\mathcal { T } I ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            632,
+            772,
+            678
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we have used the fact that there are $\\tau - | d |$ ways to write $d$ as $t - s$ for $t , s \\in \\{ 1 , 2 , \\ldots , T \\}$ ",
+        "bbox": [
+            171,
+            685,
+            821,
+            702
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Temporal homogeneity is a pretty mild approximation, and is analogous to the frequently used “steady-state assumption” from dynamical systems. Essentially, it is the assumption that the Markov chain defined by the system “mixes” and reaches its equilibrium distribution. If the system has any randomness, and its external inputs reach steady-state, the steady-state assumption is quite accurate for states sufficiently far from the beginning of the sequence (which will be most of them). ",
+        "bbox": [
+            173,
+            707,
+            825,
+            779
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.3 INDEPENDENCE BETWEEN THE $a _ { t }$ ’S AND THE $g _ { t }$ ’ S ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            794,
+            575,
+            809
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If we have that $a _ { t }$ and $g _ { s }$ are pair-wise independent for each $t$ and $s$ , which is the obvious generalization of the basic approximation used to derive the K-FAC approach, then following a similar derivation to the one from Section 2.3 we have ",
+        "bbox": [
+            173,
+            818,
+            825,
+            861
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1ff9f400e0d0386ca44b393751edc10f3090d0687df8cc0391db3a9c5307be5.jpg",
+        "text": "$$\nV _ { t , s } = \\mathbb { E } [ ( a _ { t } a _ { s } ^ { \\top } ) \\otimes ( g _ { t } g _ { s } ^ { \\top } ) ] = \\mathbb { E } [ ( a _ { t } a _ { s } ^ { \\top } ) \\otimes ( { \\mathcal { D } } s _ { t } { \\mathcal { D } } s _ { s } ^ { \\top } ) ] = A _ { t , s } \\otimes G _ { t , s } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            867,
+            732,
+            887
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we have defined ",
+        "bbox": [
+            174,
+            893,
+            326,
+            907
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6356bb289a15c6423d1271ffe96c19134d4ccc2f9edd4ed0ea82c9d5c55d9fab.jpg",
+        "text": "$$\nA _ { t , s } = \\mathbb { E } [ a _ { t } a _ { s } ^ { \\top } ] \\quad \\mathrm { a n d } \\quad G _ { t , s } = \\mathbb { E } [ g _ { t } g _ { s } ^ { \\top } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            906,
+            638,
+            926
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Extending our temporal homogeneity assumption from the $w _ { t }$ ’s to the $a _ { t }$ ’s and $g _ { t }$ ’s (which is natural to do since $w _ { t } = \\dot { \\mathrm { v e c } } ( g _ { t } a _ { t } ^ { \\top } ) )$ ), the following notation becomes well-defined: ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3f3e72c65685b648a60a7141c7c1ba87921a734dad85b63a4781e6bc2caebe54.jpg",
+        "text": "$$\nA _ { t - s } = A _ { t , s } \\quad \\mathrm { a n d } \\quad G _ { t - s } = G _ { t , s } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            138,
+            617,
+            156
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "which allows us to write ",
+        "bbox": [
+            174,
+            161,
+            334,
+            175
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce8e9b851920b9a33290e1deb0ed86aadd62d85b83c9768012c4135557ade271.jpg",
+        "text": "$$\nV _ { d } = A _ { d } \\otimes G _ { d } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            174,
+            552,
+            190
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 AN INITIAL ATTEMPT TO OBTAIN A TRACTABLE FISHER APPROXIMATION ",
+        "bbox": [
+            174,
+            207,
+            718,
+            222
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Given the approximating assumptions made in the previous subsections we have ",
+        "bbox": [
+            174,
+            232,
+            700,
+            247
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b098a98747f88d1c618f1f61be2a2d8ac40b238208957947ae39bfa20ba02a0.jpg",
+        "text": "$$\nF _ { \\mathcal { T } } = \\sum _ { d = - \\mathcal { T } } ^ { \\mathcal { T } } ( \\mathcal { T } - | d | ) V _ { d } = \\sum _ { d = - \\mathcal { T } } ^ { \\mathcal { T } } ( \\mathcal { T } - | d | ) ( A _ { d } \\otimes G _ { d } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            252,
+            679,
+            299
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assuming for the moment that all of the training sequences have the same length, so that $F = F _ { \\mathcal { T } _ { 0 } }$ for some $\\mathcal { T } _ { 0 }$ , we have that $F$ will be the sum of $2 \\mathcal { T } _ { 0 } + 1$ Kronecker products. ",
+        "bbox": [
+            171,
+            303,
+            821,
+            332
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Without assuming any additional structure, such as a relationship between the various $A _ { d }$ ’s or $G _ { d }$ ’s, there doesn’t appear to be any efficient way to invert such a sum. One can use the elementary identity $( B \\otimes C ) ^ { - 1 } = \\bar { B } ^ { - 1 } \\otimes C ^ { - 1 }$ to invert a single Kronecker product, and there exists decomposition-based methods to efficiently invert sums of two Kronecker products (see Martens & Grosse (2015)), however there is no known efficient algorithm for inverting sums of three or more Kronecker products. Thus is appears that we must make additional approximating assumptions in order to proceed. ",
+        "bbox": [
+            173,
+            338,
+            825,
+            422
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 ASSUMING INDEPENDENCE OF THE $w _ { t }$ ’S ACROSS TIME ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            439,
+            593,
+            454
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "If we assume that the contributions to the gradient (the $w _ { t }$ ’s) are independent across time, or at least uncorrelated, this means that $V _ { d } = 0$ for $d \\neq 0$ . This is analogous to the “spatially uncorrelated derivatives” assumption of KFC. ",
+        "bbox": [
+            176,
+            464,
+            825,
+            507
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this case eqn. 5 simplifies to ",
+        "bbox": [
+            173,
+            513,
+            379,
+            529
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46b78d4a31ce0f237b01dc616474d46f0ab02a1694210c1a022475d13ba418d4.jpg",
+        "text": "$$\nF _ { \\mathcal { T } } = \\sum _ { d = - \\mathcal { T } } ^ { \\mathcal { T } } ( \\mathcal { T } - | d | ) V _ { d } = ( \\mathcal { T } - 0 ) V _ { 0 } = \\mathcal { T } V _ { 0 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            534,
+            655,
+            579
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "so that ",
+        "bbox": [
+            173,
+            585,
+            220,
+            599
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/358adf27967813c75f5623d2abcf48881425daace9c0b1354b6b358bfae04604.jpg",
+        "text": "$$\nF = \\mathbb { E } _ { \\mathcal { T } } [ F _ { \\mathcal { T } } ] = \\mathbb { E } _ { \\mathcal { T } } [ \\mathcal { T } V _ { 0 } ] = \\mathbb { E } _ { \\mathcal { T } } [ \\mathcal { T } ] V _ { 0 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            597,
+            632,
+            614
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using the identities in eqn. 2, and the symmetry of $A _ { 0 }$ and $G _ { 0 }$ , we can thus efficiently multiply $F ^ { - 1 }$ by a vector $z = \\operatorname { v e c } ( Z )$ using the formula ",
+        "bbox": [
+            173,
+            625,
+            821,
+            654
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a601149e6e60979795465a39723d18c77d132a83db3d4cd23643a0cdbc65910.jpg",
+        "text": "$$\nF ^ { - 1 } z = { \\frac { 1 } { \\mathbb { E } _ { T } [ T ] } } \\operatorname { v e c } ( G _ { 0 } ^ { - 1 } Z A _ { 0 } ^ { - 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            659,
+            614,
+            693
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This is, up to normalization by $\\mathbb { E } \\tau [ \\mathcal { T } ]$ , identical to the inverse multiplication formula used in the original K-FAC approximation for fully-connected layers. ",
+        "bbox": [
+            174,
+            698,
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+            727
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We note that $\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { T } } [ \\mathcal { T } ] = \\sum _ { i } \\omega _ { i } \\mathcal { T } _ { i } } \\end{array}$ , where $\\mathcal { T } _ { i }$ are the different values of $\\tau$ , and $\\omega _ { i } \\geqslant 0$ are normalized weights (with $\\textstyle \\sum _ { i } \\omega _ { i } = 1 $ ) that measure their proportions in the training set. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.5 MODELING THE RELATIONSHIPS BETWEEN THE $w _ { t }$ ’S USING AN LGGM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            779,
+            707,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As we saw in Section 3.3, the approximation assumptions made in Section 3.2 (independence of $\\tau$ , temporal homogeneity, and independence between the $a _ { t }$ ’s and the $g _ { t }$ ’s), aren’t sufficient to yield a tractable formula for $\\scriptstyle { \\dot { F } } ^ { - 1 }$ . And while additionally assuming independence across time of the $w _ { t }$ ’s is sufficient (as shown in Section 3.4), it seems like an overly severe approximation to make. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section we consider a less severe approximation which we will show still produces a tractable $F ^ { - 1 }$ . In particular, we will assume that the statistical relationship of the $w _ { t }$ ’s is described by a simple linear Gaussian graphical model (LGGM) with a compact parameterization (whose size is independent of $\\tau$ ). Such an approach to computing a tractable Fisher approximations was first explored by Grosse & Salakhutdinov (2015) for RBMs, although our use of it here is substantially different, and requires additional mathematical machinery. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The model we will use is a fairly natural one. It is a linear Gaussian graphical model with a onedimensional chain structure corresponding to time. The graphical structure of our model is given by the following picture: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b5321b39344150c6ed0d2d609369eeef29164dc9baac7830ff39342112dba98.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            282
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Variables in the model evolve forward in time according to the following equation: ",
+        "bbox": [
+            173,
+            295,
+            714,
+            310
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a75cb929e9b0974d353d2675534dd3a010d3989a8936c46cdd71a114ab66a747.jpg",
+        "text": "$$\nw _ { t } = \\Psi w _ { t - 1 } + \\epsilon _ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            318,
+            560,
+            333
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\Psi$ is a square matrix and $\\epsilon _ { t }$ are i.i.d. from $\\mathcal { N } ( 0 , \\Sigma )$ for some positive definite matrix $\\Sigma$ (which is the conditional covariance of $w _ { t }$ given $w _ { t - 1 }$ ). ",
+        "bbox": [
+            173,
+            339,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Due to the well-known equivalence between directed and undirected Gaussian graphical models for tree-structured graphs like this one, the decision of whether to make the edges directed or undirected, and whether to have them point forwards or backwards in time, are irrelevant from a modeling perspective (and thus to the Fisher approximation we eventually compute). We will use a directed representation purely for mathematical convenience. ",
+        "bbox": [
+            173,
+            375,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We will assume that our model extends infinitely in both directions, with indices in the range $( - \\infty , \\infty )$ , so that the $w _ { t }$ ’s are all in their stationary distribution (with respect to time). For this to yield a well-defined model we require that $\\Psi$ has spectral radius $< 1$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The intuition behind this model structure is clear. The correlations between gradient contributions (the $w _ { t }$ ’s) at two different time-steps should be reasonably well explained by the gradient contributions made at time-steps between them. In other words, they should be approximately Markovian. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We know that the gradient computations are generated by a process, Back-prop Through Time (BPTT), where information flows only between consecutive time-steps (forwards through time during the “forward pass”, and backwards during the “backwards pass”). This process involves temporal quantities which are external to the $w _ { t }$ ’s, such as the inputs $x$ and activations for other layers, which essentially act as “hidden variables”. The evolution of these external quantities may be described by their own separate temporal dynamics (e.g. the unknown process which generates the true $x$ ’s), and thus the $w _ { t }$ ’s won’t be Markovian in general. But insofar as the $w _ { t }$ ’s (or equivalently the $a _ { t }$ ’s and $g _ { t }$ ’s) encode the relevant information contained in these external variables, they should be approximately Markovian. (If they contained all of the information they would be exactly Markovian.) ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A similar approximation across consecutive layers was made in the “block-tridiagonal” version of the original K-FAC approach. It was shown by Martens & Grosse (2015) that this approximation was a pretty reasonable one. The linear-Gaussian assumption meanwhile is a more severe one to make, but it seems necessary for there to be any hope that the required expectations remain tractable. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.5.1 INITIAL COMPUTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            753,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Define the following “transformed” versions of $F _ { T }$ and $\\Psi$ : ",
+        "bbox": [
+            173,
+            777,
+            558,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dbe0e197f6c49b46ae96150dd2deb25a1144b64774c1db75c8788dfcdd229cb1.jpg",
+        "text": "$$\n\\hat { F } _ { \\mathcal { T } } = V _ { 0 } ^ { 1 / 2 } F _ { \\mathcal { T } } V _ { 0 } ^ { 1 / 2 } \\quad \\mathrm { a n d } \\quad \\hat { \\Psi } = \\hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } \\Psi V _ { 0 } ^ { 1 / 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            797,
+            679,
+            820
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As shown in Section A.1 of the appendix we have ",
+        "bbox": [
+            173,
+            833,
+            501,
+            848
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d04e84052445dcc9a29e586976b085a603737fb422798d3e92c50159f2b16067.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { F } _ { \\mathcal { T } } = { \\displaystyle \\sum _ { d = 0 } ^ { \\mathcal { T } } } ( \\mathcal { T } - d ) \\hat { \\Psi } ^ { d } + \\left( { \\displaystyle \\sum _ { d = 0 } ^ { \\mathcal { T } } } ( \\mathcal { T } - d ) \\hat { \\Psi } ^ { d } \\right) ^ { \\top } - \\mathcal { T } I } \\\\ & { ~ = \\zeta \\tau ( \\hat { \\Psi } ) + \\zeta \\tau ( \\hat { \\Psi } ^ { \\top } ) - \\mathcal { T } I } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            853,
+            671,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            174,
+            104,
+            217,
+            117
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96b04df38bfe0378c61b5374fd89f549138e4580d5fc00f7f68e8c265e6755a0.jpg",
+        "text": "$$\n\\zeta \\tau ( x ) = \\frac { \\mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \\mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            113,
+            612,
+            150
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(Note that rational functions can be evaluated with matrix arguments in this way, as discussed in Section A.1.) ",
+        "bbox": [
+            173,
+            152,
+            825,
+            181
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to compute $\\hat { F } ^ { - 1 }$ , from which we can recover $F ^ { - 1 }$ via the simple relation $F ^ { - 1 } =$ $ { V _ { 0 } } ^ { - 1 / 2 }  { \\hat { F } } ^ { - 1 }  { V _ { 0 } } ^ { - 1 / 2 }$ ",
+        "bbox": [
+            174,
+            188,
+            823,
+            222
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately it doesn’t appear to be possible to simplify this formula sufficiently enough to allow for the efficient computation of $\\hat { F } ^ { - 1 } = \\mathbb { E } _ { T } [ \\hat { F } _ { T } ] ^ { - 1 }$ when $\\hat { \\Psi }$ is a Kronecker product (which it will be when $V _ { 0 }$ and $V _ { 1 }$ are). The difficulty is due to both the appearance of $\\hat { \\Psi }$ and its transpose (which are not codiagonalizable/commutative in general), and various higher powers of $\\hat { \\Psi }$ . ",
+        "bbox": [
+            174,
+            228,
+            825,
+            292
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To proceed from this point and obtain a formula which can be efficiently evaluated when $\\hat { \\Psi }$ is a Kronecker product, we will make one of two simplifying assumptions/approximations, which we call “Option $1 ^ { \\circ }$ and “Option $2 ^ { \\circ }$ respectively. These are explained in the next two subsections. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.5.2 OPTION 1: $V _ { 1 }$ IS SYMMETRIC ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            358,
+            429,
+            373
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "If $V _ { 1 }$ (the cross-moment over time) is symmetric, this implies that $\\hat { \\Psi } = \\hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 }$ is also symmetric. Thus by eqn. 7 we have ",
+        "bbox": [
+            173,
+            383,
+            823,
+            415
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e343f7087a803436c0e65a1b54b4d4640052fbc1151716fff1e1d1e33dd4b866.jpg",
+        "text": "$$\n\\hat { F } _ { T } = \\zeta _ { T } ( \\hat { \\Psi } ) + \\zeta _ { T } ( \\hat { \\Psi } ) - \\mathcal { T } I = \\eta _ { T } ( \\hat { \\Psi } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            421,
+            632,
+            441
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            174,
+            449,
+            217,
+            462
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/16ffa88a3e9cbeddac2c554d0155df02d75309d11ac850ee3164074787efff98.jpg",
+        "text": "$$\n\\eta _ { T } ( x ) = 2 \\zeta \\tau ( x ) - \\mathcal { T } = \\frac { 2 ( \\mathcal { T } ( 1 - x ) - x ( 1 - x ^ { T } ) ) } { ( 1 - x ) ^ { 2 } } - \\mathcal { T } = \\frac { \\mathcal { T } ( 1 - x ^ { 2 } ) - 2 x ( 1 - x ^ { T } ) } { ( 1 - x ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            467,
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+            503
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Let $U \\mathrm { d i a g } ( \\hat { \\psi } ) U ^ { \\top } = \\hat { \\Psi }$ be the eigen-decomposition of $\\hat { \\Psi }$ . By the above expression for $\\hat { F } _ { T }$ we have that $\\hat { F } _ { T } = U \\mathrm { d i a g } ( \\eta _ { T } ( \\hat { \\psi } ) ) U ^ { \\top }$ , where $f ( b )$ denotes the component-wise evaluation of a function $f$ for each component of the vector $b$ , i.e. $[ { \\dot { f } } ( b ) ] _ { i } = f ( [ b ] _ { i } )$ . We thus have ",
+        "bbox": [
+            174,
+            517,
+            825,
+            565
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3548a54afec7cbdbbb3978caafddf6927cb4b6dab4b07523edbd930a04ad6335.jpg",
+        "text": "$$\n\\hat { F } = \\mathbb { E } _ { \\mathcal { T } } [ \\hat { F } _ { \\mathcal { T } } ] = \\mathbb { E } _ { \\mathcal { T } } [ U \\operatorname { d i a g } ( \\eta _ { \\mathcal { T } } ( \\hat { \\psi } ) ) U ^ { \\top } ] = U \\operatorname { d i a g } ( \\mathbb { E } _ { \\mathcal { T } } [ \\eta _ { \\mathcal { T } } ( \\hat { \\psi } ) ] ) U ^ { \\top } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            571,
+            727,
+            592
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Inverting both sides of this yields ",
+        "bbox": [
+            174,
+            606,
+            393,
+            621
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f5644f9f2371c4e9a750efa8d89c2e18921b55d74c71a1654e2fd4fe1e0fd9da.jpg",
+        "text": "$$\n\\hat { F } ^ { - 1 } = U \\mathrm { d i a g } ( \\gamma ( \\hat { \\psi } ) ) U ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            628,
+            584,
+            648
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where we have defined $\\gamma ( x ) = 1 / \\mathbb { E } _ { T } [ \\eta _ { T } ( x ) ]$ ",
+        "bbox": [
+            176,
+            655,
+            477,
+            670
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This expression can be efficiently evaluated when $\\hat { \\Psi }$ is a Kronecker product since the eigendecomposition of a Kronecker product can be easily obtained from the eigendecomposition of the factors. Evaluation of $\\gamma ( \\hat { \\psi } )$ is done component-wise (i.e. $[ \\gamma ( \\hat { \\psi } ) ] _ { i } = \\gamma ( [ \\hat { \\psi } ] _ { i } ) )$ and is thus easy to perform. See Section 3.5.5 for further details. ",
+        "bbox": [
+            174,
+            678,
+            823,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "$V _ { 1 }$ is symmetric if and only if $\\hat { \\Psi }$ is symmetric. And as shown in the proof of Proposition 1 (see Appendix A.1) $\\hat { \\Psi }$ has the interpretation of being the transition matrix of an LGGM which describes the evolution of “whitened” versions of the wt’s (given by wˆt = V −1/20 w . Linear dynamical systems with symmetric transition matrices arise frequently in machine learning and related areas (Huang et al., 2016; Hazan et al., 2017), particularly because of the algorithmic techniques they enable. Intuitively, a symmetric transition matrix allows allows one to model exponential decay of different basis components of the signal over time, but not rotations between these components (which are required to model sinusoidal/oscillating signals). ",
+        "bbox": [
+            173,
+            742,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that the observed/measured $V _ { 1 }$ may or may not be exactly symmetric up to numerical precision, even if it well approximated as symmetric. For these calculations to make sense it must be exactly symmetric, and so even if it turns out to be approximately symmetric one should ensure that it is exactly so by using the symmetrized version $( \\dot { V _ { 1 } } + V _ { 1 } ^ { \\top } ) / 2$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.5.3 OPTION 2: COMPUTING THE LIMITING VALUE INSTEAD ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            614,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "If $V _ { 1 }$ is not well approximated as symmetric, another option is to approximate ",
+        "bbox": [
+            173,
+            127,
+            684,
+            142
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0676cb188da2c781b65c59072b60f8236db59a5aeb42116a4809e535017baf92.jpg",
+        "text": "$$\n\\hat { F } = \\mathbb { E } _ { T } [ \\hat { F } _ { T } ^ { ( \\infty ) } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            148,
+            553,
+            171
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(instead of $\\hat { F } = \\mathbb { E } _ { T } [ \\hat { F } _ { T } ] )$ , where we define ",
+        "bbox": [
+            174,
+            178,
+            460,
+            195
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dac5270dbd5540fca621293993ee3eeb3310d86ea9f0fcc9500276ce1ea2138a.jpg",
+        "text": "$$\n{ \\hat { F } } _ { \\mathcal { T } } ^ { ( \\infty ) } \\equiv \\operatorname* { l i m } _ { { \\mathcal { T } } ^ { \\prime } \\to \\infty } { \\frac { \\mathcal { T } } { { \\mathcal { T } } ^ { \\prime } } } { \\hat { F } } _ { { \\mathcal { T } } ^ { \\prime } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            200,
+            578,
+            232
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This is essentially equivalent to the assumption that the training sequences are all infinitely long, which may be a reasonable one to make in practice. We re-scale by the factor $\\frac { \\mathcal { T } } { \\mathcal { T } ^ { \\prime } }$ to achieve the proper scaling characteristics of $\\hat { F } _ { T }$ , and to ensure that the limit actually exists. ",
+        "bbox": [
+            173,
+            237,
+            825,
+            284
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As shown in Section A.2 of the appendix this yields the following remarkably simple expression for $\\hat { F } ^ { - 1 }$ : ",
+        "bbox": [
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+            318
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3081eb344c9b7249d1003be2d55087ed6c950fd3a3780c739cf12c860a7ff47.jpg",
+        "text": "$$\n\\hat { F } ^ { - 1 } = \\frac { 1 } { \\mathbb { E } _ { T } [ \\mathcal { T } ] } ( I - \\hat { \\Psi } ) ( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 } ( I - \\hat { \\Psi } ^ { \\top } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            315,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Despite the fact that it includes both $\\hat { \\Psi }$ and $\\hat { \\Psi } ^ { \\top }$ , this formula can be efficiently evaluated when $\\hat { \\Psi }$ is a Kronecker product due to the existance of decomposition-based techniques for inverting matrices of the form $A \\otimes B + C \\otimes D$ . See Section 3.5.5 for further details. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This approximation can break down if some of the linear components of $\\hat { w } _ { t }$ have temporal autocorrelations close to 1 (i.e. $[ \\hat { \\psi } ] _ { i } \\approx 1$ for some $i$ ) and $\\tau$ is relatively small. In such a case we will have that $[ \\hat { \\psi } ] _ { i } ^ { \\mathcal { T } }$ is large for some $i$ (despite being raised to the $\\tau$ -th power) so that ${ \\hat { F } } _ { T } ^ { ( \\infty ) }$ may essentially “overcount” the amount of temporal correlation that contributes to the sum. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This can be made more concrete by noting that the approximation is essentially equivalent to taking $\\begin{array} { r } { \\zeta _ { \\mathcal { T } } ( x ) \\approx \\operatorname* { l i m } _ { \\mathcal { T } ^ { \\prime }  \\infty } \\frac { \\mathcal { T } } { \\mathcal { T } ^ { \\prime } } \\zeta _ { \\mathcal { T } ^ { \\prime } } ( x ) \\equiv \\kappa ( x ) } \\end{array}$ for each $x = [ \\hat { \\psi } ] _ { i }$ . We can express the error of this as ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d45e48f2b208ac68b59e322ff14b75ae1e349d44d37d5d1debe776d33bd29f67.jpg",
+        "text": "$$\n| \\kappa ( x ) - \\zeta \\tau ( x ) | = \\left| { \\frac { \\mathcal { T } } { 1 - x } } - { \\frac { \\mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \\mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } } \\right| = \\left| { \\frac { x ( 1 - x ^ { \\mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } } \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is easy to see how this expression, when evaluated at $x = [ \\hat { \\psi } ] _ { i }$ , might be large when $[ \\hat { \\psi } ] _ { i }$ is close to 1, and $\\tau$ is relatively small. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3.5.4 ESTIMATING $\\hat { \\Psi }$ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The formulae for $\\hat { F } ^ { - 1 }$ from the previous sections depend on the quantity $\\hat { \\Psi } = \\hat { V } _ { 1 } = { V _ { 0 } ^ { - 1 / 2 } \\Psi { V _ { 0 } ^ { 1 / 2 } } }$ and so it remains to compute $\\Psi$ . We observe that ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a6e5dd9b1e5a0241953d484a5fb69b3a01f19da49c6abad5fec26b4c5e713520.jpg",
+        "text": "$$\nV _ { 1 } = V _ { 1 , 0 } = \\mathbb { E } [ w _ { 1 } w _ { 0 } ^ { \\top } ] = \\mathbb { E } [ ( \\Psi w _ { 0 } + \\epsilon _ { 1 } ) w _ { 0 } ^ { \\top } ] = \\Psi \\mathbb { E } [ w _ { 0 } w _ { 0 } ^ { \\top } ] + \\mathbb { E } [ \\epsilon _ { 1 } w _ { 0 } ^ { \\top } ] = \\Psi V _ { 0 } + 0 = \\Psi V _ { 0 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Right-multiplying both sides by $V _ { 0 }$ yields $\\Psi = { \\cal V } _ { 1 } { \\cal V } _ { 0 } ^ { - 1 }$ . Thus, given estimates of $V _ { 0 }$ and $V _ { 1 }$ , we may compute an estimate of $\\hat { \\Psi }$ as ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a54554e090e043ea2ff4a13298180f7e530d3103df96d9550f9036c91d04939.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\Psi } = V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In practice we estimate $V _ { 0 }$ and $V _ { 1 }$ by forming estimates of their Kronecker factors and taking the product. The factors themselves are estimated using exponentially decayed averages over mini-batch estimates. And the mini-batch estimates are in turn computed by averaging over cases and summing across time-steps, before divide by the expected number of time-steps. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For example, for $A _ { 0 }$ and $A _ { 1 }$ these the mini-batch estimates are averages of 1ET [ ] PTt=1 ata>t and 1ET [ ] PT −1t=1 at+1a>t , respectively. Note that as long as V0 is computed as the 2nd-order moment of some empirical data, and $V _ { 1 }$ computed as the 2nd-order moment between that same data and a temporally shifted version, the spectral radius of $\\hat { \\Psi } = { V _ { 0 } ^ { - 1 / 2 } } V _ { 1 } V _ { 0 } ^ { - 1 / 2 }$ (and similarly $\\Psi = V _ { 1 } V _ { 0 } ^ { - 1 } \\dag ,$ ) will indeed be less than or equal to , as we prove in Section B.2 of the appendix. This bound on the spectral radius is a necessary condition for our infinite chain-structured Gaussian graphical model to be well-defined, and for our calculations to make sense. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The sufficient condition that the spectral radius is actually less than 1 will most often be satisfied too, except in the unlikely event that some eigen component remains perfectly constant across time. But even if this somehow happens, the inclusion within the given $V _ { 0 }$ of some damping/regularization term such as $\\lambda I$ will naturally deal with this problem. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "3.5.5 EFFICIENT IMPLEMENTATION ASSUMING $V _ { 0 }$ AND $V _ { 1 }$ ARE KRONECKER-FACTORED ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "It remains to show that the approximations developed in Section 3.5 can be combined with the Kronecker-factored approximations for $V _ { 0 }$ and $V _ { 1 }$ from Section 3.2.3 to yield an efficient algorithm for computing $F ^ { - 1 } z$ for an arbitrary vector $z = \\mathrm { v e c } ( Z )$ . This is a straightforward although very long computation which we leave to Section C of the appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Full pseudo-code for the resulting algorithms is given in Section C.3. As they only involve symmetric eigen-decomposition and matrix-matrix products with matrices the size of $A _ { 0 }$ and $G _ { 0 }$ they are only several times more expensive to compute than eqn. 6. This extra overhead will often be negligible since the gradient computation via BPTT, whose costs scales with the sequence length $\\tau$ , tends to dominate all the other costs. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the benefit of our novel curvature matrix approximations for RNNs, we empirically evaluated them within the standard “distributed K-FAC” framework (Ba et al., 2017) on two different RNN training tasks. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The 2nd-order statistics (i.e. the Kronecker factors $A _ { 0 }$ , $A _ { 1 }$ , $G _ { 0 }$ , and $G _ { 1 }$ ) are accumulated through an exponential moving average during training. When computing our approximate inverse Fisher, factored Tikhonov damping (Martens & Grosse, 2015) was applied to $V _ { 0 } = G _ { 0 } \\otimes A _ { 0 }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We used a single machine with 16 CPU cores and a Nvidia K40 GPU for all the experiments. The additional computations required to get the approximate Fisher inverse from these statistics (i.e. the “pre-processing steps” described in Section C.3) are performed asynchronously on the CPUs, while the GPU is used for the usual forward evaluation and back-propagation to compute the gradient. Updates are computed using the most recently computed values of these (which are allowed to be stale), so there is minimal per-iteration computational overhead compared SGD. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We adopted the step-size selection technique described in Section 5 of Ba et al. (2017), as we found it let us use larger learning rates without compromising the stability of the optimization. The hyperparameters of our approach, which include the max learning rate and trust-region size for the aforementioned step-size selection procedure, as well as the momentum, damping constants, and the decay-rate for the second-order statistics, as well as the hyper-parameters of the baseline methods, were tuned using a grid search. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Word-level language model: We start by applying our method to a two-layer RNN based on the well-studied Long Short-Term Memory (LSTM) architecture (Hochreiter & Schmidhuber, 1997) for a word-level language modeling task on the Penn-TreeBank (PTB) dataset (Marcus et al., 1993) following the experimental setup in Zaremba et al. (2014). The gradients are computed using a fixed sequence length truncated back-propagation scheme in which the initial states of the recurrent hidden units are inherited from the final state of the preceding sequence. The truncation length used in the experiments is 35 timesteps. The learning rate is given by a carefully tuned decaying schedule (whose base value we tune along with the other hyperparamters). ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In our experiments we simply substitute their optimizer with our modified distributed K-FAC optimizer that uses our proposed RNN Fisher approximations. We performed experiments on two different sizes of the same architecture, which use two-layer 650 and 1024 LSTM units respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "LSTMs have 4 groups of internal units: input gates, output gates, forget gates, and update candidates. We treat the 4 weight matrices that compute the pre-activations to each of these as distinct for the purposes of defining Fisher blocks (whereas many LSTM implementations treat them as one big matrix). This results in smaller Kronecker factors that are cheaper to compute and invert. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad3c90338cf7847de32bfd1df5c5edabca76a0d435c7d9a88c3e87b2b1f8ae9b.jpg",
+        "image_caption": [
+            "Figure 1: Optimization performance of our method compared to the baselines in perplexity-perword on length-35 word sequences from Penn-TreeBank. All the methods used a mini-batch size of 200. K-FAC indep. uses the update in eqn. 6, K-FAC option1 uses eqn. 8, and K-FAC option2 uses eqn. 9. (left) Training perplexity v.s. the number of updates. Dashed lines denote the training curves for RNNs with 1024 LSTM units and solid lines denote the training curves for RNNs with 650 LSTM units. (right) Training perplexity v.s. the wall-clock time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "4.1 LANGUAGE MODELING WITH LONG SHORT-TERM MEMORY UNITS",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Because the typical vocabulary size used for PTB is 10,000, the Fisher blocks for the input embedding layer and output layer (computing the logits to the softmax) each contain a 10,000 by 10,000 sizes Kronecker factor, which is too large to be inverted with any reasonable frequency. Given that the input vector uses a one-hot encoding it is easy to see its associated factor is actually diagonal, and so we can store and invert it as such. Meanwhile the large factor associated with the output isn’t diagonal, but we nonetheless approximate it as such for the sake of efficiency. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In our experiments we found that each parameter update of our method required about $80 \\%$ more wall-clock time than an SGD update (using mini-batch size of 200) although the updates made more much progress. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In Figure 1, we plot the training progress as a function of the number of parameter updates. While Adam outperforms SGD in the first few epochs, SGD obtains a lower loss at the end of training. We found the recent layer-normalization technique (Ba et al., 2016) helps speed up Adam considerably, but it hurts the SGD performance. Such an observation is consistent with previous findings. In comparison, our proposed method still significantly outperform both the Adam and the SGD baselines even with the help of layer-normalization. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "While optimization performance, not generalization performance, is the focus of this paper, we have included validation performance data in the appendix for the sake of completeness. (See Figure 4 in Appendix D.) Not surprisingly, we found that the 2nd-order methods, including our approach and diagonal ones like Adam, tended to overfit more than SGD on these tasks. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The tendency for SGD w/ early-stopping to self-regularize is well-documented, and there are many compelling theories about why this happens (e.g. Duvenaud et al., 2016; Hardt et al., 2015). It is also well-known that 2nd-order methods, including K-FAC and diagonal methods like Adam/RMSprop, dont self-regularize nearly as much (e.g. Wilson et al., 2017; Keskar & Socher, 2017). We feel that this problem can likely be addressed through the careful application of additional explicit regularization (e.g. increased weight decay, drop-out, etc) and/or model modifications, but that exploring this is outside of the scope of this paper. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Character-level model: To further investigate the optimization performance of our proposed Fisher approximation, we use a small two layer LSTM with 128 units to model the character sequences on the Penn-TreeBank (PTB) dataset (Marcus et al., 1993). We employ the same data partition in Mikolov et al. (2012). We plotted the bits-per-character vs the number of parameter updates and the wall-clock times in Figure 2. The K-FAC updates were roughly twice as time-consuming to compute as the Adam updates in our implementation. Despite this, our results demonstrate that K-FAC has a significant advantage over the Adam baseline in terms of wall-clock time. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/195985476ed79c9ed7a96907def60b68725ef3c2df313969a5ef4933221affdf.jpg",
+        "image_caption": [
+            "Figure 2: Optimization performance in bit-per-character on length-100 character sequences from Penn-TreeBank. batchsize indicates the mini-batch size used to train the baseline methods (our method always used a mini-batch size of 200). K-FAC indep. uses the update in eqn. 6, $K – F A C$ option1 uses eqn. 8, and K-FAC option2 uses eqn. 9. (left) Training perplexity v.s. the number of updates. (right) Training perplexity v.s. the wall-clock time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            106,
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+            244
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/2dcddfdcf44fcf48896aeb8b94df1bf2448c115c90055b93b77f229570a76702.jpg",
+        "image_caption": [
+            "Figure 3: Optimization performance for differentiable Neural Computers (DNC) on a repeated copy task. $K – F A C$ indep. uses the update in eqn. 6, K-FAC option1 uses eqn. 8, and $K -$ -FAC option2 uses eqn. 9. (left) Training cross entropy loss v.s. the number of updates. (right) Training cross entropy loss v.s. the wall-clock time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            348,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "4.2 LEARNING DIFFERENTIABLE NEURAL COMPUTERS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            561,
+            568,
+            575
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To further investigate the potential benefits of using our approach over existing methods, we applied it to the Differentiable Neural Computer (DNC) model (Graves et al., 2016) for learning simple algorithmic programs. Recently, there have been several attempts (Weston et al., 2014; Graves et al., 2016) to extend the existing RNN models to incorporate more long-term memory storage devices in order to help solve problems beyond simple sequence prediction tasks. Although these extended RNNs could potentially be more powerful than simple LSTMs, they often require thousands of parameter updates to learn simple copy tasks (Graves et al., 2016). Both the complexity of these models and the difficulty of the learning tasks have posed a significant challenge to commonly used optimization methods. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The DNC model is designed to solve structured algorithmic tasks by using an LSTM to control an external read-write memory. We applied the Fisher-based precondition to compute the updates for both the weights in the LSTM controller and the read-write weight matrices used to interface with the memory. We trained the model on a simple repeated copy task in which the DNC needs to recreate a series of two random binary sequences after they are presented as inputs. The total length of the sequence is fixed to 22 time-steps. From Figure 3, we see that our method significantly outperforms the Adam baseline in terms of update count, although only provides a modest improvement in wallclock time. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This gap is explained by the fact that the iterations were significantly more time-consuming to compute relative to the gradient computations than they were in previous two experiments on language models. This is likely due to a different trade-off in terms of the gradient computation vs the overheads specific to our method owing to smallness of the model and dataset. With more careful engineering to reduce the communication costs, and/or a larger model and dataset, we would expect to see a bigger improvement in wall-clock time. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            318,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We have presented a new family of approximations to the Fisher information matrix of recurrent neural networks (RNNs), extending previous work on Kronecker-factored approximations. With this contribution, recurrent networks can now finally be trained with the K-FAC optimization method. We have demonstrated that our new approximations substantially reduce the required number of iterations for convergence vs standard baseline optimizers on several realistic tasks. And we have also shown that in a modern distributed training setup this results in a substantial savings in wallclock time as well. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            253,
+            287,
+            268
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Shun-Ichi Amari. Natural gradient works efficiently in learning. Neural Computation, 10(2):251– 276, 1998. ",
+        "bbox": [
+            174,
+            277,
+            823,
+            306
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jimmy Ba and Diederik Kingma. Adam: A method for stochastic optimization. In ICLR, 2015. ",
+        "bbox": [
+            176,
+            315,
+            797,
+            332
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jimmy Ba, Roger Grosse, and James Martens. Distributed second-order optimization using kronecker-factored approximations. In International Conference on Learning Representations (ICLR’2017), 2017. ",
+        "bbox": [
+            173,
+            340,
+            823,
+            383
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. ",
+        "bbox": [
+            171,
+            393,
+            823,
+            422
+        ],
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+    {
+        "type": "text",
+        "text": "A SUPPLEMENTARY COMPUTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            493,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.1 PROOFS FOR SECTION 3.5.1 ",
+        "bbox": [
+            174,
+            132,
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+            147
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 Given ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            159,
+            316,
+            174
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/feb3941abbf1a02c038e9c46dd0f2f247902dcca3c87b8a00f463a90bb46992f.jpg",
+        "text": "$$\n\\hat { F } _ { \\mathcal { T } } = V _ { 0 } ^ { 1 / 2 } F _ { \\mathcal { T } } V _ { 0 } ^ { 1 / 2 } \\quad a n d \\quad \\hat { \\Psi } = \\hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } \\Psi V _ { 0 } ^ { 1 / 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            179,
+            678,
+            200
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "we have ",
+        "bbox": [
+            173,
+            214,
+            230,
+            228
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            173,
+            246,
+            215,
+            260
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d893365544f1369c79baa11a842bfd3b05109beeb1245c67f5e5571e8199536a.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\hat { F } _ { \\mathcal { T } } = \\zeta \\tau ( \\hat { \\Psi } ) + \\zeta \\tau ( \\hat { \\Psi } ^ { \\top } ) - \\mathcal { T } I } } \\\\ { { { } } } \\\\ { { \\zeta \\tau ( x ) = \\displaystyle \\frac { \\mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \\mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            222,
+            612,
+            291
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Proof ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            301,
+            217,
+            315
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For $d > 0$ we have that $V _ { d } = \\Psi V _ { d - 1 }$ , which can be seen as follows: ",
+        "bbox": [
+            173,
+            321,
+            614,
+            338
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/149fda31c91985a075a891b3258d5617c1e87b353a040e9f272a6481a9efab2a.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { V _ { d } } } & { { = } } & { { V _ { d , 0 } = \\mathbb { E } [ w _ { d } w _ { 0 } ^ { \\top } ] = \\mathbb { E } [ \\Psi w _ { d - 1 } + \\epsilon _ { d } w _ { 0 } ^ { \\top } ] } } \\\\ { { } } & { { = } } & { { \\Psi \\mathbb { E } [ w _ { d - 1 } w _ { 0 } ^ { \\top } ] + \\mathbb { E } [ \\epsilon _ { d } w _ { 0 } ^ { \\top } ] } } \\\\ { { } } & { { = } } & { { \\Psi V _ { d - 1 } + 0 = \\Psi V _ { d - 1 } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            342,
+            665,
+            401
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Applying $V _ { d } = \\Psi V _ { d - 1 }$ recursively yields ",
+        "bbox": [
+            174,
+            405,
+            446,
+            421
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c4fdd0c6fe058c6b2e084baaacbd1cad8877d23e21cebfffe5aa96b26682c704.jpg",
+        "text": "$$\nV _ { d } = \\Psi ^ { d } V _ { 0 } \\quad \\mathrm { f o r } \\quad d \\geqslant 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            426,
+            588,
+            444
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "And using $V _ { - d } = V _ { d } ^ { \\top }$ it also follows that ",
+        "bbox": [
+            176,
+            449,
+            446,
+            465
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6407bfc18f62b1d765219ea6b21618756994433fd4ea934fd4fe51c01aa5dc7f.jpg",
+        "text": "$$\nV _ { d } = V _ { 0 } ( \\Psi ^ { d } ) ^ { \\top } \\quad \\mathrm { f o r } \\quad d \\leqslant 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            470,
+            601,
+            491
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Setting $d = 1$ and multiplying both sides by $V _ { 0 }$ (which is assumed to be invertible) one can also derive the following simple formula for $\\Psi$ : ",
+        "bbox": [
+            174,
+            496,
+            823,
+            523
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96b21b6b03589e250050795f296c19865fceb97f7cb987707e4382f6ecef4dd5.jpg",
+        "text": "$$\n\\Psi = { \\cal V } _ { 1 } { \\cal V } _ { 0 } ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            529,
+            544,
+            547
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To proceed from here we define a “transformed” version of the original chain-structured linearGaussian graphical model whose variables are $\\hat { w } _ { t } = V _ { 0 } ^ { - 1 / 2 } w _ { t }$ . (Here we assume that $V _ { 0 }$ is invertible – it is symmetric by definition.) All quantities related to the original model have their analogues in the transformed model, which we indicate with the hat symbol ˆ·. ",
+        "bbox": [
+            173,
+            560,
+            826,
+            621
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In the transformed model the 2nd-order moments of the $\\hat { w } _ { t }$ ’s are given by ",
+        "bbox": [
+            173,
+            627,
+            656,
+            642
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a305035d287adbdeb1246a17f962e9e0ea5aedb8923bb131772a963fa9465c7d.jpg",
+        "text": "$$\n\\hat { V } _ { d } = \\mathbb { E } \\left[ \\left( V _ { 0 } ^ { - 1 / 2 } w _ { d } \\right) \\left( V _ { 0 } ^ { - 1 / 2 } w _ { 0 } \\right) ^ { \\top } \\right] = V _ { 0 } ^ { - 1 / 2 } \\mathbb { E } \\left[ w _ { d } w _ { 0 } ^ { \\top } \\right] V _ { 0 } ^ { - 1 / 2 } = V _ { 0 } ^ { - 1 / 2 } V _ { d } V _ { 0 } ^ { - 1 / 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            647,
+            784,
+            683
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We observe that $\\hat { V } _ { 0 } = I$ ",
+        "bbox": [
+            174,
+            689,
+            334,
+            705
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Analogously to the original model, the transformed version obeys ",
+        "bbox": [
+            173,
+            712,
+            606,
+            727
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f90cd07a0c9c24795dc145aa17b2d81b1e0c5c49a2c7d84f37cee14f0dce02c6.jpg",
+        "text": "$$\n\\hat { w } _ { t } = \\hat { \\Psi } \\hat { w } _ { t - 1 } + \\hat { \\epsilon } _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            732,
+            560,
+            751
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "with $\\hat { \\epsilon } _ { t } = V _ { 0 } ^ { - 1 / 2 } \\epsilon _ { t }$ and $\\hat { \\Psi } = \\hat { V } _ { 1 } \\hat { V } _ { 0 } ^ { - 1 } = \\hat { V } _ { 1 }$ (using $\\hat { V } _ { 0 } = I$ ). This can be seen by noting that ",
+        "bbox": [
+            169,
+            758,
+            766,
+            776
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58aa6314b67e5d77e768b7ceffb19f9f6bda045e70e4970257399ea6a9cea410.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\hat { w } _ { t } } & { = } & { V _ { 0 } ^ { - 1 / 2 } w _ { t } } \\\\ & { = } & { V _ { 0 } ^ { - 1 / 2 } ( \\Psi w _ { t - 1 } + \\epsilon _ { t } ) } \\\\ & { = } & { V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 } w _ { t - 1 } + V _ { 0 } ^ { - 1 / 2 } \\epsilon _ { t } } \\\\ & { = } & { ( V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 } ) ( V _ { 0 } ^ { - 1 / 2 } w _ { t - 1 } ) + \\hat { \\epsilon } _ { t } } \\\\ & { = } & { \\hat { V } _ { 1 } \\hat { w } _ { t - 1 } + \\hat { \\epsilon } _ { t } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            781,
+            651,
+            890
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "It also remains true that the spectral radius of $\\hat { \\Psi }$ is less than 1, which can be seen in at least one of two ways: by noticing that the transformed model is well-defined in the infinite limit if and only if ",
+        "bbox": [
+            171,
+            893,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "the original one is, or that $\\hat { \\Psi } = \\hat { V } _ { 1 } = { V _ { 0 } ^ { - 1 / 2 } \\Psi { V _ { 0 } ^ { 1 / 2 } } }$ is a similar matrix to $\\Psi$ (in the technical sense) and hence has the same eigenvalues. ",
+        "bbox": [
+            174,
+            101,
+            823,
+            133
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "As the transformed model is isomorphic to the original one, all of the previously derived relationships which held for it also hold here, simply by replacing each quantity with its transformed version (denoted by the hat symbol ˆ·). ",
+        "bbox": [
+            174,
+            138,
+            825,
+            183
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Given these relations (included the transformed analogue of equation 5) we can express $\\hat { F } _ { T }$ as ",
+        "bbox": [
+            171,
+            189,
+            792,
+            205
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f44f3fdfed5824001b623465e24f777984a29aa78dec5e64fd97fa0a65ef4241.jpg",
+        "text": "$$\n\\hat { F } _ { \\mathcal { T } } = \\sum _ { d = 0 } ^ { \\mathcal { T } } ( \\mathcal { T } - d ) \\hat { \\Psi } ^ { d } + \\left( \\sum _ { d = 0 } ^ { \\mathcal { T } } ( \\mathcal { T } - d ) \\hat { \\Psi } ^ { d } \\right) ^ { \\top } - \\mathcal { T } I .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            210,
+            673,
+            258
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "It is a well-known fact that one can evaluate rational functions, and functions that are the limiting values of sequences of rational functions, with matrix arguments. This is done by replacing scalar multiplication with matrix multiplication, division with matrix inversion, and scalar constants with scalar multiples of the identity matrix, etc. Note that because sums of powers and inverses of matrices are co-diagonalizable/commutative when the matrices themselves are, there is no issue of ambiguity caused by mixing commutative and non-commutative algebra in this way. ",
+        "bbox": [
+            173,
+            262,
+            826,
+            348
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Moreover, the value of some such function $f ( x )$ , given a matrix argument $B$ , is ",
+        "bbox": [
+            174,
+            353,
+            694,
+            369
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad90ca0a2231d74286d8766316282befc346940a400b376c3f1c57e4420ac266.jpg",
+        "text": "$$\nf ( B ) = V \\mathrm { d i a g } ( f ( b ) ) V ^ { - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            375,
+            589,
+            393
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "where $V \\mathrm { d i a g } ( b ) V ^ { - 1 } = B$ is eigendecomposition of $\\mathbf { B }$ , and where $f ( b )$ denotes the componentwise evaluation of $f$ for each component of the vector $b$ , i.e. $[ f ( b ) ] _ { i } = f ( [ b ] _ { i } )$ . Note that if $[ { \\bar { f } } ( b ) ] _ { i }$ is undefined from some $i$ , either because of a division by zero, or because the limit which defines $f ( x )$ doesn’t converge for $x = [ b ] _ { i }$ , then $f ( B )$ doesn’t exist for that particular $B$ (and otherwise it does). ",
+        "bbox": [
+            173,
+            398,
+            825,
+            457
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We observe that our above expression for $\\widehat { F } _ { T }$ can be rewritten as ",
+        "bbox": [
+            173,
+            463,
+            599,
+            479
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3c790a9028bc022804d1894e0134b73269c535830efada854f27ba0d70ad909.jpg",
+        "text": "$$\n\\hat { F } _ { \\mathcal { T } } = \\zeta _ { \\mathcal { T } } ( \\hat { \\Psi } ) + \\zeta _ { \\mathcal { T } } ( \\hat { \\Psi } ^ { \\top } ) - \\mathcal { T } I ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            484,
+            604,
+            505
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "where $\\textstyle \\zeta _ { T } ( x ) = \\sum _ { d = 0 } ^ { T } ( T - d ) x ^ { d }$ . By Proposition 3 in Appendix B.1, we have for $x \\neq 1$ that ",
+        "bbox": [
+            169,
+            511,
+            782,
+            530
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7c9147f70146fdca74b8aadc8fca5a6b37fd3c4ca71bf0a7062b06e81ad0be96.jpg",
+        "text": "$$\n\\zeta \\tau ( x ) = \\frac { \\mathcal { T } ( 1 - x ) - x ( 1 - x ^ { \\mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            535,
+            612,
+            570
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Let $U \\mathrm { d i a g } ( \\hat { \\psi } ) U ^ { - 1 } = \\hat { \\Psi }$ be the eigendecomposition of $\\hat { \\Psi }$ . Because $\\hat { \\Psi }$ has a spectral radius less than 1, we have $| [ \\hat { \\psi } ] _ { i } | < 1$ for each $i$ (so that in particular $[ \\hat { \\psi } ] _ { i } \\neq 1 \\rangle$ ), and thus we can evaluate $\\zeta \\tau ( { \\hat { \\Psi } } )$ and $\\zeta \\tau ( \\hat { \\Psi } ^ { \\top } )$ according to the above formula for $\\zeta \\tau ( x )$ . ",
+        "bbox": [
+            173,
+            584,
+            825,
+            635
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.2 PROOFS FOR SECTION 3.5.3 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            679,
+            411,
+            694
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2 Suppose we approximate $\\hat { F } = \\mathbb { E } _ { \\mathcal { T } } [ \\hat { F } _ { \\mathcal { T } } ^ { ( \\infty ) } ] ,$ , where we have defined ",
+        "bbox": [
+            173,
+            702,
+            704,
+            722
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7c28b4b53e6b124b58adcb6afebe4ec761386335acb27ead2585ccdd2200a8ce.jpg",
+        "text": "$$\n{ \\hat { F } } _ { \\mathcal { T } } ^ { ( \\infty ) } \\equiv \\operatorname* { l i m } _ { { \\mathcal { T } } ^ { \\prime } \\to \\infty } { \\frac { \\mathcal { T } } { { \\mathcal { T } } ^ { \\prime } } } { \\hat { F } } _ { { \\mathcal { T } } ^ { \\prime } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            727,
+            576,
+            758
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Then we have ",
+        "bbox": [
+            174,
+            771,
+            266,
+            785
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e30e8cd7958eca3a26299fe669aaf44ae03308eaf1fef4727d0df3858e9dbb0.jpg",
+        "text": "$$\n\\hat { F } ^ { - 1 } = \\frac { 1 } { \\mathbb { E } _ { T } [ \\mathcal { T } ] } ( I - \\hat { \\Psi } ) ( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 } ( I - \\hat { \\Psi } ^ { \\top } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            780,
+            660,
+            814
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Proof ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            824,
+            217,
+            838
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "From eqn. 7 we have that ",
+        "bbox": [
+            173,
+            845,
+            343,
+            859
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7f3b6f3e28fc62ff28bd241da8a3a65c556973f365ac23c1919a5d1c140a34a.jpg",
+        "text": "$$\n\\begin{array} { l c l } { { \\hat { F } _ { 7 } ^ { ( \\infty ) } } } & { { = } } & { { \\displaystyle \\operatorname* { l i m } _ { T ^ { \\prime } \\to \\infty } \\frac { T } { T ^ { \\prime } } ( \\zeta _ { T ^ { \\prime } } ( \\hat { \\Psi } ) + \\zeta _ { T ^ { \\prime } } ( \\hat { \\Psi } ^ { \\top } ) - { T ^ { \\prime } } I ) } } \\\\ { { } } & { { = } } & { { \\displaystyle \\operatorname* { l i m } _ { T ^ { \\prime } \\to \\infty } \\frac { T } { T ^ { \\prime } } \\zeta _ { T ^ { \\prime } } ( \\hat { \\Psi } ) + \\operatorname* { l i m } _ { T ^ { \\prime } \\to \\infty } \\frac { T } { T ^ { \\prime } } \\zeta _ { T ^ { \\prime } } ( \\hat { \\Psi } ^ { \\top } ) - \\operatorname* { l i m } _ { T ^ { \\prime } \\to \\infty } \\frac { T } { T ^ { \\prime } } { T ^ { \\prime } } I . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            866,
+            727,
+            933
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "To evaluate this we first term note that ",
+        "bbox": [
+            176,
+            103,
+            428,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d8d0311ff260c8175d7e4c653f89dac749ca010de16337d3f35a4ed5a577dc3.jpg",
+        "text": "$$\n\\operatorname * { l i m } _ { T ^ { \\prime } \\to \\infty } \\frac { T } { T ^ { \\prime } } \\mathcal { T } ^ { \\prime } I = \\mathcal { T } I \\quad \\mathrm { a n d } \\quad \\operatorname * { l i m } _ { T ^ { \\prime } \\to \\infty } \\frac { T } { T ^ { \\prime } } \\zeta _ { T ^ { \\prime } } ( A ) = \\kappa ( A ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            122,
+            686,
+            154
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "where we have defined ",
+        "bbox": [
+            174,
+            160,
+            326,
+            174
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14e38e0f94b82f8f3247f34cb5c29aa53aec528fa72039ce2434f34595cbfe81.jpg",
+        "text": "$$\n\\kappa ( x ) = \\operatorname* { l i m } _ { \\mathcal { T } ^ { \\prime } \\to \\infty } \\frac { \\mathcal { T } } { \\mathcal { T } ^ { \\prime } } \\zeta \\tau ^ { \\prime } ( x ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            169,
+            584,
+            200
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "For $| x | < 1$ we have that $\\begin{array} { r } { \\operatorname* { l i m } _ { T ^ { \\prime } \\to \\infty } x ^ { T ^ { \\prime } } = 0 } \\end{array}$ , from which it follows that ",
+        "bbox": [
+            173,
+            212,
+            642,
+            228
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d97bba580c9ab5411c650bd3ed7b2b1f9610a7b16e34d5a40f7403b53bc46bfa.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\kappa ( x ) } & { = \\underset { T ^ { \\prime }  \\infty } { \\mathrm { \\operatorname* { l i m } } } \\frac { T } { T ^ { \\prime } } \\Bigg ( \\frac { T ^ { \\prime } ( 1 - x ) - x ( 1 - x ^ { T ^ { \\prime } } ) } { ( 1 - x ) ^ { 2 } } \\Bigg ) } \\\\ & { = \\underset { T ^ { \\prime }  \\infty } { \\mathrm { \\operatorname* { l i m } } } \\frac { T } { T ^ { \\prime } } \\frac { T ^ { \\prime } } { 1 - x } - \\underset { T ^ { \\prime }  \\infty } { \\mathrm { \\operatorname* { l i m } } } \\frac { T } { T ^ { \\prime } } \\frac { x ( 1 - x ^ { T ^ { \\prime } } ) } { ( 1 - x ) ^ { 2 } } } \\\\ & { = \\underset { T ^ { \\prime }  \\infty } { \\mathrm { \\operatorname* { l i m } } } \\frac { T } { 1 - x } - \\underset { T ^ { \\prime }  \\infty } { \\mathrm { \\operatorname* { l i m } } } \\frac { T } { T ^ { \\prime } } \\frac { \\sin } { r  \\infty } \\frac { x ( 1 - x ^ { T ^ { \\prime } } ) } { ( 1 - x ) ^ { 2 } } } \\\\ & { = \\underset { 1 - x } { \\mathrm { \\operatorname* { l i m } } } - \\underset { T ^ { \\prime }  \\infty } { \\mathrm { \\operatorname* { l i m } } } \\frac { T } { T ^ { \\prime } } \\frac { x ( 1 - 0 ) } { ( 1 - x ) ^ { 2 } } } \\\\ & { = \\underset { 1 - x } { \\mathrm { \\operatorname* { l i m } } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            236,
+            686,
+            421
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Let $U \\mathrm { d i a g } ( \\hat { \\psi } ) U ^ { - 1 } = \\hat { \\Psi }$ be the eigendecomposition of $\\hat { \\Psi }$ . Using the fact that $| [ \\hat { \\psi } ] _ { i } | < 1$ (as established in Section A.1) we can use the above expression to evaluate $\\kappa ( x )$ at both $x = { \\hat { \\Psi } }$ and $x = \\hat { \\Psi } ^ { \\top }$ , which yields ",
+        "bbox": [
+            173,
+            433,
+            825,
+            479
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e8cbfe4bd36a35e8922bbf40860e3312282e538c81f026fca0d9450b022f1b5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { F } _ { \\mathcal { T } } ^ { ( \\infty ) } = \\kappa ( \\hat { \\Psi } ) + \\kappa ( \\hat { \\Psi } ^ { \\top } ) = \\mathcal { T } ( ( I - \\hat { \\Psi } ) ^ { - 1 } + ( I - \\hat { \\Psi } ^ { \\top } ) ^ { - 1 } - I ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            478,
+            705,
+            500
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Pre-multiplying both sides by $I - { \\hat { \\Psi } } ^ { \\top }$ , and post-multiplying both sides by $I - { \\hat { \\Psi } }$ , we have ",
+        "bbox": [
+            171,
+            505,
+            766,
+            521
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7575decd8000fa26dfe9cbe45e0bfb4ae848c63a45692a5a6ca42bdcc2e775d8.jpg",
+        "text": "$$\n\\begin{array} { r c l } { ( I - \\hat { \\Psi } ^ { \\top } ) \\hat { F } _ { \\mathcal { T } } ^ { ( \\infty ) } ( I - \\hat { \\Psi } ) } & { = } & { \\mathcal { T } ( ( I - \\hat { \\Psi } ^ { \\top } ) + ( I - \\hat { \\Psi } ) - ( I - \\hat { \\Psi } ^ { \\top } ) ( I - \\hat { \\Psi } ) ) } \\\\ & { = } & { \\mathcal { T } ( I - \\hat { \\Psi } ^ { \\top } + I - \\hat { \\Psi } - I + \\hat { \\Psi } ^ { \\top } + \\hat { \\Psi } - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) } \\\\ & { = } & { \\mathcal { T } ( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            526,
+            751,
+            590
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Then applying the reverse operation gives ",
+        "bbox": [
+            176,
+            594,
+            449,
+            611
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00a5f9149985e98afbf6d1e33ec7f4bf3c2b75cc08bc10bfce759e2330ec2008.jpg",
+        "text": "$$\n\\widehat F _ { \\tau } ^ { ( \\infty ) } = \\mathcal T ( I - \\hat { \\Psi } ^ { \\top } ) ^ { - 1 } ( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ( I - \\hat { \\Psi } ) ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            616,
+            651,
+            640
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Taking the expectation over $\\tau$ gives ",
+        "bbox": [
+            174,
+            646,
+            413,
+            661
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/68136ed3bb42bb96e466d2b74e65fa18588325263cf5d6a10abf0f723b6829c1.jpg",
+        "text": "$$\n\\hat { F } = \\mathbb { E } _ { \\mathcal { T } } [ \\hat { F } _ { \\mathcal { T } } ^ { ( \\infty ) } ] = \\mathbb { E } _ { \\mathcal { T } } [ \\mathcal { T } ] ( I - \\hat { \\Psi } ^ { \\top } ) ^ { - 1 } ( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ( I - \\hat { \\Psi } ) ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            667,
+            702,
+            689
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Finally, inverting both sides yields ",
+        "bbox": [
+            173,
+            695,
+            401,
+            710
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a1e44e757f77e63b1d1ae4496f734e27dff2c75bb5731446dcba01c2077c9b1.jpg",
+        "text": "$$\n\\hat { F } ^ { - 1 } = \\frac { 1 } { \\mathbb { E } _ { T } [ \\mathcal { T } ] } ( I - \\hat { \\Psi } ) ( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 } ( I - \\hat { \\Psi } ^ { \\top } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            717,
+            660,
+            750
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL TECHNICAL PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            806,
+            491,
+            823
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "B.1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            838,
+            200,
+            851
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3 Suppose $x \\in \\mathbb { C }$ , $x \\neq 0$ , and $\\tau$ is a non-negative integer. We have ",
+        "bbox": [
+            171,
+            863,
+            702,
+            878
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61df8741a5dc8b8e21ef15ff7f673f041d8756947d5dba2607d5d694e0662df9.jpg",
+        "text": "$$\n\\sum _ { i = 0 } ^ { T } ( \\mathcal T - i ) x ^ { i } = \\frac { \\mathcal T ( 1 - x ) - x ( 1 - x ^ { \\mathcal T } ) } { ( 1 - x ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            886,
+            637,
+            928
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Proof Observe that ",
+        "bbox": [
+            173,
+            103,
+            307,
+            117
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/68f93ea21222d100c79bc0179e03bf524af272c14eb63b12a4340c1adbe079b5.jpg",
+        "text": "$$\nx { \\frac { \\mathrm { d } \\sum _ { i = 0 } ^ { \\mathcal { T } } x ^ { i } } { \\mathrm { d } x } } = x \\sum _ { i = 0 } ^ { \\mathcal { T } } { \\frac { \\mathrm { d } x ^ { i } } { \\mathrm { d } x } } = x \\sum _ { i = 0 } ^ { \\mathcal { T } } i x ^ { i - 1 } = \\sum _ { i = 0 } ^ { \\mathcal { T } } i x ^ { i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            119,
+            671,
+            165
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Another way to express this is to use the geometric series formula $\\textstyle \\sum _ { i = 0 } ^ { T } x ^ { i } = { \\frac { 1 - x ^ { T + 1 } } { 1 - x } }$ (which holds ",
+        "bbox": [
+            173,
+            170,
+            828,
+            203
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/930c9c479f01c280bf34aa8eec7755970679e5c013f174e5d77d3752ca62fd18.jpg",
+        "text": "$$\n: \\frac { \\mathrm { d } \\sum _ { i = 0 } ^ { T } x ^ { i } } { \\mathrm { ~ d } x } = x \\frac { \\mathrm { d } \\left( \\frac { 1 - x ^ { T + 1 } } { 1 - x } \\right) } { \\mathrm { ~ d } x } = x \\left( - \\frac { ( 1 + T ) x ^ { T } } { 1 - x } + \\frac { 1 - x ^ { T + 1 } } { ( 1 - x ) ^ { 2 } } \\right) = \\frac { x ( 1 - x ^ { T + 1 } - ( 1 + T ) ( 1 - x ) ) } { ( 1 - x ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            207,
+            839,
+            252
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Thus we have ",
+        "bbox": [
+            173,
+            256,
+            267,
+            270
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/16be47097ba496e1fcc38f67dc3c865cf00d98a1e3c738c8c851d567b3879d6f.jpg",
+        "text": "$$\n\\sum _ { i = 0 } ^ { T } i x ^ { i } = \\frac { x ( 1 - x ^ { T + 1 } - ( 1 + \\mathcal { T } ) ( 1 - x ) x ^ { \\mathcal { T } } ) } { ( 1 - x ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            265,
+            655,
+            309
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "And so ",
+        "bbox": [
+            173,
+            310,
+            223,
+            324
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/78a3f21a7ed440778691bbc0f62fd54a775206c40a9dcb485a551d7203883d6a.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\frac { T } { - \\alpha } ( T - \\epsilon ) x ^ { 2 } } & { = \\begin{array} { r l } { \\underline { \\gamma } } & { \\underline { \\gamma } } \\\\ & { - \\underline { \\gamma } } & { \\underline { \\gamma } } \\end{array} ; } \\\\ { = \\begin{array} { r l } & { \\underline { \\gamma } ( 1 - x ^ { * } ) } \\\\ & { - \\underline { \\gamma } } & { \\underline { \\gamma } ( 1 - x ^ { * } ) } \\\\ & { \\underline { \\gamma } } & { \\underline { \\gamma } } \\end{array} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            327,
+            839,
+            585
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "where we have again used the geometric series formula $\\textstyle \\sum _ { i = 0 } ^ { T } x ^ { i } = { \\frac { 1 - x ^ { T + 1 } } { 1 - x } }$ on the second line. ",
+        "bbox": [
+            169,
+            592,
+            797,
+            612
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "B.2 SPECTRAL BOUND FOR ESTIMATE OF $\\hat { \\Psi }$ ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            636,
+            490,
+            652
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In what follows all quantities are computed using their defining formulae, starting from the estimated values of $A _ { 0 } , A _ { 1 } , G _ { 0 }$ , and $G _ { 1 }$ . ",
+        "bbox": [
+            174,
+            662,
+            823,
+            693
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "First we observe that since $\\hat { \\Psi } = \\hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } \\Psi V _ { 0 } ^ { 1 / 2 }$ is similar to $\\Psi$ (in the technical sense of the word), they share the same eigenvalues. Thus it suffices bound to the spectral radius of $\\Psi = { \\cal V } _ { 1 } { \\cal V } _ { 0 } ^ { - 1 }$ . ",
+        "bbox": [
+            173,
+            698,
+            825,
+            732
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Next we observe that $V _ { 0 } = A _ { 0 } \\otimes G _ { 0 }$ and $V _ { 1 } = A _ { 1 } \\otimes G _ { 1 }$ , so that ",
+        "bbox": [
+            174,
+            737,
+            599,
+            753
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8db52c4c9d25f75558cb609b0dc4f59704b118c8e4185583f731b480cbb94914.jpg",
+        "text": "$$\nV _ { 1 } V _ { 0 } ^ { - 1 } = ( A _ { 1 } \\otimes G _ { 1 } ) ( A _ { 0 } \\otimes G _ { 0 } ) ^ { - 1 } = ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \\otimes ( G _ { 1 } G _ { 0 } ^ { - 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            757,
+            700,
+            776
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Because the eigendecomposition of a Kronecker product is the Kronecker product of the decompositions of the factors we have that $\\rho ( V _ { 1 } V _ { 0 } ^ { - 1 } ) \\stackrel { { } = } { = } \\rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \\rho ( G _ { 1 } G _ { 0 } ^ { - 1 } )$ , where $\\rho ( X )$ denotes the spectral radius of a matrix $X$ . ",
+        "bbox": [
+            173,
+            787,
+            825,
+            830
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Thus it suffices to show that $\\rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \\leq 1$ and $\\rho ( G _ { 1 } G _ { 0 } ^ { - 1 } ) \\leq 1$ for $A _ { i }$ and $G _ { i }$ as computed by the estimation scheme outlined in Section 3.5.4. Recall that this is the exponentially decayed average of mini-batch averages of estimators of the form $\\begin{array} { r } { \\frac { 1 } { \\mathbb { E } \\tau [ \\mathcal { T } ] } \\sum _ { t = 1 } ^ { \\mathcal { T } } a _ { t } a _ { t } ^ { \\top } } \\end{array}$ and $\\begin{array} { r } { \\frac { 1 } { \\mathbb { E } \\tau [ \\mathcal { T } ] } \\dot { \\sum _ { t = 1 } ^ { \\mathcal { T } - 1 } { a _ { t + 1 } } } { a _ { t } ^ { \\top } } } \\end{array}$ . ",
+        "bbox": [
+            174,
+            835,
+            825,
+            886
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In the remainder of this section we will show that $\\rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \\ \\leq \\ 1$ . The argument to show that $\\rho ( G _ { 1 } G _ { 0 } ^ { - 1 } ) \\leq 1$ is identical. ",
+        "bbox": [
+            173,
+            892,
+            823,
+            925
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Define ",
+        "bbox": [
+            173,
+            103,
+            220,
+            117
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/53b904d8c9add16807ff0091ac9cae1399319ae36ec574cbc6e89d574fcaaac5.jpg",
+        "text": "$$\nM _ { 0 } = [ 0 \\quad a _ { 1 } \\quad a _ { 2 } \\quad \\cdots \\quad a _ { T } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            122,
+            602,
+            141
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "and ",
+        "bbox": [
+            173,
+            146,
+            200,
+            160
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0980344555103df98393404aef7eef92665ca1ca1714af161592b83d31546be.jpg",
+        "text": "$$\nM _ { 1 } = [ a _ { 1 } \\quad a _ { 2 } \\quad \\cdot \\cdot \\cdot \\quad a _ { \\mathcal { T } } \\quad 0 ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            165,
+            602,
+            183
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { M _ { 0 } M _ { 0 } ^ { \\top } = M _ { 1 } M _ { 1 } ^ { \\top } = \\sum _ { t = 1 } ^ { \\mathcal T } a _ { t } a _ { t } ^ { \\top } \\mathrm { ~ a n d ~ } M _ { 1 } M _ { 0 } ^ { \\top } = \\sum _ { t = 1 } ^ { \\mathcal T - 1 } a _ { t + 1 } a _ { t } ^ { \\top } . } \\end{array}$ ",
+        "bbox": [
+            176,
+            188,
+            714,
+            208
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Provided that the exponentially decayed averages for $A _ { 0 }$ and $A _ { 1 }$ are computed in the same way and use the same normalizers (i.e. $1 / ( m \\mathbb { E } _ { \\mathcal { T } } [ \\mathcal { T } ] )  $ ) we thus have that the matrix ",
+        "bbox": [
+            173,
+            212,
+            825,
+            242
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c0a726aaf6051338cd52151c9f3d748aa1948b6a528c3205e112324b91f4ce4c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { A } = \\left[ \\begin{array} { l l } { A _ { 0 } } & { A _ { 1 } } \\\\ { A _ { 1 } ^ { \\top } } & { A _ { 0 } } \\end{array} \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            246,
+            555,
+            281
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "is a positively-weighted linear combination of terms of the form ",
+        "bbox": [
+            173,
+            285,
+            594,
+            300
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b8b99091319e256ec9fd30d7c0ea6b34cc0d8a046f78b6d5c850e96fbcd5ea98.jpg",
+        "text": "$$\n\\left[ { \\cal M } _ { 1 } \\right] \\left[ { \\cal M } _ { 1 } \\right] ^ { \\top } \\succeq 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            304,
+            563,
+            343
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "where the various $a _ { t }$ ’s are computed on different data using current and previous model parameters.   \nIt thus follows that $A \\succeq 0$ . The inequality $\\rho ( A _ { 1 } A _ { 0 } ^ { - 1 } ) \\leq 1$ now follows from the following lemma. ",
+        "bbox": [
+            173,
+            347,
+            823,
+            376
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1 Consider a real, symmetric, positive semi-definite block matrix ",
+        "bbox": [
+            173,
+            387,
+            661,
+            401
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/deb967dad583a1aa035669651ba360f8181cfcc050bcd9b2e5dc290de12d60ce.jpg",
+        "text": "$$\n{ \\left[ \\begin{array} { l l } { B } & { C } \\\\ { C ^ { \\mathsf { T } } } & { B } \\end{array} \\right] } \\succeq 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            406,
+            552,
+            441
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "If $B$ is invertible then we have $\\rho ( C B ^ { - 1 } ) \\le 1$ , and further if the block matrix is positive definite we have $\\rho ( C B ^ { - 1 } ) < 1$ . ",
+        "bbox": [
+            173,
+            445,
+            825,
+            476
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Proof Because similarity transformations preserve eigenvalues, the statement is equivalent to $\\rho ( B ^ { - \\frac { 1 } { 2 } } C B ^ { - \\frac { 1 } { 2 } } ) \\le 1$ . Define $X \\triangleq B ^ { - \\frac { 1 } { 2 } } C \\bar { B } ^ { - \\frac { 1 } { 2 } }$ . Because any induced matrix norm is an upperbound on the spectral radius, it suffices to show $\\| X \\| _ { 2 } = \\sigma _ { \\operatorname* { m a x } } ( X ) \\leq 1$ , where $\\sigma _ { \\mathrm { m a x } } ( X )$ denotes the largest singular value of $X$ . ",
+        "bbox": [
+            173,
+            487,
+            825,
+            546
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "By taking Schur complements of the block matrix (11) we have ",
+        "bbox": [
+            173,
+            553,
+            591,
+            568
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c578d2a664d3cf57bf94411ed8dab2bbee96ef9e9f08565b388e3959abff414.jpg",
+        "text": "$$\n\\begin{array} { c } { { 0 { \\preceq } B - C B ^ { - 1 } C ^ { \\mathsf { T } } = B ^ { \\frac { 1 } { 2 } } ( I - ( B ^ { - \\frac { 1 } { 2 } } C B ^ { - \\frac { 1 } { 2 } } ) ( B ^ { - \\frac { 1 } { 2 } } C B ^ { - \\frac { 1 } { 2 } } ) ) B ^ { \\frac { 1 } { 2 } } } } \\\\ { { { } } } \\\\ { { { } = B ^ { \\frac { 1 } { 2 } } ( I - X X ^ { \\mathsf { T } } ) B ^ { \\frac { 1 } { 2 } } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            571,
+            712,
+            616
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Note that the Schur complement is PSD because the original block matrix itself is (e.g. Zhang, 2006). ",
+        "bbox": [
+            176,
+            618,
+            820,
+            648
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Using the fact that $Z \\mapsto B ^ { { \\frac { 1 } { 2 } } } Z B ^ { { \\frac { 1 } { 2 } } }$ maps positive semidefinite matrices to positive semidefinite matrices, this implies ",
+        "bbox": [
+            176,
+            654,
+            823,
+            685
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a673475f36cfc9173b4535c984a9d3a9e78160f52b4ad5f35c3b5749f9b01e0d.jpg",
+        "text": "$$\n0 \\preceq I - X X ^ { \\mathsf { T } } \\Rightarrow X X ^ { \\mathsf { T } } \\preceq I \\Rightarrow \\| X \\| _ { 2 } ^ { 2 } \\leq 1 \\Rightarrow \\rho ( X ) \\leq 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            689,
+            691,
+            708
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "When the block matrix is positive definite, the inequalities become strict. ",
+        "bbox": [
+            171,
+            713,
+            650,
+            728
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "C EFFICIENT IMPLEMENTATION ASSUMING $V _ { 0 }$ AND $V _ { 1 }$ ARE KRONECKER-FACTORED ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            753,
+            676,
+            789
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "The ultimate goal of our calculations is to efficiently compute the matrix-vector product of some arbitrary vector $z$ (which will often be the gradient $\\mathrm { v e c } ( { \\mathcal { D } } W ) ,$ ) with our inverse Fisher approximation $F ^ { - 1 }$ . That is, we wish to compute $F ^ { - 1 } z$ . ",
+        "bbox": [
+            174,
+            803,
+            825,
+            847
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "It will be convenient to assume that $z$ is given as a matrix $Z$ (with the same dimensions as $\\mathcal { D } W$ ) so that $z = \\operatorname { v e c } ( Z )$ . ",
+        "bbox": [
+            171,
+            852,
+            825,
+            882
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Multiplying the vector $z$ by $F ^ { - 1 } = V _ { 0 } ^ { - 1 / 2 } \\hat { F } ^ { - 1 } V _ { 0 } ^ { - 1 / 2 }$ amounts to first multiplying by $V _ { 0 } ^ { - 1 / 2 }$ , then by $\\hat { F } ^ { - 1 }$ , and then by $V _ { 0 } ^ { - 1 / 2 }$ again. ",
+        "bbox": [
+            174,
+            887,
+            825,
+            925
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "We will suppose that we are given $A _ { 0 } , A _ { 1 } , G _ { 0 }$ , and $G _ { 1 }$ such that ",
+        "bbox": [
+            173,
+            102,
+            599,
+            119
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ae7cb99024510f31331d7f016f0c61ea291e9ceab9e8b7f771d530ce9f1be88.jpg",
+        "text": "$$\nV _ { 0 } = A _ { 0 } \\otimes G _ { 0 } \\quad { \\mathrm { a n d } } \\quad V _ { 1 } = A _ { 1 } \\otimes G _ { 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            122,
+            632,
+            138
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "We then have ",
+        "bbox": [
+            174,
+            143,
+            264,
+            159
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/709f837c21e3be31996d0edd6d5ce2727db669811d3dd6a877b51ecdb05f97b6.jpg",
+        "text": "$$\nV _ { 0 } ^ { - 1 / 2 } = A _ { 0 } ^ { - 1 / 2 } \\otimes G _ { 0 } ^ { - 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            152,
+            586,
+            174
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "and thus to multiply by $V _ { 0 } ^ { - 1 / 2 }$ we can use the identity $( C \\otimes B ) \\operatorname { v e c } ( X ) = \\operatorname { v e c } ( B X C ^ { \\top } )$ , giving ",
+        "bbox": [
+            174,
+            176,
+            802,
+            194
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23f9aa9e06d1d174a09f64f6feff8ebb4e15ffbf437e443fc341762ce6013388.jpg",
+        "text": "$$\nV _ { 0 } ^ { - 1 / 2 } z = \\mathrm { v e c } ( G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            198,
+            606,
+            219
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "In matrix form this is simply $G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 }$ . Note that $A _ { 0 } ^ { - 1 / 2 }$ and $G _ { 0 } ^ { - 1 / 2 }$ can be computed using the eigendecompositions of $A _ { 0 }$ and $G _ { 0 }$ , for example. ",
+        "bbox": [
+            173,
+            223,
+            825,
+            256
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "The procedure to efficiently multiply a vector $z$ by $\\hat { F } ^ { - 1 }$ is more involved and depends on which approximation “option” we are using. However one immediate useful insight we can make before specializing to Option 1 or Option 2 is that $\\hat { \\Psi }$ can be written as a Kronecker product as follows: ",
+        "bbox": [
+            174,
+            262,
+            825,
+            309
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/291786defd55743b7152a7fa206e10948fc539f888eac6bdf4b964ecfc46585a.jpg",
+        "text": "$$\n\\begin{array} { l c l } { { \\hat { \\Psi } } } & { { = } } & { { V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 } } } \\\\ { { } } & { { = } } & { { ( A _ { 0 } \\otimes G _ { 0 } ) ^ { - 1 / 2 } ( A _ { 1 } \\otimes G _ { 1 } ) ( A _ { 0 } \\otimes G _ { 0 } ) ^ { - 1 / 2 } } } \\\\ { { } } & { { = } } & { { ( A _ { 0 } ^ { - 1 / 2 } \\otimes G _ { 0 } ^ { - 1 / 2 } ) ( A _ { 1 } \\otimes G _ { 1 } ) ( A _ { 0 } ^ { - 1 / 2 } \\otimes G _ { 0 } ^ { - 1 / 2 } ) } } \\\\ { { } } & { { = } } & { { ( A _ { 0 } ^ { - 1 / 2 } A _ { 1 } A _ { 0 } ^ { - 1 / 2 } ) \\otimes ( G _ { 0 } ^ { - 1 / 2 } G _ { 1 } G _ { 0 } ^ { - 1 / 2 } ) } } \\\\ { { } } & { { = } } & { { \\hat { \\Psi } _ { A } \\otimes \\hat { \\Psi } _ { G } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            311,
+            686,
+            416
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where we have defined $\\hat { \\Psi } _ { A } = A _ { 0 } ^ { - 1 / 2 } A _ { 1 } A _ { 0 } ^ { - 1 / 2 }$ and $\\hat { \\Psi } _ { G } = G _ { 0 } ^ { - 1 / 2 } G _ { 1 } G _ { 0 } ^ { - 1 / 2 }$ ",
+        "bbox": [
+            174,
+            417,
+            668,
+            439
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "C.1 OPTION 1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            453,
+            285,
+            468
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "For Option 1 we assume that $V _ { 1 }$ , and hence $\\hat { \\Psi } = \\hat { V } _ { 1 } = V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 }$ , is symmetric. We note that $V _ { 1 } = A _ { 1 } \\otimes G _ { 1 }$ will be symmetric if and only if both $A _ { 1 }$ and $G _ { 1 }$ are. ",
+        "bbox": [
+            173,
+            476,
+            825,
+            508
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Our task is to compute the matrix-vector product ",
+        "bbox": [
+            176,
+            515,
+            495,
+            529
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/04159ed7e146c04a28053e5f93ebbe6f521a81d651ce0825831644b75268ccfd.jpg",
+        "text": "$$\n\\hat { F } ^ { - 1 } z = U \\mathrm { d i a g } ( \\gamma ( \\hat { \\psi } ) ) U ^ { \\top } z\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            534,
+            593,
+            553
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where $U \\mathrm { d i a g } ( \\hat { \\psi } ) U ^ { \\top }$ is the eigendecomposition of $\\hat { \\Psi }$ , and $\\gamma ( x )$ is defined as in Section 3.5.3, eqn. 8.   \nTo do this we first multiply the vector by $U ^ { \\top }$ , then by $\\mathrm { d i a g } ( \\gamma ( \\hat { \\psi } ) )$ , and then finally by $U$ . ",
+        "bbox": [
+            173,
+            558,
+            825,
+            592
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "The eigendecomposition of $\\hat { \\Psi }$ can be computed efficiently using its Kronecker product structure. In particular, we compute the eigendecompositions of each factor as ",
+        "bbox": [
+            173,
+            598,
+            826,
+            628
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02ca3808a08c85fc57057c3dd2757146d099898d79ef7c10ace85ead3744207a.jpg",
+        "text": "$$\nU _ { A } \\mathrm { d i a g } ( \\hat { \\psi } _ { A } ) U _ { A } ^ { \\top } = \\hat { \\Psi } _ { A } \\quad \\mathrm { a n d } \\quad U _ { G } \\mathrm { d i a g } ( \\hat { \\psi } _ { G } ) U _ { G } ^ { \\top } = \\hat { \\Psi } _ { G } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            632,
+            691,
+            652
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "from which we can write the eigendecomposition $U \\mathrm { d i a g } ( \\hat { \\psi } ) U ^ { - 1 }$ of $\\hat { \\Psi }$ as ",
+        "bbox": [
+            176,
+            657,
+            656,
+            674
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2882f83b7dc1bc453fe75f1f03aa143c0ddb922b786891a5f9c1425cb08ee76.jpg",
+        "text": "$$\n\\begin{array} { l c l } { { \\hat { \\Psi } } } & { { = } } & { { \\hat { \\Psi } _ { A } \\otimes \\hat { \\Psi } _ { G } } } \\\\ { { } } & { { = } } & { { ( U _ { A } \\mathrm { d i a g } ( \\hat { \\psi } _ { A } ) U _ { A } ^ { \\top } ) \\otimes ( U _ { G } \\mathrm { d i a g } ( \\hat { \\psi } _ { G } ) U _ { G } ^ { \\top } ) } } \\\\ { { } } & { { = } } & { { ( U _ { A } \\otimes U _ { G } ) ( \\mathrm { d i a g } ( \\hat { \\psi } _ { A } ) \\otimes \\mathrm { d i a g } ( \\hat { \\psi } _ { G } ) ) ( U _ { A } ^ { \\top } \\otimes U _ { G } ^ { \\top } ) } } \\\\ { { } } & { { = } } & { { ( U _ { A } \\otimes U _ { G } ) \\mathrm { d i a g } ( \\mathrm { v e c } ( \\hat { \\psi } _ { G } \\hat { \\psi } _ { A } ^ { \\top } ) ) ( U _ { A } ^ { \\top } \\otimes U _ { G } ^ { \\top } ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            676,
+            689,
+            758
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "In other words, we have $U = U _ { A } \\otimes U _ { G }$ and $\\hat { \\psi } = \\mathrm { v e c } ( \\hat { \\psi } _ { G } \\hat { \\psi } _ { A } ^ { \\top } )$ . ",
+        "bbox": [
+            173,
+            761,
+            583,
+            779
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "To multiply $z$ by $U ^ { \\top } = U _ { A } ^ { \\top } \\otimes U _ { G } ^ { \\top }$ we use the identity $( C \\otimes B ) \\operatorname { v e c } ( X ) = \\operatorname { v e c } ( B X C ^ { \\top } )$ which gives ",
+        "bbox": [
+            169,
+            784,
+            823,
+            803
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/776089b03dc536000cbcc7d9d36a2ab2378430b79d3274bd876871a310b08e40.jpg",
+        "text": "$$\nU ^ { \\top } z = ( U _ { A } ^ { \\top } \\otimes U _ { G } ^ { \\top } ) \\operatorname { v e c } ( Z ) = \\operatorname { v e c } ( U _ { G } ^ { \\top } Z U _ { A } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            805,
+            648,
+            824
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Similarly, the multiplication of $z$ by $U = U _ { A } \\otimes U _ { G }$ can be computed as $\\mathrm { v e c } ( U _ { G } Z U _ { A } ^ { \\top } )$ . ",
+        "bbox": [
+            169,
+            827,
+            746,
+            844
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Finally, multiplying $z$ by $\\mathrm { d i a g } ( \\gamma ( \\hat { \\psi } ) ) = \\mathrm { d i a g } ( \\gamma ( \\mathrm { v e c } ( \\hat { \\psi } _ { G } \\hat { \\psi } _ { A } ^ { \\top } ) ) )$ corresponds to entry-wise multiplication of $Z$ by a matrix $Y$ , where $\\mathrm { v e c } ( Y ) ~ = ~ \\gamma ( \\mathrm { v e c } ( \\hat { \\psi } _ { G } \\hat { \\psi } _ { A } ^ { \\top } ) )$ , or in other words $[ Y ] _ { i , j } ~ =$ $\\gamma ( [ \\hat { \\psi } _ { G } \\hat { \\psi } _ { A } ^ { \\top } ] _ { i , j } ) = \\gamma ( [ \\hat { \\psi } _ { G } ] _ { i } [ \\hat { \\psi } _ { A } ] _ { j } )$ . Thus we have ",
+        "bbox": [
+            173,
+            849,
+            825,
+            902
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d4b7e43a3c95f085fda588099c440743e0106435c79ee4ddc01b02d412f35d08.jpg",
+        "text": "$$\n\\mathrm { d i a g } ( \\gamma ( \\hat { \\psi } ) ) z = \\mathrm { d i a g } ( \\mathrm { v e c } ( Y ) ) \\mathrm { v e c } ( Z ) = \\mathrm { v e c } ( Z \\odot Y ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            906,
+            676,
+            925
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ denotes entry-wise multiplication of matrices. ",
+        "bbox": [
+            173,
+            103,
+            537,
+            119
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "In summary we have that $\\hat { F } ^ { - 1 } z$ can be computed in matrix form as ",
+        "bbox": [
+            173,
+            126,
+            616,
+            142
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c13202c59ffbf559074621f5a9a261b0b75659ae51feff0fdc930ede39df138.jpg",
+        "text": "$$\nU _ { G } ( ( U _ { G } ^ { \\top } Z U _ { A } ) \\odot Y ) U _ { A } ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            150,
+            581,
+            171
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "for $Y$ s.t. $[ Y ] _ { i , j } = \\gamma ( [ \\hat { \\psi } _ { G } ] _ { i } [ \\hat { \\psi } _ { A } ] _ { j } )$ . ",
+        "bbox": [
+            174,
+            180,
+            406,
+            200
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We note that computing $\\gamma ( [ \\hat { \\psi } _ { G } ] _ { i } [ \\hat { \\psi } _ { A } ] _ { j } )$ is trivial since it is just a scalar evaluation of the rational function $\\gamma ( x )$ . ",
+        "bbox": [
+            174,
+            205,
+            823,
+            237
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "C.2 OPTION 2 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            256,
+            287,
+            271
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "For Option 2 we must compute the matrix-vector product ",
+        "bbox": [
+            173,
+            284,
+            550,
+            299
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b3ef22f519246faf95315989f2dbe93d4ac29ada15bec0b0203e223c6cf2ca8f.jpg",
+        "text": "$$\n\\hat { F } ^ { - 1 } z = \\frac { 1 } { \\sum _ { i } \\omega _ { i } \\mathcal { T } _ { i } } ( I - \\hat { \\Psi } ) ( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 } ( I - \\hat { \\Psi } ^ { \\top } ) v .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            308,
+            673,
+            342
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "To do this we will first multiply by $I - { \\hat { \\Psi } } ^ { \\top }$ , then by $( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 }$ , and then by $I - { \\hat { \\Psi } }$ , before finally dividing the result by $\\sum _ { i } { \\omega _ { i } { \\bar { T } } _ { i } }$ . ",
+        "bbox": [
+            173,
+            351,
+            823,
+            383
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "To compute the matrix-vector product $( I - { \\hat { \\Psi } } ^ { \\top } ) z$ we use the identity $( C \\otimes B ) \\operatorname { v e c } ( X ) =$ $\\mathrm { v e c } ( B X C ^ { \\top } )$ while noting that $\\hat { \\Psi } ^ { \\top } = ( \\hat { \\Psi } _ { A } \\otimes \\hat { \\Psi } _ { G } ) ^ { \\top } = \\hat { \\Psi } _ { A } ^ { \\top } \\otimes \\hat { \\Psi } _ { G } ^ { \\top }$ . This gives ",
+        "bbox": [
+            173,
+            390,
+            825,
+            424
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d8729bebe4da51db4693b9877d4e4ab739c361cff4740c128247084054fd005.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { ( I - \\hat { \\Psi } ^ { \\top } ) \\mathrm { v e c } ( Z ) } } & { { = } } & { { \\mathrm { v e c } ( Z ) - ( \\hat { \\Psi } _ { A } ^ { \\top } \\otimes \\hat { \\Psi } _ { G } ^ { \\top } ) \\mathrm { v e c } ( Z ) } } \\\\ { { } } & { { = } } & { { \\mathrm { v e c } ( Z ) - \\mathrm { v e c } ( \\hat { \\Psi } _ { G } ^ { \\top } Z \\hat { \\Psi } _ { A } ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            431,
+            681,
+            474
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "The matrix form of this is simply $Z - \\hat { \\Psi } _ { G } ^ { \\top } Z \\hat { \\Psi } _ { A }$ . ",
+        "bbox": [
+            173,
+            483,
+            493,
+            501
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We may similarly compute $( I - { \\hat { \\Psi } } ) z$ in matrix form as $Z - \\hat { \\Psi } _ { G } Z \\hat { \\Psi } _ { A } ^ { \\top }$ . ",
+        "bbox": [
+            174,
+            507,
+            635,
+            525
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "The harder task is to compute $( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 } z$ , which is what we tackle next. ",
+        "bbox": [
+            174,
+            531,
+            676,
+            549
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We first observe that ",
+        "bbox": [
+            174,
+            555,
+            312,
+            569
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b60b08292d5c77148c2b9e7d472c8df37448a220a2a5ec38e9ae355fad7a47c.jpg",
+        "text": "$$\n\\begin{array} { l l l l } { { I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } } } & { { = } } & { { I \\otimes I - ( \\hat { \\Psi } _ { A } ^ { \\top } \\otimes \\hat { \\Psi } _ { G } ^ { \\top } ) ( \\hat { \\Psi } _ { A } \\otimes \\hat { \\Psi } _ { G } ) } } \\\\ { { } } & { { = } } & { { I \\otimes I - ( \\hat { \\Psi } _ { A } ^ { \\top } \\hat { \\Psi } _ { A } ) \\otimes ( \\hat { \\Psi } _ { G } ^ { \\top } \\hat { \\Psi } _ { G } ) } } \\\\ { { } } & { { \\equiv } } & { { I \\otimes I - M _ { A } \\otimes M _ { G } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            575,
+            665,
+            636
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Given the eigendecompositions $E _ { A } \\mathrm { d i a g } ( m _ { A } ) E _ { A } ^ { \\top } = M _ { A }$ and $E _ { G } \\mathrm { d i a g } ( m _ { G } ) E _ { G } ^ { \\top } = M _ { G }$ we can thus compute the larger eigendecomposition as ",
+        "bbox": [
+            173,
+            642,
+            823,
+            672
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f65b643019d8fe9db6c659a3d9b46437e13d133a33d9289f94c2611e0bef06e.jpg",
+        "text": "$$\n\\begin{array} { l l l } { { I \\otimes I - M _ { A } \\otimes M _ { G } } } & { { = } } & { { ( E _ { A } \\otimes E _ { G } ) ( I \\otimes I - \\mathrm { d i a g } ( m _ { A } ) \\otimes \\mathrm { d i a g } ( m _ { G } ) ) ( E _ { A } ^ { \\top } \\otimes E _ { G } ^ { \\top } ) } } \\\\ { { } } & { { = } } & { { ( E _ { A } \\otimes E _ { G } ) \\mathrm { d i a g } ( \\mathbb { I } \\otimes \\mathbb { I } - m _ { A } \\otimes m _ { G } ) ( E _ { A } ^ { \\top } \\otimes E _ { G } ^ { \\top } ) } } \\\\ { { } } & { { = } } & { { ( E _ { A } \\otimes E _ { G } ) \\mathrm { d i a g } ( \\mathrm { v e c } ( \\mathbb { I } \\mathbb { I } ^ { \\top } - m _ { G } m _ { A } ^ { \\top } ) ) ( E _ { A } ^ { \\top } \\otimes E _ { G } ^ { \\top } ) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            681,
+            781,
+            743
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "where l is the vector of ones. ",
+        "bbox": [
+            174,
+            750,
+            364,
+            765
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Using the eigendecomposition the inverse can then be easily computed as ",
+        "bbox": [
+            174,
+            771,
+            656,
+            786
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3779d77f960ae1b076bf947785f10f0a355ba439cd2e34f3e4ab2ef2e2f0c949.jpg",
+        "text": "$$\n( I \\otimes I - M _ { A } \\otimes M _ { G } ) ^ { - 1 } = ( E _ { A } \\otimes E _ { G } ) \\mathrm { d i a g } ( \\mathrm { v e c } ( \\mathbb { 1 } \\mathbb { I } ^ { \\top } - m _ { G } m _ { A } ^ { \\top } ) ) ^ { - 1 } ( E _ { A } ^ { \\top } \\otimes E _ { G } ^ { \\top } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            227,
+            795,
+            767,
+            815
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Thus $( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 } z$ may be computed by first multiplying by $( E _ { A } ^ { \\top } \\otimes E _ { G } ^ { \\top } )$ , then by $\\mathrm { d i a g ( v e c ( I I ^ { \\top } - }$ $m _ { G } m _ { A } ^ { \\top } ) ) ^ { - 1 }$ (which in matrix form corresponds to element-wise division by $\\mathbb { I } \\mathbb { I } ^ { \\top } - m _ { G } m _ { A } ^ { \\top } )$ , and then by $E _ { A } \\otimes E _ { G }$ . The matrix form of this is ",
+        "bbox": [
+            173,
+            825,
+            826,
+            871
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08e4b2f68b80d78a42ccc0eb0cb287a8e45271bc61975e66a14f1eadf8d33ee8.jpg",
+        "text": "$$\nE _ { G } ( ( E _ { G } ^ { \\top } Z E _ { A } ) \\oslash ( \\mathbb { I I } ^ { \\top } - m _ { G } m _ { A } ^ { \\top } ) ) E _ { A } ^ { \\top } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            880,
+            629,
+            900
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "where $B \\oslash C$ denotes element-wise division of the matrix $B$ by the matrix $C$ . ",
+        "bbox": [
+            173,
+            909,
+            684,
+            924
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "C.3 PSEUDO-CODE FOR THE COMPUTATION OF $F ^ { - 1 } z$ ",
+        "bbox": [
+            173,
+            102,
+            555,
+            118
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Given $A _ { 0 }$ , $A _ { 1 }$ , $G _ { 0 }$ , and $G _ { 1 }$ such that $V _ { 0 } = A _ { 0 } \\otimes G _ { 0 }$ and $V _ { 1 } = A _ { 1 } \\otimes G _ { 1 }$ , the procedure to compute $F ^ { - 1 } z$ for an arbitrary vector $z = \\operatorname { v e c } ( Z )$ is as follows. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            159
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Pre-processing for Option 1 only: ",
+        "bbox": [
+            174,
+            165,
+            397,
+            180
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Ensure that $A _ { 1 }$ and $G _ { 1 }$ are exactly symmetric, and if not, symmertrize them via: ",
+        "bbox": [
+            214,
+            190,
+            758,
+            207
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6924fc8722990e4a6dedc2d8783d2dec65e12558d7e12c4802a63c1b6f3a1434.jpg",
+        "text": "$$\nA _ { 1 }  { \\frac { A _ { 1 } + A _ { 1 } ^ { \\top } } { 2 } } \\quad { \\mathrm { a n d } } \\quad G _ { 1 }  { \\frac { G _ { 1 } + G _ { 1 } ^ { \\top } } { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            212,
+            674,
+            246
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Pre-processing steps (both options): ",
+        "bbox": [
+            174,
+            255,
+            411,
+            270
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Compute the matrix square roots of the factors of $V _ { 0 }$ (e.g. using eigendecompositions): ",
+        "bbox": [
+            215,
+            281,
+            802,
+            296
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ac6f403d010492f8b0740cf28d0d33e08eca85b61032e858f255e60cfae3578.jpg",
+        "text": "$$\nA _ { 0 } ^ { - 1 / 2 } \\quad \\mathrm { a n d } \\quad G _ { 0 } ^ { - 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            452,
+            303,
+            604,
+            324
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Compute the factors of the transformed transition matrix $\\hat { \\Psi }$ : ",
+        "bbox": [
+            215,
+            335,
+            625,
+            352
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6ce65704972651aa71c2eaca95df2f305056c501537037161f2c63d4a8e7f796.jpg",
+        "text": "$$\n\\hat { \\Psi } _ { A } = A _ { 0 } ^ { - 1 / 2 } A _ { 1 } A _ { 0 } ^ { - 1 / 2 } \\quad \\mathrm { a n d } \\quad \\hat { \\Psi } _ { G } = G _ { 0 } ^ { - 1 / 2 } G _ { 1 } G _ { 0 } ^ { - 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            357,
+            712,
+            378
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Pre-processing for Option 1 only: ",
+        "bbox": [
+            173,
+            388,
+            397,
+            404
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Compute the eigendecompositions of the factors of $\\hat { \\Psi }$ ",
+        "bbox": [
+            217,
+            416,
+            584,
+            433
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4518c13773fe72ca495316b9dde1360bd00ac6b62758b326bd7978dcd33d7d05.jpg",
+        "text": "$$\nU _ { A } \\mathrm { d i a g } ( \\hat { \\psi } _ { A } ) U _ { A } ^ { \\top } = \\hat { \\Psi } _ { A } \\quad \\mathrm { a n d } \\quad U _ { G } \\mathrm { d i a g } ( \\hat { \\psi } _ { G } ) U _ { G } ^ { \\top } = \\hat { \\Psi } _ { G }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            439,
+            720,
+            459
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Sum across the temporal correlations and then invert by performing the element-wise computation in eigenspace: ",
+        "bbox": [
+            209,
+            468,
+            823,
+            497
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e51075c9a12fc76a9bdc580ca8d9abc746d170950d1c0181e82fa74145858ac9.jpg",
+        "text": "$$\n[ Y ] _ { i , j } = \\gamma ( [ \\hat { \\psi } _ { G } ] _ { i } [ \\hat { \\psi } _ { A } ] _ { j } ) \\quad \\mathrm { w h e r e } \\quad \\gamma ( x ) = \\frac { ( 1 - x ) ^ { 2 } } { \\sum _ { i } \\omega _ { i } ( { \\mathcal T } _ { i } ( 1 - x ^ { 2 } ) - 2 x ( 1 - x ^ { { \\mathcal T } _ { i } } ) ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            503,
+            787,
+            540
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Pre-processing for Option 2 only: ",
+        "bbox": [
+            173,
+            547,
+            395,
+            564
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Compute the eigendecompositions of the factors of $\\hat { \\Psi } ^ { \\top } \\hat { \\Psi }$ (or equivalently the SVD’s of $\\hat { \\Psi } _ { A }$ and $\\hat { \\Psi } _ { G }$ ): ",
+        "bbox": [
+            214,
+            575,
+            825,
+            608
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25b2e4a449b194cccd1852be98e3251b8d10bd6f6edb2043c387c3a86ba7c20e.jpg",
+        "text": "$$\nE _ { A } \\mathrm { d i a g } ( m _ { A } ) E _ { A } ^ { \\top } = \\hat { \\Psi } _ { A } ^ { \\top } \\hat { \\Psi } _ { A } \\quad \\mathrm { a n d } \\quad E _ { G } \\mathrm { d i a g } ( m _ { G } ) E _ { G } ^ { \\top } = \\hat { \\Psi } _ { G } ^ { \\top } \\hat { \\Psi } _ { G }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            614,
+            746,
+            633
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "$z$ -dependent calculations for Option 1: ",
+        "bbox": [
+            174,
+            643,
+            428,
+            660
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Multiply the input vector $z = \\operatorname { v e c } ( Z )$ by $V _ { 0 } ^ { - 1 / 2 }$ ",
+        "bbox": [
+            215,
+            671,
+            553,
+            690
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d9ce2f317b9b28f66adf71e6903cc9db3de6725626633b20db5574ffd7e8f9e9.jpg",
+        "text": "$$\nZ _ { 0 } = G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            457,
+            695,
+            599,
+            718
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Multiply by $\\hat { F } ^ { - 1 }$ using its eigendecomposition: ",
+        "bbox": [
+            215,
+            728,
+            547,
+            744
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a133abd1337426755329ac4b99f234f4f9ca7d885dbffaa2aaaffef502674da7.jpg",
+        "text": "$$\nZ _ { 1 } = U _ { G } ( ( U _ { G } ^ { \\top } Z _ { 0 } U _ { A } ) \\odot Y ) U _ { A } ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            751,
+            633,
+            770
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "where $B \\odot C$ denotes the element-wise product between matrices $B$ and $C$ . ",
+        "bbox": [
+            232,
+            776,
+            730,
+            792
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Multiply the result by $V _ { 0 } ^ { - 1 / 2 }$ and express as a vector: ",
+        "bbox": [
+            215,
+            796,
+            583,
+            815
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6542658339bd4c0d2d2d36dc5b687331160deb56bfb1318dca20111a4ff38d5b.jpg",
+        "text": "$$\nZ _ { 2 } = G _ { 0 } ^ { - 1 / 2 } Z _ { 1 } A _ { 0 } ^ { - 1 / 2 } \\quad \\mathrm { a n d } \\quad F ^ { - 1 } z = \\mathrm { v e c } ( Z _ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            820,
+            692,
+            842
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "$z$ -dependent calculations for Option 2: ",
+        "bbox": [
+            174,
+            852,
+            428,
+            867
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Multiply the input vector $z = \\operatorname { v e c } ( Z )$ by $V _ { 0 } ^ { - 1 / 2 }$ : ",
+        "bbox": [
+            215,
+            880,
+            553,
+            898
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/81c91de40e3a5bab9356450bd949df68ded1badb8009ea6d0cd29c15a29d59dd.jpg",
+        "text": "$$\nZ _ { 0 } = G _ { 0 } ^ { - 1 / 2 } Z A _ { 0 } ^ { - 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            457,
+            904,
+            598,
+            925
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "• Multiply the result by $I - { \\hat { \\Psi } } ^ { \\top }$ : ",
+        "bbox": [
+            217,
+            102,
+            437,
+            118
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e9bcdb368feebfca47b2e58cd02a3935f59ff6a3dad6204d43d38d76f6a7358.jpg",
+        "text": "$$\nZ _ { 1 } = Z _ { 0 } - \\hat { \\Psi } _ { G } ^ { \\top } Z _ { 0 } \\hat { \\Psi } _ { A }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            125,
+            602,
+            145
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "• Multiply the previous result by $( I - \\hat { \\Psi } ^ { \\top } \\hat { \\Psi } ) ^ { - 1 }$ using its eigendecomposition: ",
+        "bbox": [
+            215,
+            154,
+            735,
+            171
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c9b09b449868694db51b6cca11d669f5fbe0135bf7dc3aebf17cd1333c58dd63.jpg",
+        "text": "$$\n\\begin{array} { r } { Z _ { 2 } = E _ { G } ( ( E _ { G } ^ { \\top } Z _ { 1 } E _ { A } ) \\oslash ( \\mathbb { I } \\mathbb { I } ^ { \\top } - m _ { G } m _ { A } ^ { \\top } ) ) E _ { A } ^ { \\top } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            178,
+            681,
+            196
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "where $X \\oslash Y$ denotes the element-wise division of $X$ by $Y$ . ",
+        "bbox": [
+            230,
+            202,
+            630,
+            218
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "• Multiply the result by $I - { \\hat { \\Psi } }$ ",
+        "bbox": [
+            217,
+            223,
+            423,
+            239
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "img_path": "images/40b5248150b206a51d62f6ea9e11677aeda0a65d5504e8e726d3569523da3137.jpg",
+        "text": "$$\nZ _ { 3 } = Z _ { 2 } - \\hat { \\Psi } _ { G } Z _ { 2 } \\hat { \\Psi } _ { A } ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            236,
+            602,
+            256
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "• Normalize the result by $\\begin{array} { r } { \\mathbb { E } \\tau [ \\mathcal { T } ] = \\sum _ { i } \\omega _ { i } \\mathcal { T } _ { i } } \\end{array}$ (for $\\mathcal { T } _ { i }$ and $\\omega _ { i }$ as defined at the bottom of Section 3.4): ",
+        "bbox": [
+            215,
+            261,
+            826,
+            289
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c02bcdd9bfca5c71b0348ab1564645729dbf1ac8ff9e883f83755b12e627bb0d.jpg",
+        "text": "$$\nZ _ { 4 } = \\frac { 1 } { \\sum _ { i } \\omega _ { i } \\mathcal { T } _ { i } } Z _ { 3 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            467,
+            286,
+            589,
+            320
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "• Multiply the result by $V _ { 0 } ^ { - 1 / 2 }$ and express as a vector: ",
+        "bbox": [
+            215,
+            327,
+            583,
+            345
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5793a7c23b85b4ea363014cbedc80964cd2277cde42ea3fbf206ff40b0514676.jpg",
+        "text": "$$\nZ _ { 5 } = G _ { 0 } ^ { - 1 / 2 } Z _ { 4 } A _ { 0 } ^ { - 1 / 2 } \\quad \\mathrm { a n d } \\quad F ^ { - 1 } z = \\mathrm { v e c } ( Z _ { 5 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            351,
+            691,
+            372
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "image",
+        "img_path": "images/16150c71374fcf05e47bd613efbbf69dd41fdf0750837d9fd974275f9db499e0.jpg",
+        "image_caption": [
+            "Figure 4: Generalization performance of our method compared to the baselines in perplexity-perword on length-35 word sequences from Penn-TreeBank. All the methods used a mini-batch size of 200. K-FAC indep. uses the update in eqn. 6, K-FAC option1 uses eqn. 8, and K-FAC option2 uses eqn. 9. (left) Validation perplexity v.s. the number of updates. The dashed lines correspond to the experiments that used RNNs with 1024 LSTM units, and the solid lines correspond to experiments that used RNNs with 650 LSTM units. (right) Validation perplexity v.s. the wall-clock time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            103,
+            789,
+            268
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "D TEST PERFORMANCE ON PENN-TREEBANK ",
+        "bbox": [
+            173,
+            392,
+            570,
+            409
+        ],
+        "page_idx": 24
+    }
+]
\ No newline at end of file
diff --git a/parse/train/HyMTkQZAb/HyMTkQZAb_middle.json b/parse/train/HyMTkQZAb/HyMTkQZAb_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..cd3cfc2742b56bf2f741686d54519344eae64513
--- /dev/null
+++ b/parse/train/HyMTkQZAb/HyMTkQZAb_middle.json
@@ -0,0 +1,76443 @@
+{
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+                                    "content": "\\begin{array} { r } { \\zeta _ { \\mathcal { T } } ( x ) \\approx \\operatorname* { l i m } _ { \\mathcal { T } ^ { \\prime }  \\infty } \\frac { \\mathcal { T } } { \\mathcal { T } ^ { \\prime } } \\zeta _ { \\mathcal { T } ^ { \\prime } } ( x ) \\equiv \\kappa ( x ) } \\end{array}",
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+                                    "content": ". We can express the error of this as",
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+                                    "score": 0.93,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "following the experimental setup in Zaremba et al. (2014). The gradients are computed using a",
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+                                    "content": "(whose base value we tune along with the other hyperparamters).",
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+                                    "score": 1.0,
+                                    "content": "an exponential moving average during training. When computing our approximate inverse Fisher,",
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+                                    "content": "factored Tikhonov damping (Martens & Grosse, 2015) was applied to",
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+                                    "score": 0.93,
+                                    "content": "V _ { 0 } = G _ { 0 } \\otimes A _ { 0 }",
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+                                    "content": "Updates are computed using the most recently computed values of these (which are allowed to be",
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+                                    "content": "stale), so there is minimal per-iteration computational overhead compared SGD.",
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+                                    "content": "for a word-level language modeling task on the Penn-TreeBank (PTB) dataset (Marcus et al., 1993)",
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+                                    "score": 1.0,
+                                    "content": "following the experimental setup in Zaremba et al. (2014). The gradients are computed using a",
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+                                            "score": 1.0,
+                                            "content": "Figure 1: Optimization performance of our method compared to the baselines in perplexity-per-",
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+                                            "content": "of 200. K-FAC indep. uses the update in eqn. 6, K-FAC option1 uses eqn. 8, and K-FAC option2",
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+                                            "content": "uses eqn. 9. (left) Training perplexity v.s. the number of updates. Dashed lines denote the training",
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+                                            "content": "curves for RNNs with 1024 LSTM units and solid lines denote the training curves for RNNs with",
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+                                    "content": "tion in Mikolov et al. (2012). We plotted the bits-per-character vs the number of parameter updates",
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+                                    "content": "sizes Kronecker factor, which is too large to be inverted with any reasonable frequency. Given that",
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+                                    "score": 1.0,
+                                    "content": "and so we can store and invert it as such. Meanwhile the large factor associated with the output isn’t",
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+                                    "content": "Appendix D.) Not surprisingly, we found that the 2nd-order methods, including our approach and",
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+                                    "content": "diagonal ones like Adam, tended to overfit more than SGD on these tasks.",
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+                                    "content": "ization (e.g. increased weight decay, drop-out, etc) and/or model modifications, but that exploring",
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+                                    "score": 1.0,
+                                    "content": "this is outside of the scope of this paper.",
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+                                    "content": "Character-level model: To further investigate the optimization performance of our proposed",
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+                                    "score": 1.0,
+                                    "content": "quences on the Penn-TreeBank (PTB) dataset (Marcus et al., 1993). We employ the same data parti-",
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+                                    "score": 1.0,
+                                    "content": "tion in Mikolov et al. (2012). We plotted the bits-per-character vs the number of parameter updates",
+                                    "type": "text"
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+                        },
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+                                    "content": "\\begin{array} { r c l } { \\hat { w } _ { t } } & { = } & { V _ { 0 } ^ { - 1 / 2 } w _ { t } } \\\\ & { = } & { V _ { 0 } ^ { - 1 / 2 } ( \\Psi w _ { t - 1 } + \\epsilon _ { t } ) } \\\\ & { = } & { V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 } w _ { t - 1 } + V _ { 0 } ^ { - 1 / 2 } \\epsilon _ { t } } \\\\ & { = } & { ( V _ { 0 } ^ { - 1 / 2 } V _ { 1 } V _ { 0 } ^ { - 1 / 2 } ) ( V _ { 0 } ^ { - 1 / 2 } w _ { t - 1 } ) + \\hat { \\epsilon } _ { t } } \\\\ & { = } & { \\hat { V } _ { 1 } \\hat { w } _ { t - 1 } + \\hat { \\epsilon } _ { t } . } \\end{array}",
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diff --git a/parse/train/JrsfBJtDFdI/JrsfBJtDFdI.md b/parse/train/JrsfBJtDFdI/JrsfBJtDFdI.md
new file mode 100644
index 0000000000000000000000000000000000000000..61643de4345cac140a765c7aa961b89dc1361e56
--- /dev/null
+++ b/parse/train/JrsfBJtDFdI/JrsfBJtDFdI.md
@@ -0,0 +1,251 @@
+# What Matters in Learning from Offline Human Demonstrations for Robot Manipulation
+
+Ajay Mandlekar1, Danfei $\mathbf { X } \mathbf { u } ^ { 1 }$ , Josiah $\mathbf { W o n g } ^ { 1 }$ , Soroush Nasiriany2, Chen Wang1,
+
+Rohun Kulkarni1, Li Fei-Fei1, Silvio Savarese1, Yuke $\mathbf { Z } \mathbf { h } \mathbf { u } ^ { 2 }$ , Roberto Martín-Martín
+
+1Stanford University, 2The University of Texas at Austin
+
+Abstract: Imitating human demonstrations is a promising approach to endow robots with various manipulation capabilities. While recent advances have been made in imitation learning and batch (offline) reinforcement learning, a lack of open-source human datasets and reproducible learning methods make assessing the state of the field difficult. In this paper, we conduct an extensive study of six offline learning algorithms for robot manipulation on five simulated and three real-world multi-stage manipulation tasks of varying complexity, and with datasets of varying quality. Our study analyzes the most critical challenges when learning from offline human data for manipulation. Based on the study, we derive a series of lessons including the sensitivity to different algorithmic design choices, the dependence on the quality of the demonstrations, and the variability based on the stopping criteria due to the different objectives in training and evaluation. We also highlight opportunities for learning from human datasets, such as the ability to learn proficient policies on challenging, multi-stage tasks beyond the scope of current reinforcement learning methods, and the ability to easily scale to natural, real-world manipulation scenarios where only raw sensory signals are available. We have open-sourced our datasets and all algorithm implementations to facilitate future research and fair comparisons in learning from human demonstration data at https://arise-initiative.github.io/robomimic-web/
+
+Keywords: Imitation Learning, Offline Reinforcement Learning, Manipulation
+
+# 1 Introduction
+
+Human supervision has been at the heart of the most significant recent advances in several domains such as computer vision [1–4] and natural language processing [5–7]. By intelligently extracting information from large-scale human-labeled datasets, autonomous machines have been able to reach near- or even super-human performance on decades-old problems such as image recognition and question answering. Roboticists have also attempted to tackle robot manipulation through learning from human datasets, using the paradigms of Imitation Learning [8–10] and Batch (Offline) Reinforcement Learning [11–13], where datasets consisting of robot arm trajectories, action labels at each timestep, and possibly reward labels, are used to train closed-loop policies.
+
+As in other domains, large offline datasets offer several benefits such as scale, portability, and reproducible evaluations to measure progress. Recently, there has been considerable progress in offline learning for robot manipulation from human demonstrations [9, 14, 10]. Despite these advances, the offline learning paradigm has not been nearly as disruptive in robotics as in other disciplines – there is a large gap between autonomous robot manipulation capabilities and the wide range of tasks that humans can solve effortlessly using physical and cognitive intelligence. What has inhibited the use of large human-provided datasets to address this gap?
+
+In contrast to other domains where supervised learning has been successful, robotic manipulation is a time-evolving dynamical system, requiring fine-grained real-time control to guide robot arms successfully through tasks – consequently, data collection can present technical challenges requiring specialized systems [15], which can explain why large-scale human-provided datasets [16, 17] have not been very prevalent. Learning from such datasets can also present several challenges. Human demonstrations can differ from machine-generated datasets (a recent trend in benchmarks for offline policy learning [18, 19]) due to a non-Markovian decision process, since humans may not act purely based on the current observation. There can also be significant variance in both data quality and solution strategy when collecting data from multiple humans [20]. Differences from classic supervised learning, such as a mismatch between training and evaluation objectives (task success rate), can make selecting a final policy challenging [21, 22], especially in real-world settings where evaluating each policy on a robot can be infeasible. Finally, offline learning is sensitive to state and action space coverage (dataset size) and agent design decisions.
+
+![](images/a97aa5e831187c50c760167fe83e2a45178cff321359a918d96c1a01d1af822e.jpg)  
+Figure 1: Tasks. We collect datasets across 6 operators of varying proficiency and evaluate offline policy learning methods on 8 challenging manipulation tasks that test a wide range of manipulation capabilities including pick-and-place, multi-arm coordination, and high-precision insertion and assembly.
+
+Studying these challenges in the context of robot manipulation and human-provided datasets could be a stepping stone to closing the gap between robot and human manipulation capabilities. Unfortunately, a lack of suitable benchmark and human datasets have made studying this setting difficult. Prior works are either limited to studying simple 2D environments [23] or using data generated from hard-coded policies [24, 25]. In this paper, we address this need by presenting a study of data-driven offline policy learning methods on several human-provided robot manipulation datasets. We collect task demonstrations from human teleoperators across a broad range of simulated and real world manipulation tasks and investigate several factors that play a role in learning from such data.
+
+From our results, we point out several lessons to guide future research in leveraging human supervision for robot manipulation effectively. We find that history-dependent models can be extremely effective in learning from single and multi-human datasets while state-of-the-art batch RL algorithms struggle to learn from such datasets, and that the choice of observation space and hyperparameters play a substantial role in training proficient policies. We also find that there is substantial promise for solving more complex tasks using large-scale human datasets and that our insights directly transfer to real-world scenarios, making this an important setting to explore further.
+
+# 2 Challenges in Offline Learning from Human Datasets
+
+In this section, we outline five challenges in offline learning from human datasets that motivate different factors that we investigate in our study.
+
+(C1) Data from Non-Markovian Decision Process. Human demonstrations can differ substantially from machine-generated demonstrations because humans may not act purely based on a single current observation. External factors (teleoperation device, past actions, history of episode) may all play a role. Prior work [20] has noted substantial benefits from leveraging models that are history-dependent and / or with temporal abstraction to learn from human demonstrations. We investigate various design choices related to such architectures in this study.
+
+Table 1: Results on Low-Dimensional Observations. We present success rates averaged over 3 seeds for each method across the low-dim Machine-Generated (MG), Proficient-Human (PH), and Multi-Human (MH) datasets. The results show that methods that model temporal correlations (BC-RNN, HBC, IRIS) exhibit strong performance on human datasets. Furthermore, while Batch RL algorithms like BCQ are proficient on machine-generated data, they perform poorly on human datasets.   
+
+<table><tr><td>Dataset</td><td>BC</td><td>BC-RNN</td><td>BCQ</td><td>CQL</td><td>HBC</td><td>IRIS</td></tr><tr><td>Lift (MG)</td><td>65.3 ±2.5</td><td>70.7±3.4</td><td>91.3±1.9</td><td>64.0±2.8</td><td>47.3±4.1</td><td>96.0±1.6</td></tr><tr><td>Can (MG)</td><td>64.7±3.4</td><td>68.7±2.5</td><td>75.3±0.9</td><td>1.3±0.9</td><td>40.7±3.4</td><td>48.0±6.5</td></tr><tr><td>Lift (PH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>100.0±0.0</td><td>92.7±5.0</td><td>100.0±0.0</td><td>100.0±0.0</td></tr><tr><td>Can (PH)</td><td>95.3±0.9</td><td>100.0±0.0</td><td>88.7±0.9</td><td>38.0±7.5</td><td>100.0±0.0</td><td>100.0±0.0</td></tr><tr><td>Square (PH)</td><td>78.7±1.9</td><td>84.0±0.0</td><td>50.0±4.9</td><td>5.3±2.5</td><td>82.6±0.9</td><td>78.7±2.5</td></tr><tr><td>Transport (PH)</td><td>17.3±2.5</td><td>71.3±6.6</td><td>7.3±3.3</td><td>0.0±0.0</td><td>48.6±3.8</td><td>41.3 ±3.4</td></tr><tr><td>Tool Hang (PH)</td><td>29.3±0.9</td><td>19.3±5.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>30.0±7.1</td><td>11.3±2.5</td></tr><tr><td>Lift (MH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>100.0±0.0</td><td>56.7±40.3</td><td>100.0±0.0</td><td>100.0±0.0</td></tr><tr><td>Can (MH)</td><td>86.0±4.3</td><td>100.0±0.0</td><td>62.7±8.2</td><td>22.0±5.7</td><td>91.3±2.5</td><td>92.7±0.9</td></tr><tr><td>Square (MH)</td><td>52.7±6.6</td><td>78.0±4.3</td><td>14.0±4.3</td><td>0.7±0.9</td><td>60.7±5.0</td><td>52.7±5.0</td></tr><tr><td>Transport (MH)</td><td>11.3 ± 2.5</td><td>65.3±7.4</td><td>2.6±0.9</td><td>0.0±0.0</td><td>14.0±1.6</td><td>10.7±0.9</td></tr></table>
+
+(C2) Variance in Demonstration Quality from Multiple Humans. Prior work [20, 17] has found that data collected from several humans can differ substantially in both demonstration proficiency and solution strategy. Differences in supervisor proficiency can manifest in many ways, such as large variations in trajectory length and noise in robot movement or mistakes (e.g. missed grasps). In our study, we evaluate offline policy learning algorithms on such datasets. While recent batch RL algorithms have shown an excellent ability to learn from mixed quality machine-generated datasets [26, 27], we empirically find that they fail to learn well from mixed quality human data.
+
+(C3) Dependence on Dataset Size. Offline policy learning is sensitive to the state and action space coverage in the dataset, and by extension, the size of the dataset itself. In our study, we investigate how dataset sizes affect policy performance. This analysis is useful to understand the value of adding more data – an important consideration since collecting human demonstrations can be costly.
+
+(C4) Mismatch between Training and Evaluation Objectives. Unlike traditional supervised learning, where model selection can be achieved by using the model with the lowest validation loss [21], offline policy learning often suffers from the fact that the training objective is only a surrogate for the true objective of interest (e.g. task success rate), and policy performance can change significantly from epoch to epoch. This makes it difficult to select the best trained model [19, 28, 29]. In our study, we evaluate each policy checkpoint online in the environment in simulation, and report the best policy success rate per training run. We use these ground-truth values to understand the effectiveness of different selection criteria, and confirm that offline policy selection is an important problem, especially in real-world scenarios where large-scale empirical evaluation is difficult.
+
+(C5) High Sensitivity to Agent Design Decisions. Prior studies on machine-generated datasets have shown that offline policy learning can be extremely sensitive to hyperparameter choices [19, 28]. In our study, we explore how agent design decisions affect policy performances, including the choice of agent architecture, agent observation space, and hyperparameter choices per algorithm. This results in several practical conclusions that should prove useful to researchers and practitioners alike. We further show that important design decisions made through our study in simulation directly translate to effective policy learning on real world tasks and datasets.
+
+# 3 Study Design
+
+# 3.1 Tasks
+
+We conducted our study across 5 simulated and 3 real world tasks. The tasks were chosen to test a broad range of manipulation capabilities. See Fig 1 and Appendix E for more details.
+
+Lift $\mathbf { \ s } \mathbf { i } \mathbf { m } + \mathbf { r e } \mathbf { a } \mathbf { l } )$ ). The robot arm must lift a small cube. This is the simplest task.
+
+Can $\mathbf { \ s } \mathbf { i } \mathbf { m } + \mathbf { r e } \mathbf { a } \mathbf { l } )$ . The robot must place a coke can from a large bin into a smaller target bin. Slightly more challenging than Lift, since picking the can is harder than picking the cube, and the can must also be placed into the bin.
+
+Square (sim). The robot must pick a square nut and place it on a rod. Substantially more difficult than Lift and Pick Place Can due to the precision needed to pick up the nut and insert it on the rod.
+
+<table><tr><td>Dataset</td><td>BC</td><td>BC-RNN</td><td>BCQ</td><td>CQL</td><td>HBC</td><td>IRIS</td></tr><tr><td>Can-Worse</td><td>56.7±2.5</td><td>92.0±1.6</td><td>29.3 ±10.9</td><td>4.0±3.3</td><td>78.7±3.4</td><td>77.3±1.9</td></tr><tr><td>Can-Okay</td><td>72.0±2.8</td><td>95.3±1.9</td><td>58.0±8.6</td><td>22.0±4.3</td><td>97.3± 0.9</td><td>96.0±0.0</td></tr><tr><td>Can-Better</td><td>83.3±2.5</td><td>99.3± 0.9</td><td>62.0±5.9</td><td>20.7±7.4</td><td>96.7±0.9</td><td>96.0±0.0</td></tr><tr><td>Can-Worse-Okay</td><td>74.7±5.7</td><td>98.7±1.9</td><td>50.7±3.8</td><td>18.7±2.5</td><td>88.0±1.6</td><td>87.3±1.9</td></tr><tr><td>Can-Worse-Better</td><td>76.0±4.3</td><td>100.0±0.0</td><td>48.0±4.9</td><td>20.7±5.7</td><td>90.0±1.6</td><td>91.3±2.5</td></tr><tr><td>Can-Okay-Better</td><td>90.7±1.9</td><td>100.0±0.0</td><td>68.7±2.5</td><td>30.7±7.7</td><td>99.3± 0.9</td><td>98.0±1.6</td></tr><tr><td>Square-Worse</td><td>22.0±4.3</td><td>39.3±3.8</td><td>5.3±1.9</td><td>0.0±0.0</td><td>44.7±6.8</td><td>38.7±0.9</td></tr><tr><td>Square-Okay</td><td>27.3±3.4</td><td>45.3 ±2.5</td><td>6.7±1.9</td><td>0.0±0.0</td><td>52.0±2.8</td><td>42.0±3.3</td></tr><tr><td>Square-Better</td><td>58.7±2.5</td><td>66.0±2.8</td><td>32.0±4.3</td><td>0.7±0.9</td><td>61.3±1.9</td><td>60.0±1.6</td></tr><tr><td>Square-Worse-Okay</td><td>28.7±2.5</td><td>55.3±0.9</td><td>8.7±1.9</td><td>2.7±1.9</td><td>50.7±4.1</td><td>43.3±2.5</td></tr><tr><td>Square-Worse-Better</td><td>46.7±5.7</td><td>73.3±6.2</td><td>15.3± 2.5</td><td>1.3±0.9</td><td>65.3±3.4</td><td>56.7±3.4</td></tr><tr><td>Square-Okay-Better</td><td>56.7±4.1</td><td>74.0±2.8</td><td>22.0±4.3</td><td>1.3±0.9</td><td>63.3±4.1</td><td>56.7±3.8</td></tr><tr><td>Can-Paired</td><td>64.0±9.1</td><td>70.0±4.3</td><td>44.7±1.9</td><td>6.0±1.6</td><td>70.7±5.2</td><td>75.3±1.9</td></tr></table>
+
+Table 2: Results on Suboptimal Human Data. We present success rates averaged over 3 seeds for each method across different subsets of the Multi-Human datasets, corresponding to mixtures of demonstrations from “Better”, “Adequate”, and “Worse” human operators, and finally on a diagnostic dataset with paired success and failure human trajectories for each starting initialization. Results indicate that BC-RNN is a strong baseline, and that Batch RL methods perform poorly across all datasets, even on the simple diagnostic dataset.
+
+Transport (sim). Two robot arms must transfer a hammer from a closed container on a shelf to a target bin on another shelf. One robot arm must retrieve the hammer from the container, while the other arm must clear the target bin by moving a piece of trash to the nearby receptacle. Finally, one arm must hand the hammer over to the other, which must place the hammer in the target bin.
+
+Tool Hang $( \sin + \mathrm { r e a l } )$ ). A robot arm must assemble a frame consisting of a base piece and hook piece by inserting the hook into the base, and hang a wrench on the hook. This is the most difficult task due to the multiple stages that each require precise, and dexterous, rotation-heavy movements.
+
+# 3.2 Data Collection
+
+To study the effect of dataset source, we collected data from three different sources – MachineGenerated, Proficient-Human, and Multi-Human (more details in Appendix B).
+
+Machine-Generated (MG). We collected these datasets by first training a state-of-the-art RL algorithm [30] on the Lift and Can task, taking agent checkpoints that are saved regularly during training, and collecting 300 rollout trajectories from each checkpoint. Consequently, these datasets are comprised of mixtures of expert and suboptimal data, and resemble datasets from common offline RL benchmarks [18, 19]. We excluded other tasks because they could not be solved by the RL algorithm even with substantial tuning. See the appendix for more details.
+
+Proficient-Human (PH) and Multi-Human (MH). Datasets are collected by humans through RoboTurk [15, 17], a remote teleoperation platform. The PH datasets consist of 200 demonstrations collected by a single, experienced teleoperator, while the MH datasets consist of 300 demonstrations, collected by 6 teleoperators of varying proficiency, each of which provided 50 demonstrations. The 6 teleoperators consisted of a “better” group of 2 experienced operators, an “okay” group of 2 adequate operators, and a “worse” group of 2 inexperienced operators. These data subsets in the Multi-Human data allowed us to investigate the ability of algorithms to deal with mixed quality human data.
+
+Observation Modalities. To study the effect of observation modalities, we capture a diverse set of sensor streams when collecting the dataset, including end-effector, gripper fingers, and joints, groundtruth object poses, and images from an external camera and wrist-mounted camera per robot arm (see Appendix E). We have two observation spaces – “low-dim” and “image”. Both include end-effector poses and gripper finger positions, and only differ in whether ground-truth object information is used (low-dim) or whether that information is replaced by the available camera observations (image).
+
+# 3.3 Training and Evaluation Protocols
+
+There are several approaches to offline imitation learning [31–34, 9, 10, 25, 35, 36] and offline reinforcement learning [26, 27, 37–43] (see Appendix A for more discussion on related work). We chose to evaluate 6 algorithms in this study – Behavioral Cloning (BC), BC with an RNN policy (BC-RNN), Hierarchical Behavioral Cloning (HBC) [10], Batch-Constrained Q-Learning (BCQ) [26], Conservative Q-Learning (CQL) [27], and IRIS [20]. BC-RNN, HBC, and IRIS have all been used in prior work to learn offline from teleoperated human demonstrations, while BCQ and CQL are commonly-used offline RL algorithms (see Appendix C). We use binary task completion rewards for all our experiments. Each agent is trained for $N$ epochs, where each epoch consists of $M$ gradient steps, and evaluated every $E$ epochs, by running 50 rollouts in the environment and reporting the success rate over a maximum horizon. For each agent, we report the maximum success rate over the coarse of training, and average over 3 seeds. For low-dim agents, $N = 2 0 0 0$ , $M = 1 0 0$ , and $E = 5 0$ and for image agents, $N = 6 0 0$ , $M = 5 0 0$ , and $E = 2 0$ (see Appendix B.2).
+
+![](images/1708792fdb110bc98cd5d0650c4c89add2bdc9f0187ca36dae6a23079cb20e3f.jpg)  
+Figure 2: Effect of Observation Space and Hyperparameter Choice. We show how the success rate that BC-RNN obtains can drop drastically due to changes to the observation space and hyperparameter settings.
+
+![](images/82b04fc1c61417f99d9329cd1130e9d29aca7a4583ed31afd6f35cd001eea4a5.jpg)  
+Figure 3: Effect of Dataset Size. We study how the BC-RNN success rate changes when lowering the quantity of data to $20 \%$ and $50 \%$ . Results show that less complex tasks (Lift, Can) be learned with a fraction of the data, while more complex tasks might benefit from even larger human datasets.
+
+# 4 Experiments
+
+In this section, we present each factor that we explored in our study, and note the relevant challenges from Sec. 2 that each pertains to.
+
+# 4.1 Algorithm Comparison on Single and Multi-Human Demonstrations (C1, C2)
+
+We trained and evaluated all algorithms on the Proficient-Human (PH) and Multi-Human (MH) datasets and report the average success rates across 3 seeds in Table 1.
+
+Observation history is crucial for good performance. There is a substantial performance gap between BC-RNN and BC, which highlights the benefits of history-dependence. The performance gap is larger for longer-horizon tasks (e.g. $\sim 5 5 \%$ for Transport (PH) compared to $\sim 5 \%$ for Square (PH)) and for multi-human data compared to single-human data (e.g. $\sim 2 5 \%$ for Square (MH) compared to $\sim 5 \%$ for Square (PH)). Interestingly, results are lower for MH datasets compared to PH datasets, even though the MH datasets contain 100 more demos (300 demos vs. 200 demos). This most likely stems from the presence of suboptimal and multimodal data in the MH datasets.
+
+Batch RL algorithms perform poorly on Human Datasets. Recent batch (offline) RL algorithms such as BCQ and CQL have demonstrated excellent results in learning from suboptimal and multimodal agent-generated datasets. Our results confirm the capacity of such algorithms to work well – BCQ in particular performs strongly on our agent-generated MG datasets that consist of a diverse mixture of good and poor policies. Surprisingly though, neither BCQ nor CQL performs particularly well on these human-generated datasets. This puts the ability of such algorithms to learn from more
+
+![](images/140d11bc2f4e30457b0a83d0ae53d710a9f13fa72e757ac1e7503c8582e0cd21.jpg)
+
+Table 3: Results on Image Observations.   
+
+<table><tr><td>Dataset</td><td>BC</td><td>BC-RNN</td><td>BCQ</td><td>CQL</td></tr><tr><td>Lift (PH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>98.0±1.6</td><td>52.0±13.0</td></tr><tr><td>Can (PH)</td><td>97.3±1.9</td><td>98.0±0.9</td><td>86.7±2.5</td><td>0.7±0.9</td></tr><tr><td>Square (PH)</td><td>62.0±4.9</td><td>82.0±0.0</td><td>41.3±4.1</td><td></td></tr><tr><td>Transport (PH)</td><td>55.3±6.2</td><td>72.0±4.3</td><td>0.7±0.9</td><td></td></tr><tr><td>Tool Hang (PH)</td><td>20.0±5.9</td><td>67.3±4.1</td><td>3.3±0.9</td><td>=</td></tr><tr><td>Lift (MH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>93.3±0.9</td><td>11.3 ±9.3</td></tr><tr><td>Can (MH)</td><td>85.3±0.9</td><td>96.0±1.6</td><td>77.3±6.8</td><td>0.0±0.0</td></tr><tr><td>Square (MH)</td><td>46.0±1.6</td><td>76.7±3.4</td><td>17.3±7.5</td><td></td></tr><tr><td>Transport (MH)</td><td>18.7±2.5</td><td>42.0±1.6</td><td>0.0±0.0</td><td>=</td></tr></table>
+
+Figure 5: (left) Effect of Policy Selection Criteria. We compare how performance decreases when choosing the policy to evaluate by using the lowest validation loss, or when using the final trained checkpoint, with respect to the best policy performance. (right) Results on Image Observations. We present success rates for each method across the image observation human datasets. BC-RNN maintains nearly the same performance as learning from ground-truth observations, providing an optimistic view for learning with real-world raw sensory observations. natural dataset distributions into question (instead of those collected via RL exploration or pre-trained agents). There is an opportunity for future work in batch RL to resolve this gap.
+
+# 4.2 Learning from Suboptimal Human Data (C2)
+
+To further investigate how algorithms deal with suboptimal human data, we split our MH datasets into smaller subsets based on the proficiency of the human operators. The MH-Better, MH-Okay, and MH-Worse are the 100 demo subsets corresponding to the 2 “better”, 2 “okay”, and 2 “worse” operators respectively, while MH Worse-Okay, MH Worse-Better, and MH Okay-Better are the 200 demo subsets corresponding to the mixture of the previous subsets. Similar data mixtures have been used for evaluations in batch RL [18]. Appendix B shows the average trajectory lengths in each data subset – lower quality datasets contain demonstrations that take more time to solve the task.
+
+BC-RNN is a strong baseline on suboptimal human data, but there is room for improvement. Table 2 shows that BC exhibits a large performance gap between the Better and Worse 100-demo subsets (roughly $27 \%$ and $3 5 \%$ for Can and Square respectively). Interestingly, BC-RNN is able to nearly eliminate this gap in performance on the Can task, but not on the Square task. However, BC-RNN outperforms BC on all datasets $7 \% - 3 5 \%$ improvement). Comparing results on the 100 Better demonstrations and 100 Okay demonstrations to the 200 Worse-Better demonstrations and 200 Worse-Okay demonstrations further allows us to analyze how adding 100 “worse” demonstrations impacts the performance of each algorithm. Most algorithms decline in performance while BC-RNN is able to uniformly improve from the added data. Comparing the performance of BC-RNN on the 200-demo Square mixture datasets $( 5 5 . 3 \%$ , $7 3 . 3 \%$ , $7 4 . 0 \%$ ) to the high-quality 200-demo Square (PH) dataset $( 8 4 . 0 \% )$ shows that there is still room for algorithms to improve on the use of this data.
+
+Diagnostic dataset shows that Batch RL struggles in simpler settings as well. The final row of Table 2 shows additional results on a diagnostic dataset termed Can-Paired, where a single operator collected 2 demonstrations for each of 100 task initializations – one successful demonstration, and one where the can is tossed outside of the bin (task failure), for a total of 200 demonstrations. There is a strong expectation for batch RL algorithms to be able to distinguish between actions leading to successful placement and actions leading to task failure, but even in this simple setting, most algorithms suffer, providing a pessimistic view of the state-of-the-art. The $5 \%$ improvement that IRIS provides over BC-RNN suggests that introducing history-dependence into state-of-the-art batch RL algorithms might be a promising direction for future work.
+
+# 4.3 Effect of Observation Space (C5)
+
+Learning from image observations can match low-dim agent performance. In Table 3, we present policy learning results when using image observations instead of ground-truth object locations – an important setting for real-world policy learning. BC-RNN still maintains superior performance improvements over BC on the complex Square and Transport tasks, and with the exception of Transport (MH), maintains nearly the same performance as learning from ground-truth observations. This result provides an optimistic view for learning with real-world raw sensory observations.
+
+Features used for robot proprioception can matter. In Fig 2a, we study the effect of adding end effector velocities to the observations $^ +$ EEF Vel), and joint positions and velocities to the observations $\left( + \operatorname { J o i n t } \right)$ . Surprisingly, we find that including end effector velocity information, and joint information hurts agents trained on low-dim observations substantially $( 4 9 \% - 8 8 \%$ relative performance drop), while image-based agents are more tolerant to the inclusion of this extra information $2 \% - 2 9 \%$ relative performance drop). We hypothesize that performance drops might be due to overfitting to the presence of this extra information not needed for solving these tasks. Thus, practitioners should take care to engineer the robot observation space and exclude possibly irrelevant information – information-hiding can be a powerful paradigm for training proficient robots [44].
+
+Image randomization and wrist observations can be crucial for manipulation tasks. In Fig 2a, we report performance drops from removing pixel shift image randomization (- Rand) and the wrist camera (- Wrist) from image-based agents to understand their importance. We see that removing randomization results in $47 \%$ and $3 5 \%$ relative performance drops on Square and Transport respectively, and removing wrist images results in $9 \%$ and $43 \%$ relative drops. Consequently, both wrist camera images and image randomization play a substantial role in producing performant policies. We confirm the importance of each for visuomotor imitation in the real world as well (see Sec 4.7). Wrist observations likely help the robot improve gripper alignment for grasping and randomization helps the policy develop invariance for portions of the image that are not important for action prediction.
+
+# 4.4 Effect of Hyperparameter Choice (C5)
+
+In this section, we take our default hyperparameters for BC-RNN and study the effect of changing a subset of them to report practical recommendations for learning from human datasets (see Appendix I for BCQ and CQL). We present our results in Fig 2b (low-dim) and Fig 2c (image).
+
+(larger LR) Increasing the learning rate from 1e-4 to 1e-3 affects the performance of image-agents substantially (drop of $3 5 \% - 6 3 \%$ ), while low-dim agents are more tolerant to the change. (no GMM) Using a deterministic policy instead of learning a GMM action distribution results in significant relative performance drops on the MH datasets (especially low-dim Transport, with a drop of $58 \%$ ). (larger MLP) Using a larger MLP size at each RNN timestep reduces performance uniformly, suggesting that it is possible to overfit to dataset actions if network architectures are too large. (shallow Conv) Using a shallow convolutional network [45] instead of the ResNet backbone [46] for encoding image observations reduces performance significantly – with relative drops of $2 5 \% { - } 6 2 \%$ , suggesting that large-capacity visual encoders are crucial for visuomotor imitation. (smaller RNN dim) Reducing the size of the RNN hidden dimension from 400 to 100 (low-dim) and 1000 to 400 (image) uniformly decreases performance (drops of $3 \% . 5 8 \%$ ), showing the importance of a large RNN hidden dimension. (Recommendations) We recommend tuning the LR (especially for image agents) and network structure (MLP size, size of RNN dim) carefully. Opting to use a GMM policy and a ResNet encoder appears to be uniformly better.
+
+# 4.5 Selecting a Policy to Evaluate (C4)
+
+Model selection in offline policy learning can be challenging – for this reason, in our simulation experiments, we evaluated every policy checkpoint online and reported the best one. This is not feasible for real-world settings, making offline policy selection desirable. In Fig 4a, we show that this can be non-trivial, by showing the relative performance drop when selecting the policy using the best loss on validation data (common in supervised learning), and when using the final training checkpoint as well (common in offline RL [47, 18, 27]) – in both cases, the selected policy is significantly worse than the best one ( $10 \%$ to $100 \%$ decrease). See Appendix G for more detailed results and discussion. This motivates the need for better offline evaluation metrics.
+
+# 4.6 Effect of Dataset Size (C3)
+
+To study how dataset size impacts performance, we formed smaller $20 \%$ and $50 \%$ subsets of our human datasets by sampling trajectories. We evaluate low-dim and image BC-RNN agents across these subsets in Table 27 and Table 28. There are several promising results here. We first note that less complex tasks (Lift, Can) can yield proficient policies $7 5 \% - 1 0 0 \%$ success rate) using a small fraction of the data $( 2 0 \% )$ . Second, while policies trained on more complex tasks (Square, Transport) suffer substantially when using $50 \%$ or $20 \%$ of the data, the converse is also true – adding more data (e.g. moving from $20 \%$ to $50 \%$ or $50 \%$ to $100 \%$ size) can result in significant policy improvement. This confirms the value of using large human datasets as a means to obtain proficient policies for challenging and complex manipulation tasks.
+
+# 4.7 Applicability to Real-World Settings
+
+Here, we show that design decisions made in simulation can potentially transfer to real world settings. We collected 3 additional real-world datasets with a Franka robotic arm – Lift (Real), Can (Real), and Tool Hang (Real). Each consists of 200 trajectories collected by one operator. We train BC-RNN and report the final policy checkpoint success rate, over 30 rollouts, due to the time-consuming nature of real world policy evaluation. We also emphasize that no real-world hyperparameter tuning took place, so our results are a lower bound. We were able to train proficient Lift $( 9 6 . 7 \% )$ and Can $( 7 3 . 3 \% )$ policies, and the Tool Hang $( 3 . 3 \% )$ policy is able to generate some task successes, despite the extremely difficult nature of the task. Furthermore, as in Sec. 4.3, we validate the importance of pixel shift randomization and the wrist camera by ablating each component on the Can task, and show that including both is the difference between a proficient and non-proficient real-world policy – Can (- Rand) $( 2 6 . 7 \% )$ , Can (- Wrist) $( 4 3 . 3 \% )$ .
+
+# 5 Discussion
+
+In this section, we summarize the lessons from our study and make recommendations for future work.
+
+(L1) Models with temporal abstraction can be extremely effective in learning from human datasets. In $\mathrm { S e c 4 . 1 }$ and Sec 4.2, we demonstrated that history-dependent models (BC-RNN, HBC, and IRIS) are particularly effective in learning from human datasets compared to algorithms that do not take temporal context into account.
+
+(L2) Need to improve the ability of batch (offline) RL to learn from suboptimal human datasets. $\mathrm { S e c } ~ 4 . 2$ and Appendix I demonstrated that state-of-the-art batch RL algorithms are excellent at learning from suboptimal machine-generated datasets but much worse at learning from suboptimal human datasets. They even struggled with a diagnostic dataset with paired good and bad human demonstration trajectories while IRIS was able to improve slightly on BC-RNN, suggesting that combining history-dependence with value learning might be a good place to start for improving batch RL methods [48–51]. This also demonstrates a need to start benchmarking new batch RL algorithms on human datasets instead of purely on machine-generated datasets.
+
+(L3) Improving offline policy selection is important for real world settings. Sec 4.5 demonstrated the need for better ways to select an evaluation policy in an offline manner. We hope that our datasets can help supplement other efforts [29].
+
+(L4) Observation space plays a large role and hyperparameters matter. Sec 4.3 demonstrates that policies trained on low-dim observations can be very sensitive to the choice of robot proprioception, while pixel shift randomization and wrist camera images are critical for effective visuomotor policy learning. The choice of observation space for imitation merits careful consideration – other work has also confirmed the importance of feature representations used for offline policy learning [52, 53]. Sec 4.4 and Appendix I made practical recommendations for choosing hyperparameters to learn from human data.
+
+(L5) There is substantial promise for solving more complex tasks using large-scale human datasets. Sec 4.6 showed that adding more data can result in significant policy improvement on complex tasks. Table 3 and Sec 4.7 shows that we could learn proficient policies on the Tool Hang task, our most complex task, without any hyperparameter tuning on the task or dataset. Together, these results show the potential of large human datasets as a means to solve challenging and complex manipulation tasks.
+
+(L6) Study results transfer to real-world settings. In Sec 4.7, we showed that we could directly apply hyperparameters that were tuned on simulated tasks directly to real-world datasets and tasks. This provides promise for using our tasks, datasets, and codebase to enable reproducible evaluation in simulation, while also being confident that conclusions can transfer to real-world settings.
+
+Going forward, we hope that the datasets, tasks, code, and subsequent insights of our study will serve researchers and practitioners alike.
+
+# Acknowledgments
+
+We would like to thank Albert Tung for helping with the RoboTurk data collection system, Jim Fan for providing timely lab cluster support, and Helen Roman for helping order items for the physical robot tasks. Ajay Mandlekar acknowledges the support of the Department of Defense (DoD) through the NDSEG program. We acknowledge the support of Toyota Research Institute (“TRI”); this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity. We acknowledge the support of the US Army Research Office (award W911NF-15-1-0479) and the National Science Foundation (award CNS-1955523). This work relates to Department of Navy award N00014-14-1-0671 issued by the Office of Naval Research.
+
+# References
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diff --git a/parse/train/JrsfBJtDFdI/JrsfBJtDFdI_content_list.json b/parse/train/JrsfBJtDFdI/JrsfBJtDFdI_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..19db3620f295e50294e4241de4a7906b99341cc5
--- /dev/null
+++ b/parse/train/JrsfBJtDFdI/JrsfBJtDFdI_content_list.json
@@ -0,0 +1,993 @@
+[
+    {
+        "type": "text",
+        "text": "What Matters in Learning from Offline Human Demonstrations for Robot Manipulation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ajay Mandlekar1, Danfei $\\mathbf { X } \\mathbf { u } ^ { 1 }$ , Josiah $\\mathbf { W o n g } ^ { 1 }$ , Soroush Nasiriany2, Chen Wang1, ",
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+    },
+    {
+        "type": "text",
+        "text": "Rohun Kulkarni1, Li Fei-Fei1, Silvio Savarese1, Yuke $\\mathbf { Z } \\mathbf { h } \\mathbf { u } ^ { 2 }$ , Roberto Martín-Martín ",
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+        "type": "text",
+        "text": "1Stanford University, 2The University of Texas at Austin ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Abstract: Imitating human demonstrations is a promising approach to endow robots with various manipulation capabilities. While recent advances have been made in imitation learning and batch (offline) reinforcement learning, a lack of open-source human datasets and reproducible learning methods make assessing the state of the field difficult. In this paper, we conduct an extensive study of six offline learning algorithms for robot manipulation on five simulated and three real-world multi-stage manipulation tasks of varying complexity, and with datasets of varying quality. Our study analyzes the most critical challenges when learning from offline human data for manipulation. Based on the study, we derive a series of lessons including the sensitivity to different algorithmic design choices, the dependence on the quality of the demonstrations, and the variability based on the stopping criteria due to the different objectives in training and evaluation. We also highlight opportunities for learning from human datasets, such as the ability to learn proficient policies on challenging, multi-stage tasks beyond the scope of current reinforcement learning methods, and the ability to easily scale to natural, real-world manipulation scenarios where only raw sensory signals are available. We have open-sourced our datasets and all algorithm implementations to facilitate future research and fair comparisons in learning from human demonstration data at https://arise-initiative.github.io/robomimic-web/ ",
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+    {
+        "type": "text",
+        "text": "Keywords: Imitation Learning, Offline Reinforcement Learning, Manipulation ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Human supervision has been at the heart of the most significant recent advances in several domains such as computer vision [1–4] and natural language processing [5–7]. By intelligently extracting information from large-scale human-labeled datasets, autonomous machines have been able to reach near- or even super-human performance on decades-old problems such as image recognition and question answering. Roboticists have also attempted to tackle robot manipulation through learning from human datasets, using the paradigms of Imitation Learning [8–10] and Batch (Offline) Reinforcement Learning [11–13], where datasets consisting of robot arm trajectories, action labels at each timestep, and possibly reward labels, are used to train closed-loop policies. ",
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+        "type": "text",
+        "text": "As in other domains, large offline datasets offer several benefits such as scale, portability, and reproducible evaluations to measure progress. Recently, there has been considerable progress in offline learning for robot manipulation from human demonstrations [9, 14, 10]. Despite these advances, the offline learning paradigm has not been nearly as disruptive in robotics as in other disciplines – there is a large gap between autonomous robot manipulation capabilities and the wide range of tasks that humans can solve effortlessly using physical and cognitive intelligence. What has inhibited the use of large human-provided datasets to address this gap? ",
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+        "text": "In contrast to other domains where supervised learning has been successful, robotic manipulation is a time-evolving dynamical system, requiring fine-grained real-time control to guide robot arms successfully through tasks – consequently, data collection can present technical challenges requiring specialized systems [15], which can explain why large-scale human-provided datasets [16, 17] have not been very prevalent. Learning from such datasets can also present several challenges. Human demonstrations can differ from machine-generated datasets (a recent trend in benchmarks for offline policy learning [18, 19]) due to a non-Markovian decision process, since humans may not act purely based on the current observation. There can also be significant variance in both data quality and solution strategy when collecting data from multiple humans [20]. Differences from classic supervised learning, such as a mismatch between training and evaluation objectives (task success rate), can make selecting a final policy challenging [21, 22], especially in real-world settings where evaluating each policy on a robot can be infeasible. Finally, offline learning is sensitive to state and action space coverage (dataset size) and agent design decisions. ",
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+        "image_caption": [
+            "Figure 1: Tasks. We collect datasets across 6 operators of varying proficiency and evaluate offline policy learning methods on 8 challenging manipulation tasks that test a wide range of manipulation capabilities including pick-and-place, multi-arm coordination, and high-precision insertion and assembly. "
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+        "text": "Studying these challenges in the context of robot manipulation and human-provided datasets could be a stepping stone to closing the gap between robot and human manipulation capabilities. Unfortunately, a lack of suitable benchmark and human datasets have made studying this setting difficult. Prior works are either limited to studying simple 2D environments [23] or using data generated from hard-coded policies [24, 25]. In this paper, we address this need by presenting a study of data-driven offline policy learning methods on several human-provided robot manipulation datasets. We collect task demonstrations from human teleoperators across a broad range of simulated and real world manipulation tasks and investigate several factors that play a role in learning from such data. ",
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+        "text": "From our results, we point out several lessons to guide future research in leveraging human supervision for robot manipulation effectively. We find that history-dependent models can be extremely effective in learning from single and multi-human datasets while state-of-the-art batch RL algorithms struggle to learn from such datasets, and that the choice of observation space and hyperparameters play a substantial role in training proficient policies. We also find that there is substantial promise for solving more complex tasks using large-scale human datasets and that our insights directly transfer to real-world scenarios, making this an important setting to explore further. ",
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+    {
+        "type": "text",
+        "text": "2 Challenges in Offline Learning from Human Datasets ",
+        "text_level": 1,
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+        "text": "In this section, we outline five challenges in offline learning from human datasets that motivate different factors that we investigate in our study. ",
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+        "text": "(C1) Data from Non-Markovian Decision Process. Human demonstrations can differ substantially from machine-generated demonstrations because humans may not act purely based on a single current observation. External factors (teleoperation device, past actions, history of episode) may all play a role. Prior work [20] has noted substantial benefits from leveraging models that are history-dependent and / or with temporal abstraction to learn from human demonstrations. We investigate various design choices related to such architectures in this study. ",
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+        "table_caption": [
+            "Table 1: Results on Low-Dimensional Observations. We present success rates averaged over 3 seeds for each method across the low-dim Machine-Generated (MG), Proficient-Human (PH), and Multi-Human (MH) datasets. The results show that methods that model temporal correlations (BC-RNN, HBC, IRIS) exhibit strong performance on human datasets. Furthermore, while Batch RL algorithms like BCQ are proficient on machine-generated data, they perform poorly on human datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>BC</td><td>BC-RNN</td><td>BCQ</td><td>CQL</td><td>HBC</td><td>IRIS</td></tr><tr><td>Lift (MG)</td><td>65.3 ±2.5</td><td>70.7±3.4</td><td>91.3±1.9</td><td>64.0±2.8</td><td>47.3±4.1</td><td>96.0±1.6</td></tr><tr><td>Can (MG)</td><td>64.7±3.4</td><td>68.7±2.5</td><td>75.3±0.9</td><td>1.3±0.9</td><td>40.7±3.4</td><td>48.0±6.5</td></tr><tr><td>Lift (PH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>100.0±0.0</td><td>92.7±5.0</td><td>100.0±0.0</td><td>100.0±0.0</td></tr><tr><td>Can (PH)</td><td>95.3±0.9</td><td>100.0±0.0</td><td>88.7±0.9</td><td>38.0±7.5</td><td>100.0±0.0</td><td>100.0±0.0</td></tr><tr><td>Square (PH)</td><td>78.7±1.9</td><td>84.0±0.0</td><td>50.0±4.9</td><td>5.3±2.5</td><td>82.6±0.9</td><td>78.7±2.5</td></tr><tr><td>Transport (PH)</td><td>17.3±2.5</td><td>71.3±6.6</td><td>7.3±3.3</td><td>0.0±0.0</td><td>48.6±3.8</td><td>41.3 ±3.4</td></tr><tr><td>Tool Hang (PH)</td><td>29.3±0.9</td><td>19.3±5.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>30.0±7.1</td><td>11.3±2.5</td></tr><tr><td>Lift (MH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>100.0±0.0</td><td>56.7±40.3</td><td>100.0±0.0</td><td>100.0±0.0</td></tr><tr><td>Can (MH)</td><td>86.0±4.3</td><td>100.0±0.0</td><td>62.7±8.2</td><td>22.0±5.7</td><td>91.3±2.5</td><td>92.7±0.9</td></tr><tr><td>Square (MH)</td><td>52.7±6.6</td><td>78.0±4.3</td><td>14.0±4.3</td><td>0.7±0.9</td><td>60.7±5.0</td><td>52.7±5.0</td></tr><tr><td>Transport (MH)</td><td>11.3 ± 2.5</td><td>65.3±7.4</td><td>2.6±0.9</td><td>0.0±0.0</td><td>14.0±1.6</td><td>10.7±0.9</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(C2) Variance in Demonstration Quality from Multiple Humans. Prior work [20, 17] has found that data collected from several humans can differ substantially in both demonstration proficiency and solution strategy. Differences in supervisor proficiency can manifest in many ways, such as large variations in trajectory length and noise in robot movement or mistakes (e.g. missed grasps). In our study, we evaluate offline policy learning algorithms on such datasets. While recent batch RL algorithms have shown an excellent ability to learn from mixed quality machine-generated datasets [26, 27], we empirically find that they fail to learn well from mixed quality human data. ",
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+    {
+        "type": "text",
+        "text": "(C3) Dependence on Dataset Size. Offline policy learning is sensitive to the state and action space coverage in the dataset, and by extension, the size of the dataset itself. In our study, we investigate how dataset sizes affect policy performance. This analysis is useful to understand the value of adding more data – an important consideration since collecting human demonstrations can be costly. ",
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+    {
+        "type": "text",
+        "text": "(C4) Mismatch between Training and Evaluation Objectives. Unlike traditional supervised learning, where model selection can be achieved by using the model with the lowest validation loss [21], offline policy learning often suffers from the fact that the training objective is only a surrogate for the true objective of interest (e.g. task success rate), and policy performance can change significantly from epoch to epoch. This makes it difficult to select the best trained model [19, 28, 29]. In our study, we evaluate each policy checkpoint online in the environment in simulation, and report the best policy success rate per training run. We use these ground-truth values to understand the effectiveness of different selection criteria, and confirm that offline policy selection is an important problem, especially in real-world scenarios where large-scale empirical evaluation is difficult. ",
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+        "type": "text",
+        "text": "(C5) High Sensitivity to Agent Design Decisions. Prior studies on machine-generated datasets have shown that offline policy learning can be extremely sensitive to hyperparameter choices [19, 28]. In our study, we explore how agent design decisions affect policy performances, including the choice of agent architecture, agent observation space, and hyperparameter choices per algorithm. This results in several practical conclusions that should prove useful to researchers and practitioners alike. We further show that important design decisions made through our study in simulation directly translate to effective policy learning on real world tasks and datasets. ",
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+    {
+        "type": "text",
+        "text": "3 Study Design ",
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+        "type": "text",
+        "text": "3.1 Tasks ",
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+        "type": "text",
+        "text": "We conducted our study across 5 simulated and 3 real world tasks. The tasks were chosen to test a broad range of manipulation capabilities. See Fig 1 and Appendix E for more details. ",
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+    {
+        "type": "text",
+        "text": "Lift $\\mathbf { \\ s } \\mathbf { i } \\mathbf { m } + \\mathbf { r e } \\mathbf { a } \\mathbf { l } )$ ). The robot arm must lift a small cube. This is the simplest task. ",
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+    {
+        "type": "text",
+        "text": "Can $\\mathbf { \\ s } \\mathbf { i } \\mathbf { m } + \\mathbf { r e } \\mathbf { a } \\mathbf { l } )$ . The robot must place a coke can from a large bin into a smaller target bin. Slightly more challenging than Lift, since picking the can is harder than picking the cube, and the can must also be placed into the bin. ",
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+        "type": "text",
+        "text": "Square (sim). The robot must pick a square nut and place it on a rod. Substantially more difficult than Lift and Pick Place Can due to the precision needed to pick up the nut and insert it on the rod. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>BC</td><td>BC-RNN</td><td>BCQ</td><td>CQL</td><td>HBC</td><td>IRIS</td></tr><tr><td>Can-Worse</td><td>56.7±2.5</td><td>92.0±1.6</td><td>29.3 ±10.9</td><td>4.0±3.3</td><td>78.7±3.4</td><td>77.3±1.9</td></tr><tr><td>Can-Okay</td><td>72.0±2.8</td><td>95.3±1.9</td><td>58.0±8.6</td><td>22.0±4.3</td><td>97.3± 0.9</td><td>96.0±0.0</td></tr><tr><td>Can-Better</td><td>83.3±2.5</td><td>99.3± 0.9</td><td>62.0±5.9</td><td>20.7±7.4</td><td>96.7±0.9</td><td>96.0±0.0</td></tr><tr><td>Can-Worse-Okay</td><td>74.7±5.7</td><td>98.7±1.9</td><td>50.7±3.8</td><td>18.7±2.5</td><td>88.0±1.6</td><td>87.3±1.9</td></tr><tr><td>Can-Worse-Better</td><td>76.0±4.3</td><td>100.0±0.0</td><td>48.0±4.9</td><td>20.7±5.7</td><td>90.0±1.6</td><td>91.3±2.5</td></tr><tr><td>Can-Okay-Better</td><td>90.7±1.9</td><td>100.0±0.0</td><td>68.7±2.5</td><td>30.7±7.7</td><td>99.3± 0.9</td><td>98.0±1.6</td></tr><tr><td>Square-Worse</td><td>22.0±4.3</td><td>39.3±3.8</td><td>5.3±1.9</td><td>0.0±0.0</td><td>44.7±6.8</td><td>38.7±0.9</td></tr><tr><td>Square-Okay</td><td>27.3±3.4</td><td>45.3 ±2.5</td><td>6.7±1.9</td><td>0.0±0.0</td><td>52.0±2.8</td><td>42.0±3.3</td></tr><tr><td>Square-Better</td><td>58.7±2.5</td><td>66.0±2.8</td><td>32.0±4.3</td><td>0.7±0.9</td><td>61.3±1.9</td><td>60.0±1.6</td></tr><tr><td>Square-Worse-Okay</td><td>28.7±2.5</td><td>55.3±0.9</td><td>8.7±1.9</td><td>2.7±1.9</td><td>50.7±4.1</td><td>43.3±2.5</td></tr><tr><td>Square-Worse-Better</td><td>46.7±5.7</td><td>73.3±6.2</td><td>15.3± 2.5</td><td>1.3±0.9</td><td>65.3±3.4</td><td>56.7±3.4</td></tr><tr><td>Square-Okay-Better</td><td>56.7±4.1</td><td>74.0±2.8</td><td>22.0±4.3</td><td>1.3±0.9</td><td>63.3±4.1</td><td>56.7±3.8</td></tr><tr><td>Can-Paired</td><td>64.0±9.1</td><td>70.0±4.3</td><td>44.7±1.9</td><td>6.0±1.6</td><td>70.7±5.2</td><td>75.3±1.9</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Results on Suboptimal Human Data. We present success rates averaged over 3 seeds for each method across different subsets of the Multi-Human datasets, corresponding to mixtures of demonstrations from “Better”, “Adequate”, and “Worse” human operators, and finally on a diagnostic dataset with paired success and failure human trajectories for each starting initialization. Results indicate that BC-RNN is a strong baseline, and that Batch RL methods perform poorly across all datasets, even on the simple diagnostic dataset. ",
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+        "type": "text",
+        "text": "Transport (sim). Two robot arms must transfer a hammer from a closed container on a shelf to a target bin on another shelf. One robot arm must retrieve the hammer from the container, while the other arm must clear the target bin by moving a piece of trash to the nearby receptacle. Finally, one arm must hand the hammer over to the other, which must place the hammer in the target bin. ",
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+    {
+        "type": "text",
+        "text": "Tool Hang $( \\sin + \\mathrm { r e a l } )$ ). A robot arm must assemble a frame consisting of a base piece and hook piece by inserting the hook into the base, and hang a wrench on the hook. This is the most difficult task due to the multiple stages that each require precise, and dexterous, rotation-heavy movements. ",
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+    {
+        "type": "text",
+        "text": "3.2 Data Collection ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To study the effect of dataset source, we collected data from three different sources – MachineGenerated, Proficient-Human, and Multi-Human (more details in Appendix B). ",
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+        "type": "text",
+        "text": "Machine-Generated (MG). We collected these datasets by first training a state-of-the-art RL algorithm [30] on the Lift and Can task, taking agent checkpoints that are saved regularly during training, and collecting 300 rollout trajectories from each checkpoint. Consequently, these datasets are comprised of mixtures of expert and suboptimal data, and resemble datasets from common offline RL benchmarks [18, 19]. We excluded other tasks because they could not be solved by the RL algorithm even with substantial tuning. See the appendix for more details. ",
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+    {
+        "type": "text",
+        "text": "Proficient-Human (PH) and Multi-Human (MH). Datasets are collected by humans through RoboTurk [15, 17], a remote teleoperation platform. The PH datasets consist of 200 demonstrations collected by a single, experienced teleoperator, while the MH datasets consist of 300 demonstrations, collected by 6 teleoperators of varying proficiency, each of which provided 50 demonstrations. The 6 teleoperators consisted of a “better” group of 2 experienced operators, an “okay” group of 2 adequate operators, and a “worse” group of 2 inexperienced operators. These data subsets in the Multi-Human data allowed us to investigate the ability of algorithms to deal with mixed quality human data. ",
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+    {
+        "type": "text",
+        "text": "Observation Modalities. To study the effect of observation modalities, we capture a diverse set of sensor streams when collecting the dataset, including end-effector, gripper fingers, and joints, groundtruth object poses, and images from an external camera and wrist-mounted camera per robot arm (see Appendix E). We have two observation spaces – “low-dim” and “image”. Both include end-effector poses and gripper finger positions, and only differ in whether ground-truth object information is used (low-dim) or whether that information is replaced by the available camera observations (image). ",
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+    {
+        "type": "text",
+        "text": "3.3 Training and Evaluation Protocols ",
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+    {
+        "type": "text",
+        "text": "There are several approaches to offline imitation learning [31–34, 9, 10, 25, 35, 36] and offline reinforcement learning [26, 27, 37–43] (see Appendix A for more discussion on related work). We chose to evaluate 6 algorithms in this study – Behavioral Cloning (BC), BC with an RNN policy (BC-RNN), Hierarchical Behavioral Cloning (HBC) [10], Batch-Constrained Q-Learning (BCQ) [26], Conservative Q-Learning (CQL) [27], and IRIS [20]. BC-RNN, HBC, and IRIS have all been used in prior work to learn offline from teleoperated human demonstrations, while BCQ and CQL are commonly-used offline RL algorithms (see Appendix C). We use binary task completion rewards for all our experiments. Each agent is trained for $N$ epochs, where each epoch consists of $M$ gradient steps, and evaluated every $E$ epochs, by running 50 rollouts in the environment and reporting the success rate over a maximum horizon. For each agent, we report the maximum success rate over the coarse of training, and average over 3 seeds. For low-dim agents, $N = 2 0 0 0$ , $M = 1 0 0$ , and $E = 5 0$ and for image agents, $N = 6 0 0$ , $M = 5 0 0$ , and $E = 2 0$ (see Appendix B.2). ",
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+    {
+        "type": "image",
+        "img_path": "images/1708792fdb110bc98cd5d0650c4c89add2bdc9f0187ca36dae6a23079cb20e3f.jpg",
+        "image_caption": [
+            "Figure 2: Effect of Observation Space and Hyperparameter Choice. We show how the success rate that BC-RNN obtains can drop drastically due to changes to the observation space and hyperparameter settings. "
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/82b04fc1c61417f99d9329cd1130e9d29aca7a4583ed31afd6f35cd001eea4a5.jpg",
+        "image_caption": [
+            "Figure 3: Effect of Dataset Size. We study how the BC-RNN success rate changes when lowering the quantity of data to $20 \\%$ and $50 \\%$ . Results show that less complex tasks (Lift, Can) be learned with a fraction of the data, while more complex tasks might benefit from even larger human datasets. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we present each factor that we explored in our study, and note the relevant challenges from Sec. 2 that each pertains to. ",
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+    {
+        "type": "text",
+        "text": "4.1 Algorithm Comparison on Single and Multi-Human Demonstrations (C1, C2) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We trained and evaluated all algorithms on the Proficient-Human (PH) and Multi-Human (MH) datasets and report the average success rates across 3 seeds in Table 1. ",
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+    {
+        "type": "text",
+        "text": "Observation history is crucial for good performance. There is a substantial performance gap between BC-RNN and BC, which highlights the benefits of history-dependence. The performance gap is larger for longer-horizon tasks (e.g. $\\sim 5 5 \\%$ for Transport (PH) compared to $\\sim 5 \\%$ for Square (PH)) and for multi-human data compared to single-human data (e.g. $\\sim 2 5 \\%$ for Square (MH) compared to $\\sim 5 \\%$ for Square (PH)). Interestingly, results are lower for MH datasets compared to PH datasets, even though the MH datasets contain 100 more demos (300 demos vs. 200 demos). This most likely stems from the presence of suboptimal and multimodal data in the MH datasets. ",
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+        "type": "text",
+        "text": "Batch RL algorithms perform poorly on Human Datasets. Recent batch (offline) RL algorithms such as BCQ and CQL have demonstrated excellent results in learning from suboptimal and multimodal agent-generated datasets. Our results confirm the capacity of such algorithms to work well – BCQ in particular performs strongly on our agent-generated MG datasets that consist of a diverse mixture of good and poor policies. Surprisingly though, neither BCQ nor CQL performs particularly well on these human-generated datasets. This puts the ability of such algorithms to learn from more ",
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+        "table_caption": [
+            "Table 3: Results on Image Observations. "
+        ],
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+        "table_body": "<table><tr><td>Dataset</td><td>BC</td><td>BC-RNN</td><td>BCQ</td><td>CQL</td></tr><tr><td>Lift (PH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>98.0±1.6</td><td>52.0±13.0</td></tr><tr><td>Can (PH)</td><td>97.3±1.9</td><td>98.0±0.9</td><td>86.7±2.5</td><td>0.7±0.9</td></tr><tr><td>Square (PH)</td><td>62.0±4.9</td><td>82.0±0.0</td><td>41.3±4.1</td><td></td></tr><tr><td>Transport (PH)</td><td>55.3±6.2</td><td>72.0±4.3</td><td>0.7±0.9</td><td></td></tr><tr><td>Tool Hang (PH)</td><td>20.0±5.9</td><td>67.3±4.1</td><td>3.3±0.9</td><td>=</td></tr><tr><td>Lift (MH)</td><td>100.0±0.0</td><td>100.0±0.0</td><td>93.3±0.9</td><td>11.3 ±9.3</td></tr><tr><td>Can (MH)</td><td>85.3±0.9</td><td>96.0±1.6</td><td>77.3±6.8</td><td>0.0±0.0</td></tr><tr><td>Square (MH)</td><td>46.0±1.6</td><td>76.7±3.4</td><td>17.3±7.5</td><td></td></tr><tr><td>Transport (MH)</td><td>18.7±2.5</td><td>42.0±1.6</td><td>0.0±0.0</td><td>=</td></tr></table>",
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+        "text": "Figure 5: (left) Effect of Policy Selection Criteria. We compare how performance decreases when choosing the policy to evaluate by using the lowest validation loss, or when using the final trained checkpoint, with respect to the best policy performance. (right) Results on Image Observations. We present success rates for each method across the image observation human datasets. BC-RNN maintains nearly the same performance as learning from ground-truth observations, providing an optimistic view for learning with real-world raw sensory observations. natural dataset distributions into question (instead of those collected via RL exploration or pre-trained agents). There is an opportunity for future work in batch RL to resolve this gap. ",
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+        "text": "4.2 Learning from Suboptimal Human Data (C2) ",
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+        "type": "text",
+        "text": "To further investigate how algorithms deal with suboptimal human data, we split our MH datasets into smaller subsets based on the proficiency of the human operators. The MH-Better, MH-Okay, and MH-Worse are the 100 demo subsets corresponding to the 2 “better”, 2 “okay”, and 2 “worse” operators respectively, while MH Worse-Okay, MH Worse-Better, and MH Okay-Better are the 200 demo subsets corresponding to the mixture of the previous subsets. Similar data mixtures have been used for evaluations in batch RL [18]. Appendix B shows the average trajectory lengths in each data subset – lower quality datasets contain demonstrations that take more time to solve the task. ",
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+        "type": "text",
+        "text": "BC-RNN is a strong baseline on suboptimal human data, but there is room for improvement. Table 2 shows that BC exhibits a large performance gap between the Better and Worse 100-demo subsets (roughly $27 \\%$ and $3 5 \\%$ for Can and Square respectively). Interestingly, BC-RNN is able to nearly eliminate this gap in performance on the Can task, but not on the Square task. However, BC-RNN outperforms BC on all datasets $7 \\% - 3 5 \\%$ improvement). Comparing results on the 100 Better demonstrations and 100 Okay demonstrations to the 200 Worse-Better demonstrations and 200 Worse-Okay demonstrations further allows us to analyze how adding 100 “worse” demonstrations impacts the performance of each algorithm. Most algorithms decline in performance while BC-RNN is able to uniformly improve from the added data. Comparing the performance of BC-RNN on the 200-demo Square mixture datasets $( 5 5 . 3 \\%$ , $7 3 . 3 \\%$ , $7 4 . 0 \\%$ ) to the high-quality 200-demo Square (PH) dataset $( 8 4 . 0 \\% )$ shows that there is still room for algorithms to improve on the use of this data. ",
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+        "type": "text",
+        "text": "Diagnostic dataset shows that Batch RL struggles in simpler settings as well. The final row of Table 2 shows additional results on a diagnostic dataset termed Can-Paired, where a single operator collected 2 demonstrations for each of 100 task initializations – one successful demonstration, and one where the can is tossed outside of the bin (task failure), for a total of 200 demonstrations. There is a strong expectation for batch RL algorithms to be able to distinguish between actions leading to successful placement and actions leading to task failure, but even in this simple setting, most algorithms suffer, providing a pessimistic view of the state-of-the-art. The $5 \\%$ improvement that IRIS provides over BC-RNN suggests that introducing history-dependence into state-of-the-art batch RL algorithms might be a promising direction for future work. ",
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+        "type": "text",
+        "text": "4.3 Effect of Observation Space (C5) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Learning from image observations can match low-dim agent performance. In Table 3, we present policy learning results when using image observations instead of ground-truth object locations – an important setting for real-world policy learning. BC-RNN still maintains superior performance improvements over BC on the complex Square and Transport tasks, and with the exception of Transport (MH), maintains nearly the same performance as learning from ground-truth observations. This result provides an optimistic view for learning with real-world raw sensory observations. ",
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+        "type": "text",
+        "text": "Features used for robot proprioception can matter. In Fig 2a, we study the effect of adding end effector velocities to the observations $^ +$ EEF Vel), and joint positions and velocities to the observations $\\left( + \\operatorname { J o i n t } \\right)$ . Surprisingly, we find that including end effector velocity information, and joint information hurts agents trained on low-dim observations substantially $( 4 9 \\% - 8 8 \\%$ relative performance drop), while image-based agents are more tolerant to the inclusion of this extra information $2 \\% - 2 9 \\%$ relative performance drop). We hypothesize that performance drops might be due to overfitting to the presence of this extra information not needed for solving these tasks. Thus, practitioners should take care to engineer the robot observation space and exclude possibly irrelevant information – information-hiding can be a powerful paradigm for training proficient robots [44]. ",
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+        "type": "text",
+        "text": "Image randomization and wrist observations can be crucial for manipulation tasks. In Fig 2a, we report performance drops from removing pixel shift image randomization (- Rand) and the wrist camera (- Wrist) from image-based agents to understand their importance. We see that removing randomization results in $47 \\%$ and $3 5 \\%$ relative performance drops on Square and Transport respectively, and removing wrist images results in $9 \\%$ and $43 \\%$ relative drops. Consequently, both wrist camera images and image randomization play a substantial role in producing performant policies. We confirm the importance of each for visuomotor imitation in the real world as well (see Sec 4.7). Wrist observations likely help the robot improve gripper alignment for grasping and randomization helps the policy develop invariance for portions of the image that are not important for action prediction. ",
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+        "text": "4.4 Effect of Hyperparameter Choice (C5) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we take our default hyperparameters for BC-RNN and study the effect of changing a subset of them to report practical recommendations for learning from human datasets (see Appendix I for BCQ and CQL). We present our results in Fig 2b (low-dim) and Fig 2c (image). ",
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+        "type": "text",
+        "text": "(larger LR) Increasing the learning rate from 1e-4 to 1e-3 affects the performance of image-agents substantially (drop of $3 5 \\% - 6 3 \\%$ ), while low-dim agents are more tolerant to the change. (no GMM) Using a deterministic policy instead of learning a GMM action distribution results in significant relative performance drops on the MH datasets (especially low-dim Transport, with a drop of $58 \\%$ ). (larger MLP) Using a larger MLP size at each RNN timestep reduces performance uniformly, suggesting that it is possible to overfit to dataset actions if network architectures are too large. (shallow Conv) Using a shallow convolutional network [45] instead of the ResNet backbone [46] for encoding image observations reduces performance significantly – with relative drops of $2 5 \\% { - } 6 2 \\%$ , suggesting that large-capacity visual encoders are crucial for visuomotor imitation. (smaller RNN dim) Reducing the size of the RNN hidden dimension from 400 to 100 (low-dim) and 1000 to 400 (image) uniformly decreases performance (drops of $3 \\% . 5 8 \\%$ ), showing the importance of a large RNN hidden dimension. (Recommendations) We recommend tuning the LR (especially for image agents) and network structure (MLP size, size of RNN dim) carefully. Opting to use a GMM policy and a ResNet encoder appears to be uniformly better. ",
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+    {
+        "type": "text",
+        "text": "4.5 Selecting a Policy to Evaluate (C4) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Model selection in offline policy learning can be challenging – for this reason, in our simulation experiments, we evaluated every policy checkpoint online and reported the best one. This is not feasible for real-world settings, making offline policy selection desirable. In Fig 4a, we show that this can be non-trivial, by showing the relative performance drop when selecting the policy using the best loss on validation data (common in supervised learning), and when using the final training checkpoint as well (common in offline RL [47, 18, 27]) – in both cases, the selected policy is significantly worse than the best one ( $10 \\%$ to $100 \\%$ decrease). See Appendix G for more detailed results and discussion. This motivates the need for better offline evaluation metrics. ",
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+    {
+        "type": "text",
+        "text": "4.6 Effect of Dataset Size (C3) ",
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+        "type": "text",
+        "text": "To study how dataset size impacts performance, we formed smaller $20 \\%$ and $50 \\%$ subsets of our human datasets by sampling trajectories. We evaluate low-dim and image BC-RNN agents across these subsets in Table 27 and Table 28. There are several promising results here. We first note that less complex tasks (Lift, Can) can yield proficient policies $7 5 \\% - 1 0 0 \\%$ success rate) using a small fraction of the data $( 2 0 \\% )$ . Second, while policies trained on more complex tasks (Square, Transport) suffer substantially when using $50 \\%$ or $20 \\%$ of the data, the converse is also true – adding more data (e.g. moving from $20 \\%$ to $50 \\%$ or $50 \\%$ to $100 \\%$ size) can result in significant policy improvement. This confirms the value of using large human datasets as a means to obtain proficient policies for challenging and complex manipulation tasks. ",
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+        "type": "text",
+        "text": "4.7 Applicability to Real-World Settings ",
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+    {
+        "type": "text",
+        "text": "Here, we show that design decisions made in simulation can potentially transfer to real world settings. We collected 3 additional real-world datasets with a Franka robotic arm – Lift (Real), Can (Real), and Tool Hang (Real). Each consists of 200 trajectories collected by one operator. We train BC-RNN and report the final policy checkpoint success rate, over 30 rollouts, due to the time-consuming nature of real world policy evaluation. We also emphasize that no real-world hyperparameter tuning took place, so our results are a lower bound. We were able to train proficient Lift $( 9 6 . 7 \\% )$ and Can $( 7 3 . 3 \\% )$ policies, and the Tool Hang $( 3 . 3 \\% )$ policy is able to generate some task successes, despite the extremely difficult nature of the task. Furthermore, as in Sec. 4.3, we validate the importance of pixel shift randomization and the wrist camera by ablating each component on the Can task, and show that including both is the difference between a proficient and non-proficient real-world policy – Can (- Rand) $( 2 6 . 7 \\% )$ , Can (- Wrist) $( 4 3 . 3 \\% )$ . ",
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+    {
+        "type": "text",
+        "text": "5 Discussion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we summarize the lessons from our study and make recommendations for future work. ",
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+        "type": "text",
+        "text": "(L1) Models with temporal abstraction can be extremely effective in learning from human datasets. In $\\mathrm { S e c 4 . 1 }$ and Sec 4.2, we demonstrated that history-dependent models (BC-RNN, HBC, and IRIS) are particularly effective in learning from human datasets compared to algorithms that do not take temporal context into account. ",
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+        "type": "text",
+        "text": "(L2) Need to improve the ability of batch (offline) RL to learn from suboptimal human datasets. $\\mathrm { S e c } ~ 4 . 2$ and Appendix I demonstrated that state-of-the-art batch RL algorithms are excellent at learning from suboptimal machine-generated datasets but much worse at learning from suboptimal human datasets. They even struggled with a diagnostic dataset with paired good and bad human demonstration trajectories while IRIS was able to improve slightly on BC-RNN, suggesting that combining history-dependence with value learning might be a good place to start for improving batch RL methods [48–51]. This also demonstrates a need to start benchmarking new batch RL algorithms on human datasets instead of purely on machine-generated datasets. ",
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+        "type": "text",
+        "text": "(L3) Improving offline policy selection is important for real world settings. Sec 4.5 demonstrated the need for better ways to select an evaluation policy in an offline manner. We hope that our datasets can help supplement other efforts [29]. ",
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+        "type": "text",
+        "text": "(L4) Observation space plays a large role and hyperparameters matter. Sec 4.3 demonstrates that policies trained on low-dim observations can be very sensitive to the choice of robot proprioception, while pixel shift randomization and wrist camera images are critical for effective visuomotor policy learning. The choice of observation space for imitation merits careful consideration – other work has also confirmed the importance of feature representations used for offline policy learning [52, 53]. Sec 4.4 and Appendix I made practical recommendations for choosing hyperparameters to learn from human data. ",
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+    {
+        "type": "text",
+        "text": "(L5) There is substantial promise for solving more complex tasks using large-scale human datasets. Sec 4.6 showed that adding more data can result in significant policy improvement on complex tasks. Table 3 and Sec 4.7 shows that we could learn proficient policies on the Tool Hang task, our most complex task, without any hyperparameter tuning on the task or dataset. Together, these results show the potential of large human datasets as a means to solve challenging and complex manipulation tasks. ",
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+    {
+        "type": "text",
+        "text": "(L6) Study results transfer to real-world settings. In Sec 4.7, we showed that we could directly apply hyperparameters that were tuned on simulated tasks directly to real-world datasets and tasks. This provides promise for using our tasks, datasets, and codebase to enable reproducible evaluation in simulation, while also being confident that conclusions can transfer to real-world settings. ",
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+    {
+        "type": "text",
+        "text": "Going forward, we hope that the datasets, tasks, code, and subsequent insights of our study will serve researchers and practitioners alike. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Albert Tung for helping with the RoboTurk data collection system, Jim Fan for providing timely lab cluster support, and Helen Roman for helping order items for the physical robot tasks. Ajay Mandlekar acknowledges the support of the Department of Defense (DoD) through the NDSEG program. We acknowledge the support of Toyota Research Institute (“TRI”); this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity. We acknowledge the support of the US Army Research Office (award W911NF-15-1-0479) and the National Science Foundation (award CNS-1955523). This work relates to Department of Navy award N00014-14-1-0671 issued by the Office of Naval Research. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+    {
+        "type": "text",
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+                                            "content": "Table 1: Results on Low-Dimensional Observations. We present success rates averaged over 3 seeds for",
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+                                    "content": "from epoch to epoch. This makes it difficult to select the best trained model [19, 28, 29]. In our",
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+                                    "content": "). The robot arm must lift a small cube. This is the simplest task.",
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+                                            "html": "<table><tr><td>Dataset</td><td>BC</td><td>BC-RNN</td><td>BCQ</td><td>CQL</td><td>HBC</td><td>IRIS</td></tr><tr><td>Can-Worse</td><td>56.7±2.5</td><td>92.0±1.6</td><td>29.3 ±10.9</td><td>4.0±3.3</td><td>78.7±3.4</td><td>77.3±1.9</td></tr><tr><td>Can-Okay</td><td>72.0±2.8</td><td>95.3±1.9</td><td>58.0±8.6</td><td>22.0±4.3</td><td>97.3± 0.9</td><td>96.0±0.0</td></tr><tr><td>Can-Better</td><td>83.3±2.5</td><td>99.3± 0.9</td><td>62.0±5.9</td><td>20.7±7.4</td><td>96.7±0.9</td><td>96.0±0.0</td></tr><tr><td>Can-Worse-Okay</td><td>74.7±5.7</td><td>98.7±1.9</td><td>50.7±3.8</td><td>18.7±2.5</td><td>88.0±1.6</td><td>87.3±1.9</td></tr><tr><td>Can-Worse-Better</td><td>76.0±4.3</td><td>100.0±0.0</td><td>48.0±4.9</td><td>20.7±5.7</td><td>90.0±1.6</td><td>91.3±2.5</td></tr><tr><td>Can-Okay-Better</td><td>90.7±1.9</td><td>100.0±0.0</td><td>68.7±2.5</td><td>30.7±7.7</td><td>99.3± 0.9</td><td>98.0±1.6</td></tr><tr><td>Square-Worse</td><td>22.0±4.3</td><td>39.3±3.8</td><td>5.3±1.9</td><td>0.0±0.0</td><td>44.7±6.8</td><td>38.7±0.9</td></tr><tr><td>Square-Okay</td><td>27.3±3.4</td><td>45.3 ±2.5</td><td>6.7±1.9</td><td>0.0±0.0</td><td>52.0±2.8</td><td>42.0±3.3</td></tr><tr><td>Square-Better</td><td>58.7±2.5</td><td>66.0±2.8</td><td>32.0±4.3</td><td>0.7±0.9</td><td>61.3±1.9</td><td>60.0±1.6</td></tr><tr><td>Square-Worse-Okay</td><td>28.7±2.5</td><td>55.3±0.9</td><td>8.7±1.9</td><td>2.7±1.9</td><td>50.7±4.1</td><td>43.3±2.5</td></tr><tr><td>Square-Worse-Better</td><td>46.7±5.7</td><td>73.3±6.2</td><td>15.3± 2.5</td><td>1.3±0.9</td><td>65.3±3.4</td><td>56.7±3.4</td></tr><tr><td>Square-Okay-Better</td><td>56.7±4.1</td><td>74.0±2.8</td><td>22.0±4.3</td><td>1.3±0.9</td><td>63.3±4.1</td><td>56.7±3.8</td></tr><tr><td>Can-Paired</td><td>64.0±9.1</td><td>70.0±4.3</td><td>44.7±1.9</td><td>6.0±1.6</td><td>70.7±5.2</td><td>75.3±1.9</td></tr></table>",
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+                                    "type": "text"
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+                                    "content": "Transport (sim). Two robot arms must transfer a hammer from a closed container on a shelf to a",
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+                                    "content": "other arm must clear the target bin by moving a piece of trash to the nearby receptacle. Finally, one",
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+                                    "content": "arm must hand the hammer over to the other, which must place the hammer in the target bin.",
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+                                    "score": 1.0,
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+                                    "content": "algorithm even with substantial tuning. See the appendix for more details.",
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+                                    "content": "collected by 6 teleoperators of varying proficiency, each of which provided 50 demonstrations. The 6",
+                                    "type": "text"
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+                            ],
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+                                    "type": "text"
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+                                    "content": "data allowed us to investigate the ability of algorithms to deal with mixed quality human data.",
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+                                    "score": 1.0,
+                                    "content": "Observation Modalities. To study the effect of observation modalities, we capture a diverse set of",
+                                    "type": "text"
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+                                    "content": "Appendix E). We have two observation spaces – “low-dim” and “image”. Both include end-effector",
+                                    "type": "text"
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "chose to evaluate 6 algorithms in this study – Behavioral Cloning (BC), BC with an RNN policy",
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+                                    "type": "text"
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+                                    "content": "Conservative Q-Learning (CQL) [27], and IRIS [20]. BC-RNN, HBC, and IRIS have all been used",
+                                    "type": "text"
+                                }
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+                                    "score": 1.0,
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+                                    "content": "and MH-Worse are the 100 demo subsets corresponding to the 2 “better”, 2 “okay”, and 2 “worse”",
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+                                    "score": 1.0,
+                                    "content": "used for evaluations in batch RL [18]. Appendix B shows the average trajectory lengths in each data",
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+                                    "score": 1.0,
+                                    "content": "confirm the importance of each for visuomotor imitation in the real world as well (see Sec 4.7). Wrist",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the policy develop invariance for portions of the image that are not important for action prediction.",
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+                                    "content": "4.4 Effect of Hyperparameter Choice (C5)",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "subset of them to report practical recommendations for learning from human datasets (see Appendix I",
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+                                    "score": 1.0,
+                                    "content": "for BCQ and CQL). We present our results in Fig 2b (low-dim) and Fig 2c (image).",
+                                    "type": "text"
+                                }
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+                            "index": 21
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+                                    "content": "(larger LR) Increasing the learning rate from 1e-4 to 1e-3 affects the performance of image-agents",
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+                                    "score": 1.0,
+                                    "content": "), while low-dim agents are more tolerant to the change. (no GMM)",
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+                                    "score": 1.0,
+                                    "content": "Using a deterministic policy instead of learning a GMM action distribution results in significant",
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+                                    "score": 1.0,
+                                    "content": "relative performance drops on the MH datasets (especially low-dim Transport, with a drop of",
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+                                {
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "(larger MLP) Using a larger MLP size at each RNN timestep reduces performance uniformly,",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "suggesting that it is possible to overfit to dataset actions if network architectures are too large.",
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+                                    "content": "(shallow Conv) Using a shallow convolutional network [45] instead of the ResNet backbone [46] for",
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+                                    "score": 1.0,
+                                    "content": "suggesting that large-capacity visual encoders are crucial for visuomotor imitation. (smaller RNN",
+                                    "type": "text"
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+                                    "bbox": [
+                                        106,
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+                                    "score": 1.0,
+                                    "content": "dim) Reducing the size of the RNN hidden dimension from 400 to 100 (low-dim) and 1000 to 400",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(image) uniformly decreases performance (drops of",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.85,
+                                    "content": "3 \\% . 5 8 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "), showing the importance of a large",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "RNN hidden dimension. (Recommendations) We recommend tuning the LR (especially for image",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "agents) and network structure (MLP size, size of RNN dim) carefully. Opting to use a GMM policy",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and a ResNet encoder appears to be uniformly better.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
+                        }
+                    ],
+                    "index": 28.5
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+                {
+                    "type": "title",
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+                                    "content": "4.5 Selecting a Policy to Evaluate (C4)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        }
+                    ],
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+                                    "content": "Model selection in offline policy learning can be challenging – for this reason, in our simulation",
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+                            ],
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+                                    "content": "experiments, we evaluated every policy checkpoint online and reported the best one. This is not",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "feasible for real-world settings, making offline policy selection desirable. In Fig 4a, we show that this",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "can be non-trivial, by showing the relative performance drop when selecting the policy using the best",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "loss on validation data (common in supervised learning), and when using the final training checkpoint",
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+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "as well (common in offline RL [47, 18, 27]) – in both cases, the selected policy is significantly worse",
+                                    "type": "text"
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+                            ],
+                            "index": 42
+                        },
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+                                    "score": 1.0,
+                                    "content": "than the best one (",
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+                                    "score": 0.89,
+                                    "content": "10 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 0.86,
+                                    "content": "100 \\%",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "decrease). See Appendix G for more detailed results and discussion.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "This motivates the need for better offline evaluation metrics.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 44
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+                    ],
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+                                    "content": "4.6 Effect of Dataset Size (C3)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 45
+                        }
+                    ],
+                    "index": 45
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+                                    "score": 1.0,
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+                                    "content": "subsets of our",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "human datasets by sampling trajectories. We evaluate low-dim and image BC-RNN agents across",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 47
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "these subsets in Table 27 and Table 28. There are several promising results here. We first note that",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 48
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "less complex tasks (Lift, Can) can yield proficient policies",
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+                                },
+                                {
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+                                    "score": 0.84,
+                                    "content": "7 5 \\% - 1 0 0 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "success rate) using a small",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "fraction of the data",
+                                    "type": "text"
+                                },
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+                                    "score": 0.88,
+                                    "content": "( 2 0 \\% )",
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+                                    "score": 1.0,
+                                    "content": ". Second, while policies trained on more complex tasks (Square, Transport)",
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+                            ],
+                            "index": 50
+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "of the data, the converse is also true – adding more data",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 51
+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "Features used for robot proprioception can matter. In Fig 2a, we study the effect of adding end ef-",
+                                    "type": "text"
+                                }
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+                        },
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+                                    "score": 0.35,
+                                    "content": "\\left( + \\operatorname { J o i n t } \\right)",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Surprisingly, we find that including end effector velocity information, and joint information",
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+                                    "score": 1.0,
+                                    "content": "hurts agents trained on low-dim observations substantially",
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+                                    "score": 0.8,
+                                    "content": "( 4 9 \\% - 8 8 \\%",
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+                                    "score": 1.0,
+                                    "content": "relative performance drop),",
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+                            ],
+                            "index": 3
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "while image-based agents are more tolerant to the inclusion of this extra information",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.8,
+                                    "content": "2 \\% - 2 9 \\%",
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "relative performance drop). We hypothesize that performance drops might be due to overfitting",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "to the presence of this extra information not needed for solving these tasks. Thus, practitioners",
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+                            ],
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+                        },
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+                                    "content": "should take care to engineer the robot observation space and exclude possibly irrelevant information –",
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+                                    "content": "information-hiding can be a powerful paradigm for training proficient robots [44].",
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+                                    "score": 1.0,
+                                    "content": "Image randomization and wrist observations can be crucial for manipulation tasks. In Fig 2a,",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "camera (- Wrist) from image-based agents to understand their importance. We see that removing",
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+                                    "score": 1.0,
+                                    "content": "relative performance drops on Square and Transport respec-",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "relative drops. Consequently, both wrist",
+                                    "type": "text"
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+                            ],
+                            "index": 13
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+                        {
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+                                    "score": 1.0,
+                                    "content": "camera images and image randomization play a substantial role in producing performant policies. We",
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+                            ],
+                            "index": 14
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "confirm the importance of each for visuomotor imitation in the real world as well (see Sec 4.7). Wrist",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
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+                                    "score": 1.0,
+                                    "content": "observations likely help the robot improve gripper alignment for grasping and randomization helps",
+                                    "type": "text"
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+                            ],
+                            "index": 16
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the policy develop invariance for portions of the image that are not important for action prediction.",
+                                    "type": "text"
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+                    ],
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "subset of them to report practical recommendations for learning from human datasets (see Appendix I",
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+                                    "score": 1.0,
+                                    "content": "for BCQ and CQL). We present our results in Fig 2b (low-dim) and Fig 2c (image).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "(larger LR) Increasing the learning rate from 1e-4 to 1e-3 affects the performance of image-agents",
+                                    "type": "text"
+                                }
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+                                    "score": 0.72,
+                                    "content": "3 5 \\% - 6 3 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "), while low-dim agents are more tolerant to the change. (no GMM)",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "Using a deterministic policy instead of learning a GMM action distribution results in significant",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "relative performance drops on the MH datasets (especially low-dim Transport, with a drop of",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.86,
+                                    "content": "58 \\%",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ").",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "(larger MLP) Using a larger MLP size at each RNN timestep reduces performance uniformly,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "suggesting that it is possible to overfit to dataset actions if network architectures are too large.",
+                                    "type": "text"
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+                            ],
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+                                    "content": "(shallow Conv) Using a shallow convolutional network [45] instead of the ResNet backbone [46] for",
+                                    "type": "text"
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+                                    "score": 0.8,
+                                    "content": "2 5 \\% { - } 6 2 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ",",
+                                    "type": "text"
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "suggesting that large-capacity visual encoders are crucial for visuomotor imitation. (smaller RNN",
+                                    "type": "text"
+                                }
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+                        },
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "dim) Reducing the size of the RNN hidden dimension from 400 to 100 (low-dim) and 1000 to 400",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(image) uniformly decreases performance (drops of",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.85,
+                                    "content": "3 \\% . 5 8 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "), showing the importance of a large",
+                                    "type": "text"
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "RNN hidden dimension. (Recommendations) We recommend tuning the LR (especially for image",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "agents) and network structure (MLP size, size of RNN dim) carefully. Opting to use a GMM policy",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and a ResNet encoder appears to be uniformly better.",
+                                    "type": "text"
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+                            "index": 35
+                        }
+                    ],
+                    "index": 28.5,
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+                                    "score": 1.0,
+                                    "content": "4.5 Selecting a Policy to Evaluate (C4)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
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+                    ],
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+                                    "content": "Model selection in offline policy learning can be challenging – for this reason, in our simulation",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "experiments, we evaluated every policy checkpoint online and reported the best one. This is not",
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+                                    "score": 1.0,
+                                    "content": "feasible for real-world settings, making offline policy selection desirable. In Fig 4a, we show that this",
+                                    "type": "text"
+                                }
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+                                    "score": 1.0,
+                                    "content": "can be non-trivial, by showing the relative performance drop when selecting the policy using the best",
+                                    "type": "text"
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+                            ],
+                            "index": 40
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "loss on validation data (common in supervised learning), and when using the final training checkpoint",
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+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "as well (common in offline RL [47, 18, 27]) – in both cases, the selected policy is significantly worse",
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+                            "index": 42
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+                                    "score": 1.0,
+                                    "content": "than the best one (",
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+                                    "score": 0.86,
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+                                    "score": 1.0,
+                                    "content": "decrease). See Appendix G for more detailed results and discussion.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
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+                        {
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+                                    "score": 1.0,
+                                    "content": "This motivates the need for better offline evaluation metrics.",
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+                    ],
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "human datasets by sampling trajectories. We evaluate low-dim and image BC-RNN agents across",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 47
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "these subsets in Table 27 and Table 28. There are several promising results here. We first note that",
+                                    "type": "text"
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+                            ],
+                            "index": 48
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+                        {
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+                                    "score": 0.84,
+                                    "content": "7 5 \\% - 1 0 0 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "success rate) using a small",
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+                            ],
+                            "index": 49
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+                        {
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+                                    "score": 1.0,
+                                    "content": "fraction of the data",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.88,
+                                    "content": "( 2 0 \\% )",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". Second, while policies trained on more complex tasks (Square, Transport)",
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+                            ],
+                            "index": 50
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+                                    "score": 1.0,
+                                    "content": "suffer substantially when using",
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+                                    "score": 0.87,
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "of the data, the converse is also true – adding more data",
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+                            "index": 51
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+                                    "cross_page": true
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+                                    "score": 1.0,
+                                    "content": "size) can result in significant policy improvement.",
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+                                    "cross_page": true
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+                                    "content": "This confirms the value of using large human datasets as a means to obtain proficient policies for",
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+                                    "cross_page": true
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+                                    "score": 1.0,
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+                                    "content": "This confirms the value of using large human datasets as a means to obtain proficient policies for",
+                                    "type": "text"
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+                            ],
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+                                    "content": "4.7 Applicability to Real-World Settings",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 3
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+                    ],
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+                                    "content": "Here, we show that design decisions made in simulation can potentially transfer to real world settings.",
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+                            ],
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+                                    "content": "We collected 3 additional real-world datasets with a Franka robotic arm – Lift (Real), Can (Real), and",
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+                                    "content": "Tool Hang (Real). Each consists of 200 trajectories collected by one operator. We train BC-RNN and",
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+                            ],
+                            "index": 6
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+                        {
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+                                    "score": 1.0,
+                                    "content": "report the final policy checkpoint success rate, over 30 rollouts, due to the time-consuming nature",
+                                    "type": "text"
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+                            ],
+                            "index": 7
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+                                    "score": 1.0,
+                                    "content": "of real world policy evaluation. We also emphasize that no real-world hyperparameter tuning took",
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+                                    "score": 1.0,
+                                    "content": "place, so our results are a lower bound. We were able to train proficient Lift",
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+                                    "content": "( 9 6 . 7 \\% )",
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+                                    "score": 1.0,
+                                    "content": "policy is able to generate some task successes, despite",
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+                                    "score": 1.0,
+                                    "content": "the extremely difficult nature of the task. Furthermore, as in Sec. 4.3, we validate the importance",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "of pixel shift randomization and the wrist camera by ablating each component on the Can task, and",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "show that including both is the difference between a proficient and non-proficient real-world policy –",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": ".",
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+                    ],
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+                {
+                    "type": "title",
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+                                    "type": "text"
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+                    ],
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+                                    "score": 0.32,
+                                    "content": "\\mathrm { S e c 4 . 1 }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "and Sec 4.2, we demonstrated that history-dependent models (BC-RNN, HBC,",
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+                                    "score": 1.0,
+                                    "content": "and IRIS) are particularly effective in learning from human datasets compared to algorithms that do",
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "not take temporal context into account.",
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+                    ],
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+                                    "content": "(L2) Need to improve the ability of batch (offline) RL to learn from suboptimal human datasets.",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "score": 1.0,
+                                    "content": "and Appendix I demonstrated that state-of-the-art batch RL algorithms are excellent at",
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+                                    "score": 1.0,
+                                    "content": "human datasets. They even struggled with a diagnostic dataset with paired good and bad human",
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+                                    "score": 1.0,
+                                    "content": "demonstration trajectories while IRIS was able to improve slightly on BC-RNN, suggesting that",
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+                                    "content": "RL methods [48–51]. This also demonstrates a need to start benchmarking new batch RL algorithms",
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+                                    "score": 1.0,
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+                                    "content": "(L3) Improving offline policy selection is important for real world settings. Sec 4.5 demonstrated",
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+                                    "content": "policy learning. The choice of observation space for imitation merits careful consideration – other",
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+                                    "score": 1.0,
+                                    "content": "ing [52, 53]. Sec 4.4 and Appendix I made practical recommendations for choosing hyperparameters",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "complex tasks. Table 3 and Sec 4.7 shows that we could learn proficient policies on the Tool Hang",
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+                                }
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+                                    "score": 1.0,
+                                    "content": "task, our most complex task, without any hyperparameter tuning on the task or dataset. Together,",
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+                                    "score": 1.0,
+                                    "content": "these results show the potential of large human datasets as a means to solve challenging and complex",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "manipulation tasks.",
+                                    "type": "text"
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "(L6) Study results transfer to real-world settings. In Sec 4.7, we showed that we could directly",
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+                                    "score": 1.0,
+                                    "content": "and Sec 4.2, we demonstrated that history-dependent models (BC-RNN, HBC,",
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+                                    "score": 1.0,
+                                    "content": "and Appendix I demonstrated that state-of-the-art batch RL algorithms are excellent at",
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+                                    "type": "text"
+                                }
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diff --git a/parse/train/b36m4ZYG1gD/b36m4ZYG1gD.md b/parse/train/b36m4ZYG1gD/b36m4ZYG1gD.md
new file mode 100644
index 0000000000000000000000000000000000000000..d21d5cc6d953d35e5dbe3357df69daae538cd292
--- /dev/null
+++ b/parse/train/b36m4ZYG1gD/b36m4ZYG1gD.md
@@ -0,0 +1,382 @@
+# ML4C: Seeing Causality Through Latent Vicinity
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Supervised Causal Learning (SCL) aims to learn causal relations from observational   
+2 data by accessing previously seen datasets associated with ground truth causal   
+3 relations. This paper presents a first attempt at addressing a fundamental question:   
+4 What are the benefits from supervision and how does it benefit? Starting from seeing   
+5 that SCL is not better than random guessing if the learning target is non-identifiable   
+6 a priori, we propose a two-phase paradigm for SCL by explicitly considering   
+7 structure identifiability. Following this paradigm, we tackle the problem of SCL on   
+8 discrete data and propose ML4C. The core of ML4C is a binary classifier with a   
+9 novel learning target: it classifies whether an Unshielded Triple (UT) is a v-structure   
+10 or not. Starting from an input dataset with the corresponding skeleton provided,   
+11 ML4C orients each UT once it is classified as a v-structure. These v-structures are   
+12 together used to construct the final output. To address the fundamental question   
+13 of SCL, we propose a principled method for ML4C featurization: we exploit the   
+14 vicinity of a given UT (i.e., the neighbors of UT in skeleton), and derive features by   
+15 considering the conditional dependencies and structural entanglement within the   
+16 vicinity. We further prove that ML4C is asymptotically perfect. Last but foremost,   
+17 thorough experiments conducted on benchmark datasets demonstrate that ML4C   
+18 remarkably outperforms other state-of-the-art algorithms in terms of accuracy,   
+19 robustness and transferability. In summary, ML4C shows promising results on   
+20 validating the effectiveness of supervision for causal learning.
+
+# 21 1 Introduction
+
+22 The problem of causal learning is to learn causal relations from observational data [13]. The learned   
+23 causal relations are typically represented in the form of a Directed Acyclic Graph (DAG), where each   
+24 edge in the DAG indicates direct cause-effect relation between the parent node and child node.   
+25 The methods of causal learning mostly fall into four categories: constraint-based, score-based,   
+26 continuous optimization method and functional causal models. Each of these methods takes a given   
+27 dataset as input and outputs a DAG but with different criteria. For instance, the DAG should be   
+28 consistent with conditional independencies in the data (constraint-based); or it is optimal w.r.t. a   
+29 pre-defined score function under either combinatorial constraint (score-based) or continuous equality   
+30 constraint (continuous optimization). In a nutshell, these methods can be viewed as unsupervised   
+31 since they do not access additional datasets associated with ground truth causal relations.   
+32 A new line of research called Supervised Causal Learning (SCL), on the other hand, aims to learn   
+33 causal relations in the supervised fashion: the algorithm has access to datasets associated with   
+34 ground truth causal relations, in the hope that such supervision is beneficial to learning causal   
+35 relations on newly unseen datasets. Despite several existing works on this direction (see Related   
+36 Work), a fundamental question remains unanswered: How is supervised causal learning possible?   
+37 Specifically, compared with unsupervised causal learning methods, can we gain additional benefits   
+38 from supervision? If the answer is positive, then what are the benefits and how does it benefit?   
+39 We tackle the problem by first seeing crucial connection between SCL and causal structure identi  
+40 fiability. Considering the problem of causal learning on discrete data, theorem in [24] states that,   
+41 under standard assumptions (i.e., Markov assumption, faithfulness and causal sufficiency), we can   
+42 only identify a graph up to its Markov equivalence class. Markov equivalence class is the set of   
+43 DAGs having same skeleton and same v-structures, which can be represented by CPDAG (Com  
+44 pleted Partially Directed Acyclic Graph). Thus, the (un)directed edges in the CPDAG indicate   
+45 (non-)identifiable causal relations. Each non-identifiable edge in CPDAG can be oriented by either   
+46 direction to equivalently fit the observational data. Given an SCL algorithm with learning target as   
+47 the orientation of an edge, we see that it is not better than random guessing (or could be worse due to   
+48 sample bias in training data) to predict any non-identifiable edge since we can assign either $X  Y$   
+49 or $X  Y$ with same input dataset. This statement is applicable to general learning target since an   
+50 SCL algorithm can take different target such as orientation of an edge, the whole DAG, or others.
+
+![](images/b27c9954d5c1984f08d48654bd07fafec8c6793034aeffc73f282a018f11ef7b.jpg)  
+Figure 1: (a) Two-phase paradigm for supervised causal learning. (b) ML4C’s workflow.
+
+Proposition 1. If the learning target is non-identifiable (i.e., every edge in the target is nonidentifiable) a priori, then SCL is not better than random guessing.
+
+53 Consequently, we propose and advocate a two-phase paradigm for SCL, as depicted in Figure 1(a):   
+54 phase one corresponds to a binary classification task, where an SCL algorithm needs to classify   
+55 whether a specific learning target is identifiable or not; only if it is classified as identifiable, then we   
+56 go to phase two to classify the specific orientation of the learning target. Following this paradigm,   
+57 we tackle the problem of SCL on discrete data and propose an algorithm ML4C. The core of ML4C   
+58 is a binary classifier with a novel learning target: it classifies whether an Unshielded Triple (UT: a   
+59 triple of variables $\langle X , T , Y \rangle$ where $X$ and $Y$ are adjacent to $T$ but are not adjacent to each other) is a   
+60 v-structure or not. Starting from an input dataset with the corresponding skeleton provided, ML4C   
+61 orients each UT once it is classified as a v-structure. These v-structures are further used to construct   
+62 a CPDAG as output. Such a single classifier facilitates both learning tasks in the two phases, since an   
+63 identifiable UT implies that it is a v-structure [32] (i.e., up to the partial DAG before applying Meek   
+64 rules [23] which is a standard post processing).   
+65 To address the fundamental question of SCL, we propose a principled method for ML4C featurization.   
+66 Specifically, we exploit the vicinity of a given UT (i.e., the neighbors of UT in skeleton), and derive   
+67 features by considering the conditional dependencies and structural entanglement within the vicinity.   
+68 We further define discriminative predicate (i.e., a binary predicate function with domain as ML4C’s   
+69 feature set) and prove that there exist weak discriminative predicates and strong discriminative   
+70 predicates (i.e., values of the predicates are one-to-one correspondence with ground truth labels).   
+71 We further prove that ML4C is asymptotically perfect. Last but foremost, thorough experiments on   
+72 benchmark datasets demonstrate that ML4C remarkably outperforms other state-of-the-art algorithms   
+73 w.r.t. accuracy, robustness and transferability. Our main contributions are summarized as follows:
+
+74 1. We advocate the two-phase paradigm for SCL with consideration of causal structure identifiability.
+
+75 2. We propose an SCL algorithm ML4C, with the following novelties: i) Learning Target: The core   
+76 of ML4C is a binary classifier with the orientation of a UT as its learning target to address the two  
+77 phase tasks simultaneously. ii) Featurization: A principled method to exploit vicinity information   
+78 in terms of dependencies and entanglement of a given UT. iii) Learnability: We prove that ML4C   
+79 is asymptotically perfect. iv) Empirical Performance: Experiments conducted on benchmark   
+80 datasets demonstrate that ML4C remarkably outperforms other state-of-the-art algorithms.
+
+# 81 2 Related Work
+
+82 We divide literature on causal learning into supervised and unsupervised approaches, depending on   
+83 whether additional datasets (associated with ground truth causal relations) are accessed (supervised)   
+84 or not (unsupervised). In the literature of unsupervised causal learning, constraint-based methods aim   
+85 to identify a DAG which is consistent with conditional independencies. The learning procedure of   
+86 constraint-based methods first identifies the corresponding skeleton and then conducts orientation   
+87 based on v-structure identification [34]. The typical algorithm is PC [31], and there are also PC  
+88 derived algorithms such as Conservative-PC [26], PC-stable [7] and Consistent-PC [19] which   
+89 improve the robustness on v-structure identification. Score-based methods aim to find the DAG which   
+90 is optimal w.r.t. a pre-defined score function under combinatorial constraint by a specific search   
+91 procedure, such as forward-backward search GES [6], hill-climbing [16], integer programming [8], or   
+92 by approximate algorithms based on order search [27]. Continuous optimization methods transform   
+93 the discrete search procedure into continuous equality constraint: NOTEARS [36] formulates the   
+94 acyclic constraint as a continuous equality constraint, it is further extended by DAG-GNN [35] to   
+95 support learning non-linear causal relations.   
+96 SCL emerges from the task of orienting edge in the continuous, non-linear bivariate case under   
+97 Functional Causal Model (FCM) formalism. Given a collection of cause-effect samples (dataset   
+98 $\sim$ binary label indicating whether $X  Y$ or $X  Y$ ), supervised approaches such as RCC [20],   
+99 NCC [21], D2C [4] and Jarfo [12] achieve better performance on predicting pairwise relations (i.e.,   
+100 orientation of an edge) than unsupervised approaches such as ANM [14] or IGCI [15]. Differently,   
+101 [18] sets the learning target as the whole DAG structure instead of pairwise relation and it is applied on   
+102 data which is generated by linear Structural Equation Model (SEM). We summarize the differences in   
+103 problem space between ML4C and the other SCL approaches as follows: i) We advocate a two-phase   
+104 learning paradigm and emphasize the relationship between identifiability and learnability. Specifically,   
+105 presuming additive noise model [14] or linear SEM with non-Gaussian noise [29] provides license   
+106 to identifiability thus the aforementioned approaches can be viewed as specific tasks in phase two.   
+107 ii) We tackle SCL’s learnability not only via empirical evaluation but also by theoretical analysis to   
+108 shed light on the fundamental question of learnability. iii) ML4C deals with discrete data while other   
+109 approaches mainly focus on continuous data.
+
+# 110 3 Background
+
+# 3.1 Basic Notations
+
+A discrete dataset $D _ { i }$ consists of $n _ { i }$ records and $d _ { i }$ categorical columns, which represents $n _ { i }$ instances drawn i.i.d. from $d _ { i }$ discrete variables $X _ { 1 } , X _ { 2 } , \cdots , X _ { d _ { i } }$ by a joint probability distribution $P _ { i }$ , which is entailed by an underlying data generating process, denoted as DAG $G _ { i }$ .
+
+Markov factorization property: Given a joint probability distribution $P$ and a DAG $G$ , $P$ is said to satisfy Markov factorization property w.r.t. $G$ if $\begin{array} { r } { P : = P \left( X _ { 1 } , X _ { 2 } , \cdots , X _ { d } \right) = \prod _ { i = 1 } ^ { d } P \left( X _ { i } | \mathsf { p a } _ { i } ^ { G } \right) } \end{array}$ where $\mathrm { p a } _ { i } ^ { G }$ is the parent set of $X _ { i }$ in $G$ .
+
+Markov assumption: $P$ is said to satisfy Markov assumption (or Markovian) w.r.t. a DAG $G$ if $X \bot _ { G } Y | Z \Rightarrow X \bot Y | Z$ . Here $\perp _ { G }$ denotes $\mathrm { d }$ -separation, and $\perp$ denotes statistical independence. Markov assumption indicates that any d-separation in graph $G$ implies conditional independence in distribution $P$ . Markov assumption is equivalent to Markov factorization property [17].
+
+Faithfulness: Distribution $P$ is faithful w.r.t. a DAG $G$ if $X \bot Y | Z \Rightarrow X \bot _ { G } Y | Z$ .
+
+123 Canonical dataset: We say a discrete dataset $D$ is canonical if its underlying probability distribution   
+124 $P$ is Markovian and faithful w.r.t. some DAG $G$ .
+
+# 25 3.2 Causal Structure Identifiability
+
+Identifiability discusses which parts of the causal structure can in principle be inferred from the distribution. Below we present the established theory of identifiability on discrete data.
+
+Causal sufficiency: There are no latent common causes of any of the variables in the graph.
+
+Definition 1 (Markov equivalence). Two graphs are Markov equivalent if and only if they have same skeleton and same $\nu$ -structures. A Markov equivalence class can be represented by a CPDAG having both directed and undirected edges. A CPDAG can be derived from a DAG G, denoted as $C P D A G ( G )$ . The theorem of Markov completeness in [24] states that, under causal sufficiency, we can only identify a causal graph up to its Markov equivalence class on canonical data. Therefore, the (non-)identifiable causal relations are the (un)directed edges in the CPDAG. Formally,
+
+Definition 2 (Identifiability). Assuming $P$ is Markovian and faithful w.r.t. a DAG G and causal sufficiency, then each (un)directed edge in $C P D A G ( G )$ indicates (non-)identifiable causal relation.
+
+# 3.3 ML4C Related Notations
+
+Definition 3 (Skeleton). A skeleton $E$ defined over distribution $P \left( X _ { 1 } , X _ { 2 } , \cdots , X _ { d } \right)$ is an undirected graph such that there is an edge between $X _ { i }$ and $X _ { j }$ if and only if $X _ { i }$ and $X _ { j }$ are always dependent, i.e., $\nexists Z \ \subseteq \ \{ X _ { 1 } , X _ { 2 } , \cdot \cdot \cdot \ , X _ { d } \}$ s.t. $X _ { i } \bot X _ { j } | Z$ . Skeleton is a statistical concept, which can be obtained prior to facilitating various downstream tasks. Recently, there have been some novel skeleton learning algorithms such as [10]. In particular, skeleton can be used for causal learning: theorem in [32] states that if distribution $P$ is Markovian and faithful w.r.t. a DAG $G$ , then skeleton $E$ is the same as the undirected graph of $G$ .
+
+145 Definition 4 (UT). A triple of variables $\langle X , T , Y \rangle$ in a skeleton is an unshielded triple, or short   
+146 for UT, if $X$ and $Y$ are adjacent to $T$ but are not adjacent to each other. $\langle X , T , Y \rangle$ can be further   
+47 oriented to become a $\nu$ -structure $X \right. T \left. Y$ , in which $T$ is called the collider.   
+148 Definition 5 (PC). Denote the set of parents and children of $X$ in a skeleton as $P C _ { X }$ , in other words,   
+49 $P C _ { X }$ are the neighbors of $X$ in the skeleton. For convenience, if we discuss $P C _ { X }$ in the context of $a$   
+50 $U T \left. X , T , Y \right.$ , we intentionally mean the set of parents and children of $X$ but exclude $T$ .
+
+Definition 6 (Vicinity). We define the vicinity of a51 $U T \left. X , T , Y \right.$ as $V _ { \langle X , T , Y \rangle } : = \{ X , T , Y \} \cup P C _ { X } \cup$ 52 $P C _ { Y } \cup P C _ { T }$ . Vicinity is a generalized version of $P C$ , i.e., the neighbors of $\{ X , T , Y \}$ in the skeleton.
+
+153 ML4C’s training set: The training set is a collection of discrete datasets $D _ { 1 } , \cdots , D _ { n }$ , where each   
+154 dataset $D _ { i }$ is associated with a ground truth DAG $G _ { i }$ , such that $D _ { i }$ is sampled from $G _ { i }$ . $G _ { i }$ derives   
+155 labels (depends on the chosen learning target), thus $\{ D _ { i } , G _ { i } \} _ { i \in \{ 1 , \cdots , n \} }$ form ML4C’s training set.   
+156 We can sample graphs from DAG space and generate corresponding datasets, thus obtaining training   
+157 set in our problem is straightforward.
+
+# 4 Approach
+
+# 4.1 Overview
+
+The core of ML4C is a binary classifier called ML4C-Learner, which takes the orientation of a UT as its learning target, i.e., it classifies whether an input UT $\langle X , T , Y \rangle$ is a v-structure (orientation: $X \right. T \left. Y$ ) or not (orientation remains unknown). Figure 1(b) depicts the overall workflow of ML4C, which is composed of ML4C-Learner with other important inductive biases. Starting from an input dataset $D _ { i }$ with corresponding skeleton $E _ { i }$ , we first obtain all the UTs from $E _ { i }$ . Featurization is then conducted to represent each UT as an embedded vector, which is further fed into ML4C-Learner. In the inference stage, we obtain all the v-structures which are classified by ML4C-Learner and reconstruct a partial DAG and then, a CPDAG is output by applying Meek rules on the partial DAG. In the training stage, the label of each UT is obtained by querying from ground truth DAG $G _ { i }$ . We collect labeled data from multiple datasets in ML4C’s training set.
+
+Proposition 2. If ML4C-Learner is a perfect classifier, then ML4C outputs correct CPDAG of a canonical dataset (i.e., ML4C is perfect).
+
+172 By Markov completeness, the set of v-structures is invariant across all Markov equivalent DAGs for   
+173 a canonical dataset, and it can fully recover the CPDAG, provided that the skeleton is given. Thus,   
+74 besides its dedicated role in phase 2, ML4C-Learner also facilitates learning task in phase 1 since an   
+75 identifiable UT implies that it is a v-structure (up to the partial DAG before applying Meek rules).
+
+# 4.2 Featurization
+
+We propose a principled method for ML4C-Learner’s featurization, which avoids the need of handcrafted features. More importantly, we further prove that ML4C-Learner is asymptotically perfect.
+
+Design Principles: Our key aspect of featurization is to broaden focus from a specific UT $\langle X , T , Y \rangle$ to its vicinity and seeking conditional dependencies and structural entanglement within the vicinity, to reveal reliable and robust asymmetry to distinguish v-structure and non-v-structure UTs. Specifically, conditional dependencies are the key materials for traditional causal learning methods (e.g., conditional independences for constraint-based methods), and structural entanglement (e.g., $P C _ { X } = P C _ { T } .$ ) are relevant to identifiability: higher entanglement makes the UT less likely to be identifiable.
+
+# • Dependencies within Vicinity
+
+Conditional dependency: Denoted as $X \sim Y | \mathbf { Z }$ , which is a non-negative scalar that measures the dependence between two random variables $X$ and $Y$ given variable set $\mathbf { Z }$ . Operationally, $X \sim Y | \mathbf { Z }$ is composed of two parts, bivariable $X \sim Y$ , and conditional Z. We further extend the definition to allow a set of variables in bivariable, and an ensemble (i.e., a set of set) as conditional:
+
+Extended conditional dependency: Denoted as $\mathbf { A } \sim \mathbf { B } | { \mathcal { Z } } { : = \{ X \sim Y | \mathbf { Z } : X \in \mathbf { A } , Y \in \mathbf { B } , \mathbf { Z } \in } $ ${ \mathcal { Z } } \}$ , where A and $\mathbf { B }$ are set of variables, and $\mathcal { Z }$ is an ensemble. Thus, extended conditional dependency is a set of scalars.
+
+Within the vicinity of $\langle X , T , Y \rangle$ , we start from measuring dependencies between $\{ X , P C _ { X } \}$ and $\{ Y , P C _ { Y } \}$ by conditioning on $\{ T , P C _ { T } \}$ . Intuitively, if $\langle X , T , Y \rangle$ is a v-structure, conditioning on $T$ or $T$ ’s descendants tends to strengthen the dependency between $P C _ { X }$ and $P C _ { Y }$ since the paths passing $X - T - Y$ are unblocked; otherwise, conditioning on $T$ tends to weaken the dependency between $P C _ { X }$ and $P C _ { Y }$ because $T$ blocks the paths passing $X - T - Y$ . Therefore, such conditional dependencies reflect potential asymmetry to distinguish v-structure and non-v-structure. Formally,
+
+Definition 7 (Domain of bivariable). Denoted as $\mathbb { B } \ : = \ \{ X , P C _ { X } \} \ \times \ \{ Y , P C _ { Y } \} \ \equiv$ $\{ X \sim Y , X \sim P C _ { Y } , P C _ { X } \sim Y , P C _ { X } \sim P C _ { Y } \}$ , here symbol $^ { \prime } \times$ is Cartesian product.
+
+Definition 8 (Sepsets). Denoted as $\begin{array} { r } { \mathcal { S } : = \{ S : X \bot Y | S , S \subset P C _ { X } \cup T } \end{array}$ , or $S \subset P C _ { Y } \cup T \}$ . Under faithfulness assumption, sepsets $s$ is an ensemble where each item is a subset of variables within the vicinity that d-separates $X$ and $Y$ .
+
+Definition 9 (Domain of conditional). Denoted as $\begin{array} { r c l } { \mathbb { C } } & { \colon = } & { \{ \emptyset , T , \mathcal { P C } _ { T } \} \ \vee \ \{ \emptyset , S \} \quad \equiv } \end{array}$ $\{ \emptyset , T , \mathcal { P } \mathcal { C } _ { T } , S , S \vee T , S \vee \mathcal { P } \mathcal { C } _ { T } \}$ , where $\mathcal { P C } _ { T } : = \{ \{ I \} : I \in P C _ { T } \}$ which is an ensemble version of $P C _ { T }$ , and $\mathcal { S } \vee \mathcal { P } \mathcal { C } _ { T } : = \left\{ S \cup I : S \in \mathcal { S } , I \in \mathcal { P } \mathcal { C } _ { T } \right\}$ . Here symbol ∨ is element-wise union.
+
+We exploit the extended conditional dependencies from $\mathbb { B } \times \mathbb { C }$ , i.e., we pick a bivariable from $\mathbb { B }$ and a conditional from $\mathbb { C }$ , and calculate the extended conditional dependency. There are in total $\lvert \mathbb { B } \rvert \times \lvert \mathbb { C } \rvert = 2 4$ extended conditional dependencies.
+
+Lemma 1. Sepsets $s$ of any UT of a canonical dataset is non-empty. All proofs are available in the supplementary material.
+
+Remark 1. We intend to restrict the sepsets within the vicinity of $\langle X , T , Y \rangle$ . Lemma 1 shows the existence of such $d$ -separation sets within vicinity. Furthermore, searching for all $d$ -separation sets is highly time-consuming, thus the computational cost can also be saved drastically.
+
+# • Entanglement within Vicinity
+
+Structural entanglement reflects complex structure within the vicinity of $\langle X , T , Y \rangle$ . Variables $X , Y$ and $T$ can mutually share common neighbors, and their neighbors may also overlap with sepsets $s$ . We call such overlaps structural entanglement. Intuitively, stronger entanglement indicates denser structure of vicinity thus making the UT less likely to be identifiable. Therefore, structural entanglement is an important aspect for featurization. Specifically, we exploit the overlap coefficient [33] to measure the entanglement:
+
+Definition 10 (Overlap coefficient). OLP $( \mathbf { A } , \mathbf { B } ) : = | \mathbf { A } \cap \mathbf { B } | / \operatorname* { m i n } \left( | \mathbf { A } | , | \mathbf { B } | \right)$ , where A and $\mathbf { B }$ are 223 two sets of variables. We extend this formula to support ensemble as input:
+
+(Extended) Overlap coefficient: OLP 224 $( \mathbf { A } , S ) : = \textstyle \sum _ { i = 1 } ^ { | S | }$ OLP $\left( \mathbf { A } , S _ { i } \right) / \left| S \right|$ . Naturally, we consider the entanglement in terms of overlap coefficient on each pair of items in domain Thus, we use 6 scalars to represent the entanglement within the vicinity of a UT.
+
+# • Embedding
+
+We aim to represent the dependencies and entanglement by a feature vector with fixed dimensionality, which can be used to train ML4C-Learner. Regarding each extended conditional dependency $\mathbf { A } \sim$ $\mathbf { B } | \mathcal { Z } : \mathbf { A } \sim \mathbf { B } \in \mathbb { B } , \mathcal { Z } \in \mathbb { C }$ , it consists of a set of scalars with varied set size across UTs, we adopt the kernel mean embedding technique in [30] to represent each $\mathbf { A } \sim \mathbf { B } | \mathcal { Z }$ as a vector with fixed dimensionality. We further modify the embedding algorithm by adding min $\{ \mathbf { A } \sim \mathbf { B } | \mathcal { Z } \}$ and max $\{ \mathbf { A } \sim \mathbf { B } | \mathcal { Z } \}$ as two additional features. We directly use the 6 scalars to represent structural entanglement without further transformation. We concatenate all the embedded vectors to form the final feature vector, as input for ML4C-Learner.
+
+# 36 4.3 Learnability
+
+We have presented ML4C’s featurization and started seeing that conditional dependencies and structural entanglement have potential to reveal asymmetry to distinguish v-structure and non-vstructure UTs. Now we provide rigorous analysis to show that, for a canonical dataset with sufficient samples, ML4C-Learner tends to a perfect classifier. To prove this, we first propose a surrogate object called discriminative predicate:
+
+Definition 11 (Discriminative predicate). A discriminative predicate is a binary predicate function with domain as ML4C’s feature set. A discriminative predicate can be viewed as a special classifier with pre-specified form of mechanism (i.e., not learned from data).
+
+Definition 12 (Weak / Strong discriminative predicate). Whenever a discriminative predicate takes the feature vector of a UT as input, a weak discriminative predicate satisfies one of the following two criteria; a strong discriminative predicate satisfies both: i) it is evaluated to TRUE if the UT is a v-structure; ii) it is evaluated to FALSE if the UT is not a v-structure.
+
+By definition, a weak discriminative predicate exhibits discriminative power since it is evaluated false implies the UT is a non-v-structure (or true implies v-structure). A strong discriminative predicate can be viewed as a perfect classifier. Denote $\{ \mathbf { A } \sim \mathbf { B } | \mathcal { Z } \} > \delta : = X \sim Y | \mathbf { Z } > \delta : \forall X \in \mathbf { A } , Y \in$ $\mathbf { B } , \mathbf { Z } \in { \mathcal { Z } }$ , then we have:
+
+Lemma 2 (Existence of weak discriminative predicate). For a canonical dataset with infinite samples, the following are three weak discriminative predicates: i) $\{ X \sim Y | T \} > 0$ , ii) $\{ X \sim Y | \mathcal { P } \mathcal { C } _ { T } \} = 0$ , iii) $\{ P C _ { X } \overset { \cdot } { \sim } P C _ { Y } | S \cup T \} > 0 .$ .Take $\{ X \sim Y | T \} > 0$ as an example, $\langle X , T , Y \rangle$ is a v-structure $\Rightarrow T$ is a collider $\Rightarrow T$ unblocks $X$ and $Y$ through path $X - T - Y \Rightarrow \{ X \sim Y | T \} > 0 \Rightarrow$ $\operatorname* { m i n } { \{ X \sim Y | T \} } > 0$ , where $\operatorname* { m i n } { \{ X \sim Y | T \} }$ is a feature of ML4C-Learner since $X \sim Y \in$ B $\ S , \{ T \} \in \mathbb { C }$ .
+
+Lemma 3 (Existence of strong discriminative predicate). For a canonical dataset with infinite samples, the following are three strong discriminative predicates: $i$ ) $\begin{array} { r } { \mathrm { O L P } ( T , S ) = 0 , } \end{array}$ ii) $\begin{array} { r } { \mathrm { o L P } ( T , S ) < } \end{array}$ 0.5, iii) $\operatorname { o L P } ( T , S ) < 1 \land \operatorname* { m i n } \left\{ X \sim Y | T \cup S \right\} > 0$ .
+
+CPC/MPC/GLL-MB as special cases of ML4C-Learner: Predicate OLP $( T , S ) = 0 \Longleftrightarrow \forall S \in$ $s$ , $T \not \in S$ , which states that the predicate is TRUE if $T$ is not in any $\mathrm { d }$ -separation set of $X$ and $Y$ . Having correct skeleton provided, this is the criterion of Conservative PC algorithm (CPC) [25] for identifying v-structures. Thus, CPC can be viewed as a special case of ML4C by replacing ML4CLearner with such a pre-specified logic; OLP $( T , S ) < 0 . { \bar { 5 } }$ indicates that if more than half of the dseparation sets do not contain $T$ , then the UT is oriented as a v-structure, which is called majority rule PC algorithm (MPC) [9]; predicate OLP ( $T , S ) < 1 \land \operatorname* { m i n } \left\{ X \sim Y | T \cup S \right\} > 0 \Rightarrow \exists S \in S$ , $T \not \in S$ and $X$ and $Y$ are dependent when conditioning on $T \cup S$ , which is used for GLL-MB [2] to more securely identify v-structures. These predicates are with suboptimal performance because only a small portion of features are exploited and the overall loss function of training data is disregarded, thus in practice when an appropriate machine learning model is adopted, ML4C-Learner achieves better performance.
+
+4 Theorem 1. ML4C-Learner tends to a perfect classifier on classifying a canonical dataset with   
+5 sufficient samples.
+
+Benchmark Datasets We use discrete datasets sampled by all 24 networks from bnlearn repository [28] for evaluation. For each network, we sample 1k, 5k, 10k, 15k, 20k records for use.
+
+ML4C’s Training and Inference We generate ML4C’s training data synthetically (which is also used for other SCL competitors). Specifically, 400 unique DAGs are randomly generated by two models: Erdos-Rényi (ER) model [ ˝ 11] and Scale-Free (SF) model [1], with the number of nodes ranging from 10 to 1,000. A standard random forward data generation process is applied to obtain 10k observational samples for each graph. We further extract UTs from the $4 0 0 \mathrm { D A G s }$ , consisting of 97,010 v-structures (label $= 1$ ) and 195,691 non-v-structures $\mathrm { | a b e l = 0 }$ ). We use these instances to train ML4C-Learner, which is implemented by a XGBoost [5] binary classifier with default hyperparameters and we use binary cross-entropy as the loss function. Details on our synthesis procedure, configurations and implementation of ML4C-Learner are available in the supplementary material.
+
+Competitors We categorize state-of-the-art causal learning algorithms from two aspects, supervised vs. unsupervised, and can or cannot take skeleton as input. We choose Jarfo [12], D2C [4], RCC [20], and NCC [21] as SCL competitors. Same as ML4C, all these algorithms can and do require skeleton as input. All these algorithms use ML4C’s training set for training but with different learning target extracted. Regarding unsupervised algorithms, we choose PC [31], Conservative-PC (CPC) [26], Majority-rule PC (MPC) [7], GLL-MB (GMB) [2], GES [6], Grow-Shrink (GS) [22], Hill-Climbing (HC) [16], and Conditional Distribution Similarity (CDS) [12]. which can also take skeleton as input. Lastly, we also compare with DAG-GNN (DGNN) [35], BLIP [27], and GOBNILP (GNIP) [8], which are unsupervised algorithms but cannot take skeleton as input. All these competitors are capable of dealing with discrete data. All experiments are done in a Windows Server with 2.8GHz Intel E5-2680 CPU and 256G RAM. Details are in the supplementary material.
+
+299 Design Our evaluation mainly consists of two parts: end-to-end comparison with competitors on   
+300 benchmark datasets, and in-depth experiments on ML4C’s learnability. The latter is further divided   
+301 into four aspects: i) Towards a perfect classifier. As stated in proposition proposition 2, ML4C  
+302 Learner is the core component and we would like to know how far it is from a perfect classifier.   
+303 ii) Reliability (against weak / strong discriminative predicates). As stated in lemma 2 and 3, there   
+304 exist weak and strong discriminative predicates, which have discriminative power and thus are helpful   
+305 for ML4C-Learner. Some strong discriminative predicates are equivalent to specific logics of existing   
+306 work such as CPC or GLL-MB. Thus, we would like to see how ML4C-Learner takes the advantage   
+307 of machine learning, to learn a more reliable classification mechanism (which is also latent and   
+308 more sophisticated) than individual weak / strong discriminative predicates. iii) Robustness (against   
+309 varied sample size). It is known that many causal learning algorithms lack robustness w.r.t sample   
+310 noise for finite datasets [20], especially CI tests are error-prone on small samples for constraint  
+311 based algorithms. We would like to evaluate the robustness of ML4C (i.e., the latent classification   
+312 mechanism) against varied sample sizes. iv) Transferability. It’s important for a machine learning   
+313 model to generalize well to various types of testing data which are different from training data, such   
+314 as different scale (#nodes), graph sparsity, different generating mechanisms, etc.   
+315 Metrics We use two standard metrics for performance evaluation: Structural Hamming Distance   
+316 (SHD) and F1-score. For each dataset, we measure the SHD / F1-score of the output CPDAG (learned   
+317 by a specific algorithm) against the ground truth CPDAG. Specifically, SHD is calculated at CPDAG   
+318 level, which is the smallest number of edge additions, deletions, direction reversals and type changes   
+319 (directed vs. undirected) to convert the output CPDAG to ground truth CPDAG. F1-score is calculated   
+320 over identifiable edges. Roughly, F1-score can be viewed as a normalized version of SHD. Now we   
+321 present the experiment results:
+
+End-to-End Comparison Due to page limit, we report SHD and F1-score of all algorithms on 19 large-scale datasets (full results including other 5 smallest and trivial datasets are available in the supplementary material), as depicted in Table 1. ‘-’ means the algorithm fails on the dataset (either out-of-memory / exceeds 24 hours execution time / break caused by unknown errors). ML4C significantly outperforms all other competitors. The average F1-score of ML4C is the highest (0.92, first column in Table 2). Moreover, ML4C exhibits the most stable performance across all datasets, its average ranking is $1 . 5 \pm 0 . 7$ , while the second best is GLL-MB (GMB), with average ranking
+
+Table 1: Experiment results for end-to-end comparison with SOTA causal learning algorithms on benchmark datasets. Algorithm names are abbreviated. SHD and F1-score are reported. The last two rows show statistics of rank by SHD and F1-score for all competitors (Note: F1-score is at UT level).   
+
+<table><tr><td rowspan="2">Datasets #nodes/#edges</td><td rowspan="2"></td><td colspan="4">supervised</td><td colspan="6">unsupervised</td><td colspan="4">no skeleton input</td></tr><tr><td>ML4C</td><td>Jarfo</td><td>D2C RCC</td><td>NCC</td><td>PC CPC</td><td>MPC</td><td>GMB</td><td>GES</td><td>GS</td><td>HC</td><td>CDS</td><td>DGNN</td><td>BLIP 0</td><td>GNIP</td></tr><tr><td>child 20/25</td><td>SHD F1</td><td>.</td><td>8</td><td>16</td><td>8 </td><td>22 </td><td>.94</td><td></td><td>1</td><td>3</td><td>3</td><td></td><td>2</td><td>1.0</td><td>0 1.0</td></tr><tr><td>insurance 27/52</td><td>SHD F1</td><td>9</td><td>6</td><td>30 3 .44</td><td>28 .44</td><td>36 </td><td>6</td><td>3</td><td>44</td><td>38</td><td>1</td><td>36</td><td>8</td><td>3</td><td>1</td></tr><tr><td>water 32/66</td><td>SHD F1</td><td>4</td><td>3</td><td>34 36</td><td>0 1.0</td><td>4 60 .97 .00</td><td>3</td><td>8</td><td>38</td><td>3</td><td>38 .46</td><td>18 .76</td><td>.8</td><td></td><td>3</td></tr><tr><td>mildew 35/46</td><td>SHD F1</td><td>8</td><td>：</td><td>18 30</td><td>3</td><td>36 ：</td><td>：</td><td>35</td><td>3</td><td>8</td><td>3</td><td>8</td><td></td><td>36</td><td>：</td></tr><tr><td>alarm 37/46</td><td>SHD F1</td><td>.8</td><td>3</td><td>4 18 .64</td><td>39</td><td>29 29</td><td>品</td><td>4</td><td>8</td><td>4</td><td>26</td><td>8</td><td></td><td></td><td>8</td></tr><tr><td>barley 48/84</td><td>SHD F1</td><td>5</td><td>48</td><td>58 5</td><td>L</td><td>6 1</td><td></td><td>8</td><td>3</td><td>：</td><td>34</td><td>5</td><td>8</td><td>60 .48</td><td>务</td></tr><tr><td>hailfinder 56/66</td><td>SHD F1</td><td>11 .80</td><td>3</td><td>41 43 .45 .42</td><td>0 1.0</td><td>17 .85 1</td><td></td><td>86</td><td>2</td><td>1</td><td>2</td><td>44 .42</td><td>76</td><td>111 .18</td><td>1</td></tr><tr><td>hepar2 70/123</td><td>SHD F1</td><td>0. 1.0</td><td>54 .59</td><td>81 59 .34 .54</td><td>0 1.0</td><td>35 27 .72 .81</td><td>37 .70</td><td>14 .89</td><td>46 .75</td><td>40 .70</td><td>35 .81</td><td>75 .39</td><td>123 .00</td><td>79 .54</td><td>61 .68</td></tr><tr><td>win95pts 76/112</td><td>SHD F1</td><td>1 .99</td><td>65 .43</td><td>51 3 .54</td><td>0 1.0</td><td>8 42 .95 .64</td><td>7 .95</td><td>品</td><td>3</td><td>21 .85</td><td>16 .91</td><td>50 .57</td><td>112 .00</td><td>103 .47</td><td>1 1</td></tr><tr><td>pathfinder 109/195</td><td>SHD F1</td><td>2</td><td>157 .21</td><td>145 151 .21 .29</td><td>0 1.0</td><td>150 .29 -</td><td>1 ： -</td><td>147 .30</td><td>158 .29</td><td>1 -</td><td>168 .28</td><td>148 .31</td><td>196 .00</td><td>241 .07</td><td>- 1</td></tr><tr><td>munin1 186/273</td><td>SHD F1</td><td>1</td><td>169 .42</td><td>154 153 .47 .46</td><td>3</td><td>8</td><td>7 ：</td><td>8</td><td>189</td><td>1</td><td>233</td><td>15</td><td>1</td><td>27</td><td>- -</td></tr><tr><td>andes 223/338</td><td>SHD F1</td><td>.0</td><td>226 .35</td><td>209 246 .41 .29</td><td>0 1.0</td><td>9</td><td>8 9</td><td>8</td><td>3</td><td>15 .96</td><td>38</td><td>149 .60</td><td>1 -</td><td>175</td><td>-</td></tr><tr><td>diabetes 413/602</td><td>SHD F1</td><td>3</td><td>2</td><td>38 237</td><td>8</td><td>10</td><td>1 </td><td>284</td><td></td><td>146</td><td>1 58</td><td>368 .43</td><td>1 1</td><td>534</td><td>1 1</td></tr><tr><td>pigs 441/592</td><td>SHD F1</td><td>0 1.0</td><td>350 .44</td><td>332 263 .46 .59</td><td>400 .35</td><td>400 .35</td><td>1 - - 1</td><td>268 .56</td><td></td><td>0 1 1.0 -</td><td>532 .18</td><td>316 .50</td><td>- -</td><td>6 1.0</td><td>-</td></tr><tr><td>link 724/1125</td><td>SHD F1</td><td>0 1.0</td><td>731 .38</td><td>630 .45</td><td>638 749 .45 .39</td><td>737 .40</td><td>1 - = -</td><td>204 .81</td><td></td><td>324 .80</td><td>1047 - .14 1</td><td>400 .64</td><td>- -</td><td>947 .49</td><td>= =</td></tr><tr><td>munin 1041/1397</td><td>SHD F1</td><td>3</td><td>97</td><td>790 .48</td><td>816 0 .44 1.0</td><td>156 .89</td><td>1 1 -</td><td>458 .69</td><td>661</td><td>.62 1</td><td>1397 1.00 -</td><td>7</td><td>1 -</td><td>1599 .29</td><td>- =</td></tr><tr><td>munin2 1003/1244</td><td>SHD F1</td><td>118 .92</td><td>554 .60</td><td>611 .56</td><td>646 1052 .55 .19</td><td>898 .30</td><td>- - - -</td><td>539</td><td></td><td>632 .58</td><td>1240 - .01 -</td><td>753 .49</td><td>- -</td><td>1321 .46</td><td>- -</td></tr><tr><td>munin3 1041/1306</td><td>SHD F1</td><td>113 .92</td><td>616 .58</td><td>629 59</td><td>688 1048 .54 .25</td><td>860 .37</td><td>1 - 1 -</td><td>544 .60</td><td></td><td>56</td><td>- 1306 - 1.00</td><td>819 .46</td><td>- 1</td><td>1539 .26</td><td>- -</td></tr><tr><td>munin4 1038/1388</td><td>SHD F1</td><td>126 .93</td><td>696 .54</td><td>658 .56</td><td>776 1058 .50 .29</td><td>876 .39</td><td>- - - -</td><td></td><td></td><td>618 .64</td><td>1388 1 1 1.00</td><td>812 49</td><td>1 -</td><td>17</td><td>- -</td></tr><tr><td>rank(SHD)</td><td>mean 士stdd</td><td>1</td><td>3</td><td>3</td><td>38 51</td><td>3</td><td>10.8 2.9</td><td>8 24</td><td></td><td>28</td><td>36</td><td>27 2</td><td>1</td><td>15</td><td>17</td></tr><tr><td>UT-F1</td><td>mean ±stdd</td><td></td><td>:22</td><td>:1</td><td>46</td><td>4</td><td>33</td><td>:</td><td>3</td><td></td><td>:7</td><td>3</td><td>2</td><td>09 36</td><td>3</td></tr></table>
+
+Table 2: Reliability: average F1-score of ML4C vs. 8 discriminative predicates extracted from ML4C features on benchmark datasets.   
+
+<table><tr><td rowspan="2">ML4C</td><td rowspan="2"></td><td colspan="4">strong predicates</td><td colspan="4">weak predicates</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>F1</td><td>.92±.20</td><td>.77±.31</td><td>.52±.27</td><td>.38±.25</td><td></td><td>.66±.27 .72±.25 .61±.29</td><td></td><td>.73±.30</td><td>）.55±.27</td></tr></table>
+
+329 $4 . 4 \pm 2 . 4$ . Among the competitors, NCC ranks #1 on 8 datasets (note that ML4C ranks #1 on 11   
+330 datasets), but its performance fluctuates. Overall it only ranks $5 . 1 \pm 4 . 2$ . Last but not least, ML4C   
+331 shows high accuracy $( \mathrm { F } 1 { > } 0 . 9 )$ on very large-scale datasets (e.g., medicine datasets ‘munin\*’ [3])   
+332 while max(others) $\sim 0 . 6$ .   
+333 Towards a Perfect Classifier The last row of Table 1 shows the performance of ML4C-Learner   
+334 component at UT level by UT-F1 (i.e., F1-score of classifying UTs): such UT level accuracy is   
+335 crucial for causal learning on discrete data, since the set of v-structures is invariant across all Markov   
+336 equivalent DAGs and it can fully recover the CPDAG. The average F1-score of ML4C-Learner is   
+337 $0 . 9 0 \pm 0 . 1 3$ , which shows promising results towards a perfect classifier.
+
+Table 3: Robustness: ML4C is trained on synthetic datasets with sample size $= 1 0 \mathrm { k }$ , but tested on benchmark datasets with different sample sizes $\in \{ 1 \mathrm { k } , 5 \mathrm { k } , 1 0 \mathrm { k } , 1 5 \mathrm { k } , 2 0 \mathrm { k } \}$ .   
+
+<table><tr><td>Ⅱ</td><td>size</td><td>1k</td><td>5k</td><td>10k</td><td></td><td>15k 20k</td><td>size</td><td>1k</td><td>5k</td><td>10k</td><td>15k20k|</td><td></td><td>size</td><td>1k</td><td>5k</td><td>10k</td><td>15k 20k</td><td></td></tr><tr><td>SHD</td><td>insurance</td><td>11</td><td>1</td><td>5</td><td>1</td><td>0</td><td>water</td><td>12</td><td>11</td><td>5</td><td>8</td><td>6</td><td>mildew</td><td>8</td><td>5</td><td>3</td><td>6</td><td>1</td></tr><tr><td>F1</td><td>27/52</td><td>.81</td><td>.97</td><td>.89</td><td>.97</td><td>1.0</td><td>32/66</td><td>.86</td><td>.87</td><td>.94</td><td>.89</td><td>.93</td><td>35/46</td><td>.83</td><td>.89</td><td>.93</td><td>.87</td><td>.98</td></tr><tr><td>SHD</td><td>alarm</td><td>5</td><td>4</td><td>0</td><td>1</td><td>5</td><td>barley</td><td>13</td><td>9</td><td>4</td><td>8</td><td>6</td><td>hailfinder</td><td>15</td><td>15</td><td>6</td><td>15</td><td>13</td></tr><tr><td>F1</td><td>37/46</td><td>.93</td><td>.95</td><td>1.0</td><td>.98</td><td>.93</td><td>48/84</td><td>.88</td><td>.93</td><td>.97</td><td>.92</td><td>.94</td><td>56/66</td><td>.74</td><td>.72</td><td>.90</td><td>.72</td><td>.76</td></tr><tr><td>SHD</td><td>hepar2</td><td>8</td><td>2</td><td>0</td><td>1</td><td>2</td><td>win95pts</td><td>7</td><td>1</td><td>0</td><td>1</td><td>1</td><td>pathfinder</td><td>1</td><td>7</td><td>25</td><td>7</td><td>1</td></tr><tr><td>F1</td><td>70/123</td><td>.96</td><td>.99</td><td>1.0</td><td>.99</td><td>.99</td><td>76/112</td><td>.96</td><td>.99</td><td>1.0</td><td>.99</td><td>.99</td><td>109/195</td><td>.99</td><td>.92</td><td>.77</td><td>.92</td><td>.99</td></tr><tr><td>SHD</td><td>munin1</td><td>32</td><td>7</td><td>10</td><td>9</td><td>15</td><td>andes</td><td>3</td><td>2</td><td>0</td><td>2</td><td>0</td><td>diabetes</td><td>18</td><td>28</td><td>4</td><td>26</td><td>27</td></tr><tr><td>F1</td><td>186/273</td><td>.89</td><td>.98</td><td>.97</td><td>.97</td><td>.95</td><td>223/338</td><td>.99</td><td>.99</td><td>1.0</td><td>.99</td><td>1.0</td><td>413/602</td><td>.97</td><td>.95</td><td>.99</td><td>.96</td><td>.96</td></tr><tr><td>SHD</td><td>pigs</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>link</td><td>88</td><td>13</td><td>0</td><td>0</td><td>0</td><td>munin</td><td>107</td><td>76</td><td>71</td><td>93</td><td>87</td></tr><tr><td>F1</td><td>441/592</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>724/1125</td><td>.93</td><td>.99</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1041/1397</td><td>.93</td><td>.95</td><td>.96</td><td>.94</td><td>.94</td></tr><tr><td>SHD</td><td>munin2</td><td>117</td><td>95</td><td>120 110</td><td></td><td>97</td><td>munin3</td><td>151</td><td>119</td><td>113</td><td>99</td><td>62</td><td>munin4</td><td>165</td><td>130</td><td>123</td><td>146</td><td>5133</td></tr><tr><td>F1</td><td>1003/1244</td><td>.92.93</td><td></td><td>.92</td><td>.93</td><td>.93</td><td>1041/1306</td><td>.90</td><td>.92</td><td>.92</td><td>.94</td><td>.96</td><td>1038/1388</td><td>.90</td><td>.92</td><td>.93</td><td>.91</td><td>.93</td></tr></table>
+
+Table 4: Transferability: ML4C trains/tests both on synthetic datasets with different configurations.   
+
+<table><tr><td></td><td>train</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td></tr><tr><td></td><td>10</td><td>10</td><td>1.2±2.4</td><td>.94±.12</td><td>50</td><td>4.8±3.4</td><td>.95±.03</td><td>100</td><td>6.6±4.7</td><td>.97±.02</td><td>1k</td><td>50.6±8.4</td><td>.97±.00</td></tr><tr><td></td><td>50</td><td>10</td><td>0.4±0.8</td><td>.97±.05</td><td>50</td><td>0.8±1.0</td><td>.99±.01</td><td>100</td><td>4.4±4.7</td><td>.98±.02</td><td>1k</td><td>23.2±5.7</td><td>.99±.00</td></tr><tr><td>əpou #</td><td>100</td><td>10</td><td>0.0±0.0</td><td>1.0±.00</td><td>50</td><td>1.2±1.6</td><td>.99±.01</td><td>100</td><td>4.0±4.6</td><td>.98±.02</td><td>1k</td><td>21.6±4.8</td><td>.99±.00</td></tr><tr><td></td><td>1k</td><td>10</td><td>0.4±0.8</td><td>.97±.05</td><td>50</td><td>0.8±1.0</td><td>.99±.01</td><td>100</td><td>1.4±2.3</td><td>.99±.01</td><td>1k</td><td>14.8±8.2</td><td>.99±.00</td></tr><tr><td>sreds</td><td>1</td><td>1</td><td>0.8±1.6</td><td>.99±.02</td><td>2</td><td>3.4±2.9</td><td>.97±.02</td><td>3</td><td>3.0±2.5</td><td>.98±.01</td><td>4</td><td>11.4±3.9</td><td>.95±.02</td></tr><tr><td></td><td>2</td><td>1</td><td>1.8±1.6</td><td>.98±.02</td><td>2</td><td>2.2±1.7</td><td>.98±.01</td><td>3</td><td>2.2±2.0</td><td>.99±.01</td><td>4</td><td>8.2±2.5</td><td>.97±.01</td></tr><tr><td></td><td>3</td><td>1</td><td>1.0±1.3</td><td>.98±.02</td><td>2</td><td>2.2±1.3</td><td>.98±.01</td><td>3</td><td>4.4±3.6</td><td>.97±.02</td><td>4</td><td>4.0±3.2</td><td>.98±.01</td></tr><tr><td></td><td>4</td><td>1</td><td>2.4±2.3</td><td>.97±.03</td><td>2</td><td>2.2±1.9</td><td>.98±.01</td><td>3</td><td>3.2±2.7</td><td>.98±.02</td><td>4</td><td>4.8±3.7</td><td>.98±.01</td></tr><tr><td></td><td>1k</td><td>1k</td><td>2.8±2.3</td><td>.97±.02</td><td>5k</td><td>2.0±2.2</td><td>.98±.02</td><td>10k</td><td>1.6±2.3</td><td>.98±.02</td><td>20k</td><td>1.0±1.3</td><td>.99±.01</td></tr><tr><td>srrsrstte</td><td>5k</td><td>1k</td><td>5.2±2.9</td><td>.95±.03</td><td>5k</td><td>1.0±2.0</td><td>.99±.02</td><td>10k</td><td>2.2±3.5</td><td>.98±.04</td><td>20k</td><td>0.6±0.8</td><td>.99±.01</td></tr><tr><td></td><td>10k</td><td>1k</td><td>5.2±4.8</td><td>.95±.05</td><td>5k</td><td>1.8±2.7</td><td>.98±.02</td><td>10k</td><td>2.0±3.1</td><td>.98±.03</td><td>20k</td><td>0.6±0.8</td><td>.99±.01</td></tr><tr><td></td><td>20k</td><td>1k</td><td>4.8±3.3</td><td>.95±.03</td><td>5k</td><td>2.4±2.6</td><td>.98±.02</td><td>10k</td><td>1.2±1.6</td><td>.99±.02</td><td>20k</td><td>1.0±1.3</td><td>.99±.02</td></tr><tr><td>R</td><td>ER</td><td>ER</td><td>1.0±2.0</td><td>.99±.02</td><td>SF</td><td>2.2±1.6</td><td>.98±.01</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>SF</td><td>ER</td><td>1.6±1.9</td><td>.98±.02</td><td>SF</td><td>2.2±2.4</td><td>.98±.02</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Reliability We manually identify 4 strong discriminative predicates and 4 weak discriminative predicates and treat each one as a replacement of ML4C-Learner. Table 2 shows the performance of these predicates. Although most predicates show value on discriminating UTs (e.g., 5/8 predicates are with ${ > } 0 . 6$ F1-score), ML4C-Learner has higher performance (average F1-score $= 0 . 9 2$ ) than each individual predicate (best average F1-score $= 0 . 7 7$ ). Thus, it is evident that ML4C-Learner learns a more reliable classification mechanism, by taking advantage of machine learning techniques.
+
+Robustness To evaluate robustness, ML4C is trained on synthetic datasets with sample size $=$ $1 0 \mathrm { k }$ , but it is tested on benchmark datasets with different sample sizes: 1k, 5k, 10k, $1 5 \mathrm { k }$ and $2 0 \mathrm { k }$ respectively. Table 3 shows that ML4C exhibits satisfactory robustness (decrease of F1-score is less than 0.1) against sample size on most datasets (17/18, except for ‘hailfinder’).
+
+Transferability To evaluate whether ML4C generalizes well to various types of testing data, we vary scale (#nodes), graph sparsity, generating mechanism and sample size. ML4C is trained on a fixed configuration but it is tested with different domains (i.e., data generated under different configuration). Result is depicted in Table 4, ML4C transfers well on different domains, e.g., even if it is trained on 10 nodes but tested on 1,000 nodes (last column of the first row in Table 4), the F1-score only drops 0.02.
+
+# 54 6 Conclusion and Future Work
+
+We have proposed a supervised causal learning algorithm ML4C, with theoretical guarantee on learnability and remarkable empirical performance. More importantly, ML4C shows promising results on validating the effectiveness of supervision. To make SCL practical in real-world scenarios, one important direction for future work is to identify reliable and accurate skeleton from data, considering ML4C requires skeleton as additional input.
+
+References   
+[1] Réka Albert and Albert-László Barabási. Statistical mechanics of complex networks. Reviews of modern physics, 74(1):47, 2002.   
+[2] Constantin F Aliferis, Alexander Statnikov, Ioannis Tsamardinos, Subramani Mani, and Xenofon D Koutsoukos. Local causal and markov blanket induction for causal discovery and feature selection for classification part i: algorithms and empirical evaluation. Journal of Machine Learning Research, 11(1), 2010.   
+[3] Steen Andreassen, Marianne Woldbye, Bjørn Falck, and Stig K Andersen. Munin: A causal probabilistic network for interpretation of electromyographic findings. In Proceedings of the 10th international joint conference on Artificial intelligence-Volume 1, pages 366–372, 1987.   
+[4] Gianluca Bontempi and Maxime Flauder. From dependency to causality: a machine learning approach. J. Mach. Learn. Res., 16(1):2437–2457, 2015.   
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+[6] David Maxwell Chickering. Optimal structure identification with greedy search. Journal of machine learning research, 3(Nov):507–554, 2002.   
+[7] Diego Colombo and Marloes H Maathuis. Order-independent constraint-based causal structure learning. J. Mach. Learn. Res., 15(1):3741–3782, 2014.   
+[8] James Cussens. Bayesian network learning with cutting planes. arXiv preprint arXiv:1202.3713, 2012.   
+[9] Colombo Diego and H. Maathuis Marloes. Order-independent constraint-based causal structure learning. Journal of Machine Learning Research, 15(116):3921–3962, 2014.   
+[10] Rui Ding, Yanzhi Liu, Jingjing Tian, Zhouyu Fu, Shi Han, and Dongmei Zhang. Reliable and efficient anytime skeleton learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, 2020.   
+[11] P. Erdos and Rényi A. On random graphs. ˝ Publicationes, Mathematicae, 6:290–297, 1959.   
+[12] José AR Fonollosa. Conditional distribution variability measures for causality detection. In Cause Effect Pairs in Machine Learning, pages 339–347. Springer, 2019.   
+[13] Clark Glymour, Kun Zhang, and Peter Spirtes. Review of causal discovery methods based on graphical models. Frontiers in genetics, 10:524, 2019.   
+[14] Patrik Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlinear causal discovery with additive noise models. Advances in neural information processing systems, 21:689–696, 2008.   
+[15] Dominik Janzing, Joris Mooij, Kun Zhang, Jan Lemeire, Jakob Zscheischler, Povilas Daniušis, Bastian Steudel, and Bernhard Schölkopf. Information-geometric approach to inferring causal directions. Artificial Intelligence, 182:1–31, 2012.   
+[16] Daphne Koller and Nir Friedman. Probabilistic graphical models: principles and techniques. MIT press, 2009.   
+[17] Steffen L Lauritzen. Graphical models, volume 17. Clarendon Press, 1996.   
+[18] Hebi Li, Qi Xiao, and Jin Tian. Supervised whole dag causal discovery. arXiv preprint arXiv:2006.04697, 2020.   
+[19] Honghao Li, Vincent Cabeli, Nadir Sella, and Hervé Isambert. Constraint-based causal structure learning with consistent separating sets. In 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), 2019.   
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+[22] Dimitris Margaritis. Learning bayesian network model structure from data. Technical report, CarnegieMellon Univ Pittsburgh Pa School of Computer Science, 2003.   
+[23] Christopher Meek. Causal inference and causal explanation with background knowledge. arXiv preprint arXiv:1302.4972, 2013.   
+[24] Christopher Meek. Strong completeness and faithfulness in bayesian networks. arXiv preprint arXiv:1302.4973, 2013.   
+[25] Joseph Ramsey, Jiji Zhang, and Peter Spirtes. Adjacency-faithfulness and conservative causal inference. arXiv preprint arXiv:1206.6843, 2012.   
+[26] Joseph Ramsey, Jiji Zhang, and Peter L Spirtes. Adjacency-faithfulness and conservative causal inference. arXiv preprint arXiv:1206.6843, 2012.   
+[27] Mauro Scanagatta, Cassio Polpo de Campos, Giorgio Corani, and Marco Zaffalon. Learning bayesian networks with thousands of variables. In NIPS, pages 1864–1872, 2015.   
+[28] Marco Scutari. Bayesian network repository. URL http://www. bnlearn. com, 2012.   
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+[31] Peter Spirtes and Clark Glymour. An algorithm for fast recovery of sparse causal graphs. Social science computer review, 9(1):62–72, 1991.   
+[32] Peter Spirtes, Clark N Glymour, Richard Scheines, and David Heckerman. Causation, prediction, and search. MIT press, 2000.   
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+[35] Yue Yu, Jie Chen, Tian Gao, and Mo Yu. Dag-gnn: Dag structure learning with graph neural networks. In International Conference on Machine Learning, pages 7154–7163. PMLR, 2019.   
+[36] Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P Xing. Dags with no tears: Continuous optimization for structure learning. arXiv preprint arXiv:1803.01422, 2018.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See $\ S 6$ . ML4C requires skeleton as additional input, thus we put identifying reliable and accurate skeleton from data as future work.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] Details of proofs are in the supplementary material due to page limit.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We have included the main functionalities of ML4C, synthetic data generator, and ML4C’s training data (include instructions) for reproducibility. Details are in the supplementary material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Key information is in content. Details are in the supplementary material.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We carefully design experiments, by running experiments multiple times, including error bars in report to ensure reproducibility. For instance, Table 4 contains error bar information.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We compare our algorithm with 15 SOTA competitors and we have cite these work properly. We also use open source benchmark for evaluation and we also properly cite the creators.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We share new assets, include main functionalities of our algorithm, and our training data. A private URL is provided. See details in the supplementary material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] Due to page limit, the detailed discussion is available in the supplementary material.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We double-checked that the data we are using/curating contains no personally identifiable information or offensive content.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
\ No newline at end of file
diff --git a/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_content_list.json b/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..660b94d179c68f97b470fa55a008e53a4ebc811c
--- /dev/null
+++ b/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_content_list.json
@@ -0,0 +1,1176 @@
+[
+    {
+        "type": "text",
+        "text": "ML4C: Seeing Causality Through Latent Vicinity ",
+        "text_level": 1,
+        "bbox": [
+            197,
+            122,
+            800,
+            148
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            200,
+            580,
+            256
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+            309
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Supervised Causal Learning (SCL) aims to learn causal relations from observational   \n2 data by accessing previously seen datasets associated with ground truth causal   \n3 relations. This paper presents a first attempt at addressing a fundamental question:   \n4 What are the benefits from supervision and how does it benefit? Starting from seeing   \n5 that SCL is not better than random guessing if the learning target is non-identifiable   \n6 a priori, we propose a two-phase paradigm for SCL by explicitly considering   \n7 structure identifiability. Following this paradigm, we tackle the problem of SCL on   \n8 discrete data and propose ML4C. The core of ML4C is a binary classifier with a   \n9 novel learning target: it classifies whether an Unshielded Triple (UT) is a v-structure   \n10 or not. Starting from an input dataset with the corresponding skeleton provided,   \n11 ML4C orients each UT once it is classified as a v-structure. These v-structures are   \n12 together used to construct the final output. To address the fundamental question   \n13 of SCL, we propose a principled method for ML4C featurization: we exploit the   \n14 vicinity of a given UT (i.e., the neighbors of UT in skeleton), and derive features by   \n15 considering the conditional dependencies and structural entanglement within the   \n16 vicinity. We further prove that ML4C is asymptotically perfect. Last but foremost,   \n17 thorough experiments conducted on benchmark datasets demonstrate that ML4C   \n18 remarkably outperforms other state-of-the-art algorithms in terms of accuracy,   \n19 robustness and transferability. In summary, ML4C shows promising results on   \n20 validating the effectiveness of supervision for causal learning. ",
+        "bbox": [
+            147,
+            327,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "21 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            642,
+            310,
+            659
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "22 The problem of causal learning is to learn causal relations from observational data [13]. The learned   \n23 causal relations are typically represented in the form of a Directed Acyclic Graph (DAG), where each   \n24 edge in the DAG indicates direct cause-effect relation between the parent node and child node.   \n25 The methods of causal learning mostly fall into four categories: constraint-based, score-based,   \n26 continuous optimization method and functional causal models. Each of these methods takes a given   \n27 dataset as input and outputs a DAG but with different criteria. For instance, the DAG should be   \n28 consistent with conditional independencies in the data (constraint-based); or it is optimal w.r.t. a   \n29 pre-defined score function under either combinatorial constraint (score-based) or continuous equality   \n30 constraint (continuous optimization). In a nutshell, these methods can be viewed as unsupervised   \n31 since they do not access additional datasets associated with ground truth causal relations.   \n32 A new line of research called Supervised Causal Learning (SCL), on the other hand, aims to learn   \n33 causal relations in the supervised fashion: the algorithm has access to datasets associated with   \n34 ground truth causal relations, in the hope that such supervision is beneficial to learning causal   \n35 relations on newly unseen datasets. Despite several existing works on this direction (see Related   \n36 Work), a fundamental question remains unanswered: How is supervised causal learning possible?   \n37 Specifically, compared with unsupervised causal learning methods, can we gain additional benefits   \n38 from supervision? If the answer is positive, then what are the benefits and how does it benefit?   \n39 We tackle the problem by first seeing crucial connection between SCL and causal structure identi  \n40 fiability. Considering the problem of causal learning on discrete data, theorem in [24] states that,   \n41 under standard assumptions (i.e., Markov assumption, faithfulness and causal sufficiency), we can   \n42 only identify a graph up to its Markov equivalence class. Markov equivalence class is the set of   \n43 DAGs having same skeleton and same v-structures, which can be represented by CPDAG (Com  \n44 pleted Partially Directed Acyclic Graph). Thus, the (un)directed edges in the CPDAG indicate   \n45 (non-)identifiable causal relations. Each non-identifiable edge in CPDAG can be oriented by either   \n46 direction to equivalently fit the observational data. Given an SCL algorithm with learning target as   \n47 the orientation of an edge, we see that it is not better than random guessing (or could be worse due to   \n48 sample bias in training data) to predict any non-identifiable edge since we can assign either $X  Y$   \n49 or $X  Y$ with same input dataset. This statement is applicable to general learning target since an   \n50 SCL algorithm can take different target such as orientation of an edge, the whole DAG, or others. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/b27c9954d5c1984f08d48654bd07fafec8c6793034aeffc73f282a018f11ef7b.jpg",
+        "image_caption": [
+            "Figure 1: (a) Two-phase paradigm for supervised causal learning. (b) ML4C’s workflow. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Proposition 1. If the learning target is non-identifiable (i.e., every edge in the target is nonidentifiable) a priori, then SCL is not better than random guessing. ",
+        "bbox": [
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+            487,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "53 Consequently, we propose and advocate a two-phase paradigm for SCL, as depicted in Figure 1(a):   \n54 phase one corresponds to a binary classification task, where an SCL algorithm needs to classify   \n55 whether a specific learning target is identifiable or not; only if it is classified as identifiable, then we   \n56 go to phase two to classify the specific orientation of the learning target. Following this paradigm,   \n57 we tackle the problem of SCL on discrete data and propose an algorithm ML4C. The core of ML4C   \n58 is a binary classifier with a novel learning target: it classifies whether an Unshielded Triple (UT: a   \n59 triple of variables $\\langle X , T , Y \\rangle$ where $X$ and $Y$ are adjacent to $T$ but are not adjacent to each other) is a   \n60 v-structure or not. Starting from an input dataset with the corresponding skeleton provided, ML4C   \n61 orients each UT once it is classified as a v-structure. These v-structures are further used to construct   \n62 a CPDAG as output. Such a single classifier facilitates both learning tasks in the two phases, since an   \n63 identifiable UT implies that it is a v-structure [32] (i.e., up to the partial DAG before applying Meek   \n64 rules [23] which is a standard post processing).   \n65 To address the fundamental question of SCL, we propose a principled method for ML4C featurization.   \n66 Specifically, we exploit the vicinity of a given UT (i.e., the neighbors of UT in skeleton), and derive   \n67 features by considering the conditional dependencies and structural entanglement within the vicinity.   \n68 We further define discriminative predicate (i.e., a binary predicate function with domain as ML4C’s   \n69 feature set) and prove that there exist weak discriminative predicates and strong discriminative   \n70 predicates (i.e., values of the predicates are one-to-one correspondence with ground truth labels).   \n71 We further prove that ML4C is asymptotically perfect. Last but foremost, thorough experiments on   \n72 benchmark datasets demonstrate that ML4C remarkably outperforms other state-of-the-art algorithms   \n73 w.r.t. accuracy, robustness and transferability. Our main contributions are summarized as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
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+    {
+        "type": "text",
+        "text": "74 1. We advocate the two-phase paradigm for SCL with consideration of causal structure identifiability. ",
+        "bbox": [
+            153,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "75 2. We propose an SCL algorithm ML4C, with the following novelties: i) Learning Target: The core   \n76 of ML4C is a binary classifier with the orientation of a UT as its learning target to address the two  \n77 phase tasks simultaneously. ii) Featurization: A principled method to exploit vicinity information   \n78 in terms of dependencies and entanglement of a given UT. iii) Learnability: We prove that ML4C   \n79 is asymptotically perfect. iv) Empirical Performance: Experiments conducted on benchmark   \n80 datasets demonstrate that ML4C remarkably outperforms other state-of-the-art algorithms. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "81 2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            166,
+            320,
+            183
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "82 We divide literature on causal learning into supervised and unsupervised approaches, depending on   \n83 whether additional datasets (associated with ground truth causal relations) are accessed (supervised)   \n84 or not (unsupervised). In the literature of unsupervised causal learning, constraint-based methods aim   \n85 to identify a DAG which is consistent with conditional independencies. The learning procedure of   \n86 constraint-based methods first identifies the corresponding skeleton and then conducts orientation   \n87 based on v-structure identification [34]. The typical algorithm is PC [31], and there are also PC  \n88 derived algorithms such as Conservative-PC [26], PC-stable [7] and Consistent-PC [19] which   \n89 improve the robustness on v-structure identification. Score-based methods aim to find the DAG which   \n90 is optimal w.r.t. a pre-defined score function under combinatorial constraint by a specific search   \n91 procedure, such as forward-backward search GES [6], hill-climbing [16], integer programming [8], or   \n92 by approximate algorithms based on order search [27]. Continuous optimization methods transform   \n93 the discrete search procedure into continuous equality constraint: NOTEARS [36] formulates the   \n94 acyclic constraint as a continuous equality constraint, it is further extended by DAG-GNN [35] to   \n95 support learning non-linear causal relations.   \n96 SCL emerges from the task of orienting edge in the continuous, non-linear bivariate case under   \n97 Functional Causal Model (FCM) formalism. Given a collection of cause-effect samples (dataset   \n98 $\\sim$ binary label indicating whether $X  Y$ or $X  Y$ ), supervised approaches such as RCC [20],   \n99 NCC [21], D2C [4] and Jarfo [12] achieve better performance on predicting pairwise relations (i.e.,   \n100 orientation of an edge) than unsupervised approaches such as ANM [14] or IGCI [15]. Differently,   \n101 [18] sets the learning target as the whole DAG structure instead of pairwise relation and it is applied on   \n102 data which is generated by linear Structural Equation Model (SEM). We summarize the differences in   \n103 problem space between ML4C and the other SCL approaches as follows: i) We advocate a two-phase   \n104 learning paradigm and emphasize the relationship between identifiability and learnability. Specifically,   \n105 presuming additive noise model [14] or linear SEM with non-Gaussian noise [29] provides license   \n106 to identifiability thus the aforementioned approaches can be viewed as specific tasks in phase two.   \n107 ii) We tackle SCL’s learnability not only via empirical evaluation but also by theoretical analysis to   \n108 shed light on the fundamental question of learnability. iii) ML4C deals with discrete data while other   \n109 approaches mainly focus on continuous data. ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "110 3 Background ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            630,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Basic Notations ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            667,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A discrete dataset $D _ { i }$ consists of $n _ { i }$ records and $d _ { i }$ categorical columns, which represents $n _ { i }$ instances drawn i.i.d. from $d _ { i }$ discrete variables $X _ { 1 } , X _ { 2 } , \\cdots , X _ { d _ { i } }$ by a joint probability distribution $P _ { i }$ , which is entailed by an underlying data generating process, denoted as DAG $G _ { i }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Markov factorization property: Given a joint probability distribution $P$ and a DAG $G$ , $P$ is said to satisfy Markov factorization property w.r.t. $G$ if $\\begin{array} { r } { P : = P \\left( X _ { 1 } , X _ { 2 } , \\cdots , X _ { d } \\right) = \\prod _ { i = 1 } ^ { d } P \\left( X _ { i } | \\mathsf { p a } _ { i } ^ { G } \\right) } \\end{array}$ where $\\mathrm { p a } _ { i } ^ { G }$ is the parent set of $X _ { i }$ in $G$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Markov assumption: $P$ is said to satisfy Markov assumption (or Markovian) w.r.t. a DAG $G$ if $X \\bot _ { G } Y | Z \\Rightarrow X \\bot Y | Z$ . Here $\\perp _ { G }$ denotes $\\mathrm { d }$ -separation, and $\\perp$ denotes statistical independence. Markov assumption indicates that any d-separation in graph $G$ implies conditional independence in distribution $P$ . Markov assumption is equivalent to Markov factorization property [17]. ",
+        "bbox": [
+            163,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Faithfulness: Distribution $P$ is faithful w.r.t. a DAG $G$ if $X \\bot Y | Z \\Rightarrow X \\bot _ { G } Y | Z$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "123 Canonical dataset: We say a discrete dataset $D$ is canonical if its underlying probability distribution   \n124 $P$ is Markovian and faithful w.r.t. some DAG $G$ . ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "25 3.2 Causal Structure Identifiability ",
+        "text_level": 1,
+        "bbox": [
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+            92,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Identifiability discusses which parts of the causal structure can in principle be inferred from the distribution. Below we present the established theory of identifiability on discrete data. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Causal sufficiency: There are no latent common causes of any of the variables in the graph. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Markov equivalence). Two graphs are Markov equivalent if and only if they have same skeleton and same $\\nu$ -structures. A Markov equivalence class can be represented by a CPDAG having both directed and undirected edges. A CPDAG can be derived from a DAG G, denoted as $C P D A G ( G )$ . The theorem of Markov completeness in [24] states that, under causal sufficiency, we can only identify a causal graph up to its Markov equivalence class on canonical data. Therefore, the (non-)identifiable causal relations are the (un)directed edges in the CPDAG. Formally, ",
+        "bbox": [
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+            169,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2 (Identifiability). Assuming $P$ is Markovian and faithful w.r.t. a DAG G and causal sufficiency, then each (un)directed edge in $C P D A G ( G )$ indicates (non-)identifiable causal relation. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 ML4C Related Notations ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            299,
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+            314
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 3 (Skeleton). A skeleton $E$ defined over distribution $P \\left( X _ { 1 } , X _ { 2 } , \\cdots , X _ { d } \\right)$ is an undirected graph such that there is an edge between $X _ { i }$ and $X _ { j }$ if and only if $X _ { i }$ and $X _ { j }$ are always dependent, i.e., $\\nexists Z \\ \\subseteq \\ \\{ X _ { 1 } , X _ { 2 } , \\cdot \\cdot \\cdot \\ , X _ { d } \\}$ s.t. $X _ { i } \\bot X _ { j } | Z$ . Skeleton is a statistical concept, which can be obtained prior to facilitating various downstream tasks. Recently, there have been some novel skeleton learning algorithms such as [10]. In particular, skeleton can be used for causal learning: theorem in [32] states that if distribution $P$ is Markovian and faithful w.r.t. a DAG $G$ , then skeleton $E$ is the same as the undirected graph of $G$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "145 Definition 4 (UT). A triple of variables $\\langle X , T , Y \\rangle$ in a skeleton is an unshielded triple, or short   \n146 for UT, if $X$ and $Y$ are adjacent to $T$ but are not adjacent to each other. $\\langle X , T , Y \\rangle$ can be further   \n47 oriented to become a $\\nu$ -structure $X \\right. T \\left. Y$ , in which $T$ is called the collider.   \n148 Definition 5 (PC). Denote the set of parents and children of $X$ in a skeleton as $P C _ { X }$ , in other words,   \n49 $P C _ { X }$ are the neighbors of $X$ in the skeleton. For convenience, if we discuss $P C _ { X }$ in the context of $a$   \n50 $U T \\left. X , T , Y \\right.$ , we intentionally mean the set of parents and children of $X$ but exclude $T$ . ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 6 (Vicinity). We define the vicinity of a51 $U T \\left. X , T , Y \\right.$ as $V _ { \\langle X , T , Y \\rangle } : = \\{ X , T , Y \\} \\cup P C _ { X } \\cup$ 52 $P C _ { Y } \\cup P C _ { T }$ . Vicinity is a generalized version of $P C$ , i.e., the neighbors of $\\{ X , T , Y \\}$ in the skeleton. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "153 ML4C’s training set: The training set is a collection of discrete datasets $D _ { 1 } , \\cdots , D _ { n }$ , where each   \n154 dataset $D _ { i }$ is associated with a ground truth DAG $G _ { i }$ , such that $D _ { i }$ is sampled from $G _ { i }$ . $G _ { i }$ derives   \n155 labels (depends on the chosen learning target), thus $\\{ D _ { i } , G _ { i } \\} _ { i \\in \\{ 1 , \\cdots , n \\} }$ form ML4C’s training set.   \n156 We can sample graphs from DAG space and generate corresponding datasets, thus obtaining training   \n157 set in our problem is straightforward. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 Approach ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Overview ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The core of ML4C is a binary classifier called ML4C-Learner, which takes the orientation of a UT as its learning target, i.e., it classifies whether an input UT $\\langle X , T , Y \\rangle$ is a v-structure (orientation: $X \\right. T \\left. Y$ ) or not (orientation remains unknown). Figure 1(b) depicts the overall workflow of ML4C, which is composed of ML4C-Learner with other important inductive biases. Starting from an input dataset $D _ { i }$ with corresponding skeleton $E _ { i }$ , we first obtain all the UTs from $E _ { i }$ . Featurization is then conducted to represent each UT as an embedded vector, which is further fed into ML4C-Learner. In the inference stage, we obtain all the v-structures which are classified by ML4C-Learner and reconstruct a partial DAG and then, a CPDAG is output by applying Meek rules on the partial DAG. In the training stage, the label of each UT is obtained by querying from ground truth DAG $G _ { i }$ . We collect labeled data from multiple datasets in ML4C’s training set. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. If ML4C-Learner is a perfect classifier, then ML4C outputs correct CPDAG of a canonical dataset (i.e., ML4C is perfect). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "172 By Markov completeness, the set of v-structures is invariant across all Markov equivalent DAGs for   \n173 a canonical dataset, and it can fully recover the CPDAG, provided that the skeleton is given. Thus,   \n74 besides its dedicated role in phase 2, ML4C-Learner also facilitates learning task in phase 1 since an   \n75 identifiable UT implies that it is a v-structure (up to the partial DAG before applying Meek rules). ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 Featurization ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We propose a principled method for ML4C-Learner’s featurization, which avoids the need of handcrafted features. More importantly, we further prove that ML4C-Learner is asymptotically perfect. ",
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+    },
+    {
+        "type": "text",
+        "text": "Design Principles: Our key aspect of featurization is to broaden focus from a specific UT $\\langle X , T , Y \\rangle$ to its vicinity and seeking conditional dependencies and structural entanglement within the vicinity, to reveal reliable and robust asymmetry to distinguish v-structure and non-v-structure UTs. Specifically, conditional dependencies are the key materials for traditional causal learning methods (e.g., conditional independences for constraint-based methods), and structural entanglement (e.g., $P C _ { X } = P C _ { T } .$ ) are relevant to identifiability: higher entanglement makes the UT less likely to be identifiable. ",
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• Dependencies within Vicinity ",
+        "text_level": 1,
+        "bbox": [
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+            287,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Conditional dependency: Denoted as $X \\sim Y | \\mathbf { Z }$ , which is a non-negative scalar that measures the dependence between two random variables $X$ and $Y$ given variable set $\\mathbf { Z }$ . Operationally, $X \\sim Y | \\mathbf { Z }$ is composed of two parts, bivariable $X \\sim Y$ , and conditional Z. We further extend the definition to allow a set of variables in bivariable, and an ensemble (i.e., a set of set) as conditional: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Extended conditional dependency: Denoted as $\\mathbf { A } \\sim \\mathbf { B } | { \\mathcal { Z } } { : = \\{ X \\sim Y | \\mathbf { Z } : X \\in \\mathbf { A } , Y \\in \\mathbf { B } , \\mathbf { Z } \\in } $ ${ \\mathcal { Z } } \\}$ , where A and $\\mathbf { B }$ are set of variables, and $\\mathcal { Z }$ is an ensemble. Thus, extended conditional dependency is a set of scalars. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Within the vicinity of $\\langle X , T , Y \\rangle$ , we start from measuring dependencies between $\\{ X , P C _ { X } \\}$ and $\\{ Y , P C _ { Y } \\}$ by conditioning on $\\{ T , P C _ { T } \\}$ . Intuitively, if $\\langle X , T , Y \\rangle$ is a v-structure, conditioning on $T$ or $T$ ’s descendants tends to strengthen the dependency between $P C _ { X }$ and $P C _ { Y }$ since the paths passing $X - T - Y$ are unblocked; otherwise, conditioning on $T$ tends to weaken the dependency between $P C _ { X }$ and $P C _ { Y }$ because $T$ blocks the paths passing $X - T - Y$ . Therefore, such conditional dependencies reflect potential asymmetry to distinguish v-structure and non-v-structure. Formally, ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 7 (Domain of bivariable). Denoted as $\\mathbb { B } \\ : = \\ \\{ X , P C _ { X } \\} \\ \\times \\ \\{ Y , P C _ { Y } \\} \\ \\equiv$ $\\{ X \\sim Y , X \\sim P C _ { Y } , P C _ { X } \\sim Y , P C _ { X } \\sim P C _ { Y } \\}$ , here symbol $^ { \\prime } \\times$ is Cartesian product. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 8 (Sepsets). Denoted as $\\begin{array} { r } { \\mathcal { S } : = \\{ S : X \\bot Y | S , S \\subset P C _ { X } \\cup T } \\end{array}$ , or $S \\subset P C _ { Y } \\cup T \\}$ . Under faithfulness assumption, sepsets $s$ is an ensemble where each item is a subset of variables within the vicinity that d-separates $X$ and $Y$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 9 (Domain of conditional). Denoted as $\\begin{array} { r c l } { \\mathbb { C } } & { \\colon = } & { \\{ \\emptyset , T , \\mathcal { P C } _ { T } \\} \\ \\vee \\ \\{ \\emptyset , S \\} \\quad \\equiv } \\end{array}$ $\\{ \\emptyset , T , \\mathcal { P } \\mathcal { C } _ { T } , S , S \\vee T , S \\vee \\mathcal { P } \\mathcal { C } _ { T } \\}$ , where $\\mathcal { P C } _ { T } : = \\{ \\{ I \\} : I \\in P C _ { T } \\}$ which is an ensemble version of $P C _ { T }$ , and $\\mathcal { S } \\vee \\mathcal { P } \\mathcal { C } _ { T } : = \\left\\{ S \\cup I : S \\in \\mathcal { S } , I \\in \\mathcal { P } \\mathcal { C } _ { T } \\right\\}$ . Here symbol ∨ is element-wise union. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We exploit the extended conditional dependencies from $\\mathbb { B } \\times \\mathbb { C }$ , i.e., we pick a bivariable from $\\mathbb { B }$ and a conditional from $\\mathbb { C }$ , and calculate the extended conditional dependency. There are in total $\\lvert \\mathbb { B } \\rvert \\times \\lvert \\mathbb { C } \\rvert = 2 4$ extended conditional dependencies. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Sepsets $s$ of any UT of a canonical dataset is non-empty. All proofs are available in the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. We intend to restrict the sepsets within the vicinity of $\\langle X , T , Y \\rangle$ . Lemma 1 shows the existence of such $d$ -separation sets within vicinity. Furthermore, searching for all $d$ -separation sets is highly time-consuming, thus the computational cost can also be saved drastically. ",
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• Entanglement within Vicinity ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Structural entanglement reflects complex structure within the vicinity of $\\langle X , T , Y \\rangle$ . Variables $X , Y$ and $T$ can mutually share common neighbors, and their neighbors may also overlap with sepsets $s$ . We call such overlaps structural entanglement. Intuitively, stronger entanglement indicates denser structure of vicinity thus making the UT less likely to be identifiable. Therefore, structural entanglement is an important aspect for featurization. Specifically, we exploit the overlap coefficient [33] to measure the entanglement: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 10 (Overlap coefficient). OLP $( \\mathbf { A } , \\mathbf { B } ) : = | \\mathbf { A } \\cap \\mathbf { B } | / \\operatorname* { m i n } \\left( | \\mathbf { A } | , | \\mathbf { B } | \\right)$ , where A and $\\mathbf { B }$ are 223 two sets of variables. We extend this formula to support ensemble as input: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(Extended) Overlap coefficient: OLP 224 $( \\mathbf { A } , S ) : = \\textstyle \\sum _ { i = 1 } ^ { | S | }$ OLP $\\left( \\mathbf { A } , S _ { i } \\right) / \\left| S \\right|$ . Naturally, we consider the entanglement in terms of overlap coefficient on each pair of items in domain Thus, we use 6 scalars to represent the entanglement within the vicinity of a UT. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Embedding ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We aim to represent the dependencies and entanglement by a feature vector with fixed dimensionality, which can be used to train ML4C-Learner. Regarding each extended conditional dependency $\\mathbf { A } \\sim$ $\\mathbf { B } | \\mathcal { Z } : \\mathbf { A } \\sim \\mathbf { B } \\in \\mathbb { B } , \\mathcal { Z } \\in \\mathbb { C }$ , it consists of a set of scalars with varied set size across UTs, we adopt the kernel mean embedding technique in [30] to represent each $\\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z }$ as a vector with fixed dimensionality. We further modify the embedding algorithm by adding min $\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\}$ and max $\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\}$ as two additional features. We directly use the 6 scalars to represent structural entanglement without further transformation. We concatenate all the embedded vectors to form the final feature vector, as input for ML4C-Learner. ",
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+    {
+        "type": "text",
+        "text": "36 4.3 Learnability ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We have presented ML4C’s featurization and started seeing that conditional dependencies and structural entanglement have potential to reveal asymmetry to distinguish v-structure and non-vstructure UTs. Now we provide rigorous analysis to show that, for a canonical dataset with sufficient samples, ML4C-Learner tends to a perfect classifier. To prove this, we first propose a surrogate object called discriminative predicate: ",
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+        "type": "text",
+        "text": "Definition 11 (Discriminative predicate). A discriminative predicate is a binary predicate function with domain as ML4C’s feature set. A discriminative predicate can be viewed as a special classifier with pre-specified form of mechanism (i.e., not learned from data). ",
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+        "type": "text",
+        "text": "Definition 12 (Weak / Strong discriminative predicate). Whenever a discriminative predicate takes the feature vector of a UT as input, a weak discriminative predicate satisfies one of the following two criteria; a strong discriminative predicate satisfies both: i) it is evaluated to TRUE if the UT is a v-structure; ii) it is evaluated to FALSE if the UT is not a v-structure. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "By definition, a weak discriminative predicate exhibits discriminative power since it is evaluated false implies the UT is a non-v-structure (or true implies v-structure). A strong discriminative predicate can be viewed as a perfect classifier. Denote $\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\} > \\delta : = X \\sim Y | \\mathbf { Z } > \\delta : \\forall X \\in \\mathbf { A } , Y \\in$ $\\mathbf { B } , \\mathbf { Z } \\in { \\mathcal { Z } }$ , then we have: ",
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+    {
+        "type": "text",
+        "text": "Lemma 2 (Existence of weak discriminative predicate). For a canonical dataset with infinite samples, the following are three weak discriminative predicates: i) $\\{ X \\sim Y | T \\} > 0$ , ii) $\\{ X \\sim Y | \\mathcal { P } \\mathcal { C } _ { T } \\} = 0$ , iii) $\\{ P C _ { X } \\overset { \\cdot } { \\sim } P C _ { Y } | S \\cup T \\} > 0 .$ .Take $\\{ X \\sim Y | T \\} > 0$ as an example, $\\langle X , T , Y \\rangle$ is a v-structure $\\Rightarrow T$ is a collider $\\Rightarrow T$ unblocks $X$ and $Y$ through path $X - T - Y \\Rightarrow \\{ X \\sim Y | T \\} > 0 \\Rightarrow$ $\\operatorname* { m i n } { \\{ X \\sim Y | T \\} } > 0$ , where $\\operatorname* { m i n } { \\{ X \\sim Y | T \\} }$ is a feature of ML4C-Learner since $X \\sim Y \\in$ B $\\ S , \\{ T \\} \\in \\mathbb { C }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "Lemma 3 (Existence of strong discriminative predicate). For a canonical dataset with infinite samples, the following are three strong discriminative predicates: $i$ ) $\\begin{array} { r } { \\mathrm { O L P } ( T , S ) = 0 , } \\end{array}$ ii) $\\begin{array} { r } { \\mathrm { o L P } ( T , S ) < } \\end{array}$ 0.5, iii) $\\operatorname { o L P } ( T , S ) < 1 \\land \\operatorname* { m i n } \\left\\{ X \\sim Y | T \\cup S \\right\\} > 0$ . ",
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+    {
+        "type": "text",
+        "text": "CPC/MPC/GLL-MB as special cases of ML4C-Learner: Predicate OLP $( T , S ) = 0 \\Longleftrightarrow \\forall S \\in$ $s$ , $T \\not \\in S$ , which states that the predicate is TRUE if $T$ is not in any $\\mathrm { d }$ -separation set of $X$ and $Y$ . Having correct skeleton provided, this is the criterion of Conservative PC algorithm (CPC) [25] for identifying v-structures. Thus, CPC can be viewed as a special case of ML4C by replacing ML4CLearner with such a pre-specified logic; OLP $( T , S ) < 0 . { \\bar { 5 } }$ indicates that if more than half of the dseparation sets do not contain $T$ , then the UT is oriented as a v-structure, which is called majority rule PC algorithm (MPC) [9]; predicate OLP ( $T , S ) < 1 \\land \\operatorname* { m i n } \\left\\{ X \\sim Y | T \\cup S \\right\\} > 0 \\Rightarrow \\exists S \\in S$ , $T \\not \\in S$ and $X$ and $Y$ are dependent when conditioning on $T \\cup S$ , which is used for GLL-MB [2] to more securely identify v-structures. These predicates are with suboptimal performance because only a small portion of features are exploited and the overall loss function of training data is disregarded, thus in practice when an appropriate machine learning model is adopted, ML4C-Learner achieves better performance. ",
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+    {
+        "type": "text",
+        "text": "4 Theorem 1. ML4C-Learner tends to a perfect classifier on classifying a canonical dataset with   \n5 sufficient samples. ",
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+    {
+        "type": "text",
+        "text": "Benchmark Datasets We use discrete datasets sampled by all 24 networks from bnlearn repository [28] for evaluation. For each network, we sample 1k, 5k, 10k, 15k, 20k records for use. ",
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+    {
+        "type": "text",
+        "text": "ML4C’s Training and Inference We generate ML4C’s training data synthetically (which is also used for other SCL competitors). Specifically, 400 unique DAGs are randomly generated by two models: Erdos-Rényi (ER) model [ ˝ 11] and Scale-Free (SF) model [1], with the number of nodes ranging from 10 to 1,000. A standard random forward data generation process is applied to obtain 10k observational samples for each graph. We further extract UTs from the $4 0 0 \\mathrm { D A G s }$ , consisting of 97,010 v-structures (label $= 1$ ) and 195,691 non-v-structures $\\mathrm { | a b e l = 0 }$ ). We use these instances to train ML4C-Learner, which is implemented by a XGBoost [5] binary classifier with default hyperparameters and we use binary cross-entropy as the loss function. Details on our synthesis procedure, configurations and implementation of ML4C-Learner are available in the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "Competitors We categorize state-of-the-art causal learning algorithms from two aspects, supervised vs. unsupervised, and can or cannot take skeleton as input. We choose Jarfo [12], D2C [4], RCC [20], and NCC [21] as SCL competitors. Same as ML4C, all these algorithms can and do require skeleton as input. All these algorithms use ML4C’s training set for training but with different learning target extracted. Regarding unsupervised algorithms, we choose PC [31], Conservative-PC (CPC) [26], Majority-rule PC (MPC) [7], GLL-MB (GMB) [2], GES [6], Grow-Shrink (GS) [22], Hill-Climbing (HC) [16], and Conditional Distribution Similarity (CDS) [12]. which can also take skeleton as input. Lastly, we also compare with DAG-GNN (DGNN) [35], BLIP [27], and GOBNILP (GNIP) [8], which are unsupervised algorithms but cannot take skeleton as input. All these competitors are capable of dealing with discrete data. All experiments are done in a Windows Server with 2.8GHz Intel E5-2680 CPU and 256G RAM. Details are in the supplementary material. ",
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+        "type": "text",
+        "text": "299 Design Our evaluation mainly consists of two parts: end-to-end comparison with competitors on   \n300 benchmark datasets, and in-depth experiments on ML4C’s learnability. The latter is further divided   \n301 into four aspects: i) Towards a perfect classifier. As stated in proposition proposition 2, ML4C  \n302 Learner is the core component and we would like to know how far it is from a perfect classifier.   \n303 ii) Reliability (against weak / strong discriminative predicates). As stated in lemma 2 and 3, there   \n304 exist weak and strong discriminative predicates, which have discriminative power and thus are helpful   \n305 for ML4C-Learner. Some strong discriminative predicates are equivalent to specific logics of existing   \n306 work such as CPC or GLL-MB. Thus, we would like to see how ML4C-Learner takes the advantage   \n307 of machine learning, to learn a more reliable classification mechanism (which is also latent and   \n308 more sophisticated) than individual weak / strong discriminative predicates. iii) Robustness (against   \n309 varied sample size). It is known that many causal learning algorithms lack robustness w.r.t sample   \n310 noise for finite datasets [20], especially CI tests are error-prone on small samples for constraint  \n311 based algorithms. We would like to evaluate the robustness of ML4C (i.e., the latent classification   \n312 mechanism) against varied sample sizes. iv) Transferability. It’s important for a machine learning   \n313 model to generalize well to various types of testing data which are different from training data, such   \n314 as different scale (#nodes), graph sparsity, different generating mechanisms, etc.   \n315 Metrics We use two standard metrics for performance evaluation: Structural Hamming Distance   \n316 (SHD) and F1-score. For each dataset, we measure the SHD / F1-score of the output CPDAG (learned   \n317 by a specific algorithm) against the ground truth CPDAG. Specifically, SHD is calculated at CPDAG   \n318 level, which is the smallest number of edge additions, deletions, direction reversals and type changes   \n319 (directed vs. undirected) to convert the output CPDAG to ground truth CPDAG. F1-score is calculated   \n320 over identifiable edges. Roughly, F1-score can be viewed as a normalized version of SHD. Now we   \n321 present the experiment results: ",
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+        "text": "End-to-End Comparison Due to page limit, we report SHD and F1-score of all algorithms on 19 large-scale datasets (full results including other 5 smallest and trivial datasets are available in the supplementary material), as depicted in Table 1. ‘-’ means the algorithm fails on the dataset (either out-of-memory / exceeds 24 hours execution time / break caused by unknown errors). ML4C significantly outperforms all other competitors. The average F1-score of ML4C is the highest (0.92, first column in Table 2). Moreover, ML4C exhibits the most stable performance across all datasets, its average ranking is $1 . 5 \\pm 0 . 7$ , while the second best is GLL-MB (GMB), with average ranking ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d462d1e83f7115507e5eb293a7f6b66df44077f8f1ef2c53048eb0c08f073370.jpg",
+        "table_caption": [
+            "Table 1: Experiment results for end-to-end comparison with SOTA causal learning algorithms on benchmark datasets. Algorithm names are abbreviated. SHD and F1-score are reported. The last two rows show statistics of rank by SHD and F1-score for all competitors (Note: F1-score is at UT level). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Datasets #nodes/#edges</td><td rowspan=\"2\"></td><td colspan=\"4\">supervised</td><td colspan=\"6\">unsupervised</td><td colspan=\"4\">no skeleton input</td></tr><tr><td>ML4C</td><td>Jarfo</td><td>D2C RCC</td><td>NCC</td><td>PC CPC</td><td>MPC</td><td>GMB</td><td>GES</td><td>GS</td><td>HC</td><td>CDS</td><td>DGNN</td><td>BLIP 0</td><td>GNIP</td></tr><tr><td>child 20/25</td><td>SHD F1</td><td>.</td><td>8</td><td>16</td><td>8 </td><td>22 </td><td>.94</td><td></td><td>1</td><td>3</td><td>3</td><td></td><td>2</td><td>1.0</td><td>0 1.0</td></tr><tr><td>insurance 27/52</td><td>SHD F1</td><td>9</td><td>6</td><td>30 3 .44</td><td>28 .44</td><td>36 </td><td>6</td><td>3</td><td>44</td><td>38</td><td>1</td><td>36</td><td>8</td><td>3</td><td>1</td></tr><tr><td>water 32/66</td><td>SHD F1</td><td>4</td><td>3</td><td>34 36</td><td>0 1.0</td><td>4 60 .97 .00</td><td>3</td><td>8</td><td>38</td><td>3</td><td>38 .46</td><td>18 .76</td><td>.8</td><td></td><td>3</td></tr><tr><td>mildew 35/46</td><td>SHD F1</td><td>8</td><td>：</td><td>18 30</td><td>3</td><td>36 ：</td><td>：</td><td>35</td><td>3</td><td>8</td><td>3</td><td>8</td><td></td><td>36</td><td>：</td></tr><tr><td>alarm 37/46</td><td>SHD F1</td><td>.8</td><td>3</td><td>4 18 .64</td><td>39</td><td>29 29</td><td>品</td><td>4</td><td>8</td><td>4</td><td>26</td><td>8</td><td></td><td></td><td>8</td></tr><tr><td>barley 48/84</td><td>SHD F1</td><td>5</td><td>48</td><td>58 5</td><td>L</td><td>6 1</td><td></td><td>8</td><td>3</td><td>：</td><td>34</td><td>5</td><td>8</td><td>60 .48</td><td>务</td></tr><tr><td>hailfinder 56/66</td><td>SHD F1</td><td>11 .80</td><td>3</td><td>41 43 .45 .42</td><td>0 1.0</td><td>17 .85 1</td><td></td><td>86</td><td>2</td><td>1</td><td>2</td><td>44 .42</td><td>76</td><td>111 .18</td><td>1</td></tr><tr><td>hepar2 70/123</td><td>SHD F1</td><td>0. 1.0</td><td>54 .59</td><td>81 59 .34 .54</td><td>0 1.0</td><td>35 27 .72 .81</td><td>37 .70</td><td>14 .89</td><td>46 .75</td><td>40 .70</td><td>35 .81</td><td>75 .39</td><td>123 .00</td><td>79 .54</td><td>61 .68</td></tr><tr><td>win95pts 76/112</td><td>SHD F1</td><td>1 .99</td><td>65 .43</td><td>51 3 .54</td><td>0 1.0</td><td>8 42 .95 .64</td><td>7 .95</td><td>品</td><td>3</td><td>21 .85</td><td>16 .91</td><td>50 .57</td><td>112 .00</td><td>103 .47</td><td>1 1</td></tr><tr><td>pathfinder 109/195</td><td>SHD F1</td><td>2</td><td>157 .21</td><td>145 151 .21 .29</td><td>0 1.0</td><td>150 .29 -</td><td>1 ： -</td><td>147 .30</td><td>158 .29</td><td>1 -</td><td>168 .28</td><td>148 .31</td><td>196 .00</td><td>241 .07</td><td>- 1</td></tr><tr><td>munin1 186/273</td><td>SHD F1</td><td>1</td><td>169 .42</td><td>154 153 .47 .46</td><td>3</td><td>8</td><td>7 ：</td><td>8</td><td>189</td><td>1</td><td>233</td><td>15</td><td>1</td><td>27</td><td>- -</td></tr><tr><td>andes 223/338</td><td>SHD F1</td><td>.0</td><td>226 .35</td><td>209 246 .41 .29</td><td>0 1.0</td><td>9</td><td>8 9</td><td>8</td><td>3</td><td>15 .96</td><td>38</td><td>149 .60</td><td>1 -</td><td>175</td><td>-</td></tr><tr><td>diabetes 413/602</td><td>SHD F1</td><td>3</td><td>2</td><td>38 237</td><td>8</td><td>10</td><td>1 </td><td>284</td><td></td><td>146</td><td>1 58</td><td>368 .43</td><td>1 1</td><td>534</td><td>1 1</td></tr><tr><td>pigs 441/592</td><td>SHD F1</td><td>0 1.0</td><td>350 .44</td><td>332 263 .46 .59</td><td>400 .35</td><td>400 .35</td><td>1 - - 1</td><td>268 .56</td><td></td><td>0 1 1.0 -</td><td>532 .18</td><td>316 .50</td><td>- -</td><td>6 1.0</td><td>-</td></tr><tr><td>link 724/1125</td><td>SHD F1</td><td>0 1.0</td><td>731 .38</td><td>630 .45</td><td>638 749 .45 .39</td><td>737 .40</td><td>1 - = -</td><td>204 .81</td><td></td><td>324 .80</td><td>1047 - .14 1</td><td>400 .64</td><td>- -</td><td>947 .49</td><td>= =</td></tr><tr><td>munin 1041/1397</td><td>SHD F1</td><td>3</td><td>97</td><td>790 .48</td><td>816 0 .44 1.0</td><td>156 .89</td><td>1 1 -</td><td>458 .69</td><td>661</td><td>.62 1</td><td>1397 1.00 -</td><td>7</td><td>1 -</td><td>1599 .29</td><td>- =</td></tr><tr><td>munin2 1003/1244</td><td>SHD F1</td><td>118 .92</td><td>554 .60</td><td>611 .56</td><td>646 1052 .55 .19</td><td>898 .30</td><td>- - - -</td><td>539</td><td></td><td>632 .58</td><td>1240 - .01 -</td><td>753 .49</td><td>- -</td><td>1321 .46</td><td>- -</td></tr><tr><td>munin3 1041/1306</td><td>SHD F1</td><td>113 .92</td><td>616 .58</td><td>629 59</td><td>688 1048 .54 .25</td><td>860 .37</td><td>1 - 1 -</td><td>544 .60</td><td></td><td>56</td><td>- 1306 - 1.00</td><td>819 .46</td><td>- 1</td><td>1539 .26</td><td>- -</td></tr><tr><td>munin4 1038/1388</td><td>SHD F1</td><td>126 .93</td><td>696 .54</td><td>658 .56</td><td>776 1058 .50 .29</td><td>876 .39</td><td>- - - -</td><td></td><td></td><td>618 .64</td><td>1388 1 1 1.00</td><td>812 49</td><td>1 -</td><td>17</td><td>- -</td></tr><tr><td>rank(SHD)</td><td>mean 士stdd</td><td>1</td><td>3</td><td>3</td><td>38 51</td><td>3</td><td>10.8 2.9</td><td>8 24</td><td></td><td>28</td><td>36</td><td>27 2</td><td>1</td><td>15</td><td>17</td></tr><tr><td>UT-F1</td><td>mean ±stdd</td><td></td><td>:22</td><td>:1</td><td>46</td><td>4</td><td>33</td><td>:</td><td>3</td><td></td><td>:7</td><td>3</td><td>2</td><td>09 36</td><td>3</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 2: Reliability: average F1-score of ML4C vs. 8 discriminative predicates extracted from ML4C features on benchmark datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">ML4C</td><td rowspan=\"2\"></td><td colspan=\"4\">strong predicates</td><td colspan=\"4\">weak predicates</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>F1</td><td>.92±.20</td><td>.77±.31</td><td>.52±.27</td><td>.38±.25</td><td></td><td>.66±.27 .72±.25 .61±.29</td><td></td><td>.73±.30</td><td>）.55±.27</td></tr></table>",
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+        "text": "329 $4 . 4 \\pm 2 . 4$ . Among the competitors, NCC ranks #1 on 8 datasets (note that ML4C ranks #1 on 11   \n330 datasets), but its performance fluctuates. Overall it only ranks $5 . 1 \\pm 4 . 2$ . Last but not least, ML4C   \n331 shows high accuracy $( \\mathrm { F } 1 { > } 0 . 9 )$ on very large-scale datasets (e.g., medicine datasets ‘munin\\*’ [3])   \n332 while max(others) $\\sim 0 . 6$ .   \n333 Towards a Perfect Classifier The last row of Table 1 shows the performance of ML4C-Learner   \n334 component at UT level by UT-F1 (i.e., F1-score of classifying UTs): such UT level accuracy is   \n335 crucial for causal learning on discrete data, since the set of v-structures is invariant across all Markov   \n336 equivalent DAGs and it can fully recover the CPDAG. The average F1-score of ML4C-Learner is   \n337 $0 . 9 0 \\pm 0 . 1 3$ , which shows promising results towards a perfect classifier. ",
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+        "table_caption": [
+            "Table 3: Robustness: ML4C is trained on synthetic datasets with sample size $= 1 0 \\mathrm { k }$ , but tested on benchmark datasets with different sample sizes $\\in \\{ 1 \\mathrm { k } , 5 \\mathrm { k } , 1 0 \\mathrm { k } , 1 5 \\mathrm { k } , 2 0 \\mathrm { k } \\}$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Ⅱ</td><td>size</td><td>1k</td><td>5k</td><td>10k</td><td></td><td>15k 20k</td><td>size</td><td>1k</td><td>5k</td><td>10k</td><td>15k20k|</td><td></td><td>size</td><td>1k</td><td>5k</td><td>10k</td><td>15k 20k</td><td></td></tr><tr><td>SHD</td><td>insurance</td><td>11</td><td>1</td><td>5</td><td>1</td><td>0</td><td>water</td><td>12</td><td>11</td><td>5</td><td>8</td><td>6</td><td>mildew</td><td>8</td><td>5</td><td>3</td><td>6</td><td>1</td></tr><tr><td>F1</td><td>27/52</td><td>.81</td><td>.97</td><td>.89</td><td>.97</td><td>1.0</td><td>32/66</td><td>.86</td><td>.87</td><td>.94</td><td>.89</td><td>.93</td><td>35/46</td><td>.83</td><td>.89</td><td>.93</td><td>.87</td><td>.98</td></tr><tr><td>SHD</td><td>alarm</td><td>5</td><td>4</td><td>0</td><td>1</td><td>5</td><td>barley</td><td>13</td><td>9</td><td>4</td><td>8</td><td>6</td><td>hailfinder</td><td>15</td><td>15</td><td>6</td><td>15</td><td>13</td></tr><tr><td>F1</td><td>37/46</td><td>.93</td><td>.95</td><td>1.0</td><td>.98</td><td>.93</td><td>48/84</td><td>.88</td><td>.93</td><td>.97</td><td>.92</td><td>.94</td><td>56/66</td><td>.74</td><td>.72</td><td>.90</td><td>.72</td><td>.76</td></tr><tr><td>SHD</td><td>hepar2</td><td>8</td><td>2</td><td>0</td><td>1</td><td>2</td><td>win95pts</td><td>7</td><td>1</td><td>0</td><td>1</td><td>1</td><td>pathfinder</td><td>1</td><td>7</td><td>25</td><td>7</td><td>1</td></tr><tr><td>F1</td><td>70/123</td><td>.96</td><td>.99</td><td>1.0</td><td>.99</td><td>.99</td><td>76/112</td><td>.96</td><td>.99</td><td>1.0</td><td>.99</td><td>.99</td><td>109/195</td><td>.99</td><td>.92</td><td>.77</td><td>.92</td><td>.99</td></tr><tr><td>SHD</td><td>munin1</td><td>32</td><td>7</td><td>10</td><td>9</td><td>15</td><td>andes</td><td>3</td><td>2</td><td>0</td><td>2</td><td>0</td><td>diabetes</td><td>18</td><td>28</td><td>4</td><td>26</td><td>27</td></tr><tr><td>F1</td><td>186/273</td><td>.89</td><td>.98</td><td>.97</td><td>.97</td><td>.95</td><td>223/338</td><td>.99</td><td>.99</td><td>1.0</td><td>.99</td><td>1.0</td><td>413/602</td><td>.97</td><td>.95</td><td>.99</td><td>.96</td><td>.96</td></tr><tr><td>SHD</td><td>pigs</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>link</td><td>88</td><td>13</td><td>0</td><td>0</td><td>0</td><td>munin</td><td>107</td><td>76</td><td>71</td><td>93</td><td>87</td></tr><tr><td>F1</td><td>441/592</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>724/1125</td><td>.93</td><td>.99</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1041/1397</td><td>.93</td><td>.95</td><td>.96</td><td>.94</td><td>.94</td></tr><tr><td>SHD</td><td>munin2</td><td>117</td><td>95</td><td>120 110</td><td></td><td>97</td><td>munin3</td><td>151</td><td>119</td><td>113</td><td>99</td><td>62</td><td>munin4</td><td>165</td><td>130</td><td>123</td><td>146</td><td>5133</td></tr><tr><td>F1</td><td>1003/1244</td><td>.92.93</td><td></td><td>.92</td><td>.93</td><td>.93</td><td>1041/1306</td><td>.90</td><td>.92</td><td>.92</td><td>.94</td><td>.96</td><td>1038/1388</td><td>.90</td><td>.92</td><td>.93</td><td>.91</td><td>.93</td></tr></table>",
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+            "Table 4: Transferability: ML4C trains/tests both on synthetic datasets with different configurations. "
+        ],
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+        "table_body": "<table><tr><td></td><td>train</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td></tr><tr><td></td><td>10</td><td>10</td><td>1.2±2.4</td><td>.94±.12</td><td>50</td><td>4.8±3.4</td><td>.95±.03</td><td>100</td><td>6.6±4.7</td><td>.97±.02</td><td>1k</td><td>50.6±8.4</td><td>.97±.00</td></tr><tr><td></td><td>50</td><td>10</td><td>0.4±0.8</td><td>.97±.05</td><td>50</td><td>0.8±1.0</td><td>.99±.01</td><td>100</td><td>4.4±4.7</td><td>.98±.02</td><td>1k</td><td>23.2±5.7</td><td>.99±.00</td></tr><tr><td>əpou #</td><td>100</td><td>10</td><td>0.0±0.0</td><td>1.0±.00</td><td>50</td><td>1.2±1.6</td><td>.99±.01</td><td>100</td><td>4.0±4.6</td><td>.98±.02</td><td>1k</td><td>21.6±4.8</td><td>.99±.00</td></tr><tr><td></td><td>1k</td><td>10</td><td>0.4±0.8</td><td>.97±.05</td><td>50</td><td>0.8±1.0</td><td>.99±.01</td><td>100</td><td>1.4±2.3</td><td>.99±.01</td><td>1k</td><td>14.8±8.2</td><td>.99±.00</td></tr><tr><td>sreds</td><td>1</td><td>1</td><td>0.8±1.6</td><td>.99±.02</td><td>2</td><td>3.4±2.9</td><td>.97±.02</td><td>3</td><td>3.0±2.5</td><td>.98±.01</td><td>4</td><td>11.4±3.9</td><td>.95±.02</td></tr><tr><td></td><td>2</td><td>1</td><td>1.8±1.6</td><td>.98±.02</td><td>2</td><td>2.2±1.7</td><td>.98±.01</td><td>3</td><td>2.2±2.0</td><td>.99±.01</td><td>4</td><td>8.2±2.5</td><td>.97±.01</td></tr><tr><td></td><td>3</td><td>1</td><td>1.0±1.3</td><td>.98±.02</td><td>2</td><td>2.2±1.3</td><td>.98±.01</td><td>3</td><td>4.4±3.6</td><td>.97±.02</td><td>4</td><td>4.0±3.2</td><td>.98±.01</td></tr><tr><td></td><td>4</td><td>1</td><td>2.4±2.3</td><td>.97±.03</td><td>2</td><td>2.2±1.9</td><td>.98±.01</td><td>3</td><td>3.2±2.7</td><td>.98±.02</td><td>4</td><td>4.8±3.7</td><td>.98±.01</td></tr><tr><td></td><td>1k</td><td>1k</td><td>2.8±2.3</td><td>.97±.02</td><td>5k</td><td>2.0±2.2</td><td>.98±.02</td><td>10k</td><td>1.6±2.3</td><td>.98±.02</td><td>20k</td><td>1.0±1.3</td><td>.99±.01</td></tr><tr><td>srrsrstte</td><td>5k</td><td>1k</td><td>5.2±2.9</td><td>.95±.03</td><td>5k</td><td>1.0±2.0</td><td>.99±.02</td><td>10k</td><td>2.2±3.5</td><td>.98±.04</td><td>20k</td><td>0.6±0.8</td><td>.99±.01</td></tr><tr><td></td><td>10k</td><td>1k</td><td>5.2±4.8</td><td>.95±.05</td><td>5k</td><td>1.8±2.7</td><td>.98±.02</td><td>10k</td><td>2.0±3.1</td><td>.98±.03</td><td>20k</td><td>0.6±0.8</td><td>.99±.01</td></tr><tr><td></td><td>20k</td><td>1k</td><td>4.8±3.3</td><td>.95±.03</td><td>5k</td><td>2.4±2.6</td><td>.98±.02</td><td>10k</td><td>1.2±1.6</td><td>.99±.02</td><td>20k</td><td>1.0±1.3</td><td>.99±.02</td></tr><tr><td>R</td><td>ER</td><td>ER</td><td>1.0±2.0</td><td>.99±.02</td><td>SF</td><td>2.2±1.6</td><td>.98±.01</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>SF</td><td>ER</td><td>1.6±1.9</td><td>.98±.02</td><td>SF</td><td>2.2±2.4</td><td>.98±.02</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "type": "text",
+        "text": "Reliability We manually identify 4 strong discriminative predicates and 4 weak discriminative predicates and treat each one as a replacement of ML4C-Learner. Table 2 shows the performance of these predicates. Although most predicates show value on discriminating UTs (e.g., 5/8 predicates are with ${ > } 0 . 6$ F1-score), ML4C-Learner has higher performance (average F1-score $= 0 . 9 2$ ) than each individual predicate (best average F1-score $= 0 . 7 7$ ). Thus, it is evident that ML4C-Learner learns a more reliable classification mechanism, by taking advantage of machine learning techniques. ",
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+        "type": "text",
+        "text": "Robustness To evaluate robustness, ML4C is trained on synthetic datasets with sample size $=$ $1 0 \\mathrm { k }$ , but it is tested on benchmark datasets with different sample sizes: 1k, 5k, 10k, $1 5 \\mathrm { k }$ and $2 0 \\mathrm { k }$ respectively. Table 3 shows that ML4C exhibits satisfactory robustness (decrease of F1-score is less than 0.1) against sample size on most datasets (17/18, except for ‘hailfinder’). ",
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+        "type": "text",
+        "text": "Transferability To evaluate whether ML4C generalizes well to various types of testing data, we vary scale (#nodes), graph sparsity, generating mechanism and sample size. ML4C is trained on a fixed configuration but it is tested with different domains (i.e., data generated under different configuration). Result is depicted in Table 4, ML4C transfers well on different domains, e.g., even if it is trained on 10 nodes but tested on 1,000 nodes (last column of the first row in Table 4), the F1-score only drops 0.02. ",
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+        "type": "text",
+        "text": "54 6 Conclusion and Future Work ",
+        "text_level": 1,
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+        "text": "We have proposed a supervised causal learning algorithm ML4C, with theoretical guarantee on learnability and remarkable empirical performance. More importantly, ML4C shows promising results on validating the effectiveness of supervision. To make SCL practical in real-world scenarios, one important direction for future work is to identify reliable and accurate skeleton from data, considering ML4C requires skeleton as additional input. ",
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+        "text": "References   \n[1] Réka Albert and Albert-László Barabási. Statistical mechanics of complex networks. Reviews of modern physics, 74(1):47, 2002.   \n[2] Constantin F Aliferis, Alexander Statnikov, Ioannis Tsamardinos, Subramani Mani, and Xenofon D Koutsoukos. Local causal and markov blanket induction for causal discovery and feature selection for classification part i: algorithms and empirical evaluation. Journal of Machine Learning Research, 11(1), 2010.   \n[3] Steen Andreassen, Marianne Woldbye, Bjørn Falck, and Stig K Andersen. Munin: A causal probabilistic network for interpretation of electromyographic findings. In Proceedings of the 10th international joint conference on Artificial intelligence-Volume 1, pages 366–372, 1987.   \n[4] Gianluca Bontempi and Maxime Flauder. From dependency to causality: a machine learning approach. J. Mach. Learn. Res., 16(1):2437–2457, 2015.   \n[5] Tianqi Chen and Carlos Guestrin. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pages 785–794, 2016.   \n[6] David Maxwell Chickering. Optimal structure identification with greedy search. Journal of machine learning research, 3(Nov):507–554, 2002.   \n[7] Diego Colombo and Marloes H Maathuis. Order-independent constraint-based causal structure learning. J. Mach. Learn. Res., 15(1):3741–3782, 2014.   \n[8] James Cussens. Bayesian network learning with cutting planes. arXiv preprint arXiv:1202.3713, 2012.   \n[9] Colombo Diego and H. Maathuis Marloes. Order-independent constraint-based causal structure learning. Journal of Machine Learning Research, 15(116):3921–3962, 2014.   \n[10] Rui Ding, Yanzhi Liu, Jingjing Tian, Zhouyu Fu, Shi Han, and Dongmei Zhang. Reliable and efficient anytime skeleton learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, 2020.   \n[11] P. Erdos and Rényi A. On random graphs. ˝ Publicationes, Mathematicae, 6:290–297, 1959.   \n[12] José AR Fonollosa. Conditional distribution variability measures for causality detection. In Cause Effect Pairs in Machine Learning, pages 339–347. Springer, 2019.   \n[13] Clark Glymour, Kun Zhang, and Peter Spirtes. Review of causal discovery methods based on graphical models. Frontiers in genetics, 10:524, 2019.   \n[14] Patrik Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlinear causal discovery with additive noise models. Advances in neural information processing systems, 21:689–696, 2008.   \n[15] Dominik Janzing, Joris Mooij, Kun Zhang, Jan Lemeire, Jakob Zscheischler, Povilas Daniušis, Bastian Steudel, and Bernhard Schölkopf. Information-geometric approach to inferring causal directions. Artificial Intelligence, 182:1–31, 2012.   \n[16] Daphne Koller and Nir Friedman. Probabilistic graphical models: principles and techniques. MIT press, 2009.   \n[17] Steffen L Lauritzen. Graphical models, volume 17. Clarendon Press, 1996.   \n[18] Hebi Li, Qi Xiao, and Jin Tian. Supervised whole dag causal discovery. arXiv preprint arXiv:2006.04697, 2020.   \n[19] Honghao Li, Vincent Cabeli, Nadir Sella, and Hervé Isambert. Constraint-based causal structure learning with consistent separating sets. In 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), 2019.   \n[20] David Lopez-Paz, Krikamol Muandet, and Benjamin Recht. The randomized causation coefficient. J. Mach. Learn. Res., 16:2901–2907, 2015.   \n[21] David Lopez-Paz, Robert Nishihara, Soumith Chintala, Bernhard Scholkopf, and Léon Bottou. Discovering causal signals in images. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 6979–6987, 2017.   \n[22] Dimitris Margaritis. Learning bayesian network model structure from data. Technical report, CarnegieMellon Univ Pittsburgh Pa School of Computer Science, 2003.   \n[23] Christopher Meek. Causal inference and causal explanation with background knowledge. arXiv preprint arXiv:1302.4972, 2013.   \n[24] Christopher Meek. Strong completeness and faithfulness in bayesian networks. arXiv preprint arXiv:1302.4973, 2013.   \n[25] Joseph Ramsey, Jiji Zhang, and Peter Spirtes. Adjacency-faithfulness and conservative causal inference. arXiv preprint arXiv:1206.6843, 2012.   \n[26] Joseph Ramsey, Jiji Zhang, and Peter L Spirtes. Adjacency-faithfulness and conservative causal inference. arXiv preprint arXiv:1206.6843, 2012.   \n[27] Mauro Scanagatta, Cassio Polpo de Campos, Giorgio Corani, and Marco Zaffalon. Learning bayesian networks with thousands of variables. In NIPS, pages 1864–1872, 2015.   \n[28] Marco Scutari. Bayesian network repository. URL http://www. bnlearn. com, 2012.   \n[29] Shohei Shimizu, Patrik O Hoyer, Aapo Hyvärinen, Antti Kerminen, and Michael Jordan. A linear non-gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7(10), 2006.   \n[30] Alex Smola, Arthur Gretton, Le Song, and Bernhard Schölkopf. A hilbert space embedding for distributions. In International Conference on Algorithmic Learning Theory, pages 13–31. Springer, 2007.   \n[31] Peter Spirtes and Clark Glymour. An algorithm for fast recovery of sparse causal graphs. Social science computer review, 9(1):62–72, 1991.   \n[32] Peter Spirtes, Clark N Glymour, Richard Scheines, and David Heckerman. Causation, prediction, and search. MIT press, 2000.   \n[33] M.K. Vijaymeena and K. Kavitha. A survey on similarity measures in text mining. Machine Learning and Applications: An International Journal, 3(2):19–28.   \n[34] Kui Yu, Jiuyong Li, and Lin Liu. A review on algorithms for constraint-based causal discovery. arXiv preprint arXiv:1611.03977, 2016.   \n[35] Yue Yu, Jie Chen, Tian Gao, and Mo Yu. Dag-gnn: Dag structure learning with graph neural networks. In International Conference on Machine Learning, pages 7154–7163. PMLR, 2019.   \n[36] Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P Xing. Dags with no tears: Continuous optimization for structure learning. arXiv preprint arXiv:1803.01422, 2018. ",
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+        "type": "text",
+        "text": "Checklist ",
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+        "type": "text",
+        "text": "1. For all authors... ",
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+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See $\\ S 6$ . ML4C requires skeleton as additional input, thus we put identifying reliable and accurate skeleton from data as future work.   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "2. If you are including theoretical results... ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] Details of proofs are in the supplementary material due to page limit. ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We have included the main functionalities of ML4C, synthetic data generator, and ML4C’s training data (include instructions) for reproducibility. Details are in the supplementary material.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Key information is in content. Details are in the supplementary material.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We carefully design experiments, by running experiments multiple times, including error bars in report to ensure reproducibility. For instance, Table 4 contains error bar information.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
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+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We compare our algorithm with 15 SOTA competitors and we have cite these work properly. We also use open source benchmark for evaluation and we also properly cite the creators.   \n(b) Did you mention the license of the assets? [N/A]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We share new assets, include main functionalities of our algorithm, and our training data. A private URL is provided. See details in the supplementary material.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] Due to page limit, the detailed discussion is available in the supplementary material.   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We double-checked that the data we are using/curating contains no personally identifiable information or offensive content. ",
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+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
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diff --git a/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_middle.json b/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..ee57bc09e60bb14e0ba7b7570040ca9c37cdff04
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+                                    "score": 1.0,
+                                    "content": "with same input dataset. This statement is applicable to general learning target since an",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "content": "presuming additive noise model [14] or linear SEM with non-Gaussian noise [29] provides license",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "content": "-structures. A Markov equivalence class can be represented by a CPDAG",
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+                                    "score": 1.0,
+                                    "content": "can only identify a causal graph up to its Markov equivalence class on canonical data. Therefore, the",
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+                                    "score": 1.0,
+                                    "content": "as its learning target, i.e., it classifies whether an input UT",
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+                                    "score": 0.92,
+                                    "content": "\\langle X , T , Y \\rangle",
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+                                    "score": 1.0,
+                                    "content": "is a v-structure (orientation:",
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+                        {
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+                                    "score": 0.89,
+                                    "content": "X \\right. T \\left. Y",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ") or not (orientation remains unknown). Figure 1(b) depicts the overall workflow of",
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "ML4C, which is composed of ML4C-Learner with other important inductive biases. Starting from an",
+                                    "type": "text"
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+                                    "content": "input dataset",
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+                                    "content": "D _ { i }",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "with corresponding skeleton",
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+                                    "score": 0.88,
+                                    "content": "E _ { i }",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ", we first obtain all the UTs from",
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+                                    "content": "E _ { i }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Featurization is",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "then conducted to represent each UT as an embedded vector, which is further fed into ML4C-Learner.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "In the inference stage, we obtain all the v-structures which are classified by ML4C-Learner and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "reconstruct a partial DAG and then, a CPDAG is output by applying Meek rules on the partial DAG.",
+                                    "type": "text"
+                                }
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+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "In the training stage, the label of each UT is obtained by querying from ground truth DAG",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "content": "G _ { i }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": ". We",
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+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "collect labeled data from multiple datasets in ML4C’s training set.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 44
+                        }
+                    ],
+                    "index": 39.5
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+                                    "score": 1.0,
+                                    "content": "Proposition 2. If ML4C-Learner is a perfect classifier, then ML4C outputs correct CPDAG of a",
+                                    "type": "text"
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+                        },
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+                                    "score": 1.0,
+                                    "content": "172 By Markov completeness, the set of v-structures is invariant across all Markov equivalent DAGs for",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "score": 1.0,
+                                    "content": "173 a canonical dataset, and it can fully recover the CPDAG, provided that the skeleton is given. Thus,",
+                                    "type": "text"
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+                                    "content": "25 3.2 Causal Structure Identifiability",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "score": 1.0,
+                                    "content": "Identifiability discusses which parts of the causal structure can in principle be inferred from the",
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+                            ],
+                            "index": 1
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "distribution. Below we present the established theory of identifiability on discrete data.",
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+                            ],
+                            "index": 2
+                        }
+                    ],
+                    "index": 1.5,
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+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Causal sufficiency: There are no latent common causes of any of the variables in the graph.",
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+                                }
+                            ],
+                            "index": 3
+                        }
+                    ],
+                    "index": 3,
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+                },
+                {
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+                    ],
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "Definition 1 (Markov equivalence). Two graphs are Markov equivalent if and only if they have",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                        157
+                                    ],
+                                    "score": 1.0,
+                                    "content": "same skeleton and same",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        205,
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+                                    "score": 0.47,
+                                    "content": "\\nu",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        212,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "-structures. A Markov equivalence class can be represented by a CPDAG",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        106,
+                                        156,
+                                        505,
+                                        168
+                                    ],
+                                    "score": 1.0,
+                                    "content": "having both directed and undirected edges. A CPDAG can be derived from a DAG G, denoted as",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        107,
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+                                    ],
+                                    "score": 0.91,
+                                    "content": "C P D A G ( G )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        155,
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+                                    "score": 1.0,
+                                    "content": ". The theorem of Markov completeness in [24] states that, under causal sufficiency, we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                                189
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        178,
+                                        505,
+                                        189
+                                    ],
+                                    "score": 1.0,
+                                    "content": "can only identify a causal graph up to its Markov equivalence class on canonical data. Therefore, the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                                201
+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(non-)identifiable causal relations are the (un)directed edges in the CPDAG. Formally,",
+                                    "type": "text"
+                                }
+                            ],
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+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "Definition 2 (Identifiability). Assuming",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.74,
+                                    "content": "P",
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+                                    "content": "is Markovian and faithful w.r.t. a DAG G and causal",
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+                            ],
+                            "index": 10
+                        },
+                        {
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "sufficiency, then each (un)directed edge in",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        277,
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+                                        326,
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+                                    ],
+                                    "score": 0.81,
+                                    "content": "C P D A G ( G )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "indicates (non-)identifiable causal relation.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "index": 10.5,
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+                                    "score": 1.0,
+                                    "content": "3.3 ML4C Related Notations",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "index": 12
+                },
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+                    "type": "text",
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+                                    "content": "Definition 3 (Skeleton). A skeleton",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.91,
+                                    "content": "P \\left( X _ { 1 } , X _ { 2 } , \\cdots , X _ { d } \\right)",
+                                    "type": "inline_equation"
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+                                    "content": "P C _ { T }",
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+                                    "score": 0.89,
+                                    "content": "\\mathcal { S } \\vee \\mathcal { P } \\mathcal { C } _ { T } : = \\left\\{ S \\cup I : S \\in \\mathcal { S } , I \\in \\mathcal { P } \\mathcal { C } _ { T } \\right\\}",
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+                                    "content": ". Here symbol ∨ is element-wise union.",
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+                                    "score": 1.0,
+                                    "content": ", and calculate the extended conditional dependency. There are in total",
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+                                    "content": "\\lvert \\mathbb { B } \\rvert \\times \\lvert \\mathbb { C } \\rvert = 2 4",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "-separation sets within vicinity. Furthermore, searching for all",
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+                                    "score": 1.0,
+                                    "content": "highly time-consuming, thus the computational cost can also be saved drastically.",
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+                    ],
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+                                    "content": "• Entanglement within Vicinity",
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+                                    "score": 1.0,
+                                    "content": "and",
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+                                    "score": 1.0,
+                                    "content": "can mutually share common neighbors, and their neighbors may also overlap with sepsets",
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+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 0.58,
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+                                    "score": 1.0,
+                                    "content": ". We call such overlaps structural entanglement. Intuitively, stronger entanglement indicates",
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+                                }
+                            ],
+                            "index": 44
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "denser structure of vicinity thus making the UT less likely to be identifiable. Therefore, structural",
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+                            ],
+                            "index": 45
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "entanglement is an important aspect for featurization. Specifically, we exploit the overlap coefficient",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "[33] to measure the entanglement:",
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+                                    "content": "Definition 10 (Overlap coefficient). OLP",
+                                    "type": "text"
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+                                    "score": 0.91,
+                                    "content": "( \\mathbf { A } , \\mathbf { B } ) : = | \\mathbf { A } \\cap \\mathbf { B } | / \\operatorname* { m i n } \\left( | \\mathbf { A } | , | \\mathbf { B } | \\right)",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "223 two sets of variables. We extend this formula to support ensemble as input:",
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+                                    "score": 0.85,
+                                    "content": "( \\mathbf { A } , S ) : = \\textstyle \\sum _ { i = 1 } ^ { | S | }",
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+                                    "score": 1.0,
+                                    "content": ". Naturally, we consider the",
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+                                    "score": 1.0,
+                                    "content": "entanglement in terms of overlap coefficient on each pair of items in domain",
+                                    "type": "text"
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+                            "index": 1
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Thus, we use 6 scalars to represent the entanglement within the vicinity of a UT.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "score": 1.0,
+                                    "content": "• Embedding",
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+                                    "content": "We aim to represent the dependencies and entanglement by a feature vector with fixed dimensionality,",
+                                    "type": "text"
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+                            "index": 4
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+                        {
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+                                    "score": 1.0,
+                                    "content": "which can be used to train ML4C-Learner. Regarding each extended conditional dependency",
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+                                    "score": 0.9,
+                                    "content": "\\mathbf { B } | \\mathcal { Z } : \\mathbf { A } \\sim \\mathbf { B } \\in \\mathbb { B } , \\mathcal { Z } \\in \\mathbb { C }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", it consists of a set of scalars with varied set size across UTs, we",
+                                    "type": "text"
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+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "adopt the kernel mean embedding technique in [30] to represent each",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        160,
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+                                    "score": 0.91,
+                                    "content": "\\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                        160,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "as a vector with",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        105,
+                                        170,
+                                        433,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "fixed dimensionality. We further modify the embedding algorithm by adding min",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.88,
+                                    "content": "\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\}",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "max",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        179,
+                                        194
+                                    ],
+                                    "score": 0.91,
+                                    "content": "\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\}",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        179,
+                                        182,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "as two additional features. We directly use the 6 scalars to represent structural",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                205
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "entanglement without further transformation. We concatenate all the embedded vectors to form the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        203,
+                                        299,
+                                        216
+                                    ],
+                                    "score": 1.0,
+                                    "content": "final feature vector, as input for ML4C-Learner.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "index": 7.5
+                },
+                {
+                    "type": "title",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "36 4.3 Learnability",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "type": "text",
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "We have presented ML4C’s featurization and started seeing that conditional dependencies and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "structural entanglement have potential to reveal asymmetry to distinguish v-structure and non-v-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "structure UTs. Now we provide rigorous analysis to show that, for a canonical dataset with sufficient",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "samples, ML4C-Learner tends to a perfect classifier. To prove this, we first propose a surrogate object",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                                293,
+                                233,
+                                305
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        293,
+                                        233,
+                                        305
+                                    ],
+                                    "score": 1.0,
+                                    "content": "called discriminative predicate:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "Definition 11 (Discriminative predicate). A discriminative predicate is a binary predicate function",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "with domain as ML4C’s feature set. A discriminative predicate can be viewed as a special classifier",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                                342
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        330,
+                                        375,
+                                        342
+                                    ],
+                                    "score": 1.0,
+                                    "content": "with pre-specified form of mechanism (i.e., not learned from data).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Definition 12 (Weak / Strong discriminative predicate). Whenever a discriminative predicate takes",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the feature vector of a UT as input, a weak discriminative predicate satisfies one of the following",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "two criteria; a strong discriminative predicate satisfies both: i) it is evaluated to TRUE if the UT is a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "v-structure; ii) it is evaluated to FALSE if the UT is not a v-structure.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        }
+                    ],
+                    "index": 22.5
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "By definition, a weak discriminative predicate exhibits discriminative power since it is evaluated false",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "implies the UT is a non-v-structure (or true implies v-structure). A strong discriminative predicate",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "can be viewed as a perfect classifier. Denote",
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+                                    "content": "\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\} > \\delta : = X \\sim Y | \\mathbf { Z } > \\delta : \\forall X \\in \\mathbf { A } , Y \\in",
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+                            "index": 27
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+                        {
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+                                    "bbox": [
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+                                    "score": 0.89,
+                                    "content": "\\mathbf { B } , \\mathbf { Z } \\in { \\mathcal { Z } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", then we have:",
+                                    "type": "text"
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+                            ],
+                            "index": 28
+                        }
+                    ],
+                    "index": 26.5
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Lemma 2 (Existence of weak discriminative predicate). For a canonical dataset with infinite samples,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
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+                                    "score": 0.88,
+                                    "content": "\\{ P C _ { X } \\overset { \\cdot } { \\sim } P C _ { Y } | S \\cup T \\} > 0 .",
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+                                    "score": 0.86,
+                                    "content": "\\Rightarrow T",
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+                                    "score": 1.0,
+                                    "content": "is a collider",
+                                    "type": "text"
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+                                    "score": 0.85,
+                                    "content": "\\Rightarrow T",
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+                                    "score": 1.0,
+                                    "content": "unblocks",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "and",
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+                                    "score": 1.0,
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+                                    "bbox": [
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+                                    "score": 0.84,
+                                    "content": "\\left( \\mathbf { A } , S _ { i } \\right) / \\left| S \\right|",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". Naturally, we consider the",
+                                    "type": "text"
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+                            ],
+                            "index": 0
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+                                    "score": 1.0,
+                                    "content": "entanglement in terms of overlap coefficient on each pair of items in domain",
+                                    "type": "text"
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+                            ],
+                            "index": 1
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Thus, we use 6 scalars to represent the entanglement within the vicinity of a UT.",
+                                    "type": "text"
+                                }
+                            ],
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+                    ],
+                    "index": 1,
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+                                    "score": 1.0,
+                                    "content": "• Embedding",
+                                    "type": "text"
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+                        }
+                    ],
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+                },
+                {
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+                                    "content": "We aim to represent the dependencies and entanglement by a feature vector with fixed dimensionality,",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "which can be used to train ML4C-Learner. Regarding each extended conditional dependency",
+                                    "type": "text"
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+                                    "content": "\\mathbf { A } \\sim",
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+                            "index": 5
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+                                    "score": 0.9,
+                                    "content": "\\mathbf { B } | \\mathcal { Z } : \\mathbf { A } \\sim \\mathbf { B } \\in \\mathbb { B } , \\mathcal { Z } \\in \\mathbb { C }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", it consists of a set of scalars with varied set size across UTs, we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "adopt the kernel mean embedding technique in [30] to represent each",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.91,
+                                    "content": "\\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "as a vector with",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "fixed dimensionality. We further modify the embedding algorithm by adding min",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.88,
+                                    "content": "\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\}",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        182,
+                                        127,
+                                        194
+                                    ],
+                                    "score": 1.0,
+                                    "content": "max",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
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+                                        179,
+                                        194
+                                    ],
+                                    "score": 0.91,
+                                    "content": "\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\}",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        179,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "as two additional features. We directly use the 6 scalars to represent structural",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                                205
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "entanglement without further transformation. We concatenate all the embedded vectors to form the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                        216
+                                    ],
+                                    "score": 1.0,
+                                    "content": "final feature vector, as input for ML4C-Learner.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "index": 7.5,
+                    "bbox_fs": [
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+                    ]
+                },
+                {
+                    "type": "title",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "36 4.3 Learnability",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We have presented ML4C’s featurization and started seeing that conditional dependencies and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "structural entanglement have potential to reveal asymmetry to distinguish v-structure and non-v-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "structure UTs. Now we provide rigorous analysis to show that, for a canonical dataset with sufficient",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "samples, ML4C-Learner tends to a perfect classifier. To prove this, we first propose a surrogate object",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                        293,
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+                                        305
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+                                    "score": 1.0,
+                                    "content": "called discriminative predicate:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "index": 15,
+                    "bbox_fs": [
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+                    ]
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Definition 11 (Discriminative predicate). A discriminative predicate is a binary predicate function",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "with domain as ML4C’s feature set. A discriminative predicate can be viewed as a special classifier",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                                342
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
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+                                        342
+                                    ],
+                                    "score": 1.0,
+                                    "content": "with pre-specified form of mechanism (i.e., not learned from data).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        }
+                    ],
+                    "index": 19,
+                    "bbox_fs": [
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+                    ]
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Definition 12 (Weak / Strong discriminative predicate). Whenever a discriminative predicate takes",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        369
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+                                    "score": 1.0,
+                                    "content": "the feature vector of a UT as input, a weak discriminative predicate satisfies one of the following",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "two criteria; a strong discriminative predicate satisfies both: i) it is evaluated to TRUE if the UT is a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "v-structure; ii) it is evaluated to FALSE if the UT is not a v-structure.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        }
+                    ],
+                    "index": 22.5,
+                    "bbox_fs": [
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+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "By definition, a weak discriminative predicate exhibits discriminative power since it is evaluated false",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "implies the UT is a non-v-structure (or true implies v-structure). A strong discriminative predicate",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "can be viewed as a perfect classifier. Denote",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.88,
+                                    "content": "\\{ \\mathbf { A } \\sim \\mathbf { B } | \\mathcal { Z } \\} > \\delta : = X \\sim Y | \\mathbf { Z } > \\delta : \\forall X \\in \\mathbf { A } , Y \\in",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        107,
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+                                        147,
+                                        443
+                                    ],
+                                    "score": 0.89,
+                                    "content": "\\mathbf { B } , \\mathbf { Z } \\in { \\mathcal { Z } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        148,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": ", then we have:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        }
+                    ],
+                    "index": 26.5,
+                    "bbox_fs": [
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+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Lemma 2 (Existence of weak discriminative predicate). For a canonical dataset with infinite samples,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the following are three weak discriminative predicates: i)",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.91,
+                                    "content": "\\{ X \\sim Y | T \\} > 0",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", ii)",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.92,
+                                    "content": "\\{ X \\sim Y | \\mathcal { P } \\mathcal { C } _ { T } \\} = 0",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", iii)",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "thus in practice when an appropriate machine learning model is adopted, ML4C-Learner achieves",
+                                    "type": "text"
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "better performance.",
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+                                    "score": 1.0,
+                                    "content": "4 Theorem 1. ML4C-Learner tends to a perfect classifier on classifying a canonical dataset with",
+                                    "type": "text"
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+                            ],
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+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "5 sufficient samples.",
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+                            "is_list_start_line": true
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+                    ],
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+            "preproc_blocks": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Benchmark Datasets We use discrete datasets sampled by all 24 networks from bnlearn reposi-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "tory [28] for evaluation. For each network, we sample 1k, 5k, 10k, 15k, 20k records for use.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 1
+                        }
+                    ],
+                    "index": 0.5
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+                                    "score": 1.0,
+                                    "content": "ML4C’s Training and Inference We generate ML4C’s training data synthetically (which is also",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        },
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+                                    "score": 1.0,
+                                    "content": "used for other SCL competitors). Specifically, 400 unique DAGs are randomly generated by two",
+                                    "type": "text"
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "models: Erdos-Rényi (ER) model [ ˝ 11] and Scale-Free (SF) model [1], with the number of nodes",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "ranging from 10 to 1,000. A standard random forward data generation process is applied to obtain",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "10k observational samples for each graph. We further extract UTs from the",
+                                    "type": "text"
+                                },
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+                                    "bbox": [
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+                                    "score": 0.29,
+                                    "content": "4 0 0 \\mathrm { D A G s }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", consisting of",
+                                    "type": "text"
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+                            "index": 6
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "97,010 v-structures (label",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.68,
+                                    "content": "= 1",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ") and 195,691 non-v-structures",
+                                    "type": "text"
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+                                    "bbox": [
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+                                    "score": 0.27,
+                                    "content": "\\mathrm { | a b e l = 0 }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        393,
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+                                    "score": 1.0,
+                                    "content": "). We use these instances to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "train ML4C-Learner, which is implemented by a XGBoost [5] binary classifier with default hyper-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "parameters and we use binary cross-entropy as the loss function. Details on our synthesis procedure,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                106,
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+                                499,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        216,
+                                        499,
+                                        229
+                                    ],
+                                    "score": 1.0,
+                                    "content": "configurations and implementation of ML4C-Learner are available in the supplementary material.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                        359
+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Competitors We categorize state-of-the-art causal learning algorithms from two aspects, supervised",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        249,
+                                        507,
+                                        263
+                                    ],
+                                    "score": 1.0,
+                                    "content": "vs. unsupervised, and can or cannot take skeleton as input. We choose Jarfo [12], D2C [4], RCC [20],",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                106,
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+                                273
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        261,
+                                        506,
+                                        273
+                                    ],
+                                    "score": 1.0,
+                                    "content": "and NCC [21] as SCL competitors. Same as ML4C, all these algorithms can and do require skeleton",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                272,
+                                507,
+                                285
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        272,
+                                        507,
+                                        285
+                                    ],
+                                    "score": 1.0,
+                                    "content": "as input. All these algorithms use ML4C’s training set for training but with different learning target",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        282,
+                                        507,
+                                        296
+                                    ],
+                                    "score": 1.0,
+                                    "content": "extracted. Regarding unsupervised algorithms, we choose PC [31], Conservative-PC (CPC) [26],",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
+                                104,
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+                                307
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        506,
+                                        307
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Majority-rule PC (MPC) [7], GLL-MB (GMB) [2], GES [6], Grow-Shrink (GS) [22], Hill-Climbing",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                317
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        317
+                                    ],
+                                    "score": 1.0,
+                                    "content": "(HC) [16], and Conditional Distribution Similarity (CDS) [12]. which can also take skeleton as input.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        329
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Lastly, we also compare with DAG-GNN (DGNN) [35], BLIP [27], and GOBNILP (GNIP) [8],",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                339
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        326,
+                                        506,
+                                        339
+                                    ],
+                                    "score": 1.0,
+                                    "content": "which are unsupervised algorithms but cannot take skeleton as input. All these competitors are",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                351
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        351
+                                    ],
+                                    "score": 1.0,
+                                    "content": "capable of dealing with discrete data. All experiments are done in a Windows Server with 2.8GHz",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Intel E5-2680 CPU and 256G RAM. Details are in the supplementary material.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "299",
+                                    "type": "text"
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Design Our evaluation mainly consists of two parts: end-to-end comparison with competitors on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "content": "300",
+                                    "type": "text"
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "benchmark datasets, and in-depth experiments on ML4C’s learnability. The latter is further divided",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "301",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "into four aspects: i) Towards a perfect classifier. As stated in proposition proposition 2, ML4C-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "302",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        416
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Learner is the core component and we would like to know how far it is from a perfect classifier.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
+                            "bbox": [
+                                86,
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+                                427
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        86,
+                                        417,
+                                        99,
+                                        426
+                                    ],
+                                    "score": 1.0,
+                                    "content": "303",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "ii) Reliability (against weak / strong discriminative predicates). As stated in lemma 2 and 3, there",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
+                                86,
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+                                438
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        86,
+                                        428,
+                                        99,
+                                        437
+                                    ],
+                                    "score": 1.0,
+                                    "content": "304",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        425,
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+                                        438
+                                    ],
+                                    "score": 1.0,
+                                    "content": "exist weak and strong discriminative predicates, which have discriminative power and thus are helpful",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
+                                86,
+                                435,
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+                                450
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        86,
+                                        438,
+                                        100,
+                                        449
+                                    ],
+                                    "score": 1.0,
+                                    "content": "305",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        435,
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+                                        450
+                                    ],
+                                    "score": 1.0,
+                                    "content": "for ML4C-Learner. Some strong discriminative predicates are equivalent to specific logics of existing",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
+                                86,
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+                                460
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        86,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "306",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        460
+                                    ],
+                                    "score": 1.0,
+                                    "content": "work such as CPC or GLL-MB. Thus, we would like to see how ML4C-Learner takes the advantage",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "more sophisticated) than individual weak / strong discriminative predicates. iii) Robustness (against",
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+                                    "score": 1.0,
+                                    "content": "varied sample size). It is known that many causal learning algorithms lack robustness w.r.t sample",
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+                                    "score": 1.0,
+                                    "content": "noise for finite datasets [20], especially CI tests are error-prone on small samples for constraint-",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "based algorithms. We would like to evaluate the robustness of ML4C (i.e., the latent classification",
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+                                    "score": 1.0,
+                                    "content": "mechanism) against varied sample sizes. iv) Transferability. It’s important for a machine learning",
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+                                    "score": 1.0,
+                                    "content": "model to generalize well to various types of testing data which are different from training data, such",
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+                                    "score": 1.0,
+                                    "content": "as different scale (#nodes), graph sparsity, different generating mechanisms, etc.",
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+                    "index": 29.5
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+                                    "score": 1.0,
+                                    "content": "Metrics We use two standard metrics for performance evaluation: Structural Hamming Distance",
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+                            ],
+                            "index": 38
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+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(SHD) and F1-score. For each dataset, we measure the SHD / F1-score of the output CPDAG (learned",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "by a specific algorithm) against the ground truth CPDAG. Specifically, SHD is calculated at CPDAG",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
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+                                    "score": 1.0,
+                                    "content": "level, which is the smallest number of edge additions, deletions, direction reversals and type changes",
+                                    "type": "text"
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+                            "index": 41
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(directed vs. undirected) to convert the output CPDAG to ground truth CPDAG. F1-score is calculated",
+                                    "type": "text"
+                                }
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+                            "index": 42
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "over identifiable edges. Roughly, F1-score can be viewed as a normalized version of SHD. Now we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "present the experiment results:",
+                                    "type": "text"
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+                            ],
+                            "index": 44
+                        }
+                    ],
+                    "index": 41
+                },
+                {
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+                                    "score": 1.0,
+                                    "content": "End-to-End Comparison Due to page limit, we report SHD and F1-score of all algorithms on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 45
+                        },
+                        {
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+                                    "content": "19 large-scale datasets (full results including other 5 smallest and trivial datasets are available in",
+                                    "type": "text"
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+                            ],
+                            "index": 46
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the supplementary material), as depicted in Table 1. ‘-’ means the algorithm fails on the dataset",
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+                        {
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+                                    "content": "(either out-of-memory / exceeds 24 hours execution time / break caused by unknown errors). ML4C",
+                                    "type": "text"
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+                            "index": 48
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+                                    "score": 1.0,
+                                    "content": "significantly outperforms all other competitors. The average F1-score of ML4C is the highest (0.92,",
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+                                    "score": 1.0,
+                                    "content": "first column in Table 2). Moreover, ML4C exhibits the most stable performance across all datasets,",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 0.88,
+                                    "content": "1 . 5 \\pm 0 . 7",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", while the second best is GLL-MB (GMB), with average ranking",
+                                    "type": "text"
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+                            "index": 51
+                        }
+                    ],
+                    "index": 48
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+                                    "score": 1.0,
+                                    "content": "Benchmark Datasets We use discrete datasets sampled by all 24 networks from bnlearn reposi-",
+                                    "type": "text"
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+                            ],
+                            "index": 0
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "tory [28] for evaluation. For each network, we sample 1k, 5k, 10k, 15k, 20k records for use.",
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+                            "index": 1
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+                    ],
+                    "index": 0.5,
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+                                    "score": 1.0,
+                                    "content": "ML4C’s Training and Inference We generate ML4C’s training data synthetically (which is also",
+                                    "type": "text"
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+                            ],
+                            "index": 2
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "used for other SCL competitors). Specifically, 400 unique DAGs are randomly generated by two",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "models: Erdos-Rényi (ER) model [ ˝ 11] and Scale-Free (SF) model [1], with the number of nodes",
+                                    "type": "text"
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+                            ],
+                            "index": 4
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "ranging from 10 to 1,000. A standard random forward data generation process is applied to obtain",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "10k observational samples for each graph. We further extract UTs from the",
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+                                    "score": 0.29,
+                                    "content": "4 0 0 \\mathrm { D A G s }",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "97,010 v-structures (label",
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+                                    "score": 0.68,
+                                    "content": "= 1",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ") and 195,691 non-v-structures",
+                                    "type": "text"
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+                                    "score": 0.27,
+                                    "content": "\\mathrm { | a b e l = 0 }",
+                                    "type": "inline_equation"
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "). We use these instances to",
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+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "train ML4C-Learner, which is implemented by a XGBoost [5] binary classifier with default hyper-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "parameters and we use binary cross-entropy as the loss function. Details on our synthesis procedure,",
+                                    "type": "text"
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+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "configurations and implementation of ML4C-Learner are available in the supplementary material.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "index": 6,
+                    "bbox_fs": [
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+                },
+                {
+                    "type": "text",
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+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "Competitors We categorize state-of-the-art causal learning algorithms from two aspects, supervised",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "vs. unsupervised, and can or cannot take skeleton as input. We choose Jarfo [12], D2C [4], RCC [20],",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "and NCC [21] as SCL competitors. Same as ML4C, all these algorithms can and do require skeleton",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        272,
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+                                        285
+                                    ],
+                                    "score": 1.0,
+                                    "content": "as input. All these algorithms use ML4C’s training set for training but with different learning target",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "extracted. Regarding unsupervised algorithms, we choose PC [31], Conservative-PC (CPC) [26],",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "score": 1.0,
+                                    "content": "Majority-rule PC (MPC) [7], GLL-MB (GMB) [2], GES [6], Grow-Shrink (GS) [22], Hill-Climbing",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(HC) [16], and Conditional Distribution Similarity (CDS) [12]. which can also take skeleton as input.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Lastly, we also compare with DAG-GNN (DGNN) [35], BLIP [27], and GOBNILP (GNIP) [8],",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        339
+                                    ],
+                                    "score": 1.0,
+                                    "content": "which are unsupervised algorithms but cannot take skeleton as input. All these competitors are",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "capable of dealing with discrete data. All experiments are done in a Windows Server with 2.8GHz",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Intel E5-2680 CPU and 256G RAM. Details are in the supplementary material.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
+                    "index": 16,
+                    "bbox_fs": [
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+                                    "score": 1.0,
+                                    "content": "Design Our evaluation mainly consists of two parts: end-to-end comparison with competitors on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "benchmark datasets, and in-depth experiments on ML4C’s learnability. The latter is further divided",
+                                    "type": "text"
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+                            "index": 23,
+                            "is_list_start_line": true
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "into four aspects: i) Towards a perfect classifier. As stated in proposition proposition 2, ML4C-",
+                                    "type": "text"
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+                            "index": 24,
+                            "is_list_start_line": true
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "302",
+                                    "type": "text"
+                                },
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Learner is the core component and we would like to know how far it is from a perfect classifier.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "ii) Reliability (against weak / strong discriminative predicates). As stated in lemma 2 and 3, there",
+                                    "type": "text"
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+                            ],
+                            "index": 26,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "exist weak and strong discriminative predicates, which have discriminative power and thus are helpful",
+                                    "type": "text"
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+                            ],
+                            "index": 27,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "for ML4C-Learner. Some strong discriminative predicates are equivalent to specific logics of existing",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "work such as CPC or GLL-MB. Thus, we would like to see how ML4C-Learner takes the advantage",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "of machine learning, to learn a more reliable classification mechanism (which is also latent and",
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+                            ],
+                            "index": 30,
+                            "is_list_start_line": true
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "more sophisticated) than individual weak / strong discriminative predicates. iii) Robustness (against",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "varied sample size). It is known that many causal learning algorithms lack robustness w.r.t sample",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "noise for finite datasets [20], especially CI tests are error-prone on small samples for constraint-",
+                                    "type": "text"
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+                            ],
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+                            "is_list_start_line": true
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "based algorithms. We would like to evaluate the robustness of ML4C (i.e., the latent classification",
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+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "mechanism) against varied sample sizes. iv) Transferability. It’s important for a machine learning",
+                                    "type": "text"
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+                            "index": 35,
+                            "is_list_start_line": true
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "model to generalize well to various types of testing data which are different from training data, such",
+                                    "type": "text"
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+                            "index": 36,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "as different scale (#nodes), graph sparsity, different generating mechanisms, etc.",
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+                            ],
+                            "index": 37,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "Metrics We use two standard metrics for performance evaluation: Structural Hamming Distance",
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+                            ],
+                            "index": 38,
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+                        },
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+                                    "score": 1.0,
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+                                    "type": "text"
+                                },
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(SHD) and F1-score. For each dataset, we measure the SHD / F1-score of the output CPDAG (learned",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "317",
+                                    "type": "text"
+                                },
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "by a specific algorithm) against the ground truth CPDAG. Specifically, SHD is calculated at CPDAG",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40,
+                            "is_list_start_line": true
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "318",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "level, which is the smallest number of edge additions, deletions, direction reversals and type changes",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "319",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(directed vs. undirected) to convert the output CPDAG to ground truth CPDAG. F1-score is calculated",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "320",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "over identifiable edges. Roughly, F1-score can be viewed as a normalized version of SHD. Now we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "321",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "present the experiment results:",
+                                    "type": "text"
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+                            ],
+                            "index": 44,
+                            "is_list_start_line": true
+                        }
+                    ],
+                    "index": 29.5,
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+                                    "score": 1.0,
+                                    "content": "End-to-End Comparison Due to page limit, we report SHD and F1-score of all algorithms on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 45
+                        },
+                        {
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+                                    "content": "19 large-scale datasets (full results including other 5 smallest and trivial datasets are available in",
+                                    "type": "text"
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "the supplementary material), as depicted in Table 1. ‘-’ means the algorithm fails on the dataset",
+                                    "type": "text"
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+                            ],
+                            "index": 47
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(either out-of-memory / exceeds 24 hours execution time / break caused by unknown errors). ML4C",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 48
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "significantly outperforms all other competitors. The average F1-score of ML4C is the highest (0.92,",
+                                    "type": "text"
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+                            "index": 49
+                        },
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+                                    "score": 1.0,
+                                    "content": "first column in Table 2). Moreover, ML4C exhibits the most stable performance across all datasets,",
+                                    "type": "text"
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+                            ],
+                            "index": 50
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+                                    "score": 1.0,
+                                    "content": "its average ranking is",
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+                                    "score": 0.88,
+                                    "content": "1 . 5 \\pm 0 . 7",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", while the second best is GLL-MB (GMB), with average ranking",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 51
+                        }
+                    ],
+                    "index": 48,
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+                                            "content": "benchmark datasets. Algorithm names are abbreviated. SHD and F1-score are reported. The last two",
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+                                            "score": 1.0,
+                                            "content": "rows show statistics of rank by SHD and F1-score for all competitors (Note: F1-score is at UT level).",
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+                                    "content": "Towards a Perfect Classifier The last row of Table 1 shows the performance of ML4C-Learner",
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+                                            "content": "Table 4: Transferability: ML4C trains/tests both on synthetic datasets with different configurations.",
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+                                            "html": "<table><tr><td></td><td>train</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td><td>test</td><td>SHD</td><td>F1</td></tr><tr><td></td><td>10</td><td>10</td><td>1.2±2.4</td><td>.94±.12</td><td>50</td><td>4.8±3.4</td><td>.95±.03</td><td>100</td><td>6.6±4.7</td><td>.97±.02</td><td>1k</td><td>50.6±8.4</td><td>.97±.00</td></tr><tr><td></td><td>50</td><td>10</td><td>0.4±0.8</td><td>.97±.05</td><td>50</td><td>0.8±1.0</td><td>.99±.01</td><td>100</td><td>4.4±4.7</td><td>.98±.02</td><td>1k</td><td>23.2±5.7</td><td>.99±.00</td></tr><tr><td>əpou #</td><td>100</td><td>10</td><td>0.0±0.0</td><td>1.0±.00</td><td>50</td><td>1.2±1.6</td><td>.99±.01</td><td>100</td><td>4.0±4.6</td><td>.98±.02</td><td>1k</td><td>21.6±4.8</td><td>.99±.00</td></tr><tr><td></td><td>1k</td><td>10</td><td>0.4±0.8</td><td>.97±.05</td><td>50</td><td>0.8±1.0</td><td>.99±.01</td><td>100</td><td>1.4±2.3</td><td>.99±.01</td><td>1k</td><td>14.8±8.2</td><td>.99±.00</td></tr><tr><td>sreds</td><td>1</td><td>1</td><td>0.8±1.6</td><td>.99±.02</td><td>2</td><td>3.4±2.9</td><td>.97±.02</td><td>3</td><td>3.0±2.5</td><td>.98±.01</td><td>4</td><td>11.4±3.9</td><td>.95±.02</td></tr><tr><td></td><td>2</td><td>1</td><td>1.8±1.6</td><td>.98±.02</td><td>2</td><td>2.2±1.7</td><td>.98±.01</td><td>3</td><td>2.2±2.0</td><td>.99±.01</td><td>4</td><td>8.2±2.5</td><td>.97±.01</td></tr><tr><td></td><td>3</td><td>1</td><td>1.0±1.3</td><td>.98±.02</td><td>2</td><td>2.2±1.3</td><td>.98±.01</td><td>3</td><td>4.4±3.6</td><td>.97±.02</td><td>4</td><td>4.0±3.2</td><td>.98±.01</td></tr><tr><td></td><td>4</td><td>1</td><td>2.4±2.3</td><td>.97±.03</td><td>2</td><td>2.2±1.9</td><td>.98±.01</td><td>3</td><td>3.2±2.7</td><td>.98±.02</td><td>4</td><td>4.8±3.7</td><td>.98±.01</td></tr><tr><td></td><td>1k</td><td>1k</td><td>2.8±2.3</td><td>.97±.02</td><td>5k</td><td>2.0±2.2</td><td>.98±.02</td><td>10k</td><td>1.6±2.3</td><td>.98±.02</td><td>20k</td><td>1.0±1.3</td><td>.99±.01</td></tr><tr><td>srrsrstte</td><td>5k</td><td>1k</td><td>5.2±2.9</td><td>.95±.03</td><td>5k</td><td>1.0±2.0</td><td>.99±.02</td><td>10k</td><td>2.2±3.5</td><td>.98±.04</td><td>20k</td><td>0.6±0.8</td><td>.99±.01</td></tr><tr><td></td><td>10k</td><td>1k</td><td>5.2±4.8</td><td>.95±.05</td><td>5k</td><td>1.8±2.7</td><td>.98±.02</td><td>10k</td><td>2.0±3.1</td><td>.98±.03</td><td>20k</td><td>0.6±0.8</td><td>.99±.01</td></tr><tr><td></td><td>20k</td><td>1k</td><td>4.8±3.3</td><td>.95±.03</td><td>5k</td><td>2.4±2.6</td><td>.98±.02</td><td>10k</td><td>1.2±1.6</td><td>.99±.02</td><td>20k</td><td>1.0±1.3</td><td>.99±.02</td></tr><tr><td>R</td><td>ER</td><td>ER</td><td>1.0±2.0</td><td>.99±.02</td><td>SF</td><td>2.2±1.6</td><td>.98±.01</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>SF</td><td>ER</td><td>1.6±1.9</td><td>.98±.02</td><td>SF</td><td>2.2±2.4</td><td>.98±.02</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "Reliability We manually identify 4 strong discriminative predicates and 4 weak discriminative",
+                                    "type": "text"
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+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                                    "content": "predicates and treat each one as a replacement of ML4C-Learner. Table 2 shows the performance of",
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+                            "index": 10
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+                                    "score": 1.0,
+                                    "content": "these predicates. Although most predicates show value on discriminating UTs (e.g., 5/8 predicates",
+                                    "type": "text"
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+                                    "score": 0.76,
+                                    "content": "{ > } 0 . 6",
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+                                {
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+                                    "content": "F1-score), ML4C-Learner has higher performance (average F1-score",
+                                    "type": "text"
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+                                    "score": 0.77,
+                                    "content": "= 0 . 9 2",
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+                                {
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+                                    "score": 1.0,
+                                    "content": ") than each",
+                                    "type": "text"
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+                        {
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+                                    "content": "individual predicate (best average F1-score",
+                                    "type": "text"
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+                                    "bbox": [
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+                                    "score": 0.82,
+                                    "content": "= 0 . 7 7",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "). Thus, it is evident that ML4C-Learner learns a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "more reliable classification mechanism, by taking advantage of machine learning techniques.",
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+                            "index": 14
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+                                    "score": 1.0,
+                                    "content": "Robustness To evaluate robustness, ML4C is trained on synthetic datasets with sample size",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.37,
+                                    "content": "1 0 \\mathrm { k }",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", but it is tested on benchmark datasets with different sample sizes: 1k, 5k, 10k,",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.41,
+                                    "content": "1 5 \\mathrm { k }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
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+                                {
+                                    "bbox": [
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+                                    "score": 0.5,
+                                    "content": "2 0 \\mathrm { k }",
+                                    "type": "inline_equation"
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+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "respectively. Table 3 shows that ML4C exhibits satisfactory robustness (decrease of F1-score is less",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "than 0.1) against sample size on most datasets (17/18, except for ‘hailfinder’).",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Transferability To evaluate whether ML4C generalizes well to various types of testing data, we",
+                                    "type": "text"
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+                            ],
+                            "index": 19
+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "content": "if it is trained on 10 nodes but tested on 1,000 nodes (last column of the first row in Table 4), the",
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+                                    "score": 1.0,
+                                    "content": "F1-score only drops 0.02.",
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+                            ],
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+                                    "type": "text"
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+                            ],
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+                                    "type": "text"
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+                                    "type": "text"
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+                                    "type": "text"
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+}
\ No newline at end of file
diff --git a/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_model.json b/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..f3c2f5373dbb0bbb71077f8cbe08c226e6aeff5b
--- /dev/null
+++ b/parse/train/b36m4ZYG1gD/b36m4ZYG1gD_model.json
@@ -0,0 +1,22836 @@
+[
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diff --git a/parse/train/rkgW0oA9FX/rkgW0oA9FX.md b/parse/train/rkgW0oA9FX/rkgW0oA9FX.md
new file mode 100644
index 0000000000000000000000000000000000000000..521d6b4310d0d1d2a72e0bfa6c1cfefc6582710e
--- /dev/null
+++ b/parse/train/rkgW0oA9FX/rkgW0oA9FX.md
@@ -0,0 +1,379 @@
+# GRAPH HYPERNETWORKS FOR NEURAL ARCHITECTURE SEARCH
+
+Chris Zhang1,2, Mengye $\mathbf { R e n } ^ { 1 , 3 }$ & Raquel Urtasun1,3 1Uber Advanced Technologies Group, 2University of Waterloo, 3University of Toronto cjzhang@edu.uwaterloo.ca, {mren3,urtasun}@uber.com
+
+# ABSTRACT
+
+Neural architecture search (NAS) automatically finds the best task-specific neural network topology, outperforming many manual architecture designs. However, it can be prohibitively expensive as the search requires training thousands of different networks, while each can last for hours. In this work, we propose the Graph HyperNetwork (GHN) to amortize the search cost: given an architecture, it directly generates the weights by running inference on a graph neural network. GHNs model the topology of an architecture and therefore can predict network performance more accurately than regular hypernetworks and premature early stopping. To perform NAS, we randomly sample architectures and use the validation accuracy of networks with GHN generated weights as the surrogate search signal. GHNs are fast – they can search nearly $1 0 \times$ faster than other random search methods on CIFAR-10 and ImageNet. GHNs can be further extended to the anytime prediction setting, where they have found networks with better speed-accuracy tradeoff than the state-of-the-art manual designs.
+
+# 1 INTRODUCTION
+
+The success of deep learning marks the transition from manual feature engineering to automated feature learning. However, designing effective neural network architectures requires expert domain knowledge and repetitive trial and error. Recently, there has been a surge of interest in neural architecture search (NAS), where neural network architectures are automatically optimized.
+
+One approach for architecture search is to consider it as a nested optimization problem, where the inner loop finds the optimal parameters $w ^ { * }$ for a given architecture $a$ w.r.t. the training loss $\mathcal { L } _ { t r a i n }$ , and the outer loop searches the optimal architecture w.r.t. a validation loss $\mathcal { L } _ { v a l }$ :
+
+$$
+w ^ { * } ( a ) = \arg \operatorname* { m i n } _ { w } \mathcal { L } _ { t r a i n } ( w , a )
+$$
+
+$$
+\boldsymbol { a } ^ { * } = \arg \operatorname* { m i n } _ { \boldsymbol { a } } \mathcal { L } _ { v a l } ( \boldsymbol { w } ^ { * } ( \boldsymbol { a } ) , \boldsymbol { a } )
+$$
+
+Traditional NAS is expensive since solving the inner optimization in Eq. 1 requires a lengthy optimization process (e.g. stochastic gradient descent (SGD)). Instead, we propose to learn a parametric function approximation referred to as a hypernetwork (Ha et al., 2017; Brock et al., 2018), which attempts to generate the network weights directly. Learning a hypernetwork is an amortization of the cost of solving Eq. 1 repeatedly for multiple architectures. A trained hypernetwork is well correlated with SGD and can act as a much faster substitute.
+
+Yet, the architecture of the hypernet itself is still to be determined. Existing methods have explored a variety of tactics to represent architectures, such as an ingenious 3D tensor encoding scheme (Brock et al., 2018), or a string serialization processed by an LSTM (Zoph & Le, 2017; Zoph et al., 2018; Pham et al., 2018). In this work, we advocate for a computation graph representation as it allows for the topology of an architecture to be explicitly modeled. Furthermore, it is intuitive to understand and can be easily extensible to various graph sizes.
+
+To this end, in this paper we propose the Graph HyperNetwork (GHN), which can aggregate graph level information by directly learning on the graph representation. Using a hypernetwork to guide architecture search, our approach requires significantly less computation when compared to state-ofthe-art methods. The computation graph representation allows GHNs to be the first hypernetwork to generate all the weights of arbitrary CNNs rather than a subset (e.g. Brock et al. (2018)), achieving stronger correlation and thus making the search more efficient and accurate.
+
+While the validation accuracy is often the primary goal in architecture search, networks must also be resource aware in real-world applications. Towards this goal, we exploit the flexibility of the GHN by extending it to the problem of anytime prediction. Models capable of anytime prediction progressively update their predictions, allowing for a prediction at any time. This is desirable in settings such as real-time systems, where the computational budget available for each test case may vary greatly and cannot be known ahead of time. Although anytime models have non-trivial differences to classical models, we show the GHN is amenable to these changes.
+
+We summarize our main contributions of this work:
+
+1. We propose Graph HyperNetwork that predicts the parameters of unseen neural networks by directly operating on their computational graph representations.   
+2. Our approach achieves highly competitive results with state-of-the-art NAS methods on both CIFAR-10 and ImageNet-mobile and is $1 0 \times$ faster than other random search methods.   
+3. We demonstrate that our approach can be generalized and applied in the domain of anytimeprediction, previously unexplored by NAS programs, outperforming the existing manually designed state-of-the-art models.
+
+# 2 RELATED WORK
+
+Various search methods such as reinforcement learning (Zoph & Le, 2017; Baker et al., 2017a; Zoph et al., 2018), evolutionary methods (Real et al., 2017; Miikkulainen et al., 2017; Xie & Yuille, 2017; Liu et al., 2018b; Real et al., 2018) and gradient-based methods (Liu et al., 2018c; Luo et al., 2018) have been proposed to address the outer optimization (Eq. 2) of NAS, where an agent learns to sample architectures that are more likely to achieve higher accuracy. Different from these methods, this paper places its focus on the inner-loop: inferring the parameters of a given network (Eq. 1). Following Brock et al. (2018); Bender et al. (2018), we opt for a simple random search algorithm to complete the outer loop.
+
+While initial NAS methods simply train candidate architectures for a brief period with SGD to obtain the search signal, recent approaches have proposed alternatives in the interest of computational cost. Baker et al. (2017b) propose directly predicting performance from the learning curve, and Deng et al. (2017) propose to predict performance directly from the architecture without learning curve information. However, training a performance predictor requires a ground truth, thus the expensive process of computing the inner optimization is not avoided. Pham et al. (2018); Bender et al. (2018); Liu et al. (2018c) use parameter sharing, where a “one-shot” model containing all possible architectures in the search space is trained. Individual architectures are sampled by deactivating some nodes or edges in the one-shot model. In this case, predicting $w ^ { * } ( a )$ can be seen as using a selection function from the set of parameters in the one-shot model.
+
+Prior work has shown the feasibility of predicting parameters in a network with a function approximator (Denil et al., 2013). Schmidhuber (1992; 1993) proposed “fast-weights”, where one network produces weight changes for another. HyperNetworks (Ha et al., 2017) generate the weights of another network and show strong results in large-scale language modeling and image classification experiments. SMASH (Brock et al., 2018) applied HyperNetworks to perform NAS, where an architecture is encoded as a 3D tensor using a memory channel scheme. In contrast, we encode a network as a computation graph and use a graph neural network. While SMASH predicts a subset of the weights, our graph model is able to predict all the free weights.
+
+While earlier NAS methods focused on standard image classification and language modeling, recent literature has extended NAS to search for architectures that are computationally efficient (Tan et al., 2018; Dong et al., 2018; Hsu et al., 2018; Elsken et al., 2018; Zhou et al., 2018). In this work, we applied our GHN based search program on the task of anytime prediction, where we not only optimize for the final speed but the entire speed-accuracy trade-off curve.
+
+# 3 BACKGROUND
+
+We review the two major building blocks of our model: graph neural networks and hypernetworks.
+
+Graph Neural Network: A graph neural network (Scarselli et al., 2009; Li et al., 2016; Kipf & Welling, 2017) is a collection of nodes and edges $( \nu , \mathcal { E } )$ , where each node is a recurrent neural network (RNN) that individually sends and receives messages along the edges, spanning over the
+
+![](images/0ef3cc913230ebdf5240c3fb99335e7264d17c45c39a7135f8a534a643296895.jpg)  
+Figure 1: Our system diagram. A: A neural network architecture is randomly sampled, forming a GHN. B: After graph propagation, each node in the GHN generates its own weight parameters. C: The GHN is trained to minimize the training loss of the sampled network with the generated weights. Random networks are ranked according to their performance using GHN generated weights.
+
+horizon of message passing. Each node $v$ stores an internal node embedding vector $\pmb { h } _ { v } ^ { ( t ) } \in \mathbb { R } ^ { D }$ , and is updated recurrently:
+
+$$
+\pmb { h } _ { v } ^ { ( t + 1 ) } = \left\{ \begin{array} { l l } { U \left( \pmb { h } _ { v } ^ { ( t ) } , \pmb { m } _ { v } ^ { ( t ) } \right) \ } & { \mathrm { i f ~ n o d e ~ } v \mathrm { ~ i s ~ a c t i v e } , } \\ { \pmb { h } _ { v } ^ { ( t ) } \ } & { \mathrm { o t h e r w i s e } , } \end{array} \right.
+$$
+
+where $U$ is a recurrent cell function and $\mathbf { \Delta } _ { m _ { v } } ^ { ( t ) }$ is the message received by $v$ at time step $t$
+
+$$
+\pmb { m } _ { v } ^ { ( t ) } = \sum _ { u \in N _ { i n } ( v ) } M \left( \pmb { h } _ { u } ^ { ( t ) } \right) ,
+$$
+
+with $M$ the message function and $N _ { i n } ( v )$ the set of neighbors with incoming edges pointing towards $v$ . $U$ is often modeled with a long short-term memory (LSTM) unit (Hochreiter & Schmidhuber, 1997) or gated recurrent unit (GRU) (Cho et al., 2014), and $M$ with an MLP. Given a graph $\mathcal { A }$ , we define the GNN operator $G _ { A }$ to be a mapping from a set of initial node embeddings $\{ \bar { h _ { v } ^ { ( 0 ) } } \}$ to a set of different node embeddings $\{ h _ { v } ^ { ( t ) } \}$ , parameterized by some learnable parameters $\phi$ :
+
+$$
+\left\{ \pmb { h } _ { v } ^ { ( t ) } | v \in \mathcal { V } \right\} = G _ { \mathcal { A } } ^ { ( t ) } \left( \left\{ \pmb { h } _ { v } ^ { ( 0 ) } | v \in \mathcal { V } \right\} ; \phi \right) .
+$$
+
+Throughout propagation the node embeddings $\boldsymbol { h } _ { v } ^ { ( t ) }$ continuously aggregate graph level information, which can be used for tasks such as node prediction and graph prediction by further aggregation. Similar to RNNs, GNNs are typically learned using backpropagation through time (BPTT) (Werbos, 1990).
+
+Hypernetwork: A hypernetwork (Ha et al., 2017) is a neural network that generates the parameters of another network. For a typical deep feedforward network with $D$ layers, the parameters of the $j$ -th layer $W _ { j }$ can be generated by a learned function $H$ :
+
+$$
+W _ { j } = H ( z _ { j } ) , \thinspace \thinspace \forall j = 1 , . . . , D ,
+$$
+
+where $z _ { j }$ is the layer embedding, and $H$ is shared for all layers. The output dimensionality of the hypernetwork is fixed, but it’s possible to accommodate predicting weights for layers of varying kernel sizes by concatenating multiple kernels of the fixed size. Varying spatial sizes can also be accommodated by slicing in the spatial dimensions. Hypernetworks have been found effective in standard image recognition and text classification problems, and can be viewed as a relaxed weight sharing mechanism. Recently, they have shown to be effective in accelerating architecture search (Brock et al., 2018).
+
+# 4 GRAPH HYPERNETWORKS FOR NEURAL ARCHITECTURAL SEARCH
+
+Our proposed Graph HyperNetwork (GHN) is a composition of a graph neural network and a hypernetwork. It takes in a computation graph (CG) and generates all free parameters in the graph. During evaluation, the generated parameters are used to evaluate the fitness of a random architecture, and the top performer architecture on a separate validation set is then selected. This allows us to search over a large number of architectures at the cost of training a single GHN. We refer the reader to Figure 1 for a high level system overview.
+
+# 4.1 GRAPHICAL REPRESENTATION
+
+We represent a given architecture as a directed acyclic graph $\mathcal { A } = ( \mathcal { V } , \mathcal { E } )$ , where each node $v \in \mathcal V$ has an associated computational operator $f _ { v }$ parametrized by $w _ { v }$ , which produces an output activation tensor $x _ { v }$ . Edges $\bar { e _ { u \mapsto v } } = ( u , \bar { v } ) \in \mathcal { E }$ represent the flow of activation tensors from node $u$ to node $v$ . $x _ { v }$ is computed by applying its associated computational operator on each of its inputs and taking summation as follows
+
+$$
+x _ { v } = \sum _ { e _ { u \mapsto v } \in \mathcal { E } } f _ { v } ( x _ { u } ; w _ { v } ) , \forall v \in \mathcal { V } .
+$$
+
+# 4.2 GRAPH HYPERNETWORK
+
+Our proposed Graph Hypernetwork is defined as a composition of a GNN and a hypernetwork. First, given an input architecture, we used the graphical representation discussed above to form a graph $\mathcal { A }$ . A parallel GNN $G _ { A }$ is then constructed to be homomorphic to $\mathcal { A }$ with the exact same topology. Node embeddings are initialized to one-hot vectors representing the node’s computational operator. After graph message-passing steps, a hypernet uses the node embeddings to generate each node’s associated parameters. Let ${ h _ { v } ^ { ( T ) } }$ be the embedding of node $v$ after $T$ steps of GNN propagation, and let $H \left( \cdot ; \varphi \right)$ be a hypernetwork parametrized by $\varphi$ , the generated parameters $\tilde { \pmb { w } } _ { v }$ are:
+
+$$
+\tilde { \pmb { w } } _ { v } = H \left( \pmb { h } _ { v } ^ { ( T ) } ; \varphi \right) .
+$$
+
+For simplicity, we implement $H$ with a multilayer perceptron (MLP). It is important to note that $H$ is shared across all nodes, which can be viewed as an output prediction branch in each node of the GNN. Thus the final set of generated weights of the entire architecture $\tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } ^ { \tilde { \mathbf { \Gamma } } } \tilde { \mathbf { \Gamma } } ^ { \tilde { \mathbf { \Gamma } } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde \mathrm { \Gamma }$ is found by applying $H$ on all the nodes and their respective embeddings which are computed by $G _ { A }$ :
+
+$$
+\begin{array} { r l } & { \tilde { w } = \left\{ \tilde { w } _ { v } | v \in \mathcal { V } \right\} = \left\{ H \left( h _ { v } ^ { ( T ) } ; \varphi \right) \big | \ v \in \mathcal { V } \right\} } \\ & { \qquad = \Big \{ H \left( h ; \varphi \right) \big | \ h \in G _ { \mathcal { A } } ^ { ( T ) } \left( \Big \{ h _ { v } ^ { ( 0 ) } \big | v \in \mathcal { V } \Big \} ; \phi \right) \Big \} } \\ & { \qquad = G H N \left( \mathcal { A } ; \phi , \varphi \right) . } \end{array}
+$$
+
+# 4.3 ARCHITECTURAL MOTIFS AND STACKED GNNS
+
+The computation graph of some popular CNN architectures often spans over hundreds of nodes (He et al., 2016a; Huang et al., 2017), which makes the search problem scale poorly. Repeated architecture motifs are originally exploited in those architectures where the computation of each computation block at different resolutions is the same, e.g. ResNet (He et al., 2016b). Recently, the use of architectural motifs also became popular in the context of neural architecture search, e.g. (Zoph et al., 2018; Pham et al., 2018), where a small graph module with a fewer number of computation nodes is searched, and the final architecture is formed by repeatedly stacking the same module. Zoph et al.
+
+![](images/378dc46434c74ff338ba0122f8c13779193cff9ae2746933e320170561f1e2b3.jpg)  
+Figure 2: Stacked GHN along the depth dimension.
+
+(2018) showed that this leads to stronger performance due to a reduced search space; the module can also be transferred to larger datasets by adopting a different repeating pattern.
+
+Our proposed method scales naturally with the design of repeated modules by stacking the same graph hypernetwork along the depth dimension. Let $\mathcal { A }$ be a graph composed of a chain of repeated modules embeddi $\textstyle { \bar { \{ A _ { i } } \} } _ { i = 1 } ^ { N }$ . A graph level embedding  a full propagation of the cur $h _ { \mathcal { A } _ { i } }$ is computed by taking an average over all nodemodule, and passed onwards to the input node of the next module as a message before graph propagation continues to the next module.
+
+$$
+\begin{array} { r l } & { \pmb { h } _ { \mathcal { A } _ { 0 } } = 0 , } \\ & { \pmb { h } _ { \mathcal { A } _ { i } } = \frac { 1 } { | \mathscr { V } _ { i } | } \displaystyle \sum _ { v \in \mathscr { V } _ { i } } \left\{ \pmb { h } _ { v } ^ { ( T ) } | v \in \mathscr { V } _ { i } \right\} } \\ & { \quad = \frac { 1 } { | \mathscr { V } _ { i } | } \displaystyle \sum _ { \sigma } G _ { \mathcal { A } _ { i } } ^ { ( T ) } \left( \left\{ \pmb { h } _ { v } ^ { ( 0 ) } | v \in \mathscr { V } _ { i } \right\} , \pmb { h } _ { \mathcal { A } _ { i - 1 } } ; \phi \right) \forall i > 0 } \end{array}
+$$
+
+Note that $G _ { \mathcal { A } _ { i } }$ share parameters for all $\mathbf { \mathcal { A } } _ { i }$ . Please see Figure 2 for an overview.
+
+Table 1: Comparison with image classifiers found by state-of-the-art NAS methods which employ a random search on CIFAR-10. Results shown are mean $\pm$ standard deviation.   
+
+<table><tr><td>Method</td><td>Search Cost (GPU days)</td><td>Param ×106</td><td>Accuracy</td></tr><tr><td>SMASHv1 (Brock et al.,2018)</td><td>？</td><td>4.6</td><td>94.5</td></tr><tr><td>SMASHv2 (Brock et al.,2018)</td><td>3</td><td>16.0</td><td>96.0</td></tr><tr><td>One-Shot Top (F=32) (Bender et al.,2018)</td><td>4</td><td>2.7 ±0.3</td><td>95.5 ± 0.1</td></tr><tr><td>One-Shot Top (F=64) (Bender et al.,2018)</td><td>4</td><td>10.4 ± 1.0</td><td>95.9 ± 0.2</td></tr><tr><td>Random (F=32)</td><td>-</td><td>4.6± 0.6</td><td>94.6± 0.3</td></tr><tr><td>GHN Top (F=32)</td><td>0.42</td><td>5.1±0.6</td><td>95.7 ± 0.1</td></tr></table>
+
+# 4.4 FORWARD-BACKWARD GNN MESSAGE PASSING
+
+Standard GNNs employ the synchronous propagation scheme (Li et al., 2016), where the node embeddings of all nodes are updated simultaneously at every step (see Equation 3). Recently, Liao et al. (2018) found that such propagation scheme is inefficient in passing long-range messages and suffers from the vanishing gradient problem as do regular RNNs. To mitigate these shortcomings they proposed asynchronous propagation using graph partitions. In our application domain, deep neural architectures are chain-like graphs with a long diameter; This can make synchronous message passing difficult. Inspired by the backpropagation algorithm, we propose another variant of asynchronous propagation scheme, which we called forward-backward propagation, that directly mimics the order of node execution in a backpropagation algorithm. Specifically, let $s$ be a topological sort of the nodes in the computation graph in a forward pass,
+
+$$
+\begin{array} { r } { h _ { v } ^ { ( t + 1 ) } = \left\{ \begin{array} { l l } { U \left( h _ { v } ^ { ( t ) } , \pmb { m } _ { v } ^ { ( t ) } \right) \ } & { \mathrm { ~ i f ~ } s ( t ) = v \mathrm { ~ a n d ~ } 1 \leq t \leq | \mathcal { V } | } \\ & { \mathrm { ~ o r ~ i f ~ } s ( 2 | \mathcal { V } | - t ) = v \mathrm { ~ a n d ~ } | \mathcal { V } | + 1 \leq t < 2 | \mathcal { V } | , } \\ { h _ { v } ^ { ( t ) } \ } & { \mathrm { ~ o t h e r w i s e . } } \end{array} \right. } \end{array}
+$$
+
+The total number of propagation steps $T$ for a full forward-backward pass will then become $2 | \mathcal { V } | - 1$ Under the synchronous scheme, propagating information across a graph with diameter $| \nu |$ would require $O ( | \mathcal { V } | ^ { 2 } )$ messages. This is reduced to $O ( | \nu | )$ under the forward-backward scheme.
+
+# 4.5 LEARNING
+
+Learning a graph hypernetwork is straightforward since $\tilde { w }$ are directly generated by a differentiable network. We compute gradients of the graph hypernetwork parameters $\phi , \varphi$ using the chain rule:
+
+$$
+\nabla _ { \phi , \varphi } \mathcal { L } _ { t r a i n } ( \tilde { \boldsymbol { w } } ) = \nabla _ { \tilde { \boldsymbol { w } } } \mathcal { L } _ { t r a i n } ( \tilde { \boldsymbol { w } } ) \cdot \nabla _ { \phi , \varphi } \tilde { \boldsymbol { w } }
+$$
+
+The first term is the gradients of standard network parameters, the second term is decomposed as
+
+$$
+\begin{array} { r l } & { \nabla _ { \phi } \tilde { w } = \left\{ \nabla _ { h } H ( h ; \varphi ) \cdot \nabla _ { \phi } h \big | h \in G ^ { ( T ) } \left( \{ h _ { v } ^ { ( 0 ) } \} , \mathcal { A } , \phi \right) \right\} , } \\ & { \nabla _ { \varphi } \tilde { w } = \left\{ \nabla _ { \varphi } H ( h _ { v } ^ { ( T ) } ; \varphi ) \big | v \in \mathcal { V } \right\} } \end{array}
+$$
+
+where (Eq. 17) is the contribution from GNN module $G$ and (Eq. 18) is the contribution from the hypernet module $H$ . Both $G$ and $H$ are jointly learned throughout training.
+
+# 5 EXPERIMENTS
+
+In this section, we use our proposed GHN to search for the best CNN architecture for image classification. First, we evaluate the GHN on the standard CIFAR (Krizhevsky & Hinton, 2009) and ImageNet (Russakovsky et al., 2015) architecture search benchmarks. Next, we apply GHN on an “anytime prediction” task where we optimize the speed-accuracy tradeoff that is key for many real-time applications. Finally, we benchmark the GHN’s predicted-performance correlation and explore various factors in an ablation study.
+
+# 5.1 NAS BENCHMARKS
+
+# 5.1.1 CIFAR-10
+
+We conduct our initial set of experiments on CIFAR-10 (Krizhevsky & Hinton, 2009), which contains 10 object classes and 50,000 training images and 10,000 test images of size $3 2 \times 3 2 \times 3$ . We use 5,000 images split from the training set as our validation set.
+
+Table 2: Comparison with image classifiers found by state-of-the-art NAS methods which employ advanced search methods on CIFAR-10. Results shown are mean $\pm$ standard deviation.   
+
+<table><tr><td>Method</td><td>Search Cost (GPU days)</td><td>Param ×106</td><td>Accuracy</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>1800</td><td>3.3</td><td>97.35</td></tr><tr><td>ENAS Cell search (Pham et al.,2018)</td><td>0.45</td><td>4.6</td><td>97.11</td></tr><tr><td>DARTS (first order) (Liu et al.,2018c)</td><td>1.5</td><td>2.9</td><td>97.06</td></tr><tr><td>DARTS (second order) (Liu et al.,2018c)</td><td>4</td><td>3.4</td><td>97.17 ± 0.06</td></tr><tr><td>GHN Top-Best,1K (F=32)</td><td>0.84</td><td>5.7</td><td>97.16± 0.07</td></tr></table>
+
+Table 3: Comparison with image classifiers found by state-of-the-art NAS methods which employ advanced search methods on ImageNet-Mobile.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Search Cost (GPU days)</td><td rowspan="2">Param ×106</td><td rowspan="2">FLOPs ×106</td><td colspan="2">Accuracy</td></tr><tr><td>Top1</td><td>Top 5</td></tr><tr><td>NASNet-A (Zoph et al., 2018)</td><td>1800</td><td>5.3</td><td>564</td><td>74.0</td><td>91.6</td></tr><tr><td>NASNet-C (Zoph etal., 2018)</td><td>1800</td><td>4.9</td><td>558</td><td>72.5</td><td>91.0</td></tr><tr><td>AmoebaNet-A (Real et al.,2018)</td><td>3150</td><td>5.1</td><td>555</td><td>74.5</td><td>92.0</td></tr><tr><td>AmoebaNet-C (Real et al., 2018)</td><td>3150</td><td>6.4</td><td>570</td><td>75.7</td><td>92.4</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>225</td><td>5.1</td><td>588</td><td>74.2</td><td>91.9</td></tr><tr><td>DARTS (second order) (Liu et al.,2018c)</td><td>4</td><td>4.9</td><td>595</td><td>73.1</td><td>91.0</td></tr><tr><td>GHN Top-Best, 1K</td><td>0.84</td><td>6.1</td><td>569</td><td>73.0</td><td>91.3</td></tr></table>
+
+Search space: Following existing NAS methods, we choose to search for optimal blocks rather than the entire network. Each block contains 17 nodes, with 8 possible operations. The final architecture is formed by stacking 18 blocks. The spatial size is halved and the number of channels is doubled after blocks 6 and 12. These settings are all chosen following recent NAS methods (Zoph & Le, 2017; Pham et al., 2018; Liu et al., 2018c), with details in the Appendix.
+
+Training: For the GNN module, we use a standard GRU cell (Cho et al., 2014) with hidden size 32 and 2 layer MLP with hidden size 32 as the recurrent cell function $U$ and message function $M$ respectively. The shared hypernetwork $H \left( \cdot ; \varphi \right)$ is a 2-layer MLP with hidden size 64. From the results of ablations studies in Section 5.4, the GHN is trained with blocks with $N = 7$ nodes and $T = 5$ propagations under the forward-backward scheme, using the ADAM optimizer (Kingma $\&$ Ba, 2015). Training details of the final selected architectures are chosen to follow existing works and can be found in the Appendix.
+
+Evaluation: First, we compare to similar methods that use random search with a hypernetwork or a one-shot model as a surrogate search signal. We randomly sample 10 architectures and train until convergence for our random baseline. Next, we randomly sample 1000 architectures, and select the top 10 performing architectures with GHN generated weights, which we refer to as GHN Top. Our reported search cost includes both the GHN training and evaluation phase. Shown in Table 1, the GHN achieves competitive results with nearly an order of magnitude reduction in search cost.
+
+In Table 2, we compare with methods which use more advanced search methods, such as reinforcement learning and evolution. Once again, we sample 1000 architectures and use the GHN to select the top 10. To make a fair comparison for random search, we train the top 10 for a short period before selecting the best to train until convergence. The accuracy reported for GHN Top-Best is the average of 5 runs of the same final architecture. Note that all methods in Table 2 use CutOut (Devries & Taylor, 2017). GHN achieves very competitive results with a simple random search algorithm, while only using a fraction of the total search cost. Using advanced search methods with GHNs may bring further gains.
+
+# 5.1.2 IMAGENET-MOBILE
+
+We also run our GHN algorithm on the ImageNet dataset (Russakovsky et al., 2015), which contains 1.28 million training images. We report the top-1 accuracy on the 50,000 validation images. Following existing literature, we conduct the ImageNet experiments in the mobile setting, where the model is constrained to be under 600M FLOPS. We directly transfer the best architecture block found in the CIFAR-10 experiments, using an initial convolution layer of stride 2 before stacking 14 blocks with scale reduction at blocks 1, 2, 6 and 10. The total number of flops is constrained by choosing the initial number of channels. We follow existing NAS methods on the training procedure of the final architecture; details can be found in the Appendix. As shown in Table 3 the transferred block is competitive with other NAS methods which require a far greater search cost.
+
+![](images/78fcd6d0d3038bc13cc432a354ae7b4bb6db04e1b236ac49f50a1cea720ecd3b.jpg)  
+Figure 3: Comparison with state-of-the-art human-designed networks on CIFAR-10.
+
+![](images/bd12ef32546905a588ef2bda3f48a1e44e0de4ed5a56497ac8d1eb7be1c1f97f.jpg)  
+Figure 4: Comparison between random 10 and top 10 networks on CIFAR-10.
+
+Table 4: Benchmarking the correlation between the predicted and true performance of the GHN against SGD and a one-shot model baselines. Results are on CIFAR-10.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Computation cost</td><td colspan="2">Correlation</td></tr><tr><td>Initial (GPU hours)</td><td>Per arch.(GPU seconds)</td><td>Random-100</td><td>Top-50</td></tr><tr><td>SGD 10 Steps</td><td>-</td><td>0.9</td><td>0.26</td><td>-0.05</td></tr><tr><td>SGD 100 Steps</td><td>=</td><td>9</td><td>0.59</td><td>0.06</td></tr><tr><td>SGD 200 Steps</td><td>=</td><td>18</td><td>0.62</td><td>0.20</td></tr><tr><td>SGD 1000 Steps</td><td></td><td>90</td><td>0.77</td><td>0.26</td></tr><tr><td>One-Shot</td><td>9.8</td><td>0.06</td><td>0.58</td><td>0.31</td></tr><tr><td>GHN</td><td>6.1</td><td>0.08</td><td>0.68</td><td>0.48</td></tr></table>
+
+# 5.2 ANYTIME PREDICTION
+
+In the real-time setting, the computational budget available can vary for each test case and cannot be known ahead of time. This is formalized in anytime prediction, (Grubb & Bagnell, 2012) the setting in which for each test example $\mathbf { x }$ , there is non-deterministic computational budget $B$ drawn from the joint distribution $P ( \bar { \bf x } , B )$ . The goal is then to minimize the expected loss $L ( f ) = \mathbb { E } \left[ L \left( f ( \mathbf { x } ) , \mathbf { \bar { \theta } } \right) \right] _ { P ( \mathbf { x } , B ) }$ , where $\dot { f } ( \cdot )$ is the model and $L ( \cdot )$ is the loss for an $f ( \cdot )$ that must produce a prediction within the budget $B$ .
+
+We conduct experiments on CIFAR-10. Our anytime search space consists of networks with 3 cells containing 24, 16, and 8 nodes. Each node is given the additional properties: 1) the spatial size it operates at and 2) if an early-exit classifier is attached to it. A node enforces its spatial size by pooling or upsampling any input feature maps inputs that are of different scale. Note that while a naive one-shot model would triple its size to include three different parameter sets at three different scales, the GHN is negligibly affected by such a change. The GHN uses the area under the predicted accuracy-FLOPS curve as its selection criteria. The search space, contains various convolution and pooling operators. Training methodology of the final architectures are chosen to match Huang et al. (2018) and can be found in the Appendix.
+
+Figure 3 shows a comparison with the various methods presented by Huang et al. (2018). Our experiments show that the best searched architectures can outperform the current state-of-the-art human designed networks. We see the GHN is amenable to the changes proposed above, and can find efficient architectures with a random search when used with a strong search space.
+
+# 5.3 PREDICTED PERFORMANCE CORRELATION (CIFAR-10)
+
+In this section, we evaluate whether the parameters generated from GHN can be indicative of the final performance. Our metric is the correlation between the accuracy of a model with trained weights vs. GHN generated weights. We use a fixed set of 100 random architectures that have not been seen by the GHN during training, and we train them for 50 epochs to obtain our “ground-truth” accuracy, and finally compare with the accuracy obtained from GHN generated weights. We report the Pearson’s R score on all 100 random architectures and the top 50 performing architectures (i.e. above average architectures). Since we are interested in searching for the best architecture, obtaining a higher correlation on top performing architectures is more meaningful.
+
+To evaluate the effectiveness of GHN, we further consider two baselines: 1) training a network with SGD from scratch for a varying number of steps, and 2) our own implementation of the oneshot model proposed by Pham et al. (2018), where nodes store a set of shared parameters for each possible operation. Unlike GHN, which is compatible with varying number of nodes, the one-shot model must be trained with $N = 1 7$ nodes to match the evaluation. The GHN is trained with $N = 7$ , $T = 5$ using forward-backward propagation. These GHN parameters are selected based on the results found in Section 5.4.
+
+![](images/c442ba59a9b67d0fb2d019ca2b4e202c6143ceb067eaab349e5eb4b2f2b43be6.jpg)  
+Figure 5: GHN when varying the number of nodes and propagation scheme
+
+Table 4 shows performance correlation and search cost of SGD, the one-shot model, and our GHN. Note that GHN clearly outperforms the one-shot model, showing the effectiveness of dynamically predicting parameters based on graph topology. While it takes 1000 SGD steps to surpasses GHN in the “Random- $1 0 0 ^ { \circ }$ setting, GHN is still the strongest in the “Top- $. 5 0 ^ { \circ }$ setting, which is more important for architecture search. Moreover, compared to GHN, running 1000 SGD steps for every random architecture is over 1000 times more computationally expensive. In contrast, GHN only requires a pre-training stage of 6 hours, and afterwards, the trained GHN can be used to efficiently evaluate a massive number of random architectures of different sizes.
+
+# 5.4 ABLATION STUDIES (CIFAR-10)
+
+Number of graph nodes: The GHN is compatible with varying number of nodes - graphs used in training need not be the same size as the graphs used for evaluation. Figure 5a shows how GHN performance varies as a function of the number of nodes employed during training - fewer nodes generally produces better performance. While the GHN has difficulty learning on larger graphs, likely due to the vanishing gradient problem, it can generalize well from just learning on smaller graphs. Note that all GHNs are tested with the full graph size $N = 1 7$ nodes).
+
+Number of propagation steps: We now compare the forward-backward propagation scheme with the regular synchronous propagation scheme. Note that $T = 1$ synchronous step corresponds to one full forward-backward phase. As shown in Figure 5b, the forward-backward scheme consistently outperforms the synchronous scheme. More propagation steps also help improving the performance, with a diminishing return. While the forward-backward scheme is less amenable to acceleration from parallelization due to its sequential nature, it is possible to parallelize the evaluation phase across multiple GHNs when testing the fitness of candidate architectures.
+
+Stacked GHN for architectural motifs: We also evaluate different design choices of GHNs on representing architectural motifs. We compare 1) individual GHNs, each predicting one block independently, 2) a stacked GHN where individual GHN’s pass on their graph embedding without sharing parameters, 3) a stacked GHN with shared parameters (our proposed approach). As shown in Table 5, passing messages between GHN’s is crucial, and sharing parameters produces better performance.
+
+![](images/86911b051eb44ada06562af49bc7e412933719155cd2388801c1af90cd25a5ab.jpg)  
+Table 5: Stacked GHN Correlation. SP denotes sharing parameters and PE denotes passing embeddings
+
+# 6 CONCLUSION
+
+In this work, we propose the Graph HyperNetwork (GHN), a composition of graph neural networks and hypernetworks that generates the weights of any architecture by operating directly on their computation graph representation. We demonstrate a strong correlation between the performance with the generated weights and the fully-trained weights. Using our GHN to form a surrogate search signal, we achieve competitive results on CIFAR-10 and ImageNet mobile with nearly $\bar { 1 } 0 \times$ faster speed compared to other random search methods. Furthermore, we show that our proposed method can be extended to outperform the best human-designed architectures in setting of anytime prediction, greatly reducing the computation cost of real-time neural networks.
+
+# REFERENCES
+
+Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. Designing neural network architectures using reinforcement learning. In Proceedings of the 5th International Conference on Learning Representations (ICLR), 2017a.
+
+Bowen Baker, Otkrist Gupta, Ramesh Raskar, and Nikhil Naik. Accelerating neural architecture search using performance prediction. In NIPS Workshop on Meta-Learning, 2017b.
+
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+Jurgen Schmidhuber. A ‘self-referential’weight matrix. In ¨ International Conference on Artificial Neural Networks (ICANN). Springer, 1993.
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+Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. In Proceedings of the 5th International Conference on Learning Representations (ICLR), 2017.   
+Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures for scalable image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.
+
+# 7 APPENDIX
+
+# 7.1 SEARCH SPACE
+
+Standard image classification on CIFAR-10 and ImageNet The search space for CIFAR-10 and ImageNet classification experiments includes the following operations:
+
+• identity   
+• $1 \times 1$ convolution   
+• $3 \times 3$ separable convolution   
+• $5 \times 5$ separable convolution   
+• $3 \times 3$ dilated separable convolution   
+• $5 \times 5$ dilated separable convolution   
+• $1 \times 7$ convolution followed $7 \times 1$ convolution   
+• $3 \times 3$ max pooling   
+• $3 \times 3$ average pooling
+
+A block forms an output by concatenating all leaf nodes in the graph. Blocks have 2 input nodes which ingest the output of block $i - 1$ and block $i - 2$ respectively. The input nodes are bottleneck layers, and can reduce the spatial size by using stride 2.
+
+Note that while ENAS supports only 5 operators due to memory constraints, GHNs can search for more operators. This is because ENAS (and other methods which use one-shot models) must store all the parameters in memory because it finds paths in a larger model. Thus the memory requirements are ${ \cal O } ( K N )$ where $K$ is the number of operations and $\bar { N }$ is the number of nodes in the candidate architecture. In contrast, the memory requirement for GHNs is $O ( N ) + O ( K )$ for the candidate architecture and GHN respectively.
+
+Anytime prediction on CIFAR-10 The search space for the CIFAR-10 anytime prediction experiments includes the following operations:
+
+• $1 \times 1$ convolution • $3 \times 3$ convolution • $5 \times 5$ convolution • $3 \times 3$ max pooling • $3 \times 3$ average pooling
+
+In the anytime setting, nodes concatenate their inputs rather than sum. Thus, the identity operator was removed as it would be redundant. The search space does not include separable convolutions so that it is comparable with our baselines (Huang et al., 2018). Block 1 contains nodes which may operate on any of the 3 scales $( 3 2 \times 3 2 , 1 6 \times 1 \bar { 6 } , 8 \times 8 )$ . Block 2 contains nodes which can only operate on scales $1 6 \times 1 6$ and $8 \times 8$ . Block 3 only contains nodes which operate on the scale $8 \times 8$ . We fix the number of exit nodes. These choices are inspired by Huang et al. (2018)
+
+# 7.2 GRAPH HYPERNETWORK DETAILS
+
+Standard image classification on CIFAR-10 and ImageNet While node embeddings are initialized to a one-hot vector representing computational operator of the node, we found it helpful to pass the sparse vector through a learned embedding matrix prior to graph propagation. The GHN is trained for 200 epochs with batch size 64 using the ADAM optimizer with an initial learning rate 1e-3 that is divided by 2 at epoch 100 and 150. A naive hypernet would have a separate output branch for each possible node type, and simply ignore branches that aren’t applicable to the specific node. In this manner, the number of parameters of the hypernetwork scale according to the number of possible node computations. In contrast, the number of parameters for a one-shot model scale according to the number of nodes in the graph. We further reduce number of parameters by obtaining smaller sized convolutions kernels through the slicing of larger sized kernels.
+
+Anytime prediction In the anytime prediction setting, two one-hot vectors representing the node’s scale and presence of an early exit classifier are additionally concatenated to the first initialized node embedding. We found it helpful to train the GHN with a random number of nodes per block, with maximum number of allowed nodes being the evaluation block size. Because nodes concatenate their inputs, a bottleneck layer is required. The hypernetwork can predict bottleneck parameters for a varying number of input nodes by generating weights based on edge activations rather than node activations. We form edge activations by concatenating the node activations of the parent and child. Edge weights generated this way can be concatenated, allowing the dimensionality of the predicted bottleneck weights the be proportional to the number of incoming edges.
+
+# 7.3 FINAL ARCHITECTURE TRAINING DETAILS
+
+CIFAR-10 Following existing NAS methods (Zoph et al., 2018; Real et al., 2018), the final candidates are trained for 600 epochs using SGD with momentum 0.9, a single period cosine schedule with $l _ { m a x } = 0 . 0 2 5$ , and batch size 64. For regularization, we use scheduled drop-path with a final dropout probability of 0.4. We use an auxiliary head located at 2/3 of the network weighted by 0.5. We accelerate training by performing distributed training across 32 GPUs; the learning rate is multiplied by 32 with an initial linear warmup of 5 epochs.
+
+ImageNet Mobile For ImageNet mobile experiments, we use an image size of $2 2 4 \times 2 2 4$ . Following existing NAS methods (Zoph et al., 2018; Real et al., 2018), the final candidates are trained for 250 epochs using SGD with momentum 0.9, initial learning rate 0.1 multiplied by 0.97 every epoch. We use an auxiliary head located at 2/3 of the network weighted by 0.5. We use the same regularization techniques, and similarly accelerate training in a distributed fashion.
+
+Anytime Following Huang et al. (2018), the final candidates are trained using SGD with momentum 0.9. We train the models for 300 epochs use an initial learning rate of 0.1, which is divided by 10 after 150 and 225 epochs using a batch size of 64. We accelerate training with distributed training in a similar fashion as the CIFAR-10 classification and ImageNet mobile experiments. The number of filters for the final architecture is chosen such that the number of FLOPS is comparable to existing baselines.
+
+# 7.4 INVESTIGATING ACCURACY DROP OFF
+
+Figure 6 shows a plot comparing the accuracy of an architecture that is trained for 50 epochs and the accuracy of the same architecture using GHN generated weights.
+
+![](images/435dcfb054894d6e6b35f95f2f81768ec54bceb0941b0d207a1d2b08007782df.jpg)  
+Figure 6: Comparison for 100 randomly sampled architectures.
+
+7.5 VISUALIZATION OF FINAL ARCHITECTURES
+
+7.5.1 CIFAR-10 AND IMAGENET CLASSIFICATION
+
+Figure 7 shows the best found block in the CIFAR-10 Experiments.
+
+![](images/6e53da1af3ef571884aeb6868911e9f36757f2a006d81ee6cd231f10f3a837b8.jpg)  
+Figure 7: Best block found for classification
+
+# 7.5.2 ANYTIME PREDICTION
+
+Figures 8, 9 and 10 show blocks 1 2 and 3 of the best architecture found in the anytime experiments.   
+The color red denotes that an early exit is attached to the output of the node.
+
+![](images/73c5c098227fa57a76f03bdc53d9a5640b6a37364f0f65d5ef23d5d5fa4ac452.jpg)  
+Figure 8: Block 1 for anytime network. Red color denotes early exit.
+
+![](images/b4ea6ef23a76b8c3479f473823c4ef089219d805c71f114dc19ce269c6da4292.jpg)  
+Figure 9: Block 2 for anytime network. Red color denotes early exit.
+
+![](images/470a3b4b93cfc85d75d378477a15e47c60b2d1b6ac057277a04dbb55a90961fe.jpg)  
+Figure 10: Block 3 for anytime network. Red color denotes early exit.
\ No newline at end of file
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new file mode 100644
index 0000000000000000000000000000000000000000..b6722477495286fd9e80a0ff8aa968080d20ec71
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@@ -0,0 +1,1898 @@
+[
+    {
+        "type": "text",
+        "text": "GRAPH HYPERNETWORKS FOR NEURAL ARCHITECTURE SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
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+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chris Zhang1,2, Mengye $\\mathbf { R e n } ^ { 1 , 3 }$ & Raquel Urtasun1,3 1Uber Advanced Technologies Group, 2University of Waterloo, 3University of Toronto cjzhang@edu.uwaterloo.ca, {mren3,urtasun}@uber.com ",
+        "bbox": [
+            183,
+            167,
+            754,
+            210
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            247,
+            544,
+            262
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural architecture search (NAS) automatically finds the best task-specific neural network topology, outperforming many manual architecture designs. However, it can be prohibitively expensive as the search requires training thousands of different networks, while each can last for hours. In this work, we propose the Graph HyperNetwork (GHN) to amortize the search cost: given an architecture, it directly generates the weights by running inference on a graph neural network. GHNs model the topology of an architecture and therefore can predict network performance more accurately than regular hypernetworks and premature early stopping. To perform NAS, we randomly sample architectures and use the validation accuracy of networks with GHN generated weights as the surrogate search signal. GHNs are fast – they can search nearly $1 0 \\times$ faster than other random search methods on CIFAR-10 and ImageNet. GHNs can be further extended to the anytime prediction setting, where they have found networks with better speed-accuracy tradeoff than the state-of-the-art manual designs. ",
+        "bbox": [
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+            276,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The success of deep learning marks the transition from manual feature engineering to automated feature learning. However, designing effective neural network architectures requires expert domain knowledge and repetitive trial and error. Recently, there has been a surge of interest in neural architecture search (NAS), where neural network architectures are automatically optimized. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One approach for architecture search is to consider it as a nested optimization problem, where the inner loop finds the optimal parameters $w ^ { * }$ for a given architecture $a$ w.r.t. the training loss $\\mathcal { L } _ { t r a i n }$ , and the outer loop searches the optimal architecture w.r.t. a validation loss $\\mathcal { L } _ { v a l }$ : ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fc4398d712a82a69d9900e7cbbf958b3f5de3eff550d0d95075744c2574c4ee4.jpg",
+        "text": "$$\nw ^ { * } ( a ) = \\arg \\operatorname* { m i n } _ { w } \\mathcal { L } _ { t r a i n } ( w , a )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "equation",
+        "img_path": "images/6f5e4515569fb5b4439b1a2fca1843290c09783d45125b0936b013cb8b0f4efb.jpg",
+        "text": "$$\n\\boldsymbol { a } ^ { * } = \\arg \\operatorname* { m i n } _ { \\boldsymbol { a } } \\mathcal { L } _ { v a l } ( \\boldsymbol { w } ^ { * } ( \\boldsymbol { a } ) , \\boldsymbol { a } )\n$$",
+        "text_format": "latex",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Traditional NAS is expensive since solving the inner optimization in Eq. 1 requires a lengthy optimization process (e.g. stochastic gradient descent (SGD)). Instead, we propose to learn a parametric function approximation referred to as a hypernetwork (Ha et al., 2017; Brock et al., 2018), which attempts to generate the network weights directly. Learning a hypernetwork is an amortization of the cost of solving Eq. 1 repeatedly for multiple architectures. A trained hypernetwork is well correlated with SGD and can act as a much faster substitute. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yet, the architecture of the hypernet itself is still to be determined. Existing methods have explored a variety of tactics to represent architectures, such as an ingenious 3D tensor encoding scheme (Brock et al., 2018), or a string serialization processed by an LSTM (Zoph & Le, 2017; Zoph et al., 2018; Pham et al., 2018). In this work, we advocate for a computation graph representation as it allows for the topology of an architecture to be explicitly modeled. Furthermore, it is intuitive to understand and can be easily extensible to various graph sizes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To this end, in this paper we propose the Graph HyperNetwork (GHN), which can aggregate graph level information by directly learning on the graph representation. Using a hypernetwork to guide architecture search, our approach requires significantly less computation when compared to state-ofthe-art methods. The computation graph representation allows GHNs to be the first hypernetwork to generate all the weights of arbitrary CNNs rather than a subset (e.g. Brock et al. (2018)), achieving stronger correlation and thus making the search more efficient and accurate. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While the validation accuracy is often the primary goal in architecture search, networks must also be resource aware in real-world applications. Towards this goal, we exploit the flexibility of the GHN by extending it to the problem of anytime prediction. Models capable of anytime prediction progressively update their predictions, allowing for a prediction at any time. This is desirable in settings such as real-time systems, where the computational budget available for each test case may vary greatly and cannot be known ahead of time. Although anytime models have non-trivial differences to classical models, we show the GHN is amenable to these changes. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We summarize our main contributions of this work: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We propose Graph HyperNetwork that predicts the parameters of unseen neural networks by directly operating on their computational graph representations.   \n2. Our approach achieves highly competitive results with state-of-the-art NAS methods on both CIFAR-10 and ImageNet-mobile and is $1 0 \\times$ faster than other random search methods.   \n3. We demonstrate that our approach can be generalized and applied in the domain of anytimeprediction, previously unexplored by NAS programs, outperforming the existing manually designed state-of-the-art models. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            343,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Various search methods such as reinforcement learning (Zoph & Le, 2017; Baker et al., 2017a; Zoph et al., 2018), evolutionary methods (Real et al., 2017; Miikkulainen et al., 2017; Xie & Yuille, 2017; Liu et al., 2018b; Real et al., 2018) and gradient-based methods (Liu et al., 2018c; Luo et al., 2018) have been proposed to address the outer optimization (Eq. 2) of NAS, where an agent learns to sample architectures that are more likely to achieve higher accuracy. Different from these methods, this paper places its focus on the inner-loop: inferring the parameters of a given network (Eq. 1). Following Brock et al. (2018); Bender et al. (2018), we opt for a simple random search algorithm to complete the outer loop. ",
+        "bbox": [
+            173,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While initial NAS methods simply train candidate architectures for a brief period with SGD to obtain the search signal, recent approaches have proposed alternatives in the interest of computational cost. Baker et al. (2017b) propose directly predicting performance from the learning curve, and Deng et al. (2017) propose to predict performance directly from the architecture without learning curve information. However, training a performance predictor requires a ground truth, thus the expensive process of computing the inner optimization is not avoided. Pham et al. (2018); Bender et al. (2018); Liu et al. (2018c) use parameter sharing, where a “one-shot” model containing all possible architectures in the search space is trained. Individual architectures are sampled by deactivating some nodes or edges in the one-shot model. In this case, predicting $w ^ { * } ( a )$ can be seen as using a selection function from the set of parameters in the one-shot model. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Prior work has shown the feasibility of predicting parameters in a network with a function approximator (Denil et al., 2013). Schmidhuber (1992; 1993) proposed “fast-weights”, where one network produces weight changes for another. HyperNetworks (Ha et al., 2017) generate the weights of another network and show strong results in large-scale language modeling and image classification experiments. SMASH (Brock et al., 2018) applied HyperNetworks to perform NAS, where an architecture is encoded as a 3D tensor using a memory channel scheme. In contrast, we encode a network as a computation graph and use a graph neural network. While SMASH predicts a subset of the weights, our graph model is able to predict all the free weights. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While earlier NAS methods focused on standard image classification and language modeling, recent literature has extended NAS to search for architectures that are computationally efficient (Tan et al., 2018; Dong et al., 2018; Hsu et al., 2018; Elsken et al., 2018; Zhou et al., 2018). In this work, we applied our GHN based search program on the task of anytime prediction, where we not only optimize for the final speed but the entire speed-accuracy trade-off curve. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We review the two major building blocks of our model: graph neural networks and hypernetworks. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Graph Neural Network: A graph neural network (Scarselli et al., 2009; Li et al., 2016; Kipf & Welling, 2017) is a collection of nodes and edges $( \\nu , \\mathcal { E } )$ , where each node is a recurrent neural network (RNN) that individually sends and receives messages along the edges, spanning over the ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0ef3cc913230ebdf5240c3fb99335e7264d17c45c39a7135f8a534a643296895.jpg",
+        "image_caption": [
+            "Figure 1: Our system diagram. A: A neural network architecture is randomly sampled, forming a GHN. B: After graph propagation, each node in the GHN generates its own weight parameters. C: The GHN is trained to minimize the training loss of the sampled network with the generated weights. Random networks are ranked according to their performance using GHN generated weights. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "horizon of message passing. Each node $v$ stores an internal node embedding vector $\\pmb { h } _ { v } ^ { ( t ) } \\in \\mathbb { R } ^ { D }$ , and is updated recurrently: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46bc8f8de9c02a61a1493d067a7f44394f1db3084233e72b6fe1a050dde87ebb.jpg",
+        "text": "$$\n\\pmb { h } _ { v } ^ { ( t + 1 ) } = \\left\\{ \\begin{array} { l l } { U \\left( \\pmb { h } _ { v } ^ { ( t ) } , \\pmb { m } _ { v } ^ { ( t ) } \\right) \\ } & { \\mathrm { i f ~ n o d e ~ } v \\mathrm { ~ i s ~ a c t i v e } , } \\\\ { \\pmb { h } _ { v } ^ { ( t ) } \\ } & { \\mathrm { o t h e r w i s e } , } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            311,
+            663,
+            357
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $U$ is a recurrent cell function and $\\mathbf { \\Delta } _ { m _ { v } } ^ { ( t ) }$ is the message received by $v$ at time step $t$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8465baa381f508ee99d128322d536239383bd84ea6c1f52dec41bd666d792eb.jpg",
+        "text": "$$\n\\pmb { m } _ { v } ^ { ( t ) } = \\sum _ { u \\in N _ { i n } ( v ) } M \\left( \\pmb { h } _ { u } ^ { ( t ) } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            390,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with $M$ the message function and $N _ { i n } ( v )$ the set of neighbors with incoming edges pointing towards $v$ . $U$ is often modeled with a long short-term memory (LSTM) unit (Hochreiter & Schmidhuber, 1997) or gated recurrent unit (GRU) (Cho et al., 2014), and $M$ with an MLP. Given a graph $\\mathcal { A }$ , we define the GNN operator $G _ { A }$ to be a mapping from a set of initial node embeddings $\\{ \\bar { h _ { v } ^ { ( 0 ) } } \\}$ to a set of different node embeddings $\\{ h _ { v } ^ { ( t ) } \\}$ , parameterized by some learnable parameters $\\phi$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1bb3ad8f3ad6224bedc201620659eca7f8e1cadf86dd5b1322861982cd5841b1.jpg",
+        "text": "$$\n\\left\\{ \\pmb { h } _ { v } ^ { ( t ) } | v \\in \\mathcal { V } \\right\\} = G _ { \\mathcal { A } } ^ { ( t ) } \\left( \\left\\{ \\pmb { h } _ { v } ^ { ( 0 ) } | v \\in \\mathcal { V } \\right\\} ; \\phi \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Throughout propagation the node embeddings $\\boldsymbol { h } _ { v } ^ { ( t ) }$ continuously aggregate graph level information, which can be used for tasks such as node prediction and graph prediction by further aggregation. Similar to RNNs, GNNs are typically learned using backpropagation through time (BPTT) (Werbos, 1990). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hypernetwork: A hypernetwork (Ha et al., 2017) is a neural network that generates the parameters of another network. For a typical deep feedforward network with $D$ layers, the parameters of the $j$ -th layer $W _ { j }$ can be generated by a learned function $H$ : ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77e5c888d5576428a8917a0e29e31f54a706565b92ca2ed898c95f5c19e6b1f3.jpg",
+        "text": "$$\nW _ { j } = H ( z _ { j } ) , \\thinspace \\thinspace \\forall j = 1 , . . . , D ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { j }$ is the layer embedding, and $H$ is shared for all layers. The output dimensionality of the hypernetwork is fixed, but it’s possible to accommodate predicting weights for layers of varying kernel sizes by concatenating multiple kernels of the fixed size. Varying spatial sizes can also be accommodated by slicing in the spatial dimensions. Hypernetworks have been found effective in standard image recognition and text classification problems, and can be viewed as a relaxed weight sharing mechanism. Recently, they have shown to be effective in accelerating architecture search (Brock et al., 2018). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 GRAPH HYPERNETWORKS FOR NEURAL ARCHITECTURAL SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our proposed Graph HyperNetwork (GHN) is a composition of a graph neural network and a hypernetwork. It takes in a computation graph (CG) and generates all free parameters in the graph. During evaluation, the generated parameters are used to evaluate the fitness of a random architecture, and the top performer architecture on a separate validation set is then selected. This allows us to search over a large number of architectures at the cost of training a single GHN. We refer the reader to Figure 1 for a high level system overview. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 GRAPHICAL REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We represent a given architecture as a directed acyclic graph $\\mathcal { A } = ( \\mathcal { V } , \\mathcal { E } )$ , where each node $v \\in \\mathcal V$ has an associated computational operator $f _ { v }$ parametrized by $w _ { v }$ , which produces an output activation tensor $x _ { v }$ . Edges $\\bar { e _ { u \\mapsto v } } = ( u , \\bar { v } ) \\in \\mathcal { E }$ represent the flow of activation tensors from node $u$ to node $v$ . $x _ { v }$ is computed by applying its associated computational operator on each of its inputs and taking summation as follows ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d47aa928b2aefb7e7382af7fc7dc0e6d2c1b6bd27339f21f1e925947b88edca0.jpg",
+        "text": "$$\nx _ { v } = \\sum _ { e _ { u \\mapsto v } \\in \\mathcal { E } } f _ { v } ( x _ { u } ; w _ { v } ) , \\forall v \\in \\mathcal { V } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 GRAPH HYPERNETWORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our proposed Graph Hypernetwork is defined as a composition of a GNN and a hypernetwork. First, given an input architecture, we used the graphical representation discussed above to form a graph $\\mathcal { A }$ . A parallel GNN $G _ { A }$ is then constructed to be homomorphic to $\\mathcal { A }$ with the exact same topology. Node embeddings are initialized to one-hot vectors representing the node’s computational operator. After graph message-passing steps, a hypernet uses the node embeddings to generate each node’s associated parameters. Let ${ h _ { v } ^ { ( T ) } }$ be the embedding of node $v$ after $T$ steps of GNN propagation, and let $H \\left( \\cdot ; \\varphi \\right)$ be a hypernetwork parametrized by $\\varphi$ , the generated parameters $\\tilde { \\pmb { w } } _ { v }$ are: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f4dc936f3835e196af0e5be0b335625f8404b6d99bea8029a567b90faaea8bf.jpg",
+        "text": "$$\n\\tilde { \\pmb { w } } _ { v } = H \\left( \\pmb { h } _ { v } ^ { ( T ) } ; \\varphi \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            367,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For simplicity, we implement $H$ with a multilayer perceptron (MLP). It is important to note that $H$ is shared across all nodes, which can be viewed as an output prediction branch in each node of the GNN. Thus the final set of generated weights of the entire architecture $\\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } ^ { \\tilde { \\mathbf { \\Gamma } } } \\tilde { \\mathbf { \\Gamma } } ^ { \\tilde { \\mathbf { \\Gamma } } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde { \\mathbf { \\Gamma } } \\tilde \\mathrm { \\Gamma }$ is found by applying $H$ on all the nodes and their respective embeddings which are computed by $G _ { A }$ : ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/901b1f6f8517feb937be6a79e14971c6c5d74117c769bacf0d1c37434b43f3c5.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\tilde { w } = \\left\\{ \\tilde { w } _ { v } | v \\in \\mathcal { V } \\right\\} = \\left\\{ H \\left( h _ { v } ^ { ( T ) } ; \\varphi \\right) \\big | \\ v \\in \\mathcal { V } \\right\\} } \\\\ & { \\qquad = \\Big \\{ H \\left( h ; \\varphi \\right) \\big | \\ h \\in G _ { \\mathcal { A } } ^ { ( T ) } \\left( \\Big \\{ h _ { v } ^ { ( 0 ) } \\big | v \\in \\mathcal { V } \\Big \\} ; \\phi \\right) \\Big \\} } \\\\ & { \\qquad = G H N \\left( \\mathcal { A } ; \\phi , \\varphi \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 ARCHITECTURAL MOTIFS AND STACKED GNNS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            540,
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+            555
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The computation graph of some popular CNN architectures often spans over hundreds of nodes (He et al., 2016a; Huang et al., 2017), which makes the search problem scale poorly. Repeated architecture motifs are originally exploited in those architectures where the computation of each computation block at different resolutions is the same, e.g. ResNet (He et al., 2016b). Recently, the use of architectural motifs also became popular in the context of neural architecture search, e.g. (Zoph et al., 2018; Pham et al., 2018), where a small graph module with a fewer number of computation nodes is searched, and the final architecture is formed by repeatedly stacking the same module. Zoph et al. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/378dc46434c74ff338ba0122f8c13779193cff9ae2746933e320170561f1e2b3.jpg",
+        "image_caption": [
+            "Figure 2: Stacked GHN along the depth dimension. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            606,
+            568,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(2018) showed that this leads to stronger performance due to a reduced search space; the module can also be transferred to larger datasets by adopting a different repeating pattern. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our proposed method scales naturally with the design of repeated modules by stacking the same graph hypernetwork along the depth dimension. Let $\\mathcal { A }$ be a graph composed of a chain of repeated modules embeddi $\\textstyle { \\bar { \\{ A _ { i } } \\} } _ { i = 1 } ^ { N }$ . A graph level embedding  a full propagation of the cur $h _ { \\mathcal { A } _ { i } }$ is computed by taking an average over all nodemodule, and passed onwards to the input node of the next module as a message before graph propagation continues to the next module. ",
+        "bbox": [
+            174,
+            744,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6928b0440c112f6ddef6efbccd77d3be544173a9035bc4c01618d93d5c75009d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\pmb { h } _ { \\mathcal { A } _ { 0 } } = 0 , } \\\\ & { \\pmb { h } _ { \\mathcal { A } _ { i } } = \\frac { 1 } { | \\mathscr { V } _ { i } | } \\displaystyle \\sum _ { v \\in \\mathscr { V } _ { i } } \\left\\{ \\pmb { h } _ { v } ^ { ( T ) } | v \\in \\mathscr { V } _ { i } \\right\\} } \\\\ & { \\quad = \\frac { 1 } { | \\mathscr { V } _ { i } | } \\displaystyle \\sum _ { \\sigma } G _ { \\mathcal { A } _ { i } } ^ { ( T ) } \\left( \\left\\{ \\pmb { h } _ { v } ^ { ( 0 ) } | v \\in \\mathscr { V } _ { i } \\right\\} , \\pmb { h } _ { \\mathcal { A } _ { i - 1 } } ; \\phi \\right) \\forall i > 0 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            815,
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+            907
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that $G _ { \\mathcal { A } _ { i } }$ share parameters for all $\\mathbf { \\mathcal { A } } _ { i }$ . Please see Figure 2 for an overview. ",
+        "bbox": [
+            168,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/8edaae587bfb52a5fef695ae6333e5ae19ad497adb95756d72faf4923d747164.jpg",
+        "table_caption": [
+            "Table 1: Comparison with image classifiers found by state-of-the-art NAS methods which employ a random search on CIFAR-10. Results shown are mean $\\pm$ standard deviation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Search Cost (GPU days)</td><td>Param ×106</td><td>Accuracy</td></tr><tr><td>SMASHv1 (Brock et al.,2018)</td><td>？</td><td>4.6</td><td>94.5</td></tr><tr><td>SMASHv2 (Brock et al.,2018)</td><td>3</td><td>16.0</td><td>96.0</td></tr><tr><td>One-Shot Top (F=32) (Bender et al.,2018)</td><td>4</td><td>2.7 ±0.3</td><td>95.5 ± 0.1</td></tr><tr><td>One-Shot Top (F=64) (Bender et al.,2018)</td><td>4</td><td>10.4 ± 1.0</td><td>95.9 ± 0.2</td></tr><tr><td>Random (F=32)</td><td>-</td><td>4.6± 0.6</td><td>94.6± 0.3</td></tr><tr><td>GHN Top (F=32)</td><td>0.42</td><td>5.1±0.6</td><td>95.7 ± 0.1</td></tr></table>",
+        "bbox": [
+            189,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.4 FORWARD-BACKWARD GNN MESSAGE PASSING ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Standard GNNs employ the synchronous propagation scheme (Li et al., 2016), where the node embeddings of all nodes are updated simultaneously at every step (see Equation 3). Recently, Liao et al. (2018) found that such propagation scheme is inefficient in passing long-range messages and suffers from the vanishing gradient problem as do regular RNNs. To mitigate these shortcomings they proposed asynchronous propagation using graph partitions. In our application domain, deep neural architectures are chain-like graphs with a long diameter; This can make synchronous message passing difficult. Inspired by the backpropagation algorithm, we propose another variant of asynchronous propagation scheme, which we called forward-backward propagation, that directly mimics the order of node execution in a backpropagation algorithm. Specifically, let $s$ be a topological sort of the nodes in the computation graph in a forward pass, ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eee1caf2241fc741a1952a2d4da2b50e2866197a9c02d7fb5b9e2bb4deaf2357.jpg",
+        "text": "$$\n\\begin{array} { r } { h _ { v } ^ { ( t + 1 ) } = \\left\\{ \\begin{array} { l l } { U \\left( h _ { v } ^ { ( t ) } , \\pmb { m } _ { v } ^ { ( t ) } \\right) \\ } & { \\mathrm { ~ i f ~ } s ( t ) = v \\mathrm { ~ a n d ~ } 1 \\leq t \\leq | \\mathcal { V } | } \\\\ & { \\mathrm { ~ o r ~ i f ~ } s ( 2 | \\mathcal { V } | - t ) = v \\mathrm { ~ a n d ~ } | \\mathcal { V } | + 1 \\leq t < 2 | \\mathcal { V } | , } \\\\ { h _ { v } ^ { ( t ) } \\ } & { \\mathrm { ~ o t h e r w i s e . } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            400,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The total number of propagation steps $T$ for a full forward-backward pass will then become $2 | \\mathcal { V } | - 1$ Under the synchronous scheme, propagating information across a graph with diameter $| \\nu |$ would require $O ( | \\mathcal { V } | ^ { 2 } )$ messages. This is reduced to $O ( | \\nu | )$ under the forward-backward scheme. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.5 LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Learning a graph hypernetwork is straightforward since $\\tilde { w }$ are directly generated by a differentiable network. We compute gradients of the graph hypernetwork parameters $\\phi , \\varphi$ using the chain rule: ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f64134729617c751d13e606ced28b3dda00cb80f412fe3ffee27a9a0cd5c551f.jpg",
+        "text": "$$\n\\nabla _ { \\phi , \\varphi } \\mathcal { L } _ { t r a i n } ( \\tilde { \\boldsymbol { w } } ) = \\nabla _ { \\tilde { \\boldsymbol { w } } } \\mathcal { L } _ { t r a i n } ( \\tilde { \\boldsymbol { w } } ) \\cdot \\nabla _ { \\phi , \\varphi } \\tilde { \\boldsymbol { w } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The first term is the gradients of standard network parameters, the second term is decomposed as ",
+        "bbox": [
+            176,
+            594,
+            805,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e845c170446d69ea69e6804e973fc654df3e72f22394fcbef0c201ee484a8aa.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\nabla _ { \\phi } \\tilde { w } = \\left\\{ \\nabla _ { h } H ( h ; \\varphi ) \\cdot \\nabla _ { \\phi } h \\big | h \\in G ^ { ( T ) } \\left( \\{ h _ { v } ^ { ( 0 ) } \\} , \\mathcal { A } , \\phi \\right) \\right\\} , } \\\\ & { \\nabla _ { \\varphi } \\tilde { w } = \\left\\{ \\nabla _ { \\varphi } H ( h _ { v } ^ { ( T ) } ; \\varphi ) \\big | v \\in \\mathcal { V } \\right\\} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            611,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where (Eq. 17) is the contribution from GNN module $G$ and (Eq. 18) is the contribution from the hypernet module $H$ . Both $G$ and $H$ are jointly learned throughout training. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we use our proposed GHN to search for the best CNN architecture for image classification. First, we evaluate the GHN on the standard CIFAR (Krizhevsky & Hinton, 2009) and ImageNet (Russakovsky et al., 2015) architecture search benchmarks. Next, we apply GHN on an “anytime prediction” task where we optimize the speed-accuracy tradeoff that is key for many real-time applications. Finally, we benchmark the GHN’s predicted-performance correlation and explore various factors in an ablation study. ",
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+    {
+        "type": "text",
+        "text": "5.1 NAS BENCHMARKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1.1 CIFAR-10 ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We conduct our initial set of experiments on CIFAR-10 (Krizhevsky & Hinton, 2009), which contains 10 object classes and 50,000 training images and 10,000 test images of size $3 2 \\times 3 2 \\times 3$ . We use 5,000 images split from the training set as our validation set. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ad36e39562f213ea51cbbc265e78c702538ac847ff06cffbc1736918f2b8d1ad.jpg",
+        "table_caption": [
+            "Table 2: Comparison with image classifiers found by state-of-the-art NAS methods which employ advanced search methods on CIFAR-10. Results shown are mean $\\pm$ standard deviation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Search Cost (GPU days)</td><td>Param ×106</td><td>Accuracy</td></tr><tr><td>NASNet-A (Zoph et al.,2018)</td><td>1800</td><td>3.3</td><td>97.35</td></tr><tr><td>ENAS Cell search (Pham et al.,2018)</td><td>0.45</td><td>4.6</td><td>97.11</td></tr><tr><td>DARTS (first order) (Liu et al.,2018c)</td><td>1.5</td><td>2.9</td><td>97.06</td></tr><tr><td>DARTS (second order) (Liu et al.,2018c)</td><td>4</td><td>3.4</td><td>97.17 ± 0.06</td></tr><tr><td>GHN Top-Best,1K (F=32)</td><td>0.84</td><td>5.7</td><td>97.16± 0.07</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/730f5b080cd442f33af41ffecffb49c02bd7add167a922c68a3e891357c72217.jpg",
+        "table_caption": [
+            "Table 3: Comparison with image classifiers found by state-of-the-art NAS methods which employ advanced search methods on ImageNet-Mobile. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Search Cost (GPU days)</td><td rowspan=\"2\">Param ×106</td><td rowspan=\"2\">FLOPs ×106</td><td colspan=\"2\">Accuracy</td></tr><tr><td>Top1</td><td>Top 5</td></tr><tr><td>NASNet-A (Zoph et al., 2018)</td><td>1800</td><td>5.3</td><td>564</td><td>74.0</td><td>91.6</td></tr><tr><td>NASNet-C (Zoph etal., 2018)</td><td>1800</td><td>4.9</td><td>558</td><td>72.5</td><td>91.0</td></tr><tr><td>AmoebaNet-A (Real et al.,2018)</td><td>3150</td><td>5.1</td><td>555</td><td>74.5</td><td>92.0</td></tr><tr><td>AmoebaNet-C (Real et al., 2018)</td><td>3150</td><td>6.4</td><td>570</td><td>75.7</td><td>92.4</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>225</td><td>5.1</td><td>588</td><td>74.2</td><td>91.9</td></tr><tr><td>DARTS (second order) (Liu et al.,2018c)</td><td>4</td><td>4.9</td><td>595</td><td>73.1</td><td>91.0</td></tr><tr><td>GHN Top-Best, 1K</td><td>0.84</td><td>6.1</td><td>569</td><td>73.0</td><td>91.3</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Search space: Following existing NAS methods, we choose to search for optimal blocks rather than the entire network. Each block contains 17 nodes, with 8 possible operations. The final architecture is formed by stacking 18 blocks. The spatial size is halved and the number of channels is doubled after blocks 6 and 12. These settings are all chosen following recent NAS methods (Zoph & Le, 2017; Pham et al., 2018; Liu et al., 2018c), with details in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "Training: For the GNN module, we use a standard GRU cell (Cho et al., 2014) with hidden size 32 and 2 layer MLP with hidden size 32 as the recurrent cell function $U$ and message function $M$ respectively. The shared hypernetwork $H \\left( \\cdot ; \\varphi \\right)$ is a 2-layer MLP with hidden size 64. From the results of ablations studies in Section 5.4, the GHN is trained with blocks with $N = 7$ nodes and $T = 5$ propagations under the forward-backward scheme, using the ADAM optimizer (Kingma $\\&$ Ba, 2015). Training details of the final selected architectures are chosen to follow existing works and can be found in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "Evaluation: First, we compare to similar methods that use random search with a hypernetwork or a one-shot model as a surrogate search signal. We randomly sample 10 architectures and train until convergence for our random baseline. Next, we randomly sample 1000 architectures, and select the top 10 performing architectures with GHN generated weights, which we refer to as GHN Top. Our reported search cost includes both the GHN training and evaluation phase. Shown in Table 1, the GHN achieves competitive results with nearly an order of magnitude reduction in search cost. ",
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+    {
+        "type": "text",
+        "text": "In Table 2, we compare with methods which use more advanced search methods, such as reinforcement learning and evolution. Once again, we sample 1000 architectures and use the GHN to select the top 10. To make a fair comparison for random search, we train the top 10 for a short period before selecting the best to train until convergence. The accuracy reported for GHN Top-Best is the average of 5 runs of the same final architecture. Note that all methods in Table 2 use CutOut (Devries & Taylor, 2017). GHN achieves very competitive results with a simple random search algorithm, while only using a fraction of the total search cost. Using advanced search methods with GHNs may bring further gains. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1.2 IMAGENET-MOBILE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also run our GHN algorithm on the ImageNet dataset (Russakovsky et al., 2015), which contains 1.28 million training images. We report the top-1 accuracy on the 50,000 validation images. Following existing literature, we conduct the ImageNet experiments in the mobile setting, where the model is constrained to be under 600M FLOPS. We directly transfer the best architecture block found in the CIFAR-10 experiments, using an initial convolution layer of stride 2 before stacking 14 blocks with scale reduction at blocks 1, 2, 6 and 10. The total number of flops is constrained by choosing the initial number of channels. We follow existing NAS methods on the training procedure of the final architecture; details can be found in the Appendix. As shown in Table 3 the transferred block is competitive with other NAS methods which require a far greater search cost. ",
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+    {
+        "type": "image",
+        "img_path": "images/78fcd6d0d3038bc13cc432a354ae7b4bb6db04e1b236ac49f50a1cea720ecd3b.jpg",
+        "image_caption": [
+            "Figure 3: Comparison with state-of-the-art human-designed networks on CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/bd12ef32546905a588ef2bda3f48a1e44e0de4ed5a56497ac8d1eb7be1c1f97f.jpg",
+        "image_caption": [
+            "Figure 4: Comparison between random 10 and top 10 networks on CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 4: Benchmarking the correlation between the predicted and true performance of the GHN against SGD and a one-shot model baselines. Results are on CIFAR-10. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4372fb8f330be3e16de8eb10183a30ca2e0f964639b8f9dcc1d7ea192a37459c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Computation cost</td><td colspan=\"2\">Correlation</td></tr><tr><td>Initial (GPU hours)</td><td>Per arch.(GPU seconds)</td><td>Random-100</td><td>Top-50</td></tr><tr><td>SGD 10 Steps</td><td>-</td><td>0.9</td><td>0.26</td><td>-0.05</td></tr><tr><td>SGD 100 Steps</td><td>=</td><td>9</td><td>0.59</td><td>0.06</td></tr><tr><td>SGD 200 Steps</td><td>=</td><td>18</td><td>0.62</td><td>0.20</td></tr><tr><td>SGD 1000 Steps</td><td></td><td>90</td><td>0.77</td><td>0.26</td></tr><tr><td>One-Shot</td><td>9.8</td><td>0.06</td><td>0.58</td><td>0.31</td></tr><tr><td>GHN</td><td>6.1</td><td>0.08</td><td>0.68</td><td>0.48</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 ANYTIME PREDICTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the real-time setting, the computational budget available can vary for each test case and cannot be known ahead of time. This is formalized in anytime prediction, (Grubb & Bagnell, 2012) the setting in which for each test example $\\mathbf { x }$ , there is non-deterministic computational budget $B$ drawn from the joint distribution $P ( \\bar { \\bf x } , B )$ . The goal is then to minimize the expected loss $L ( f ) = \\mathbb { E } \\left[ L \\left( f ( \\mathbf { x } ) , \\mathbf { \\bar { \\theta } } \\right) \\right] _ { P ( \\mathbf { x } , B ) }$ , where $\\dot { f } ( \\cdot )$ is the model and $L ( \\cdot )$ is the loss for an $f ( \\cdot )$ that must produce a prediction within the budget $B$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on CIFAR-10. Our anytime search space consists of networks with 3 cells containing 24, 16, and 8 nodes. Each node is given the additional properties: 1) the spatial size it operates at and 2) if an early-exit classifier is attached to it. A node enforces its spatial size by pooling or upsampling any input feature maps inputs that are of different scale. Note that while a naive one-shot model would triple its size to include three different parameter sets at three different scales, the GHN is negligibly affected by such a change. The GHN uses the area under the predicted accuracy-FLOPS curve as its selection criteria. The search space, contains various convolution and pooling operators. Training methodology of the final architectures are chosen to match Huang et al. (2018) and can be found in the Appendix. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 3 shows a comparison with the various methods presented by Huang et al. (2018). Our experiments show that the best searched architectures can outperform the current state-of-the-art human designed networks. We see the GHN is amenable to the changes proposed above, and can find efficient architectures with a random search when used with a strong search space. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 PREDICTED PERFORMANCE CORRELATION (CIFAR-10) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we evaluate whether the parameters generated from GHN can be indicative of the final performance. Our metric is the correlation between the accuracy of a model with trained weights vs. GHN generated weights. We use a fixed set of 100 random architectures that have not been seen by the GHN during training, and we train them for 50 epochs to obtain our “ground-truth” accuracy, and finally compare with the accuracy obtained from GHN generated weights. We report the Pearson’s R score on all 100 random architectures and the top 50 performing architectures (i.e. above average architectures). Since we are interested in searching for the best architecture, obtaining a higher correlation on top performing architectures is more meaningful. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To evaluate the effectiveness of GHN, we further consider two baselines: 1) training a network with SGD from scratch for a varying number of steps, and 2) our own implementation of the oneshot model proposed by Pham et al. (2018), where nodes store a set of shared parameters for each possible operation. Unlike GHN, which is compatible with varying number of nodes, the one-shot model must be trained with $N = 1 7$ nodes to match the evaluation. The GHN is trained with $N = 7$ , $T = 5$ using forward-backward propagation. These GHN parameters are selected based on the results found in Section 5.4. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c442ba59a9b67d0fb2d019ca2b4e202c6143ceb067eaab349e5eb4b2f2b43be6.jpg",
+        "image_caption": [
+            "Figure 5: GHN when varying the number of nodes and propagation scheme "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4 shows performance correlation and search cost of SGD, the one-shot model, and our GHN. Note that GHN clearly outperforms the one-shot model, showing the effectiveness of dynamically predicting parameters based on graph topology. While it takes 1000 SGD steps to surpasses GHN in the “Random- $1 0 0 ^ { \\circ }$ setting, GHN is still the strongest in the “Top- $. 5 0 ^ { \\circ }$ setting, which is more important for architecture search. Moreover, compared to GHN, running 1000 SGD steps for every random architecture is over 1000 times more computationally expensive. In contrast, GHN only requires a pre-training stage of 6 hours, and afterwards, the trained GHN can be used to efficiently evaluate a massive number of random architectures of different sizes. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.4 ABLATION STUDIES (CIFAR-10) ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Number of graph nodes: The GHN is compatible with varying number of nodes - graphs used in training need not be the same size as the graphs used for evaluation. Figure 5a shows how GHN performance varies as a function of the number of nodes employed during training - fewer nodes generally produces better performance. While the GHN has difficulty learning on larger graphs, likely due to the vanishing gradient problem, it can generalize well from just learning on smaller graphs. Note that all GHNs are tested with the full graph size $N = 1 7$ nodes). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Number of propagation steps: We now compare the forward-backward propagation scheme with the regular synchronous propagation scheme. Note that $T = 1$ synchronous step corresponds to one full forward-backward phase. As shown in Figure 5b, the forward-backward scheme consistently outperforms the synchronous scheme. More propagation steps also help improving the performance, with a diminishing return. While the forward-backward scheme is less amenable to acceleration from parallelization due to its sequential nature, it is possible to parallelize the evaluation phase across multiple GHNs when testing the fitness of candidate architectures. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Stacked GHN for architectural motifs: We also evaluate different design choices of GHNs on representing architectural motifs. We compare 1) individual GHNs, each predicting one block independently, 2) a stacked GHN where individual GHN’s pass on their graph embedding without sharing parameters, 3) a stacked GHN with shared parameters (our proposed approach). As shown in Table 5, passing messages between GHN’s is crucial, and sharing parameters produces better performance. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/86911b051eb44ada06562af49bc7e412933719155cd2388801c1af90cd25a5ab.jpg",
+        "image_caption": [
+            "Table 5: Stacked GHN Correlation. SP denotes sharing parameters and PE denotes passing embeddings "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose the Graph HyperNetwork (GHN), a composition of graph neural networks and hypernetworks that generates the weights of any architecture by operating directly on their computation graph representation. We demonstrate a strong correlation between the performance with the generated weights and the fully-trained weights. Using our GHN to form a surrogate search signal, we achieve competitive results on CIFAR-10 and ImageNet mobile with nearly $\\bar { 1 } 0 \\times$ faster speed compared to other random search methods. Furthermore, we show that our proposed method can be extended to outperform the best human-designed architectures in setting of anytime prediction, greatly reducing the computation cost of real-time neural networks. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015. ",
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+        "text": "Mingxing Tan, Bo Chen, Ruoming Pang, Vijay Vasudevan, and Quoc V Le. Mnasnet: Platformaware neural architecture search for mobile. arXiv preprint arXiv:1807.11626, 2018. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Paul J Werbos. Backpropagation through time: what it does and how to do it. Proceedings of the IEEE, 78(10):1550–1560, 1990.   \nLingxi Xie and Alan L. Yuille. Genetic CNN. In IEEE International Conference on Computer Vision (ICCV), 2017.   \nYanqi Zhou, Siavash Ebrahimi, Sercan O Arık, Haonan Yu, Hairong Liu, and Greg Diamos. ¨ Resource-efficient neural architect. arXiv preprint arXiv:1806.07912, 2018.   \nBarret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. In Proceedings of the 5th International Conference on Learning Representations (ICLR), 2017.   \nBarret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures for scalable image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. ",
+        "bbox": [
+            171,
+            102,
+            826,
+            290
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            290,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7.1 SEARCH SPACE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            320,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Standard image classification on CIFAR-10 and ImageNet The search space for CIFAR-10 and ImageNet classification experiments includes the following operations: ",
+        "bbox": [
+            173,
+            159,
+            823,
+            188
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• identity   \n• $1 \\times 1$ convolution   \n• $3 \\times 3$ separable convolution   \n• $5 \\times 5$ separable convolution   \n• $3 \\times 3$ dilated separable convolution   \n• $5 \\times 5$ dilated separable convolution   \n• $1 \\times 7$ convolution followed $7 \\times 1$ convolution   \n• $3 \\times 3$ max pooling   \n• $3 \\times 3$ average pooling ",
+        "bbox": [
+            215,
+            199,
+            534,
+            369
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A block forms an output by concatenating all leaf nodes in the graph. Blocks have 2 input nodes which ingest the output of block $i - 1$ and block $i - 2$ respectively. The input nodes are bottleneck layers, and can reduce the spatial size by using stride 2. ",
+        "bbox": [
+            176,
+            381,
+            823,
+            421
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Note that while ENAS supports only 5 operators due to memory constraints, GHNs can search for more operators. This is because ENAS (and other methods which use one-shot models) must store all the parameters in memory because it finds paths in a larger model. Thus the memory requirements are ${ \\cal O } ( K N )$ where $K$ is the number of operations and $\\bar { N }$ is the number of nodes in the candidate architecture. In contrast, the memory requirement for GHNs is $O ( N ) + O ( K )$ for the candidate architecture and GHN respectively. ",
+        "bbox": [
+            174,
+            428,
+            825,
+            508
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Anytime prediction on CIFAR-10 The search space for the CIFAR-10 anytime prediction experiments includes the following operations: ",
+        "bbox": [
+            174,
+            523,
+            823,
+            551
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• $1 \\times 1$ convolution • $3 \\times 3$ convolution • $5 \\times 5$ convolution • $3 \\times 3$ max pooling • $3 \\times 3$ average pooling ",
+        "bbox": [
+            215,
+            563,
+            380,
+            656
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the anytime setting, nodes concatenate their inputs rather than sum. Thus, the identity operator was removed as it would be redundant. The search space does not include separable convolutions so that it is comparable with our baselines (Huang et al., 2018). Block 1 contains nodes which may operate on any of the 3 scales $( 3 2 \\times 3 2 , 1 6 \\times 1 \\bar { 6 } , 8 \\times 8 )$ . Block 2 contains nodes which can only operate on scales $1 6 \\times 1 6$ and $8 \\times 8$ . Block 3 only contains nodes which operate on the scale $8 \\times 8$ . We fix the number of exit nodes. These choices are inspired by Huang et al. (2018) ",
+        "bbox": [
+            173,
+            667,
+            825,
+            747
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7.2 GRAPH HYPERNETWORK DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            765,
+            454,
+            779
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Standard image classification on CIFAR-10 and ImageNet While node embeddings are initialized to a one-hot vector representing computational operator of the node, we found it helpful to pass the sparse vector through a learned embedding matrix prior to graph propagation. The GHN is trained for 200 epochs with batch size 64 using the ADAM optimizer with an initial learning rate 1e-3 that is divided by 2 at epoch 100 and 150. A naive hypernet would have a separate output branch for each possible node type, and simply ignore branches that aren’t applicable to the specific node. In this manner, the number of parameters of the hypernetwork scale according to the number of possible node computations. In contrast, the number of parameters for a one-shot model scale according to the number of nodes in the graph. We further reduce number of parameters by obtaining smaller sized convolutions kernels through the slicing of larger sized kernels. ",
+        "bbox": [
+            173,
+            790,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Anytime prediction In the anytime prediction setting, two one-hot vectors representing the node’s scale and presence of an early exit classifier are additionally concatenated to the first initialized node embedding. We found it helpful to train the GHN with a random number of nodes per block, with maximum number of allowed nodes being the evaluation block size. Because nodes concatenate their inputs, a bottleneck layer is required. The hypernetwork can predict bottleneck parameters for a varying number of input nodes by generating weights based on edge activations rather than node activations. We form edge activations by concatenating the node activations of the parent and child. Edge weights generated this way can be concatenated, allowing the dimensionality of the predicted bottleneck weights the be proportional to the number of incoming edges. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            223
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7.3 FINAL ARCHITECTURE TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            243,
+            508,
+            257
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10 Following existing NAS methods (Zoph et al., 2018; Real et al., 2018), the final candidates are trained for 600 epochs using SGD with momentum 0.9, a single period cosine schedule with $l _ { m a x } = 0 . 0 2 5$ , and batch size 64. For regularization, we use scheduled drop-path with a final dropout probability of 0.4. We use an auxiliary head located at 2/3 of the network weighted by 0.5. We accelerate training by performing distributed training across 32 GPUs; the learning rate is multiplied by 32 with an initial linear warmup of 5 epochs. ",
+        "bbox": [
+            174,
+            270,
+            825,
+            349
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "ImageNet Mobile For ImageNet mobile experiments, we use an image size of $2 2 4 \\times 2 2 4$ . Following existing NAS methods (Zoph et al., 2018; Real et al., 2018), the final candidates are trained for 250 epochs using SGD with momentum 0.9, initial learning rate 0.1 multiplied by 0.97 every epoch. We use an auxiliary head located at 2/3 of the network weighted by 0.5. We use the same regularization techniques, and similarly accelerate training in a distributed fashion. ",
+        "bbox": [
+            174,
+            368,
+            825,
+            435
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Anytime Following Huang et al. (2018), the final candidates are trained using SGD with momentum 0.9. We train the models for 300 epochs use an initial learning rate of 0.1, which is divided by 10 after 150 and 225 epochs using a batch size of 64. We accelerate training with distributed training in a similar fashion as the CIFAR-10 classification and ImageNet mobile experiments. The number of filters for the final architecture is chosen such that the number of FLOPS is comparable to existing baselines. ",
+        "bbox": [
+            174,
+            454,
+            825,
+            534
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7.4 INVESTIGATING ACCURACY DROP OFF ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            554,
+            482,
+            568
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows a plot comparing the accuracy of an architecture that is trained for 50 epochs and the accuracy of the same architecture using GHN generated weights. ",
+        "bbox": [
+            173,
+            580,
+            823,
+            609
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/435dcfb054894d6e6b35f95f2f81768ec54bceb0941b0d207a1d2b08007782df.jpg",
+        "image_caption": [
+            "Figure 6: Comparison for 100 randomly sampled architectures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            346,
+            643,
+            629,
+            792
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7.5 VISUALIZATION OF FINAL ARCHITECTURES ",
+        "bbox": [
+            174,
+            858,
+            517,
+            872
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7.5.1 CIFAR-10 AND IMAGENET CLASSIFICATION ",
+        "bbox": [
+            174,
+            883,
+            544,
+            898
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows the best found block in the CIFAR-10 Experiments. ",
+        "bbox": [
+            174,
+            909,
+            614,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/6e53da1af3ef571884aeb6868911e9f36757f2a006d81ee6cd231f10f3a837b8.jpg",
+        "image_caption": [
+            "Figure 7: Best block found for classification "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            128,
+            787,
+            390
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.5.2 ANYTIME PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            467,
+            388,
+            481
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figures 8, 9 and 10 show blocks 1 2 and 3 of the best architecture found in the anytime experiments.   \nThe color red denotes that an early exit is attached to the output of the node. ",
+        "bbox": [
+            171,
+            489,
+            823,
+            518
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/73c5c098227fa57a76f03bdc53d9a5640b6a37364f0f65d5ef23d5d5fa4ac452.jpg",
+        "image_caption": [
+            "Figure 8: Block 1 for anytime network. Red color denotes early exit. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            418,
+            171,
+            575,
+            839
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b4ea6ef23a76b8c3479f473823c4ef089219d805c71f114dc19ce269c6da4292.jpg",
+        "image_caption": [
+            "Figure 9: Block 2 for anytime network. Red color denotes early exit. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            367,
+            145,
+            637,
+            873
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/470a3b4b93cfc85d75d378477a15e47c60b2d1b6ac057277a04dbb55a90961fe.jpg",
+        "image_caption": [
+            "Figure 10: Block 3 for anytime network. Red color denotes early exit. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            363,
+            787,
+            627
+        ],
+        "page_idx": 16
+    }
+]
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new file mode 100644
index 0000000000000000000000000000000000000000..bf1a3d7a77cabf4f0bdb61c2e6dc4b5e74b79f55
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+                                    "content": "propagations under the forward-backward scheme, using the ADAM optimizer (Kingma",
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+                                    "content": "Ba, 2015). Training details of the final selected architectures are chosen to follow existing works",
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+                            "index": 20
+                        },
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+                                    "score": 1.0,
+                                    "content": "and can be found in the Appendix.",
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+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "Evaluation: First, we compare to similar methods that use random search with a hypernetwork or",
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+                                    "content": "top 10 performing architectures with GHN generated weights, which we refer to as GHN Top. Our",
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+                                    "score": 1.0,
+                                    "content": "reported search cost includes both the GHN training and evaluation phase. Shown in Table 1, the",
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+                                    "content": "GHN achieves competitive results with nearly an order of magnitude reduction in search cost.",
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+                    ],
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+                                    "content": "In Table 2, we compare with methods which use more advanced search methods, such as",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "select the top 10. To make a fair comparison for random search, we train the top 10 for a short period",
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+                        },
+                        {
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+                                    "content": "before selecting the best to train until convergence. The accuracy reported for GHN Top-Best is the",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "average of 5 runs of the same final architecture. Note that all methods in Table 2 use CutOut (Devries",
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+                                    "score": 1.0,
+                                    "content": "& Taylor, 2017). GHN achieves very competitive results with a simple random search algorithm,",
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+                                    "content": "while only using a fraction of the total search cost. Using advanced search methods with GHNs may",
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+                                    "score": 1.0,
+                                    "content": "bring further gains.",
+                                    "type": "text"
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+                                    "content": "5.1.2 IMAGENET-MOBILE",
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+                                    "content": "We also run our GHN algorithm on the ImageNet dataset (Russakovsky et al., 2015), which contains",
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+                                    "content": "1.28 million training images. We report the top-1 accuracy on the 50,000 validation images.",
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+                                    "score": 1.0,
+                                    "content": "Following existing literature, we conduct the ImageNet experiments in the mobile setting, where",
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+                                    "content": "14 blocks with scale reduction at blocks 1, 2, 6 and 10. The total number of flops is constrained by",
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+                                    "score": 1.0,
+                                    "content": "choosing the initial number of channels. We follow existing NAS methods on the training procedure",
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+                                    "score": 1.0,
+                                    "content": "of the final architecture; details can be found in the Appendix. As shown in Table 3 the transferred",
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+                                            "html": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Search Cost (GPU days)</td><td rowspan=\"2\">Param ×106</td><td rowspan=\"2\">FLOPs ×106</td><td colspan=\"2\">Accuracy</td></tr><tr><td>Top1</td><td>Top 5</td></tr><tr><td>NASNet-A (Zoph et al., 2018)</td><td>1800</td><td>5.3</td><td>564</td><td>74.0</td><td>91.6</td></tr><tr><td>NASNet-C (Zoph etal., 2018)</td><td>1800</td><td>4.9</td><td>558</td><td>72.5</td><td>91.0</td></tr><tr><td>AmoebaNet-A (Real et al.,2018)</td><td>3150</td><td>5.1</td><td>555</td><td>74.5</td><td>92.0</td></tr><tr><td>AmoebaNet-C (Real et al., 2018)</td><td>3150</td><td>6.4</td><td>570</td><td>75.7</td><td>92.4</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>225</td><td>5.1</td><td>588</td><td>74.2</td><td>91.9</td></tr><tr><td>DARTS (second order) (Liu et al.,2018c)</td><td>4</td><td>4.9</td><td>595</td><td>73.1</td><td>91.0</td></tr><tr><td>GHN Top-Best, 1K</td><td>0.84</td><td>6.1</td><td>569</td><td>73.0</td><td>91.3</td></tr></table>",
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+                                    "content": "In Table 2, we compare with methods which use more advanced search methods, such as",
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+                                    "score": 1.0,
+                                    "content": "select the top 10. To make a fair comparison for random search, we train the top 10 for a short period",
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+                                    "score": 1.0,
+                                    "content": "before selecting the best to train until convergence. The accuracy reported for GHN Top-Best is the",
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+                                    "score": 1.0,
+                                    "content": "average of 5 runs of the same final architecture. Note that all methods in Table 2 use CutOut (Devries",
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+                                    "score": 1.0,
+                                    "content": "& Taylor, 2017). GHN achieves very competitive results with a simple random search algorithm,",
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+                                    "content": "We also run our GHN algorithm on the ImageNet dataset (Russakovsky et al., 2015), which contains",
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+                                    "score": 1.0,
+                                    "content": "Following existing literature, we conduct the ImageNet experiments in the mobile setting, where",
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+                                    "content": "naive one-shot model would triple its size to include three different parameter sets at three different",
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+                                    "score": 1.0,
+                                    "content": "scales, the GHN is negligibly affected by such a change. The GHN uses the area under the predicted",
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+                                    "score": 1.0,
+                                    "content": "pooling operators. Training methodology of the final architectures are chosen to match Huang et al.",
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+                                    "score": 1.0,
+                                    "content": "(2018) and can be found in the Appendix.",
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+                                    "content": "Figure 3 shows a comparison with the various methods presented by Huang et al. (2018). Our",
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+                                    "score": 1.0,
+                                    "content": "the Pearson’s R score on all 100 random architectures and the top 50 performing architectures (i.e.",
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+                                    "score": 1.0,
+                                    "content": "above average architectures). Since we are interested in searching for the best architecture, obtaining",
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+                                    "content": ". The goal is then to minimize the expected loss",
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+                                    "score": 1.0,
+                                    "content": "pooling or upsampling any input feature maps inputs that are of different scale. Note that while a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "naive one-shot model would triple its size to include three different parameter sets at three different",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "scales, the GHN is negligibly affected by such a change. The GHN uses the area under the predicted",
+                                    "type": "text"
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+                            ],
+                            "index": 25
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "accuracy-FLOPS curve as its selection criteria. The search space, contains various convolution and",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "pooling operators. Training methodology of the final architectures are chosen to match Huang et al.",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "(2018) and can be found in the Appendix.",
+                                    "type": "text"
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+                            ],
+                            "index": 28
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "Figure 3 shows a comparison with the various methods presented by Huang et al. (2018). Our",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "experiments show that the best searched architectures can outperform the current state-of-the-art",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "human designed networks. We see the GHN is amenable to the changes proposed above, and can",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "find efficient architectures with a random search when used with a strong search space.",
+                                    "type": "text"
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+                    ],
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+                    ],
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+                                    "content": "In this section, we evaluate whether the parameters generated from GHN can be indicative of the",
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+                                    "score": 1.0,
+                                    "content": "final performance. Our metric is the correlation between the accuracy of a model with trained",
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+                                    "score": 1.0,
+                                    "content": "weights vs. GHN generated weights. We use a fixed set of 100 random architectures that have not",
+                                    "type": "text"
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+                            ],
+                            "index": 36
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+                        {
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+                                    "score": 1.0,
+                                    "content": "been seen by the GHN during training, and we train them for 50 epochs to obtain our “ground-truth”",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "accuracy, and finally compare with the accuracy obtained from GHN generated weights. We report",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the Pearson’s R score on all 100 random architectures and the top 50 performing architectures (i.e.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "above average architectures). Since we are interested in searching for the best architecture, obtaining",
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+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "a higher correlation on top performing architectures is more meaningful.",
+                                    "type": "text"
+                                }
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+                                    "score": 1.0,
+                                    "content": "To evaluate the effectiveness of GHN, we further consider two baselines: 1) training a network",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "with SGD from scratch for a varying number of steps, and 2) our own implementation of the one-",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "shot model proposed by Pham et al. (2018), where nodes store a set of shared parameters for each",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "possible operation. Unlike GHN, which is compatible with varying number of nodes, the one-shot",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 5
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+                                    "score": 1.0,
+                                    "content": "model must be trained with",
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+                                    "cross_page": true
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+                                    "content": "requires a pre-training stage of 6 hours, and afterwards, the trained GHN can be used to efficiently",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "evaluate a massive number of random architectures of different sizes.",
+                                    "type": "text"
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+                    ],
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+                                    "content": "5.4 ABLATION STUDIES (CIFAR-10)",
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+                                }
+                            ],
+                            "index": 17
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+                    ],
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+                            ],
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+                        {
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+                                    "content": "in training need not be the same size as the graphs used for evaluation. Figure 5a shows how GHN",
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+                                    "content": "performance varies as a function of the number of nodes employed during training - fewer nodes",
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+                            ],
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+                        {
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+                                    "content": "generally produces better performance. While the GHN has difficulty learning on larger graphs,",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "likely due to the vanishing gradient problem, it can generalize well from just learning on smaller",
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+                                    "score": 1.0,
+                                    "content": "graphs. Note that all GHNs are tested with the full graph size",
+                                    "type": "text"
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+                                {
+                                    "bbox": [
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+                                    "score": 0.87,
+                                    "content": "N = 1 7",
+                                    "type": "inline_equation"
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+                                {
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+                                    "content": "nodes).",
+                                    "type": "text"
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+                                    "content": "Number of propagation steps: We now compare the forward-backward propagation scheme with",
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+                            ],
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+                        {
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+                        {
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+                                    "score": 1.0,
+                                    "content": "with a diminishing return. While the forward-backward scheme is less amenable to acceleration",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "from parallelization due to its sequential nature, it is possible to parallelize the evaluation phase",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "across multiple GHNs when testing the fitness of candidate architectures.",
+                                    "type": "text"
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+                            "index": 30
+                        }
+                    ],
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+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "Stacked GHN for architectural motifs: We also eval-",
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+                            "index": 31
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+                        {
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+                                    "score": 1.0,
+                                    "content": "uate different design choices of GHNs on representing ar-",
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+                                    "score": 1.0,
+                                    "content": "chitectural motifs. We compare 1) individual GHNs, each",
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+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                            "index": 35
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+                                    "score": 1.0,
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+                            "index": 36
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+                                    "score": 1.0,
+                                    "content": "without sharing parameters, 3) a stacked GHN with shared",
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+                            "index": 37
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+                                    "score": 1.0,
+                                    "content": "parameters (our proposed approach). As shown in Table 5,",
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+                            ],
+                            "index": 38
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+                        {
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+                                    "score": 1.0,
+                                    "content": "passing messages between GHN’s is crucial, and sharing",
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+                            "index": 39
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+                                            "content": "denotes passing embeddings",
+                                            "type": "text"
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+                                    "content": "6 CONCLUSION",
+                                    "type": "text"
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+                {
+                    "type": "text",
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+                                    "content": "In this work, we propose the Graph HyperNetwork (GHN), a composition of graph neural networks",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "and hypernetworks that generates the weights of any architecture by operating directly on their",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "computation graph representation. We demonstrate a strong correlation between the performance",
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+                                    "score": 1.0,
+                                    "content": "with the generated weights and the fully-trained weights. Using our GHN to form a surrogate",
+                                    "type": "text"
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "search signal, we achieve competitive results on CIFAR-10 and ImageNet mobile with nearly",
+                                    "type": "text"
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+                                {
+                                    "bbox": [
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+                                    "score": 0.86,
+                                    "content": "\\bar { 1 } 0 \\times",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 50
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "faster speed compared to other random search methods. Furthermore, we show that our proposed",
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+                                    "content": "smaller sized convolutions kernels through the slicing of larger sized kernels.",
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+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Block 2 contains nodes which can only",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "operate on scales",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.9,
+                                    "content": "1 6 \\times 1 6",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.9,
+                                    "content": "8 \\times 8",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Block 3 only contains nodes which operate on the scale",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.89,
+                                    "content": "8 \\times 8",
+                                    "type": "inline_equation"
+                                },
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+                                    "score": 1.0,
+                                    "content": ".",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We fix the number of exit nodes. These choices are inspired by Huang et al. (2018)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        }
+                    ],
+                    "index": 31.5,
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Standard image classification on CIFAR-10 and ImageNet While node embeddings are",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "initialized to a one-hot vector representing computational operator of the node, we found it helpful",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "to pass the sparse vector through a learned embedding matrix prior to graph propagation. The GHN",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is trained for 200 epochs with batch size 64 using the ADAM optimizer with an initial learning rate",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "1e-3 that is divided by 2 at epoch 100 and 150. A naive hypernet would have a separate output",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "branch for each possible node type, and simply ignore branches that aren’t applicable to the specific",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "node. In this manner, the number of parameters of the hypernetwork scale according to the number",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                        712
+                                    ],
+                                    "score": 1.0,
+                                    "content": "of possible node computations. In contrast, the number of parameters for a one-shot model scale",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "score": 1.0,
+                                    "content": "according to the number of nodes in the graph. We further reduce number of parameters by obtaining",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 44
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        416,
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+                                    "score": 1.0,
+                                    "content": "smaller sized convolutions kernels through the slicing of larger sized kernels.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 45
+                        }
+                    ],
+                    "index": 40.5,
+                    "bbox_fs": [
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+                }
+            ]
+        },
+        {
+            "preproc_blocks": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Anytime prediction In the anytime prediction setting, two one-hot vectors representing the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "node’s scale and presence of an early exit classifier are additionally concatenated to the first",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 1
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
+                                        106,
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+                                        505,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "initialized node embedding. We found it helpful to train the GHN with a random number of nodes",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        126
+                                    ],
+                                    "score": 1.0,
+                                    "content": "per block, with maximum number of allowed nodes being the evaluation block size. Because nodes",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
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+                                137
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        124,
+                                        506,
+                                        137
+                                    ],
+                                    "score": 1.0,
+                                    "content": "concatenate their inputs, a bottleneck layer is required. The hypernetwork can predict bottleneck",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        505,
+                                        147
+                                    ],
+                                    "score": 1.0,
+                                    "content": "parameters for a varying number of input nodes by generating weights based on edge activations",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        105,
+                                        145,
+                                        505,
+                                        158
+                                    ],
+                                    "score": 1.0,
+                                    "content": "rather than node activations. We form edge activations by concatenating the node activations of the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        104,
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+                                        169
+                                    ],
+                                    "score": 1.0,
+                                    "content": "parent and child. Edge weights generated this way can be concatenated, allowing the dimensionality",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        463,
+                                        179
+                                    ],
+                                    "score": 1.0,
+                                    "content": "of the predicted bottleneck weights the be proportional to the number of incoming edges.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "type": "title",
+                    "bbox": [
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+                    ],
+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "7.3 FINAL ARCHITECTURE TRAINING DETAILS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "type": "text",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "CIFAR-10 Following existing NAS methods (Zoph et al., 2018; Real et al., 2018), the final",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "candidates are trained for 600 epochs using SGD with momentum 0.9, a single period cosine",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "schedule with",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.9,
+                                    "content": "l _ { m a x } = 0 . 0 2 5",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": ", and batch size 64. For regularization, we use scheduled drop-path",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
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+                                        505,
+                                        257
+                                    ],
+                                    "score": 1.0,
+                                    "content": "with a final dropout probability of 0.4. We use an auxiliary head located at 2/3 of the network",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        255,
+                                        506,
+                                        269
+                                    ],
+                                    "score": 1.0,
+                                    "content": "weighted by 0.5. We accelerate training by performing distributed training across 32 GPUs; the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        267,
+                                        405,
+                                        279
+                                    ],
+                                    "score": 1.0,
+                                    "content": "learning rate is multiplied by 32 with an initial linear warmup of 5 epochs.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        }
+                    ],
+                    "index": 12.5
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "ImageNet Mobile For ImageNet mobile experiments, we use an image size of",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.88,
+                                    "content": "2 2 4 \\times 2 2 4",
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+                                    "score": 1.0,
+                                    "content": ".",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        302,
+                                        505,
+                                        315
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Following existing NAS methods (Zoph et al., 2018; Real et al., 2018), the final candidates are",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        313,
+                                        505,
+                                        326
+                                    ],
+                                    "score": 1.0,
+                                    "content": "trained for 250 epochs using SGD with momentum 0.9, initial learning rate 0.1 multiplied by 0.97",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        336
+                                    ],
+                                    "score": 1.0,
+                                    "content": "every epoch. We use an auxiliary head located at 2/3 of the network weighted by 0.5. We use the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        347
+                                    ],
+                                    "score": 1.0,
+                                    "content": "same regularization techniques, and similarly accelerate training in a distributed fashion.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    "lines": [
+                        {
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "Anytime Following Huang et al. (2018), the final candidates are trained using SGD with",
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+                                    "score": 1.0,
+                                    "content": "rather than node activations. We form edge activations by concatenating the node activations of the",
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+                                    "content": "with a final dropout probability of 0.4. We use an auxiliary head located at 2/3 of the network",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "ImageNet Mobile For ImageNet mobile experiments, we use an image size of",
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diff --git a/parse/train/rkgW0oA9FX/rkgW0oA9FX_model.json b/parse/train/rkgW0oA9FX/rkgW0oA9FX_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..81983956dd3641e324754a22c8f1bb946074b40d
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