diff --git a/parse/train/9CPc4EIr2t1/9CPc4EIr2t1.md b/parse/train/9CPc4EIr2t1/9CPc4EIr2t1.md new file mode 100644 index 0000000000000000000000000000000000000000..cea7c99514097ade315ef775ed77c20336dc26c5 --- /dev/null +++ b/parse/train/9CPc4EIr2t1/9CPc4EIr2t1.md @@ -0,0 +1,294 @@ +# Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks + +# Qiyu Kang∗ + +Continental-NTU Corporate Lab Nanyang Technological University 50 Nanyang Avenue, 639798, Singapore kang0080@e.ntu.edu.sg + +Yang Song∗ School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, 639798, Singapore songy@ntu.edu.sg + +# Qinxu Ding + +School of Business +Singapore University of Social Sciences +463 Clementi Road, 599494, Singapore qinxuding@suss.edu.sg + +Wee Peng Tay School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, 639798, Singapore wptay@ntu.edu.sg + +# Abstract + +Deep neural networks (DNNs) are well-known to be vulnerable to adversarial attacks, where malicious human-imperceptible perturbations are included in the input to the deep network to fool it into making a wrong classification. Recent studies have demonstrated that neural Ordinary Differential Equations (ODEs) are intrinsically more robust against adversarial attacks compared to vanilla DNNs. In this work, we propose a stable neural ODE with Lyapunov-stable equilibrium points for defending against adversarial attacks (SODEF). By ensuring that the equilibrium points of the ODE solution used as part of SODEF is Lyapunov-stable, the ODE solution for an input with a small perturbation converges to the same solution as the unperturbed input. We provide theoretical results that give insights into the stability of SODEF as well as the choice of regularizers to ensure its stability. Our analysis suggests that our proposed regularizers force the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium points of the ODE. SODEF is compatible with many defense methods and can be applied to any neural network’s final regressor layer to enhance its stability against adversarial attacks. + +# 1 Introduction + +Although deep learning has found successful applications in many tasks such as image classification [1, 2], speech recognition [3], and natural language processing [4], the vulnerability of deep learning to adversarial attacks (e.g., see [5]) has limited its real-world applications due to performance and safety concerns in critical applications. Inputs corrupted with human-imperceptible perturbations can easily fool many vanilla deep neural networks (DNNs) into mis-classifying them and thus significantly impact their performance. + +Recent studies [6–8] have applied neural Ordinary Differential Equations (ODEs) [9] to defend against adversarial attacks. Some works like [6] have revealed interesting intrinsic properties of + +ODEs that make them more stable than conventional convolutional neural networks (CNNs). The paper [6] proposes a time-invariant steady neural ODE (TisODE) using the property that the integral curves from a ODE solution starting from different initial points (inputs) do not intersect and always preserve uniqueness in the solution function space. However, this does not guarantee that small perturbations of the initial point lead to small perturbations of the integral curve output at a later time $T$ . The authors thus proposed a regularizer to limit the evolution of the curves by forcing the integrand to be close to zero. However, neither the non-intersecting property nor the steady-state constraint used in TisODE can guarantee robustness against input perturbations since these constraints do not ensure that the inputs are within a neighborhood of Lyapunov-stable equilibrium points. An example is an ODE that serves as an identity mapping is not robust to input perturbations but satisfies all the constraints proposed in [6]. + +In this paper, our objective is to design a neural ODE such that the features extracted are within a neighborhood of the Lyapunov-stable equilibrium points of the ODE. We first develop a diversity promoting technique applied in the final fully connected (FC) layer to improve the ODE’s stability and analyze the reasons why. We then propose a stable neural ODE with Lyapunov-stable equilibrium points to eliminate the effects of perturbations in the input. From linear control theory [10], a linear time-invariant system $\mathrm { d } { \mathbf { z } ( t ) } / \mathrm { d } t = { \mathbf { A } } { \mathbf { z } ( t ) }$ , where $\mathbf { A }$ is a constant matrix, is exponentially stable if all eigenvalues of $\mathbf { A }$ have negative real parts. Specifically, we propose to force the Jacobian matrix of the ODE used in the neural ODE to have eigenvalues with negative real parts. Instead of directly imposing constraints on the eigenvalues of the matrix, which lead to high computational complexity when the Jacobian matrix is large, we instead add constraints to the matrix elements to implicitly force the real parts of its eigenvalues to be negative. + +Our main contributions are summarized as follows: + +1. Based on the concept of Lyapunov-stable equilibrium points, we propose a simple yet effective technique to improve the robustness of neural ODE networks by fixing the final FC layer to be a matrix whose rows have unit norm and such that the maximum cosine similarity between any two rows is minimized. Such a FC layer can be constructed off-line. +2. We propose a stable neural ODE for deFending against adversarial attacks (SODEF) to suppress the input perturbations. We derive an optimization formulation for SODEF to force the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium points of the SODEF ODE. We provide sufficient conditions for learning a robust feature representation under SODEF. +3. We test SODEF on several widely used datasets MNIST [11], CIFAR-10 and CIFAR-100 [12] under well-known adversarial attacks. We demonstrate that SODEF is robust against adversarial white-box attacks with improvement in classification accuracy of adversarial examples under PGD attack [13] of up to $4 4 . 0 2 \%$ , $5 2 . 5 4 \%$ and $1 8 . 9 1 \%$ percentage points compared to another current state-of-the-art neural ODE network TisODE [6] on MNIST, CIFAR-10 and CIFAR-100, respectively. Similar improvements in classification accuracy of adversarial examples of up to $\bar { 4 3 . 6 9 \% }$ , $5 2 . 3 8 \%$ and $\mathrm { \bar { 1 8 . 9 9 \% } }$ percentage points compared to ODE net [9] are also obtained. + +The rest of this paper is organized as follows. We provide essential preliminaries on neural ODE and its stability analysis in Section 2. In Section 3, we present SODEF model architecture and its training method. We show how to maximize the distance between stable equilibrium points of neural ODEs. We propose an optimization and present theoretical results on its stability properties. We summarize experimental results in Section 4 and conclude the paper in Section 5. The proofs for all lemmas and theorems proposed in this paper are given in the supplementary material. We also refer interested readers to the supplementary material for a more detailed account of related works [14–16, 6] and some popular adversarial attacks [17, 13] that are used to verify the robustness of our proposed SODEF. In the paper, we use lowercase boldface characters like $\mathbf { z }$ to denote vectors in $\mathbb { R } ^ { n }$ , capital boldface characters like $\mathbf { A }$ to denote matrices in $\mathbb { R } ^ { n \times n }$ , and normal characters like $z$ to denote scalars except that the notation $( x , y )$ are normal characters reserved to denote the input and label pairs. A vector $\mathbf { z } \in \mathbb { R } ^ { n }$ is represented as $( \mathbf { z } ^ { ( 1 ) } , \mathbf { z } ^ { ( 2 ) } , \ldots , \mathbf { z } ^ { ( n ) } )$ . The $( i , j )$ -th element of a matrix $\mathbf { A }$ is $\mathbf { A } _ { i j }$ or $[ \mathbf { A } ] _ { i j }$ . The Jacobian matrix of a function $f : \mathbb { R } ^ { n } \mapsto \mathbb { R } ^ { n }$ evaluated at $\mathbf { z }$ is denoted as $\nabla f ( \mathbf { z } )$ The set of functions $\mathbb { R } ^ { n } \mapsto \mathbb { R } ^ { n }$ with continuous first derivatives is denoted as $C ^ { 1 } ( \mathbb { R } ^ { n } , \mathbb { R } ^ { n } )$ . + +# 2 Preliminaries: Neural ODE and Stability + +In a neural ODE layer, the relation between the layer input ${ \bf z } ( 0 )$ and output ${ \mathbf z } ( T )$ is described as the following differential equation: + +$$ +\frac { \mathrm { d } \mathbf { z } ( t ) } { \mathrm { d } t } = f _ { \pmb { \theta } } ( \mathbf { z } ( t ) , t ) +$$ + +where $f _ { \pmb \theta } : \mathbb { R } ^ { n } \times [ 0 , \infty ) \mapsto \mathbb { R } ^ { n }$ denotes the non-linear trainable layers that are parameterized by weights $\pmb { \theta }$ and $\mathbf { z } : [ 0 , \infty ) \mapsto \mathbb { R } ^ { n }$ represents the $n$ -dimensional state of the neural ODE. Neural ODEs are the continuous analog of residual networks where the hidden layers of residual networks can be regarded as discrete-time difference equations ${ \bf z } ( t + 1 ) = { \bf z } ( t ) + { f _ { \theta } } ( { \bf z } ( t ) , t )$ . In this work, for simplicity, we only consider the time-invariant (autonomous) case $f _ { \pmb \theta } ( \mathbf { z } ( t ) , t ) = f _ { \pmb \theta } ( \mathbf { z } ( t ) )$ , where the dynamical system does not explicitly depend on $t$ . For such non-linear dynamical systems, the following theorem shows that under mild conditions, its behaviour can be studied via linearization near special points called hyperbolic equilibrium points. + +Theorem 1 (Hartman–Grobman Theorem [18]). Consider a system evolving in time with state $\mathbf { z } ( t ) \in \mathbb { R } ^ { n }$ that satisfies the differential equation $\frac { \mathrm { d } { \bf z } ( t ) } { \mathrm { d } t } = f ( { \bf z } ( t ) )$ for some $f \in C ^ { 1 } ( \mathbb { R } ^ { n } , \mathbb { R } ^ { n } ) ;$ , $f ( \mathbf { z } ) = ( f ^ { ( 1 ) } ( \mathbf { z } ) , \ldots , f ^ { ( n ) } ( \mathbf { z } ) )$ . Suppose the map has a hyperbolic equilibrium state $\mathbf { z } ^ { \ast } \in \mathbb { R } ^ { n }$ , i.e., $f ( { \bf z } ^ { * } ) = 0$ and the Jacobian matrix with real part equal to zer $\nabla f = [ \partial f ^ { ( i ) } / \partial \mathbf { z } ^ { ( j ) } ] _ { i , j = 1 } ^ { n }$ of hb $f$ evalurhood at of $\textbf { z } = \textbf { z } ^ { * }$ has nolibrium $N _ { \mathbf { z } ^ { * } }$ point $\mathbf { z } ^ { \ast }$ and a homeomorphism $g : N _ { \mathbf { z } ^ { * } } \mapsto \mathbb { R } ^ { n }$ , such that $g ( \mathbf { z } ^ { * } ) = 0$ and in the neighbourhood $N _ { \mathbf { z } ^ { * } }$ , the flow of $\frac { \mathrm { d } { \bf z } ( t ) } { \mathrm { d } t } = f ( { \bf z } ( t ) )$ is topologically conjugate by the continuous map $\bar { \mathbf { z } } ( t ) = g ( \mathbf { z } ( t ) )$ to the flow of its linearization $\frac { \mathrm { d } \bar { \mathbf { z } } ( t ) } { \mathrm { d } t } = \nabla f ( \mathbf { z } ^ { * } ) \cdot \bar { \mathbf { z } } ( t ) .$ + +The theorem states that when the Jacobian matrix at the zeros of $f$ has no eigenvalue with zero real part, the behaviour of the original dynamical system can be studied using the simpler linearization of the system around those zeros. We next review some definitions and theorems from linear control theory [10]. + +Definition 1 (Lyapunov Stability [10]). The linear time-invariant system $\frac { \mathrm { d } \bar { \mathbf { z } } ( t ) } { \mathrm { d } t } = \mathbf { A } \bar { \mathbf { z } } ( t )$ with constant matrix A is marginally stable or stable in the sense of Lyapunov if every finite initial state $\bar { \mathbf { z } } ( 0 )$ excites a bounded response. It is asymptotically stable if every finite initial state excites $a$ bounded response, which, in addition, approaches 0 as $t \to \infty$ . + +Theorem 2 (Lyapunov Stability Theorem [10]). a) The equation $\frac { \mathrm { d } \bar { \mathbf { z } } ( t ) } { \mathrm { d } t } = \mathbf { A } \bar { \mathbf { z } } ( t ) )$ is marginally stable if and only if all eigenvalues of A have zero or negative real parts and those with zero real parts are simple roots of the minimal polynomial of A. $^ b$ ) The equation $\frac { \mathrm { d } \bar { \mathbf { z } } ( t ) } { \mathrm { d } t } = \mathbf { A } \bar { \mathbf { z } } ( t )$ is asymptotically stable if and only if all eigenvalues of A have negative real parts. + +In Theorem 1, we say that a hyperbolic equilibrium point is Lyapunov-stable if all eigenvalues of the Jacobian matrix evaluated at it have negative real parts. From Theorems 1 and 2, we see that a small perturbation around the Lyapunov-stable equilibrium point ${ \bf z } ( 0 )$ leads to $\tilde { \mathbf { z } } ( t ) \to \mathbf { z } ( 0 )$ as $t \to \infty$ , i.e., $\exists \delta > 0$ such that for all $\tilde { \mathbf { z } } ( 0 )$ with $\lVert { \mathbf z } ( 0 ) \dot { - } \tilde { { \mathbf z } } ( 0 ) \rVert _ { 2 } \dot { < } \delta$ , we have $\| \widetilde { \mathbf { z } } ( t ) - \mathbf { z } ( 0 ) \| _ { 2 } \to \dot { 0 }$ as $t \to \infty$ , where $\tilde { \mathbf { z } } ( t )$ is the ODE solution for the perturbed input $\tilde { \mathbf { z } } ( 0 )$ . In the context of neural network adversarial attacks, if the malicious perturbations around the ODE input ${ \bf z } ( 0 )$ is small, then the output ${ \mathbf z } ( T )$ for large enough $T$ will not be affected significantly by the perturbation. Consequently, the succeeding network layers after the neural ODE layer can still perform well without being affected by the input perturbation. The perturbation weakening phenomenon around Lyapunov-stable equilibrium points works like a noise filter and acts as a defense against adversarial attacks. + +We require the following definition and result in our stability analysis. + +Definition 2 (Strictly diagonally dominant [19]). Let $\mathbf { A } \in \mathbb { C } ^ { n \times n }$ . We say that A is strictly diagonally dominant if $\begin{array} { r } { \left| { { { \mathbf { A } } _ { i i } } } \right| > \sum _ { j \neq i } \left| { { { \mathbf { A } } _ { i j } } } \right| } \end{array}$ for all $i = 1 , . . . , n$ . + +Theorem 3 (Levy–Desplanques theorem [19]). If $\mathbf { A } \in \mathbb { C } ^ { n \times n }$ is strictly diagonally dominant and if every main diagonal entry of A is real and negative, then $A$ is non-singular and every eigenvalue of A has negative real part. + +![](images/96641a4aad1ec2c9f3138777e3d67fc9ff9849146d1c1b15becd36b84d50c353.jpg) +Fig. 1: SODEF model architecture. + +Lemma 1. Given $k$ distinct points $\mathbf { z } _ { i } \in \mathbb { R } ^ { n }$ and matrices $\mathbf { A } _ { i } \in \mathbb { R } ^ { n \times n }$ , $i = 1 , . . . , k$ , there exists a function $f \in C ^ { 1 } ( \mathbb { R } ^ { n } , \mathbb { R } ^ { n } )$ such that $f ( \mathbf { z } _ { i } ) = 0$ and $\nabla f _ { \pmb { \theta } } ( \mathbf { z } _ { i } ) = \mathbf { A } _ { i }$ . + +# 3 SODEF Architecture + +We consider a classification problem with $L$ classes. The proposed SODEF model architecture is shown in Fig. 1. The input $x \in X$ (e.g., an image) is first passed through a feature extractor $h _ { \phi } : X \mapsto \mathbb { R } ^ { n }$ to obtain an embedding feature representation ${ \bf z } ( 0 )$ . A neural ODE layer $f _ { \theta }$ follows as a nonlinear feature mapping to stabilize the feature representation output ${ \bf z } ( 0 )$ from $h _ { \phi }$ . The final FC layer $\mathbf { V }$ serves as a linear mapping to generate a prediction vector based on the output $\mathbf { \dot { z } } ( T )$ of the neural ODE layer. The parameters $\phi , \theta$ and $\mathbf { V }$ are parameterized weights for the feature extractor, neural ODE layer and FC layer, respectively. + +We provide motivation and design guidance for the FC layer V in Section 3.1, which attempts to separate Lyapunov-stable equilibrium points implicitly by maximizing the similarity distance between feature representations corresponding to the $L$ different classes. Experimental results demonstrate the advantages of our diversity promoting FC layer in Section 3.1 with comparisons to traditional neural ODEs without diversity promoting. + +However, the embedded features after using diversity promoting are not guaranteed to locate near the Lyapunov-stable equilibrium points. In Section 3.2, we formulate an optimization problem to force embedding features to locate near the Lyapunov-stable equilibrium points. We introduce optimization constraints to force the Jacobian matrix of the ODE in our neural ODE layer to have eigenvalues with negative real parts at the Lyapunov-stable equilibrium points. Instead of directly imposing constraints on the eigenvalue of the matrix, which may be computationally complex especially when the matrix is large, we add constraints to the matrix elements instead. + +# 3.1 Maximizing the Distance between Lyapunov-Stable Equilibrium Points + +From Section 2, we observe that points in a small neighbourhood of a Lyapunov-stable equilibrium point is robust against adversarial perturbations. We call this neighborhood a stable neighborhood. However Lyapunov-stable equilibrium points for different classes may very well locate near each other and therefore each stable neighborhood may be very small, leading to poor adversarial defense. In this section, we propose to add a FC layer after the neural ODE layer given by (1) to avoid this scenario. The purpose of the FC layer is to map the output of the neural ODE layer to a feature vector $\mathbf { v } _ { l }$ if the input $x$ belongs to the class $l = 1 , \ldots , L$ . We design the FC layer so that the cosine similarities between different $\mathbf { v } _ { l }$ ’s are minimized. + +Lemma 2. Given a set of $k$ unit vectors $\mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { k }$ in $\mathbb { R } ^ { n }$ , where $n \geq k$ , let $a ( \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { k } ) =$ $\mathrm { m a x } _ { i \neq j } { \mathbf { \Delta v } _ { i } ^ { \mathsf { T } } } { \mathbf { v } _ { j } }$ . Then $\operatorname* { m i n } a ( \mathbf { v } _ { 1 } , . . . , \mathbf { v } _ { k } ) = 1 / ( 1 - k )$ , where the minimum is taken over all possible sets of $k$ unit vectors $\mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { k }$ . + +Corollary 1. Consider a $k \times k$ matrix $\mathbf { B } = [ b _ { i j } ] _ { i , j = 1 } ^ { k }$ with $b _ { i i } = 1$ and $b _ { i j } = 1 / ( 1 - k )$ , $\forall i \ne j$ Let the eigen decomposition of $\mathbf { B }$ be $\mathbf { B } = \mathbf { U } \pmb { \Sigma } \bar { \mathbf { U } } ^ { \dag }$ . For any $n \geq k$ and $i = 1 , \ldots , k$ , let $\mathbf { v } _ { i }$ be the $i$ -th column of $\mathbf { Q } \mathbf { \Sigma } ^ { \mathrm { { X } ^ { 1 / 2 } \bar { U } ^ { \top } } }$ , where $\mathbf { Q }$ is any $n \times k$ matrix such that $\mathbf { Q } ^ { \intercal } \mathbf { Q } = \mathbf { I } _ { k }$ . Then, $a ( \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { k } ) = \operatorname* { m a x } _ { i \neq j } \mathbf { v } _ { i } ^ { \mathsf { T } } \mathbf { v } _ { j } = { 1 } / ( 1 - k )$ . + +Corollary 1 suggests a diversity promoting scheme to maximally separate the equilibrium points of the neural ODE layer. The FC layer is represented by an $n \times L$ matrix $\mathbf { V } = \left[ \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { L } \right]$ , where $n$ is the dimension of ${ \mathbf z } ( T )$ , the output from the neural ODE layer. If ${ \mathbf z } ( T )$ is generated from an input from class $l$ , it is mapped to $\mathbf { v } _ { l }$ . By minimizing the maximum cosine similarity $a ( \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { k } ) = \operatorname* { m a x } _ { i \neq j } \mathbf { v } _ { i } ^ { \mathsf { T } } \mathbf { v } _ { j }$ between the representations from two different classes, we ensure that the output of SODEF is robust to perturbations in the input. Corollary 1 provides a way to choose the FC layer weights $\mathbf { V }$ . + +To validate our observations, we conduct experiments to compare the robustness of ODE net [9] and TisODE [6] with and without our proposed FC layer V, on two standard datasets: MNIST [2] and CIFAR10 [12] 2. On the MNIST dataset, all models consist of four convolutional layers and one fully-connected layer. On the CIFAR10 dataset, the networks are similar to those for MNIST except the down-sampling network is a stack of 2 ResNet blocks. In practice, the neural ODE can be solved with different numerical solvers such as the Euler method and the Runge-Kutta methods [9]. Here, we use Runge-Kutta of order 5 in our experiments. Our implementation builds on the open-source neural ODE codes.3 During training, no Gaussian noise or adversarial examples are augmented into the training set. We test the performance of our model in defending against white-box attacks FGSM [17] and PGD [13] . The parameters for different attack methods used in this paper are given in the supplementary material. From Tables 1 and 2, we observe that for both datasets, our fixed FC layer improves each network’s defense ability by a significant margin. We visualize the features before the final FC layer using t-SNE [20] in Figs. 2 and 3. We observe that with the FC layer, the features for different classes are well separated even under attacks. + +Table 1: Classification accuracy $( \% )$ on adversarial MNIST examples, where the superscript + indicates the last FC layer is fixed to be $\mathbf { V }$ . + +
AttackPara.ODEODE+TisODETisODE+
None-99.699.799.599.7
FGSM∈ = 0.331.452.845.963.5
PGD∈=0.30.290.300.420.20
+ +Table 2: Classification accuracy $( \% )$ on adversarial CIFAR10 examples, where the superscript + indicates the last FC layer is fixed to be $\mathbf { V }$ . + +
AttackPara.ODE ODE+TisODETisODE+
None-87.0 85.087.481.8
FGSM∈ = 0.112.9 47.613.141.9
PGD∈=0.17.8 14.77.416.2
+ +![](images/2298e75b6e25d3394e76c905205ab4b066d9a6a5a3e690669fcd410610b0c475.jpg) +Fig. 2: t-SNE visualization results on the features before the final FC layer. The input is the test set of MNIST. Left: trained with TisODE, middle: TisODE using a randomly chosen orthogonal matrix as the final FC, right: TisODE using proposed $\mathbf { V }$ as the final FC. + +# 3.2 Objective Formulation and Stability + +In this subsection, we formulate an optimization framework for SODEF to force output features to locate within the stable neighborhood of Lyapunov-stable equilibrium points. We make the following assumption. + +Assumption 1. The input x takes values in a compact metric space $X$ and has probability distribution $\mu$ . The feature extractor $h _ { \phi }$ is injective and continuous. + +![](images/7a95cfded3451c7f092fca178fd0f289e9f1e03180a9b9200b577ccb2fbf9ae9.jpg) +Fig. 3: t-SNE visualization results on the features before the final FC layer. The input is the adversarial examples of the test set of MNIST generated using FGSM method at $\epsilon = 0 . 3$ . Left: trained with TisODE, middle: TisODE using a randomly chosen orthogonal matrix as the final FC, right: TisODE using proposed $\mathbf { V }$ as the final FC. + +The above assumption is satisfied if the input $x$ (e.g., an image) resides in a bounded and closed set of a Euclidean space. We denote the pushforward measure (still a probability distribution) of $\mu$ under the continuous feature extractor mapping $h _ { \phi }$ as $\nu _ { \phi } = \mu \circ h _ { \phi } ^ { - 1 }$ , where $\circ$ denotes function composition. The conditional probability distribution for the embedding of each class $l \in \{ 1 , . . . , L \}$ has compact support $E _ { l } \subset \mathbb { R } ^ { n }$ since $E _ { l }$ is closed and $h _ { \phi } ( X )$ is bounded in $\mathbb { R } ^ { n }$ . In Section 3.1, the FC layer $\mathbf { V }$ tries to maximize the distance between $E _ { l }$ , $l = 1 , \ldots , L$ . In this section for analysis purposes, we also assume the following. + +Assumption 2. We have $E _ { l } \bigcap E _ { l ^ { \prime } } = \varnothing i f l \neq l ^ { \prime }$ , i.e., the supports of each class are pairwise disjoint. + +Our objective function is formulated as follows, which is explained in detail in the sequel: + +$$ +\begin{array} { r l } & { \underset { \theta , \phi } { \operatorname* { m i n } } \mathbb { E } _ { \boldsymbol { \mu } } \ell ( { \mathbf { V } } ^ { \top } ( { \mathbf { z } } ( T ) ) , y _ { i } ) } \\ & { \mathrm { s . t . } \ \mathbb { E } _ { \boldsymbol { \nu } _ { \phi } } \| f _ { \theta } ( { \mathbf { z } } ( 0 ) ) \| _ { 2 } < \epsilon , \ f _ { \theta } \in C ^ { 1 } ( \mathbb { R } ^ { n } , \mathbb { R } ^ { n } ) , } \\ & { \quad \quad \quad \mathbb { E } _ { \boldsymbol { \nu } _ { \phi } } \left[ \nabla f _ { \theta } ( { \mathbf { z } } ( 0 ) ) \right] _ { i i } < 0 , \ \forall i = 1 , \ldots , n , } \\ & { \quad \quad \mathbb { E } _ { \boldsymbol { \nu } _ { \phi } } \left[ | [ \nabla f _ { \theta } ( { \mathbf { z } } ( 0 ) ) ] _ { i i } | - \sum _ { j \neq i } | [ \nabla f _ { \theta } ( { \mathbf { z } } ( 0 ) ) ] _ { i j } | \right] > 0 , \ \forall i = 1 , \ldots , n , } \end{array} +$$ + +$\mathbf { z } ( 0 ) = h _ { \phi } ( x )$ , and ${ \mathbf z } ( T )$ is the output of (1) with input ${ \bf z } ( 0 )$ . + +Here, $\ell$ is a loss function and $\epsilon > 0$ is a positive constant. The constraints (3) to (5) force ${ \bf z } ( 0 )$ to be near the Lyapunov-stable equilibrium points with strictly diagonally dominant derivatives. We limit the $f _ { \theta }$ to be in $C ^ { 1 } ( \mathbb { R } ^ { n } , \mathbb { R } ^ { n } )$ to satisfy the condition in Theorem 1. From [21], we also know that standard multi-layer feed forward networks with as few as a single hidden layer and arbitrary bounded and non-constant activation function are universal approximators for $C ^ { 1 ^ { \prime } } ( \mathbb { R } ^ { n } , \mathbb { R } ^ { n } )$ functions with respect to some performance criteria provided only that sufficiently many hidden units are available. + +As a comparison, TisODE [6] only includes a constraint similar to (3), which in general provides no guarantee to force ${ \bf z } ( 0 )$ near the Lyapunov-stable equilibrium points. In the extreme case with parameters $\theta = 0$ for $f _ { \theta }$ such that $f _ { \pmb { \theta } } = 0$ , the ODE degenerates to an identity mapping. No $\mathbf { z } ( 0 ) \in \mathbb { R } ^ { n }$ can now be a Lyapunov-stable equilibrium point, and no stability can therefore be guaranteed to defend against adversarial attacks even though the ODE curves still possess the nonintersecting property and steady-state constraint, which were cited as reasons for the stability of TisODE. + +Instead of directly optimizing the above objective function, in our implementation, we optimize the following empirical Lagrangian with a training set $\left\{ \left( x _ { k } , y _ { k } \right) : k = 1 , { \overset { - } { \ldots } } , N \right\}$ : + +$$ +\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { \theta , \phi } \frac { 1 } { N } \sum _ { k = 0 } ^ { N - 1 } \bigg ( \ell \big ( \mathbf { V } ^ { \top } \mathbf { z } _ { k } ( T ) , y _ { k } \big ) + \alpha _ { 1 } \| f _ { \theta } \big ( \mathbf { z } _ { k } ( 0 ) \big ) \| _ { 2 } + \alpha _ { 2 } g _ { 1 } \Big ( \displaystyle \sum _ { i = 1 } ^ { n } [ \nabla f _ { \theta } ( \mathbf { z } _ { k } ( 0 ) ) ] _ { i i } \Big ) } \\ { \displaystyle \quad \quad + \alpha _ { 3 } g _ { 2 } \Big ( \displaystyle \sum _ { i = 1 } ^ { n } ( - | [ \nabla f _ { \theta } ( \mathbf { z } _ { k } ( 0 ) ) ] _ { i i } | + \displaystyle \sum _ { j \neq i } \vert [ \nabla f _ { \theta } ( \mathbf { z } _ { k } ( 0 ) ) ] _ { i j } | ) \Big ) \bigg ) } \end{array} +$$ + +s. t. $\mathbf { z } _ { k } ( 0 ) = h _ { \phi } ( x _ { k } )$ , and ${ \mathbf z } _ { k } ( T )$ is the output of (1) with input $\mathbf { z } _ { k } ( 0 ) , \forall k = 1 , . . . , N$ (8) + +where $\alpha _ { 1 } , \alpha _ { 2 }$ and $\alpha _ { 3 }$ are hyperparameter weights, $g _ { 1 }$ and $g _ { 2 }$ are chosen monotonically increasing functions bounded below to eliminate the unbounded impact of the two regularizers that can otherwise dominate the loss. In this paper, we set $g _ { 1 } ( \cdot ) = g _ { 2 } ( \cdot ) \bar { = } \exp ( \cdot )$ . We call these two latter terms the SODEF regularizers. + +Suppose for each class $l = 1 , \ldots , L$ , the embedding feature set $E _ { l } = \{ \mathbf { z } _ { 1 } ^ { ( l ) } , \dots , \mathbf { z } _ { k } ^ { ( l ) } \}$ z(l)k } is finite. For each $i = 1 , \ldots , k$ , let $\mathbf { A } _ { i } \in \mathbb { R } ^ { n \times n }$ be strictly diagonally dominant matrix with every main diagonal entry be negative such that the eigenvalues for $\mathbf { A } _ { i }$ all have negative real part. From Theorem 3, each $\mathbf { A } _ { i }$ is non-singular and every eigenvalue of $\mathbf { A } _ { i }$ has negative real part. Therefore, from Theorem 2 and Lemma 1, there exists a function $f _ { \theta }$ such that all $\mathbf { z } _ { i } ^ { ( l ) }$ are Lyapunov-stable equilibrium points with corresponding first derivative $\nabla f _ { \pmb \theta } ( \mathbf z _ { i } ^ { ( l ) } ) = \mathbf { A } _ { i }$ . This shows that if there exist only finite representation points for each class, we can find a function $f _ { \theta }$ such that all inputs to the neural ODE layer are Lyapunov-stable equilibrium points for $f _ { \theta }$ and + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \boldsymbol \nu _ { \phi } } \left\| f _ { \boldsymbol \theta } ( \mathbf { z } ( 0 ) ) \right\| _ { 2 } = 0 , } \\ & { \mathbb { E } _ { \boldsymbol \nu _ { \phi } } \left[ \nabla f _ { \boldsymbol \theta } ( \mathbf { z } ( 0 ) ) \right] _ { i i } < 0 , \forall i = 1 , \ldots , n , } \\ & { \mathbb { E } _ { \boldsymbol \nu _ { \phi } } \left[ | [ \nabla f _ { \boldsymbol \theta } ( \mathbf { z } ( 0 ) ) ] _ { i i } | - \sum _ { j \neq i } \vert [ \nabla f _ { \boldsymbol \theta } ( \mathbf { z } ( 0 ) ) ] _ { i j } \vert \right] > 0 , \forall i = 1 , \ldots , n . } \end{array} +$$ + +If the input space $X$ has infinite cardinality, then an injective and continuous feature extractor $h _ { \phi }$ results in a $\nu _ { \phi }$ with non-finite support, i.e., at least one $E _ { l }$ , $l = 1 , \ldots , L$ , is infinite. It is not obvious whether we can obtain a $f _ { \theta }$ where every point in $\textstyle E = \bigcup _ { l } E _ { l }$ is a stable equilibrium point. The following result gives a negative answer if $\nu _ { \phi }$ is a continuous measure (i.e., absolutely continuous with respect to (w.r.t.) Lebesgue measure) on some subset. + +Lemma 3. If the restriction of $\nu _ { \phi }$ to some open set $E ^ { \prime } \subset E$ is a continuous measure, there is no continuous function fθ such that for $\nu _ { \phi }$ -almost surely all $\mathbf { z } \in E$ , $f _ { \pmb \theta } ( \mathbf z ) = 0$ and all the eigenvalues of $\nabla f _ { \boldsymbol { \theta } } ( \mathbf { z } )$ have negative real parts. In other words, there is no continuous function fθ such that almost surely all $\mathbf { z }$ in $E$ are Lyapunov-stable equilibrium points. + +Lemma 3 indicates that it is too much to hope for all points in $E$ to be Lyapunov-stable equilibrium points. In the following, we relax this requirement and show that under mild conditions, for all $\epsilon > 0$ , we can find a continuous function $f _ { \theta }$ with finitely many stable equilibrium points such that conditions (b) and (c) above hold and condition (a) is replaced by $\mathbb { E } _ { \boldsymbol { \nu } _ { \phi } } \bar { \| } f _ { \theta } ( \mathbf { z } ( 0 ) ) \| _ { 2 } ^ { - } < \epsilon$ . This motivates the optimization constraints in (3) to (5). + +Theorem 4. Suppose Assumptions $I$ and 2. If $\nu _ { \phi }$ is not a continuous uniform measure on $E _ { l }$ for each $l = 1 , \ldots , L$ , then the following holds: 1) The function space satisfying the constraints in (3) to (5) is non-empty for all $\epsilon > 0$ . 2) If additionally the restriction of $\nu _ { \phi }$ to any open set $O \subset E _ { l }$ is not a continuous uniform measure, there exist functions in this space such that each support $E _ { l }$ contains at least one Lyapunov-stable equilibrium point. + +# 4 Experiments + +In this section, we evaluate the robustness of SODEF under adversarial attacks with different attack parameters. We conduct experiments to compare the robustness of ODE net [9] and TisODE net [6] on three standard datasets: MNIST [2], CIFAR10 and CIFAR100 [3]. Since SODEF is compatible with many defense methods, it can be applied to any neural network’s final regressor layer to enhance its stability against adversarial attacks. Our experiment codes are provided in https://github.com/KANGQIYU/SODEF. + +# 4.1 Setup + +We use open-source pre-trained models that achieve the top accuracy on each dataset as the feature extractor $h _ { \phi }$ . Specifically for simple MNIST task, we use the ResNet18 model provided in Pytorch. We use the model provided by [22], which obtains nearly $8 8 \%$ clean accuracy on CIFAR100 using EfficientNet [23] and the model provided by [24], which has nearly $9 5 \%$ clean accuracy on CIFAR10. In the neural ODE layer, $f _ { \theta }$ consists of 2 FC layers. During the trainings of SODEF (except in the experiment included in Section 4.2), we train the neural network with the fixed FC introduced in Section 3.1. In the first 30 epochs, we fixed $f _ { \theta }$ to let the feature extractor $h _ { \phi }$ learn a feature representation with only the cross-entropy loss $\ell$ , and in the remaining 120 epochs, we release $h _ { \phi }$ to further train $f _ { \theta }$ using (7) with $\alpha _ { 1 } = 1$ and $\alpha _ { 2 } = \alpha _ { 3 } = 0 . 0 5$ . For CIFAR10 and CIFAR100, the pixel values are normalized by $( x - \mu ) / \sigma$ where $\mu = [ 0 . 4 9 1 4 , 0 . 4 8 2 2 , 0 . 4 4 6 5 ]$ and $\sigma = [ 0 . 2 0 2 3 , 0 . 1 9 9 4 , 0 . 2 0 1 0 ] ^ { 4 }$ . To show that our SODEF is compatible with many defense methods and can be applied to any neural network’s final regression layer, we conduct an experiment where we use a recently proposed robust network TRADES [25] as the feature extractor in our SODEF. The pretrained model is provided here 5, and we choose the model with architecture "WRN- $3 4 \mathrm { - } 1 0 "$ to conduct our experiments. Besides the two vanilla white-box attacks FGSM and PGD as metioned in Section 3.1, we also include a strong ensemble attack AutoAttack [26], which sequentially performs attack using all of the following four individual attacks: three white-box attacks APGDCE, APGDTDLR and FABT[27], and one black-box Square attack [28]. We refer the reader to the the supplementary material for more details of the attacks used in this paper, where, in additional, more experiments are included. + +# 4.2 Compatibility of SODEF + +Adversarial training (AT) is one of the most effective strategies for defending adversarial attacks. TRADES [25] is one of the adversarial training defense methods with combinations of tricks of warmup, early stopping, weight decay, batch size and other hyper parameter settings. In this experiment we fix the pretained TRADES model (except the final FC layer (size $6 4 0 \mathrm { x } 1 0 )$ ) as our feature extractor $h _ { \phi }$ . We then append our (trainable) SODEF with integration time $T = 5$ to the output of the feature extractor. To evaluate model robustness, we use AutoAttack and attack the models using both the $\mathcal { L } _ { 2 }$ norm $\epsilon = 0 . 5$ ) and $\mathcal { L } _ { \infty }$ norm $( \epsilon = 8 / 2 5 5 )$ ). The results are shown in Table 3. We clearly observe that our SODEF can enhance TRADES’s robustness under all the four individual attacks and the strongest ensemble AutoAttack. For the strong $\mathcal { L } _ { 2 }$ AutoAttack, our SODEF have improved the model robustness from $5 9 . 4 2 \%$ to $6 7 . 7 5 \%$ . Our experiment show that SODEF can be applied to many defense models’ regression layer to enhance their stability against attacks. + +Table 3: Classification accuracy $( \% )$ using TRADES (w/ and w/o SODEF) under AutoAttack on adversarial CIFAR10 examples with $\mathcal { L } _ { 2 }$ norm $\epsilon = 0 . 5 )$ and $\mathcal { L } _ { \infty }$ norm $( \epsilon = 8 / 2 5 5 )$ . + +
Attack /ModelTRADES LTRADES+SODEF LTRADES L2TRADES+SODEF L2
Clean85.4885.1885.48 85.18
APGDcE56.0870.9061.74 74.35
APGDDLR53.7064.1559.22 68.55
FABT54.1882.9260.31 83.15
Square59.1262.2172.65 76.02
AutoAttack53.6957.7659.42 67.75
+ +# 4.3 Influence of Integration Time $T$ + +From the discussion after Theorems 1 and 2, we know if the malicious perturbations around the ODE input Lyapunov-stable equilibrium point ${ \bf z } ( 0 )$ is small, then the output ${ \mathbf z } ( T )$ for large enough $T$ will not be affected significantly by the perturbation: $\| \tilde { { \mathbf z } } ( t ) - { \mathbf z } ( 0 ) \| _ { 2 } \dot { \to } 0$ as $t \to \infty$ . Consequently, the succeeding network layers after the neural ODE layer can still perform well without being affected by the input perturbation. In this section, we test the influence of the SODEF integration time $T$ using CIFAR100. We use the model EfficientNet provided by [23] as $h _ { \phi }$ (Note, unlike Section 4.2, $h _ { \phi }$ is trainable in this experiments). We use AutoAttack with $\mathcal { L } _ { 2 }$ norm $\epsilon = 0 . 5$ ). We observe that for all the four individual attacks and the strongest ensemble AutoAttack, SODEF performs generally better for large integration time $T$ . We also test larger integration time $T > 1 0$ , but do not see any obvious improvements. + +# 4.4 Performance Comparison Under AutoAttack + +For a comparison, we provide the results of applying AutoAttack to other baseline models mentioned in the paper. We set the same integration time for ODE, TisODE and SODEF. We observe that for the + +Table 4: Classification accuracy $( \% )$ under AutoAttack on adversarial CIFAR100 examples with $\mathcal { L } _ { 2 }$ norm, $\epsilon = 0 . 5$ and different integration time $T$ for SODEF. + +
Attack/T135678910
Clean88.0088.1288.1588.0087.9288.0088.0588.10
APGDcE17.2021.3321.0523.6769.6785.3387.1086.88
APGDLR21.0221.0022.0026.0063.3086.9086.2086.54
FABT86.3385.1086.3687.7087.6786.5586.2285.93
Square84.6786.2287.0587.2086.9086.3387.0586.75
AutoAttack2.003.534.874.3330.6678.8078.9779.10
+ +strongest AutoAttack, our SODEF outperforms the other baseline models by a significant margin. In this case, SODEF achieves $7 9 . 1 0 \%$ accuracy while other models only get less than $3 \%$ accuracy. + +Table 5: Classification accuracy $( \% )$ under AutoAttack on adversarial CIFAR100 examples with $\mathcal { L } _ { 2 }$ norm, $\epsilon = 0 . 5$ and $T = 1 0$ . + +
Attack/ModelNoODEODETisODESODEF
Clean88.0087.9088.0088.10
APGDCE23.306.7514.3286.88
7.3322.0024.2086.54
FAB79.3078.6777.1685.93
Square84.5285.6786.3286.75
AutoAttack0.001.334.0679.10
+ +# 4.5 Performance Under PGD and FGSM Attacks + +White-box adversaries have knowledge of the classifier models, including training data, model architectures and parameters. We test the performance of our model in defending against the whitebox attacks, PGD and FGSM. We set $T = 5$ as the integration time for the neural ODE layer. The parameters for different attack methods used are given in the supplementary material. The subsequent experiments use these settings by default, unless otherwise stated. + +Table 6: Classification accuracy $( \% )$ on adversarial MNIST examples. + +
AttackPara.no odeODETisODESODEF
None-99.4599.4299.4399.44
FGSM∈=0.310.0329.636.7063.36
PGD∈= 0.30.311.561.8245.25
+ +The classification results on MNIST are shown in Table 6. We observe that while maintaining the state-of-the-art accuracy on normal images, SODEF improves the adversarial robustness as compared to the other two methods. For the most effective attack in this experiment, i.e., PGD attack, SODEF shows a $4 5 . 2 5 \% - 1 . 5 6 \% = 4 3 . 6 9 \%$ improvement over ODE and a $4 5 . 2 5 \% - 1 . 2 3 \% = 4 4 . 0 2 \%$ improvement over TisODE. + +Table 7: Classification accuracy $( \% )$ on adversarial CIFAR10 examples. + +
AttackPara.no odeODETisODESODEF
None-95.294.995.195.0
FGSM∈=0.147.3145.2343.2868.05
PGD∈=0.13.093.213.8055.59
+ +For CIFAR-10, we see from Table 7 that SODEF maintains high accuracy on normal examples and makes the best predictions under adversarial attacks. In particular, SODEF achieves an absolute percentage point improvement over ODE net up to $5 2 . 3 8 \%$ and over TisODE up to $5 2 . 5 4 \%$ for PGD attack. + +For CIFAR-100, the results in the supplementary material shows that the most effective attack causes the classification accuracy to drop relatively by $\begin{array} { r } { 7 4 . 6 \% = \frac { 8 8 . 0 - 2 2 . 3 5 } { 8 8 . 0 } } \end{array}$ 88.0−22.35 for SODEF and by $\begin{array} { r } { 9 7 . 3 \% = \frac { 8 8 . 3 - 2 . 3 9 } { 8 8 . 3 } } \end{array}$ for vanilla EfficientNet, which is pre-trained on ImageNet to obtain a top clean accuracy. Neither ODE net nor TisODE net can improve the classification accuracy under PGD attack by a big margin, e.g. TisODE net only improves the classification accuracy from $2 . 3 9 \%$ to $3 . 4 4 \%$ , while SODEF still shows clear defense capability in this scenario. + +# 4.6 Ablation Studies + +The impact of the ODE with and without the SODEF regularizers in (7) has been presented in the above comparisons between SODEF and ODE. In this section, we show the necessity of diversity promoting using the FC introduced in Section 3.1 and conduct transferability study. + +# 4.6.1 Impact of Diversity Promotion + +Table 8: Classification accuracy $( \% )$ on adversarial MNIST examples, where the superscript indicates the last FC layer is not fixed to be $\mathbf { V }$ and is set to be a trainable layer. + +
AttackPara.SODEFSODEF-
None-95.095.1
FGSM∈=0.163.3651.6
PGD∈=0.145.2534.9
+ +Table 8 shows the difference of the defense performance when fixing the final FC be $\mathbf { V }$ or setting it to a trainable linear layer. It can be seen that having diversity control improves the robustness. One possible reason for this phenomenon given in Section 3 is that diversity promotion with a fixed designed FC attempts to make the embedding feature support $E _ { l }$ of each class $l$ disjoint to each other and therefore the Lyapunov-stable equilibrium points for each $E _ { l }$ are well separated. + +# 4.6.2 Transferability Study + +Transferability study is carried out on CIFAR-10, where the adversarial examples are generated using FGSM and PGD attacks using ResNet18 without any ODEs. The classification accuracy drops from $6 8 . 0 5 \%$ to $5 9 \%$ for FGSM with $\epsilon = 0 . 3$ , and from $5 5 . 5 9 \%$ to $3 4 \%$ for PGD with $\epsilon = 0 . 1$ . One possible reason for this phenomenon is that ODEs have obfuscated gradient masking effect as discussed in [15], and a transfer attack may deteriorate the defense effect. However, as we observe from Table 7, even with a transfer attack on SODEF, it still performs better than other ODEs without transfer attacks. + +# 5 Conclusion + +In this paper, we have developed a new neural ODE network, SODEF, to suppress input perturbations. SODEF is compatible with any existing neural networks and can thus be appended to the state-of-theart networks to increase their robustness to adversarial attacks. We demonstrated empirically and theoretically that the robustness of SODEF mainly derives from its stability and proposed a training method that imposes constraints to ensure all eigenvalues of the Jacobian matrix of the neural ODE layer have negative real parts. When each classification class converges to its own equilibrium points, we showed that the last FC layer can be designed in such a way that the distance between the stable equilibrium points is maximized, which further improves the network’s robustness. The effectiveness of SODEF has been verified under several popular while-box attacks. + +# Acknowledgments and Disclosure of Funding + +This research is supported in part by A\*STAR under its RIE2020 Advanced Manufacturing and Engineering (AME) Industry Alignment Fund – Pre Positioning (IAF-PP) (Grant No. A19D6a0053) and the RIE2020 Industry Alignment Fund – Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s). The computational work for this article was partially performed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg). + +# Broader Impact + +Our work, which contributes to more robust DNNs, is supposed to mitigate the threat of adversarial attacks. However, on the hand, the reliable deployment of DNNs in automation of tasks will potentially bring mass-scale unemployment and social unrest. As DNNs become more robust and more tasks, especially those whose failures will bring high risks to human lives or large property losses under adversarial attacks, fall into the automatic task category, massive jobs could disappear. + +# References + +[1] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet classification with deep convolutional neural networks,” in Proc. Advances Neural Inf. Process. Syst., 2012. +[2] Y. Lecun, L. Bottou, Y. Bengio, and P. 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Hein, “Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks,” in Proc. Int. Conf. Mach. Learning, 2020, pp. 2206–2216. +[27] ——, “Minimally distorted adversarial examples with a fast adaptive boundary attack,” in Proc. Int. Conf. Mach. Learning, 2020, pp. 2196–2205. +[28] M. Andriushchenko, F. Croce, N. Flammarion, and M. Hein, “Square attack: a query-efficient black-box adversarial attack via random search,” in Proc. European Conf. Comput. Vision. Springer, 2020, pp. 484–501. + +# 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] +(c) Did you discuss any potential negative societal impacts of your work? [Yes] +(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] . See Assumption 1 and Assumption 2 +(b) Did you include complete proofs of all theoretical results? [Yes] . See supplementary material. + +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] . +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] . See Sections 3.1 and 4 +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Having ODE blocks in our model, it would be too computationally expensive to repeat experiments for many times. We repeated each experiment for 2-3 times and we observe the deviation of the classification results is within $\pm 3 \%$ , though these experimental repetitions are not enough to construct error bars. +(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 Section 3.1. + +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] See Sections 3.1 and 4 for the open-source models we have used from GitHub. +(b) Did you mention the license of the assets? [No] . Please see the licenses given in the GitHub link. +(c) Did you include any new assets either in the supplemental material or as a URL? [No] No new assets. +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] MNIST, CIFAR-10 and CIFAR-100 are all open-source datasets. +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] There is no identifiable information or offensive content in the datasets. + +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/9CPc4EIr2t1/9CPc4EIr2t1_content_list.json b/parse/train/9CPc4EIr2t1/9CPc4EIr2t1_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b47b7b751b343c65bb8d1b5aeb4be05d8eb77c10 --- /dev/null +++ b/parse/train/9CPc4EIr2t1/9CPc4EIr2t1_content_list.json @@ -0,0 +1,1372 @@ +[ + { + "type": "text", + "text": "Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks ", + "text_level": 1, + "bbox": [ + 245, + 122, + 753, + 196 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Qiyu Kang∗ ", + "text_level": 1, + "bbox": [ + 289, + 247, + 372, + 261 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Continental-NTU Corporate Lab Nanyang Technological University 50 Nanyang Avenue, 639798, Singapore kang0080@e.ntu.edu.sg ", + "bbox": [ + 196, + 262, + 460, + 316 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yang Song∗ School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, 639798, Singapore songy@ntu.edu.sg ", + "bbox": [ + 488, + 247, + 803, + 316 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Qinxu Ding ", + "text_level": 1, + "bbox": [ + 287, + 338, + 369, + 352 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "School of Business \nSingapore University of Social Sciences \n463 Clementi Road, 599494, Singapore qinxuding@suss.edu.sg ", + "bbox": [ + 196, + 353, + 459, + 407 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Wee Peng Tay School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, 639798, Singapore wptay@ntu.edu.sg ", + "bbox": [ + 488, + 338, + 802, + 407 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 443, + 535, + 459 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks (DNNs) are well-known to be vulnerable to adversarial attacks, where malicious human-imperceptible perturbations are included in the input to the deep network to fool it into making a wrong classification. Recent studies have demonstrated that neural Ordinary Differential Equations (ODEs) are intrinsically more robust against adversarial attacks compared to vanilla DNNs. In this work, we propose a stable neural ODE with Lyapunov-stable equilibrium points for defending against adversarial attacks (SODEF). By ensuring that the equilibrium points of the ODE solution used as part of SODEF is Lyapunov-stable, the ODE solution for an input with a small perturbation converges to the same solution as the unperturbed input. We provide theoretical results that give insights into the stability of SODEF as well as the choice of regularizers to ensure its stability. Our analysis suggests that our proposed regularizers force the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium points of the ODE. SODEF is compatible with many defense methods and can be applied to any neural network’s final regressor layer to enhance its stability against adversarial attacks. ", + "bbox": [ + 233, + 476, + 766, + 695 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 724, + 310, + 741 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Although deep learning has found successful applications in many tasks such as image classification [1, 2], speech recognition [3], and natural language processing [4], the vulnerability of deep learning to adversarial attacks (e.g., see [5]) has limited its real-world applications due to performance and safety concerns in critical applications. Inputs corrupted with human-imperceptible perturbations can easily fool many vanilla deep neural networks (DNNs) into mis-classifying them and thus significantly impact their performance. ", + "bbox": [ + 174, + 756, + 825, + 839 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent studies [6–8] have applied neural Ordinary Differential Equations (ODEs) [9] to defend against adversarial attacks. Some works like [6] have revealed interesting intrinsic properties of ", + "bbox": [ + 176, + 845, + 821, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ODEs that make them more stable than conventional convolutional neural networks (CNNs). The paper [6] proposes a time-invariant steady neural ODE (TisODE) using the property that the integral curves from a ODE solution starting from different initial points (inputs) do not intersect and always preserve uniqueness in the solution function space. However, this does not guarantee that small perturbations of the initial point lead to small perturbations of the integral curve output at a later time $T$ . The authors thus proposed a regularizer to limit the evolution of the curves by forcing the integrand to be close to zero. However, neither the non-intersecting property nor the steady-state constraint used in TisODE can guarantee robustness against input perturbations since these constraints do not ensure that the inputs are within a neighborhood of Lyapunov-stable equilibrium points. An example is an ODE that serves as an identity mapping is not robust to input perturbations but satisfies all the constraints proposed in [6]. ", + "bbox": [ + 174, + 90, + 825, + 243 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, our objective is to design a neural ODE such that the features extracted are within a neighborhood of the Lyapunov-stable equilibrium points of the ODE. We first develop a diversity promoting technique applied in the final fully connected (FC) layer to improve the ODE’s stability and analyze the reasons why. We then propose a stable neural ODE with Lyapunov-stable equilibrium points to eliminate the effects of perturbations in the input. From linear control theory [10], a linear time-invariant system $\\mathrm { d } { \\mathbf { z } ( t ) } / \\mathrm { d } t = { \\mathbf { A } } { \\mathbf { z } ( t ) }$ , where $\\mathbf { A }$ is a constant matrix, is exponentially stable if all eigenvalues of $\\mathbf { A }$ have negative real parts. Specifically, we propose to force the Jacobian matrix of the ODE used in the neural ODE to have eigenvalues with negative real parts. Instead of directly imposing constraints on the eigenvalues of the matrix, which lead to high computational complexity when the Jacobian matrix is large, we instead add constraints to the matrix elements to implicitly force the real parts of its eigenvalues to be negative. ", + "bbox": [ + 174, + 250, + 825, + 401 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our main contributions are summarized as follows: ", + "bbox": [ + 174, + 409, + 509, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. Based on the concept of Lyapunov-stable equilibrium points, we propose a simple yet effective technique to improve the robustness of neural ODE networks by fixing the final FC layer to be a matrix whose rows have unit norm and such that the maximum cosine similarity between any two rows is minimized. Such a FC layer can be constructed off-line. \n2. We propose a stable neural ODE for deFending against adversarial attacks (SODEF) to suppress the input perturbations. We derive an optimization formulation for SODEF to force the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium points of the SODEF ODE. We provide sufficient conditions for learning a robust feature representation under SODEF. \n3. We test SODEF on several widely used datasets MNIST [11], CIFAR-10 and CIFAR-100 [12] under well-known adversarial attacks. We demonstrate that SODEF is robust against adversarial white-box attacks with improvement in classification accuracy of adversarial examples under PGD attack [13] of up to $4 4 . 0 2 \\%$ , $5 2 . 5 4 \\%$ and $1 8 . 9 1 \\%$ percentage points compared to another current state-of-the-art neural ODE network TisODE [6] on MNIST, CIFAR-10 and CIFAR-100, respectively. Similar improvements in classification accuracy of adversarial examples of up to $\\bar { 4 3 . 6 9 \\% }$ , $5 2 . 3 8 \\%$ and $\\mathrm { \\bar { 1 8 . 9 9 \\% } }$ percentage points compared to ODE net [9] are also obtained. ", + "bbox": [ + 210, + 436, + 825, + 699 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The rest of this paper is organized as follows. We provide essential preliminaries on neural ODE and its stability analysis in Section 2. In Section 3, we present SODEF model architecture and its training method. We show how to maximize the distance between stable equilibrium points of neural ODEs. We propose an optimization and present theoretical results on its stability properties. We summarize experimental results in Section 4 and conclude the paper in Section 5. The proofs for all lemmas and theorems proposed in this paper are given in the supplementary material. We also refer interested readers to the supplementary material for a more detailed account of related works [14–16, 6] and some popular adversarial attacks [17, 13] that are used to verify the robustness of our proposed SODEF. In the paper, we use lowercase boldface characters like $\\mathbf { z }$ to denote vectors in $\\mathbb { R } ^ { n }$ , capital boldface characters like $\\mathbf { A }$ to denote matrices in $\\mathbb { R } ^ { n \\times n }$ , and normal characters like $z$ to denote scalars except that the notation $( x , y )$ are normal characters reserved to denote the input and label pairs. A vector $\\mathbf { z } \\in \\mathbb { R } ^ { n }$ is represented as $( \\mathbf { z } ^ { ( 1 ) } , \\mathbf { z } ^ { ( 2 ) } , \\ldots , \\mathbf { z } ^ { ( n ) } )$ . The $( i , j )$ -th element of a matrix $\\mathbf { A }$ is $\\mathbf { A } _ { i j }$ or $[ \\mathbf { A } ] _ { i j }$ . The Jacobian matrix of a function $f : \\mathbb { R } ^ { n } \\mapsto \\mathbb { R } ^ { n }$ evaluated at $\\mathbf { z }$ is denoted as $\\nabla f ( \\mathbf { z } )$ The set of functions $\\mathbb { R } ^ { n } \\mapsto \\mathbb { R } ^ { n }$ with continuous first derivatives is denoted as $C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } )$ . ", + "bbox": [ + 173, + 714, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Preliminaries: Neural ODE and Stability ", + "text_level": 1, + "bbox": [ + 173, + 88, + 547, + 107 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In a neural ODE layer, the relation between the layer input ${ \\bf z } ( 0 )$ and output ${ \\mathbf z } ( T )$ is described as the following differential equation: ", + "bbox": [ + 171, + 118, + 823, + 147 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/40fc799ca8fac2cda94627f8fa8e3909cd3c435bfc40eb885acf21aca9021058.jpg", + "text": "$$\n\\frac { \\mathrm { d } \\mathbf { z } ( t ) } { \\mathrm { d } t } = f _ { \\pmb { \\theta } } ( \\mathbf { z } ( t ) , t )\n$$", + "text_format": "latex", + "bbox": [ + 431, + 148, + 566, + 179 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $f _ { \\pmb \\theta } : \\mathbb { R } ^ { n } \\times [ 0 , \\infty ) \\mapsto \\mathbb { R } ^ { n }$ denotes the non-linear trainable layers that are parameterized by weights $\\pmb { \\theta }$ and $\\mathbf { z } : [ 0 , \\infty ) \\mapsto \\mathbb { R } ^ { n }$ represents the $n$ -dimensional state of the neural ODE. Neural ODEs are the continuous analog of residual networks where the hidden layers of residual networks can be regarded as discrete-time difference equations ${ \\bf z } ( t + 1 ) = { \\bf z } ( t ) + { f _ { \\theta } } ( { \\bf z } ( t ) , t )$ . In this work, for simplicity, we only consider the time-invariant (autonomous) case $f _ { \\pmb \\theta } ( \\mathbf { z } ( t ) , t ) = f _ { \\pmb \\theta } ( \\mathbf { z } ( t ) )$ , where the dynamical system does not explicitly depend on $t$ . For such non-linear dynamical systems, the following theorem shows that under mild conditions, its behaviour can be studied via linearization near special points called hyperbolic equilibrium points. ", + "bbox": [ + 173, + 180, + 825, + 291 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 1 (Hartman–Grobman Theorem [18]). Consider a system evolving in time with state $\\mathbf { z } ( t ) \\in \\mathbb { R } ^ { n }$ that satisfies the differential equation $\\frac { \\mathrm { d } { \\bf z } ( t ) } { \\mathrm { d } t } = f ( { \\bf z } ( t ) )$ for some $f \\in C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } ) ;$ , $f ( \\mathbf { z } ) = ( f ^ { ( 1 ) } ( \\mathbf { z } ) , \\ldots , f ^ { ( n ) } ( \\mathbf { z } ) )$ . Suppose the map has a hyperbolic equilibrium state $\\mathbf { z } ^ { \\ast } \\in \\mathbb { R } ^ { n }$ , i.e., $f ( { \\bf z } ^ { * } ) = 0$ and the Jacobian matrix with real part equal to zer $\\nabla f = [ \\partial f ^ { ( i ) } / \\partial \\mathbf { z } ^ { ( j ) } ] _ { i , j = 1 } ^ { n }$ of hb $f$ evalurhood at of $\\textbf { z } = \\textbf { z } ^ { * }$ has nolibrium $N _ { \\mathbf { z } ^ { * } }$ point $\\mathbf { z } ^ { \\ast }$ and a homeomorphism $g : N _ { \\mathbf { z } ^ { * } } \\mapsto \\mathbb { R } ^ { n }$ , such that $g ( \\mathbf { z } ^ { * } ) = 0$ and in the neighbourhood $N _ { \\mathbf { z } ^ { * } }$ , the flow of $\\frac { \\mathrm { d } { \\bf z } ( t ) } { \\mathrm { d } t } = f ( { \\bf z } ( t ) )$ is topologically conjugate by the continuous map $\\bar { \\mathbf { z } } ( t ) = g ( \\mathbf { z } ( t ) )$ to the flow of its linearization $\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\nabla f ( \\mathbf { z } ^ { * } ) \\cdot \\bar { \\mathbf { z } } ( t ) .$ ", + "bbox": [ + 173, + 292, + 826, + 450 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The theorem states that when the Jacobian matrix at the zeros of $f$ has no eigenvalue with zero real part, the behaviour of the original dynamical system can be studied using the simpler linearization of the system around those zeros. We next review some definitions and theorems from linear control theory [10]. ", + "bbox": [ + 173, + 457, + 825, + 512 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 1 (Lyapunov Stability [10]). The linear time-invariant system $\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t )$ with constant matrix A is marginally stable or stable in the sense of Lyapunov if every finite initial state $\\bar { \\mathbf { z } } ( 0 )$ excites a bounded response. It is asymptotically stable if every finite initial state excites $a$ bounded response, which, in addition, approaches 0 as $t \\to \\infty$ . ", + "bbox": [ + 173, + 515, + 825, + 583 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 2 (Lyapunov Stability Theorem [10]). a) The equation $\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t ) )$ is marginally stable if and only if all eigenvalues of A have zero or negative real parts and those with zero real parts are simple roots of the minimal polynomial of A. $^ b$ ) The equation $\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t )$ is asymptotically stable if and only if all eigenvalues of A have negative real parts. ", + "bbox": [ + 173, + 585, + 825, + 667 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In Theorem 1, we say that a hyperbolic equilibrium point is Lyapunov-stable if all eigenvalues of the Jacobian matrix evaluated at it have negative real parts. From Theorems 1 and 2, we see that a small perturbation around the Lyapunov-stable equilibrium point ${ \\bf z } ( 0 )$ leads to $\\tilde { \\mathbf { z } } ( t ) \\to \\mathbf { z } ( 0 )$ as $t \\to \\infty$ , i.e., $\\exists \\delta > 0$ such that for all $\\tilde { \\mathbf { z } } ( 0 )$ with $\\lVert { \\mathbf z } ( 0 ) \\dot { - } \\tilde { { \\mathbf z } } ( 0 ) \\rVert _ { 2 } \\dot { < } \\delta$ , we have $\\| \\widetilde { \\mathbf { z } } ( t ) - \\mathbf { z } ( 0 ) \\| _ { 2 } \\to \\dot { 0 }$ as $t \\to \\infty$ , where $\\tilde { \\mathbf { z } } ( t )$ is the ODE solution for the perturbed input $\\tilde { \\mathbf { z } } ( 0 )$ . In the context of neural network adversarial attacks, if the malicious perturbations around the ODE input ${ \\bf z } ( 0 )$ is small, then the output ${ \\mathbf z } ( T )$ for large enough $T$ will not be affected significantly by the perturbation. Consequently, the succeeding network layers after the neural ODE layer can still perform well without being affected by the input perturbation. The perturbation weakening phenomenon around Lyapunov-stable equilibrium points works like a noise filter and acts as a defense against adversarial attacks. ", + "bbox": [ + 173, + 674, + 825, + 814 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We require the following definition and result in our stability analysis. ", + "bbox": [ + 174, + 819, + 637, + 834 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2 (Strictly diagonally dominant [19]). Let $\\mathbf { A } \\in \\mathbb { C } ^ { n \\times n }$ . We say that A is strictly diagonally dominant if $\\begin{array} { r } { \\left| { { { \\mathbf { A } } _ { i i } } } \\right| > \\sum _ { j \\neq i } \\left| { { { \\mathbf { A } } _ { i j } } } \\right| } \\end{array}$ for all $i = 1 , . . . , n$ . ", + "bbox": [ + 174, + 835, + 823, + 866 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 3 (Levy–Desplanques theorem [19]). If $\\mathbf { A } \\in \\mathbb { C } ^ { n \\times n }$ is strictly diagonally dominant and if every main diagonal entry of A is real and negative, then $A$ is non-singular and every eigenvalue of A has negative real part. ", + "bbox": [ + 174, + 869, + 826, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/96641a4aad1ec2c9f3138777e3d67fc9ff9849146d1c1b15becd36b84d50c353.jpg", + "image_caption": [ + "Fig. 1: SODEF model architecture. " + ], + "image_footnote": [], + "bbox": [ + 183, + 87, + 815, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 1. Given $k$ distinct points $\\mathbf { z } _ { i } \\in \\mathbb { R } ^ { n }$ and matrices $\\mathbf { A } _ { i } \\in \\mathbb { R } ^ { n \\times n }$ , $i = 1 , . . . , k$ , there exists a function $f \\in C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } )$ such that $f ( \\mathbf { z } _ { i } ) = 0$ and $\\nabla f _ { \\pmb { \\theta } } ( \\mathbf { z } _ { i } ) = \\mathbf { A } _ { i }$ . ", + "bbox": [ + 171, + 184, + 823, + 213 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 SODEF Architecture ", + "text_level": 1, + "bbox": [ + 174, + 232, + 382, + 250 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We consider a classification problem with $L$ classes. The proposed SODEF model architecture is shown in Fig. 1. The input $x \\in X$ (e.g., an image) is first passed through a feature extractor $h _ { \\phi } : X \\mapsto \\mathbb { R } ^ { n }$ to obtain an embedding feature representation ${ \\bf z } ( 0 )$ . A neural ODE layer $f _ { \\theta }$ follows as a nonlinear feature mapping to stabilize the feature representation output ${ \\bf z } ( 0 )$ from $h _ { \\phi }$ . The final FC layer $\\mathbf { V }$ serves as a linear mapping to generate a prediction vector based on the output $\\mathbf { \\dot { z } } ( T )$ of the neural ODE layer. The parameters $\\phi , \\theta$ and $\\mathbf { V }$ are parameterized weights for the feature extractor, neural ODE layer and FC layer, respectively. ", + "bbox": [ + 173, + 265, + 825, + 362 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We provide motivation and design guidance for the FC layer V in Section 3.1, which attempts to separate Lyapunov-stable equilibrium points implicitly by maximizing the similarity distance between feature representations corresponding to the $L$ different classes. Experimental results demonstrate the advantages of our diversity promoting FC layer in Section 3.1 with comparisons to traditional neural ODEs without diversity promoting. ", + "bbox": [ + 174, + 367, + 825, + 438 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "However, the embedded features after using diversity promoting are not guaranteed to locate near the Lyapunov-stable equilibrium points. In Section 3.2, we formulate an optimization problem to force embedding features to locate near the Lyapunov-stable equilibrium points. We introduce optimization constraints to force the Jacobian matrix of the ODE in our neural ODE layer to have eigenvalues with negative real parts at the Lyapunov-stable equilibrium points. Instead of directly imposing constraints on the eigenvalue of the matrix, which may be computationally complex especially when the matrix is large, we add constraints to the matrix elements instead. ", + "bbox": [ + 173, + 444, + 825, + 541 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 Maximizing the Distance between Lyapunov-Stable Equilibrium Points ", + "text_level": 1, + "bbox": [ + 174, + 558, + 705, + 574 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "From Section 2, we observe that points in a small neighbourhood of a Lyapunov-stable equilibrium point is robust against adversarial perturbations. We call this neighborhood a stable neighborhood. However Lyapunov-stable equilibrium points for different classes may very well locate near each other and therefore each stable neighborhood may be very small, leading to poor adversarial defense. In this section, we propose to add a FC layer after the neural ODE layer given by (1) to avoid this scenario. The purpose of the FC layer is to map the output of the neural ODE layer to a feature vector $\\mathbf { v } _ { l }$ if the input $x$ belongs to the class $l = 1 , \\ldots , L$ . We design the FC layer so that the cosine similarities between different $\\mathbf { v } _ { l }$ ’s are minimized. ", + "bbox": [ + 174, + 583, + 825, + 695 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 2. Given a set of $k$ unit vectors $\\mathbf { v } _ { 1 } , \\ldots , \\mathbf { v } _ { k }$ in $\\mathbb { R } ^ { n }$ , where $n \\geq k$ , let $a ( \\mathbf { v } _ { 1 } , \\ldots , \\mathbf { v } _ { k } ) =$ $\\mathrm { m a x } _ { i \\neq j } { \\mathbf { \\Delta v } _ { i } ^ { \\mathsf { T } } } { \\mathbf { v } _ { j } }$ . Then $\\operatorname* { m i n } a ( \\mathbf { v } _ { 1 } , . . . , \\mathbf { v } _ { k } ) = 1 / ( 1 - k )$ , where the minimum is taken over all possible sets of $k$ unit vectors $\\mathbf { v } _ { 1 } , \\ldots , \\mathbf { v } _ { k }$ . ", + "bbox": [ + 173, + 700, + 825, + 742 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Corollary 1. Consider a $k \\times k$ matrix $\\mathbf { B } = [ b _ { i j } ] _ { i , j = 1 } ^ { k }$ with $b _ { i i } = 1$ and $b _ { i j } = 1 / ( 1 - k )$ , $\\forall i \\ne j$ Let the eigen decomposition of $\\mathbf { B }$ be $\\mathbf { B } = \\mathbf { U } \\pmb { \\Sigma } \\bar { \\mathbf { U } } ^ { \\dag }$ . For any $n \\geq k$ and $i = 1 , \\ldots , k$ , let $\\mathbf { v } _ { i }$ be the $i$ -th column of $\\mathbf { Q } \\mathbf { \\Sigma } ^ { \\mathrm { { X } ^ { 1 / 2 } \\bar { U } ^ { \\top } } }$ , where $\\mathbf { Q }$ is any $n \\times k$ matrix such that $\\mathbf { Q } ^ { \\intercal } \\mathbf { Q } = \\mathbf { I } _ { k }$ . Then, $a ( \\mathbf { v } _ { 1 } , \\ldots , \\mathbf { v } _ { k } ) = \\operatorname* { m a x } _ { i \\neq j } \\mathbf { v } _ { i } ^ { \\mathsf { T } } \\mathbf { v } _ { j } = { 1 } / ( 1 - k )$ . ", + "bbox": [ + 173, + 746, + 825, + 804 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Corollary 1 suggests a diversity promoting scheme to maximally separate the equilibrium points of the neural ODE layer. The FC layer is represented by an $n \\times L$ matrix $\\mathbf { V } = \\left[ \\mathbf { v } _ { 1 } , \\ldots , \\mathbf { v } _ { L } \\right]$ , where $n$ is the dimension of ${ \\mathbf z } ( T )$ , the output from the neural ODE layer. If ${ \\mathbf z } ( T )$ is generated from an input from class $l$ , it is mapped to $\\mathbf { v } _ { l }$ . By minimizing the maximum cosine similarity $a ( \\mathbf { v } _ { 1 } , \\ldots , \\mathbf { v } _ { k } ) = \\operatorname* { m a x } _ { i \\neq j } \\mathbf { v } _ { i } ^ { \\mathsf { T } } \\mathbf { v } _ { j }$ between the representations from two different classes, we ensure that the output of SODEF is robust to perturbations in the input. Corollary 1 provides a way to choose the FC layer weights $\\mathbf { V }$ . ", + "bbox": [ + 173, + 814, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To validate our observations, we conduct experiments to compare the robustness of ODE net [9] and TisODE [6] with and without our proposed FC layer V, on two standard datasets: MNIST [2] and CIFAR10 [12] 2. On the MNIST dataset, all models consist of four convolutional layers and one fully-connected layer. On the CIFAR10 dataset, the networks are similar to those for MNIST except the down-sampling network is a stack of 2 ResNet blocks. In practice, the neural ODE can be solved with different numerical solvers such as the Euler method and the Runge-Kutta methods [9]. Here, we use Runge-Kutta of order 5 in our experiments. Our implementation builds on the open-source neural ODE codes.3 During training, no Gaussian noise or adversarial examples are augmented into the training set. We test the performance of our model in defending against white-box attacks FGSM [17] and PGD [13] . The parameters for different attack methods used in this paper are given in the supplementary material. From Tables 1 and 2, we observe that for both datasets, our fixed FC layer improves each network’s defense ability by a significant margin. We visualize the features before the final FC layer using t-SNE [20] in Figs. 2 and 3. We observe that with the FC layer, the features for different classes are well separated even under attacks. ", + "bbox": [ + 173, + 90, + 826, + 285 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/0e5ba2745a4221d61b6b2b59a254f41e43e3de35524fb1614f36b886d3893236.jpg", + "table_caption": [ + "Table 1: Classification accuracy $( \\% )$ on adversarial MNIST examples, where the superscript + indicates the last FC layer is fixed to be $\\mathbf { V }$ . " + ], + "table_footnote": [], + "table_body": "
AttackPara.ODEODE+TisODETisODE+
None-99.699.799.599.7
FGSM∈ = 0.331.452.845.963.5
PGD∈=0.30.290.300.420.20
", + "bbox": [ + 294, + 327, + 704, + 412 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/a89e2702121d6d02f24b9e928e41b754195ff4c548711b8b858664ec9ba55b89.jpg", + "table_caption": [ + "Table 2: Classification accuracy $( \\% )$ on adversarial CIFAR10 examples, where the superscript + indicates the last FC layer is fixed to be $\\mathbf { V }$ . " + ], + "table_footnote": [], + "table_body": "
AttackPara.ODE ODE+TisODETisODE+
None-87.0 85.087.481.8
FGSM∈ = 0.112.9 47.613.141.9
PGD∈=0.17.8 14.77.416.2
", + "bbox": [ + 294, + 469, + 704, + 554 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/2298e75b6e25d3394e76c905205ab4b066d9a6a5a3e690669fcd410610b0c475.jpg", + "image_caption": [ + "Fig. 2: t-SNE visualization results on the features before the final FC layer. The input is the test set of MNIST. Left: trained with TisODE, middle: TisODE using a randomly chosen orthogonal matrix as the final FC, right: TisODE using proposed $\\mathbf { V }$ as the final FC. " + ], + "image_footnote": [], + "bbox": [ + 214, + 589, + 789, + 684 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 Objective Formulation and Stability ", + "text_level": 1, + "bbox": [ + 176, + 773, + 464, + 789 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this subsection, we formulate an optimization framework for SODEF to force output features to locate within the stable neighborhood of Lyapunov-stable equilibrium points. We make the following assumption. ", + "bbox": [ + 174, + 799, + 825, + 840 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 1. The input x takes values in a compact metric space $X$ and has probability distribution $\\mu$ . The feature extractor $h _ { \\phi }$ is injective and continuous. ", + "bbox": [ + 174, + 844, + 823, + 873 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/7a95cfded3451c7f092fca178fd0f289e9f1e03180a9b9200b577ccb2fbf9ae9.jpg", + "image_caption": [ + "Fig. 3: t-SNE visualization results on the features before the final FC layer. The input is the adversarial examples of the test set of MNIST generated using FGSM method at $\\epsilon = 0 . 3$ . Left: trained with TisODE, middle: TisODE using a randomly chosen orthogonal matrix as the final FC, right: TisODE using proposed $\\mathbf { V }$ as the final FC. " + ], + "image_footnote": [], + "bbox": [ + 215, + 102, + 790, + 179 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The above assumption is satisfied if the input $x$ (e.g., an image) resides in a bounded and closed set of a Euclidean space. We denote the pushforward measure (still a probability distribution) of $\\mu$ under the continuous feature extractor mapping $h _ { \\phi }$ as $\\nu _ { \\phi } = \\mu \\circ h _ { \\phi } ^ { - 1 }$ , where $\\circ$ denotes function composition. The conditional probability distribution for the embedding of each class $l \\in \\{ 1 , . . . , L \\}$ has compact support $E _ { l } \\subset \\mathbb { R } ^ { n }$ since $E _ { l }$ is closed and $h _ { \\phi } ( X )$ is bounded in $\\mathbb { R } ^ { n }$ . In Section 3.1, the FC layer $\\mathbf { V }$ tries to maximize the distance between $E _ { l }$ , $l = 1 , \\ldots , L$ . In this section for analysis purposes, we also assume the following. ", + "bbox": [ + 173, + 284, + 826, + 386 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Assumption 2. We have $E _ { l } \\bigcap E _ { l ^ { \\prime } } = \\varnothing i f l \\neq l ^ { \\prime }$ , i.e., the supports of each class are pairwise disjoint. ", + "bbox": [ + 174, + 388, + 823, + 405 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our objective function is formulated as follows, which is explained in detail in the sequel: ", + "bbox": [ + 176, + 416, + 769, + 431 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/17dad62102c6b430df4524e5a1b5fe563dc3b301d4cb21934e0dc25966c092c6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\theta , \\phi } { \\operatorname* { m i n } } \\mathbb { E } _ { \\boldsymbol { \\mu } } \\ell ( { \\mathbf { V } } ^ { \\top } ( { \\mathbf { z } } ( T ) ) , y _ { i } ) } \\\\ & { \\mathrm { s . t . } \\ \\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\| f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) \\| _ { 2 } < \\epsilon , \\ f _ { \\theta } \\in C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } ) , } \\\\ & { \\quad \\quad \\quad \\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\left[ \\nabla f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) \\right] _ { i i } < 0 , \\ \\forall i = 1 , \\ldots , n , } \\\\ & { \\quad \\quad \\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\left[ | [ \\nabla f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) ] _ { i i } | - \\sum _ { j \\neq i } | [ \\nabla f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) ] _ { i j } | \\right] > 0 , \\ \\forall i = 1 , \\ldots , n , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 438, + 730, + 541 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\mathbf { z } ( 0 ) = h _ { \\phi } ( x )$ , and ${ \\mathbf z } ( T )$ is the output of (1) with input ${ \\bf z } ( 0 )$ . ", + "bbox": [ + 299, + 542, + 700, + 559 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here, $\\ell$ is a loss function and $\\epsilon > 0$ is a positive constant. The constraints (3) to (5) force ${ \\bf z } ( 0 )$ to be near the Lyapunov-stable equilibrium points with strictly diagonally dominant derivatives. We limit the $f _ { \\theta }$ to be in $C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } )$ to satisfy the condition in Theorem 1. From [21], we also know that standard multi-layer feed forward networks with as few as a single hidden layer and arbitrary bounded and non-constant activation function are universal approximators for $C ^ { 1 ^ { \\prime } } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } )$ functions with respect to some performance criteria provided only that sufficiently many hidden units are available. ", + "bbox": [ + 173, + 566, + 826, + 651 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As a comparison, TisODE [6] only includes a constraint similar to (3), which in general provides no guarantee to force ${ \\bf z } ( 0 )$ near the Lyapunov-stable equilibrium points. In the extreme case with parameters $\\theta = 0$ for $f _ { \\theta }$ such that $f _ { \\pmb { \\theta } } = 0$ , the ODE degenerates to an identity mapping. No $\\mathbf { z } ( 0 ) \\in \\mathbb { R } ^ { n }$ can now be a Lyapunov-stable equilibrium point, and no stability can therefore be guaranteed to defend against adversarial attacks even though the ODE curves still possess the nonintersecting property and steady-state constraint, which were cited as reasons for the stability of TisODE. ", + "bbox": [ + 173, + 656, + 825, + 753 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Instead of directly optimizing the above objective function, in our implementation, we optimize the following empirical Lagrangian with a training set $\\left\\{ \\left( x _ { k } , y _ { k } \\right) : k = 1 , { \\overset { - } { \\ldots } } , N \\right\\}$ : ", + "bbox": [ + 174, + 760, + 825, + 790 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0b4d301939d37a92afda712785de51f0bc1a3534c657b3defe6e57b3e127faa3.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { m i n } _ { \\theta , \\phi } \\frac { 1 } { N } \\sum _ { k = 0 } ^ { N - 1 } \\bigg ( \\ell \\big ( \\mathbf { V } ^ { \\top } \\mathbf { z } _ { k } ( T ) , y _ { k } \\big ) + \\alpha _ { 1 } \\| f _ { \\theta } \\big ( \\mathbf { z } _ { k } ( 0 ) \\big ) \\| _ { 2 } + \\alpha _ { 2 } g _ { 1 } \\Big ( \\displaystyle \\sum _ { i = 1 } ^ { n } [ \\nabla f _ { \\theta } ( \\mathbf { z } _ { k } ( 0 ) ) ] _ { i i } \\Big ) } \\\\ { \\displaystyle \\quad \\quad + \\alpha _ { 3 } g _ { 2 } \\Big ( \\displaystyle \\sum _ { i = 1 } ^ { n } ( - | [ \\nabla f _ { \\theta } ( \\mathbf { z } _ { k } ( 0 ) ) ] _ { i i } | + \\displaystyle \\sum _ { j \\neq i } \\vert [ \\nabla f _ { \\theta } ( \\mathbf { z } _ { k } ( 0 ) ) ] _ { i j } | ) \\Big ) \\bigg ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 796, + 750, + 891 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "s. t. $\\mathbf { z } _ { k } ( 0 ) = h _ { \\phi } ( x _ { k } )$ , and ${ \\mathbf z } _ { k } ( T )$ is the output of (1) with input $\\mathbf { z } _ { k } ( 0 ) , \\forall k = 1 , . . . , N$ (8) ", + "bbox": [ + 204, + 892, + 823, + 909 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\alpha _ { 1 } , \\alpha _ { 2 }$ and $\\alpha _ { 3 }$ are hyperparameter weights, $g _ { 1 }$ and $g _ { 2 }$ are chosen monotonically increasing functions bounded below to eliminate the unbounded impact of the two regularizers that can otherwise dominate the loss. In this paper, we set $g _ { 1 } ( \\cdot ) = g _ { 2 } ( \\cdot ) \\bar { = } \\exp ( \\cdot )$ . We call these two latter terms the SODEF regularizers. ", + "bbox": [ + 174, + 92, + 825, + 147 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Suppose for each class $l = 1 , \\ldots , L$ , the embedding feature set $E _ { l } = \\{ \\mathbf { z } _ { 1 } ^ { ( l ) } , \\dots , \\mathbf { z } _ { k } ^ { ( l ) } \\}$ z(l)k } is finite. For each $i = 1 , \\ldots , k$ , let $\\mathbf { A } _ { i } \\in \\mathbb { R } ^ { n \\times n }$ be strictly diagonally dominant matrix with every main diagonal entry be negative such that the eigenvalues for $\\mathbf { A } _ { i }$ all have negative real part. From Theorem 3, each $\\mathbf { A } _ { i }$ is non-singular and every eigenvalue of $\\mathbf { A } _ { i }$ has negative real part. Therefore, from Theorem 2 and Lemma 1, there exists a function $f _ { \\theta }$ such that all $\\mathbf { z } _ { i } ^ { ( l ) }$ are Lyapunov-stable equilibrium points with corresponding first derivative $\\nabla f _ { \\pmb \\theta } ( \\mathbf z _ { i } ^ { ( l ) } ) = \\mathbf { A } _ { i }$ . This shows that if there exist only finite representation points for each class, we can find a function $f _ { \\theta }$ such that all inputs to the neural ODE layer are Lyapunov-stable equilibrium points for $f _ { \\theta }$ and ", + "bbox": [ + 173, + 154, + 826, + 277 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/3bbe106347beefc10dd2696ad5328d308d46fd72b887ecee504a3bbcb4dd165d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\boldsymbol \\nu _ { \\phi } } \\left\\| f _ { \\boldsymbol \\theta } ( \\mathbf { z } ( 0 ) ) \\right\\| _ { 2 } = 0 , } \\\\ & { \\mathbb { E } _ { \\boldsymbol \\nu _ { \\phi } } \\left[ \\nabla f _ { \\boldsymbol \\theta } ( \\mathbf { z } ( 0 ) ) \\right] _ { i i } < 0 , \\forall i = 1 , \\ldots , n , } \\\\ & { \\mathbb { E } _ { \\boldsymbol \\nu _ { \\phi } } \\left[ | [ \\nabla f _ { \\boldsymbol \\theta } ( \\mathbf { z } ( 0 ) ) ] _ { i i } | - \\sum _ { j \\neq i } \\vert [ \\nabla f _ { \\boldsymbol \\theta } ( \\mathbf { z } ( 0 ) ) ] _ { i j } \\vert \\right] > 0 , \\forall i = 1 , \\ldots , n . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 284, + 679, + 353 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "If the input space $X$ has infinite cardinality, then an injective and continuous feature extractor $h _ { \\phi }$ results in a $\\nu _ { \\phi }$ with non-finite support, i.e., at least one $E _ { l }$ , $l = 1 , \\ldots , L$ , is infinite. It is not obvious whether we can obtain a $f _ { \\theta }$ where every point in $\\textstyle E = \\bigcup _ { l } E _ { l }$ is a stable equilibrium point. The following result gives a negative answer if $\\nu _ { \\phi }$ is a continuous measure (i.e., absolutely continuous with respect to (w.r.t.) Lebesgue measure) on some subset. ", + "bbox": [ + 174, + 359, + 825, + 429 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lemma 3. If the restriction of $\\nu _ { \\phi }$ to some open set $E ^ { \\prime } \\subset E$ is a continuous measure, there is no continuous function fθ such that for $\\nu _ { \\phi }$ -almost surely all $\\mathbf { z } \\in E$ , $f _ { \\pmb \\theta } ( \\mathbf z ) = 0$ and all the eigenvalues of $\\nabla f _ { \\boldsymbol { \\theta } } ( \\mathbf { z } )$ have negative real parts. In other words, there is no continuous function fθ such that almost surely all $\\mathbf { z }$ in $E$ are Lyapunov-stable equilibrium points. ", + "bbox": [ + 173, + 433, + 823, + 488 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lemma 3 indicates that it is too much to hope for all points in $E$ to be Lyapunov-stable equilibrium points. In the following, we relax this requirement and show that under mild conditions, for all $\\epsilon > 0$ , we can find a continuous function $f _ { \\theta }$ with finitely many stable equilibrium points such that conditions (b) and (c) above hold and condition (a) is replaced by $\\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\bar { \\| } f _ { \\theta } ( \\mathbf { z } ( 0 ) ) \\| _ { 2 } ^ { - } < \\epsilon$ . This motivates the optimization constraints in (3) to (5). ", + "bbox": [ + 173, + 497, + 825, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 4. Suppose Assumptions $I$ and 2. If $\\nu _ { \\phi }$ is not a continuous uniform measure on $E _ { l }$ for each $l = 1 , \\ldots , L$ , then the following holds: 1) The function space satisfying the constraints in (3) to (5) is non-empty for all $\\epsilon > 0$ . 2) If additionally the restriction of $\\nu _ { \\phi }$ to any open set $O \\subset E _ { l }$ is not a continuous uniform measure, there exist functions in this space such that each support $E _ { l }$ contains at least one Lyapunov-stable equilibrium point. ", + "bbox": [ + 174, + 570, + 825, + 641 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 659, + 312, + 676 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we evaluate the robustness of SODEF under adversarial attacks with different attack parameters. We conduct experiments to compare the robustness of ODE net [9] and TisODE net [6] on three standard datasets: MNIST [2], CIFAR10 and CIFAR100 [3]. Since SODEF is compatible with many defense methods, it can be applied to any neural network’s final regressor layer to enhance its stability against adversarial attacks. Our experiment codes are provided in https://github.com/KANGQIYU/SODEF. ", + "bbox": [ + 173, + 689, + 825, + 773 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Setup", + "text_level": 1, + "bbox": [ + 174, + 789, + 253, + 804 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use open-source pre-trained models that achieve the top accuracy on each dataset as the feature extractor $h _ { \\phi }$ . Specifically for simple MNIST task, we use the ResNet18 model provided in Pytorch. We use the model provided by [22], which obtains nearly $8 8 \\%$ clean accuracy on CIFAR100 using EfficientNet [23] and the model provided by [24], which has nearly $9 5 \\%$ clean accuracy on CIFAR10. In the neural ODE layer, $f _ { \\theta }$ consists of 2 FC layers. During the trainings of SODEF (except in the experiment included in Section 4.2), we train the neural network with the fixed FC introduced in Section 3.1. In the first 30 epochs, we fixed $f _ { \\theta }$ to let the feature extractor $h _ { \\phi }$ learn a feature representation with only the cross-entropy loss $\\ell$ , and in the remaining 120 epochs, we release $h _ { \\phi }$ to further train $f _ { \\theta }$ using (7) with $\\alpha _ { 1 } = 1$ and $\\alpha _ { 2 } = \\alpha _ { 3 } = 0 . 0 5$ . For CIFAR10 and CIFAR100, the pixel values are normalized by $( x - \\mu ) / \\sigma$ where $\\mu = [ 0 . 4 9 1 4 , 0 . 4 8 2 2 , 0 . 4 4 6 5 ]$ and $\\sigma = [ 0 . 2 0 2 3 , 0 . 1 9 9 4 , 0 . 2 0 1 0 ] ^ { 4 }$ . To show that our SODEF is compatible with many defense methods and can be applied to any neural network’s final regression layer, we conduct an experiment where we use a recently proposed robust network TRADES [25] as the feature extractor in our SODEF. The pretrained model is provided here 5, and we choose the model with architecture \"WRN- $3 4 \\mathrm { - } 1 0 \"$ to conduct our experiments. Besides the two vanilla white-box attacks FGSM and PGD as metioned in Section 3.1, we also include a strong ensemble attack AutoAttack [26], which sequentially performs attack using all of the following four individual attacks: three white-box attacks APGDCE, APGDTDLR and FABT[27], and one black-box Square attack [28]. We refer the reader to the the supplementary material for more details of the attacks used in this paper, where, in additional, more experiments are included. ", + "bbox": [ + 174, + 814, + 826, + 911 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 257 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 Compatibility of SODEF ", + "text_level": 1, + "bbox": [ + 174, + 273, + 385, + 287 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Adversarial training (AT) is one of the most effective strategies for defending adversarial attacks. TRADES [25] is one of the adversarial training defense methods with combinations of tricks of warmup, early stopping, weight decay, batch size and other hyper parameter settings. In this experiment we fix the pretained TRADES model (except the final FC layer (size $6 4 0 \\mathrm { x } 1 0 )$ ) as our feature extractor $h _ { \\phi }$ . We then append our (trainable) SODEF with integration time $T = 5$ to the output of the feature extractor. To evaluate model robustness, we use AutoAttack and attack the models using both the $\\mathcal { L } _ { 2 }$ norm $\\epsilon = 0 . 5$ ) and $\\mathcal { L } _ { \\infty }$ norm $( \\epsilon = 8 / 2 5 5 )$ ). The results are shown in Table 3. We clearly observe that our SODEF can enhance TRADES’s robustness under all the four individual attacks and the strongest ensemble AutoAttack. For the strong $\\mathcal { L } _ { 2 }$ AutoAttack, our SODEF have improved the model robustness from $5 9 . 4 2 \\%$ to $6 7 . 7 5 \\%$ . Our experiment show that SODEF can be applied to many defense models’ regression layer to enhance their stability against attacks. ", + "bbox": [ + 173, + 297, + 825, + 450 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/72bb27c9394cce1e70f688ac4fde16bcef7caea32833005fb4cf6ed7459bf413.jpg", + "table_caption": [ + "Table 3: Classification accuracy $( \\% )$ using TRADES (w/ and w/o SODEF) under AutoAttack on adversarial CIFAR10 examples with $\\mathcal { L } _ { 2 }$ norm $\\epsilon = 0 . 5 )$ and $\\mathcal { L } _ { \\infty }$ norm $( \\epsilon = 8 / 2 5 5 )$ . " + ], + "table_footnote": [], + "table_body": "
Attack /ModelTRADES LTRADES+SODEF LTRADES L2TRADES+SODEF L2
Clean85.4885.1885.48 85.18
APGDcE56.0870.9061.74 74.35
APGDDLR53.7064.1559.22 68.55
FABT54.1882.9260.31 83.15
Square59.1262.2172.65 76.02
AutoAttack53.6957.7659.42 67.75
", + "bbox": [ + 235, + 492, + 761, + 597 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 Influence of Integration Time $T$ ", + "text_level": 1, + "bbox": [ + 174, + 627, + 431, + 642 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "From the discussion after Theorems 1 and 2, we know if the malicious perturbations around the ODE input Lyapunov-stable equilibrium point ${ \\bf z } ( 0 )$ is small, then the output ${ \\mathbf z } ( T )$ for large enough $T$ will not be affected significantly by the perturbation: $\\| \\tilde { { \\mathbf z } } ( t ) - { \\mathbf z } ( 0 ) \\| _ { 2 } \\dot { \\to } 0$ as $t \\to \\infty$ . Consequently, the succeeding network layers after the neural ODE layer can still perform well without being affected by the input perturbation. In this section, we test the influence of the SODEF integration time $T$ using CIFAR100. We use the model EfficientNet provided by [23] as $h _ { \\phi }$ (Note, unlike Section 4.2, $h _ { \\phi }$ is trainable in this experiments). We use AutoAttack with $\\mathcal { L } _ { 2 }$ norm $\\epsilon = 0 . 5$ ). We observe that for all the four individual attacks and the strongest ensemble AutoAttack, SODEF performs generally better for large integration time $T$ . We also test larger integration time $T > 1 0$ , but do not see any obvious improvements. ", + "bbox": [ + 173, + 652, + 825, + 791 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 Performance Comparison Under AutoAttack ", + "text_level": 1, + "bbox": [ + 173, + 808, + 526, + 823 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For a comparison, we provide the results of applying AutoAttack to other baseline models mentioned in the paper. We set the same integration time for ODE, TisODE and SODEF. We observe that for the ", + "bbox": [ + 176, + 833, + 823, + 861 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/221ec33a77e3f820b4d08617a24a5d36fff411ef1966cd3186ed841be8ed6a35.jpg", + "table_caption": [ + "Table 4: Classification accuracy $( \\% )$ under AutoAttack on adversarial CIFAR100 examples with $\\mathcal { L } _ { 2 }$ norm, $\\epsilon = 0 . 5$ and different integration time $T$ for SODEF. " + ], + "table_footnote": [], + "table_body": "
Attack/T135678910
Clean88.0088.1288.1588.0087.9288.0088.0588.10
APGDcE17.2021.3321.0523.6769.6785.3387.1086.88
APGDLR21.0221.0022.0026.0063.3086.9086.2086.54
FABT86.3385.1086.3687.7087.6786.5586.2285.93
Square84.6786.2287.0587.2086.9086.3387.0586.75
AutoAttack2.003.534.874.3330.6678.8078.9779.10
", + "bbox": [ + 243, + 116, + 756, + 220 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "strongest AutoAttack, our SODEF outperforms the other baseline models by a significant margin. In this case, SODEF achieves $7 9 . 1 0 \\%$ accuracy while other models only get less than $3 \\%$ accuracy. ", + "bbox": [ + 173, + 246, + 825, + 275 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/22e650d3db41d1dda7808ccc6f63fd084fecc876451293b266e6f81cd54c2ab8.jpg", + "table_caption": [ + "Table 5: Classification accuracy $( \\% )$ under AutoAttack on adversarial CIFAR100 examples with $\\mathcal { L } _ { 2 }$ norm, $\\epsilon = 0 . 5$ and $T = 1 0$ . " + ], + "table_footnote": [], + "table_body": "
Attack/ModelNoODEODETisODESODEF
Clean88.0087.9088.0088.10
APGDCE23.306.7514.3286.88
7.3322.0024.2086.54
FAB79.3078.6777.1685.93
Square84.5285.6786.3286.75
AutoAttack0.001.334.0679.10
", + "bbox": [ + 313, + 314, + 683, + 419 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 Performance Under PGD and FGSM Attacks ", + "text_level": 1, + "bbox": [ + 174, + 440, + 527, + 457 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "White-box adversaries have knowledge of the classifier models, including training data, model architectures and parameters. We test the performance of our model in defending against the whitebox attacks, PGD and FGSM. We set $T = 5$ as the integration time for the neural ODE layer. The parameters for different attack methods used are given in the supplementary material. The subsequent experiments use these settings by default, unless otherwise stated. ", + "bbox": [ + 173, + 467, + 826, + 536 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/08b150433d3576a1490d72d07775e57a65e7a73c041808639a034b1d4b92953f.jpg", + "table_caption": [ + "Table 6: Classification accuracy $( \\% )$ on adversarial MNIST examples. " + ], + "table_footnote": [], + "table_body": "
AttackPara.no odeODETisODESODEF
None-99.4599.4299.4399.44
FGSM∈=0.310.0329.636.7063.36
PGD∈= 0.30.311.561.8245.25
", + "bbox": [ + 310, + 564, + 684, + 631 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The classification results on MNIST are shown in Table 6. We observe that while maintaining the state-of-the-art accuracy on normal images, SODEF improves the adversarial robustness as compared to the other two methods. For the most effective attack in this experiment, i.e., PGD attack, SODEF shows a $4 5 . 2 5 \\% - 1 . 5 6 \\% = 4 3 . 6 9 \\%$ improvement over ODE and a $4 5 . 2 5 \\% - 1 . 2 3 \\% = 4 4 . 0 2 \\%$ improvement over TisODE. ", + "bbox": [ + 173, + 643, + 826, + 713 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/3c17b7a629d8b072e122c9a92c1b4da9ce1ce3169872ef2f345d05c365d9d807.jpg", + "table_caption": [ + "Table 7: Classification accuracy $( \\% )$ on adversarial CIFAR10 examples. " + ], + "table_footnote": [], + "table_body": "
AttackPara.no odeODETisODESODEF
None-95.294.995.195.0
FGSM∈=0.147.3145.2343.2868.05
PGD∈=0.13.093.213.8055.59
", + "bbox": [ + 310, + 741, + 686, + 808 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For CIFAR-10, we see from Table 7 that SODEF maintains high accuracy on normal examples and makes the best predictions under adversarial attacks. In particular, SODEF achieves an absolute percentage point improvement over ODE net up to $5 2 . 3 8 \\%$ and over TisODE up to $5 2 . 5 4 \\%$ for PGD attack. ", + "bbox": [ + 173, + 820, + 826, + 877 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For CIFAR-100, the results in the supplementary material shows that the most effective attack causes the classification accuracy to drop relatively by $\\begin{array} { r } { 7 4 . 6 \\% = \\frac { 8 8 . 0 - 2 2 . 3 5 } { 8 8 . 0 } } \\end{array}$ 88.0−22.35 for SODEF and by $\\begin{array} { r } { 9 7 . 3 \\% = \\frac { 8 8 . 3 - 2 . 3 9 } { 8 8 . 3 } } \\end{array}$ for vanilla EfficientNet, which is pre-trained on ImageNet to obtain a top clean accuracy. Neither ODE net nor TisODE net can improve the classification accuracy under PGD attack by a big margin, e.g. TisODE net only improves the classification accuracy from $2 . 3 9 \\%$ to $3 . 4 4 \\%$ , while SODEF still shows clear defense capability in this scenario. ", + "bbox": [ + 173, + 882, + 825, + 914 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 89, + 825, + 147 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.6 Ablation Studies ", + "text_level": 1, + "bbox": [ + 174, + 165, + 328, + 179 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The impact of the ODE with and without the SODEF regularizers in (7) has been presented in the above comparisons between SODEF and ODE. In this section, we show the necessity of diversity promoting using the FC introduced in Section 3.1 and conduct transferability study. ", + "bbox": [ + 176, + 190, + 823, + 233 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.6.1 Impact of Diversity Promotion ", + "text_level": 1, + "bbox": [ + 174, + 248, + 437, + 263 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/d06c699467442a5ac71624b7467860c1c3db93486d0d6f42604ef4497a5112f5.jpg", + "table_caption": [ + "Table 8: Classification accuracy $( \\% )$ on adversarial MNIST examples, where the superscript indicates the last FC layer is not fixed to be $\\mathbf { V }$ and is set to be a trainable layer. " + ], + "table_footnote": [], + "table_body": "
AttackPara.SODEFSODEF-
None-95.095.1
FGSM∈=0.163.3651.6
PGD∈=0.145.2534.9
", + "bbox": [ + 352, + 316, + 643, + 402 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Table 8 shows the difference of the defense performance when fixing the final FC be $\\mathbf { V }$ or setting it to a trainable linear layer. It can be seen that having diversity control improves the robustness. One possible reason for this phenomenon given in Section 3 is that diversity promotion with a fixed designed FC attempts to make the embedding feature support $E _ { l }$ of each class $l$ disjoint to each other and therefore the Lyapunov-stable equilibrium points for each $E _ { l }$ are well separated. ", + "bbox": [ + 174, + 417, + 826, + 488 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.6.2 Transferability Study ", + "text_level": 1, + "bbox": [ + 174, + 503, + 374, + 518 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Transferability study is carried out on CIFAR-10, where the adversarial examples are generated using FGSM and PGD attacks using ResNet18 without any ODEs. The classification accuracy drops from $6 8 . 0 5 \\%$ to $5 9 \\%$ for FGSM with $\\epsilon = 0 . 3$ , and from $5 5 . 5 9 \\%$ to $3 4 \\%$ for PGD with $\\epsilon = 0 . 1$ . One possible reason for this phenomenon is that ODEs have obfuscated gradient masking effect as discussed in [15], and a transfer attack may deteriorate the defense effect. However, as we observe from Table 7, even with a transfer attack on SODEF, it still performs better than other ODEs without transfer attacks. ", + "bbox": [ + 174, + 527, + 825, + 626 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 646, + 299, + 662 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we have developed a new neural ODE network, SODEF, to suppress input perturbations. SODEF is compatible with any existing neural networks and can thus be appended to the state-of-theart networks to increase their robustness to adversarial attacks. We demonstrated empirically and theoretically that the robustness of SODEF mainly derives from its stability and proposed a training method that imposes constraints to ensure all eigenvalues of the Jacobian matrix of the neural ODE layer have negative real parts. When each classification class converges to its own equilibrium points, we showed that the last FC layer can be designed in such a way that the distance between the stable equilibrium points is maximized, which further improves the network’s robustness. The effectiveness of SODEF has been verified under several popular while-box attacks. ", + "bbox": [ + 174, + 678, + 825, + 803 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 823, + 553, + 840 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This research is supported in part by A\\*STAR under its RIE2020 Advanced Manufacturing and Engineering (AME) Industry Alignment Fund – Pre Positioning (IAF-PP) (Grant No. A19D6a0053) and the RIE2020 Industry Alignment Fund – Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s). The computational work for this article was partially performed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg). ", + "bbox": [ + 176, + 854, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 825, + 119 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Broader Impact ", + "text_level": 1, + "bbox": [ + 174, + 138, + 310, + 156 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Our work, which contributes to more robust DNNs, is supposed to mitigate the threat of adversarial attacks. However, on the hand, the reliable deployment of DNNs in automation of tasks will potentially bring mass-scale unemployment and social unrest. As DNNs become more robust and more tasks, especially those whose failures will bring high risks to human lives or large property losses under adversarial attacks, fall into the automatic task category, massive jobs could disappear. 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Vision. Springer, 2020, pp. 484–501. ", + "bbox": [ + 178, + 280, + 826, + 914 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 87, + 830, + 590 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 174, + 612, + 254, + 628 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 638, + 339, + 652 + ], + "page_idx": 11 + }, + { + "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] \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 657, + 825, + 748 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 753, + 493, + 767 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] . See Assumption 1 and Assumption 2 \n(b) Did you include complete proofs of all theoretical results? [Yes] . See supplementary material. ", + "bbox": [ + 238, + 771, + 825, + 830 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 834, + 393, + 849 + ], + "page_idx": 11 + }, + { + "type": "text", + "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] . \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] . See Sections 3.1 and 4 \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Having ODE blocks in our model, it would be too computationally expensive to repeat experiments for many times. We repeated each experiment for 2-3 times and we observe the deviation of the classification results is within $\\pm 3 \\%$ , though these experimental repetitions are not enough to construct error bars. \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 Section 3.1. ", + "bbox": [ + 238, + 853, + 825, + 911 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 235, + 92, + 825, + 205 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 214, + 209, + 823, + 224 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] See Sections 3.1 and 4 for the open-source models we have used from GitHub. \n(b) Did you mention the license of the assets? [No] . Please see the licenses given in the GitHub link. \n(c) Did you include any new assets either in the supplemental material or as a URL? [No] No new assets. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] MNIST, CIFAR-10 and CIFAR-100 are all open-source datasets. \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] There is no identifiable information or offensive content in the datasets. ", + "bbox": [ + 238, + 228, + 825, + 390 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 395, + 705, + 410 + ], + "page_idx": 12 + }, + { + "type": "text", + "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] ", + "bbox": [ + 238, + 412, + 825, + 502 + ], + "page_idx": 12 + } +] \ No newline at end of file diff --git a/parse/train/9CPc4EIr2t1/9CPc4EIr2t1_middle.json b/parse/train/9CPc4EIr2t1/9CPc4EIr2t1_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..a81c5c40d9d059043071b82cf9c4f8ad36b6381c --- /dev/null +++ b/parse/train/9CPc4EIr2t1/9CPc4EIr2t1_middle.json @@ -0,0 +1,41789 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 150, + 97, + 461, + 156 + ], + "lines": [ + { + "bbox": [ + 148, + 96, + 462, + 118 + ], + "spans": [ + { + "bbox": [ + 148, + 96, + 462, + 118 + ], + "score": 1.0, + "content": "Stable Neural ODE with Lyapunov-Stable", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 149, + 116, + 462, + 138 + ], + "spans": [ + { + "bbox": [ + 149, + 116, + 462, + 138 + ], + "score": 1.0, + "content": "Equilibrium Points for Defending Against", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": 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By ensuring that the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 453, + 470, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 470, + 466 + ], + "score": 1.0, + "content": "equilibrium points of the ODE solution used as part of SODEF is Lyapunov-stable,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 464, + 469, + 477 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 469, + 477 + ], + "score": 1.0, + "content": "the ODE solution for an input with a small perturbation converges to the same", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 475, + 469, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 469, + 488 + ], + "score": 1.0, + "content": "solution as the unperturbed input. We provide theoretical results that give insights", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 486, + 470, + 499 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 470, + 499 + ], + "score": 1.0, + "content": "into the stability of SODEF as well as the choice of regularizers to ensure its", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 496, + 470, + 511 + ], + "spans": [ + { + "bbox": [ + 141, + 496, + 470, + 511 + ], + "score": 1.0, + "content": "stability. Our analysis suggests that our proposed regularizers force the extracted", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 508, + 469, + 520 + ], + "spans": [ + { + "bbox": [ + 141, + 508, + 469, + 520 + ], + "score": 1.0, + "content": "feature points to be within a neighborhood of the Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 519, + 470, + 531 + ], + "spans": [ + { + "bbox": [ + 141, + 519, + 470, + 531 + ], + "score": 1.0, + "content": "points of the ODE. SODEF is compatible with many defense methods and can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 529, + 470, + 543 + ], + "spans": [ + { + "bbox": [ + 141, + 529, + 470, + 543 + ], + "score": 1.0, + "content": "applied to any neural network’s final regressor layer to enhance its stability against", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 142, + 541, + 221, + 552 + ], + "spans": [ + { + "bbox": [ + 142, + 541, + 221, + 552 + ], + "score": 1.0, + "content": "adversarial attacks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 31.5, + "bbox_fs": [ + 141, + 377, + 471, + 552 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 574, + 190, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 192, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 192, + 590 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "Although deep learning has found successful applications in many tasks such as image classification", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "[1, 2], speech recognition [3], and natural language processing [4], the vulnerability of deep learning", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "to adversarial attacks (e.g., see [5]) has limited its real-world applications due to performance and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "safety concerns in critical applications. Inputs corrupted with human-imperceptible perturbations can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "easily fool many vanilla deep neural networks (DNNs) into mis-classifying them and thus significantly", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 212, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 212, + 667 + ], + "score": 1.0, + "content": "impact their performance.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 599, + 505, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 670, + 503, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "Recent studies [6–8] have applied neural Ordinary Differential Equations (ODEs) [9] to defend", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "against adversarial attacks. Some works like [6] have revealed interesting intrinsic properties of", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5, + "bbox_fs": [ + 106, + 670, + 505, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "ODEs that make them more stable than conventional convolutional neural networks (CNNs). The", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "score": 1.0, + "content": "paper [6] proposes a time-invariant steady neural ODE (TisODE) using the property that the integral", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "curves from a ODE solution starting from different initial points (inputs) do not intersect and always", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "preserve uniqueness in the solution function space. However, this does not guarantee that small", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "perturbations of the initial point lead to small perturbations of the integral curve output at a later time", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 107, + 128, + 115, + 137 + ], + "score": 0.64, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 127, + 506, + 140 + ], + "score": 1.0, + "content": ". The authors thus proposed a regularizer to limit the evolution of the curves by forcing the integrand", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "to be close to zero. However, neither the non-intersecting property nor the steady-state constraint", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "used in TisODE can guarantee robustness against input perturbations since these constraints do not", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "ensure that the inputs are within a neighborhood of Lyapunov-stable equilibrium points. An example", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "is an ODE that serves as an identity mapping is not robust to input perturbations but satisfies all the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 218, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 218, + 195 + ], + "score": 1.0, + "content": "constraints proposed in [6].", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "In this paper, our objective is to design a neural ODE such that the features extracted are within a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "neighborhood of the Lyapunov-stable equilibrium points of the ODE. We first develop a diversity", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "score": 1.0, + "content": "promoting technique applied in the final fully connected (FC) layer to improve the ODE’s stability", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "and analyze the reasons why. We then propose a stable neural ODE with Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "points to eliminate the effects of perturbations in the input. From linear control theory [10], a linear", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 198, + 265 + ], + "score": 1.0, + "content": "time-invariant system", + "type": "text" + }, + { + "bbox": [ + 199, + 253, + 276, + 265 + ], + "score": 0.92, + "content": "\\mathrm { d } { \\mathbf { z } ( t ) } / \\mathrm { d } t = { \\mathbf { A } } { \\mathbf { z } ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 253, + 307, + 265 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 308, + 253, + 317, + 263 + ], + "score": 0.26, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "is a constant matrix, is exponentially stable if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 179, + 276 + ], + "score": 1.0, + "content": "all eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 180, + 264, + 189, + 274 + ], + "score": 0.26, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "have negative real parts. Specifically, we propose to force the Jacobian matrix", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "of the ODE used in the neural ODE to have eigenvalues with negative real parts. Instead of directly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "imposing constraints on the eigenvalues of the matrix, which lead to high computational complexity", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "when the Jacobian matrix is large, we instead add constraints to the matrix elements to implicitly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 315, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 315, + 320 + ], + "score": 1.0, + "content": "force the real parts of its eigenvalues to be negative.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 312, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 313, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 313, + 336 + ], + "score": 1.0, + "content": "Our main contributions are summarized as follows:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 129, + 346, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 129, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 129, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "1. Based on the concept of Lyapunov-stable equilibrium points, we propose a simple yet", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "effective technique to improve the robustness of neural ODE networks by fixing the final FC", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "layer to be a matrix whose rows have unit norm and such that the maximum cosine similarity", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 380, + 468, + 392 + ], + "spans": [ + { + "bbox": [ + 142, + 380, + 468, + 392 + ], + "score": 1.0, + "content": "between any two rows is minimized. Such a FC layer can be constructed off-line.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 128, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 128, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "2. We propose a stable neural ODE for deFending against adversarial attacks (SODEF) to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 413, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 141, + 413, + 505, + 424 + ], + "score": 1.0, + "content": "suppress the input perturbations. We derive an optimization formulation for SODEF to force", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 142, + 424, + 505, + 435 + ], + "score": 1.0, + "content": "the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 142, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "points of the SODEF ODE. We provide sufficient conditions for learning a robust feature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 446, + 262, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 446, + 262, + 457 + ], + "score": 1.0, + "content": "representation under SODEF.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 129, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 129, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "3. We test SODEF on several widely used datasets MNIST [11], CIFAR-10 and CIFAR-100", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 142, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "[12] under well-known adversarial attacks. We demonstrate that SODEF is robust against", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 142, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "adversarial white-box attacks with improvement in classification accuracy of adversarial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 140, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 140, + 498, + 311, + 513 + ], + "score": 1.0, + "content": "examples under PGD attack [13] of up to", + "type": "text" + }, + { + "bbox": [ + 311, + 499, + 343, + 510 + ], + "score": 0.87, + "content": "4 4 . 0 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 498, + 347, + 513 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 347, + 499, + 380, + 510 + ], + "score": 0.85, + "content": "5 2 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 498, + 398, + 513 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 398, + 499, + 430, + 510 + ], + "score": 0.92, + "content": "1 8 . 9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "percentage points", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 510, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 142, + 510, + 506, + 522 + ], + "score": 1.0, + "content": "compared to another current state-of-the-art neural ODE network TisODE [6] on MNIST,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 141, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "CIFAR-10 and CIFAR-100, respectively. Similar improvements in classification accuracy of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 141, + 532, + 261, + 545 + ], + "score": 1.0, + "content": "adversarial examples of up to", + "type": "text" + }, + { + "bbox": [ + 262, + 532, + 294, + 543 + ], + "score": 0.87, + "content": "\\bar { 4 3 . 6 9 \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 532, + 297, + 545 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 297, + 532, + 330, + 543 + ], + "score": 0.88, + "content": "5 2 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 532, + 348, + 545 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 532, + 380, + 543 + ], + "score": 0.88, + "content": "\\mathrm { \\bar { 1 8 . 9 9 \\% } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "percentage points compared to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 142, + 543, + 267, + 555 + ], + "spans": [ + { + "bbox": [ + 142, + 543, + 267, + 555 + ], + "score": 1.0, + "content": "ODE net [9] are also obtained.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "The rest of this paper is organized as follows. We provide essential preliminaries on neural ODE", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "and its stability analysis in Section 2. In Section 3, we present SODEF model architecture and its", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "training method. We show how to maximize the distance between stable equilibrium points of neural", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "ODEs. We propose an optimization and present theoretical results on its stability properties. We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "summarize experimental results in Section 4 and conclude the paper in Section 5. The proofs for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "all lemmas and theorems proposed in this paper are given in the supplementary material. We also", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "refer interested readers to the supplementary material for a more detailed account of related works", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "[14–16, 6] and some popular adversarial attacks [17, 13] that are used to verify the robustness of our", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 400, + 666 + ], + "score": 1.0, + "content": "proposed SODEF. 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The", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "score": 1.0, + "content": "paper [6] proposes a time-invariant steady neural ODE (TisODE) using the property that the integral", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "curves from a ODE solution starting from different initial points (inputs) do not intersect and always", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "preserve uniqueness in the solution function space. However, this does not guarantee that small", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "perturbations of the initial point lead to small perturbations of the integral curve output at a later time", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 107, + 128, + 115, + 137 + ], + "score": 0.64, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 127, + 506, + 140 + ], + "score": 1.0, + "content": ". The authors thus proposed a regularizer to limit the evolution of the curves by forcing the integrand", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "to be close to zero. However, neither the non-intersecting property nor the steady-state constraint", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "used in TisODE can guarantee robustness against input perturbations since these constraints do not", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "ensure that the inputs are within a neighborhood of Lyapunov-stable equilibrium points. An example", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "is an ODE that serves as an identity mapping is not robust to input perturbations but satisfies all the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 218, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 218, + 195 + ], + "score": 1.0, + "content": "constraints proposed in [6].", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 72, + 506, + 195 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "In this paper, our objective is to design a neural ODE such that the features extracted are within a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "neighborhood of the Lyapunov-stable equilibrium points of the ODE. We first develop a diversity", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "score": 1.0, + "content": "promoting technique applied in the final fully connected (FC) layer to improve the ODE’s stability", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "and analyze the reasons why. We then propose a stable neural ODE with Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "points to eliminate the effects of perturbations in the input. From linear control theory [10], a linear", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 198, + 265 + ], + "score": 1.0, + "content": "time-invariant system", + "type": "text" + }, + { + "bbox": [ + 199, + 253, + 276, + 265 + ], + "score": 0.92, + "content": "\\mathrm { d } { \\mathbf { z } ( t ) } / \\mathrm { d } t = { \\mathbf { A } } { \\mathbf { z } ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 253, + 307, + 265 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 308, + 253, + 317, + 263 + ], + "score": 0.26, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "is a constant matrix, is exponentially stable if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 179, + 276 + ], + "score": 1.0, + "content": "all eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 180, + 264, + 189, + 274 + ], + "score": 0.26, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "have negative real parts. Specifically, we propose to force the Jacobian matrix", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "of the ODE used in the neural ODE to have eigenvalues with negative real parts. Instead of directly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "imposing constraints on the eigenvalues of the matrix, which lead to high computational complexity", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "when the Jacobian matrix is large, we instead add constraints to the matrix elements to implicitly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 315, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 315, + 320 + ], + "score": 1.0, + "content": "force the real parts of its eigenvalues to be negative.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 198, + 506, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 312, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 313, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 313, + 336 + ], + "score": 1.0, + "content": "Our main contributions are summarized as follows:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 322, + 313, + 336 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 346, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 129, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 129, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "1. Based on the concept of Lyapunov-stable equilibrium points, we propose a simple yet", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "effective technique to improve the robustness of neural ODE networks by fixing the final FC", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "layer to be a matrix whose rows have unit norm and such that the maximum cosine similarity", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 380, + 468, + 392 + ], + "spans": [ + { + "bbox": [ + 142, + 380, + 468, + 392 + ], + "score": 1.0, + "content": "between any two rows is minimized. Such a FC layer can be constructed off-line.", + "type": "text" + } + ], + "index": 26, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 128, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "2. We propose a stable neural ODE for deFending against adversarial attacks (SODEF) to", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 413, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 141, + 413, + 505, + 424 + ], + "score": 1.0, + "content": "suppress the input perturbations. We derive an optimization formulation for SODEF to force", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 142, + 424, + 505, + 435 + ], + "score": 1.0, + "content": "the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 142, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "points of the SODEF ODE. We provide sufficient conditions for learning a robust feature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 446, + 262, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 446, + 262, + 457 + ], + "score": 1.0, + "content": "representation under SODEF.", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 129, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "3. We test SODEF on several widely used datasets MNIST [11], CIFAR-10 and CIFAR-100", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 142, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "[12] under well-known adversarial attacks. 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We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "summarize experimental results in Section 4 and conclude the paper in Section 5. The proofs for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "all lemmas and theorems proposed in this paper are given in the supplementary material. 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In this work, for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 380, + 199 + ], + "score": 1.0, + "content": "simplicity, we only consider the time-invariant (autonomous) case", + "type": "text" + }, + { + "bbox": [ + 381, + 186, + 473, + 199 + ], + "score": 0.9, + "content": "f _ { \\pmb \\theta } ( \\mathbf { z } ( t ) , t ) = f _ { \\pmb \\theta } ( \\mathbf { z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 186, + 505, + 199 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 317, + 210 + ], + "score": 1.0, + "content": "the dynamical system does not explicitly depend on", + "type": "text" + }, + { + "bbox": [ + 318, + 199, + 322, + 207 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 197, + 505, + 210 + ], + "score": 1.0, + "content": ". For such non-linear dynamical systems, the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "following theorem shows that under mild conditions, its behaviour can be studied via linearization", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 333, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 333, + 232 + ], + "score": 1.0, + "content": "near special points called hyperbolic equilibrium points.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 506, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 245 + ], + "score": 1.0, + "content": "Theorem 1 (Hartman–Grobman Theorem [18]). Consider a system evolving in time with state", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 152, + 262 + ], + "score": 0.91, + "content": "\\mathbf { z } ( t ) \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 248, + 312, + 263 + ], + "score": 1.0, + "content": "that satisfies the differential equation", + "type": "text" + }, + { + "bbox": [ + 313, + 243, + 385, + 266 + ], + "score": 0.89, + "content": "\\frac { \\mathrm { d } { \\bf z } ( t ) } { \\mathrm { d } t } = f ( { \\bf z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 248, + 430, + 262 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 430, + 249, + 503, + 262 + ], + "score": 0.91, + "content": "f \\in C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } ) ;", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 248, + 506, + 262 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 507, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 229, + 278 + ], + "score": 0.92, + "content": "f ( \\mathbf { z } ) = ( f ^ { ( 1 ) } ( \\mathbf { z } ) , \\ldots , f ^ { ( n ) } ( \\mathbf { z } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 264, + 449, + 280 + ], + "score": 1.0, + "content": ". Suppose the map has a hyperbolic equilibrium state", + "type": "text" + }, + { + "bbox": [ + 449, + 266, + 485, + 277 + ], + "score": 0.9, + "content": "\\mathbf { z } ^ { \\ast } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 264, + 507, + 280 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 271, + 509, + 302 + ], + "spans": [ + { + "bbox": [ + 107, + 279, + 153, + 290 + ], + "score": 0.91, + "content": "f ( { \\bf z } ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 271, + 259, + 302 + ], + "score": 1.0, + "content": "and the Jacobian matrix with real part equal to zer", + "type": "text" + }, + { + "bbox": [ + 259, + 277, + 364, + 291 + ], + "score": 0.92, + "content": "\\nabla f = [ \\partial f ^ { ( i ) } / \\partial \\mathbf { z } ^ { ( j ) } ] _ { i , j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 271, + 377, + 302 + ], + "score": 1.0, + "content": "of hb", + "type": "text" + }, + { + "bbox": [ + 377, + 279, + 384, + 290 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 271, + 411, + 302 + ], + "score": 1.0, + "content": "evalurhood", + "type": "text" + }, + { + "bbox": [ + 428, + 271, + 440, + 302 + ], + "score": 1.0, + "content": "at of", + "type": "text" + }, + { + "bbox": [ + 440, + 279, + 473, + 289 + ], + "score": 0.9, + "content": "\\textbf { z } = \\textbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 271, + 509, + 302 + ], + "score": 1.0, + "content": "has nolibrium", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 411, + 290, + 428, + 300 + ], + "spans": [ + { + "bbox": [ + 411, + 290, + 428, + 300 + ], + "score": 0.9, + "content": "N _ { \\mathbf { z } ^ { * } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 299, + 507, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 299, + 129, + 314 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 301, + 140, + 310 + ], + "score": 0.84, + "content": "\\mathbf { z } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 299, + 235, + 314 + ], + "score": 1.0, + "content": "and a homeomorphism", + "type": "text" + }, + { + "bbox": [ + 235, + 300, + 294, + 312 + ], + "score": 0.91, + "content": "g : N _ { \\mathbf { z } ^ { * } } \\mapsto \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 299, + 336, + 314 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 337, + 300, + 379, + 312 + ], + "score": 0.93, + "content": "g ( \\mathbf { z } ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 299, + 485, + 314 + ], + "score": 1.0, + "content": "and in the neighbourhood", + "type": "text" + }, + { + "bbox": [ + 485, + 301, + 502, + 311 + ], + "score": 0.89, + "content": "N _ { \\mathbf { z } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 299, + 507, + 314 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 151, + 331 + ], + "score": 1.0, + "content": "the flow of", + "type": "text" + }, + { + "bbox": [ + 151, + 311, + 219, + 334 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } { \\bf z } ( t ) } { \\mathrm { d } t } = f ( { \\bf z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 315, + 419, + 333 + ], + "score": 1.0, + "content": "is topologically conjugate by the continuous map", + "type": "text" + }, + { + "bbox": [ + 419, + 318, + 479, + 330 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { z } } ( t ) = g ( \\mathbf { z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 315, + 506, + 333 + ], + "score": 1.0, + "content": "to the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 297, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 201, + 352 + ], + "score": 1.0, + "content": "flow of its linearization", + "type": "text" + }, + { + "bbox": [ + 201, + 333, + 297, + 357 + ], + "score": 0.88, + "content": "\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\nabla f ( \\mathbf { z } ^ { * } ) \\cdot \\bar { \\mathbf { z } } ( t ) .", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 365, + 374 + ], + "score": 1.0, + "content": "The theorem states that when the Jacobian matrix at the zeros of", + "type": "text" + }, + { + "bbox": [ + 366, + 363, + 373, + 374 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "has no eigenvalue with zero real", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 384 + ], + "score": 1.0, + "content": "part, the behaviour of the original dynamical system can be studied using the simpler linearization of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "the system around those zeros. We next review some definitions and theorems from linear control", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 157, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 157, + 408 + ], + "score": 1.0, + "content": "theory [10].", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 408, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 414, + 428 + ], + "score": 1.0, + "content": "Definition 1 (Lyapunov Stability [10]). The linear time-invariant system", + "type": "text" + }, + { + "bbox": [ + 415, + 407, + 483, + 431 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 411, + 505, + 428 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "constant matrix A is marginally stable or stable in the sense of Lyapunov if every finite initial state", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 438, + 504, + 454 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 126, + 452 + ], + "score": 0.78, + "content": "\\bar { \\mathbf { z } } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 438, + 497, + 454 + ], + "score": 1.0, + "content": "excites a bounded response. It is asymptotically stable if every finite initial state excites", + "type": "text" + }, + { + "bbox": [ + 497, + 442, + 504, + 450 + ], + "score": 0.37, + "content": "a", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 450, + 361, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 327, + 464 + ], + "score": 1.0, + "content": "bounded response, which, in addition, approaches 0 as", + "type": "text" + }, + { + "bbox": [ + 327, + 452, + 357, + 461 + ], + "score": 0.87, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 450, + 361, + 464 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 377, + 483 + ], + "score": 1.0, + "content": "Theorem 2 (Lyapunov Stability Theorem [10]). a) The equation", + "type": "text" + }, + { + "bbox": [ + 377, + 463, + 446, + 487 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 506, + 485 + ], + "score": 1.0, + "content": "is marginally", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "stable if and only if all eigenvalues of A have zero or negative real parts and those with zero real parts", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 495, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 304, + 514 + ], + "score": 1.0, + "content": "are simple roots of the minimal polynomial of A.", + "type": "text" + }, + { + "bbox": [ + 304, + 503, + 311, + 512 + ], + "score": 0.39, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 495, + 370, + 514 + ], + "score": 1.0, + "content": ") The equation", + "type": "text" + }, + { + "bbox": [ + 371, + 496, + 434, + 518 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 498, + 506, + 517 + ], + "score": 1.0, + "content": "is asymptotically", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 516, + 367, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 367, + 529 + ], + "score": 1.0, + "content": "stable if and only if all eigenvalues of A have negative real parts.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "In Theorem 1, we say that a hyperbolic equilibrium point is Lyapunov-stable if all eigenvalues of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "the Jacobian matrix evaluated at it have negative real parts. From Theorems 1 and 2, we see that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 383, + 570 + ], + "score": 1.0, + "content": "a small perturbation around the Lyapunov-stable equilibrium point", + "type": "text" + }, + { + "bbox": [ + 383, + 557, + 402, + 569 + ], + "score": 0.89, + "content": "{ \\bf z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 556, + 439, + 570 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 439, + 557, + 492, + 568 + ], + "score": 0.9, + "content": "\\tilde { \\mathbf { z } } ( t ) \\to \\mathbf { z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 137, + 578 + ], + "score": 0.88, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 566, + 157, + 581 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 157, + 568, + 189, + 578 + ], + "score": 0.9, + "content": "\\exists \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 566, + 253, + 581 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 253, + 568, + 272, + 580 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { z } } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 566, + 293, + 581 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 294, + 568, + 374, + 580 + ], + "score": 0.92, + "content": "\\lVert { \\mathbf z } ( 0 ) \\dot { - } \\tilde { { \\mathbf z } } ( 0 ) \\rVert _ { 2 } \\dot { < } \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 566, + 412, + 581 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 412, + 568, + 493, + 580 + ], + "score": 0.9, + "content": "\\| \\widetilde { \\mathbf { z } } ( t ) - \\mathbf { z } ( 0 ) \\| _ { 2 } \\to \\dot { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 137, + 589 + ], + "score": 0.88, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 578, + 166, + 591 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 167, + 579, + 184, + 590 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { z } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 578, + 354, + 591 + ], + "score": 1.0, + "content": "is the ODE solution for the perturbed input", + "type": "text" + }, + { + "bbox": [ + 355, + 579, + 373, + 590 + ], + "score": 0.88, + "content": "\\tilde { \\mathbf { z } } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 578, + 506, + 591 + ], + "score": 1.0, + "content": ". In the context of neural network", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 389, + 603 + ], + "score": 1.0, + "content": "adversarial attacks, if the malicious perturbations around the ODE input", + "type": "text" + }, + { + "bbox": [ + 389, + 590, + 408, + 601 + ], + "score": 0.9, + "content": "{ \\bf z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "is small, then the output", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 128, + 613 + ], + "score": 0.91, + "content": "{ \\mathbf z } ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 601, + 199, + 613 + ], + "score": 1.0, + "content": "for large enough", + "type": "text" + }, + { + "bbox": [ + 200, + 601, + 208, + 610 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "will not be affected significantly by the perturbation. Consequently, the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "succeeding network layers after the neural ODE layer can still perform well without being affected by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "the input perturbation. The perturbation weakening phenomenon around Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 424, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 424, + 646 + ], + "score": 1.0, + "content": "points works like a noise filter and acts as a defense against adversarial attacks.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 390, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 388, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 388, + 663 + ], + "score": 1.0, + "content": "We require the following definition and result in our stability analysis.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 504, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 315, + 676 + ], + "score": 1.0, + "content": "Definition 2 (Strictly diagonally dominant [19]). Let", + "type": "text" + }, + { + "bbox": [ + 315, + 662, + 361, + 673 + ], + "score": 0.89, + "content": "\\mathbf { A } \\in \\mathbb { C } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 659, + 506, + 676 + ], + "score": 1.0, + "content": ". We say that A is strictly diagonally", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 672, + 315, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 155, + 687 + ], + "score": 1.0, + "content": "dominant if", + "type": "text" + }, + { + "bbox": [ + 155, + 673, + 236, + 688 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\left| { { { \\mathbf { A } } _ { i i } } } \\right| > \\sum _ { j \\neq i } \\left| { { { \\mathbf { A } } _ { i j } } } \\right| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 672, + 264, + 687 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 265, + 674, + 310, + 685 + ], + "score": 0.88, + "content": "i = 1 , . . . , n", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 672, + 315, + 687 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 308, + 702 + ], + "score": 1.0, + "content": "Theorem 3 (Levy–Desplanques theorem [19]). If", + "type": "text" + }, + { + "bbox": [ + 308, + 689, + 354, + 700 + ], + "score": 0.88, + "content": "\\mathbf { A } \\in \\mathbb { C } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 687, + 507, + 702 + ], + "score": 1.0, + "content": "is strictly diagonally dominant and if", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 330, + 712 + ], + "score": 1.0, + "content": "every main diagonal entry of A is real and negative, then", + "type": "text" + }, + { + "bbox": [ + 331, + 701, + 339, + 710 + ], + "score": 0.59, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "is non-singular and every eigenvalue of A", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 198, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 198, + 724 + ], + "score": 1.0, + "content": "has negative real part.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 70, + 335, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 335, + 87 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 335, + 87 + ], + "score": 1.0, + "content": "2 Preliminaries: Neural ODE and Stability", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 94, + 504, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 342, + 107 + ], + "score": 1.0, + "content": "In a neural ODE layer, the relation between the layer input", + "type": "text" + }, + { + "bbox": [ + 342, + 95, + 361, + 107 + ], + "score": 0.91, + "content": "{ \\bf z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 94, + 407, + 107 + ], + "score": 1.0, + "content": "and output", + "type": "text" + }, + { + "bbox": [ + 407, + 95, + 428, + 107 + ], + "score": 0.92, + "content": "{ \\mathbf z } ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "is described as the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 233, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 233, + 119 + ], + "score": 1.0, + "content": "following differential equation:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 94, + 505, + 119 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 118, + 347, + 142 + ], + "lines": [ + { + "bbox": [ 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142, + 505, + 156 + ], + "score": 1.0, + "content": "denotes the non-linear trainable layers that are parameterized by", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 139, + 165 + ], + "score": 1.0, + "content": "weights", + "type": "text" + }, + { + "bbox": [ + 139, + 155, + 146, + 164 + ], + "score": 0.65, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 154, + 164, + 165 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 164, + 154, + 233, + 166 + ], + "score": 0.9, + "content": "\\mathbf { z } : [ 0 , \\infty ) \\mapsto \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 154, + 291, + 165 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + }, + { + "bbox": [ + 291, + 155, + 298, + 164 + ], + "score": 0.8, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ 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Neural ODEs", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "are the continuous analog of residual networks where the hidden layers of residual networks can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 309, + 188 + ], + "score": 1.0, + "content": "be regarded as discrete-time difference equations", + "type": "text" + }, + { + "bbox": [ + 310, + 175, + 431, + 187 + ], + "score": 0.91, + "content": "{ \\bf z } ( t + 1 ) = { \\bf z } ( t ) + { f _ { \\theta } } ( { \\bf z } ( t ) , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 175, + 506, + 188 + ], + "score": 1.0, + "content": ". In this work, for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 380, + 199 + ], + "score": 1.0, + "content": "simplicity, we only consider the time-invariant (autonomous) case", + "type": "text" + }, + { + "bbox": [ + 381, + 186, + 473, + 199 + ], + "score": 0.9, + "content": "f _ { \\pmb \\theta } ( \\mathbf { z } ( t ) , t ) = f _ { \\pmb \\theta } ( \\mathbf { z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 186, + 505, + 199 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 317, + 210 + ], + "score": 1.0, + "content": "the dynamical system does not explicitly depend on", + "type": "text" + }, + { + "bbox": [ + 318, + 199, + 322, + 207 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 197, + 505, + 210 + ], + "score": 1.0, + "content": ". For such non-linear dynamical systems, the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "following theorem shows that under mild conditions, its behaviour can be studied via linearization", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 333, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 333, + 232 + ], + "score": 1.0, + "content": "near special points called hyperbolic equilibrium points.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 142, + 506, + 232 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 506, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 245 + ], + "score": 1.0, + "content": "Theorem 1 (Hartman–Grobman Theorem [18]). Consider a system evolving in time with state", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 152, + 262 + ], + "score": 0.91, + "content": "\\mathbf { z } ( t ) \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 248, + 312, + 263 + ], + "score": 1.0, + "content": "that satisfies the differential equation", + "type": "text" + }, + { + "bbox": [ + 313, + 243, + 385, + 266 + ], + "score": 0.89, + "content": "\\frac { \\mathrm { d } { \\bf z } ( t ) } { \\mathrm { d } t } = f ( { \\bf z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 248, + 430, + 262 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 430, + 249, + 503, + 262 + ], + "score": 0.91, + "content": "f \\in C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } ) ;", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 248, + 506, + 262 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 507, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 229, + 278 + ], + "score": 0.92, + "content": "f ( \\mathbf { z } ) = ( f ^ { ( 1 ) } ( \\mathbf { z } ) , \\ldots , f ^ { ( n ) } ( \\mathbf { z } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 264, + 449, + 280 + ], + "score": 1.0, + "content": ". Suppose the map has a hyperbolic equilibrium state", + "type": "text" + }, + { + "bbox": [ + 449, + 266, + 485, + 277 + ], + "score": 0.9, + "content": "\\mathbf { z } ^ { \\ast } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 264, + 507, + 280 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 271, + 509, + 302 + ], + "spans": [ + { + "bbox": [ + 107, + 279, + 153, + 290 + ], + "score": 0.91, + "content": "f ( { \\bf z } ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 271, + 259, + 302 + ], + "score": 1.0, + "content": "and the Jacobian matrix with real part equal to zer", + "type": "text" + }, + { + "bbox": [ + 259, + 277, + 364, + 291 + ], + "score": 0.92, + "content": "\\nabla f = [ \\partial f ^ { ( i ) } / \\partial \\mathbf { z } ^ { ( j ) } ] _ { i , j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 271, + 377, + 302 + ], + "score": 1.0, + "content": "of hb", + "type": "text" + }, + { + "bbox": [ + 377, + 279, + 384, + 290 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 271, + 411, + 302 + ], + "score": 1.0, + "content": "evalurhood", + "type": "text" + }, + { + "bbox": [ + 428, + 271, + 440, + 302 + ], + "score": 1.0, + "content": "at of", + "type": "text" + }, + { + "bbox": [ + 440, + 279, + 473, + 289 + ], + "score": 0.9, + "content": "\\textbf { z } = \\textbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 271, + 509, + 302 + ], + "score": 1.0, + "content": "has nolibrium", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 411, + 290, + 428, + 300 + ], + "spans": [ + { + "bbox": [ + 411, + 290, + 428, + 300 + ], + "score": 0.9, + "content": "N _ { \\mathbf { z } ^ { * } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 299, + 507, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 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\\mathbf { z } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 299, + 507, + 314 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 151, + 331 + ], + "score": 1.0, + "content": "the flow of", + "type": "text" + }, + { + "bbox": [ + 151, + 311, + 219, + 334 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } { \\bf z } ( t ) } { \\mathrm { d } t } = f ( { \\bf z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 315, + 419, + 333 + ], + "score": 1.0, + "content": "is topologically conjugate by the continuous map", + "type": "text" + }, + { + "bbox": [ + 419, + 318, + 479, + 330 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { z } } ( t ) = g ( \\mathbf { z } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 315, + 506, + 333 + ], + "score": 1.0, + "content": "to the", + "type": "text" + } + 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We next review some definitions and theorems from linear control", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 157, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 157, + 408 + ], + "score": 1.0, + "content": "theory [10].", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 361, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 408, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 414, + 428 + ], + "score": 1.0, + "content": "Definition 1 (Lyapunov Stability [10]). The linear time-invariant system", + "type": "text" + }, + { + "bbox": [ + 415, + 407, + 483, + 431 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 411, + 505, + 428 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "constant matrix A is marginally stable or stable in the sense of Lyapunov if every finite initial state", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 438, + 504, + 454 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 126, + 452 + ], + "score": 0.78, + "content": "\\bar { \\mathbf { z } } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 438, + 497, + 454 + ], + "score": 1.0, + "content": "excites a bounded response. It is asymptotically stable if every finite initial state excites", + "type": "text" + }, + { + "bbox": [ + 497, + 442, + 504, + 450 + ], + "score": 0.37, + "content": "a", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 450, + 361, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 327, + 464 + ], + "score": 1.0, + "content": "bounded response, which, in addition, approaches 0 as", + "type": "text" + }, + { + "bbox": [ + 327, + 452, + 357, + 461 + ], + "score": 0.87, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 450, + 361, + 464 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 405, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 377, + 483 + ], + "score": 1.0, + "content": "Theorem 2 (Lyapunov Stability Theorem [10]). a) The equation", + "type": "text" + }, + { + "bbox": [ + 377, + 463, + 446, + 487 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 506, + 485 + ], + "score": 1.0, + "content": "is marginally", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "stable if and only if all eigenvalues of A have zero or negative real parts and those with zero real parts", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 495, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 304, + 514 + ], + "score": 1.0, + "content": "are simple roots of the minimal polynomial of A.", + "type": "text" + }, + { + "bbox": [ + 304, + 503, + 311, + 512 + ], + "score": 0.39, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 495, + 370, + 514 + ], + "score": 1.0, + "content": ") The equation", + "type": "text" + }, + { + "bbox": [ + 371, + 496, + 434, + 518 + ], + "score": 0.92, + "content": "\\frac { \\mathrm { d } \\bar { \\mathbf { z } } ( t ) } { \\mathrm { d } t } = \\mathbf { A } \\bar { \\mathbf { z } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 498, + 506, + 517 + ], + "score": 1.0, + "content": "is asymptotically", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 516, + 367, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 367, + 529 + ], + "score": 1.0, + "content": "stable if and only if all eigenvalues of A have negative real parts.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 461, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "In Theorem 1, we say that a hyperbolic equilibrium point is Lyapunov-stable if all eigenvalues of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "the Jacobian matrix evaluated at it have negative real parts. From Theorems 1 and 2, we see that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 383, + 570 + ], + "score": 1.0, + "content": "a small perturbation around the Lyapunov-stable equilibrium point", + "type": "text" + }, + { + "bbox": [ + 383, + 557, + 402, + 569 + ], + "score": 0.89, + "content": "{ \\bf z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 556, + 439, + 570 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 439, + 557, + 492, + 568 + ], + "score": 0.9, + "content": "\\tilde { \\mathbf { z } } ( t ) \\to \\mathbf { z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 137, + 578 + ], + "score": 0.88, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 566, + 157, + 581 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 157, + 568, + 189, + 578 + ], + "score": 0.9, + "content": "\\exists \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 566, + 253, + 581 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 253, + 568, + 272, + 580 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { z } } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 566, + 293, + 581 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 294, + 568, + 374, + 580 + ], + "score": 0.92, + "content": "\\lVert { \\mathbf z } ( 0 ) \\dot { - } \\tilde { { \\mathbf z } } ( 0 ) \\rVert _ { 2 } \\dot { < } \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 566, + 412, + 581 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 412, + 568, + 493, + 580 + ], + "score": 0.9, + "content": "\\| \\widetilde { \\mathbf { z } } ( t ) - \\mathbf { z } ( 0 ) \\| _ { 2 } \\to \\dot { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 137, + 589 + ], + "score": 0.88, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 578, + 166, + 591 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 167, + 579, + 184, + 590 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { z } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 578, + 354, + 591 + ], + "score": 1.0, + "content": "is the ODE solution for the perturbed input", + "type": "text" + }, + { + "bbox": [ + 355, + 579, + 373, + 590 + ], + "score": 0.88, + "content": "\\tilde { \\mathbf { z } } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 578, + 506, + 591 + ], + "score": 1.0, + "content": ". In the context of neural network", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 389, + 603 + ], + "score": 1.0, + "content": "adversarial attacks, if the malicious perturbations around the ODE input", + "type": "text" + }, + { + "bbox": [ + 389, + 590, + 408, + 601 + ], + "score": 0.9, + "content": "{ \\bf z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "is small, then the output", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 128, + 613 + ], + "score": 0.91, + "content": "{ \\mathbf z } ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 601, + 199, + 613 + ], + "score": 1.0, + "content": "for large enough", + "type": "text" + }, + { + "bbox": [ + 200, + 601, + 208, + 610 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "will not be affected significantly by the perturbation. Consequently, the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "succeeding network layers after the neural ODE layer can still perform well without being affected by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "the input perturbation. The perturbation weakening phenomenon around Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 424, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 424, + 646 + ], + "score": 1.0, + "content": "points works like a noise filter and acts as a defense against adversarial attacks.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 534, + 506, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 390, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 388, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 388, + 663 + ], + "score": 1.0, + "content": "We require the following definition and result in our stability analysis.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 648, + 388, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 504, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 315, + 676 + ], + "score": 1.0, + "content": "Definition 2 (Strictly diagonally dominant [19]). Let", + "type": "text" + }, + { + "bbox": [ + 315, + 662, + 361, + 673 + ], + "score": 0.89, + "content": "\\mathbf { A } \\in \\mathbb { C } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 659, + 506, + 676 + ], + "score": 1.0, + "content": ". We say that A is strictly diagonally", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 672, + 315, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 155, + 687 + ], + "score": 1.0, + "content": "dominant if", + "type": "text" + }, + { + "bbox": [ + 155, + 673, + 236, + 688 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\left| { { { \\mathbf { A } } _ { i i } } } \\right| > \\sum _ { j \\neq i } \\left| { { { \\mathbf { A } } _ { i j } } } \\right| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 672, + 264, + 687 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 265, + 674, + 310, + 685 + ], + "score": 0.88, + "content": "i = 1 , . . . , n", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 672, + 315, + 687 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 659, + 506, + 688 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 308, + 702 + ], + "score": 1.0, + "content": "Theorem 3 (Levy–Desplanques theorem [19]). If", + "type": "text" + }, + { + "bbox": [ + 308, + 689, + 354, + 700 + ], + "score": 0.88, + "content": "\\mathbf { A } \\in \\mathbb { C } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 687, + 507, + 702 + ], + "score": 1.0, + "content": "is strictly diagonally dominant and if", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 330, + 712 + ], + "score": 1.0, + "content": "every main diagonal entry of A is real and negative, then", + "type": "text" + }, + { + "bbox": [ + 331, + 701, + 339, + 710 + ], + "score": 0.59, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "is non-singular and every eigenvalue of A", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 198, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 198, + 724 + ], + "score": 1.0, + "content": "has negative real part.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 687, + 507, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 69, + 499, + 106 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 69, + 499, + 106 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 69, + 499, + 106 + ], + "spans": [ + { + "bbox": [ + 112, + 69, + 499, + 106 + ], + "score": 0.94, + "type": "image", + "image_path": "96641a4aad1ec2c9f3138777e3d67fc9ff9849146d1c1b15becd36b84d50c353.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 69, + 499, + 81.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 81.33333333333333, + 499, + 93.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 93.66666666666666, + 499, + 105.99999999999999 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 234, + 113, + 376, + 126 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 234, + 113, + 377, + 126 + ], + "spans": [ + { + "bbox": [ + 234, + 113, + 377, + 126 + ], + "score": 1.0, + "content": "Fig. 1: SODEF model architecture.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 105, + 146, + 504, + 169 + ], + "lines": [ + { + "bbox": [ + 104, + 142, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 181, + 160 + ], + "score": 1.0, + "content": "Lemma 1. Given", + "type": "text" + }, + { + "bbox": [ + 182, + 147, + 189, + 156 + ], + "score": 0.56, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 142, + 250, + 160 + ], + "score": 1.0, + "content": "distinct points", + "type": "text" + }, + { + "bbox": [ + 250, + 147, + 286, + 157 + ], + "score": 0.91, + "content": "\\mathbf { z } _ { i } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 142, + 343, + 160 + ], + "score": 1.0, + "content": "and matrices", + "type": "text" + }, + { + "bbox": [ + 344, + 146, + 393, + 157 + ], + "score": 0.85, + "content": "\\mathbf { A } _ { i } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 142, + 398, + 160 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 398, + 146, + 444, + 158 + ], + "score": 0.85, + "content": "i = 1 , . . . , k", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 142, + 507, + 160 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 156, + 375, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 141, + 171 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 157, + 210, + 169 + ], + "score": 0.93, + "content": "f \\in C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 156, + 251, + 171 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 251, + 158, + 292, + 169 + ], + "score": 0.92, + "content": "f ( \\mathbf { z } _ { i } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 156, + 311, + 171 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 158, + 371, + 169 + ], + "score": 0.92, + "content": "\\nabla f _ { \\pmb { \\theta } } ( \\mathbf { z } _ { i } ) = \\mathbf { A } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 156, + 375, + 171 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 184, + 234, + 198 + ], + "lines": [ + { + "bbox": [ + 104, + 183, + 235, + 200 + ], + "spans": [ + { + "bbox": [ + 104, + 183, + 235, + 200 + ], + "score": 1.0, + "content": "3 SODEF Architecture", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 283, + 223 + ], + "score": 1.0, + "content": "We consider a classification problem with", + "type": "text" + }, + { + "bbox": [ + 284, + 211, + 292, + 220 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "classes. 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Instead of directly imposing constraints", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "on the eigenvalue of the matrix, which may be computationally complex especially when the matrix", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 342, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 342, + 430 + ], + "score": 1.0, + "content": "is large, we add constraints to the matrix elements instead.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 352, + 506, + 430 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 442, + 432, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 433, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 433, + 456 + ], + "score": 1.0, + "content": "3.1 Maximizing the Distance between Lyapunov-Stable Equilibrium Points", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "From Section 2, we observe that points in a small neighbourhood of a Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "point is robust against adversarial perturbations. 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On the MNIST dataset, all models consist of four convolutional layers and one", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "fully-connected layer. On the CIFAR10 dataset, the networks are similar to those for MNIST except", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "the down-sampling network is a stack of 2 ResNet blocks. In practice, the neural ODE can be solved", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "with different numerical solvers such as the Euler method and the Runge-Kutta methods [9]. Here,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "we use Runge-Kutta of order 5 in our experiments. Our implementation builds on the open-source", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "neural ODE codes.3 During training, no Gaussian noise or adversarial examples are augmented into", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "the training set. We test the performance of our model in defending against white-box attacks FGSM", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "[17] and PGD [13] . The parameters for different attack methods used in this paper are given in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "supplementary material. From Tables 1 and 2, we observe that for both datasets, our fixed FC layer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "improves each network’s defense ability by a significant margin. We visualize the features before the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "final FC layer using t-SNE [20] in Figs. 2 and 3. We observe that with the FC layer, the features for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 214, + 326, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 326, + 227 + ], + "score": 1.0, + "content": "different classes are well separated even under attacks.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "table", + "bbox": [ + 180, + 259, + 431, + 327 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 236, + 504, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 234, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 104, + 234, + 244, + 249 + ], + "score": 1.0, + "content": "Table 1: Classification accuracy", + "type": "text" + }, + { + "bbox": [ + 244, + 237, + 261, + 247 + ], + "score": 0.78, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 234, + 506, + 249 + ], + "score": 1.0, + "content": "on adversarial MNIST examples, where the superscript +", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 248, + 279, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 266, + 259 + ], + "score": 1.0, + "content": "indicates the last FC layer is fixed to be", + "type": "text" + }, + { + "bbox": [ + 266, + 248, + 276, + 257 + ], + "score": 0.33, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 248, + 279, + 259 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "table_body", + "bbox": [ + 180, + 259, + 431, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 180, + 259, + 431, + 327 + ], + "spans": [ + { + "bbox": [ + 180, + 259, + 431, + 327 + ], + "score": 0.978, + "html": "
AttackPara.ODEODE+TisODETisODE+
None-99.699.799.599.7
FGSM∈ = 0.331.452.845.963.5
PGD∈=0.30.290.300.420.20
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AttackPara.ODE ODE+TisODETisODE+
None-87.0 85.087.481.8
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AttackPara.ODEODE+TisODETisODE+
None-99.699.799.599.7
FGSM∈ = 0.331.452.845.963.5
PGD∈=0.30.290.300.420.20
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AttackPara.ODE ODE+TisODETisODE+
None-87.0 85.087.481.8
FGSM∈ = 0.112.9 47.613.141.9
PGD∈=0.17.8 14.77.416.2
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The input is the adversarial", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 392, + 182 + ], + "score": 1.0, + "content": "examples of the test set of MNIST generated using FGSM method at", + "type": "text" + }, + { + "bbox": [ + 392, + 171, + 424, + 181 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 172, + 505, + 182 + ], + "score": 1.0, + "content": ". Left: trained with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "TisODE, middle: TisODE using a randomly chosen orthogonal matrix as the final FC, right: TisODE", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 193, + 244, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 169, + 204 + ], + "score": 1.0, + "content": "using proposed", + "type": "text" + }, + { + "bbox": [ + 169, + 193, + 180, + 203 + ], + "score": 0.27, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 193, + 244, + 204 + ], + "score": 1.0, + "content": "as the final FC.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 506, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 289, + 238 + ], + "score": 1.0, + "content": "The above assumption is satisfied if the input", + "type": "text" + }, + { + "bbox": [ + 289, + 227, + 296, + 235 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 225, + 506, + 238 + ], + "score": 1.0, + "content": "(e.g., an image) resides in a bounded and closed set", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 471, + 249 + ], + "score": 1.0, + "content": "of a Euclidean space. 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In Section 3.1, the FC layer", + "type": "text" + }, + { + "bbox": [ + 494, + 272, + 505, + 282 + ], + "score": 0.53, + "content": "\\mathbf { V }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 267, + 296 + ], + "score": 1.0, + "content": "tries to maximize the distance between", + "type": "text" + }, + { + "bbox": [ + 267, + 284, + 279, + 294 + ], + "score": 0.81, + "content": "E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 282, + 282, + 296 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 284, + 335, + 294 + ], + "score": 0.91, + "content": "l = 1 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 282, + 506, + 296 + ], + "score": 1.0, + "content": ". In this section for analysis purposes, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 293, + 215, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 215, + 307 + ], + "score": 1.0, + "content": "also assume the following.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 504, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 207, + 323 + ], + "score": 1.0, + "content": "Assumption 2. We have", + "type": "text" + }, + { + "bbox": [ + 208, + 309, + 295, + 321 + ], + "score": 0.9, + "content": "E _ { l } \\bigcap E _ { l ^ { \\prime } } = \\varnothing i f l \\neq l ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 308, + 506, + 323 + ], + "score": 1.0, + "content": ", i.e., the supports of each class are pairwise disjoint.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 330, + 471, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 468, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 468, + 343 + ], + "score": 1.0, + "content": "Our objective function is formulated as follows, which is explained in detail in the sequel:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 347, + 447, + 429 + ], + "lines": [ + { + "bbox": [ + 161, + 347, + 447, + 429 + ], + "spans": [ + { + "bbox": [ + 161, + 347, + 447, + 429 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\underset { \\theta , \\phi } { \\operatorname* { m i n } } \\mathbb { E } _ { \\boldsymbol { \\mu } } \\ell ( { \\mathbf { V } } ^ { \\top } ( { \\mathbf { z } } ( T ) ) , y _ { i } ) } \\\\ & { \\mathrm { s . t . } \\ \\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\| f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) \\| _ { 2 } < \\epsilon , \\ f _ { \\theta } \\in C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } ) , } \\\\ & { \\quad \\quad \\quad \\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\left[ \\nabla f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) \\right] _ { i i } < 0 , \\ \\forall i = 1 , \\ldots , n , } \\\\ & { \\quad \\quad \\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\left[ | [ \\nabla f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) ] _ { i i } | - \\sum _ { j \\neq i } | [ \\nabla f _ { \\theta } ( { \\mathbf { z } } ( 0 ) ) ] _ { i j } | \\right] > 0 , \\ \\forall i = 1 , \\ldots , n , } \\end{array}", + "type": "interline_equation", + "image_path": "17dad62102c6b430df4524e5a1b5fe563dc3b301d4cb21934e0dc25966c092c6.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 161, + 347, + 447, + 374.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 161, + 374.3333333333333, + 447, + 401.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 161, + 401.66666666666663, + 447, + 428.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 183, + 430, + 429, + 443 + ], + "lines": [ + { + "bbox": [ + 184, + 429, + 430, + 446 + ], + "spans": [ + { + "bbox": [ + 184, + 430, + 242, + 443 + ], + "score": 0.86, + "content": "\\mathbf { z } ( 0 ) = h _ { \\phi } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 429, + 264, + 446 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 264, + 430, + 286, + 443 + ], + "score": 0.92, + "content": "{ \\mathbf z } ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 429, + 407, + 446 + ], + "score": 1.0, + "content": "is the output of (1) with input", + "type": "text" + }, + { + "bbox": [ + 407, + 430, + 426, + 443 + ], + "score": 0.84, + "content": "{ \\bf z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 429, + 430, + 446 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 506, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 131, + 462 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 132, + 450, + 137, + 460 + ], + "score": 0.76, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 448, + 227, + 462 + ], + "score": 1.0, + "content": "is a loss function and", + "type": "text" + }, + { + "bbox": [ + 228, + 450, + 253, + 460 + ], + "score": 0.9, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 448, + 474, + 462 + ], + "score": 1.0, + "content": "is a positive constant. 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We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 141, + 485 + ], + "score": 1.0, + "content": "limit the", + "type": "text" + }, + { + "bbox": [ + 142, + 471, + 153, + 483 + ], + "score": 0.88, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 469, + 186, + 485 + ], + "score": 1.0, + "content": "to be in", + "type": "text" + }, + { + "bbox": [ + 186, + 471, + 236, + 483 + ], + "score": 0.95, + "content": "C ^ { 1 } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "to satisfy the condition in Theorem 1. From [21], we also know that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "standard multi-layer feed forward networks with as few as a single hidden layer and arbitrary bounded", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 391, + 506 + ], + "score": 1.0, + "content": "and non-constant activation function are universal approximators for", + "type": "text" + }, + { + "bbox": [ + 392, + 493, + 442, + 505 + ], + "score": 0.93, + "content": "C ^ { 1 ^ { \\prime } } ( \\mathbb { R } ^ { n } , \\mathbb { R } ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "functions with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 504, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 516 + ], + "score": 1.0, + "content": "respect to some performance criteria provided only that sufficiently many hidden units are available.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "As a comparison, TisODE [6] only includes a constraint similar to (3), which in general provides", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 195, + 544 + ], + "score": 1.0, + "content": "no guarantee to force", + "type": "text" + }, + { + "bbox": [ + 195, + 532, + 214, + 543 + ], + "score": 0.89, + "content": "{ \\bf z } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "near the Lyapunov-stable equilibrium points. In the extreme case with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 154, + 555 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 155, + 542, + 184, + 552 + ], + "score": 0.9, + "content": "\\theta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 542, + 201, + 555 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 202, + 543, + 213, + 554 + ], + "score": 0.86, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 542, + 256, + 555 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 257, + 542, + 290, + 554 + ], + "score": 0.91, + "content": "f _ { \\pmb { \\theta } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 542, + 506, + 555 + ], + "score": 1.0, + "content": ", the ODE degenerates to an identity mapping. 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The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 280, + 331 + ], + "score": 1.0, + "content": "following result gives a negative answer if", + "type": "text" + }, + { + "bbox": [ + 280, + 319, + 292, + 330 + ], + "score": 0.88, + "content": "\\nu _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "is a continuous measure (i.e., absolutely continuous", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 341, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 341, + 341 + ], + "score": 1.0, + "content": "with respect to (w.r.t.) 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In the following, we relax this requirement and show that under mild conditions, for all", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 133, + 427 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 417, + 284, + 429 + ], + "score": 1.0, + "content": ", we can find a continuous function", + "type": "text" + }, + { + "bbox": [ + 284, + 417, + 295, + 428 + ], + "score": 0.85, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 417, + 504, + 429 + ], + "score": 1.0, + "content": "with finitely many stable equilibrium points such", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 394, + 441 + ], + "score": 1.0, + "content": "that conditions (b) and (c) above hold and condition (a) is replaced by", + "type": "text" + }, + { + "bbox": [ + 394, + 427, + 480, + 441 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\bar { \\| } f _ { \\theta } ( \\mathbf { z } ( 0 ) ) \\| _ { 2 } ^ { - } < \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 427, + 506, + 441 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 439, + 311, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 311, + 451 + ], + "score": 1.0, + "content": "motivates the optimization constraints in (3) to (5).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 251, + 466 + ], + "score": 1.0, + "content": "Theorem 4. Suppose Assumptions", + "type": "text" + }, + { + "bbox": [ + 252, + 453, + 257, + 462 + ], + "score": 0.26, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 451, + 298, + 466 + ], + "score": 1.0, + "content": "and 2. If", + "type": "text" + }, + { + "bbox": [ + 298, + 453, + 310, + 465 + ], + "score": 0.86, + "content": "\\nu _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 451, + 477, + 466 + ], + "score": 1.0, + "content": "is not a continuous uniform measure on", + "type": "text" + }, + { + "bbox": [ + 478, + 453, + 489, + 464 + ], + "score": 0.85, + "content": "E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 128, + 475 + ], + "score": 1.0, + "content": "each", + "type": "text" + }, + { + "bbox": [ + 128, + 464, + 180, + 475 + ], + "score": 0.9, + "content": "l = 1 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 463, + 505, + 475 + ], + "score": 1.0, + "content": ", then the following holds: 1) The function space satisfying the constraints in (3)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 209, + 487 + ], + "score": 1.0, + "content": "to (5) is non-empty for all", + "type": "text" + }, + { + "bbox": [ + 209, + 475, + 233, + 485 + ], + "score": 0.77, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 474, + 372, + 487 + ], + "score": 1.0, + "content": ". 2) If additionally the restriction of", + "type": "text" + }, + { + "bbox": [ + 373, + 475, + 385, + 486 + ], + "score": 0.87, + "content": "\\nu _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 474, + 447, + 487 + ], + "score": 1.0, + "content": "to any open set", + "type": "text" + }, + { + "bbox": [ + 447, + 475, + 480, + 485 + ], + "score": 0.91, + "content": "O \\subset E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "is not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 456, + 498 + ], + "score": 1.0, + "content": "a continuous uniform measure, there exist functions in this space such that each support", + "type": "text" + }, + { + "bbox": [ + 456, + 486, + 468, + 496 + ], + "score": 0.86, + "content": "E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "contains", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 496, + 297, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 297, + 510 + ], + "score": 1.0, + "content": "at least one Lyapunov-stable equilibrium point.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 191, + 536 + ], + "lines": [ + { + "bbox": [ + 104, + 521, + 193, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 193, + 538 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 546, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "In this section, we evaluate the robustness of SODEF under adversarial attacks with different attack", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "parameters. 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Our experiment codes are provided in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 284, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 284, + 614 + ], + "score": 1.0, + "content": "https://github.com/KANGQIYU/SODEF.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 107, + 625, + 155, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 157, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 157, + 640 + ], + "score": 1.0, + "content": "4.1 Setup", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "We use open-source pre-trained models that achieve the top accuracy on each dataset as the feature", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 144, + 668 + ], + "score": 1.0, + "content": "extractor", + "type": "text" + }, + { + "bbox": [ + 144, + 657, + 157, + 668 + ], + "score": 0.89, + "content": "h _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 656, + 506, + 668 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 280, + 331 + ], + "score": 1.0, + "content": "following result gives a negative answer if", + "type": "text" + }, + { + "bbox": [ + 280, + 319, + 292, + 330 + ], + "score": 0.88, + "content": "\\nu _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "is a continuous measure (i.e., absolutely continuous", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 341, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 341, + 341 + ], + "score": 1.0, + "content": "with respect to (w.r.t.) 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In other words, there is no continuous function fθ such that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 365, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 175, + 388 + ], + "score": 1.0, + "content": "almost surely all", + "type": "text" + }, + { + "bbox": [ + 175, + 377, + 182, + 385 + ], + "score": 0.44, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 375, + 192, + 388 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 193, + 376, + 202, + 385 + ], + "score": 0.7, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 375, + 365, + 388 + ], + "score": 1.0, + "content": "are Lyapunov-stable equilibrium points.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 342, + 506, + 388 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 354, + 408 + ], + "score": 1.0, + "content": "Lemma 3 indicates that it is too much to hope for all points in", + "type": "text" + }, + { + "bbox": [ + 355, + 396, + 364, + 405 + ], + "score": 0.82, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "to be Lyapunov-stable equilibrium", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "points. In the following, we relax this requirement and show that under mild conditions, for all", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 133, + 427 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 417, + 284, + 429 + ], + "score": 1.0, + "content": ", we can find a continuous function", + "type": "text" + }, + { + "bbox": [ + 284, + 417, + 295, + 428 + ], + "score": 0.85, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 417, + 504, + 429 + ], + "score": 1.0, + "content": "with finitely many stable equilibrium points such", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 394, + 441 + ], + "score": 1.0, + "content": "that conditions (b) and (c) above hold and condition (a) is replaced by", + "type": "text" + }, + { + "bbox": [ + 394, + 427, + 480, + 441 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\boldsymbol { \\nu } _ { \\phi } } \\bar { \\| } f _ { \\theta } ( \\mathbf { z } ( 0 ) ) \\| _ { 2 } ^ { - } < \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 427, + 506, + 441 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 439, + 311, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 311, + 451 + ], + "score": 1.0, + "content": "motivates the optimization constraints in (3) to (5).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 394, + 506, + 451 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 251, + 466 + ], + "score": 1.0, + "content": "Theorem 4. Suppose Assumptions", + "type": "text" + }, + { + "bbox": [ + 252, + 453, + 257, + 462 + ], + "score": 0.26, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 451, + 298, + 466 + ], + "score": 1.0, + "content": "and 2. If", + "type": "text" + }, + { + "bbox": [ + 298, + 453, + 310, + 465 + ], + "score": 0.86, + "content": "\\nu _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 451, + 477, + 466 + ], + "score": 1.0, + "content": "is not a continuous uniform measure on", + "type": "text" + }, + { + "bbox": [ + 478, + 453, + 489, + 464 + ], + "score": 0.85, + "content": "E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 128, + 475 + ], + "score": 1.0, + "content": "each", + "type": "text" + }, + { + "bbox": [ + 128, + 464, + 180, + 475 + ], + "score": 0.9, + "content": "l = 1 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 463, + 505, + 475 + ], + "score": 1.0, + "content": ", then the following holds: 1) The function space satisfying the constraints in (3)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 209, + 487 + ], + "score": 1.0, + "content": "to (5) is non-empty for all", + "type": "text" + }, + { + "bbox": [ + 209, + 475, + 233, + 485 + ], + "score": 0.77, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 474, + 372, + 487 + ], + "score": 1.0, + "content": ". 2) If additionally the restriction of", + "type": "text" + }, + { + "bbox": [ + 373, + 475, + 385, + 486 + ], + "score": 0.87, + "content": "\\nu _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 474, + 447, + 487 + ], + "score": 1.0, + "content": "to any open set", + "type": "text" + }, + { + "bbox": [ + 447, + 475, + 480, + 485 + ], + "score": 0.91, + "content": "O \\subset E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "is not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 456, + 498 + ], + "score": 1.0, + "content": "a continuous uniform measure, there exist functions in this space such that each support", + "type": "text" + }, + { + "bbox": [ + 456, + 486, + 468, + 496 + ], + "score": 0.86, + "content": "E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "contains", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 496, + 297, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 297, + 510 + ], + "score": 1.0, + "content": "at least one Lyapunov-stable equilibrium point.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 451, + 506, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 191, + 536 + ], + "lines": [ + { + "bbox": [ + 104, + 521, + 193, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 193, + 538 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 546, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "In this section, we evaluate the robustness of SODEF under adversarial attacks with different attack", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "parameters. We conduct experiments to compare the robustness of ODE net [9] and TisODE", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "net [6] on three standard datasets: MNIST [2], CIFAR10 and CIFAR100 [3]. Since SODEF is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "compatible with many defense methods, it can be applied to any neural network’s final regressor", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "layer to enhance its stability against adversarial attacks. Our experiment codes are provided in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 284, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 284, + 614 + ], + "score": 1.0, + "content": "https://github.com/KANGQIYU/SODEF.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 547, + 506, + 614 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 625, + 155, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 157, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 157, + 640 + ], + "score": 1.0, + "content": "4.1 Setup", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "We use open-source pre-trained models that achieve the top accuracy on each dataset as the feature", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 144, + 668 + ], + "score": 1.0, + "content": "extractor", + "type": "text" + }, + { + "bbox": [ + 144, + 657, + 157, + 668 + ], + "score": 0.89, + "content": "h _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 656, + 506, + 668 + ], + "score": 1.0, + "content": ". Specifically for simple MNIST task, we use the ResNet18 model provided in Pytorch.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 665, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 339, + 681 + ], + "score": 1.0, + "content": "We use the model provided by [22], which obtains nearly", + "type": "text" + }, + { + "bbox": [ + 339, + 667, + 358, + 678 + ], + "score": 0.88, + "content": "8 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 665, + 506, + 681 + ], + "score": 1.0, + "content": "clean accuracy on CIFAR100 using", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 677, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 371, + 691 + ], + "score": 1.0, + "content": "EfficientNet [23] and the model provided by [24], which has nearly", + "type": "text" + }, + { + "bbox": [ + 371, + 678, + 391, + 689 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 677, + 507, + 691 + ], + "score": 1.0, + "content": "clean accuracy on CIFAR10.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 208, + 701 + ], + "score": 1.0, + "content": "In the neural ODE layer,", + "type": "text" + }, + { + "bbox": [ + 208, + 689, + 219, + 700 + ], + "score": 0.87, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "consists of 2 FC layers. During the trainings of SODEF (except in the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "experiment included in Section 4.2), we train the neural network with the fixed FC introduced in Sec-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 262, + 724 + ], + "score": 1.0, + "content": "tion 3.1. In the first 30 epochs, we fixed", + "type": "text" + }, + { + "bbox": [ + 262, + 711, + 273, + 722 + ], + "score": 0.88, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 710, + 376, + 724 + ], + "score": 1.0, + "content": "to let the feature extractor", + "type": "text" + }, + { + "bbox": [ + 376, + 711, + 389, + 723 + ], + "score": 0.9, + "content": "h _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 710, + 506, + 724 + ], + "score": 1.0, + "content": "learn a feature representation", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 237, + 85 + ], + "score": 1.0, + "content": "with only the cross-entropy loss", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 238, + 73, + 244, + 83 + ], + "score": 0.68, + "content": "\\ell", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 244, + 72, + 429, + 85 + ], + "score": 1.0, + "content": ", and in the remaining 120 epochs, we release", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 429, + 73, + 442, + 85 + ], + "score": 0.89, + "content": "h _ { \\phi }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 442, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "to further train", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 107, + 84, + 117, + 95 + ], + "score": 0.83, + "content": "f _ { \\theta }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 118, + 83, + 178, + 96 + ], + "score": 1.0, + "content": "using (7) with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 178, + 84, + 209, + 94 + ], + "score": 0.9, + "content": "\\alpha _ { 1 } = 1", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 209, + 83, + 227, + 96 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 228, + 84, + 295, + 95 + ], + "score": 0.89, + "content": "\\alpha _ { 2 } = \\alpha _ { 3 } = 0 . 0 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 295, + 83, + 506, + 96 + ], + "score": 1.0, + "content": ". For CIFAR10 and CIFAR100, the pixel values are", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 168, + 108 + ], + "score": 1.0, + "content": "normalized by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 168, + 95, + 212, + 106 + ], + "score": 0.92, + "content": "( x - \\mu ) / \\sigma", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 212, + 93, + 240, + 108 + ], + "score": 1.0, + "content": "where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 241, + 95, + 358, + 106 + ], + "score": 0.8, + "content": "\\mu = [ 0 . 4 9 1 4 , 0 . 4 8 2 2 , 0 . 4 4 6 5 ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 359, + 93, + 378, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 378, + 95, + 501, + 106 + ], + "score": 0.83, + "content": "\\sigma = [ 0 . 2 0 2 3 , 0 . 1 9 9 4 , 0 . 2 0 1 0 ] ^ { 4 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 502, + 93, + 506, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "To show that our SODEF is compatible with many defense methods and can be applied to any neural", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "network’s final regression layer, we conduct an experiment where we use a recently proposed robust", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "network TRADES [25] as the feature extractor in our SODEF. The pretrained model is provided here", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 316, + 150 + ], + "score": 1.0, + "content": "5, and we choose the model with architecture \"WRN-", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 316, + 138, + 344, + 149 + ], + "score": 0.3, + "content": "3 4 \\mathrm { - } 1 0 \"", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 345, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "to conduct our experiments. Besides the", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 146, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 506, + 163 + ], + "score": 1.0, + "content": "two vanilla white-box attacks FGSM and PGD as metioned in Section 3.1, we also include a strong", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "ensemble attack AutoAttack [26], which sequentially performs attack using all of the following four", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 103, + 167, + 507, + 186 + ], + "spans": [ + { + "bbox": [ + 103, + 167, + 507, + 186 + ], + "score": 1.0, + "content": "individual attacks: three white-box attacks APGDCE, APGDTDLR and FABT[27], and one black-box", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "Square attack [28]. 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The pretrained model is provided here", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 316, + 150 + ], + "score": 1.0, + "content": "5, and we choose the model with architecture \"WRN-", + "type": "text" + }, + { + "bbox": [ + 316, + 138, + 344, + 149 + ], + "score": 0.3, + "content": "3 4 \\mathrm { - } 1 0 \"", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "to conduct our experiments. 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We refer the reader to the the supplementary material for more details of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 193, + 423, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 423, + 205 + ], + "score": 1.0, + "content": "attacks used in this paper, where, in additional, more experiments are included.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 217, + 236, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 238, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 238, + 232 + ], + "score": 1.0, + "content": "4.2 Compatibility of SODEF", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "Adversarial training (AT) is one of the most effective strategies for defending adversarial attacks.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "TRADES [25] is one of the adversarial training defense methods with combinations of tricks of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "warmup, early stopping, weight decay, batch size and other hyper parameter settings. 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We set the same integration time for ODE, TisODE and SODEF. 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We also test larger integration time", + "type": "text" + }, + { + "bbox": [ + 362, + 605, + 393, + 615 + ], + "score": 0.9, + "content": "T > 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 604, + 506, + 619 + ], + "score": 1.0, + "content": ", but do not see any obvious", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 616, + 168, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 168, + 628 + ], + "score": 1.0, + "content": "improvements.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 517, + 506, + 628 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 640, + 322, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 322, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 322, + 653 + ], + "score": 1.0, + "content": "4.4 Performance Comparison Under AutoAttack", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "For a comparison, we provide the results of applying AutoAttack to other baseline models mentioned", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "in the paper. We set the same integration time for ODE, TisODE and SODEF. We observe that for the", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 660, + 505, + 683 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 149, + 92, + 463, + 175 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 69, + 503, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 68, + 504, + 82 + ], + "spans": [ + { + "bbox": [ + 105, + 68, + 236, + 82 + ], + "score": 1.0, + "content": "Table 4: Classification accuracy", + "type": "text" + }, + { + "bbox": [ + 236, + 70, + 252, + 81 + ], + "score": 0.76, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 68, + 491, + 82 + ], + "score": 1.0, + "content": "under AutoAttack on adversarial CIFAR100 examples with", + "type": "text" + }, + { + "bbox": [ + 492, + 70, + 504, + 81 + ], + "score": 0.82, + "content": "\\mathcal { L } _ { 2 }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 80, + 343, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 132, + 92 + ], + "score": 1.0, + "content": "norm,", + "type": "text" + }, + { + "bbox": [ + 132, + 81, + 164, + 91 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 80, + 284, + 92 + ], + "score": 1.0, + "content": "and different integration time", + "type": "text" + }, + { + "bbox": [ + 284, + 81, + 293, + 90 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 80, + 343, + 92 + ], + "score": 1.0, + "content": "for SODEF.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 149, + 92, + 463, + 175 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 92, + 463, + 175 + ], + "spans": [ + { + "bbox": [ + 149, + 92, + 463, + 175 + ], + "score": 0.98, + "html": "
Attack/T135678910
Clean88.0088.1288.1588.0087.9288.0088.0588.10
APGDcE17.2021.3321.0523.6769.6785.3387.1086.88
APGDLR21.0221.0022.0026.0063.3086.9086.2086.54
FABT86.3385.1086.3687.7087.6786.5586.2285.93
Square84.6786.2287.0587.2086.9086.3387.0586.75
AutoAttack2.003.534.874.3330.6678.8078.9779.10
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Attack/ModelNoODEODETisODESODEF
Clean88.0087.9088.0088.10
APGDCE23.306.7514.3286.88
7.3322.0024.2086.54
FAB79.3078.6777.1685.93
Square84.5285.6786.3286.75
AutoAttack0.001.334.0679.10
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AttackPara.no odeODETisODESODEF
None-99.4599.4299.4399.44
FGSM∈=0.310.0329.636.7063.36
PGD∈= 0.30.311.561.8245.25
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AttackPara.no odeODETisODESODEF
None-95.294.995.195.0
FGSM∈=0.147.3145.2343.2868.05
PGD∈=0.13.093.213.8055.59
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In particular, SODEF achieves an absolute", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 308, + 684 + ], + "score": 1.0, + "content": "percentage point improvement over ODE net up to", + "type": "text" + }, + { + "bbox": [ + 308, + 672, + 341, + 684 + ], + "score": 0.9, + "content": "5 2 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 673, + 435, + 684 + ], + "score": 1.0, + "content": "and over TisODE up to", + "type": "text" + }, + { + "bbox": [ + 435, + 672, + 468, + 683 + ], + "score": 0.89, + "content": "5 2 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 673, + 505, + 684 + ], + "score": 1.0, + "content": "for PGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 684, + 135, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 135, + 694 + ], + "score": 1.0, + "content": "attack.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 505, + 724 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "For CIFAR-100, the results in the supplementary material shows that the most effective attack", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 336, + 723 + ], + "score": 1.0, + "content": "causes the classification accuracy to drop relatively by", + "type": "text" + }, + { + "bbox": [ + 337, + 710, + 421, + 725 + ], + "score": 0.92, + "content": "\\begin{array} { r } { 7 4 . 6 \\% = \\frac { 8 8 . 0 - 2 2 . 3 5 } { 8 8 . 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "88.0−22.35 for SODEF and by", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 149, + 92, + 463, + 175 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 69, + 503, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 68, + 504, + 82 + ], + "spans": [ + { + "bbox": [ + 105, + 68, + 236, + 82 + ], + "score": 1.0, + "content": "Table 4: Classification accuracy", + "type": "text" + }, + { + "bbox": [ + 236, + 70, + 252, + 81 + ], + "score": 0.76, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 68, + 491, + 82 + ], + "score": 1.0, + "content": "under AutoAttack on adversarial CIFAR100 examples with", + "type": "text" + }, + { + "bbox": [ + 492, + 70, + 504, + 81 + ], + "score": 0.82, + "content": "\\mathcal { L } _ { 2 }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 80, + 343, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 132, + 92 + ], + "score": 1.0, + "content": "norm,", + "type": "text" + }, + { + "bbox": [ + 132, + 81, + 164, + 91 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 80, + 284, + 92 + ], + "score": 1.0, + "content": "and different integration time", + "type": "text" + }, + { + "bbox": [ + 284, + 81, + 293, + 90 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 80, + 343, + 92 + ], + "score": 1.0, + "content": "for SODEF.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 149, + 92, + 463, + 175 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 92, + 463, + 175 + ], + "spans": [ + { + "bbox": [ + 149, + 92, + 463, + 175 + ], + "score": 0.98, + "html": "
Attack/T135678910
Clean88.0088.1288.1588.0087.9288.0088.0588.10
APGDcE17.2021.3321.0523.6769.6785.3387.1086.88
APGDLR21.0221.0022.0026.0063.3086.9086.2086.54
FABT86.3385.1086.3687.7087.6786.5586.2285.93
Square84.6786.2287.0587.2086.9086.3387.0586.75
AutoAttack2.003.534.874.3330.6678.8078.9779.10
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Attack/ModelNoODEODETisODESODEF
Clean88.0087.9088.0088.10
APGDCE23.306.7514.3286.88
7.3322.0024.2086.54
FAB79.3078.6777.1685.93
Square84.5285.6786.3286.75
AutoAttack0.001.334.0679.10
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AttackPara.no odeODETisODESODEF
None-99.4599.4299.4399.44
FGSM∈=0.310.0329.636.7063.36
PGD∈= 0.30.311.561.8245.25
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AttackPara.no odeODETisODESODEF
None-95.294.995.195.0
FGSM∈=0.147.3145.2343.2868.05
PGD∈=0.13.093.213.8055.59
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Neither ODE net nor TisODE net can improve the classification accuracy under PGD attack", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 436, + 107 + ], + "score": 1.0, + "content": "by a big margin, e.g. TisODE net only improves the classification accuracy from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 436, + 94, + 464, + 105 + ], + "score": 0.86, + "content": "2 . 3 9 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 464, + 95, + 475, + 107 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 475, + 94, + 502, + 105 + ], + "score": 0.88, + "content": "3 . 4 4 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 503, + 95, + 506, + 107 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 371, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 371, + 118 + ], + "score": 1.0, + "content": "while SODEF still shows clear defense capability in this scenario.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 700, + 506, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 71, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 71, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 71, + 184, + 86 + ], + "score": 0.92, + "content": "\\begin{array} { r } { 9 7 . 3 \\% = \\frac { 8 8 . 3 - 2 . 3 9 } { 8 8 . 3 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 71, + 506, + 87 + ], + "score": 1.0, + "content": "for vanilla EfficientNet, which is pre-trained on ImageNet to obtain a top clean", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "accuracy. Neither ODE net nor TisODE net can improve the classification accuracy under PGD attack", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 436, + 107 + ], + "score": 1.0, + "content": "by a big margin, e.g. TisODE net only improves the classification accuracy from", + "type": "text" + }, + { + "bbox": [ + 436, + 94, + 464, + 105 + ], + "score": 0.86, + "content": "2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 95, + 475, + 107 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 475, + 94, + 502, + 105 + ], + "score": 0.88, + "content": "3 . 4 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 95, + 506, + 107 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 371, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 371, + 118 + ], + "score": 1.0, + "content": "while SODEF still shows clear defense capability in this scenario.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 131, + 201, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 203, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 203, + 144 + ], + "score": 1.0, + "content": "4.6 Ablation Studies", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 108, + 151, + 504, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "score": 1.0, + "content": "The impact of the ODE with and without the SODEF regularizers in (7) has been presented in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 176 + ], + "score": 1.0, + "content": "above comparisons between SODEF and ODE. 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AttackPara.SODEFSODEF-
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It can be seen that having diversity control improves the robustness.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "One possible reason for this phenomenon given in Section 3 is that diversity promotion with a fixed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 349, + 377 + ], + "score": 1.0, + "content": "designed FC attempts to make the embedding feature support", + "type": "text" + }, + { + "bbox": [ + 349, + 365, + 361, + 375 + ], + "score": 0.86, + "content": "E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 364, + 414, + 377 + ], + "score": 1.0, + "content": "of each class", + "type": "text" + }, + { + "bbox": [ + 415, + 365, + 419, + 374 + ], + "score": 0.57, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "disjoint to each other", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 374, + 446, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 356, + 389 + ], + "score": 1.0, + "content": "and therefore the Lyapunov-stable equilibrium points for each", + "type": "text" + }, + { + "bbox": [ + 356, + 376, + 367, + 387 + ], + "score": 0.88, + "content": "E _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 374, + 446, + 389 + ], + "score": 1.0, + "content": "are well separated.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 107, + 399, + 229, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 230, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 230, + 415 + ], + "score": 1.0, + "content": "4.6.2 Transferability Study", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "score": 1.0, + "content": "Transferability study is carried out on CIFAR-10, where the adversarial examples are generated", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "using FGSM and PGD attacks using ResNet18 without any ODEs. The classification accuracy drops", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 129, + 452 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 440, + 162, + 451 + ], + "score": 0.87, + "content": "6 8 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 441, + 174, + 452 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 174, + 440, + 195, + 451 + ], + "score": 0.88, + "content": "5 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 441, + 263, + 452 + ], + "score": 1.0, + "content": "for FGSM with", + "type": "text" + }, + { + "bbox": [ + 263, + 441, + 297, + 451 + ], + "score": 0.88, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 441, + 342, + 452 + ], + "score": 1.0, + "content": ", and from", + "type": "text" + }, + { + "bbox": [ + 343, + 440, + 375, + 451 + ], + "score": 0.89, + "content": "5 5 . 5 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 441, + 387, + 452 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 388, + 440, + 408, + 451 + ], + "score": 0.88, + "content": "3 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 441, + 469, + 452 + ], + "score": 1.0, + "content": "for PGD with", + "type": "text" + }, + { + "bbox": [ + 469, + 441, + 503, + 451 + ], + "score": 0.88, + "content": "\\epsilon = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 441, + 506, + 452 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "One possible reason for this phenomenon is that ODEs have obfuscated gradient masking effect as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "discussed in [15], and a transfer attack may deteriorate the defense effect. However, as we observe", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "score": 1.0, + "content": "from Table 7, even with a transfer attack on SODEF, it still performs better than other ODEs without", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 484, + 172, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 172, + 496 + ], + "score": 1.0, + "content": "transfer attacks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 512, + 183, + 525 + ], + "lines": [ + { + "bbox": [ + 104, + 510, + 185, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 185, + 528 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 537, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "In this paper, we have developed a new neural ODE network, SODEF, to suppress input perturbations.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 560 + ], + "score": 1.0, + "content": "SODEF is compatible with any existing neural networks and can thus be appended to the state-of-the-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "art networks to increase their robustness to adversarial attacks. We demonstrated empirically and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 584 + ], + "score": 1.0, + "content": "theoretically that the robustness of SODEF mainly derives from its stability and proposed a training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "method that imposes constraints to ensure all eigenvalues of the Jacobian matrix of the neural ODE", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "layer have negative real parts. 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The classification accuracy drops", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 129, + 452 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 440, + 162, + 451 + ], + "score": 0.87, + "content": "6 8 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 441, + 174, + 452 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 174, + 440, + 195, + 451 + ], + "score": 0.88, + "content": "5 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 441, + 263, + 452 + ], + "score": 1.0, + "content": "for FGSM with", + "type": "text" + }, + { + "bbox": [ + 263, + 441, + 297, + 451 + ], + "score": 0.88, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 441, + 342, + 452 + ], + "score": 1.0, + "content": ", and from", + "type": "text" + }, + { + "bbox": [ + 343, + 440, + 375, + 451 + ], + "score": 0.89, + "content": "5 5 . 5 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 441, + 387, + 452 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 388, + 440, + 408, + 451 + ], + "score": 0.88, + "content": "3 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 441, + 469, + 452 + ], + "score": 1.0, + "content": "for PGD with", + "type": "text" + }, + { + "bbox": [ + 469, + 441, + 503, + 451 + ], + "score": 0.88, + "content": "\\epsilon = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 441, + 506, + 452 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "One possible reason for this phenomenon is that ODEs have obfuscated gradient masking effect as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "discussed in [15], and a transfer attack may deteriorate the defense effect. However, as we observe", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "score": 1.0, + "content": "from Table 7, even with a transfer attack on SODEF, it still performs better than other ODEs without", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 484, + 172, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 172, + 496 + ], + "score": 1.0, + "content": "transfer attacks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 418, + 506, + 496 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 512, + 183, + 525 + ], + "lines": [ + { + "bbox": [ + 104, + 510, + 185, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 185, + 528 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 537, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "In this paper, we have developed a new neural ODE network, SODEF, to suppress input perturbations.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 560 + ], + "score": 1.0, + "content": "SODEF is compatible with any existing neural networks and can thus be appended to the state-of-the-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "art networks to increase their robustness to adversarial attacks. We demonstrated empirically and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 584 + ], + "score": 1.0, + "content": "theoretically that the robustness of SODEF mainly derives from its stability and proposed a training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "method that imposes constraints to ensure all eigenvalues of the Jacobian matrix of the neural ODE", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "layer have negative real parts. 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Square84.6786.2287.0587.2086.9086.3387.0586.75
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Attack/ModelNoODEODETisODESODEF
Clean88.0087.9088.0088.10
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AttackPara.no odeODETisODESODEF
None-95.294.995.195.0
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None-99.4599.4299.4399.44
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"page_no": 12, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/B1g5sA4twr/B1g5sA4twr.md b/parse/train/B1g5sA4twr/B1g5sA4twr.md new file mode 100644 index 0000000000000000000000000000000000000000..e7c0556e31a41dbc3e1bbfff0fd11a331d5e86ae --- /dev/null +++ b/parse/train/B1g5sA4twr/B1g5sA4twr.md @@ -0,0 +1,442 @@ +# DEEP DOUBLE DESCENT: WHERE BIGGER MODELS AND MORE DATA HURT + +Preetum Nakkiran∗ Harvard University + +Gal Kaplun† Harvard University + +Yamini Bansal† Harvard University + +Tristan Yang Harvard University + +Boaz Barak Harvard University + +Ilya Sutskever OpenAI + +# ABSTRACT + +We show that a variety of modern deep learning tasks exhibit a “double-descent” phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a function of model size, but also as a function of the number of training epochs. We unify the above phenomena by defining a new complexity measure we call the effective model complexity and conjecture a generalized double descent with respect to this measure. Furthermore, our notion of model complexity allows us to identify certain regimes where increasing (even quadrupling) the number of train samples actually hurts test performance. + +# 1 INTRODUCTION + +![](images/c4a62cbdb7fbeb5034e735ae46b0639e991da1ee0ce7c9ca2623db843dff96c3.jpg) +Figure 1: Left: Train and test error as a function of model size, for ResNet18s of varying width on CIFAR-10 with $15 \%$ label noise. Right: Test error, shown for varying train epochs. All models trained using Adam for 4K epochs. The largest model (width 64) corresponds to standard ResNet18. + +The bias-variance trade-off is a fundamental concept in classical statistical learning theory (e.g., Hastie et al. (2005)). The idea is that models of higher complexity have lower bias but higher variance. According to this theory, once model complexity passes a certain threshold, models “overfit” with the variance term dominating the test error, and hence from this point onward, increasing model complexity will only decrease performance (i.e., increase test error). Hence conventional wisdom in classical statistics is that, once we pass a certain threshold, “larger models are worse.” + +However, modern neural networks exhibit no such phenomenon. Such networks have millions of parameters, more than enough to fit even random labels (Zhang et al. (2016)), and yet they perform much better on many tasks than smaller models. Indeed, conventional wisdom among practitioners is that “larger models are better’’ (Krizhevsky et al. (2012), Huang et al. (2018), Szegedy et al. + +![](images/08ba3538956dd39cc42384ec2f1e07bf08c6fc91c71879c2a0bb15a89998beb1.jpg) +Figure 2: Left: Test error as a function of model size and train epochs. The horizontal line corresponds to model-wise double descent–varying model size while training for as long as possible. The vertical line corresponds to epoch-wise double descent, with test error undergoing double-descent as train time increases. Right Train error of the corresponding models. All models are Resnet18s trained on CIFAR-10 with $15 \%$ label noise, data-augmentation, and Adam for up to 4K epochs. + +(2015), Radford et al. (2019)). The effect of training time on test performance is also up for debate. In some settings, “early stopping” improves test performance, while in other settings training neural networks to zero training error only improves performance. Finally, if there is one thing both classical statisticians and deep learning practitioners agree on is “more data is always better”. + +In this paper, we present empirical evidence that both reconcile and challenge some of the above “conventional wisdoms.” We show that many deep learning settings have two different regimes. In the under-parameterized regime, where the model complexity is small compared to the number of samples, the test error as a function of model complexity follows the U-like behavior predicted by the classical bias/variance tradeoff. However, once model complexity is sufficiently large to interpolate i.e., achieve (close to) zero training error, then increasing complexity only decreases test error, following the modern intuition of “bigger models are better”. Similar behavior was previously observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al. (2018), and Geiger et al. (2019b). This phenomenon was first postulated in generality by Belkin et al. (2018) who named it “double descent”, and demonstrated it for decision trees, random features, and 2-layer neural networks with $\ell _ { 2 }$ loss, on a variety of learning tasks including MNIST and CIFAR-10. + +Main contributions. We show that double descent is a robust phenomenon that occurs in a variety of tasks, architectures, and optimization methods (see Figure 1 and Section 5; our experiments are summarized in Table A). Moreover, we propose a much more general notion of “double descent” that goes beyond varying the number of parameters. We define the effective model complexity (EMC) of a training procedure as the maximum number of samples on which it can achieve close to zero training error. The EMC depends not just on the data distribution and the architecture of the classifier but also on the training procedure—and in particular increasing training time will increase the EMC. + +We hypothesize that for many natural models and learning algorithms, double descent occurs as a function of the EMC. Indeed we observe “epoch-wise double descent” when we keep the model fixed and increase the training time, with performance following a classical U-like curve in the underfitting stage (when the EMC is smaller than the number of samples) and then improving with training time once the EMC is sufficiently larger than the number of samples (see Figure 2). As a corollary, early stopping only helps in the relatively narrow parameter regime of critically parameterized models. + +Sample non-monotonicity. Finally, our results shed light on test performance as a function of the number of train samples. Since the test error peaks around the point where EMC matches the number of samples (the transition from the under- to over-parameterization), increasing the number of samples has the effect of shifting this peak to the right. While in most settings increasing the number of samples decreases error, this shifting effect can sometimes result in a setting where more data is worse! For example, Figure 3 demonstrates cases in which increasing the number of samples by a factor of 4.5 results in worse test performance. + +![](images/5c137bc12053fdd4c20394058fe9f6b4430c5f548f5d67abd01978eb2d8cc676.jpg) +Figure 3: Test loss (per-token perplexity) as a function of Transformer model size (embedding dimension $d _ { m o d e l . }$ ) on language translation (IWSLT‘14 German-to-English). The curve for $1 8 \mathrm { k }$ samples is generally lower than the one for $4 \mathrm { k }$ samples, but also shifted to the right, since fitting 18k samples requires a larger model. Thus, for some models, the performance for $1 8 \mathrm { k }$ samples is worse than for $4 \mathrm { k \Omega }$ samples. + +# 2 OUR RESULTS + +To state our hypothesis more precisely, we define the notion of effective model complexity. We define a training procedure $\tau$ to be any procedure that takes as input a set $S = \{ ( x _ { 1 } , y _ { 1 } ) , \dots , ( x _ { n } , y _ { n } ) \}$ of labeled training samples and outputs a classifier $\mathcal { T } ( S )$ mapping data to labels. We define the effective model complexity of $\tau$ (w.r.t. distribution $\mathcal { D }$ ) to be the maximum number of samples $n$ on which $\tau$ achieves on average $\approx 0$ training error. + +Definition 1 (Effective Model Complexity) The Effective Model Complexity (EMC) of a training procedure $\tau$ , with respect to distribution $\mathcal { D }$ and parameter $\epsilon > 0$ , is defined as: + +$$ +\operatorname { E M C } _ { { \mathscr { D } } , \epsilon } ( { \mathscr { T } } ) : = \operatorname* { m a x } \left\{ n \mid \mathbb { E } _ { S \sim { \mathscr { D } } ^ { n } } [ \operatorname { E r r o r } _ { S } ( { \mathscr { T } } ( S ) ) ] \leq \epsilon \right\} +$$ + +where Error $s ( M )$ is the mean error of model $M$ on train samples $S$ . + +Our main hypothesis can be informally stated as follows: + +Hypothesis 1 (Generalized Double Descent hypothesis, informal) For any natural data distribution $\mathcal { D }$ , neural-network-based training procedure $\tau$ , and small $\epsilon > 0$ , if we consider the task of predicting labels based on n samples from $\mathcal { D }$ then: + +Under-paremeterized regime. If $\cdot _ { \mathrm { E M C } _ { { \mathscr D } , \epsilon } } ( \mathcal T )$ is sufficiently smaller than $n _ { \ast }$ , any perturbation of $\tau$ that increases its effective complexity will decrease the test error. + +Over-parameterized regime. If $\mathrm { E M C } _ { { \cal D } , \epsilon } ( T )$ is sufficiently larger than $n$ , any perturbation of $\tau$ that increases its effective complexity will decrease the test error. + +Critically parameterized regime. I $f \mathrm { E M C } _ { \mathcal { D } , \epsilon } ( \mathcal { T } ) \approx n$ , then a perturbation of $\tau$ that increases its effective complexity might decrease or increase the test error. + +Hypothesis 1 is informal in several ways. We do not have a principled way to choose the parameter $\epsilon$ (and currently heuristically use $\epsilon = 0 . 1$ ). We also are yet to have a formal specification for “sufficiently smaller” and “sufficiently larger”. Our experiments suggest that there is a critical interval around the interpolation threshold when $\mathrm { E M C } _ { \mathcal { D } , \epsilon } ( \mathcal { T } ) = n$ : below and above this interval increasing complexity helps performance, while within this interval it may hurt performance. The width of the critical interval depends on both the distribution and the training procedure in ways we do not yet completely understand. + +We believe Hypothesis 1 sheds light on the interaction between optimization algorithms, model size, and test performance and helps reconcile some of the competing intuitions about them. The main result of this paper is an experimental validation of Hypothesis 1 under a variety of settings, where we considered several natural choices of datasets, architectures, and optimization algorithms, and we changed the “interpolation threshold” by varying the number of model parameters, the length of training, the amount of label noise in the distribution, and the number of train samples. + +Model-wise Double Descent. In Section 5, we study the test error of models of increasing size, for a fixed large number of optimization steps. We show that “model-wise double-descent” occurs for various modern datasets (CIFAR-10, CIFAR-100, IWSLT‘14 de-en, with varying amounts of label noise), model architectures (CNNs, ResNets, Transformers), optimizers (SGD, Adam), number of train samples, and training procedures (data-augmentation, and regularization). Moreover, the peak in test error systematically occurs at the interpolation threshold. In particular, we demonstrate realistic settings in which bigger models are worse. + +Epoch-wise Double Descent. In Section 6, we study the test error of a fixed, large architecture over the course of training. We demonstrate, in similar settings as above, a corresponding peak in test performance when models are trained just long enough to reach $\approx 0$ train error. The test error of a large model first decreases (at the beginning of training), then increases (around the critical regime), then decreases once more (at the end of training)—that is, training longer can correct overfitting. + +Sample-wise Non-monotonicity. In Section 7, we study the test error of a fixed model and training procedure, for varying number of train samples. Consistent with our generalized double-descent hypothesis, we observe distinct test behavior in the “critical regime”, when the number of samples is near the maximum that the model can fit. This often manifests as a long plateau region, in which taking significantly more data might not help when training to completion (as is the case for CNNs on CIFAR-10). Moreover, we show settings (Transformers on IWSLT‘14 en-de), where this manifests as a peak—and for a fixed architecture and training procedure, more data actually hurts. + +Remarks on Label Noise. We observe all forms of double descent most strongly in settings with label noise in the train set (as is often the case when collecting train data in the real-world). However, we also show several realistic settings with a test-error peak even without label noise: ResNets (Figure 4a) and CNNs (Figure 20) on CIFAR-100; Transformers on IWSLT‘14 (Figure 8). Moreover, all our experiments demonstrate distinctly different test behavior in the critical regime— often manifesting as a “plateau” in the test error in the noiseless case which develops into a peak with added label noise. See Section 8 for further discussion. + +# 3 RELATED WORK + +Model-wise double descent was first proposed as a general phenomenon by Belkin et al. (2018). Similar behavior had been observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al. (2018), and Geiger et al. (2019b). Subsequently, there has been a large body of work studying the double descent phenomenon. A growing list of papers that theoretically analyze it in the tractable setting of linear least squares regression includes Belkin et al. (2019); Hastie et al. (2019); Bartlett et al. (2019); Muthukumar et al. (2019); Bibas et al. (2019); Mitra (2019); Mei & Montanari (2019). Moreover, Geiger et al. (2019a) provide preliminary results for model-wise double descent in convolutional networks trained on CIFAR-10. Our work differs from the above papers in two crucial aspects: First, we extend the idea of double-descent beyond the number of parameters to incorporate the training procedure under a unified notion of “Effective Model Complexity”, leading to novel insights like epoch-wise double descent and sample non-monotonicity. The notion that increasing train time corresponds to increasing complexity was also presented in Nakkiran et al. (2019). Second, we provide an extensive and rigorous demonstration of double-descent in modern deep learning, spanning a variety of architectures, datasets, and optimization procedures. An extended discussion of the related work is provided in Appendix C. + +# 4 EXPERIMENTAL SETUP + +We briefly describe the experimental setup here; full details are in Appendix B 1. We consider three families of architectures: ResNets, standard CNNs, and Transformers. ResNets: We parameterize a family of ResNet18s (He et al. (2016)) by scaling the width (number of filters) of convolutional layers. Specifically, we use layer widths $[ k , 2 k , \bar { 4 } k , 8 k ]$ for varying $k$ . The standard ResNet18 corresponds to $k = 6 4$ . Standard CNNs: We consider a simple family of 5-layer CNNs, with 4 convolutional layers of widths $[ k , 2 k , 4 k , 8 k ]$ for varying $k$ , and a fully-connected layer. For context, the CNN with width $k = 6 4$ , can reach over $9 0 \%$ test accuracy on CIFAR-10 with dataaugmentation. Transformers: We consider the 6 layer encoder-decoder from Vaswani et al. (2017), as implemented by Ott et al. (2019). We scale the size of the network by modifying the embedding dimension $d _ { \mathrm { m o d e l } }$ , and setting the width of the fully-connected layers proportionally $( d _ { \mathrm { f f } } = 4 \cdot d _ { \mathrm { m o d e l } } )$ . + +For ResNets and CNNs, we train with cross-entropy loss, and the following optimizers: (1) Adam with learning-rate 0.0001 for 4K epochs; (2) SGD with learning rate $\propto \frac { 1 } { \sqrt { T } }$ for 500K gradient steps. We train Transformers for 80K gradient steps, with $10 \%$ label smoothing and no drop-out. + +Label Noise. In our experiments, label noise of probability $p$ refers to training on a samples which have the correct label with probability $( 1 - p )$ , and a uniformly random incorrect label otherwise (label noise is sampled only once and not per epoch). Figure 1 plots test error on the noisy distribution, while the remaining figures plot test error with respect to the clean distribution (the two curves are just linear rescaling of one another). + +# 5 MODEL-WISE DOUBLE DESCENT + +![](images/faae61f59becd773c511ccf37b8969226757057fa9adbb5ab757470a3e69071b.jpg) +(a) CIFAR-100. There is a peak in test error even with no label noise. +Figure 4: Model-wise double descent for ResNet18s. Trained on CIFAR-100 and CIFAR-10, with varying label noise. Optimized using Adam with LR 0.0001 for 4K epochs, and data-augmentation. + +![](images/e3fd45bf969858879ad23096c5a3e4c6e094f04bd2b0a277a1d81691dbc44055.jpg) +(b) CIFAR-10. There is a “plateau” in test error around the interpolation point with no label noise, which develops into a peak for added label noise. + +In this section, we study the test error of models of increasing size, when training to completion (for a fixed large number of optimization steps). We demonstrate model-wise double descent across different architectures, datasets, optimizers, and training procedures. The critical region exhibits distinctly different test behavior around the interpolation point and there is often a peak in test error that becomes more prominent in settings with label noise. + +For the experiments in this section (Figures 4, 5, 6, 7, 8), notice that all modifications which increase the interpolation threshold (such as adding label noise, using data augmentation, and increasing the number of train samples) also correspondingly shift the peak in test error towards larger models. Additional plots showing the early-stopping behavior of these models, and additional experiments showing double descent in settings with no label noise (e.g. Figure 19) are in Appendix E.2. We also observed model-wise double descent for adversarial training, with a prominent robust test error peak even in settings without label noise. See Figure 26 in Appendix E.2. + +Discussion. Fully understanding the mechanisms behind model-wise double descent in deep neural networks remains an important open question. However, an analog of model-wise double descent occurs even for linear models. A recent stream of theoretical works analyzes this setting (Bartlett et al. (2019); Muthukumar et al. (2019); Belkin et al. (2019); Mei & Montanari (2019); Hastie et al. (2019)). We believe similar mechanisms may be at work in deep neural networks. + +Informally, our intuition is that for model-sizes at the interpolation threshold, there is effectively only one model that fits the train data and this interpolating model is very sensitive to noise in the train set and/or model mis-specification. That is, since the model is just barely able to fit the train data, forcing it to fit even slightly-noisy or mis-specified labels will destroy its global structure, and result in high test error. (See Figure 28 in the Appendix for an experiment demonstrating this noise sensitivity, by showing that ensembling helps significantly in the critically-parameterized regime). However for over-parameterized models, there are many interpolating models that fit the train set, and SGD is able to find one that “memorizes” (or “absorbs”) the noise while still performing well on the distribution. + +![](images/44d6b8438c0d02d0c76c80ad346b4c82268cb44163651e6cd9d1516b78b9044e.jpg) +Figure 5: Effect of Data Augmentation. 5-layer CNNs on CIFAR10, with and without dataaugmentation. Data-augmentation shifts the interpolation threshold to the right, shifting the test error peak accordingly. Optimized using SGD for 500K steps. See Figure 27 for larger models. + +![](images/8100cdf1068362adc8d85ddd7ddddf0ee7d65cdfb04eb08ce1bab31e8561ac9f.jpg) + +![](images/f4e2caa4b4e7287a955aced3cc3d523468cc0ef98464146881383b6d073e6465.jpg) +Figure 6: SGD vs. Adam. 5-Layer CNNs on CIFAR-10 with no label noise, and no data augmentation. Optimized using SGD for 500K gradient steps, and Adam for 4K epochs. + +![](images/c8160258a06627d94e03682b0cb3a5ea98fb3a6cc2efc061ae766953b89696b3.jpg) +Figure 7: Noiseless settings. 5-layer CNNs on CIFAR-100 with no label noise; note the peak in test error. Trained with SGD and no data augmentation. See Figure 20 for the early-stopping behavior of these models. + +The above intuition is theoretically justified for linear models. In general, this situation manifests even without label noise for linear models (Mei & Montanari (2019)), and occurs whenever there is model mis-specification between the structure of the true distribution and the model family. We believe this intuition extends to deep learning as well, and it is consistent with our experiments. + +![](images/2b473a4c748ee138162c045888f88cd9f7560771b9e3d4fa5dd16e01c2027a36.jpg) +Figure 8: Transformers on language translation tasks: Multi-head-attention encoderdecoder Transformer model trained for $8 0 \mathrm { k }$ gradient steps with labeled smoothed cross-entropy loss on IWSLT‘14 Germanto-English (160K sentences) and WMT‘14 English-to-French (subsampled to 200K sentences) dataset. Test loss is measured as pertoken perplexity. + +# 6 EPOCH-WISE DOUBLE DESCENT + +In this section, we demonstrate a novel form of double-descent with respect to training epochs, which is consistent with our unified view of effective model complexity (EMC) and the generalized double descent hypothesis. Increasing the train time increases the EMC—and thus a sufficiently large model transitions from under- to over-parameterized over the course of training. + +![](images/bdca8d3385fd5d02c8443adb0acbb61793b4152837145966b74e283d6ca4c7db.jpg) +Figure 9: Left: Training dynamics for models in three regimes. Models are ResNet18s on CIFAR10 with $20 \%$ label noise, trained using Adam with learning rate 0.0001, and data augmentation. Right: Test error over (Model size $\times$ Epochs). Three slices of this plot are shown on the left. + +As illustrated in Figure 9, sufficiently large models can undergo a “double descent” behavior where test error first decreases then increases near the interpolation threshold, and then decreases again. In contrast, for “medium sized” models, for which training to completion will only barely reach $\approx 0$ error, the test error as a function of training time will follow a classical U-like curve where it is better to stop early. Models that are too small to reach the approximation threshold will remain in the “under parameterized” regime where increasing train time monotonically decreases test error. Our experiments (Figure 10) show that many settings of dataset and architecture exhibit epoch-wise double descent, in the presence of label noise. Further, this phenomenon is robust across optimizer variations and learning rate schedules (see additional experiments in Appendix E.1). As in modelwise double descent, the test error peak is accentuated with label noise. + +Conventional wisdom suggests that training is split into two phases: (1) In the first phase, the network learns a function with a small generalization gap (2) In the second phase, the network starts to over-fit the data leading to an increase in test error. Our experiments suggest that this is not the complete picture—in some regimes, the test error decreases again and may achieve a lower value at the end of training as compared to the first minimum (see Fig 10 for $10 \%$ label noise). + +![](images/82b708c209b3222a2c568a1bae6bfb22a279f3b426ac29068d13db60405918ac.jpg) +Figure 10: Epoch-wise double descent for ResNet18 and CNN (width $\scriptstyle 1 = 1 2 8$ ). ResNets trained using Adam with learning rate 0.0001, and CNNs trained with SGD with inverse-squareroot learning rate. + +# 7 SAMPLE-WISE NON-MONOTONICITY + +In this section, we investigate the effect of varying the number of train samples, for a fixed model and training procedure. Previously, in model-wise and epoch-wise double descent, we explored behavior in the critical regime, where $\mathrm { E M C } _ { \mathcal { D } , \epsilon } ( \mathcal { T } ) \approx n$ , by varying the EMC. Here, we explore the critical regime by varying the number of train samples $n$ . By increasing $n$ , the same training procedure $\tau$ can switch from being effectively over-parameterized to effectively under-parameterized. + +We show that increasing the number of samples has two different effects on the test error vs. model complexity graph. On the one hand, (as expected) increasing the number of samples shrinks the area under the curve. On the other hand, increasing the number of samples also has the effect of “shifting the curve to the right” and increasing the model complexity at which test error peaks. + +![](images/69cf7d93b6796ff455f7a65cbe3879b86a15c2ae223d5fc03fc1b54ea4f75d5d.jpg) + +![](images/0c5bdb0c4ceaff9263d9a08a26498be4a038c2547b0dcb1c349a3eef853ce750.jpg) + +(a) Model-wise double descent for 5-layer CNNs on CIFAR-10, for varying dataset sizes. Top: There is a range of model sizes (shaded green) where training on $2 \times$ more samples does not improve test error. Bottom: There is a range of model sizes (shaded red) where training on $4 \times$ more samples does not improve test error. + +(b) Sample-wise non-monotonicity. Test loss (per-word perplexity) as a function of number of train samples, for two transformer models trained to completion on IWSLT’14. For both model sizes, there is a regime where more samples hurt performance. Compare to Figure 3, of model-wise double-descent in the identical setting. + +Figure 11: Sample-wise non-monotonicity. + +These twin effects are shown in Figure 11a. Note that there is a range of model sizes where the effects “cancel out”—and having $4 \times$ more train samples does not help test performance when training to completion. Outside the critically-parameterized regime, for sufficiently under- or overparameterized models, having more samples helps. This phenomenon is corroborated in Figure 12, which shows test error as a function of both model and sample size, in the same setting as Figure 11a. + +![](images/67eb2031049b3e7bf4fd7bf3717f93c4f08018a552ffa050f6ba459a5d149a71.jpg) +Figure 12: Left: Test Error as a function of model size and number of train samples, for 5-layer CNNs on CIFAR- $10 + 2 0 \%$ noise. Note the ridge of high test error again lies along the interpolation threshold. Right: Three slices of the left plot, showing the effect of more data for models of different sizes. Note that, when training to completion, more data helps for small and large models, but does not help for near-critically-parameterized models (green). + +In some settings, these two effects combine to yield a regime of model sizes where more data actually hurts test performance as in Figure 3 (see also Figure 11b). Note that this phenomenon is not unique to DNNs: more data can hurt even for linear models (see Appendix D). + +# 8 CONCLUSION AND DISCUSSION + +We introduce a generalized double descent hypothesis: models and training procedures exhibit atypical behavior when their Effective Model Complexity is comparable to the number of train samples. We provide extensive evidence for our hypothesis in modern deep learning settings, and show that it is robust to choices of dataset, architecture, and training procedures. In particular, we demonstrate “model-wise double descent” for modern deep networks and characterize the regime where bigger models can perform worse. We also demonstrate “epoch-wise double descent,” which, to the best of our knowledge, has not been previously proposed. Finally, we show that the double descent phenomenon can lead to a regime where training on more data leads to worse test performance. Preliminary results suggest that double descent also holds as we vary the amount of regularization for a fixed model (see Figure 22). + +We also believe our characterization of the critical regime provides a useful way of thinking for practitioners—if a model and training procedure are just barely able to fit the train set, then small changes to the model or training procedure may yield unexpected behavior (e.g. making the model slightly larger or smaller, changing regularization, etc. may hurt test performance). + +Early stopping. We note that many of the phenomena that we highlight often do not occur with optimal early-stopping. However, this is consistent with our generalized double descent hypothesis: if early stopping prevents models from reaching 0 train error then we would not expect to see doubledescent, since the EMC does not reach the number of train samples. Further, we show at least one setting where model-wise double descent can still occur even with optimal early stopping (ResNets on CIFAR-100 with no label noise, see Figure 19). We have not observed settings where more data hurts when optimal early-stopping is used. However, we are not aware of reasons which preclude this from occurring. We leave fully understanding the optimal early stopping behavior of double descent as an important open question for future work. + +Label Noise. In our experiments, we observe double descent most strongly in settings with label noise. However, we believe this effect is not fundamentally about label noise, but rather about model mis-specification. For example, consider a setting where the label noise is not truly random, but rather pseudorandom (with respect to the family of classifiers being trained). In this setting, the performance of the Bayes optimal classifier would not change (since the pseudorandom noise is deterministic, and invertible), but we would observe an identical double descent as with truly random label noise. Thus, we view adding label noise as merely a proxy for making distributions “harder”— i.e. increasing the amount of model mis-specification. + +Other Notions of Model Complexity. Our notion of Effective Model Complexity is related to classical complexity notions such as Rademacher complexity, but differs in several crucial ways: (1) EMC depends on the true labels of the data distribution, and (2) EMC depends on the training procedure, not just the model architecture. + +Other notions of model complexity which do not incorporate features (1) and (2) would not suffice to characterize the location of the double-descent peak. Rademacher complexity, for example, is determined by the ability of a model architecture to fit a randomly-labeled train set. But Rademacher complexity and VC dimension are both insufficient to determine the model-wise double descent peak location, since they do not depend on the distribution of labels— and our experiments show that adding label noise shifts the location of the peak. + +Moreover, both Rademacher complexity and VC dimension depend only on the model family and data distribution, and not on the training procedure used to find models. Thus, they are not capable of capturing train-time double-descent effects, such as “epoch-wise” double descent, and the effect of data-augmentation on the peak location. + +# ACKNOWLEDGMENTS + +We thank Mikhail Belkin for extremely useful discussions in the early stages of this work. We thank Christopher Olah for suggesting the Model Size $\times$ Epoch visualization, which led to the investigation of epoch-wise double descent, as well as for useful discussion and feedback. We also thank Alec Radford, Jacob Steinhardt, and Vaishaal Shankar for helpful discussion and suggestions. P.N. thanks OpenAI, the Simons Institute, and the Harvard Theory Group for a research environment that enabled this kind of work. + +We thank Dimitris Kalimeris, Benjamin L. Edelman, and Sharon Qian, and Aditya Ramesh for comments on an early draft of this work. + +This work supported in part by NSF grant CAREER CCF 1452961, BSF grant 2014389, NSF USICCS proposal 1540428, a Google Research award, a Facebook research award, a Simons Investigator Award, a Simons Investigator Fellowship, and NSF Awards CCF 1715187, CCF 1565264, CCF 1301976, IIS 1409097, and CNS 1618026. Y.B. would like to thank the MIT-IBM Watson AI Lab for contributing computational resources for experiments. + +# REFERENCES + +Madhu S Advani and Andrew M Saxe. 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DatasetArchitectureOpt.Aug.% NoiseDouble-DescentFigure(s)
ModelEpoch
CIFAR 10CNNSGD√0X×5,27
√10广广5,27,6
√205,27
0Xx√5,25
105
20√√5
SGD + w.d.√20√√21
ResNetAdam0√25
Adam√0XX4,10
√5√4
√104,10
√15√4,2
Various√20√4,9,10
√20√16, 17, 18
(subsampled)CNNSGD SGD√10√11a
√20√11a, 12
(adversarial)ResNetSGD0Robust err.26
CIFAR 100ResNetAdam0√X4,19,10
>>10√√4,10
CNNSGD√20√√4,10
0√X20
IWSLT '14 de-enTransformerAdam0√X8,24
(subsampled)TransformerAdam0√X11b,23
WMT'14 en-frTransformerAdam0√X8,24
+ +# B APPENDIX: EXPERIMENTAL DETAILS + +# B.1 MODELS + +We use the following families of architectures. The PyTorch Paszke et al. (2017) specification of our ResNets and CNNs are available at https://gitlab.com/ harvard-machine-learning/double-descent/tree/master. + +ResNets. We define a family of ResNet18s of increasing size as follows. We follow the Preactivation ResNet18 architecture of He et al. (2016), using 4 ResNet blocks, each consisting of two BatchNorm-ReLU-Convolution layers. The layer widths for the 4 blocks are $[ k , 2 k , 4 k , 8 k ]$ for varying $k \in \mathbb N$ and the strides are [1, 2, 2, 2]. The standard ResNet18 corresponds to $k = 6 4$ convolutional channels in the first layer. The scaling of model size with $k$ is shown in Figure 13b. Our implementation is adapted from https://github.com/kuangliu/pytorch-cifar. + +Standard CNNs. We consider a simple family of 5-layer CNNs, with four Conv-BatchNormReLU-MaxPool layers and a fully-connected output layer. We scale the four convolutional layer widths as $[ k , 2 k , 4 k , 8 k ]$ . The MaxPool is [1, 2, 2, 8]. For all the convolution layers, the kernel size $= 3$ , stride $= 1$ and padding $^ { = 1 }$ . This architecture is based on the “backbone” architecture from Page (2018). For $k = 6 4$ , this CNN has 1558026 parameters and can reach $> 9 0 \%$ test accuracy on CIFAR-10 (Krizhevsky (2009)) with data-augmentation. The scaling of model size with $k$ is shown in Figure 13a. + +Transformers. We consider the encoder-decoder Transformer model from Vaswani et al. (2017) with 6 layers and 8 attention heads per layer, as implemented by fairseq Ott et al. (2019). We scale the size of the network by modifying the embedding dimension $( d _ { \mathrm { m o d e l } } )$ , and scale the width of the fully-connected layers proportionally $\begin{array} { r } { \dot { \mathcal { d } } _ { \mathrm { f f } } = 4 { d } _ { \mathrm { m o d e l } } ) } \end{array}$ . We train with $10 \%$ label smoothing and no drop-out, for 80 gradient steps. + +![](images/d8516fb233b07ebcdc7f08ef74910030a4bc584afa359c5fbd7737999a56c148.jpg) +Figure 13: Scaling of model size with our parameterization of width & embedding dimension. + +B.2 IMAGE CLASSIFICATION: EXPERIMENTAL SETUP + +We describe the details of training for CNNs and ResNets below. + +Loss function: Unless stated otherwise, we use the cross-entropy loss for all the experiments. + +Data-augmentation: In experiments where data-augmentation was used, we apply RandomCrop(32, padding ${ \cdot = } 4$ ) and RandomHorizontalFlip. In experiments with added label noise, the label for all augmentations of a given training sample are given the same label. + +Regularization: No explicit regularization like weight decay or dropout was applied unless explicitly stated. + +Initialization: We use the default initialization provided by PyTorch for all the layers. + +# Optimization: + +• Adam: Unless specified otherwise, learning rate was set at constant to 1e−4 and all other parameters were set to their default PyTorch values. +SGD: Unless specified otherwise, learning rate schedule inverse-square root (defined below) was used with initial learning rate $\gamma _ { 0 } = 0 . 1$ and updates every $L = 5 1 2$ gradient steps. No momentum was used. + +We found our results are robust to various other natural choices of optimizers and learning rate schedule. We used the above settings because (1) they optimize well, and (2) they do not require experiment-specific hyperparameter tuning, and allow us to use the same optimization across many experiments. + +Batch size: All experiments use a batchsize of 128. + +# Learning rate schedule descriptions: + +• Inverse-square root $( \gamma _ { 0 } , L )$ : At gradient step $t$ , the learning rate is set to $\gamma ( t ) : =$ $\frac { \gamma _ { 0 } } { \sqrt { 1 + \lfloor t / 5 1 2 \rfloor } }$ . We set learning-rate with respect to number of gradient steps, and not epochs, in order to allow comparison between experiments with varying train-set sizes. • Dynamic drop ( $\mathrm { \Delta } \cdot \mathrm { \Delta } \gamma _ { 0 }$ , drop, patience): Starts with an initial learning rate of $\gamma _ { 0 }$ and drops by a factor of ’drop’ if the training loss has remained constant or become worse for ’patience’ number of gradient steps. + +B.3 NEURAL MACHINE TRANSLATION: EXPERIMENTAL SETUP + +Here we describe the experimental setup for the neural machine translation experiments. + +# Training procedure. + +In this setting, the distribution $\mathcal { D }$ consists of triples + +$$ +( x , y , i ) : x \in V _ { s r c } ^ { * } , y \in V _ { t g t } ^ { * } , i \in \{ 0 , \ldots , | y | \} +$$ + +where $V _ { s r c }$ and $V _ { t g t }$ are the source and target vocabularies, the string $x$ is a sentence in the source language, $y$ is its translation in the target language, and $i$ is the index of the token to be predicted by the model. We assume that $i | x , y$ is distributed uniformly on $\{ 0 , \ldots , | y | \}$ . + +A standard probabilistic model defines an autoregressive factorization of the likelihood: + +$$ +p _ { M } ( y | x ) = \prod _ { i = 1 } ^ { | y | } p _ { M } ( y _ { i } | y _ { < i } , x ) . +$$ + +Given a set of training samples $S$ , we define + +$$ +\operatorname { E r r o r } _ { S } ( M ) = { \frac { 1 } { | S | } } \sum _ { ( x , y , i ) \in S } - \log p _ { M } ( y _ { i } | y _ { < i } , x ) . +$$ + +In practice, $S$ is not constructed from independent samples from $D$ , but rather by first sampling $( x , y )$ and then including all $( x , y , 0 ) , \dotsc , ( x , y , | y | )$ in $S$ . + +For training transformers, we replicate the optimization procedure specified in Vaswani et al. (2017) section 5.3, where the learning rate schedule consists of a “warmup” phase with linearly increasing learning rate followed by a phase with inverse square-root decay. We preprocess the data using byte pair encoding (BPE) as described in Sennrich et al. (2015). We use the implementation provided by fairseq (https://github.com/pytorch/fairseq). + +Datasets. The IWSLT ’14 German to English dataset contains TED Talks as described in Cettolo et al. (2012). The WMT ’14 English to French dataset is taken from http://www.statmt. org/wmt14/translation-task.html. + +# B.4 PER-SECTION EXPERIMENTAL DETAILS + +Here we provide full details for experiments in the body, when not otherwise provided. + +Introduction: Experimental Details Figure 1: All models were trained using Adam with learningrate 0.0001 for 4K epochs. Plotting means and standard deviations for 5 trials, with random network initialization. + +Model-wise Double Descent: Experimental Details Figure 7: Plotting means and standard deviations for 5 trials, with random network initialization. + +Sample-wise Nonmonotonicity: Experimental Details Figure 11a: All models are trained with SGD for 500K epochs, and data-augmentation. Bottom: Means and standard deviations from 5 trials with random initialization, and random subsampling of the train set. + +# C EXTENDED DISCUSSION OF RELATED WORK + +Belkin et al. (2018): This paper proposed, in very general terms, that the apparent contradiction between traditional notions of the bias-variance trade-off and empirically successful practices in deep learning can be reconciled under a double-descent curve—as model complexity increases, the test error follows the traditional “U-shaped curve”, but beyond the point of interpolation, the error starts to decrease. This work provides empirical evidence for the double-descent curve with fully connected networks trained on subsets of MNIST, CIFAR10, SVHN and TIMIT datasets. They use the $l _ { 2 }$ loss for their experiments. They demonstrate that neural networks are not an aberration in this regard—double-descent is a general phenomenon observed also in linear regression with random features and random forests. + +Theoretical works on linear least squares regression: A variety of papers have attempted to theoretically analyze this behavior in restricted settings, particularly the case of least squares regression under various assumptions on the training data, feature spaces and regularization method. + +1. Advani & Saxe (2017); Hastie et al. (2019) both consider the linear regression problem stated above and analyze the generalization behavior in the asymptotic limit $N , D \to \infty$ using random matrix theory. Hastie et al. (2019) highlight that when the model is misspecified, the minimum of training error can occur for over-parameterized models +2. Belkin et al. (2019) Linear least squares regression for two data models, where the input data is sampled from a Gaussian and a Fourier series model for functions on a circle. They provide a finite-sample analysis for these two cases +3. Bartlett et al. (2019) provides generalization bounds for the minimum $l _ { 2 }$ -norm interpolant for Gaussian features +4. Muthukumar et al. (2019) characterize the fundamental limit of of any interpolating solution in the presence of noise and provide some interesting Fourier-theoretic interpretations. +5. Mei & Montanari (2019): This work provides asymptotic analysis for ridge regression over random features + +Similar double descent behavior, in restricted settings, was investigated in Trunk (1979); Opper (1995; 2001); Skurichina & Duin (2002). + +Neal et al. (2018) conducts a study of bias and variance in modern neural networks, observing that both bias and variance can decrease with increasing model size, contrary to conventional wisdom. + +Geiger et al. (2019b) showed that deep fully connected networks trained on the MNIST dataset with hinge loss exhibit a “jamming transition” when the number of parameters exceeds a threshold that allows training to near-zero train loss. Geiger et al. (2019a) provide further experiments on CIFAR10 with a convolutional network. They also highlight interesting behavior with ensembling around the critical regime, which is consistent with our informal intuitions in Section 5 and our experiments in Figures 28, 29. + +Advani & Saxe (2017); Geiger et al. (2019b;a) also point out that double-descent is not observed when optimal early-stopping is used. + +The study of sample non-monotonicity in learning algorithms had also existed prior to double descent, including in Duin (1995; 2000); Opper (2001); Loog & Duin (2012). + +![](images/228a2efe993d5a5b8de88a94c4213a4344f8bf8c59e0d4b46ffa29d81cf9c5ed.jpg) +Figure 14: Random Fourier Features on the Fashion MNIST dataset. The setting is equivalent to two-layer neural network with $e ^ { - i x }$ activation, with randomly-initialized first layer that is fixed throughout training. The second layer is trained using gradient flow. + +In this section, for completeness sake, we show that both the model- and sample-wise double descent phenomena are not unique to deep neural networks—they exist even in the setting of Random Fourier Features of Rahimi & Recht (2008). This setting is equivalent to a two-layer neural network with $e ^ { - i x }$ activation. The first layer is initialized with a $\textstyle { \hat { \mathcal { N } } } ( 0 , { \frac { 1 } { d } } )$ Gaussian distribution and then fixed throughout training. The width (or embedding dimension) $\dot { d }$ of the first layer parameterizes the model size. The second layer is initialized with 0s and trained with MSE loss. + +Figure 14 shows the grid of Test Error as a function of both number of samples $n$ and model size $d$ . Note that in this setting $\mathrm { E M C } = d$ (the embedding dimension). As a result, as demonstrated in the figure, the peak follows the path of $n = d$ . Both model-wise and sample-wise (see figure 15) double descent phenomena are captured, by horizontally and vertically crossing the grid, respectively. + +![](images/d3b8ed60787a0868c3f5be52fec305d08ff718efc3f20f3f4f5d527b1d16e3f6.jpg) +Figure 15: Sample-wise double-descent slice for Random Fourier Features on the Fashion MNIST dataset. In this figure the embedding dimension (number of random features) is 1000. + +# E APPENDIX: ADDITIONAL EXPERIMENTS + +E.1 EPOCH-WISE DOUBLE DESCENT: ADDITIONAL RESULTS + +Here, we provide a rigorous evaluation of epoch-wise double descent for a variety of optimizers and learning rate schedules. We train ResNet18 on CIFAR-10 with data-augmentation and $20 \%$ label noise with three different optimizers—Adam, SGD, SGD $^ +$ Momentum (momentum set to 0.9) and three different learning rate schedules—constant, inverse-square root, dynamic drop for differnet values of initial learning rate. We observe that double-descent occurs reliably for all optimizers and learning rate schedules and the peak of the double descent curve shifts with the interpolation point. + +![](images/58c1328435c48b2aab09317e06f4603fa0a9e59b7a92dde853e7a8bf3b824794.jpg) +Figure 16: Epoch-wise double descent for ResNet18 trained with Adam and multiple learning rate schedules + +A practical recommendation resulting from epoch-wise double descent is that stopping the training when the test error starts to increase may not always be the best strategy. In some cases, the test error may decrease again after reaching a maximum, and the final value may be lower than the minimum earlier in training. + +![](images/1008ca2f366058ba1435cf28efaa611f9c9131e1be26e429651a63223e79e1f5.jpg) +Figure 17: Epoch-wise double descent for ResNet18 trained with SGD and multiple learning rate schedules + +![](images/488e30e35257cba7429094f8ccf62b9894c5a64f45fd96a4c9ac9360e41020b6.jpg) +Figure 18: Epoch-wise double descent for ResNet18 trained with $\mathrm { S G D + I }$ Momentum and multiple learning rate schedules + +E.2 MODEL-WISE DOUBLE DESCENT: ADDITIONAL RESULTS + +# E.2.1 CLEAN SETTINGS WITH MODEL-WISE DOUBLE DESCENT + +CIFAR100, ResNet18 + +![](images/5210617de1e3afc414e784be266f4144de44d71b2df97b8df3a125339670d3a0.jpg) +Figure 19: Top: Train and test performance as a function of both model size and train epochs. Bottom: Test error dynamics of the same model (ResNet18, on CIFAR-100 with no label noise, data-augmentation and Adam optimizer trained for $4 \mathrm { k }$ epochs with learning rate 0.0001). Note that even with optimal early stopping this setting exhibits double descent. + +![](images/fb960e5034da77700c7a7fd448a34568a0bb5b897f622766da0a04bb8ae57ec4.jpg) +Figure 20: Top: Train and test performance as a function of both model size and train epochs. Bottom: Test error dynamics of the same models. 5-Layer CNNs, CIFAR-100 with no label noise, no data-augmentation Trained with SGD for 1e6 steps. Same experiment as Figure 7. + +![](images/81c77025d124fca35d30178bb0aa53f7ec90c4883061d61bfdb0b4ef4870d8cc.jpg) +Figure 21: Left: Test error dynamics with weight decay of 5e-4 (bottom left) and without weight decay (top left). Right: Test and train error and test loss for models with varying amounts of weight decay. All models are 5-Layer CNNs on CIFAR-10 with $10 \%$ label noise, trained with data-augmentation and SGD for 500K steps. + +Here, we now study the effect of varying the level of regularization on test error. We train CIFAR10 with data-augmentation and $20 \%$ label noise on ResNet18 for weight decay co-efficients $\lambda$ ranging from 0 to 0.1. We train the networks using $\mathrm { S G D + }$ inverse-square root learning rate. Figure below shows a picture qualitatively very similar to that observed for model-wise double descent wherein ”model complexity” is now controlled by the regularization parameter. This confirms our generalized double descent hypothesis along yet another axis of Effective Model Complexity. + +![](images/2d026f82df05ae4f47eb2f36c18491bb8ff9cd5ddf873d309f4c7bc3b3b3c258.jpg) +Figure 22: Generalized double descent for weight decay. We found that using the same initial learning rate for all weight decay values led to training instabilities. This resulted in some noise in the Test Error (Weight Decay $\times$ Epochs) plot shown above. + +Language models + +![](images/ceaf06ca3257454a126e72ecfe328fbca78241c7e0b1b27a9af01838536e68eb.jpg) +Figure 23: Model-wise test error dynamics for a subsampled IWSLT‘14 dataset. Left: 4k samples, Right: 18k samples. Note that with optimal early-stopping, more samples is always better. + +![](images/548a288bc91ae0609a7c35054d158b13e6149f1b900a2a3870f131fa4fc14015.jpg) +Figure 24: Model-wise test error dynamics for a IWSLT‘14 de-en and subsampled WMT‘14 en-fr datasets. Left: IWSLT‘14, Right: subsampled (200k samples) WMT‘14. Note that with optimal early-stopping, the test error is much lower for this task. + +CIFAR10, $10 \%$ noise, SGD + +![](images/329ab7efcf3819680ed22e9d04f97aecbd6fd2941d5217d8d45d006ce1c162c0.jpg) +Figure 25: Top: Train and test performance as a function of both model size and train epochs. Bottom: Test error dynamics of the same model (CNN, on CIFAR-10 with $10 \%$ label noise, dataaugmentation and SGD optimizer with learning rate $\propto 1 / \sqrt { T } )$ ). + +# E.2.4 TRAINING PROCEDURE + +![](images/cce2a82acf0147228634fdd0581f2e67796b76009678007530194689d914a4f5.jpg) +Figure 26: Model-wise double descent for adversarial training ResNet18s on CIFAR-10 (subsampled to $2 5 \mathrm { k }$ train samples) with no label noise. We train for L2 robustness of radius $\epsilon = 0 . 5$ and $\epsilon = 1 . 0$ , using 10-step PGD (Goodfellow et al. (2014); Madry et al. (2017)). Trained using SGD (batch size 128) with learning rate 0.1 for 400 epochs, then 0.01 for 400 epochs. + +![](images/3f166e910507b6f677862e70ccd9d2fcc5d745118d0ea7693bdfe992dda82c23.jpg) +Figure 27 + +![](images/92155799e3b955426cfd287a4260029a284a4655cd2cd750acbb4e3c8ec8bdcc.jpg) +Figure 28: Effect of Ensembling (ResNets, $15 \%$ label noise). Test error of an ensemble of 5 models, compared to the base models. The ensembled classifier is determined by plurality vote over the 5 base models. Note that emsembling helps most around the critical regime. All models are ResNet18s trained on CIFAR-10 with $15 \%$ label noise, using Adam for 4K epochs (same setting as Figure 1). Test error is measured against the original (not noisy) test set, and each model in the ensemble is trained using a train set with independently-sampled $15 \%$ label noise. + +![](images/b3feb5b8e7fb91ef88b906092c18d0ee5c04fae19b187b34bbe97bc9c5642f30.jpg) +Figure 29: Effect of Ensembling (CNNs, no label noise). Test error of an ensemble of 5 models, compared to the base models. All models are 5-layer CNNs trained on CIFAR-10 with no label noise, using SGD and no data augmentation. (same setting as Figure 7). \ No newline at end of file diff --git a/parse/train/B1g5sA4twr/B1g5sA4twr_content_list.json b/parse/train/B1g5sA4twr/B1g5sA4twr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e26339504707e3e7d08ee6bee8d6a059e8599d02 --- /dev/null +++ b/parse/train/B1g5sA4twr/B1g5sA4twr_content_list.json @@ -0,0 +1,2393 @@ +[ + { + "type": "text", + "text": "DEEP DOUBLE DESCENT: WHERE BIGGER MODELS AND MORE DATA HURT ", + "text_level": 1, + "bbox": [ + 174, + 99, + 777, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Preetum Nakkiran∗ Harvard University ", + "bbox": [ + 184, + 170, + 323, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Gal Kaplun† Harvard University ", + "bbox": [ + 354, + 169, + 480, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yamini Bansal† Harvard University ", + "bbox": [ + 519, + 169, + 648, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tristan Yang Harvard University ", + "bbox": [ + 686, + 170, + 813, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Boaz Barak Harvard University ", + "bbox": [ + 184, + 219, + 312, + 247 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ilya Sutskever OpenAI ", + "bbox": [ + 356, + 219, + 457, + 247 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 270, + 544, + 285 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We show that a variety of modern deep learning tasks exhibit a “double-descent” phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a function of model size, but also as a function of the number of training epochs. We unify the above phenomena by defining a new complexity measure we call the effective model complexity and conjecture a generalized double descent with respect to this measure. Furthermore, our notion of model complexity allows us to identify certain regimes where increasing (even quadrupling) the number of train samples actually hurts test performance. ", + "bbox": [ + 232, + 301, + 766, + 428 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 455, + 336, + 470 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/c4a62cbdb7fbeb5034e735ae46b0639e991da1ee0ce7c9ca2623db843dff96c3.jpg", + "image_caption": [ + "Figure 1: Left: Train and test error as a function of model size, for ResNet18s of varying width on CIFAR-10 with $15 \\%$ label noise. Right: Test error, shown for varying train epochs. All models trained using Adam for 4K epochs. The largest model (width 64) corresponds to standard ResNet18. " + ], + "image_footnote": [], + "bbox": [ + 173, + 491, + 861, + 636 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The bias-variance trade-off is a fundamental concept in classical statistical learning theory (e.g., Hastie et al. (2005)). The idea is that models of higher complexity have lower bias but higher variance. According to this theory, once model complexity passes a certain threshold, models “overfit” with the variance term dominating the test error, and hence from this point onward, increasing model complexity will only decrease performance (i.e., increase test error). Hence conventional wisdom in classical statistics is that, once we pass a certain threshold, “larger models are worse.” ", + "bbox": [ + 174, + 714, + 825, + 797 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, modern neural networks exhibit no such phenomenon. Such networks have millions of parameters, more than enough to fit even random labels (Zhang et al. (2016)), and yet they perform much better on many tasks than smaller models. Indeed, conventional wisdom among practitioners is that “larger models are better’’ (Krizhevsky et al. (2012), Huang et al. (2018), Szegedy et al. ", + "bbox": [ + 174, + 805, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/08ba3538956dd39cc42384ec2f1e07bf08c6fc91c71879c2a0bb15a89998beb1.jpg", + "image_caption": [ + "Figure 2: Left: Test error as a function of model size and train epochs. The horizontal line corresponds to model-wise double descent–varying model size while training for as long as possible. The vertical line corresponds to epoch-wise double descent, with test error undergoing double-descent as train time increases. Right Train error of the corresponding models. All models are Resnet18s trained on CIFAR-10 with $15 \\%$ label noise, data-augmentation, and Adam for up to 4K epochs. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 825, + 244 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(2015), Radford et al. (2019)). The effect of training time on test performance is also up for debate. In some settings, “early stopping” improves test performance, while in other settings training neural networks to zero training error only improves performance. Finally, if there is one thing both classical statisticians and deep learning practitioners agree on is “more data is always better”. ", + "bbox": [ + 174, + 364, + 823, + 420 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we present empirical evidence that both reconcile and challenge some of the above “conventional wisdoms.” We show that many deep learning settings have two different regimes. In the under-parameterized regime, where the model complexity is small compared to the number of samples, the test error as a function of model complexity follows the U-like behavior predicted by the classical bias/variance tradeoff. However, once model complexity is sufficiently large to interpolate i.e., achieve (close to) zero training error, then increasing complexity only decreases test error, following the modern intuition of “bigger models are better”. Similar behavior was previously observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al. (2018), and Geiger et al. (2019b). This phenomenon was first postulated in generality by Belkin et al. (2018) who named it “double descent”, and demonstrated it for decision trees, random features, and 2-layer neural networks with $\\ell _ { 2 }$ loss, on a variety of learning tasks including MNIST and CIFAR-10. ", + "bbox": [ + 173, + 428, + 825, + 580 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Main contributions. We show that double descent is a robust phenomenon that occurs in a variety of tasks, architectures, and optimization methods (see Figure 1 and Section 5; our experiments are summarized in Table A). Moreover, we propose a much more general notion of “double descent” that goes beyond varying the number of parameters. We define the effective model complexity (EMC) of a training procedure as the maximum number of samples on which it can achieve close to zero training error. The EMC depends not just on the data distribution and the architecture of the classifier but also on the training procedure—and in particular increasing training time will increase the EMC. ", + "bbox": [ + 174, + 609, + 825, + 707 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We hypothesize that for many natural models and learning algorithms, double descent occurs as a function of the EMC. Indeed we observe “epoch-wise double descent” when we keep the model fixed and increase the training time, with performance following a classical U-like curve in the underfitting stage (when the EMC is smaller than the number of samples) and then improving with training time once the EMC is sufficiently larger than the number of samples (see Figure 2). As a corollary, early stopping only helps in the relatively narrow parameter regime of critically parameterized models. ", + "bbox": [ + 174, + 714, + 825, + 797 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Sample non-monotonicity. Finally, our results shed light on test performance as a function of the number of train samples. Since the test error peaks around the point where EMC matches the number of samples (the transition from the under- to over-parameterization), increasing the number of samples has the effect of shifting this peak to the right. While in most settings increasing the number of samples decreases error, this shifting effect can sometimes result in a setting where more data is worse! For example, Figure 3 demonstrates cases in which increasing the number of samples by a factor of 4.5 results in worse test performance. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/5c137bc12053fdd4c20394058fe9f6b4430c5f548f5d67abd01978eb2d8cc676.jpg", + "image_caption": [ + "Figure 3: Test loss (per-token perplexity) as a function of Transformer model size (embedding dimension $d _ { m o d e l . }$ ) on language translation (IWSLT‘14 German-to-English). The curve for $1 8 \\mathrm { k }$ samples is generally lower than the one for $4 \\mathrm { k }$ samples, but also shifted to the right, since fitting 18k samples requires a larger model. Thus, for some models, the performance for $1 8 \\mathrm { k }$ samples is worse than for $4 \\mathrm { k \\Omega }$ samples. " + ], + "image_footnote": [], + "bbox": [ + 174, + 112, + 496, + 243 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 OUR RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 277, + 323, + 292 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To state our hypothesis more precisely, we define the notion of effective model complexity. We define a training procedure $\\tau$ to be any procedure that takes as input a set $S = \\{ ( x _ { 1 } , y _ { 1 } ) , \\dots , ( x _ { n } , y _ { n } ) \\}$ of labeled training samples and outputs a classifier $\\mathcal { T } ( S )$ mapping data to labels. We define the effective model complexity of $\\tau$ (w.r.t. distribution $\\mathcal { D }$ ) to be the maximum number of samples $n$ on which $\\tau$ achieves on average $\\approx 0$ training error. ", + "bbox": [ + 173, + 308, + 825, + 378 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 1 (Effective Model Complexity) The Effective Model Complexity (EMC) of a training procedure $\\tau$ , with respect to distribution $\\mathcal { D }$ and parameter $\\epsilon > 0$ , is defined as: ", + "bbox": [ + 173, + 390, + 821, + 420 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/4237f4b7c304235fd52c789a03ae91edad8d0b0c28a7b9bd01914f3c504673f9.jpg", + "text": "$$\n\\operatorname { E M C } _ { { \\mathscr { D } } , \\epsilon } ( { \\mathscr { T } } ) : = \\operatorname* { m a x } \\left\\{ n \\mid \\mathbb { E } _ { S \\sim { \\mathscr { D } } ^ { n } } [ \\operatorname { E r r o r } _ { S } ( { \\mathscr { T } } ( S ) ) ] \\leq \\epsilon \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 425, + 684, + 443 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where Error $s ( M )$ is the mean error of model $M$ on train samples $S$ . ", + "bbox": [ + 173, + 446, + 627, + 463 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our main hypothesis can be informally stated as follows: ", + "bbox": [ + 174, + 474, + 549, + 491 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hypothesis 1 (Generalized Double Descent hypothesis, informal) For any natural data distribution $\\mathcal { D }$ , neural-network-based training procedure $\\tau$ , and small $\\epsilon > 0$ , if we consider the task of predicting labels based on n samples from $\\mathcal { D }$ then: ", + "bbox": [ + 176, + 502, + 823, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Under-paremeterized regime. If $\\cdot _ { \\mathrm { E M C } _ { { \\mathscr D } , \\epsilon } } ( \\mathcal T )$ is sufficiently smaller than $n _ { \\ast }$ , any perturbation of $\\tau$ that increases its effective complexity will decrease the test error. ", + "bbox": [ + 171, + 555, + 823, + 585 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Over-parameterized regime. If $\\mathrm { E M C } _ { { \\cal D } , \\epsilon } ( T )$ is sufficiently larger than $n$ , any perturbation of $\\tau$ that increases its effective complexity will decrease the test error. ", + "bbox": [ + 173, + 593, + 820, + 623 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Critically parameterized regime. I $f \\mathrm { E M C } _ { \\mathcal { D } , \\epsilon } ( \\mathcal { T } ) \\approx n$ , then a perturbation of $\\tau$ that increases its effective complexity might decrease or increase the test error. ", + "bbox": [ + 176, + 631, + 820, + 660 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hypothesis 1 is informal in several ways. We do not have a principled way to choose the parameter $\\epsilon$ (and currently heuristically use $\\epsilon = 0 . 1$ ). We also are yet to have a formal specification for “sufficiently smaller” and “sufficiently larger”. Our experiments suggest that there is a critical interval around the interpolation threshold when $\\mathrm { E M C } _ { \\mathcal { D } , \\epsilon } ( \\mathcal { T } ) = n$ : below and above this interval increasing complexity helps performance, while within this interval it may hurt performance. The width of the critical interval depends on both the distribution and the training procedure in ways we do not yet completely understand. ", + "bbox": [ + 173, + 671, + 825, + 770 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We believe Hypothesis 1 sheds light on the interaction between optimization algorithms, model size, and test performance and helps reconcile some of the competing intuitions about them. The main result of this paper is an experimental validation of Hypothesis 1 under a variety of settings, where we considered several natural choices of datasets, architectures, and optimization algorithms, and we changed the “interpolation threshold” by varying the number of model parameters, the length of training, the amount of label noise in the distribution, and the number of train samples. ", + "bbox": [ + 174, + 776, + 825, + 861 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Model-wise Double Descent. In Section 5, we study the test error of models of increasing size, for a fixed large number of optimization steps. We show that “model-wise double-descent” occurs for various modern datasets (CIFAR-10, CIFAR-100, IWSLT‘14 de-en, with varying amounts of label noise), model architectures (CNNs, ResNets, Transformers), optimizers (SGD, Adam), number of train samples, and training procedures (data-augmentation, and regularization). Moreover, the peak in test error systematically occurs at the interpolation threshold. In particular, we demonstrate realistic settings in which bigger models are worse. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Epoch-wise Double Descent. In Section 6, we study the test error of a fixed, large architecture over the course of training. We demonstrate, in similar settings as above, a corresponding peak in test performance when models are trained just long enough to reach $\\approx 0$ train error. The test error of a large model first decreases (at the beginning of training), then increases (around the critical regime), then decreases once more (at the end of training)—that is, training longer can correct overfitting. ", + "bbox": [ + 174, + 152, + 825, + 222 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Sample-wise Non-monotonicity. In Section 7, we study the test error of a fixed model and training procedure, for varying number of train samples. Consistent with our generalized double-descent hypothesis, we observe distinct test behavior in the “critical regime”, when the number of samples is near the maximum that the model can fit. This often manifests as a long plateau region, in which taking significantly more data might not help when training to completion (as is the case for CNNs on CIFAR-10). Moreover, we show settings (Transformers on IWSLT‘14 en-de), where this manifests as a peak—and for a fixed architecture and training procedure, more data actually hurts. ", + "bbox": [ + 174, + 229, + 825, + 327 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Remarks on Label Noise. We observe all forms of double descent most strongly in settings with label noise in the train set (as is often the case when collecting train data in the real-world). However, we also show several realistic settings with a test-error peak even without label noise: ResNets (Figure 4a) and CNNs (Figure 20) on CIFAR-100; Transformers on IWSLT‘14 (Figure 8). Moreover, all our experiments demonstrate distinctly different test behavior in the critical regime— often manifesting as a “plateau” in the test error in the noiseless case which develops into a peak with added label noise. See Section 8 for further discussion. ", + "bbox": [ + 174, + 334, + 825, + 431 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 453, + 339, + 468 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Model-wise double descent was first proposed as a general phenomenon by Belkin et al. (2018). Similar behavior had been observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al. (2018), and Geiger et al. (2019b). Subsequently, there has been a large body of work studying the double descent phenomenon. A growing list of papers that theoretically analyze it in the tractable setting of linear least squares regression includes Belkin et al. (2019); Hastie et al. (2019); Bartlett et al. (2019); Muthukumar et al. (2019); Bibas et al. (2019); Mitra (2019); Mei & Montanari (2019). Moreover, Geiger et al. (2019a) provide preliminary results for model-wise double descent in convolutional networks trained on CIFAR-10. Our work differs from the above papers in two crucial aspects: First, we extend the idea of double-descent beyond the number of parameters to incorporate the training procedure under a unified notion of “Effective Model Complexity”, leading to novel insights like epoch-wise double descent and sample non-monotonicity. The notion that increasing train time corresponds to increasing complexity was also presented in Nakkiran et al. (2019). Second, we provide an extensive and rigorous demonstration of double-descent in modern deep learning, spanning a variety of architectures, datasets, and optimization procedures. An extended discussion of the related work is provided in Appendix C. ", + "bbox": [ + 173, + 484, + 825, + 693 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 714, + 398, + 729 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We briefly describe the experimental setup here; full details are in Appendix B 1. We consider three families of architectures: ResNets, standard CNNs, and Transformers. ResNets: We parameterize a family of ResNet18s (He et al. (2016)) by scaling the width (number of filters) of convolutional layers. Specifically, we use layer widths $[ k , 2 k , \\bar { 4 } k , 8 k ]$ for varying $k$ . The standard ResNet18 corresponds to $k = 6 4$ . Standard CNNs: We consider a simple family of 5-layer CNNs, with 4 convolutional layers of widths $[ k , 2 k , 4 k , 8 k ]$ for varying $k$ , and a fully-connected layer. For context, the CNN with width $k = 6 4$ , can reach over $9 0 \\%$ test accuracy on CIFAR-10 with dataaugmentation. Transformers: We consider the 6 layer encoder-decoder from Vaswani et al. (2017), as implemented by Ott et al. (2019). We scale the size of the network by modifying the embedding dimension $d _ { \\mathrm { m o d e l } }$ , and setting the width of the fully-connected layers proportionally $( d _ { \\mathrm { f f } } = 4 \\cdot d _ { \\mathrm { m o d e l } } )$ . ", + "bbox": [ + 173, + 746, + 825, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For ResNets and CNNs, we train with cross-entropy loss, and the following optimizers: (1) Adam with learning-rate 0.0001 for 4K epochs; (2) SGD with learning rate $\\propto \\frac { 1 } { \\sqrt { T } }$ for 500K gradient steps. We train Transformers for 80K gradient steps, with $10 \\%$ label smoothing and no drop-out. ", + "bbox": [ + 174, + 103, + 823, + 148 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Label Noise. In our experiments, label noise of probability $p$ refers to training on a samples which have the correct label with probability $( 1 - p )$ , and a uniformly random incorrect label otherwise (label noise is sampled only once and not per epoch). Figure 1 plots test error on the noisy distribution, while the remaining figures plot test error with respect to the clean distribution (the two curves are just linear rescaling of one another). ", + "bbox": [ + 173, + 164, + 825, + 234 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 MODEL-WISE DOUBLE DESCENT ", + "text_level": 1, + "bbox": [ + 174, + 253, + 480, + 270 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/faae61f59becd773c511ccf37b8969226757057fa9adbb5ab757470a3e69071b.jpg", + "image_caption": [ + "(a) CIFAR-100. There is a peak in test error even with no label noise. ", + "Figure 4: Model-wise double descent for ResNet18s. Trained on CIFAR-100 and CIFAR-10, with varying label noise. Optimized using Adam with LR 0.0001 for 4K epochs, and data-augmentation. " + ], + "image_footnote": [], + "bbox": [ + 174, + 290, + 480, + 525 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/e3fd45bf969858879ad23096c5a3e4c6e094f04bd2b0a277a1d81691dbc44055.jpg", + "image_caption": [ + "(b) CIFAR-10. There is a “plateau” in test error around the interpolation point with no label noise, which develops into a peak for added label noise. " + ], + "image_footnote": [], + "bbox": [ + 516, + 290, + 821, + 525 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we study the test error of models of increasing size, when training to completion (for a fixed large number of optimization steps). We demonstrate model-wise double descent across different architectures, datasets, optimizers, and training procedures. The critical region exhibits distinctly different test behavior around the interpolation point and there is often a peak in test error that becomes more prominent in settings with label noise. ", + "bbox": [ + 174, + 628, + 825, + 699 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For the experiments in this section (Figures 4, 5, 6, 7, 8), notice that all modifications which increase the interpolation threshold (such as adding label noise, using data augmentation, and increasing the number of train samples) also correspondingly shift the peak in test error towards larger models. Additional plots showing the early-stopping behavior of these models, and additional experiments showing double descent in settings with no label noise (e.g. Figure 19) are in Appendix E.2. We also observed model-wise double descent for adversarial training, with a prominent robust test error peak even in settings without label noise. See Figure 26 in Appendix E.2. ", + "bbox": [ + 173, + 705, + 825, + 803 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Discussion. Fully understanding the mechanisms behind model-wise double descent in deep neural networks remains an important open question. However, an analog of model-wise double descent occurs even for linear models. A recent stream of theoretical works analyzes this setting (Bartlett et al. (2019); Muthukumar et al. (2019); Belkin et al. (2019); Mei & Montanari (2019); Hastie et al. (2019)). We believe similar mechanisms may be at work in deep neural networks. ", + "bbox": [ + 174, + 819, + 823, + 888 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Informally, our intuition is that for model-sizes at the interpolation threshold, there is effectively only one model that fits the train data and this interpolating model is very sensitive to noise in the train set and/or model mis-specification. That is, since the model is just barely able to fit the train data, forcing it to fit even slightly-noisy or mis-specified labels will destroy its global structure, and result in high test error. (See Figure 28 in the Appendix for an experiment demonstrating this noise sensitivity, by showing that ensembling helps significantly in the critically-parameterized regime). However for over-parameterized models, there are many interpolating models that fit the train set, and SGD is able to find one that “memorizes” (or “absorbs”) the noise while still performing well on the distribution. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/44d6b8438c0d02d0c76c80ad346b4c82268cb44163651e6cd9d1516b78b9044e.jpg", + "image_caption": [ + "Figure 5: Effect of Data Augmentation. 5-layer CNNs on CIFAR10, with and without dataaugmentation. Data-augmentation shifts the interpolation threshold to the right, shifting the test error peak accordingly. Optimized using SGD for 500K steps. See Figure 27 for larger models. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 482, + 356 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/8100cdf1068362adc8d85ddd7ddddf0ee7d65cdfb04eb08ce1bab31e8561ac9f.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 516, + 102, + 821, + 344 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/f4e2caa4b4e7287a955aced3cc3d523468cc0ef98464146881383b6d073e6465.jpg", + "image_caption": [ + "Figure 6: SGD vs. Adam. 5-Layer CNNs on CIFAR-10 with no label noise, and no data augmentation. Optimized using SGD for 500K gradient steps, and Adam for 4K epochs. " + ], + "image_footnote": [], + "bbox": [ + 174, + 430, + 480, + 664 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/c8160258a06627d94e03682b0cb3a5ea98fb3a6cc2efc061ae766953b89696b3.jpg", + "image_caption": [ + "Figure 7: Noiseless settings. 5-layer CNNs on CIFAR-100 with no label noise; note the peak in test error. Trained with SGD and no data augmentation. See Figure 20 for the early-stopping behavior of these models. " + ], + "image_footnote": [], + "bbox": [ + 516, + 430, + 821, + 664 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 790, + 825, + 888 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The above intuition is theoretically justified for linear models. In general, this situation manifests even without label noise for linear models (Mei & Montanari (2019)), and occurs whenever there is model mis-specification between the structure of the true distribution and the model family. We believe this intuition extends to deep learning as well, and it is consistent with our experiments. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/2b473a4c748ee138162c045888f88cd9f7560771b9e3d4fa5dd16e01c2027a36.jpg", + "image_caption": [ + "Figure 8: Transformers on language translation tasks: Multi-head-attention encoderdecoder Transformer model trained for $8 0 \\mathrm { k }$ gradient steps with labeled smoothed cross-entropy loss on IWSLT‘14 Germanto-English (160K sentences) and WMT‘14 English-to-French (subsampled to 200K sentences) dataset. Test loss is measured as pertoken perplexity. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 498, + 265 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 303, + 825, + 330 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 EPOCH-WISE DOUBLE DESCENT ", + "text_level": 1, + "bbox": [ + 174, + 363, + 475, + 380 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we demonstrate a novel form of double-descent with respect to training epochs, which is consistent with our unified view of effective model complexity (EMC) and the generalized double descent hypothesis. Increasing the train time increases the EMC—and thus a sufficiently large model transitions from under- to over-parameterized over the course of training. ", + "bbox": [ + 173, + 401, + 825, + 458 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/bdca8d3385fd5d02c8443adb0acbb61793b4152837145966b74e283d6ca4c7db.jpg", + "image_caption": [ + "Figure 9: Left: Training dynamics for models in three regimes. Models are ResNet18s on CIFAR10 with $20 \\%$ label noise, trained using Adam with learning rate 0.0001, and data augmentation. Right: Test error over (Model size $\\times$ Epochs). Three slices of this plot are shown on the left. " + ], + "image_footnote": [], + "bbox": [ + 194, + 479, + 816, + 627 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As illustrated in Figure 9, sufficiently large models can undergo a “double descent” behavior where test error first decreases then increases near the interpolation threshold, and then decreases again. In contrast, for “medium sized” models, for which training to completion will only barely reach $\\approx 0$ error, the test error as a function of training time will follow a classical U-like curve where it is better to stop early. Models that are too small to reach the approximation threshold will remain in the “under parameterized” regime where increasing train time monotonically decreases test error. Our experiments (Figure 10) show that many settings of dataset and architecture exhibit epoch-wise double descent, in the presence of label noise. Further, this phenomenon is robust across optimizer variations and learning rate schedules (see additional experiments in Appendix E.1). As in modelwise double descent, the test error peak is accentuated with label noise. ", + "bbox": [ + 173, + 708, + 825, + 847 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Conventional wisdom suggests that training is split into two phases: (1) In the first phase, the network learns a function with a small generalization gap (2) In the second phase, the network starts to over-fit the data leading to an increase in test error. Our experiments suggest that this is not the complete picture—in some regimes, the test error decreases again and may achieve a lower value at the end of training as compared to the first minimum (see Fig 10 for $10 \\%$ label noise). ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/82b708c209b3222a2c568a1bae6bfb22a279f3b426ac29068d13db60405918ac.jpg", + "image_caption": [ + "Figure 10: Epoch-wise double descent for ResNet18 and CNN (width $\\scriptstyle 1 = 1 2 8$ ). ResNets trained using Adam with learning rate 0.0001, and CNNs trained with SGD with inverse-squareroot learning rate. " + ], + "image_footnote": [], + "bbox": [ + 183, + 123, + 797, + 320 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 SAMPLE-WISE NON-MONOTONICITY ", + "text_level": 1, + "bbox": [ + 174, + 387, + 509, + 404 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we investigate the effect of varying the number of train samples, for a fixed model and training procedure. Previously, in model-wise and epoch-wise double descent, we explored behavior in the critical regime, where $\\mathrm { E M C } _ { \\mathcal { D } , \\epsilon } ( \\mathcal { T } ) \\approx n$ , by varying the EMC. Here, we explore the critical regime by varying the number of train samples $n$ . By increasing $n$ , the same training procedure $\\tau$ can switch from being effectively over-parameterized to effectively under-parameterized. ", + "bbox": [ + 173, + 421, + 825, + 491 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We show that increasing the number of samples has two different effects on the test error vs. model complexity graph. On the one hand, (as expected) increasing the number of samples shrinks the area under the curve. On the other hand, increasing the number of samples also has the effect of “shifting the curve to the right” and increasing the model complexity at which test error peaks. ", + "bbox": [ + 173, + 497, + 825, + 554 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/69cf7d93b6796ff455f7a65cbe3879b86a15c2ae223d5fc03fc1b54ea4f75d5d.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 192, + 577, + 475, + 792 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/0c5bdb0c4ceaff9263d9a08a26498be4a038c2547b0dcb1c349a3eef853ce750.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 516, + 573, + 808, + 796 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(a) Model-wise double descent for 5-layer CNNs on CIFAR-10, for varying dataset sizes. Top: There is a range of model sizes (shaded green) where training on $2 \\times$ more samples does not improve test error. Bottom: There is a range of model sizes (shaded red) where training on $4 \\times$ more samples does not improve test error. ", + "bbox": [ + 187, + 804, + 482, + 893 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(b) Sample-wise non-monotonicity. Test loss (per-word perplexity) as a function of number of train samples, for two transformer models trained to completion on IWSLT’14. For both model sizes, there is a regime where more samples hurt performance. Compare to Figure 3, of model-wise double-descent in the identical setting. ", + "bbox": [ + 516, + 804, + 810, + 892 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 11: Sample-wise non-monotonicity. ", + "bbox": [ + 359, + 905, + 640, + 920 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "These twin effects are shown in Figure 11a. Note that there is a range of model sizes where the effects “cancel out”—and having $4 \\times$ more train samples does not help test performance when training to completion. Outside the critically-parameterized regime, for sufficiently under- or overparameterized models, having more samples helps. This phenomenon is corroborated in Figure 12, which shows test error as a function of both model and sample size, in the same setting as Figure 11a. ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/67eb2031049b3e7bf4fd7bf3717f93c4f08018a552ffa050f6ba459a5d149a71.jpg", + "image_caption": [ + "Figure 12: Left: Test Error as a function of model size and number of train samples, for 5-layer CNNs on CIFAR- $10 + 2 0 \\%$ noise. Note the ridge of high test error again lies along the interpolation threshold. Right: Three slices of the left plot, showing the effect of more data for models of different sizes. Note that, when training to completion, more data helps for small and large models, but does not help for near-critically-parameterized models (green). " + ], + "image_footnote": [], + "bbox": [ + 178, + 186, + 816, + 459 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In some settings, these two effects combine to yield a regime of model sizes where more data actually hurts test performance as in Figure 3 (see also Figure 11b). Note that this phenomenon is not unique to DNNs: more data can hurt even for linear models (see Appendix D). ", + "bbox": [ + 174, + 556, + 825, + 598 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 CONCLUSION AND DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 618, + 468, + 635 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We introduce a generalized double descent hypothesis: models and training procedures exhibit atypical behavior when their Effective Model Complexity is comparable to the number of train samples. We provide extensive evidence for our hypothesis in modern deep learning settings, and show that it is robust to choices of dataset, architecture, and training procedures. In particular, we demonstrate “model-wise double descent” for modern deep networks and characterize the regime where bigger models can perform worse. We also demonstrate “epoch-wise double descent,” which, to the best of our knowledge, has not been previously proposed. Finally, we show that the double descent phenomenon can lead to a regime where training on more data leads to worse test performance. Preliminary results suggest that double descent also holds as we vary the amount of regularization for a fixed model (see Figure 22). ", + "bbox": [ + 173, + 650, + 825, + 790 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We also believe our characterization of the critical regime provides a useful way of thinking for practitioners—if a model and training procedure are just barely able to fit the train set, then small changes to the model or training procedure may yield unexpected behavior (e.g. making the model slightly larger or smaller, changing regularization, etc. may hurt test performance). ", + "bbox": [ + 174, + 796, + 823, + 852 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Early stopping. We note that many of the phenomena that we highlight often do not occur with optimal early-stopping. However, this is consistent with our generalized double descent hypothesis: if early stopping prevents models from reaching 0 train error then we would not expect to see doubledescent, since the EMC does not reach the number of train samples. Further, we show at least one setting where model-wise double descent can still occur even with optimal early stopping (ResNets on CIFAR-100 with no label noise, see Figure 19). We have not observed settings where more data hurts when optimal early-stopping is used. However, we are not aware of reasons which preclude this from occurring. We leave fully understanding the optimal early stopping behavior of double descent as an important open question for future work. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 172 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Label Noise. In our experiments, we observe double descent most strongly in settings with label noise. However, we believe this effect is not fundamentally about label noise, but rather about model mis-specification. For example, consider a setting where the label noise is not truly random, but rather pseudorandom (with respect to the family of classifiers being trained). In this setting, the performance of the Bayes optimal classifier would not change (since the pseudorandom noise is deterministic, and invertible), but we would observe an identical double descent as with truly random label noise. Thus, we view adding label noise as merely a proxy for making distributions “harder”— i.e. increasing the amount of model mis-specification. ", + "bbox": [ + 174, + 190, + 825, + 303 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Other Notions of Model Complexity. Our notion of Effective Model Complexity is related to classical complexity notions such as Rademacher complexity, but differs in several crucial ways: (1) EMC depends on the true labels of the data distribution, and (2) EMC depends on the training procedure, not just the model architecture. ", + "bbox": [ + 174, + 319, + 825, + 375 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Other notions of model complexity which do not incorporate features (1) and (2) would not suffice to characterize the location of the double-descent peak. Rademacher complexity, for example, is determined by the ability of a model architecture to fit a randomly-labeled train set. But Rademacher complexity and VC dimension are both insufficient to determine the model-wise double descent peak location, since they do not depend on the distribution of labels— and our experiments show that adding label noise shifts the location of the peak. ", + "bbox": [ + 174, + 382, + 823, + 465 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Moreover, both Rademacher complexity and VC dimension depend only on the model family and data distribution, and not on the training procedure used to find models. Thus, they are not capable of capturing train-time double-descent effects, such as “epoch-wise” double descent, and the effect of data-augmentation on the peak location. ", + "bbox": [ + 174, + 473, + 825, + 529 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 546, + 326, + 559 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Mikhail Belkin for extremely useful discussions in the early stages of this work. We thank Christopher Olah for suggesting the Model Size $\\times$ Epoch visualization, which led to the investigation of epoch-wise double descent, as well as for useful discussion and feedback. We also thank Alec Radford, Jacob Steinhardt, and Vaishaal Shankar for helpful discussion and suggestions. P.N. thanks OpenAI, the Simons Institute, and the Harvard Theory Group for a research environment that enabled this kind of work. ", + "bbox": [ + 174, + 570, + 825, + 654 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Dimitris Kalimeris, Benjamin L. Edelman, and Sharon Qian, and Aditya Ramesh for comments on an early draft of this work. ", + "bbox": [ + 174, + 661, + 823, + 689 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work supported in part by NSF grant CAREER CCF 1452961, BSF grant 2014389, NSF USICCS proposal 1540428, a Google Research award, a Facebook research award, a Simons Investigator Award, a Simons Investigator Fellowship, and NSF Awards CCF 1715187, CCF 1565264, CCF 1301976, IIS 1409097, and CNS 1618026. Y.B. would like to thank the MIT-IBM Watson AI Lab for contributing computational resources for experiments. 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Bagging, boosting and the random subspace method for linear classifiers. Pattern Analysis & Applications, 5(2):121–135, 2002. ", + "bbox": [ + 173, + 611, + 825, + 640 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Stefano Spigler, Mario Geiger, Stephane d’Ascoli, Levent Sagun, Giulio Biroli, and Matthieu Wyart.´ A jamming transition from under-to over-parametrization affects loss landscape and generalization. arXiv preprint arXiv:1810.09665, 2018. ", + "bbox": [ + 173, + 657, + 825, + 700 + ], + "page_idx": 11 + }, + { + "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 Computer Vision and Pattern Recognition (CVPR), 2015. URL http://arxiv.org/abs/ 1409.4842. ", + "bbox": [ + 173, + 717, + 825, + 775 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Gerard V Trunk. A problem of dimensionality: A simple example. IEEE Transactions on pattern analysis and machine intelligence, (3):306–307, 1979. ", + "bbox": [ + 171, + 792, + 823, + 821 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. CoRR, abs/1706.03762, 2017. ", + "bbox": [ + 173, + 838, + 825, + 868 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ICLR, abs/1611.03530, 2016. ", + "bbox": [ + 174, + 886, + 823, + 915 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/242fa66a4d1cab84a413cba4e94acb583279a5793d8d553c5579194758f6ed18.jpg", + "table_caption": [ + "A SUMMARY TABLE OF EXPERIMENTAL RESULTS " + ], + "table_footnote": [], + "table_body": "
DatasetArchitectureOpt.Aug.% NoiseDouble-DescentFigure(s)
ModelEpoch
CIFAR 10CNNSGD√0X×5,27
√10广广5,27,6
√205,27
0Xx√5,25
105
20√√5
SGD + w.d.√20√√21
ResNetAdam0√25
Adam√0XX4,10
√5√4
√104,10
√15√4,2
Various√20√4,9,10
√20√16, 17, 18
(subsampled)CNNSGD SGD√10√11a
√20√11a, 12
(adversarial)ResNetSGD0Robust err.26
CIFAR 100ResNetAdam0√X4,19,10
>>10√√4,10
CNNSGD√20√√4,10
0√X20
IWSLT '14 de-enTransformerAdam0√X8,24
(subsampled)TransformerAdam0√X11b,23
WMT'14 en-frTransformerAdam0√X8,24
", + "bbox": [ + 171, + 132, + 799, + 492 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B APPENDIX: EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 518, + 521, + 535 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.1 MODELS ", + "text_level": 1, + "bbox": [ + 174, + 551, + 279, + 565 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We use the following families of architectures. The PyTorch Paszke et al. (2017) specification of our ResNets and CNNs are available at https://gitlab.com/ harvard-machine-learning/double-descent/tree/master. ", + "bbox": [ + 176, + 578, + 821, + 621 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "ResNets. We define a family of ResNet18s of increasing size as follows. We follow the Preactivation ResNet18 architecture of He et al. (2016), using 4 ResNet blocks, each consisting of two BatchNorm-ReLU-Convolution layers. The layer widths for the 4 blocks are $[ k , 2 k , 4 k , 8 k ]$ for varying $k \\in \\mathbb N$ and the strides are [1, 2, 2, 2]. The standard ResNet18 corresponds to $k = 6 4$ convolutional channels in the first layer. The scaling of model size with $k$ is shown in Figure 13b. Our implementation is adapted from https://github.com/kuangliu/pytorch-cifar. ", + "bbox": [ + 174, + 637, + 825, + 722 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Standard CNNs. We consider a simple family of 5-layer CNNs, with four Conv-BatchNormReLU-MaxPool layers and a fully-connected output layer. We scale the four convolutional layer widths as $[ k , 2 k , 4 k , 8 k ]$ . The MaxPool is [1, 2, 2, 8]. For all the convolution layers, the kernel size $= 3$ , stride $= 1$ and padding $^ { = 1 }$ . This architecture is based on the “backbone” architecture from Page (2018). For $k = 6 4$ , this CNN has 1558026 parameters and can reach $> 9 0 \\%$ test accuracy on CIFAR-10 (Krizhevsky (2009)) with data-augmentation. The scaling of model size with $k$ is shown in Figure 13a. ", + "bbox": [ + 174, + 738, + 825, + 837 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Transformers. We consider the encoder-decoder Transformer model from Vaswani et al. (2017) with 6 layers and 8 attention heads per layer, as implemented by fairseq Ott et al. (2019). We scale the size of the network by modifying the embedding dimension $( d _ { \\mathrm { m o d e l } } )$ , and scale the width of the fully-connected layers proportionally $\\begin{array} { r } { \\dot { \\mathcal { d } } _ { \\mathrm { f f } } = 4 { d } _ { \\mathrm { m o d e l } } ) } \\end{array}$ . We train with $10 \\%$ label smoothing and no drop-out, for 80 gradient steps. ", + "bbox": [ + 176, + 853, + 823, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/d8516fb233b07ebcdc7f08ef74910030a4bc584afa359c5fbd7737999a56c148.jpg", + "image_caption": [ + "Figure 13: Scaling of model size with our parameterization of width & embedding dimension. " + ], + "image_footnote": [], + "bbox": [ + 178, + 102, + 805, + 272 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.2 IMAGE CLASSIFICATION: EXPERIMENTAL SETUP ", + "bbox": [ + 174, + 327, + 558, + 342 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We describe the details of training for CNNs and ResNets below. ", + "bbox": [ + 176, + 353, + 598, + 368 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Loss function: Unless stated otherwise, we use the cross-entropy loss for all the experiments. ", + "bbox": [ + 176, + 376, + 787, + 390 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Data-augmentation: In experiments where data-augmentation was used, we apply RandomCrop(32, padding ${ \\cdot = } 4$ ) and RandomHorizontalFlip. In experiments with added label noise, the label for all augmentations of a given training sample are given the same label. ", + "bbox": [ + 174, + 397, + 825, + 453 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Regularization: No explicit regularization like weight decay or dropout was applied unless explicitly stated. ", + "bbox": [ + 173, + 459, + 823, + 488 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Initialization: We use the default initialization provided by PyTorch for all the layers. ", + "bbox": [ + 176, + 494, + 736, + 510 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Optimization: ", + "text_level": 1, + "bbox": [ + 174, + 516, + 272, + 531 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Adam: Unless specified otherwise, learning rate was set at constant to 1e−4 and all other parameters were set to their default PyTorch values. \nSGD: Unless specified otherwise, learning rate schedule inverse-square root (defined below) was used with initial learning rate $\\gamma _ { 0 } = 0 . 1$ and updates every $L = 5 1 2$ gradient steps. No momentum was used. ", + "bbox": [ + 215, + 544, + 825, + 619 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We found our results are robust to various other natural choices of optimizers and learning rate schedule. We used the above settings because (1) they optimize well, and (2) they do not require experiment-specific hyperparameter tuning, and allow us to use the same optimization across many experiments. ", + "bbox": [ + 176, + 633, + 825, + 689 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Batch size: All experiments use a batchsize of 128. ", + "bbox": [ + 174, + 695, + 513, + 710 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Learning rate schedule descriptions: ", + "text_level": 1, + "bbox": [ + 174, + 717, + 429, + 732 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Inverse-square root $( \\gamma _ { 0 } , L )$ : At gradient step $t$ , the learning rate is set to $\\gamma ( t ) : =$ $\\frac { \\gamma _ { 0 } } { \\sqrt { 1 + \\lfloor t / 5 1 2 \\rfloor } }$ . We set learning-rate with respect to number of gradient steps, and not epochs, in order to allow comparison between experiments with varying train-set sizes. • Dynamic drop ( $\\mathrm { \\Delta } \\cdot \\mathrm { \\Delta } \\gamma _ { 0 }$ , drop, patience): Starts with an initial learning rate of $\\gamma _ { 0 }$ and drops by a factor of ’drop’ if the training loss has remained constant or become worse for ’patience’ number of gradient steps. ", + "bbox": [ + 214, + 744, + 825, + 843 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.3 NEURAL MACHINE TRANSLATION: EXPERIMENTAL SETUP ", + "bbox": [ + 173, + 861, + 627, + 876 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we describe the experimental setup for the neural machine translation experiments. ", + "bbox": [ + 176, + 887, + 750, + 902 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Training procedure. ", + "text_level": 1, + "bbox": [ + 176, + 909, + 315, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this setting, the distribution $\\mathcal { D }$ consists of triples ", + "bbox": [ + 176, + 103, + 508, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ea45e78c04233762df1bd61ef159455579926d5711426246ad4dcd7a735227ec.jpg", + "text": "$$\n( x , y , i ) : x \\in V _ { s r c } ^ { * } , y \\in V _ { t g t } ^ { * } , i \\in \\{ 0 , \\ldots , | y | \\}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 125, + 658, + 143 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $V _ { s r c }$ and $V _ { t g t }$ are the source and target vocabularies, the string $x$ is a sentence in the source language, $y$ is its translation in the target language, and $i$ is the index of the token to be predicted by the model. We assume that $i | x , y$ is distributed uniformly on $\\{ 0 , \\ldots , | y | \\}$ . ", + "bbox": [ + 173, + 148, + 826, + 193 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A standard probabilistic model defines an autoregressive factorization of the likelihood: ", + "bbox": [ + 173, + 196, + 745, + 213 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ba0fac56dbf9e6b0f4370935bc97105f9414e542450e62db03e623ca09844fed.jpg", + "text": "$$\np _ { M } ( y | x ) = \\prod _ { i = 1 } ^ { | y | } p _ { M } ( y _ { i } | y _ { < i } , x ) .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 219, + 601, + 265 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Given a set of training samples $S$ , we define ", + "bbox": [ + 176, + 270, + 465, + 285 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6f7586678d50408da87e3bd9727cb839d2f2f7f6e180bad3f4f5b623a6068bb8.jpg", + "text": "$$\n\\operatorname { E r r o r } _ { S } ( M ) = { \\frac { 1 } { | S | } } \\sum _ { ( x , y , i ) \\in S } - \\log p _ { M } ( y _ { i } | y _ { < i } , x ) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 290, + 661, + 332 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In practice, $S$ is not constructed from independent samples from $D$ , but rather by first sampling $( x , y )$ and then including all $( x , y , 0 ) , \\dotsc , ( x , y , | y | )$ in $S$ . ", + "bbox": [ + 173, + 337, + 823, + 367 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For training transformers, we replicate the optimization procedure specified in Vaswani et al. (2017) section 5.3, where the learning rate schedule consists of a “warmup” phase with linearly increasing learning rate followed by a phase with inverse square-root decay. We preprocess the data using byte pair encoding (BPE) as described in Sennrich et al. (2015). We use the implementation provided by fairseq (https://github.com/pytorch/fairseq). ", + "bbox": [ + 173, + 372, + 825, + 443 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Datasets. The IWSLT ’14 German to English dataset contains TED Talks as described in Cettolo et al. (2012). The WMT ’14 English to French dataset is taken from http://www.statmt. org/wmt14/translation-task.html. ", + "bbox": [ + 174, + 457, + 825, + 500 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.4 PER-SECTION EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 517, + 493, + 531 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Here we provide full details for experiments in the body, when not otherwise provided. ", + "bbox": [ + 176, + 542, + 740, + 558 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Introduction: Experimental Details Figure 1: All models were trained using Adam with learningrate 0.0001 for 4K epochs. Plotting means and standard deviations for 5 trials, with random network initialization. ", + "bbox": [ + 174, + 564, + 821, + 606 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Model-wise Double Descent: Experimental Details Figure 7: Plotting means and standard deviations for 5 trials, with random network initialization. ", + "bbox": [ + 173, + 613, + 823, + 642 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Sample-wise Nonmonotonicity: Experimental Details Figure 11a: All models are trained with SGD for 500K epochs, and data-augmentation. Bottom: Means and standard deviations from 5 trials with random initialization, and random subsampling of the train set. ", + "bbox": [ + 174, + 648, + 825, + 690 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C EXTENDED DISCUSSION OF RELATED WORK ", + "text_level": 1, + "bbox": [ + 173, + 103, + 573, + 117 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Belkin et al. (2018): This paper proposed, in very general terms, that the apparent contradiction between traditional notions of the bias-variance trade-off and empirically successful practices in deep learning can be reconciled under a double-descent curve—as model complexity increases, the test error follows the traditional “U-shaped curve”, but beyond the point of interpolation, the error starts to decrease. This work provides empirical evidence for the double-descent curve with fully connected networks trained on subsets of MNIST, CIFAR10, SVHN and TIMIT datasets. They use the $l _ { 2 }$ loss for their experiments. They demonstrate that neural networks are not an aberration in this regard—double-descent is a general phenomenon observed also in linear regression with random features and random forests. ", + "bbox": [ + 174, + 133, + 825, + 258 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Theoretical works on linear least squares regression: A variety of papers have attempted to theoretically analyze this behavior in restricted settings, particularly the case of least squares regression under various assumptions on the training data, feature spaces and regularization method. ", + "bbox": [ + 176, + 273, + 825, + 316 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "1. Advani & Saxe (2017); Hastie et al. (2019) both consider the linear regression problem stated above and analyze the generalization behavior in the asymptotic limit $N , D \\to \\infty$ using random matrix theory. Hastie et al. (2019) highlight that when the model is misspecified, the minimum of training error can occur for over-parameterized models \n2. Belkin et al. (2019) Linear least squares regression for two data models, where the input data is sampled from a Gaussian and a Fourier series model for functions on a circle. They provide a finite-sample analysis for these two cases \n3. Bartlett et al. (2019) provides generalization bounds for the minimum $l _ { 2 }$ -norm interpolant for Gaussian features \n4. Muthukumar et al. (2019) characterize the fundamental limit of of any interpolating solution in the presence of noise and provide some interesting Fourier-theoretic interpretations. \n5. Mei & Montanari (2019): This work provides asymptotic analysis for ridge regression over random features ", + "bbox": [ + 210, + 330, + 825, + 530 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Similar double descent behavior, in restricted settings, was investigated in Trunk (1979); Opper (1995; 2001); Skurichina & Duin (2002). ", + "bbox": [ + 176, + 542, + 823, + 571 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Neal et al. (2018) conducts a study of bias and variance in modern neural networks, observing that both bias and variance can decrease with increasing model size, contrary to conventional wisdom. ", + "bbox": [ + 174, + 579, + 823, + 607 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Geiger et al. (2019b) showed that deep fully connected networks trained on the MNIST dataset with hinge loss exhibit a “jamming transition” when the number of parameters exceeds a threshold that allows training to near-zero train loss. Geiger et al. (2019a) provide further experiments on CIFAR10 with a convolutional network. They also highlight interesting behavior with ensembling around the critical regime, which is consistent with our informal intuitions in Section 5 and our experiments in Figures 28, 29. ", + "bbox": [ + 174, + 613, + 825, + 696 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Advani & Saxe (2017); Geiger et al. (2019b;a) also point out that double-descent is not observed when optimal early-stopping is used. ", + "bbox": [ + 173, + 704, + 823, + 733 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The study of sample non-monotonicity in learning algorithms had also existed prior to double descent, including in Duin (1995; 2000); Opper (2001); Loog & Duin (2012). ", + "bbox": [ + 173, + 739, + 821, + 768 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/228a2efe993d5a5b8de88a94c4213a4344f8bf8c59e0d4b46ffa29d81cf9c5ed.jpg", + "image_caption": [ + "Figure 14: Random Fourier Features on the Fashion MNIST dataset. The setting is equivalent to two-layer neural network with $e ^ { - i x }$ activation, with randomly-initialized first layer that is fixed throughout training. The second layer is trained using gradient flow. " + ], + "image_footnote": [], + "bbox": [ + 176, + 140, + 823, + 281 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section, for completeness sake, we show that both the model- and sample-wise double descent phenomena are not unique to deep neural networks—they exist even in the setting of Random Fourier Features of Rahimi & Recht (2008). This setting is equivalent to a two-layer neural network with $e ^ { - i x }$ activation. The first layer is initialized with a $\\textstyle { \\hat { \\mathcal { N } } } ( 0 , { \\frac { 1 } { d } } )$ Gaussian distribution and then fixed throughout training. The width (or embedding dimension) $\\dot { d }$ of the first layer parameterizes the model size. The second layer is initialized with 0s and trained with MSE loss. ", + "bbox": [ + 173, + 362, + 825, + 445 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Figure 14 shows the grid of Test Error as a function of both number of samples $n$ and model size $d$ . Note that in this setting $\\mathrm { E M C } = d$ (the embedding dimension). As a result, as demonstrated in the figure, the peak follows the path of $n = d$ . Both model-wise and sample-wise (see figure 15) double descent phenomena are captured, by horizontally and vertically crossing the grid, respectively. ", + "bbox": [ + 173, + 452, + 825, + 508 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/d3b8ed60787a0868c3f5be52fec305d08ff718efc3f20f3f4f5d527b1d16e3f6.jpg", + "image_caption": [ + "Figure 15: Sample-wise double-descent slice for Random Fourier Features on the Fashion MNIST dataset. In this figure the embedding dimension (number of random features) is 1000. " + ], + "image_footnote": [], + "bbox": [ + 305, + 525, + 691, + 820 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E APPENDIX: ADDITIONAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 544, + 118 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E.1 EPOCH-WISE DOUBLE DESCENT: ADDITIONAL RESULTS ", + "bbox": [ + 173, + 142, + 606, + 157 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Here, we provide a rigorous evaluation of epoch-wise double descent for a variety of optimizers and learning rate schedules. We train ResNet18 on CIFAR-10 with data-augmentation and $20 \\%$ label noise with three different optimizers—Adam, SGD, SGD $^ +$ Momentum (momentum set to 0.9) and three different learning rate schedules—constant, inverse-square root, dynamic drop for differnet values of initial learning rate. We observe that double-descent occurs reliably for all optimizers and learning rate schedules and the peak of the double descent curve shifts with the interpolation point. ", + "bbox": [ + 173, + 175, + 825, + 260 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/58c1328435c48b2aab09317e06f4603fa0a9e59b7a92dde853e7a8bf3b824794.jpg", + "image_caption": [ + "Figure 16: Epoch-wise double descent for ResNet18 trained with Adam and multiple learning rate schedules " + ], + "image_footnote": [], + "bbox": [ + 183, + 287, + 812, + 501 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A practical recommendation resulting from epoch-wise double descent is that stopping the training when the test error starts to increase may not always be the best strategy. In some cases, the test error may decrease again after reaching a maximum, and the final value may be lower than the minimum earlier in training. ", + "bbox": [ + 174, + 580, + 825, + 638 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/1008ca2f366058ba1435cf28efaa611f9c9131e1be26e429651a63223e79e1f5.jpg", + "image_caption": [ + "Figure 17: Epoch-wise double descent for ResNet18 trained with SGD and multiple learning rate schedules " + ], + "image_footnote": [], + "bbox": [ + 183, + 666, + 812, + 878 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/488e30e35257cba7429094f8ccf62b9894c5a64f45fd96a4c9ac9360e41020b6.jpg", + "image_caption": [ + "Figure 18: Epoch-wise double descent for ResNet18 trained with $\\mathrm { S G D + I }$ Momentum and multiple learning rate schedules " + ], + "image_footnote": [], + "bbox": [ + 183, + 99, + 812, + 314 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.2 MODEL-WISE DOUBLE DESCENT: ADDITIONAL RESULTS ", + "bbox": [ + 173, + 386, + 617, + 401 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.2.1 CLEAN SETTINGS WITH MODEL-WISE DOUBLE DESCENT ", + "text_level": 1, + "bbox": [ + 173, + 415, + 632, + 430 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "CIFAR100, ResNet18 ", + "bbox": [ + 174, + 441, + 325, + 457 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/5210617de1e3afc414e784be266f4144de44d71b2df97b8df3a125339670d3a0.jpg", + "image_caption": [ + "Figure 19: Top: Train and test performance as a function of both model size and train epochs. Bottom: Test error dynamics of the same model (ResNet18, on CIFAR-100 with no label noise, data-augmentation and Adam optimizer trained for $4 \\mathrm { k }$ epochs with learning rate 0.0001). Note that even with optimal early stopping this setting exhibits double descent. " + ], + "image_footnote": [], + "bbox": [ + 236, + 476, + 759, + 852 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/fb960e5034da77700c7a7fd448a34568a0bb5b897f622766da0a04bb8ae57ec4.jpg", + "image_caption": [ + "Figure 20: Top: Train and test performance as a function of both model size and train epochs. Bottom: Test error dynamics of the same models. 5-Layer CNNs, CIFAR-100 with no label noise, no data-augmentation Trained with SGD for 1e6 steps. Same experiment as Figure 7. " + ], + "image_footnote": [], + "bbox": [ + 236, + 132, + 759, + 487 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/81c77025d124fca35d30178bb0aa53f7ec90c4883061d61bfdb0b4ef4870d8cc.jpg", + "image_caption": [ + "Figure 21: Left: Test error dynamics with weight decay of 5e-4 (bottom left) and without weight decay (top left). Right: Test and train error and test loss for models with varying amounts of weight decay. All models are 5-Layer CNNs on CIFAR-10 with $10 \\%$ label noise, trained with data-augmentation and SGD for 500K steps. " + ], + "image_footnote": [], + "bbox": [ + 171, + 131, + 825, + 435 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Here, we now study the effect of varying the level of regularization on test error. We train CIFAR10 with data-augmentation and $20 \\%$ label noise on ResNet18 for weight decay co-efficients $\\lambda$ ranging from 0 to 0.1. We train the networks using $\\mathrm { S G D + }$ inverse-square root learning rate. Figure below shows a picture qualitatively very similar to that observed for model-wise double descent wherein ”model complexity” is now controlled by the regularization parameter. This confirms our generalized double descent hypothesis along yet another axis of Effective Model Complexity. ", + "bbox": [ + 173, + 521, + 826, + 606 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/2d026f82df05ae4f47eb2f36c18491bb8ff9cd5ddf873d309f4c7bc3b3b3c258.jpg", + "image_caption": [ + "Figure 22: Generalized double descent for weight decay. We found that using the same initial learning rate for all weight decay values led to training instabilities. This resulted in some noise in the Test Error (Weight Decay $\\times$ Epochs) plot shown above. " + ], + "image_footnote": [], + "bbox": [ + 173, + 621, + 825, + 866 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Language models ", + "bbox": [ + 173, + 127, + 299, + 142 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/ceaf06ca3257454a126e72ecfe328fbca78241c7e0b1b27a9af01838536e68eb.jpg", + "image_caption": [ + "Figure 23: Model-wise test error dynamics for a subsampled IWSLT‘14 dataset. Left: 4k samples, Right: 18k samples. Note that with optimal early-stopping, more samples is always better. " + ], + "image_footnote": [], + "bbox": [ + 236, + 155, + 759, + 277 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/548a288bc91ae0609a7c35054d158b13e6149f1b900a2a3870f131fa4fc14015.jpg", + "image_caption": [ + "Figure 24: Model-wise test error dynamics for a IWSLT‘14 de-en and subsampled WMT‘14 en-fr datasets. Left: IWSLT‘14, Right: subsampled (200k samples) WMT‘14. Note that with optimal early-stopping, the test error is much lower for this task. " + ], + "image_footnote": [], + "bbox": [ + 236, + 338, + 759, + 459 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "CIFAR10, $10 \\%$ noise, SGD ", + "bbox": [ + 174, + 527, + 364, + 542 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/329ab7efcf3819680ed22e9d04f97aecbd6fd2941d5217d8d45d006ce1c162c0.jpg", + "image_caption": [ + "Figure 25: Top: Train and test performance as a function of both model size and train epochs. Bottom: Test error dynamics of the same model (CNN, on CIFAR-10 with $10 \\%$ label noise, dataaugmentation and SGD optimizer with learning rate $\\propto 1 / \\sqrt { T } )$ ). " + ], + "image_footnote": [], + "bbox": [ + 267, + 554, + 728, + 840 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "E.2.4 TRAINING PROCEDURE ", + "text_level": 1, + "bbox": [ + 174, + 103, + 393, + 117 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/cce2a82acf0147228634fdd0581f2e67796b76009678007530194689d914a4f5.jpg", + "image_caption": [ + "Figure 26: Model-wise double descent for adversarial training ResNet18s on CIFAR-10 (subsampled to $2 5 \\mathrm { k }$ train samples) with no label noise. We train for L2 robustness of radius $\\epsilon = 0 . 5$ and $\\epsilon = 1 . 0$ , using 10-step PGD (Goodfellow et al. (2014); Madry et al. (2017)). Trained using SGD (batch size 128) with learning rate 0.1 for 400 epochs, then 0.01 for 400 epochs. " + ], + "image_footnote": [], + "bbox": [ + 343, + 140, + 653, + 483 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/3f166e910507b6f677862e70ccd9d2fcc5d745118d0ea7693bdfe992dda82c23.jpg", + "image_caption": [ + "Figure 27 " + ], + "image_footnote": [], + "bbox": [ + 302, + 580, + 692, + 880 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/92155799e3b955426cfd287a4260029a284a4655cd2cd750acbb4e3c8ec8bdcc.jpg", + "image_caption": [ + "Figure 28: Effect of Ensembling (ResNets, $15 \\%$ label noise). Test error of an ensemble of 5 models, compared to the base models. The ensembled classifier is determined by plurality vote over the 5 base models. Note that emsembling helps most around the critical regime. All models are ResNet18s trained on CIFAR-10 with $15 \\%$ label noise, using Adam for 4K epochs (same setting as Figure 1). Test error is measured against the original (not noisy) test set, and each model in the ensemble is trained using a train set with independently-sampled $15 \\%$ label noise. " + ], + "image_footnote": [], + "bbox": [ + 282, + 143, + 714, + 300 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/b3feb5b8e7fb91ef88b906092c18d0ee5c04fae19b187b34bbe97bc9c5642f30.jpg", + "image_caption": [ + "Figure 29: Effect of Ensembling (CNNs, no label noise). Test error of an ensemble of 5 models, compared to the base models. All models are 5-layer CNNs trained on CIFAR-10 with no label noise, using SGD and no data augmentation. (same setting as Figure 7). 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Right: Test error, shown for varying train epochs. All models", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "trained using Adam for 4K epochs. The largest model (width 64) corresponds to standard ResNet18.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "score": 1.0, + "content": "The bias-variance trade-off is a fundamental concept in classical statistical learning theory (e.g.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "Hastie et al. (2005)). The idea is that models of higher complexity have lower bias but higher vari-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "ance. According to this theory, once model complexity passes a certain threshold, models “overfit”", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "with the variance term dominating the test error, and hence from this point onward, increasing model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "complexity will only decrease performance (i.e., increase test error). Hence conventional wisdom", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 621, + 462, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 462, + 634 + ], + "score": 1.0, + "content": "in classical statistics is that, once we pass a certain threshold, “larger models are worse.”", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 649 + ], + "score": 1.0, + "content": "However, modern neural networks exhibit no such phenomenon. Such networks have millions of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "parameters, more than enough to fit even random labels (Zhang et al. (2016)), and yet they perform", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "much better on many tasks than smaller models. Indeed, conventional wisdom among practitioners", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "is that “larger models are better’’ (Krizhevsky et al. (2012), Huang et al. (2018), Szegedy et al.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 690, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 689, + 504, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 689, + 504, + 703 + ], + "score": 1.0, + "content": "∗Work performed in part while Preetum Nakkiran was interning at OpenAI, with Ilya Sutskever. We espe-", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "cially thank Mikhail Belkin and Christopher Olah for helpful discussions throughout this work. Correspondence", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 709, + 254, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 254, + 722 + ], + "score": 1.0, + "content": "Email: preetum@cs.harvard.edu", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 720, + 192, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 192, + 734 + ], + "score": 1.0, + "content": "†Equal contribution", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] 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Moreover, we show that double descent occurs not just as a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 272, + 469, + 285 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 469, + 285 + ], + "score": 1.0, + "content": "function of model size, but also as a function of the number of training epochs.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 283, + 470, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 283, + 470, + 296 + ], + "score": 1.0, + "content": "We unify the above phenomena by defining a new complexity measure we call", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 294, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 470, + 308 + ], + "score": 1.0, + "content": "the effective model complexity and conjecture a generalized double descent with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 306, + 469, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 469, + 318 + ], + "score": 1.0, + "content": "respect to this measure. Furthermore, our notion of model complexity allows us to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 317, + 469, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 469, + 329 + ], + "score": 1.0, + "content": "identify certain regimes where increasing (even quadrupling) the number of train", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 328, + 305, + 340 + ], + "spans": [ + { + "bbox": [ + 141, + 328, + 305, + 340 + ], + "score": 1.0, + "content": "samples actually hurts test performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 141, + 239, + 470, + 340 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 361, + 206, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 208, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 208, + 376 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "image", + "bbox": [ + 106, + 389, + 527, + 504 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 389, + 527, + 504 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 389, + 527, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 527, + 504 + ], + "score": 0.97, + "type": "image", + "image_path": "c4a62cbdb7fbeb5034e735ae46b0639e991da1ee0ce7c9ca2623db843dff96c3.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 106, + 389, + 527, + 427.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 106, + 427.3333333333333, + 527, + 465.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 106, + 465.66666666666663, + 527, + 503.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 511, + 506, + 545 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Figure 1: Left: Train and test error as a function of model size, for ResNet18s of varying width", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 184, + 535 + ], + "score": 1.0, + "content": "on CIFAR-10 with", + "type": "text" + }, + { + "bbox": [ + 184, + 523, + 204, + 533 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "label noise. Right: Test error, shown for varying train epochs. All models", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "trained using Adam for 4K epochs. The largest model (width 64) corresponds to standard ResNet18.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "score": 1.0, + "content": "The bias-variance trade-off is a fundamental concept in classical statistical learning theory (e.g.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "Hastie et al. (2005)). The idea is that models of higher complexity have lower bias but higher vari-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "ance. According to this theory, once model complexity passes a certain threshold, models “overfit”", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "with the variance term dominating the test error, and hence from this point onward, increasing model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "complexity will only decrease performance (i.e., increase test error). Hence conventional wisdom", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 621, + 462, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 462, + 634 + ], + "score": 1.0, + "content": "in classical statistics is that, once we pass a certain threshold, “larger models are worse.”", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 564, + 505, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 649 + ], + "score": 1.0, + "content": "However, modern neural networks exhibit no such phenomenon. Such networks have millions of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "parameters, more than enough to fit even random labels (Zhang et al. (2016)), and yet they perform", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "much better on many tasks than smaller models. Indeed, conventional wisdom among practitioners", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "is that “larger models are better’’ (Krizhevsky et al. (2012), Huang et al. 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The horizontal line corre-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 213, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 505, + 228 + ], + "score": 1.0, + "content": "sponds to model-wise double descent–varying model size while training for as long as possible. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "vertical line corresponds to epoch-wise double descent, with test error undergoing double-descent", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "as train time increases. Right Train error of the corresponding models. All models are Resnet18s", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 247, + 489, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 214, + 260 + ], + "score": 1.0, + "content": "trained on CIFAR-10 with", + "type": "text" + }, + { + "bbox": [ + 214, + 247, + 233, + 258 + ], + "score": 0.86, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 247, + 489, + 260 + ], + "score": 1.0, + "content": "label noise, data-augmentation, and Adam for up to 4K epochs.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 504, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "(2015), Radford et al. (2019)). The effect of training time on test performance is also up for debate.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "In some settings, “early stopping” improves test performance, while in other settings training neu-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "ral networks to zero training error only improves performance. Finally, if there is one thing both", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 322, + 484, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 484, + 335 + ], + "score": 1.0, + "content": "classical statisticians and deep learning practitioners agree on is “more data is always better”.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 104, + 338, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 505, + 353 + ], + "score": 1.0, + "content": "In this paper, we present empirical evidence that both reconcile and challenge some of the above", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "“conventional wisdoms.” We show that many deep learning settings have two different regimes.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "In the under-parameterized regime, where the model complexity is small compared to the number", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "of samples, the test error as a function of model complexity follows the U-like behavior predicted", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "by the classical bias/variance tradeoff. However, once model complexity is sufficiently large to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "interpolate i.e., achieve (close to) zero training error, then increasing complexity only decreases test", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "error, following the modern intuition of “bigger models are better”. Similar behavior was previously", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al. (2018), and Geiger et al.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "(2019b). This phenomenon was first postulated in generality by Belkin et al. (2018) who named", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "it “double descent”, and demonstrated it for decision trees, random features, and 2-layer neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 449, + 452, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 165, + 461 + ], + "score": 1.0, + "content": "networks with", + "type": "text" + }, + { + "bbox": [ + 165, + 449, + 175, + 460 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 449, + 452, + 461 + ], + "score": 1.0, + "content": "loss, on a variety of learning tasks including MNIST and CIFAR-10.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "Main contributions. We show that double descent is a robust phenomenon that occurs in a variety", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "of tasks, architectures, and optimization methods (see Figure 1 and Section 5; our experiments are", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "summarized in Table A). Moreover, we propose a much more general notion of “double descent”", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "that goes beyond varying the number of parameters. We define the effective model complexity (EMC)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "of a training procedure as the maximum number of samples on which it can achieve close to zero", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "score": 1.0, + "content": "training error. The EMC depends not just on the data distribution and the architecture of the classifier", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "but also on the training procedure—and in particular increasing training time will increase the EMC.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "We hypothesize that for many natural models and learning algorithms, double descent occurs as a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "function of the EMC. Indeed we observe “epoch-wise double descent” when we keep the model fixed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "score": 1.0, + "content": "and increase the training time, with performance following a classical U-like curve in the underfitting", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "stage (when the EMC is smaller than the number of samples) and then improving with training time", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "once the EMC is sufficiently larger than the number of samples (see Figure 2). As a corollary, early", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 621, + 496, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 496, + 633 + ], + "score": 1.0, + "content": "stopping only helps in the relatively narrow parameter regime of critically parameterized models.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Sample non-monotonicity. Finally, our results shed light on test performance as a function of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "the number of train samples. 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While in most settings increasing the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "number of samples decreases error, this shifting effect can sometimes result in a setting where more", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "data is worse! 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The horizontal line corre-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 213, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 505, + 228 + ], + "score": 1.0, + "content": "sponds to model-wise double descent–varying model size while training for as long as possible. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "vertical line corresponds to epoch-wise double descent, with test error undergoing double-descent", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "as train time increases. Right Train error of the corresponding models. All models are Resnet18s", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 247, + 489, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 214, + 260 + ], + "score": 1.0, + "content": "trained on CIFAR-10 with", + "type": "text" + }, + { + "bbox": [ + 214, + 247, + 233, + 258 + ], + "score": 0.86, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 247, + 489, + 260 + ], + "score": 1.0, + "content": "label noise, data-augmentation, and Adam for up to 4K epochs.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 504, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "(2015), Radford et al. (2019)). The effect of training time on test performance is also up for debate.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "In some settings, “early stopping” improves test performance, while in other settings training neu-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "ral networks to zero training error only improves performance. Finally, if there is one thing both", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 322, + 484, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 484, + 335 + ], + "score": 1.0, + "content": "classical statisticians and deep learning practitioners agree on is “more data is always better”.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 290, + 506, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 104, + 338, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 505, + 353 + ], + "score": 1.0, + "content": "In this paper, we present empirical evidence that both reconcile and challenge some of the above", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "“conventional wisdoms.” We show that many deep learning settings have two different regimes.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "In the under-parameterized regime, where the model complexity is small compared to the number", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "of samples, the test error as a function of model complexity follows the U-like behavior predicted", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "by the classical bias/variance tradeoff. However, once model complexity is sufficiently large to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "interpolate i.e., achieve (close to) zero training error, then increasing complexity only decreases test", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "error, following the modern intuition of “bigger models are better”. Similar behavior was previously", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al. (2018), and Geiger et al.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "(2019b). This phenomenon was first postulated in generality by Belkin et al. (2018) who named", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "it “double descent”, and demonstrated it for decision trees, random features, and 2-layer neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 449, + 452, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 165, + 461 + ], + "score": 1.0, + "content": "networks with", + "type": "text" + }, + { + "bbox": [ + 165, + 449, + 175, + 460 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 449, + 452, + 461 + ], + "score": 1.0, + "content": "loss, on a variety of learning tasks including MNIST and CIFAR-10.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 338, + 506, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "Main contributions. We show that double descent is a robust phenomenon that occurs in a variety", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "of tasks, architectures, and optimization methods (see Figure 1 and Section 5; our experiments are", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "summarized in Table A). Moreover, we propose a much more general notion of “double descent”", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "that goes beyond varying the number of parameters. We define the effective model complexity (EMC)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "of a training procedure as the maximum number of samples on which it can achieve close to zero", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "score": 1.0, + "content": "training error. The EMC depends not just on the data distribution and the architecture of the classifier", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "but also on the training procedure—and in particular increasing training time will increase the EMC.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 482, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "We hypothesize that for many natural models and learning algorithms, double descent occurs as a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "function of the EMC. Indeed we observe “epoch-wise double descent” when we keep the model fixed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "score": 1.0, + "content": "and increase the training time, with performance following a classical U-like curve in the underfitting", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "stage (when the EMC is smaller than the number of samples) and then improving with training time", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "once the EMC is sufficiently larger than the number of samples (see Figure 2). As a corollary, early", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 621, + 496, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 496, + 633 + ], + "score": 1.0, + "content": "stopping only helps in the relatively narrow parameter regime of critically parameterized models.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 565, + 507, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Sample non-monotonicity. Finally, our results shed light on test performance as a function of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "the number of train samples. 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We define", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 255, + 504, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 192, + 268 + ], + "score": 1.0, + "content": "a training procedure", + "type": "text" + }, + { + "bbox": [ + 192, + 256, + 201, + 266 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 255, + 383, + 268 + ], + "score": 1.0, + "content": "to be any procedure that takes as input a set", + "type": "text" + }, + { + "bbox": [ + 384, + 256, + 504, + 268 + ], + "score": 0.91, + "content": "S = \\{ ( x _ { 1 } , y _ { 1 } ) , \\dots , ( x _ { n } , y _ { n } ) \\}", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 318, + 280 + ], + "score": 1.0, + "content": "of labeled training samples and outputs a classifier", + "type": "text" + }, + { + "bbox": [ + 318, + 267, + 342, + 279 + ], + "score": 0.92, + "content": "\\mathcal { T } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 266, + 506, + 280 + ], + "score": 1.0, + "content": "mapping data to labels. 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If", + "type": "text" + }, + { + "bbox": [ + 244, + 441, + 298, + 453 + ], + "score": 0.92, + "content": "\\cdot _ { \\mathrm { E M C } _ { { \\mathscr D } , \\epsilon } } ( \\mathcal T )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 441, + 405, + 454 + ], + "score": 1.0, + "content": "is sufficiently smaller than", + "type": "text" + }, + { + "bbox": [ + 405, + 444, + 412, + 451 + ], + "score": 0.27, + "content": "n _ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 441, + 495, + 454 + ], + "score": 1.0, + "content": ", any perturbation of", + "type": "text" + }, + { + "bbox": [ + 495, + 442, + 504, + 451 + ], + "score": 0.34, + "content": "\\tau", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 453, + 403, + 465 + ], + "spans": [ + { + "bbox": [ + 142, + 453, + 403, + 465 + ], + "score": 1.0, + "content": "that increases its effective complexity will decrease the test error.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 441, + 504, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 502, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 504, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 241, + 484 + ], + "score": 1.0, + "content": "Over-parameterized regime. 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I", + "type": "text" + }, + { + "bbox": [ + 255, + 501, + 331, + 513 + ], + "score": 0.9, + "content": "f \\mathrm { E M C } _ { \\mathcal { D } , \\epsilon } ( \\mathcal { T } ) \\approx n", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 501, + 425, + 514 + ], + "score": 1.0, + "content": ", then a perturbation of", + "type": "text" + }, + { + "bbox": [ + 425, + 501, + 434, + 511 + ], + "score": 0.61, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "that increases its", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 512, + 391, + 524 + ], + "spans": [ + { + "bbox": [ + 142, + 512, + 391, + 524 + ], + "score": 1.0, + "content": "effective complexity might decrease or increase the test error.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 501, + 505, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "Hypothesis 1 is informal in several ways. 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Our experiments suggest that there is a critical", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 306, + 578 + ], + "score": 1.0, + "content": "interval around the interpolation threshold when", + "type": "text" + }, + { + "bbox": [ + 306, + 566, + 379, + 578 + ], + "score": 0.93, + "content": "\\mathrm { E M C } _ { \\mathcal { D } , \\epsilon } ( \\mathcal { T } ) = n", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 566, + 506, + 578 + ], + "score": 1.0, + "content": ": below and above this interval", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "increasing complexity helps performance, while within this interval it may hurt performance. The", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "width of the critical interval depends on both the distribution and the training procedure in ways we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 599, + 244, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 244, + 611 + ], + "score": 1.0, + "content": "do not yet completely understand.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 532, + 506, + 611 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "We believe Hypothesis 1 sheds light on the interaction between optimization algorithms, model size,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "and test performance and helps reconcile some of the competing intuitions about them. 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In particular, we demonstrate", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 314, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 314, + 117 + ], + "score": 1.0, + "content": "realistic settings in which bigger models are worse.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 52.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "of train samples, and training procedures (data-augmentation, and regularization). Moreover, the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "peak in test error systematically occurs at the interpolation threshold. In particular, we demonstrate", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 314, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 314, + 117 + ], + "score": 1.0, + "content": "realistic settings in which bigger models are worse.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "Epoch-wise Double Descent. In Section 6, we study the test error of a fixed, large architecture over", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "the course of training. We demonstrate, in similar settings as above, a corresponding peak in test", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 366, + 156 + ], + "score": 1.0, + "content": "performance when models are trained just long enough to reach", + "type": "text" + }, + { + "bbox": [ + 367, + 144, + 385, + 154 + ], + "score": 0.84, + "content": "\\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "train error. The test error of a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "large model first decreases (at the beginning of training), then increases (around the critical regime),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 496, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 496, + 179 + ], + "score": 1.0, + "content": "then decreases once more (at the end of training)—that is, training longer can correct overfitting.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "Sample-wise Non-monotonicity. In Section 7, we study the test error of a fixed model and training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "procedure, for varying number of train samples. Consistent with our generalized double-descent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "hypothesis, we observe distinct test behavior in the “critical regime”, when the number of samples", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "is near the maximum that the model can fit. This often manifests as a long plateau region, in which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "taking significantly more data might not help when training to completion (as is the case for CNNs on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "CIFAR-10). Moreover, we show settings (Transformers on IWSLT‘14 en-de), where this manifests", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 460, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 460, + 261 + ], + "score": 1.0, + "content": "as a peak—and for a fixed architecture and training procedure, more data actually hurts.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "Remarks on Label Noise. We observe all forms of double descent most strongly in settings with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "label noise in the train set (as is often the case when collecting train data in the real-world). How-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "ever, we also show several realistic settings with a test-error peak even without label noise: ResNets", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "(Figure 4a) and CNNs (Figure 20) on CIFAR-100; Transformers on IWSLT‘14 (Figure 8). More-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "over, all our experiments demonstrate distinctly different test behavior in the critical regime— often", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "manifesting as a “plateau” in the test error in the noiseless case which develops into a peak with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 328, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 328, + 342 + ], + "score": 1.0, + "content": "added label noise. See Section 8 for further discussion.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 359, + 208, + 371 + ], + "lines": [ + { + "bbox": [ + 104, + 357, + 211, + 374 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 211, + 374 + ], + "score": 1.0, + "content": "3 RELATED WORK", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "Model-wise double descent was first proposed as a general phenomenon by Belkin et al. (2018).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "Similar behavior had been observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "(2018), and Geiger et al. (2019b). Subsequently, there has been a large body of work studying the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "double descent phenomenon. A growing list of papers that theoretically analyze it in the tractable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "setting of linear least squares regression includes Belkin et al. (2019); Hastie et al. (2019); Bartlett", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 504, + 452 + ], + "score": 1.0, + "content": "et al. (2019); Muthukumar et al. (2019); Bibas et al. (2019); Mitra (2019); Mei & Montanari (2019).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "Moreover, Geiger et al. (2019a) provide preliminary results for model-wise double descent in con-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "volutional networks trained on CIFAR-10. Our work differs from the above papers in two crucial", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "aspects: First, we extend the idea of double-descent beyond the number of parameters to incorporate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "the training procedure under a unified notion of “Effective Model Complexity”, leading to novel in-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "sights like epoch-wise double descent and sample non-monotonicity. The notion that increasing train", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "time corresponds to increasing complexity was also presented in Nakkiran et al. (2019). Second, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "provide an extensive and rigorous demonstration of double-descent in modern deep learning, span-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "ning a variety of architectures, datasets, and optimization procedures. An extended discussion of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 538, + 268, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 268, + 550 + ], + "score": 1.0, + "content": "related work is provided in Appendix C.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 566, + 244, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 245, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 245, + 580 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "We briefly describe the experimental setup here; full details are in Appendix B 1. We consider three", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "families of architectures: ResNets, standard CNNs, and Transformers. ResNets: We parameterize", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "a family of ResNet18s (He et al. (2016)) by scaling the width (number of filters) of convolutional", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 281, + 637 + ], + "score": 1.0, + "content": "layers. Specifically, we use layer widths", + "type": "text" + }, + { + "bbox": [ + 281, + 624, + 338, + 636 + ], + "score": 0.94, + "content": "[ k , 2 k , \\bar { 4 } k , 8 k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 623, + 390, + 637 + ], + "score": 1.0, + "content": "for varying", + "type": "text" + }, + { + "bbox": [ + 390, + 624, + 397, + 634 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 623, + 506, + 637 + ], + "score": 1.0, + "content": ". The standard ResNet18", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 169, + 648 + ], + "score": 1.0, + "content": "corresponds to", + "type": "text" + }, + { + "bbox": [ + 169, + 636, + 204, + 646 + ], + "score": 0.91, + "content": "k = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 635, + 506, + 648 + ], + "score": 1.0, + "content": ". Standard CNNs: We consider a simple family of 5-layer CNNs, with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 244, + 659 + ], + "score": 1.0, + "content": "4 convolutional layers of widths", + "type": "text" + }, + { + "bbox": [ + 245, + 646, + 301, + 658 + ], + "score": 0.93, + "content": "[ k , 2 k , 4 k , 8 k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 646, + 353, + 659 + ], + "score": 1.0, + "content": "for varying", + "type": "text" + }, + { + "bbox": [ + 354, + 647, + 361, + 657 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 646, + 506, + 659 + ], + "score": 1.0, + "content": ", and a fully-connected layer. For", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 227, + 670 + ], + "score": 1.0, + "content": "context, the CNN with width", + "type": "text" + }, + { + "bbox": [ + 227, + 658, + 260, + 668 + ], + "score": 0.89, + "content": "k = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 656, + 326, + 670 + ], + "score": 1.0, + "content": ", can reach over", + "type": "text" + }, + { + "bbox": [ + 327, + 657, + 347, + 668 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 656, + 506, + 670 + ], + "score": 1.0, + "content": "test accuracy on CIFAR-10 with data-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "augmentation. Transformers: We consider the 6 layer encoder-decoder from Vaswani et al. (2017),", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 678, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 506, + 693 + ], + "score": 1.0, + "content": "as implemented by Ott et al. (2019). We scale the size of the network by modifying the embedding", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 150, + 704 + ], + "score": 1.0, + "content": "dimension", + "type": "text" + }, + { + "bbox": [ + 150, + 690, + 174, + 701 + ], + "score": 0.91, + "content": "d _ { \\mathrm { m o d e l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 689, + 439, + 704 + ], + "score": 1.0, + "content": ", and setting the width of the fully-connected layers proportionally", + "type": "text" + }, + { + "bbox": [ + 439, + 690, + 500, + 702 + ], + "score": 0.89, + "content": "( d _ { \\mathrm { f f } } = 4 \\cdot d _ { \\mathrm { m o d e l } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 689, + 506, + 704 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 105, + 712, + 501, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 708, + 503, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 383, + 724 + ], + "score": 1.0, + "content": "1The raw data from our experiments are available at:", + "type": "text" + }, + { + "bbox": [ + 399, + 710, + 503, + 723 + ], + "score": 1.0, + "content": "https://gitlab.com/", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 720, + 385, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 385, + 732 + ], + "score": 1.0, + "content": "harvard-machine-learning/double-descent/tree/master", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 115 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "Epoch-wise Double Descent. In Section 6, we study the test error of a fixed, large architecture over", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "the course of training. We demonstrate, in similar settings as above, a corresponding peak in test", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 366, + 156 + ], + "score": 1.0, + "content": "performance when models are trained just long enough to reach", + "type": "text" + }, + { + "bbox": [ + 367, + 144, + 385, + 154 + ], + "score": 0.84, + "content": "\\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "train error. The test error of a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "large model first decreases (at the beginning of training), then increases (around the critical regime),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 496, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 496, + 179 + ], + "score": 1.0, + "content": "then decreases once more (at the end of training)—that is, training longer can correct overfitting.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 122, + 506, + 179 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "Sample-wise Non-monotonicity. In Section 7, we study the test error of a fixed model and training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "procedure, for varying number of train samples. Consistent with our generalized double-descent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "hypothesis, we observe distinct test behavior in the “critical regime”, when the number of samples", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "is near the maximum that the model can fit. This often manifests as a long plateau region, in which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "taking significantly more data might not help when training to completion (as is the case for CNNs on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "CIFAR-10). Moreover, we show settings (Transformers on IWSLT‘14 en-de), where this manifests", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 460, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 460, + 261 + ], + "score": 1.0, + "content": "as a peak—and for a fixed architecture and training procedure, more data actually hurts.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 181, + 506, + 261 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "Remarks on Label Noise. We observe all forms of double descent most strongly in settings with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "label noise in the train set (as is often the case when collecting train data in the real-world). How-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "ever, we also show several realistic settings with a test-error peak even without label noise: ResNets", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "(Figure 4a) and CNNs (Figure 20) on CIFAR-100; Transformers on IWSLT‘14 (Figure 8). More-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "over, all our experiments demonstrate distinctly different test behavior in the critical regime— often", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "manifesting as a “plateau” in the test error in the noiseless case which develops into a peak with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 328, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 328, + 342 + ], + "score": 1.0, + "content": "added label noise. See Section 8 for further discussion.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 265, + 505, + 342 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 359, + 208, + 371 + ], + "lines": [ + { + "bbox": [ + 104, + 357, + 211, + 374 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 211, + 374 + ], + "score": 1.0, + "content": "3 RELATED WORK", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "Model-wise double descent was first proposed as a general phenomenon by Belkin et al. (2018).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "Similar behavior had been observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "(2018), and Geiger et al. (2019b). Subsequently, there has been a large body of work studying the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "double descent phenomenon. A growing list of papers that theoretically analyze it in the tractable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "setting of linear least squares regression includes Belkin et al. (2019); Hastie et al. (2019); Bartlett", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 504, + 452 + ], + "score": 1.0, + "content": "et al. (2019); Muthukumar et al. (2019); Bibas et al. (2019); Mitra (2019); Mei & Montanari (2019).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "Moreover, Geiger et al. (2019a) provide preliminary results for model-wise double descent in con-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "volutional networks trained on CIFAR-10. Our work differs from the above papers in two crucial", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "aspects: First, we extend the idea of double-descent beyond the number of parameters to incorporate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "the training procedure under a unified notion of “Effective Model Complexity”, leading to novel in-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "sights like epoch-wise double descent and sample non-monotonicity. The notion that increasing train", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "time corresponds to increasing complexity was also presented in Nakkiran et al. (2019). Second, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "provide an extensive and rigorous demonstration of double-descent in modern deep learning, span-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "ning a variety of architectures, datasets, and optimization procedures. An extended discussion of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 538, + 268, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 268, + 550 + ], + "score": 1.0, + "content": "related work is provided in Appendix C.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 384, + 506, + 550 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 566, + 244, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 245, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 245, + 580 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "We briefly describe the experimental setup here; full details are in Appendix B 1. We consider three", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "families of architectures: ResNets, standard CNNs, and Transformers. ResNets: We parameterize", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "a family of ResNet18s (He et al. (2016)) by scaling the width (number of filters) of convolutional", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 281, + 637 + ], + "score": 1.0, + "content": "layers. Specifically, we use layer widths", + "type": "text" + }, + { + "bbox": [ + 281, + 624, + 338, + 636 + ], + "score": 0.94, + "content": "[ k , 2 k , \\bar { 4 } k , 8 k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 623, + 390, + 637 + ], + "score": 1.0, + "content": "for varying", + "type": "text" + }, + { + "bbox": [ + 390, + 624, + 397, + 634 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 623, + 506, + 637 + ], + "score": 1.0, + "content": ". The standard ResNet18", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 169, + 648 + ], + "score": 1.0, + "content": "corresponds to", + "type": "text" + }, + { + "bbox": [ + 169, + 636, + 204, + 646 + ], + "score": 0.91, + "content": "k = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 635, + 506, + 648 + ], + "score": 1.0, + "content": ". 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For", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 227, + 670 + ], + "score": 1.0, + "content": "context, the CNN with width", + "type": "text" + }, + { + "bbox": [ + 227, + 658, + 260, + 668 + ], + "score": 0.89, + "content": "k = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 656, + 326, + 670 + ], + "score": 1.0, + "content": ", can reach over", + "type": "text" + }, + { + "bbox": [ + 327, + 657, + 347, + 668 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 656, + 506, + 670 + ], + "score": 1.0, + "content": "test accuracy on CIFAR-10 with data-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "augmentation. Transformers: We consider the 6 layer encoder-decoder from Vaswani et al. 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There is a “plateau” in test error", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 316, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 316, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "around the interpolation point with no label noise,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 316, + 442, + 495, + 453 + ], + "spans": [ + { + "bbox": [ + 316, + 442, + 495, + 453 + ], + "score": 1.0, + "content": "which develops into a peak for added label noise.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + } + ], + "index": 35.75 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "In this section, we study the test error of models of increasing size, when training to completion", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "(for a fixed large number of optimization steps). We demonstrate model-wise double descent across", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "score": 1.0, + "content": "different architectures, datasets, optimizers, and training procedures. The critical region exhibits", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "distinctly different test behavior around the interpolation point and there is often a peak in test error", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 542, + 338, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 338, + 554 + ], + "score": 1.0, + "content": "that becomes more prominent in settings with label noise.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 559, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "For the experiments in this section (Figures 4, 5, 6, 7, 8), notice that all modifications which increase", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "the interpolation threshold (such as adding label noise, using data augmentation, and increasing the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "number of train samples) also correspondingly shift the peak in test error towards larger models.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "Additional plots showing the early-stopping behavior of these models, and additional experiments", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "showing double descent in settings with no label noise (e.g. Figure 19) are in Appendix E.2. We", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "also observed model-wise double descent for adversarial training, with a prominent robust test error", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 625, + 402, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 402, + 638 + ], + "score": 1.0, + "content": "peak even in settings without label noise. See Figure 26 in Appendix E.2.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "Discussion. Fully understanding the mechanisms behind model-wise double descent in deep neu-", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "ral networks remains an important open question. However, an analog of model-wise double descent", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "occurs even for linear models. A recent stream of theoretical works analyzes this setting (Bartlett", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "et al. (2019); Muthukumar et al. (2019); Belkin et al. (2019); Mei & Montanari (2019); Hastie et al.", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 692, + 434, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 434, + 706 + ], + "score": 1.0, + "content": "(2019)). We believe similar mechanisms may be at work in deep neural networks.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 58 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Informally, our intuition is that for model-sizes at the interpolation threshold, there is effectively", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "only one model that fits the train data and this interpolating model is very sensitive to noise in the", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 61.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "For ResNets and CNNs, we train with cross-entropy loss, and the following optimizers: (1) Adam", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 379, + 108 + ], + "score": 1.0, + "content": "with learning-rate 0.0001 for 4K epochs; (2) SGD with learning rate", + "type": "text" + }, + { + "bbox": [ + 379, + 93, + 406, + 108 + ], + "score": 0.91, + "content": "\\propto \\frac { 1 } { \\sqrt { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "for 500K gradient steps.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 469, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 313, + 121 + ], + "score": 1.0, + "content": "We train Transformers for 80K gradient steps, with", + "type": "text" + }, + { + "bbox": [ + 313, + 106, + 332, + 117 + ], + "score": 0.84, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 105, + 469, + 121 + ], + "score": 1.0, + "content": "label smoothing and no drop-out.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 121 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 130, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 349, + 143 + ], + "score": 1.0, + "content": "Label Noise. 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There is a “plateau” in test error", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 316, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 316, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "around the interpolation point with no label noise,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 316, + 442, + 495, + 453 + ], + "spans": [ + { + "bbox": [ + 316, + 442, + 495, + 453 + ], + "score": 1.0, + "content": "which develops into a peak for added label noise.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + } + ], + "index": 35.75 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "In this section, we study the test error of models of increasing size, when training to completion", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "(for a fixed large number of optimization steps). We demonstrate model-wise double descent across", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "score": 1.0, + "content": "different architectures, datasets, optimizers, and training procedures. The critical region exhibits", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "distinctly different test behavior around the interpolation point and there is often a peak in test error", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 542, + 338, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 338, + 554 + ], + "score": 1.0, + "content": "that becomes more prominent in settings with label noise.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 498, + 505, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 559, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "For the experiments in this section (Figures 4, 5, 6, 7, 8), notice that all modifications which increase", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "the interpolation threshold (such as adding label noise, using data augmentation, and increasing the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "number of train samples) also correspondingly shift the peak in test error towards larger models.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "Additional plots showing the early-stopping behavior of these models, and additional experiments", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "showing double descent in settings with no label noise (e.g. Figure 19) are in Appendix E.2. We", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "also observed model-wise double descent for adversarial training, with a prominent robust test error", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 625, + 402, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 402, + 638 + ], + "score": 1.0, + "content": "peak even in settings without label noise. See Figure 26 in Appendix E.2.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 559, + 505, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "Discussion. Fully understanding the mechanisms behind model-wise double descent in deep neu-", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "ral networks remains an important open question. However, an analog of model-wise double descent", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "occurs even for linear models. A recent stream of theoretical works analyzes this setting (Bartlett", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "et al. (2019); Muthukumar et al. (2019); Belkin et al. (2019); Mei & Montanari (2019); Hastie et al.", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 692, + 434, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 434, + 706 + ], + "score": 1.0, + "content": "(2019)). We believe similar mechanisms may be at work in deep neural networks.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 58, + "bbox_fs": [ + 105, + 648, + 505, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Informally, our intuition is that for model-sizes at the interpolation threshold, there is effectively", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "only one model that fits the train data and this interpolating model is very sensitive to noise in the", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "train set and/or model mis-specification. That is, since the model is just barely able to fit the train", + "type": "text", + "cross_page": true + } + ], + "index": 76 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "data, forcing it to fit even slightly-noisy or mis-specified labels will destroy its global structure, and", + "type": "text", + "cross_page": true + } + ], + "index": 77 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "result in high test error. (See Figure 28 in the Appendix for an experiment demonstrating this noise", + "type": "text", + "cross_page": true + } + ], + "index": 78 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "sensitivity, by showing that ensembling helps significantly in the critically-parameterized regime).", + "type": "text", + "cross_page": true + } + ], + "index": 79 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "However for over-parameterized models, there are many interpolating models that fit the train set,", + "type": "text", + "cross_page": true + } + ], + "index": 80 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "and SGD is able to find one that “memorizes” (or “absorbs”) the noise while still performing well", + "type": "text", + "cross_page": true + } + ], + "index": 81 + }, + { + "bbox": [ + 106, + 693, + 184, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 184, + 704 + ], + "score": 1.0, + "content": "on the distribution.", + "type": "text", + "cross_page": true + } + ], + "index": 82 + } + ], + "index": 61.5, + "bbox_fs": [ + 105, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 81, + 295, + 282 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 81, + 295, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 81, + 295, + 282 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 295, + 282 + ], + "score": 0.971, + "type": "image", + "image_path": "44d6b8438c0d02d0c76c80ad346b4c82268cb44163651e6cd9d1516b78b9044e.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 107, + 81, + 295, + 93.5625 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 93.5625, + 295, + 106.125 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 106.125, + 295, + 118.6875 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 107, + 118.6875, + 295, + 131.25 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 107, + 131.25, + 295, + 143.8125 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 107, + 143.8125, + 295, + 156.375 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 107, + 156.375, + 295, + 168.9375 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 107, + 168.9375, + 295, + 181.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 107, + 181.5, + 295, + 194.0625 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 107, + 194.0625, + 295, + 206.625 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 107, + 206.625, + 295, + 219.1875 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 107, + 219.1875, + 295, + 231.75 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 107, + 231.75, + 295, + 244.3125 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 107, + 244.3125, + 295, + 256.875 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 107, + 256.875, + 295, + 269.4375 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 107, + 269.4375, + 295, + 282.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 292, + 506, + 326 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 293, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 504, + 304 + ], + "score": 1.0, + "content": "Figure 5: Effect of Data Augmentation. 5-layer CNNs on CIFAR10, with and without data-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "augmentation. 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As in model-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 660, + 392, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 392, + 671 + ], + "score": 1.0, + "content": "wise double descent, the test error peak is accentuated with label noise.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 561, + 506, + 671 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Conventional wisdom suggests that training is split into two phases: (1) In the first phase, the net-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "work learns a function with a small generalization gap (2) In the second phase, the network starts", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "to over-fit the data leading to an increase in test error. 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Note that, when training to completion, more data helps for small and large models,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 416, + 374, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 374, + 430 + ], + "score": 1.0, + "content": "but does not help for near-critically-parameterized models (green).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + } + ], + "index": 8.0 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "score": 1.0, + "content": "In some settings, these two effects combine to yield a regime of model sizes where more data actually", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "hurts test performance as in Figure 3 (see also Figure 11b). Note that this phenomenon is not unique", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 463, + 392, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 392, + 475 + ], + "score": 1.0, + "content": "to DNNs: more data can hurt even for linear models (see Appendix D).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 490, + 287, + 503 + ], + "lines": [ + { + "bbox": [ + 104, + 489, + 289, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 289, + 506 + ], + "score": 1.0, + "content": "8 CONCLUSION AND DISCUSSION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 529 + ], + "score": 1.0, + "content": "We introduce a generalized double descent hypothesis: models and training procedures exhibit atyp-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "ical behavior when their Effective Model Complexity is comparable to the number of train samples.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "We provide extensive evidence for our hypothesis in modern deep learning settings, and show that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 547, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 561 + ], + "score": 1.0, + "content": "it is robust to choices of dataset, architecture, and training procedures. In particular, we demon-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "strate “model-wise double descent” for modern deep networks and characterize the regime where", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "score": 1.0, + "content": "bigger models can perform worse. We also demonstrate “epoch-wise double descent,” which, to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "best of our knowledge, has not been previously proposed. Finally, we show that the double descent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "phenomenon can lead to a regime where training on more data leads to worse test performance.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "Preliminary results suggest that double descent also holds as we vary the amount of regularization", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 615, + 242, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 242, + 626 + ], + "score": 1.0, + "content": "for a fixed model (see Figure 22).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 504, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "We also believe our characterization of the critical regime provides a useful way of thinking for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "practitioners—if a model and training procedure are just barely able to fit the train set, then small", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "changes to the model or training procedure may yield unexpected behavior (e.g. making the model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 664, + 439, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 439, + 677 + ], + "score": 1.0, + "content": "slightly larger or smaller, changing regularization, etc. may hurt test performance).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Early stopping. We note that many of the phenomena that we highlight often do not occur with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "optimal early-stopping. However, this is consistent with our generalized double descent hypothesis:", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "if early stopping prevents models from reaching 0 train error then we would not expect to see double-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "descent, since the EMC does not reach the number of train samples. Further, we show at least one", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "These twin effects are shown in Figure 11a. Note that there is a range of model sizes where the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 247, + 107 + ], + "score": 1.0, + "content": "effects “cancel out”—and having", + "type": "text" + }, + { + "bbox": [ + 248, + 94, + 262, + 105 + ], + "score": 0.85, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "more train samples does not help test performance when", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "training to completion. 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This phenomenon is corroborated in Figure 12,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "which shows test error as a function of both model and sample size, in the same setting as Figure 11a.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 140 + ] + }, + { + "type": "image", + "bbox": [ + 109, + 148, + 500, + 364 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 148, + 500, + 364 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 148, + 500, + 364 + ], + "spans": [ + { + "bbox": [ + 109, + 148, + 500, + 364 + ], + "score": 0.733, + "type": "image", + "image_path": "67eb2031049b3e7bf4fd7bf3717f93c4f08018a552ffa050f6ba459a5d149a71.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 109, + 148, + 500, + 220.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 109, + 220.0, + 500, + 292.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 292.0, + 500, + 364.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 372, + 505, + 428 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "Figure 12: Left: Test Error as a function of model size and number of train samples, for 5-layer", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 178, + 397 + ], + "score": 1.0, + "content": "CNNs on CIFAR-", + "type": "text" + }, + { + "bbox": [ + 178, + 384, + 218, + 395 + ], + "score": 0.9, + "content": "10 + 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "noise. Note the ridge of high test error again lies along the interpolation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "threshold. Right: Three slices of the left plot, showing the effect of more data for models of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "different sizes. Note that, when training to completion, more data helps for small and large models,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 416, + 374, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 374, + 430 + ], + "score": 1.0, + "content": "but does not help for near-critically-parameterized models (green).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + } + ], + "index": 8.0 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "score": 1.0, + "content": "In some settings, these two effects combine to yield a regime of model sizes where more data actually", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "hurts test performance as in Figure 3 (see also Figure 11b). Note that this phenomenon is not unique", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 463, + 392, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 392, + 475 + ], + "score": 1.0, + "content": "to DNNs: more data can hurt even for linear models (see Appendix D).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 441, + 505, + 475 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 490, + 287, + 503 + ], + "lines": [ + { + "bbox": [ + 104, + 489, + 289, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 289, + 506 + ], + "score": 1.0, + "content": "8 CONCLUSION AND DISCUSSION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 529 + ], + "score": 1.0, + "content": "We introduce a generalized double descent hypothesis: models and training procedures exhibit atyp-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "ical behavior when their Effective Model Complexity is comparable to the number of train samples.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "We provide extensive evidence for our hypothesis in modern deep learning settings, and show that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 547, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 561 + ], + "score": 1.0, + "content": "it is robust to choices of dataset, architecture, and training procedures. In particular, we demon-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "strate “model-wise double descent” for modern deep networks and characterize the regime where", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "score": 1.0, + "content": "bigger models can perform worse. We also demonstrate “epoch-wise double descent,” which, to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "best of our knowledge, has not been previously proposed. Finally, we show that the double descent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "phenomenon can lead to a regime where training on more data leads to worse test performance.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "Preliminary results suggest that double descent also holds as we vary the amount of regularization", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 615, + 242, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 242, + 626 + ], + "score": 1.0, + "content": "for a fixed model (see Figure 22).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 514, + 506, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 504, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "We also believe our characterization of the critical regime provides a useful way of thinking for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "practitioners—if a model and training procedure are just barely able to fit the train set, then small", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "changes to the model or training procedure may yield unexpected behavior (e.g. making the model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 664, + 439, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 439, + 677 + ], + "score": 1.0, + "content": "slightly larger or smaller, changing regularization, etc. may hurt test performance).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 631, + 506, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Early stopping. We note that many of the phenomena that we highlight often do not occur with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "optimal early-stopping. However, this is consistent with our generalized double descent hypothesis:", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "if early stopping prevents models from reaching 0 train error then we would not expect to see double-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "descent, since the EMC does not reach the number of train samples. Further, we show at least one", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "setting where model-wise double descent can still occur even with optimal early stopping (ResNets", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "on CIFAR-100 with no label noise, see Figure 19). We have not observed settings where more data", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "hurts when optimal early-stopping is used. However, we are not aware of reasons which preclude", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "this from occurring. We leave fully understanding the optimal early stopping behavior of double", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 326, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 326, + 140 + ], + "score": 1.0, + "content": "descent as an important open question for future work.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "setting where model-wise double descent can still occur even with optimal early stopping (ResNets", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "on CIFAR-100 with no label noise, see Figure 19). We have not observed settings where more data", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "hurts when optimal early-stopping is used. However, we are not aware of reasons which preclude", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "this from occurring. We leave fully understanding the optimal early stopping behavior of double", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 326, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 326, + 140 + ], + "score": 1.0, + "content": "descent as an important open question for future work.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "Label Noise. In our experiments, we observe double descent most strongly in settings with label", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 174 + ], + "score": 1.0, + "content": "noise. However, we believe this effect is not fundamentally about label noise, but rather about", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "model mis-specification. For example, consider a setting where the label noise is not truly random,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "score": 1.0, + "content": "but rather pseudorandom (with respect to the family of classifiers being trained). In this setting,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "the performance of the Bayes optimal classifier would not change (since the pseudorandom noise", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "score": 1.0, + "content": "is deterministic, and invertible), but we would observe an identical double descent as with truly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "random label noise. Thus, we view adding label noise as merely a proxy for making distributions", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 370, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 370, + 242 + ], + "score": 1.0, + "content": "“harder”— i.e. increasing the amount of model mis-specification.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "Other Notions of Model Complexity. Our notion of Effective Model Complexity is related to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "classical complexity notions such as Rademacher complexity, but differs in several crucial ways:", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "(1) EMC depends on the true labels of the data distribution, and (2) EMC depends on the training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 278, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 278, + 299 + ], + "score": 1.0, + "content": "procedure, not just the model architecture.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 504, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "Other notions of model complexity which do not incorporate features (1) and (2) would not suffice", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "to characterize the location of the double-descent peak. Rademacher complexity, for example, is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "determined by the ability of a model architecture to fit a randomly-labeled train set. But Rademacher", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "score": 1.0, + "content": "complexity and VC dimension are both insufficient to determine the model-wise double descent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "peak location, since they do not depend on the distribution of labels— and our experiments show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 321, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 321, + 370 + ], + "score": 1.0, + "content": "that adding label noise shifts the location of the peak.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "Moreover, both Rademacher complexity and VC dimension depend only on the model family and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "data distribution, and not on the training procedure used to find models. Thus, they are not capable", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "of capturing train-time double-descent effects, such as “epoch-wise” double descent, and the effect", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 279, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 279, + 420 + ], + "score": 1.0, + "content": "of data-augmentation on the peak location.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 433, + 200, + 443 + ], + "lines": [ + { + "bbox": [ + 107, + 434, + 201, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 434, + 201, + 444 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "We thank Mikhail Belkin for extremely useful discussions in the early stages of this work. We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 335, + 476 + ], + "score": 1.0, + "content": "thank Christopher Olah for suggesting the Model Size", + "type": "text" + }, + { + "bbox": [ + 335, + 464, + 345, + 474 + ], + "score": 0.69, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "Epoch visualization, which led to the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "investigation of epoch-wise double descent, as well as for useful discussion and feedback. We also", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "thank Alec Radford, Jacob Steinhardt, and Vaishaal Shankar for helpful discussion and suggestions.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "P.N. thanks OpenAI, the Simons Institute, and the Harvard Theory Group for a research environment", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 507, + 231, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 231, + 519 + ], + "score": 1.0, + "content": "that enabled this kind of work.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 504, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "We thank Dimitris Kalimeris, Benjamin L. Edelman, and Sharon Qian, and Aditya Ramesh for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 536, + 270, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 270, + 546 + ], + "score": 1.0, + "content": "comments on an early draft of this work.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "This work supported in part by NSF grant CAREER CCF 1452961, BSF grant 2014389, NSF US-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "ICCS proposal 1540428, a Google Research award, a Facebook research award, a Simons Investiga-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "tor Award, a Simons Investigator Fellowship, and NSF Awards CCF 1715187, CCF 1565264, CCF", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "1301976, IIS 1409097, and CNS 1618026. Y.B. would like to thank the MIT-IBM Watson AI Lab", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 595, + 339, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 339, + 610 + ], + "score": 1.0, + "content": "for contributing computational resources for experiments.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 108, + 625, + 175, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 625, + 176, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 176, + 639 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 504, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "Madhu S Advani and Andrew M Saxe. 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Reconciling modern machine learning", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 115, + 720, + 406, + 732 + ], + "spans": [ + { + "bbox": [ + 115, + 720, + 406, + 732 + ], + "score": 1.0, + "content": "and the bias-variance trade-off. arXiv preprint arXiv:1812.11118, 2018.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 140 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "Label Noise. In our experiments, we observe double descent most strongly in settings with label", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 174 + ], + "score": 1.0, + "content": "noise. However, we believe this effect is not fundamentally about label noise, but rather about", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "model mis-specification. For example, consider a setting where the label noise is not truly random,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "score": 1.0, + "content": "but rather pseudorandom (with respect to the family of classifiers being trained). In this setting,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "the performance of the Bayes optimal classifier would not change (since the pseudorandom noise", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "score": 1.0, + "content": "is deterministic, and invertible), but we would observe an identical double descent as with truly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "random label noise. Thus, we view adding label noise as merely a proxy for making distributions", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 370, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 370, + 242 + ], + "score": 1.0, + "content": "“harder”— i.e. increasing the amount of model mis-specification.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 151, + 506, + 242 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "Other Notions of Model Complexity. Our notion of Effective Model Complexity is related to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "classical complexity notions such as Rademacher complexity, but differs in several crucial ways:", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "(1) EMC depends on the true labels of the data distribution, and (2) EMC depends on the training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 278, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 278, + 299 + ], + "score": 1.0, + "content": "procedure, not just the model architecture.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 252, + 506, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 504, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "Other notions of model complexity which do not incorporate features (1) and (2) would not suffice", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "to characterize the location of the double-descent peak. Rademacher complexity, for example, is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "determined by the ability of a model architecture to fit a randomly-labeled train set. But Rademacher", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "score": 1.0, + "content": "complexity and VC dimension are both insufficient to determine the model-wise double descent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "peak location, since they do not depend on the distribution of labels— and our experiments show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 321, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 321, + 370 + ], + "score": 1.0, + "content": "that adding label noise shifts the location of the peak.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 303, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "Moreover, both Rademacher complexity and VC dimension depend only on the model family and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "data distribution, and not on the training procedure used to find models. Thus, they are not capable", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "of capturing train-time double-descent effects, such as “epoch-wise” double descent, and the effect", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 279, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 279, + 420 + ], + "score": 1.0, + "content": "of data-augmentation on the peak location.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 375, + 506, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 433, + 200, + 443 + ], + "lines": [ + { + "bbox": [ + 107, + 434, + 201, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 434, + 201, + 444 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "We thank Mikhail Belkin for extremely useful discussions in the early stages of this work. We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 335, + 476 + ], + "score": 1.0, + "content": "thank Christopher Olah for suggesting the Model Size", + "type": "text" + }, + { + "bbox": [ + 335, + 464, + 345, + 474 + ], + "score": 0.69, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "Epoch visualization, which led to the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "investigation of epoch-wise double descent, as well as for useful discussion and feedback. We also", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "thank Alec Radford, Jacob Steinhardt, and Vaishaal Shankar for helpful discussion and suggestions.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "P.N. thanks OpenAI, the Simons Institute, and the Harvard Theory Group for a research environment", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 507, + 231, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 231, + 519 + ], + "score": 1.0, + "content": "that enabled this kind of work.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 452, + 505, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 504, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "We thank Dimitris Kalimeris, Benjamin L. Edelman, and Sharon Qian, and Aditya Ramesh for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 536, + 270, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 270, + 546 + ], + "score": 1.0, + "content": "comments on an early draft of this work.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 524, + 505, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "This work supported in part by NSF grant CAREER CCF 1452961, BSF grant 2014389, NSF US-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "ICCS proposal 1540428, a Google Research award, a Facebook research award, a Simons Investiga-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "tor Award, a Simons Investigator Fellowship, and NSF Awards CCF 1715187, CCF 1565264, CCF", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "1301976, IIS 1409097, and CNS 1618026. 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DatasetArchitectureOpt.Aug.% NoiseDouble-DescentFigure(s)
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DatasetArchitectureOpt.Aug.% NoiseDouble-DescentFigure(s)
ModelEpoch
CIFAR 10CNNSGD√0X×5,27
√10广广5,27,6
√205,27
0Xx√5,25
105
20√√5
SGD + w.d.√20√√21
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Adam√0XX4,10
√5√4
√104,10
√15√4,2
Various√20√4,9,10
√20√16, 17, 18
(subsampled)CNNSGD SGD√10√11a
√20√11a, 12
(adversarial)ResNetSGD0Robust err.26
CIFAR 100ResNetAdam0√X4,19,10
>>10√√4,10
CNNSGD√20√√4,10
0√X20
IWSLT '14 de-enTransformerAdam0√X8,24
(subsampled)TransformerAdam0√X11b,23
WMT'14 en-frTransformerAdam0√X8,24
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Our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 560, + 487, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 487, + 573 + ], + "score": 1.0, + "content": "implementation is adapted from https://github.com/kuangliu/pytorch-cifar.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 505, + 505, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "Standard CNNs. We consider a simple family of 5-layer CNNs, with four Conv-BatchNorm-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "ReLU-MaxPool layers and a fully-connected output layer. 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This architecture is based on the “backbone” architecture from", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 176, + 642 + ], + "score": 1.0, + "content": "Page (2018). 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They", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 332, + 350, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 350, + 344 + ], + "score": 1.0, + "content": "provide a finite-sample analysis for these two cases", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 130, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 130, + 345, + 425, + 359 + ], + "score": 1.0, + "content": "3. Bartlett et al. 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(2018): This paper proposed, in very general terms, that the apparent contradiction", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "between traditional notions of the bias-variance trade-off and empirically successful practices in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "deep learning can be reconciled under a double-descent curve—as model complexity increases, the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "test error follows the traditional “U-shaped curve”, but beyond the point of interpolation, the error", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 162 + ], + "score": 1.0, + "content": "starts to decrease. 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They demonstrate that neural networks are not an aberration in this", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "regard—double-descent is a general phenomenon observed also in linear regression with random", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 222, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 222, + 206 + ], + "score": 1.0, + "content": "features and random forests.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 105, + 506, + 206 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 217, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "Theoretical works on linear least squares regression: A variety of papers have attempted to the-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "oretically analyze this behavior in restricted settings, particularly the case of least squares regression", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 239, + 465, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 465, + 253 + ], + "score": 1.0, + "content": "under various assumptions on the training data, feature spaces and regularization method.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 217, + 505, + 253 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 262, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 130, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 130, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "1. Advani & Saxe (2017); Hastie et al. (2019) both consider the linear regression problem", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 272, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 453, + 285 + ], + "score": 1.0, + "content": "stated above and analyze the generalization behavior in the asymptotic limit", + "type": "text" + }, + { + "bbox": [ + 453, + 272, + 504, + 284 + ], + "score": 0.9, + "content": "N , D \\to \\infty", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 142, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "using random matrix theory. Hastie et al. (2019) highlight that when the model is mis-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 295, + 471, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 471, + 306 + ], + "score": 1.0, + "content": "specified, the minimum of training error can occur for over-parameterized models", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 130, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 130, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "2. Belkin et al. (2019) Linear least squares regression for two data models, where the input", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "data is sampled from a Gaussian and a Fourier series model for functions on a circle. They", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 332, + 350, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 350, + 344 + ], + "score": 1.0, + "content": "provide a finite-sample analysis for these two cases", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 130, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 130, + 345, + 425, + 359 + ], + "score": 1.0, + "content": "3. Bartlett et al. 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(2019) characterize the fundamental limit of of any interpolating solu-", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 141, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "tion in the presence of noise and provide some interesting Fourier-theoretic interpretations.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 129, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 129, + 397, + 505, + 411 + ], + "score": 1.0, + "content": "5. 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The setting is equivalent", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 242, + 256 + ], + "score": 1.0, + "content": "to two-layer neural network with", + "type": "text" + }, + { + "bbox": [ + 243, + 243, + 262, + 254 + ], + "score": 0.9, + "content": "e ^ { - i x }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "activation, with randomly-initialized first layer that is fixed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 255, + 380, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 380, + 267 + ], + "score": 1.0, + "content": "throughout training. 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In this figure the embedding dimension (number of random features) is 1000.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + } + ], + "index": 29.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 81, + 320, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 321, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 321, + 95 + ], + "score": 1.0, + "content": "D RANDOM FEATURES: A CASE STUDY", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 111, + 504, + 223 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 111, + 504, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 111, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 108, + 111, + 504, + 223 + ], + "score": 0.971, + "type": "image", + "image_path": "228a2efe993d5a5b8de88a94c4213a4344f8bf8c59e0d4b46ffa29d81cf9c5ed.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 111, + 504, + 148.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 148.33333333333334, + 504, + 185.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 185.66666666666669, + 504, + 223.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 232, + 506, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "Figure 14: Random Fourier Features on the Fashion MNIST dataset. The setting is equivalent", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 242, + 256 + ], + "score": 1.0, + "content": "to two-layer neural network with", + "type": "text" + }, + { + "bbox": [ + 243, + 243, + 262, + 254 + ], + "score": 0.9, + "content": "e ^ { - i x }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "activation, with randomly-initialized first layer that is fixed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 255, + 380, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 380, + 267 + ], + "score": 1.0, + "content": "throughout training. 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This setting is equivalent to a two-layer neural network", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 127, + 333 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 320, + 147, + 330 + ], + "score": 0.89, + "content": "e ^ { - i x }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 317, + 341, + 333 + ], + "score": 1.0, + "content": "activation. 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(same setting as Figure 7).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 11.5 + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/B1n8LexRZ/B1n8LexRZ.md b/parse/train/B1n8LexRZ/B1n8LexRZ.md new file mode 100644 index 0000000000000000000000000000000000000000..ce0b8e35f229d1662647930de0664a5fb363e55a --- /dev/null +++ b/parse/train/B1n8LexRZ/B1n8LexRZ.md @@ -0,0 +1,438 @@ +# GENERALIZING HAMILTONIAN MONTE CARLO WITH NEURAL NETWORKS + +Daniel Levy1∗, Matthew D. Hoffman2, Jascha Sohl-Dickstein3 1Stanford University, 2Google AI Perception , 3Google Brain danilevy@cs.stanford.edu, {mhoffman,jaschasd}@google.com + +# ABSTRACT + +We present a general-purpose method to train Markov chain Monte Carlo kernels, parameterized by deep neural networks, that converge and mix quickly to their target distribution. Our method generalizes Hamiltonian Monte Carlo and is trained to maximize expected squared jumped distance, a proxy for mixing speed. We demonstrate large empirical gains on a collection of simple but challenging distributions, for instance achieving a $1 0 6 \times$ improvement in effective sample size in one case, and mixing when standard HMC makes no measurable progress in a second. Finally, we show quantitative and qualitative gains on a real-world task: latent-variable generative modeling. We release an open source TensorFlow implementation of the algorithm. + +# 1 INTRODUCTION + +High-dimensional distributions that are only analytically tractable up to a normalizing constant are ubiquitous in many fields. For instance, they arise in protein folding (Schutte et al., 1999), physics ¨ simulations (Olsson, 1995), and machine learning (Andrieu et al., 2003). Sampling from such distributions is a critical task for learning and inference (MacKay, 2003), however it is an extremely hard problem in general. + +Markov Chain Monte Carlo (MCMC) methods promise a solution to this problem. They operate by generating a sequence of correlated samples that converge in distribution to the target. This convergence is most often guaranteed through detailed balance, a sufficient condition for the chain to have the target equilibrium distribution. In practice, for any proposal distribution, one can ensure detailed balance through a Metropolis-Hastings (Hastings, 1970) accept/reject step. + +Despite theoretical guarantees of eventual convergence, in practice convergence and mixing speed depend strongly on choosing a proposal that works well for the task at hand. What’s more, it is often more art than science to know when an MCMC chain has converged (“burned-in”), and when the chain has produced a new uncorrelated sample (“mixed”). Additionally, the reliance on detailed balance, which assigns equal probability to the forward and reverse transitions, often encourages random-walk behavior and thus slows exploration of the space (Ichiki & Ohzeki, 2013). + +For densities over continuous spaces, Hamiltonian Monte Carlo (HMC; Duane et al., 1987; Neal, 2011) introduces independent, auxiliary momentum variables, and computes a new state by integrating Hamiltonian dynamics. This method can traverse long distances in state space with a single Metropolis-Hastings test. This is the state-of-the-art method for sampling in many domains. However, HMC can perform poorly in a number of settings. While HMC mixes quickly spatially, it struggles at mixing across energy levels due to its volume-preserving dynamics. HMC also does not work well with multi-modal distributions, as the probability of sampling a large enough momentum to traverse a very low-density region is negligibly small. Furthermore, HMC struggles with ill-conditioned energy landscapes (Girolami & Calderhead, 2011) and deals poorly with rapidly changing gradients (Sohl-Dickstein et al., 2014). + +Recently, probabilistic models parameterized by deep neural networks have achieved great success at approximately sampling from highly complex, multi-modal empirical distributions (Kingma & + +Welling, 2013; Rezende et al., 2014; Goodfellow et al., 2014; Bengio et al., 2014; Sohl-Dickstein et al., 2015). Building on these successes, we present a method that, given an analytically described distribution, automatically returns an exact sampler with good convergence and mixing properties, from a class of highly expressive parametric models. The proposed family of samplers is a generalization of HMC; it transforms the HMC trajectory using parametric functions (deep networks in our experiments), while retaining theoretical guarantees with a tractable Metropolis-Hastings accept/reject step. The sampler is trained to minimize a variation on expected squared jumped distance (similar in spirit to Pasarica & Gelman (2010)). Our parameterization reduces easily to standard HMC. It is further capable of emulating several common extensions of HMC such as withintrajectory tempering (Neal, 1996) and diagonal mass matrices (Bennett, 1975). + +We evaluate our method on distributions where HMC usually struggles, as well as on a the real-world task of training latent-variable generative models. + +Our contributions are as follows: + +• We introduce a generic training procedure which takes as input a distribution defined by an energy function, and returns a fast-mixing MCMC kernel. +• We show significant empirical gains on various distributions where HMC performs poorly. +• We finally evaluate our method on the real-world task of training and sampling from a latent variable generative model, where we show improvement in the model’s log-likelihood, and greater complexity in the distribution of posterior samples. + +# 2 RELATED WORK + +Adaptively modifying proposal distributions to improve convergence and mixing has been explored in the past (Andrieu & Thoms, 2008). In the case of HMC, prior work has reduced the need to choose step size (Neal, 2011) or number of leapfrog steps (Hoffman & Gelman, 2014) by adaptively tuning those parameters. Salimans et al. (2015) proposed an alternate scheme based on variational inference. We adopt the much simpler approach of Pasarica & Gelman (2010), who show that choosing the hyperparameters of a proposal distribution to maximize expected squared jumped distance is both principled and effective in practice. + +Previous work has also explored applying models from machine learning to MCMC tasks. Kernel methods have been used both for learning a proposal distribution (Sejdinovic et al., 2014) and for approximating the gradient of the energy (Strathmann et al., 2015). In physics, Restricted and semiRestricted Boltzmann machines have been used both to build approximations of the energy function which allow more rapid sampling (Liu et al., 2017; Huang & Wang, 2017), and to motivate new hand-designed proposals (Wang, 2017). + +Most similar to our approach is recent work from Song et al. (2017), which uses adversarial training of a volume-preserving transformation, which is subsequently used as an MCMC proposal distribution. While promising, this technique has several limitations. It does not use gradient information, which is often crucial to maintaining high acceptance rates, especially in high dimensions. It also can only indirectly measure the quality of the generated sample using adversarial training, which is notoriously unstable, suffers from “mode collapse” (where only a portion of a target distribution is covered), and often requires objective modification to train in practice (Arjovsky et al., 2017). Finally, since the proposal transformation preserves volume, it can suffer from the same difficulties in mixing across energy levels as HMC, as we illustrate in Section 5. + +To compute the Metropolis-Hastings acceptance probability for a deterministic transition, the operator must be invertible and have a tractable Jacobian. Recent work (Dinh et al., 2016), introduces RNVP, an invertible transformation that operates by, at each layer, modifying only a subset of the variables by a function that depends solely on the remaining variables. This is exactly invertible with an efficiently computable Jacobian. Furthermore, by chaining enough of these layers, the model can be made arbitrarily expressive. This parameterization will directly motivate our extension of the leapfrog integrator in HMC. + +# 3 BACKGROUND + +# 3.1 MCMC METHODS AND METROPOLIS-HASTINGS + +Let $p$ be a target distribution, analytically known up to a constant, over a space $\mathcal { X }$ . Markov chain Monte Carlo (MCMC) methods (Neal, 1993) aim to provide samples from $p$ . To that end, MCMC methods construct a Markov Chain whose stationary distribution is the target distribution $p$ . Obtaining samples then corresponds to simulating a Markov Chain, i.e., given an initial distribution $\pi _ { 0 }$ and a transition kernel $K$ , constructing the following sequence of random variables: + +$$ +X _ { 0 } \sim \pi _ { 0 } , \quad X _ { t + 1 } \sim K ( \cdot | X _ { t } ) . +$$ + +In order for $p$ to be the stationary distribution of the chain, three conditions must be satisfied: $K$ must be irreducible and aperiodic (these are usually mild technical conditions) and $p$ has to be a fixed point of $K$ . This last condition can be expressed as: $\begin{array} { r } { p ( x ^ { \prime } ) = \int K ( x ^ { \prime } | x ) p ( x ) \mathrm { d } x } \end{array}$ . This condition is most often satisfied by satisfying the stronger detailed balance condition, which can be written as: $p ( x ^ { \prime } ) K ( x | x ^ { \prime } ) = p ( x ) \dot { K } ( x ^ { \prime } | \dot { x } )$ . + +Given any proposal distribution $q$ , satisfying mild conditions, we can easily construct a transition kernel that respects detailed balance using Metropolis-Hastings (Hastings, 1970) accept/reject rules. More formally, starting from $x _ { 0 } \sim \pi _ { 0 }$ , at each step $t$ , we sample $x ^ { \prime } \sim q ( \cdot | X _ { t } )$ , and with probability $\begin{array} { r } { A ( x ^ { \prime } | x _ { t } ) = \operatorname* { m i n } \left( 1 , \frac { p ( x ^ { \prime } ) q ( x _ { t } | x ^ { \prime } ) } { p ( x _ { t } ) q ( x ^ { \prime } | x _ { t } ) } \right) } \end{array}$ , accept $x ^ { \prime }$ as the next sample $x _ { t + 1 }$ in the chain. If we reject $x ^ { \prime }$ , then we retain the previous state and $x _ { t + 1 } ~ = ~ x _ { t }$ . For typical proposals this algorithm has strong asymptotic guarantees. But in practice one must often choose between very low acceptance probabilities and very cautious proposals, both of which lead to slow mixing. For continuous state spaces, Hamiltonian Monte Carlo (HMC; Neal, 2011) tackles this problem by proposing updates that move far in state space while staying roughly on iso-probability contours of $p$ . + +# 3.2 HAMILTONIAN MONTE CARLO + +Without loss of generality, we assume $p \left( x \right)$ to be defined by an energy function $U \left( x \right)$ , s.t. $p ( x ) \propto \exp ( - U ( { \bar { x } } ) )$ , and where the state $x \in \mathbb { R } ^ { n }$ . HMC extends the state space with an additional momentum vector $v \in \mathbb { R } ^ { n }$ , where $v$ is distributed independently from $x$ , as $p ( v ) \propto \exp ( - \frac { 1 } { 2 } v ^ { T } v )$ (i.e., identity-covariance Gaussian). From an augmented state $\xi \triangleq ( x , v )$ , HMC produces a proposed state $\xi ^ { \prime } = ( \dot { x } ^ { \prime } , v ^ { \prime } )$ by approximately integrating Hamiltonian dynamics jointly on $x$ and $v$ , with $U \left( x \right)$ taken to be the potential energy, and $\scriptstyle { \frac { 1 } { 2 } } v ^ { \overline { { T } } } v$ the kinetic energy. Since Hamiltonian dynamics conserve the total energy of a system, their approximate integration moves along approximate iso-probability contours of $p \bar { ( \boldsymbol { x } , \boldsymbol { v } ) } = \bar { p } ( \boldsymbol { x } ) p ( \boldsymbol { v } )$ . + +The dynamics are typically simulated using the leapfrog integrator (Hairer et al., 2003; Leimkuhler & Reich, 2004), which for a single time step consists of: + +$$ +\begin{array} { r } { v ^ { \frac { 1 } { 2 } } = v - \frac { \epsilon } { 2 } \partial _ { x } U ( x ) ; \quad x ^ { \prime } = x + \epsilon v ^ { \frac { 1 } { 2 } } ; \quad v ^ { \prime } = v - \frac { \epsilon } { 2 } \partial _ { x } U ( x ^ { \prime } ) . } \end{array} +$$ + +Following Sohl-Dickstein et al. (2014), we write the action of the leapfrog integrator in terms of an operator $\mathbf { L }$ : $\mathbf { L } \boldsymbol { \xi } \triangleq \mathbf { L } ( \boldsymbol { x } , \boldsymbol { v } ) \triangleq ( \boldsymbol { x } ^ { \prime } , \boldsymbol { v } ^ { \prime } )$ , and introduce a momentum flip operator $\mathbf { F }$ : $\mathbf { F } ( x , v ) \triangleq$ $( x , - v )$ . It is important to note two properties of these operators. First, the transformation $\mathbf { F L }$ is an involution, i.e. $\mathbf { F L F L } ( x , v ) = \mathbf { F L } ( x ^ { \prime } , - v ^ { \prime } ) = ( x , v )$ . Second, the transformations from $( x , v )$ to $( x , v ^ { \frac { 1 } { 2 } } )$ , from $( x , v ^ { \frac { 1 } { 2 } } )$ to $( x ^ { \prime } , v ^ { \frac { 1 } { 2 } } )$ , and from $( x ^ { \prime } , v ^ { \frac { 1 } { 2 } } )$ to $( x ^ { \prime } , v ^ { \prime } )$ are all volume-preserving shear transformations i.e., only one of the variables ( $x$ or $v$ ) changes, by an amount determined by the other one. The determinant of the Jacobian, $\left. \frac { \partial [ \mathbf { F } \mathbf { L } \xi ] } { \partial \xi ^ { T } } \right.$ , is thus easy to compute. For vanilla HMC $\begin{array} { r } { \left| \frac { \partial [ \mathbf { F } \mathbf { L } \xi ] } { \partial \xi ^ { T } } \right| = 1 } \end{array}$ , but we will leave it in symbolic form for use in Section 4. The Metropolis-HastingsGreen (Hastings, 1970; Green, 1995) acceptance probability for the HMC proposal is made simple by these two properties, and is + +$$ +\begin{array} { r } { A ( \mathbf { F } \mathbf { L } \xi | \xi ) = \operatorname* { m i n } \left( 1 , \frac { p ( \mathbf { F } \mathbf { L } \xi ) } { p ( \xi ) } \left| \frac { \partial [ \mathbf { F } \mathbf { L } \xi ] } { \partial \xi ^ { T } } \right| \right) . } \end{array} +$$ + +# 4 L2HMC: TRAINING MCMC SAMPLERS + +In this section, we describe our proposed method L2HMC (for ‘Learning To Hamiltonian Monte Carlo’). Given access to only an energy function $U$ (and not samples), L2HMC learns a parametric leapfrog operator $\mathbf { L } _ { \theta }$ over an augmented state space. We begin by describing what desiderata we have for $\mathbf { L } _ { \theta }$ , then go into detail on how we parameterize our sampler. Finally, we conclude this section by describing our training procedure. + +# 4.1 AUGMENTING HMC + +HMC is a powerful algorithm, but it can still struggle even on very simple problems. For example, a two-dimensional multivariate Gaussian with an ill-conditioned covariance matrix can take arbitrarily long to traverse (even if the covariance is diagonal), whereas it is trivial to sample directly from it. Another problem is that HMC can only move between energy levels via a random walk (Neal, 2011), which leads to slow mixing in some models. Finally, HMC cannot easily traverse low-density zones. For example, given a simple Gaussian mixture model, HMC cannot mix between modes without recourse to additional tricks, as illustrated in Figure 1b. These observations determine the list of desiderata for our learned MCMC kernel: fast mixing, fast burn-in, mixing across energy levels, and mixing between modes. + +While pursuing these goals, we must take care to ensure that our proposal operator retains two key features of the leapfrog operator used in HMC: it must be invertible, and the determinant of its Jacobian must be tractable. The leapfrog operator satisfies these properties by ensuring that each sub-update only affects a subset of the variables, and that no sub-update depends nonlinearly on any of the variables being updated. We are free to generalize the leapfrog operator in any way that preserves these properties. In particular, we are free to translate and rescale each sub-update of the leapfrog operator, so long as we are careful to ensure that these translation and scale terms do not depend on the variables being updated. + +# 4.1.1 STATE SPACE + +As in HMC, we begin by augmenting the current state $x \in \mathbb { R } ^ { n }$ with a continuous momentum variable $v \in \mathbb { R } ^ { n }$ drawn from a standard normal. We also introduce a binary direction variable $d \in \{ - 1 , 1 \}$ , drawn from a uniform distribution. We will denote the complete augmented state as $\xi \triangleq ( x , v , d )$ , with probability density $p ( \xi ) = p ( x ) p ( v ) p ( d )$ . Finally, to each step $t$ of the operator $\mathbf { L } _ { \theta }$ we assign a fixed random binary mask $m ^ { t } \in \{ 0 , 1 \} ^ { n }$ that will determine which variables are affected by each sub-update. We draw $m ^ { t }$ uniformly from the set of binary vectors satisfying $\begin{array} { r } { \sum _ { i = 1 } ^ { n } m _ { i } ^ { t } = \lfloor \frac { n } { 2 } \rfloor } \end{array}$ , that is, half of the entries of $m ^ { t }$ are 0 and half are 1. For convenience, we write $\bar { m } ^ { t } = 1 - m ^ { t }$ and $x _ { m ^ { t } } = x \odot m ^ { t }$ $\odot$ denotes element-wise multiplication, and $\mathbb { 1 }$ the all ones vector). + +# 4.1.2 UPDATE STEPS + +We now describe the details of our augmented leapfrog integrator $\mathbf { L } _ { \theta }$ , for a single time-step $t$ , and for direction $d = 1$ . + +We first update the momenta $v$ . This update can only depend on a subset $\zeta _ { 1 } \triangleq ( x , \partial _ { x } U ( x ) , t )$ of the full state, which excludes $v$ . It takes the form + +We have introduced three new functions of $\zeta _ { 1 } \colon T _ { v }$ , $Q _ { v }$ , and $S _ { v }$ . $T _ { v }$ is a translation, $\exp ( Q _ { v } )$ rescales the gradient, and $\exp ( \frac { \epsilon } { 2 } S _ { v } )$ rescales the momentum. The determinant of the Jacobian of this transformation is exp $, \left( \frac { \epsilon } { 2 } \mathbb { 1 } \cdot S _ { v } ( \zeta _ { 1 } ) \right)$ . Note that if $T _ { v }$ , $Q _ { v }$ , and $S _ { v }$ are all zero, then we recover the standard leapfrog momentum update. + +We now update $x$ . As hinted above, to make our transformation more expressive, we first update a subset of the coordinates of $x$ , followed by the complementary subset. The first update, which yields $x ^ { \prime }$ and affects only $x _ { m } t$ , depends on the state subset $\zeta _ { 2 } \triangleq ( x _ { \bar { m } ^ { t } } , v , t )$ . Conversely, with $x ^ { \prime }$ defined below, the second update only affects $\boldsymbol { x } _ { \bar { m } ^ { t } } ^ { \prime }$ and depends only on $\zeta _ { 3 } \triangleq ( x _ { m ^ { t } } ^ { \prime } , v , t )$ : + +$$ +\begin{array} { l } { { x ^ { \prime } = x _ { \bar { m } ^ { t } } + m ^ { t } \odot [ x \odot \exp ( \epsilon S _ { x } ( \zeta _ { 2 } ) ) + \epsilon ( v ^ { \prime } \odot \exp ( \epsilon Q _ { x } ( \zeta _ { 2 } ) ) + T _ { x } ( \zeta _ { 2 } ) ) ] } } \\ { { x ^ { \prime \prime } = x _ { m ^ { t } } ^ { \prime } + \bar { m } ^ { t } \odot [ x ^ { \prime } \odot \exp ( \epsilon S _ { x } ( \zeta _ { 3 } ) ) + \epsilon ( v ^ { \prime } \odot \exp ( \epsilon Q _ { x } ( \zeta _ { 3 } ) ) + T _ { x } ( \zeta _ { 3 } ) ) ] . } } \end{array} +$$ + +Again, $T _ { x }$ is a translation, $\exp ( Q _ { x } )$ rescales the effect of the momenta, $\exp ( \epsilon S _ { x } )$ rescales the positions $x$ , and we recover the original leapfrog position update if $T _ { x } = Q _ { x } = S _ { x } = 0$ . The determinant of the Jacobian of the first transformation is $\bar { \exp { ( \epsilon m ^ { t } \cdot S _ { x } ( \zeta _ { 2 } ) ) } }$ , and the determinant of the Jacobian of the second transformation is $\exp { ( \epsilon \bar { m } ^ { t } \cdot S _ { x } ( \zeta _ { 3 } ) ) }$ . + +Finally, we update $v$ again, based on the subset $\zeta _ { 4 } \triangleq ( x ^ { \prime \prime } , \partial _ { x } U ( x ^ { \prime \prime } ) , t )$ : + +$$ +\begin{array} { r } { v ^ { \prime \prime } = v ^ { \prime } \odot \exp ( \frac { \epsilon } { 2 } S _ { v } ( \zeta _ { 4 } ) ) - \frac { \epsilon } { 2 } ( \partial _ { x } U ( x ^ { \prime \prime } ) \odot \exp ( \epsilon Q _ { v } ( \zeta _ { 4 } ) ) + T _ { v } ( \zeta _ { 4 } ) ) . } \end{array} +$$ + +This update has the same form as the momentum update in equation 4. + +To give intuition into these terms, the scaling applied to the momentum can enable, among other things, acceleration in low-density zones, to facilitate mixing between modes. The scaling term applied to the gradient of the energy may allow better conditioning of the energy landscape (e.g., by learning a diagonal inertia tensor), or partial ignoring of the energy gradient for rapidly oscillating energies. + +The corresponding integrator for $d = - 1$ is given in Appendix A; it essentially just inverts the updates in equations 4, 5 and 6. For all experiments, the functions $Q , S , T$ are implemented using multi-layer perceptrons, with shared weights. We encode the current time step in the MLP input. + +Our leapfrog operator $\mathbf { L } _ { \theta }$ corresponds to running $M$ steps of this modified leapfrog, $\begin{array} { r l } { \mathbf { L } _ { \theta } \boldsymbol { \xi } } & { { } = } \end{array}$ ${ \bf L } _ { \theta } ( x , v , d ) \stackrel { } { = } ( x ^ { \prime \prime \times M } , v ^ { \prime \prime \times M } , d )$ , and our flip operator $\mathbf { F }$ reverses the direction variable $d$ , $\mathbf { F } \xi =$ $( x , v , - d )$ . Written in terms of these modified operators, our proposal and acceptance probability are identical to those for standard HMC. Note, however, that this parameterization enables learning non-volume-preserving transformations, as the determinant of the Jacobian is a function of $S _ { x }$ and $S _ { v }$ that does not necessarily evaluate to 1. This quantity is derived in Appendix B. + +# 4.1.3 MCMC TRANSITIONS + +For convenience, we denote by $\mathbf { R }$ an operator that re-samples the momentum and direction. I.e., given $\xi ~ = ~ ( x , v , d )$ , $\mathbf { R } \xi = ( x , v ^ { \prime } , d ^ { \prime } )$ where $v ^ { \prime } \sim \mathcal { N } ( 0 , I ) , d ^ { \prime } \sim \mathcal { U } \left( \{ - 1 , 1 \} \right)$ . Sampling thus consists of alternating application of the $\mathbf { F L } _ { \theta }$ and $\mathbf { R }$ , in the following two steps each of which is a Markov transition that satisfies detailed balance with respect to $p$ : + +1. $\boldsymbol { \xi } ^ { \prime } = \mathbf { F } \mathbf { L } _ { \boldsymbol { \theta } } \boldsymbol { \xi }$ with probability $A ( \mathbf { F L } _ { \theta } \xi | \xi )$ (Equation 3), otherwise $\xi ^ { \prime } = \xi$ +2. $\boldsymbol { \xi } ^ { \prime } = \mathbf { R } \boldsymbol { \xi }$ + +This parameterization is effectively a generalization of standard HMC as it is non-volume preserving, with learnable parameters, and easily reduces to standard HMC for $Q , S , T = 0$ . + +# 4.2 LOSS AND TRAINING PROCEDURE + +We need some criterion to train the parameters $\theta$ that control the functions $Q , S$ , and $T$ . We choose a loss designed to reduce mixing time. Specifically, we aim to minimize lag-one autocorrelation. This is equivalent to maximizing expected squared jumped distance (Pasarica & Gelman, 2010). For $\xi , \xi ^ { \prime }$ in the extended state space, we define $\delta ( \xi ^ { \bar { \prime } } , \xi ) \ = \ \delta ( ( x ^ { \prime } , v ^ { \prime } , d ^ { \prime } ) , ( x , v , d ) ) \ = \ | | x - x ^ { \prime } | | _ { 2 } ^ { 2 }$ . Expected squared jumped distance is thus $\mathbb { E } _ { \xi \sim p ( \xi ) } \left[ \delta ( \mathbf { F L } _ { \theta } \xi , \xi ) A ( \mathbf { F L } _ { \theta } \xi | \xi ) \right]$ . However, this loss need not encourage mixing across the entire state space. Indeed, maximizing this objective can lead to regions of state space where almost no mixing occurs, so long as the average squared distance traversed remains high. To optimize both for typical and worst case behavior, we include a reciprocal term in the loss, + +$$ +\begin{array} { r } { \ell _ { \lambda } ( \xi , \xi ^ { \prime } , A ( \xi ^ { \prime } | \xi ) ) = \frac { \lambda ^ { 2 } } { \delta ( \xi , \xi ^ { \prime } ) A ( \xi ^ { \prime } | \xi ) } - \frac { \delta ( \xi , \xi ^ { \prime } ) A ( \xi ^ { \prime } | \xi ) } { \lambda ^ { 2 } } , } \end{array} +$$ + +where $\lambda$ is a scale parameter, capturing the characteristic length scale of the problem. The second term encourages typical moves to be large, while the first term strongly penalizes the sampler if it is ever in a state where it cannot move effectively – $\delta ( \xi , \xi ^ { \prime } )$ being small resulting in a large loss value. We train our sampler by minimizing this loss over both the target distribution and initialization distribution. Formally, given an initial distribution $\pi _ { 0 }$ over $\mathcal { X }$ , we define $q ( \xi ) = \pi _ { 0 } ( x ) \mathcal { N } ( v ; 0 , I ) p ( d )$ , and minimize + +$$ +\begin{array} { r } { \mathcal { L } ( \boldsymbol { \theta } ) \triangleq \mathbb { E } _ { p ( \boldsymbol { \xi } ) } \left[ \ell _ { \lambda } ( \boldsymbol { \xi } , \mathbf { F } \mathbf { L } _ { \boldsymbol { \theta } } \boldsymbol { \xi } , A ( \mathbf { F } \mathbf { L } _ { \boldsymbol { \theta } } \boldsymbol { \xi } | \boldsymbol { \xi } ) ) \right] + \lambda _ { b } \mathbb { E } _ { q ( \boldsymbol { \xi } ) } \left[ \ell _ { \lambda } ( \boldsymbol { \xi } , \mathbf { F } \mathbf { L } _ { \boldsymbol { \theta } } \boldsymbol { \xi } , A ( \mathbf { F } \mathbf { L } _ { \boldsymbol { \theta } } \boldsymbol { \xi } | \boldsymbol { \xi } ) ) \right] . } \end{array} +$$ + +The first term of this loss encourages mixing as it considers our operator applied on draws from the distribution; the second term rewards fast burn-in; $\lambda _ { b }$ controls the strength of the ‘burn-in’ regularization. Given this loss, we exactly describe our training procedure in Algorithm 1. It is important to note that each training iteration can be done with only one pass through the network and can be efficiently batched. We further emphasize that this training procedure can be applied to any learnable operator whose Jacobian’s determinant is tractable, making it a general framework for training MCMC proposals. + +# Algorithm 1 Training L2HMC + +Input: Energy function $U : \mathcal { X } \mathbb { R }$ and its gradient $\nabla _ { x } U : x x$ , initial distribution over +the augmented state space $q$ , number of iterations $n _ { \mathrm { i t e r s } }$ , number of leapfrogs $M$ , learning rate +schedule (αt)t≤n , batch size $N$ , scale parameter $\lambda$ and regularization strength $\lambda _ { b }$ . +Initialize the parameters of the sampler $\theta$ . +Initialize $\{ \xi _ { p } ^ { ( i ) } \} _ { i \le N }$ from $q ( \xi )$ . +for $t = 0$ to $n _ { \mathrm { i t e r s } } - 1$ do Sample a minibatch $\{ \xi _ { q } ^ { ( i ) } \} _ { i \leq N }$ from $q ( \xi )$ . $\mathcal { L } 0$ for r $\begin{array} { r l } & { \xi _ { p } ^ { ( i ) } \gets \mathbf { R } \xi _ { p } ^ { ( i ) } } \\ & { \mathcal { L } \gets \mathcal { L } + \ell _ { \lambda } \left( \xi _ { p } ^ { ( i ) } , \mathbf { F L } _ { \theta } \xi _ { p } ^ { ( i ) } , A ( \mathbf { F L } _ { \theta } \xi _ { p } ^ { ( i ) } | \xi _ { p } ^ { ( i ) } ) \right) + \lambda _ { b } \ell _ { \lambda } \left( \xi _ { q } ^ { ( i ) } , \mathbf { F L } _ { \theta } \xi _ { q } ^ { ( i ) } , A ( \mathbf { F L } _ { \theta } \xi _ { q } ^ { ( i ) } | \xi _ { q } ^ { ( i ) } ) \right) } \\ & { } \end{array}$ $i = 1$ to $N$ do $\begin{array} { r } { \xi _ { p } ^ { ( i ) } \mathbf { F L } _ { \theta } \xi _ { p } ^ { ( i ) } } \end{array}$ with probability $A ( \mathbf { F L } _ { \theta } \xi _ { p } ^ { ( i ) } | \xi _ { p } ^ { ( i ) } )$ . end for θ θ α t θ +end for + +# 5 EXPERIMENTS + +We present an empirical evaluation of our trained sampler on a diverse set of energy functions. We first present results on a collection of toy distributions capturing common pathologies of energy landscapes, followed by results on a task from machine learning: maximum-likelihood training of deep generative models. For each, we compare against HMC with well-tuned step length and show significant gains in mixing time. Code implementing our algorithm is available online1. + +# 5.1 VARIED COLLECTION OF ENERGY FUNCTIONS + +We evaluate our L2HMC sampler on a diverse collection of energy functions, each posing different challenges for standard HMC. + +Ill-Conditioned Gaussian (ICG): Gaussian distribution with diagonal covariance spaced loglinearly between $1 0 ^ { - 2 }$ and $1 0 ^ { 2 }$ . This demonstrates that L2HMC can learn a diagonal inertia tensor. + +Strongly correlated Gaussian (SCG): We rotate a diagonal Gaussian with variances $[ 1 0 ^ { 2 } , 1 0 ^ { - 2 } ]$ by $\frac { \pi } { 4 }$ . This is an extreme version of an example from Neal (2011). This problem shows that, although our parametric sampler only applies element-wise transformations, it can adapt to structure which is not axis-aligned. + +Mixture of Gaussians (MoG): Mixture of two isotropic Gaussians with $\sigma ^ { 2 } = 0 . 1$ , and centroids separated by distance 4. The means are thus about 12 standard deviations apart, making it almost impossible for HMC to mix between modes. + +Rough Well: Similar to an example from Sohl-Dickstein et al. (2014), for a given $\eta > 0 , U ( x ) =$ $\begin{array} { r } { \frac { 1 } { 2 } x ^ { T } \overset { \smile } { x } + \eta \sum _ { i } \cos ( \frac { x _ { i } } { \eta } ) } \end{array}$ . For small $\eta$ the energy itself is altered negligibly, but its gradient is perturbed by a high frequency noise oscillating between $- 1$ and 1. In our experiments, we choose $\eta = 1 0 ^ { - 2 }$ . + +For each of these distributions, we compare against HMC with the same number of leapfrog steps and a well-tuned step-size. To compare mixing time, we plot auto-correlation for each method and report effective sample size (ESS). We compute those quantities in the same way as Sohl-Dickstein et al. (2014). We observe that samplers trained with L2HMC show greatly improved autocorrelation and ESS on the presented tasks, providing more than $1 0 6 \times$ improved ESS on the SCG task. In addition, for the MoG, we show that L2HMC can easily mix between modes while standard HMC gets stuck in a mode, unable to traverse the low density zone. Experimental details, as well as a comparison with LAHMC (Sohl-Dickstein et al., 2014), are shown in Appendix C. + +![](images/e878bce92e8dd48bdb837822419a4ea3132437ffdaebbcd93f07f493cdac4546.jpg) +Figure 1: L2HMC mixes faster than well-tuned HMC, and than A-NICE-MC, on a collection of toy distributions. + +Comparison to A-NICE-MC (Song et al., 2017) In addition to the well known challenges associated with adversarial training (Arjovsky et al., 2017), we note that parameterization using a volume-preserving operator can dramatically fail on simple energy landscapes. We build off of the mog2 experiment presented in (Song et al., 2017), which is a 2-d mixture of isotropic Gaussians separated by a distance of 10 with variances 0.5. We consider that setup but increase the ratio of variances: $\dot { \sigma } _ { 1 } ^ { 2 } = 3 , \sigma _ { 2 } ^ { 2 } = 0 . 0 5$ . We show in Figure 1d sample chains trained with L2HMC and A-NICE-MC; A-NICE-MC cannot effectively mix between the two modes as only a fraction of the volume of the large mode can be mapped to the small one, making it highly improbable to traverse. This is also an issue for HMC. On the other hand, L2HMC can both traverse the low-density region between modes, and map a larger volume in the left mode to a smaller volume in the right mode. It is important to note that the distance between both clusters is less than in the mog2 case, and it is thus a good diagnostic of the shortcomings of volume-preserving transformations. + +# 5.2 LATENT-VARIABLE GENERATIVE MODEL + +We apply our learned sampler to the task of training, and sampling from the posterior of, a latentvariable generative model. The model consists of a latent variable $z \sim p ( z )$ , where we choose $p ( z ) = \breve { \mathscr { N } } ( z ; 0 , I )$ , and a conditional distribution $p ( x | z )$ which generates the image $x$ . Given a family of parametric ‘decoders’ $\{ z \mapsto p ( x | z ; \phi ) , \phi \in \Phi \}$ , and a set of samples $\mathcal { D } = \{ x ^ { ( i ) } \} _ { i \leq N }$ training involves finding $\begin{array} { r } { \phi ^ { * } = \arg \operatorname* { m a x } _ { \phi \in \Phi } p ( \mathcal { D } ; \phi ) } \end{array}$ . However, the log-likelihood is intractable as $\begin{array} { r } { p ( x ; \boldsymbol { \phi } ) = \int p ( x | \boldsymbol { z } ; \boldsymbol { \phi } ) p ( \boldsymbol { z } ) \mathrm { d } \boldsymbol { z } } \end{array}$ . To remedy that problem, Kingma $\&$ Welling (2013) proposed jointly training an approximate posterior $q _ { \psi }$ that maximizes a tractable lower-bound on the log-likelihood: + +![](images/250475de256c4eebae15fc4fa637a5547873059d01bec521d1802c331ccf79c1.jpg) +Figure 2: Training and held-out log-likelihood for models trained with L2HMC, HMC, and the ELBO (VAE). + +$$ +\mathcal { L } _ { \mathrm { E L B O } } ( x , \phi , \psi ) = \mathbb { E } _ { q _ { \psi } ( z | x ) } \left[ p ( x | z ; \phi ) \right] - \mathrm { K L } ( q _ { \psi } ( z | x ) | | p ( z ) ) \leq p ( x ) , +$$ + +where $q _ { \psi } ( z | x )$ is a tractable conditional distribution with parameters $\psi$ , typically parameterized by a neural network. Recently, to improve upon well-known pitfalls like over-pruning (Burda et al., 2015) of the VAE, Hoffman (2017) proposed HMC-DLGM. For a data sample $x ^ { ( i ) }$ , after obtaining a sample from the approximate posterior $q _ { \psi } ( \cdot | x ^ { ( i ) } )$ , Hoffman (2017) runs a MCMC algorithm with energy function $U ( z , x ^ { ( i ) } ) = - \log p ( z ) - \log p ( x ^ { ( i ) } | z ; \phi )$ to obtain a more exact posterior sample from $p ( \boldsymbol { z } | \boldsymbol { x } ^ { ( i ) } ; \boldsymbol { \phi } )$ . Given that better posterior sample $z ^ { \prime }$ , the algorithm maximizes $\log p ( x ^ { ( i ) } | z ^ { \prime } ; \phi )$ . + +To show the benefits of L2HMC, we borrow the method from Hoffman (2017), but replace HMC by jointly training an L2HMC sampler to improve the efficiency of the posterior sampling. We call this model L2HMC-DLGM. A diagram of our model and a formal description of our training procedure are presented in Appendix D. We define, for $\xi = \{ z , v , d \} , r ( \xi | x ; \psi ) \triangleq$ $q _ { \psi } ( z | x ) \mathcal { N } ( v ; \bar { 0 , I } ) \mathcal { U } ( d ; \{ - 1 , \bar { 1 } \} )$ . + +In the subsequent sections, we compare our method to the standard VAE model from Kingma & Welling (2013) and HMC-DGLM from Hoffman (2017). It is important to note that, since our sampler is trained jointly with $p _ { \phi }$ and $q _ { \psi }$ , it performs exactly the same number of gradient computations of the energy function as HMC. We first show that training a latent variable generative model with L2HMC results in better generative models both qualitatively and quantitatively. We then show that our improved sampler enables a more expressive, non-Gaussian, posterior. + +Implementation details: Our decoder $( p _ { \phi } )$ is a neural network with 2 fully connected layers, with 1024 units each and softplus non-linearities, and outputs Bernoulli activation probabilities for each pixel. The encoder $( q _ { \psi } )$ has the same architecture, returning mean and variance for the approximate posterior. Our model was trained for 300 epochs with Adam (Kingma & Ba, 2014) and a learning rate $\alpha = 1 0 ^ { - 3 }$ . All experiments were done on the dynamically binarized MNIST dataset (LeCun). + +# 5.2.1 SAMPLE QUALITY AND DATA LIKELIHOOD + +We first present samples from decoders trained with L2HMC, HMC and the ELBO (i.e. vanilla VAE). Although higher log likelihood does not necessarily correspond to better samples (Theis et al., 2015), we can see in Figure 5, shown in the Appendix, that the decoder trained with L2HMC generates sharper samples than the compared methods. + +We now compare our method to HMC in terms of log-likelihood of the data. As we previously stated, the marginal likelihood of a data point $x \in \mathcal { X }$ is not tractable as it requires integrating $p ( x , z )$ over a high-dimensional space. However, we can estimate it using annealed importance sampling (AIS; Neal (2001)). Following Wu et al. (2016), we evaluate our generative models on both training and held-out data. In Figure 2, we plot the data’s log-likelihood against the number of gradient computation steps for both HMC-DGLM and L2HMC-DGLM. We can see that for a similar number of gradient computations, L2HMC-DGLM achieves higher likelihood for both training and held-out data. This is a strong indication that L2HMC provides significantly better posterior samples. + +![](images/fa42f98b9e8fda6e4c48a55741ecf4bec0e430aa2379cfc10ac8d84350687b92.jpg) + +(a) Block Gibbs inpainting of the top half of an MNIST digit, using (top) L2HMC as a posterior sampler, and (bottom) $q _ { \psi }$ as a posterior sampler. + +![](images/3b5a148d0cd5e54e8a2a908b2b63c8ebc00143ad9d41f371e71810f20e32da5a.jpg) +(b) Non-Gaussian posterior +Figure 3: Demonstrations of the value of a more expressive posterior approximation. + +# 5.2.2 INCREASED EXPRESSIVITY OF THE POSTERIOR + +In the standard VAE framework, approximate posteriors are often parametrized by a Gaussian, thus making a strong assumption of uni-modality. In this section, we show that using L2HMC to sample from the posterior enables learning of a richer posterior landscape. + +Block Gibbs Sampling To highlight our ability to capture more expressive posteriors, we in-paint the top of an image using Block Gibbs Sampling using the approximate posterior or L2HMC. Formally, let $x _ { 0 }$ be the starting image. We denote top or bottom-half pixels as $x _ { 0 } ^ { \mathrm { t o p } }$ and $x _ { 0 } ^ { \mathrm { b o t t o m } }$ . At each step $t$ , we sample $z ^ { ( t ) } \sim p ( z | x _ { t } ; \theta )$ , sample $\tilde { x } \sim p ( x | z _ { t } ; \theta )$ . We then set $x _ { t + 1 } ^ { \mathrm { t o p } } = \tilde { x } ^ { \mathrm { t o p } }$ and $x _ { t + 1 } ^ { \mathrm { b o t t o m } } = x _ { 0 } ^ { \mathrm { b o t t o m } }$ . We compare the results obtained by sampling from ior) vs. our trained sampler. The results are reported i $p ( z | x ; \theta )$ using a. We $q _ { \psi }$ (i.e. the see that L2HMC easily mixes between modes (3, 5, 8, and plausibly 9 in the figure) while the approximate posterior gets stuck on the same reconstructed digit (3 in the figure). + +Visualization of the posterior After training a decoder with L2HMC, we randomly choose an element $x _ { 0 } \in \mathcal { D }$ and run 512 parallel L2HMC chains for 20, 000 Metropolis-Hastings steps. We then find the direction of highest variance, project the samples along that direction and show a histogram in Figure 3b. This plot shows non-Gaussianity in the latent space for the posterior. Using our improved sampler enables the decoder to make use of a more expressive posterior, and enables the encoder to sample from this non-Gaussian posterior. + +# 6 FUTURE WORK + +The loss in Section 4.2 targets lag-one autocorrelation. It should be possible to extend this to also target lag-two and higher autocorrelations. It should also be possible to extend this loss to reward fast decay in the autocorrelation of other statistics of the samples, for instance the sample energy as well as the sample position. These additional statistics could also include learned statistics of the samples, combining benefits of the adversarial approach of (Song et al., 2017) with the current work. + +Our learned generalization of HMC should prove complementary to several other research directions related to HMC. It would be interesting to explore combining our work with the use of HMC in a minibatch setting (Chen et al., 2014); with shadow Hamiltonians (Izaguirre & Hampton, 2004); with gradient pre-conditioning approaches similar to those used in Riemannian HMC (Girolami et al., 2009; Betancourt, 2013); with the use of alternative HMC accept-reject rules (Sohl-Dickstein et al., 2014; Berger et al., 2015); with the use of non-canonical Hamiltonian dynamics (Tripuraneni et al., 2016); with variants of AIS adapted to HMC proposals (Sohl-Dickstein & Culpepper, 2012); with the extension of HMC to discrete state spaces (Zhang et al., 2012); and with the use of alternative Hamiltonian integrators (Creutz & Gocksch, 1989; Chao et al., 2015). + +Finally, our work is also complementary to other methods not utilizing gradient information. For example, we could incorporate the intuition behind Multiple Try Metropolis schemes (Martino & Read, 2013) by having several parametric operators and training each one when used. In addition, one could draw inspiration from the adaptive literature (Haario et al., 2001; Andrieu & Thoms, 2008) or component-wise strategies (Gilks & Wild, 1992). + +# 7 CONCLUSION + +In this work, we presented a general method to train expressive MCMC kernels parameterized with deep neural networks. Given a target distribution $p$ , analytically known up to a constant, our method provides a fast-mixing sampler, able to efficiently explore the state space. Our hope is that our method can be utilized in a “black-box” manner, in domains where sampling constitutes a huge bottleneck such as protein foldings (Schutte et al., 1999) or physics simulations (Olsson, 1995). ¨ + +# ACKNOWLEDGMENTS + +We would like to thank Ben Poole, Aditya Grover, David Belanger, and Colin Raffel for insightful comments on the draft, Mohammad Norouzi for support and encouragement launching the project, and Jiaming Song for discussions and help running A-NICE-MC. + +# REFERENCES + +Christophe Andrieu and Johannes Thoms. A tutorial on adaptive MCMC. Statistics and computing, 18(4): 343–373, 2008. + +Christophe Andrieu, Nando De Freitas, Arnaud Doucet, and Michael I Jordan. An introduction to MCMC for machine learning. Machine learning, 50(1-2):5–43, 2003. + +Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein GAN. ´ arXiv preprint arXiv:1701.07875, 2017. + +Yoshua Bengio, Eric Laufer, Guillaume Alain, and Jason Yosinski. 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In Advances in Neural Information Processing Systems, pp. 3194–3202, 2012. + +# Appendix + +# A REVERSE LEAPFROG OPERATOR + +Let $\xi = \{ x , v , d \}$ in the extended state space with $d = - 1$ . Here, we describe the leapfrog updates for a single time step $t$ , this consists of inverting the equations presented in the corresponding section. + +Let $\zeta _ { 1 } = \{ x , v , t \}$ , we have: + +$$ +v ^ { \prime } = \left[ v + \frac { \epsilon } { 2 } \left( \partial _ { x } U ( x ) \odot \exp ( \epsilon Q _ { v } ( \zeta _ { 1 } ) ) + T _ { v } ( \zeta _ { 1 } ) \right) \right] \odot \exp ( - S _ { v } ( \zeta _ { 1 } ) ) . +$$ + +With the notation from Section 4, let $\zeta _ { 2 } \triangleq \{ x _ { m ^ { t } } , v , t \}$ + +$$ +x ^ { \prime } = x _ { m ^ { t } } + \bar { m } ^ { t } \odot [ ( x - \epsilon ( \exp ( \epsilon Q _ { x } ( \zeta _ { 2 } ) ) \odot v ^ { \prime } + T _ { x } ( \zeta _ { 2 } ) ) ] \odot \exp ( - \epsilon S _ { v } ( \zeta _ { 2 } ) ) . +$$ + +Let us denote $\zeta _ { 3 } \triangleq \{ x _ { \bar { m } ^ { t } } ^ { \prime } , v , t \}$ : + +$$ +x ^ { \prime \prime } = x _ { \bar { m } ^ { t } } + m ^ { t } \odot [ ( x ^ { \prime } - \epsilon ( \exp ( \epsilon Q _ { x } ( \zeta _ { 3 } ) ) \odot v ^ { \prime } + T _ { x } ( \zeta _ { 3 } ) ) ] \odot \exp ( - \epsilon S _ { v } ( \zeta _ { 3 } ) ) . +$$ + +Finally, the last update, with $\zeta _ { 4 } \triangleq \{ x ^ { \prime \prime } , \partial _ { x } U ( x ^ { \prime \prime } ) , t \}$ : + +$$ +v ^ { \prime } = \left[ v + { \frac { \epsilon } { 2 } } \left( \partial _ { x } U ( x ^ { \prime \prime } ) \odot \exp ( \epsilon Q _ { v } ( \zeta _ { 4 } ) ) + T _ { v } ( \zeta _ { 4 } ) \right) \right] \odot \exp ( - S _ { v } ( \zeta _ { 4 } ) ) . +$$ + +It is important to note that to invert $\mathbf { L } _ { \theta }$ , these steps should be ran for $t$ from $M$ to 1. + +# B DETERMINANT OF THE JACOBIAN + +Given the derivations (and notations) from Section 4, for the forward operator $\mathbf { L } _ { \theta }$ , we can immediately compute the Jacobian: + +$$ +\log \left| \frac { \partial [ \mathbf { F } \mathbf { L } _ { \theta } \boldsymbol { \xi } ] } { \partial \boldsymbol { \xi } ^ { T } } \right| = d \sum _ { t \leq M } \left[ \frac { \epsilon } { 2 } \mathbf { 1 } \cdot S _ { v } ( \boldsymbol { \zeta } _ { 1 } ^ { t } ) + \epsilon m ^ { t } \cdot S _ { x } ( \boldsymbol { \zeta } _ { 2 } ^ { t } ) + \epsilon \bar { m } ^ { t } \cdot S _ { x } ( \boldsymbol { \zeta } _ { 3 } ^ { t } ) + \frac { \epsilon } { 2 } \mathbf { 1 } \cdot S _ { v } ( \boldsymbol { \zeta } _ { 4 } ^ { t } ) \right] . +$$ + +Where $\zeta _ { i } ^ { t }$ denotes the intermediary variable $\zeta _ { i }$ at time step $t$ and $d$ is the direction of $\xi$ i.e. $\xi =$ +$\{ x , v , d \}$ . + +# C EXPERIMENTAL DETAILS OF SECTION 5 + +# C.1 IMPLEMENTATION DETAILS + +First of all, we keep separate parameters for the network responsible for updating $v$ and those updating $x$ . The architectures are the same. Let us take the example of $Q _ { v } , S _ { v } , T _ { v }$ . The time step $t$ is given as input to the MLP, encoded as $\begin{array} { r } { \tau ( t ) = ( \cos ( \frac { 2 \pi t } { M } ) , \sin ( \frac { 2 \pi { \hat { t } } } { M } ) ) } \end{array}$ . $\sigma ( \cdot )$ denotes the ReLU non-linearity. + +For $n _ { h }$ hidden units per layer: + +• We first compute $h _ { 1 } = \sigma ( W _ { 1 } x + W _ { 2 } v + W _ { 3 } \tau ( t ) + b ) ( h \in \mathbb { R } ^ { n _ { h } } ) .$ · $h _ { 2 } = \sigma ( W _ { 4 } h + b _ { 4 } ) \in \mathbb { R } ^ { n _ { h } }$ • $S _ { v } = \lambda _ { s } \mathtt { t a n h } ( W _ { s } h _ { 2 } + b _ { s } ) , Q _ { v } = \lambda _ { q } \mathtt { t a n h } ( W _ { q } h _ { 2 } + b _ { q } ) , T _ { v } = W _ { t } h _ { 2 } + b _ { t } .$ + +In Section 5.1, the $Q , S , T$ are neural networks with 2 hidden layers with 10 (100 for the 50-d ICG) units and ReLU non-linearities. We train with Adam (Kingma & Ba, 2014) and a learning rate $\alpha = 1 0 ^ { - 3 }$ . We train for 5, 000 iterations with a batch size of 200. + +$\lambda _ { b }$ was set to 0 for ICG and SCG and to 1 for MoG and Rough Well. For the MoG tasks, we train our sampler with a temperature parameter that we continuously anneal; we evaluate the trained sampler without using temperature. + +![](images/ce0d3b31ad5969a1d38e2d717519f244a5cf9175ca9b22c578fb2f7627737684.jpg) +Figure 4: Diagram of our L2HMC-DGLM model. Nodes are functions of their parents. Round nodes are deterministic, diamond nodes are stochastic and the doubly-circled node is observed. + +# C.2 AUTO-CORRELATION AND ESS + +Let $( x _ { \tau } ) _ { \tau \leq T }$ be a set of correlated samples converging to the distribution $p$ with mean $\mu$ and covariance $\Sigma$ . We define auto-correlation at time $t$ as: + +$$ +\rho _ { t } \triangleq \frac { 1 } { \mathrm { T r a c e } ( \Sigma ) ( T - t ) } \sum _ { \tau \leq T - t - 1 } ( x _ { \tau } - \mu ) ^ { T } ( x _ { \tau + t } - \mu ) . +$$ + +We can now define effective sample size (ESS) as: + +$$ +\mathrm { E S S } \left( ( x _ { \tau } ) _ { \tau \leq T } \right) \triangleq \frac { 1 } { 1 + 2 \sum _ { t } \rho _ { t } } . +$$ + +Similar to Hoffman & Gelman (2014), we truncate the sum when the auto-correlation goes below 0.05. + +# C.3 COMPARISON WITH LAHMC + +We compare our trained sampler with LAHMC (Sohl-Dickstein et al., 2014). Results are reported in Table 1. L2HMC largely outperforms LAHMC on all task. LAHMC is also unable to mix between modes for the MoG task. We also note that L2HMC could be easily combined with LAHMC, by replacing the leapfrog integrator of LAHMC with the learned one of L2HMC. + +Table 1: ESS for a fixed number of gradient evaluations. + +
DistributionGradient EvaluationsESS-L2HMCESS-LAHMCRatio
50-d ICG2000156.621.47.3
Rough Well20012.58.61.5
2-d SCG500011616.714.9
MoG20,00065.0<0.53> 123.5
+ +# D L2HMC-DGLM + +# D.1 TRAINING ALGORITHM + +In this section, we present our training algorithm as well as a diagram explaining L2HMC-DGLM. For conciseness, given our operator $\mathbf { L } _ { \theta }$ , we denote by $\mathbf { K } _ { \theta } ( \cdot | x )$ the distribution over next state given sampling of a momentum and direction and the Metropolis-Hastings step. + +# D.2 IMPLEMENTATION DETAILS OF L2HMC-DGLM + +Similar to our L2HMC training on unconditional sampling, we share weights across $Q , S$ and $T$ . In addition, the auxiliary variable $x$ (here the image from MNIST) is first passed through a 2-layer neural network, with softplus non-linearities and 512 hidden units. This input is given to both + +# Algorithm 2 L2HMC for latent variable generative models + +Input: dataset $\mathcal { D }$ , number of iterations $n _ { \mathrm { i t e r s } }$ , number of Metropolis-Hastings step $J$ , number of +leapfrogs M , and learning rate schedule (αt)t≤niters . +Randomly initialize the decoder’s parameters $\phi$ and the approximate posterior $\psi$ . Initialize the +parameters of the sampler $\theta$ with $M$ leapfrog steps. +for $t = 0$ to $n _ { \mathrm { i t e r s } } - 1$ do Randomly sample a minibatch $\boldsymbol { B }$ from the dataset $\mathcal { D }$ . $\mathcal { L } _ { \mathrm { E L B O } } , \mathcal { L } _ { \mathrm { S a m p l e r } } , \mathcal { L } _ { \mathrm { D e c o d e r } } 0$ for $\boldsymbol { x } ^ { ( b ) } \in B \mathrm { d } \mathbf { 0 }$ ∈Sample $\xi _ { 0 } ^ { ( b ) } \sim r ( \cdot | x ^ { ( b ) } ; \psi )$ . $\mathcal { L } _ { \mathrm { E L B O } } p ( x ^ { ( b ) } | z _ { 0 } ^ { ( b ) } ; \phi ) - \mathrm { K L } ( q _ { \psi } ( z | x ^ { ( b ) } ) | | p ( z ) )$ . With ξ(b)0 = {z(b)0 , v(b)0 , d(b)0 } Define the energy function $U _ { x ^ { ( b ) } } ( z ) = - \log p ( x ^ { ( b ) } | z ; \theta ) - \log p ( z )$ $\mathcal { L } _ { \mathrm { S a m p l e r } } 0$ $\lambda \sqrt { \mathrm { V a r } ( q _ { \psi } ( z _ { 0 } ^ { ( b ) } | x ^ { ( b ) } ) }$ for $j = 0$ to $J - 1$ do ξ (j −b) ← Rξ(b)j LSampler ← LSampler + \`λ(ξ(b)j , FLθξ(b)j , A(FLθξ(b)j |ξ(b)j )) Set $\xi _ { j + 1 } ^ { ( b ) }$ to $\mathbf { F L } _ { \theta } \boldsymbol { \xi } _ { j } ^ { ( b ) }$ with probability $A ( \mathbf { F L } _ { \theta } \xi _ { j } ^ { ( b ) } | \xi _ { j } ^ { ( b ) } )$ . $\begin{array} { r } { \begin{array} { l l } { \frac { \operatorname { c a s s a } ^ { \ast } \operatorname { s o r } } { \mathcal { L } _ { \mathrm { D e c o d e r } } } \mathcal { L } _ { \mathrm { D e c o d e r } } + \log p ( x ^ { ( b ) } | z _ { J } ^ { ( s ) } ; \phi ) } & { \qquad \mathrm { ~ s ~ W i t h ~ } \xi _ { J } ^ { ( b ) } = \{ z _ { J } ^ { ( b ) } , v _ { J } ^ { ( b ) } , d _ { J } ^ { ( b ) } \} } \end{array} } \end{array}$ end for $\begin{array} { l } { \phi \phi + \alpha _ { t } \nabla _ { \phi } \mathcal { L } _ { \mathrm { D e c o d e r } } } \\ { \psi \psi + \alpha _ { t } \nabla _ { \psi } \mathcal { L } _ { \mathrm { E L B O } } } \\ { \theta \theta + \alpha _ { t } \nabla _ { \theta } \mathcal { L } _ { \mathrm { S a m p l e r } } } \end{array}$ +end for +$\begin{array} { r l } { 3 } & { 0 } & { 1 \leq 9 \leq 9 \leq 2 \leq 9 \leq 5 } \\ { 2 } & { 4 \times 1 \leq 2 \leq 3 \leq 6 } \\ { 6 } & { 5 \leq 2 \leq 3 \leq 5 } \\ { 3 } & { 6 \leq 8 \leq 7 \leq 3 \leq 4 } \\ { 4 \leq 9 \leq 4 \leq 9 \leq 4 } \\ { 7 \geq 5 \leq 5 \leq 5 \leq 2 \leq 4 \leq 6 } \\ { 0 \leq 6 \leq 1 \leq 2 \leq 2 \leq 4 \leq 6 } \\ { 3 \geq 6 \leq 8 \leq 5 \leq 4 } \end{array}$ 644又9 7 1 9 9 3 / 2 3 S 6 a a 8 7 2 4 一 6 8 9 9 4 D 4 B 7 5 3 6 9 7 7 9 2 4 6 7 0 8 7 ？6 4S q 1 1 23 3 7 9 1 972 076997 090q499 65『6630295147309 h0？b414593&84076 (a) L2HMC (b) HMC (c) VAE + +networks $\{ \cdot \} _ { x }$ and $\{ \cdot \} _ { v }$ . The architecture then consists of 2 hidden layers of 200 units and ReLU non-linearities. For $\lambda$ (scale parameter of the loss), we use the standard deviation of the approximate posterior. + +AIS Evaluation For each data point, we run 20 Markov Chains in parallel, 10, 000 annealing steps with 10 leapfrogs per step and choose the step size for an acceptance rate of 0.65. + +# D.3 MNIST SAMPLES + +We show in Figure 5 samples from the three evaluated models: VAE (Kingma & Welling, 2013), HMC-DGLM (Hoffman, 2017) and L2HMC-DGLM. \ No newline at end of file diff --git a/parse/train/B1n8LexRZ/B1n8LexRZ_content_list.json b/parse/train/B1n8LexRZ/B1n8LexRZ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..552cb7ae92ae5d1c83fd68c55b86238816457b73 --- /dev/null +++ b/parse/train/B1n8LexRZ/B1n8LexRZ_content_list.json @@ -0,0 +1,2181 @@ +[ + { + "type": "text", + "text": "GENERALIZING HAMILTONIAN MONTE CARLO WITH NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Daniel Levy1∗, Matthew D. Hoffman2, Jascha Sohl-Dickstein3 1Stanford University, 2Google AI Perception , 3Google Brain danilevy@cs.stanford.edu, {mhoffman,jaschasd}@google.com ", + "bbox": [ + 183, + 169, + 673, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 250, + 544, + 265 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present a general-purpose method to train Markov chain Monte Carlo kernels, parameterized by deep neural networks, that converge and mix quickly to their target distribution. Our method generalizes Hamiltonian Monte Carlo and is trained to maximize expected squared jumped distance, a proxy for mixing speed. We demonstrate large empirical gains on a collection of simple but challenging distributions, for instance achieving a $1 0 6 \\times$ improvement in effective sample size in one case, and mixing when standard HMC makes no measurable progress in a second. Finally, we show quantitative and qualitative gains on a real-world task: latent-variable generative modeling. We release an open source TensorFlow implementation of the algorithm. ", + "bbox": [ + 233, + 281, + 764, + 421 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 449, + 336, + 465 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "High-dimensional distributions that are only analytically tractable up to a normalizing constant are ubiquitous in many fields. For instance, they arise in protein folding (Schutte et al., 1999), physics ¨ simulations (Olsson, 1995), and machine learning (Andrieu et al., 2003). Sampling from such distributions is a critical task for learning and inference (MacKay, 2003), however it is an extremely hard problem in general. ", + "bbox": [ + 174, + 481, + 823, + 551 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Markov Chain Monte Carlo (MCMC) methods promise a solution to this problem. They operate by generating a sequence of correlated samples that converge in distribution to the target. This convergence is most often guaranteed through detailed balance, a sufficient condition for the chain to have the target equilibrium distribution. In practice, for any proposal distribution, one can ensure detailed balance through a Metropolis-Hastings (Hastings, 1970) accept/reject step. ", + "bbox": [ + 174, + 558, + 823, + 627 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite theoretical guarantees of eventual convergence, in practice convergence and mixing speed depend strongly on choosing a proposal that works well for the task at hand. What’s more, it is often more art than science to know when an MCMC chain has converged (“burned-in”), and when the chain has produced a new uncorrelated sample (“mixed”). Additionally, the reliance on detailed balance, which assigns equal probability to the forward and reverse transitions, often encourages random-walk behavior and thus slows exploration of the space (Ichiki & Ohzeki, 2013). ", + "bbox": [ + 174, + 635, + 825, + 718 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "For densities over continuous spaces, Hamiltonian Monte Carlo (HMC; Duane et al., 1987; Neal, 2011) introduces independent, auxiliary momentum variables, and computes a new state by integrating Hamiltonian dynamics. This method can traverse long distances in state space with a single Metropolis-Hastings test. This is the state-of-the-art method for sampling in many domains. However, HMC can perform poorly in a number of settings. While HMC mixes quickly spatially, it struggles at mixing across energy levels due to its volume-preserving dynamics. HMC also does not work well with multi-modal distributions, as the probability of sampling a large enough momentum to traverse a very low-density region is negligibly small. Furthermore, HMC struggles with ill-conditioned energy landscapes (Girolami & Calderhead, 2011) and deals poorly with rapidly changing gradients (Sohl-Dickstein et al., 2014). ", + "bbox": [ + 174, + 724, + 825, + 864 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, probabilistic models parameterized by deep neural networks have achieved great success at approximately sampling from highly complex, multi-modal empirical distributions (Kingma & ", + "bbox": [ + 176, + 871, + 821, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Welling, 2013; Rezende et al., 2014; Goodfellow et al., 2014; Bengio et al., 2014; Sohl-Dickstein et al., 2015). Building on these successes, we present a method that, given an analytically described distribution, automatically returns an exact sampler with good convergence and mixing properties, from a class of highly expressive parametric models. The proposed family of samplers is a generalization of HMC; it transforms the HMC trajectory using parametric functions (deep networks in our experiments), while retaining theoretical guarantees with a tractable Metropolis-Hastings accept/reject step. The sampler is trained to minimize a variation on expected squared jumped distance (similar in spirit to Pasarica & Gelman (2010)). Our parameterization reduces easily to standard HMC. It is further capable of emulating several common extensions of HMC such as withintrajectory tempering (Neal, 1996) and diagonal mass matrices (Bennett, 1975). ", + "bbox": [ + 174, + 103, + 825, + 242 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We evaluate our method on distributions where HMC usually struggles, as well as on a the real-world task of training latent-variable generative models. ", + "bbox": [ + 174, + 250, + 823, + 277 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions are as follows: ", + "bbox": [ + 174, + 285, + 390, + 299 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We introduce a generic training procedure which takes as input a distribution defined by an energy function, and returns a fast-mixing MCMC kernel. \n• We show significant empirical gains on various distributions where HMC performs poorly. \n• We finally evaluate our method on the real-world task of training and sampling from a latent variable generative model, where we show improvement in the model’s log-likelihood, and greater complexity in the distribution of posterior samples. ", + "bbox": [ + 207, + 316, + 825, + 430 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 460, + 344, + 477 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Adaptively modifying proposal distributions to improve convergence and mixing has been explored in the past (Andrieu & Thoms, 2008). In the case of HMC, prior work has reduced the need to choose step size (Neal, 2011) or number of leapfrog steps (Hoffman & Gelman, 2014) by adaptively tuning those parameters. Salimans et al. (2015) proposed an alternate scheme based on variational inference. We adopt the much simpler approach of Pasarica & Gelman (2010), who show that choosing the hyperparameters of a proposal distribution to maximize expected squared jumped distance is both principled and effective in practice. ", + "bbox": [ + 174, + 498, + 825, + 597 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Previous work has also explored applying models from machine learning to MCMC tasks. Kernel methods have been used both for learning a proposal distribution (Sejdinovic et al., 2014) and for approximating the gradient of the energy (Strathmann et al., 2015). In physics, Restricted and semiRestricted Boltzmann machines have been used both to build approximations of the energy function which allow more rapid sampling (Liu et al., 2017; Huang & Wang, 2017), and to motivate new hand-designed proposals (Wang, 2017). ", + "bbox": [ + 174, + 603, + 823, + 688 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Most similar to our approach is recent work from Song et al. (2017), which uses adversarial training of a volume-preserving transformation, which is subsequently used as an MCMC proposal distribution. While promising, this technique has several limitations. It does not use gradient information, which is often crucial to maintaining high acceptance rates, especially in high dimensions. It also can only indirectly measure the quality of the generated sample using adversarial training, which is notoriously unstable, suffers from “mode collapse” (where only a portion of a target distribution is covered), and often requires objective modification to train in practice (Arjovsky et al., 2017). Finally, since the proposal transformation preserves volume, it can suffer from the same difficulties in mixing across energy levels as HMC, as we illustrate in Section 5. ", + "bbox": [ + 174, + 694, + 825, + 819 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To compute the Metropolis-Hastings acceptance probability for a deterministic transition, the operator must be invertible and have a tractable Jacobian. Recent work (Dinh et al., 2016), introduces RNVP, an invertible transformation that operates by, at each layer, modifying only a subset of the variables by a function that depends solely on the remaining variables. This is exactly invertible with an efficiently computable Jacobian. Furthermore, by chaining enough of these layers, the model can be made arbitrarily expressive. This parameterization will directly motivate our extension of the leapfrog integrator in HMC. ", + "bbox": [ + 174, + 825, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 102, + 326, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 MCMC METHODS AND METROPOLIS-HASTINGS ", + "text_level": 1, + "bbox": [ + 174, + 135, + 552, + 151 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $p$ be a target distribution, analytically known up to a constant, over a space $\\mathcal { X }$ . Markov chain Monte Carlo (MCMC) methods (Neal, 1993) aim to provide samples from $p$ . To that end, MCMC methods construct a Markov Chain whose stationary distribution is the target distribution $p$ . Obtaining samples then corresponds to simulating a Markov Chain, i.e., given an initial distribution $\\pi _ { 0 }$ and a transition kernel $K$ , constructing the following sequence of random variables: ", + "bbox": [ + 173, + 162, + 825, + 233 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d705192185a850364d4b2f7d1986edb4a6abd98330b359208557454284b515a7.jpg", + "text": "$$\nX _ { 0 } \\sim \\pi _ { 0 } , \\quad X _ { t + 1 } \\sim K ( \\cdot | X _ { t } ) .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 243, + 601, + 261 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In order for $p$ to be the stationary distribution of the chain, three conditions must be satisfied: $K$ must be irreducible and aperiodic (these are usually mild technical conditions) and $p$ has to be a fixed point of $K$ . This last condition can be expressed as: $\\begin{array} { r } { p ( x ^ { \\prime } ) = \\int K ( x ^ { \\prime } | x ) p ( x ) \\mathrm { d } x } \\end{array}$ . This condition is most often satisfied by satisfying the stronger detailed balance condition, which can be written as: $p ( x ^ { \\prime } ) K ( x | x ^ { \\prime } ) = p ( x ) \\dot { K } ( x ^ { \\prime } | \\dot { x } )$ . ", + "bbox": [ + 173, + 277, + 825, + 349 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given any proposal distribution $q$ , satisfying mild conditions, we can easily construct a transition kernel that respects detailed balance using Metropolis-Hastings (Hastings, 1970) accept/reject rules. More formally, starting from $x _ { 0 } \\sim \\pi _ { 0 }$ , at each step $t$ , we sample $x ^ { \\prime } \\sim q ( \\cdot | X _ { t } )$ , and with probability $\\begin{array} { r } { A ( x ^ { \\prime } | x _ { t } ) = \\operatorname* { m i n } \\left( 1 , \\frac { p ( x ^ { \\prime } ) q ( x _ { t } | x ^ { \\prime } ) } { p ( x _ { t } ) q ( x ^ { \\prime } | x _ { t } ) } \\right) } \\end{array}$ , accept $x ^ { \\prime }$ as the next sample $x _ { t + 1 }$ in the chain. If we reject $x ^ { \\prime }$ , then we retain the previous state and $x _ { t + 1 } ~ = ~ x _ { t }$ . For typical proposals this algorithm has strong asymptotic guarantees. But in practice one must often choose between very low acceptance probabilities and very cautious proposals, both of which lead to slow mixing. For continuous state spaces, Hamiltonian Monte Carlo (HMC; Neal, 2011) tackles this problem by proposing updates that move far in state space while staying roughly on iso-probability contours of $p$ . ", + "bbox": [ + 173, + 354, + 825, + 492 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 HAMILTONIAN MONTE CARLO ", + "text_level": 1, + "bbox": [ + 176, + 511, + 429, + 526 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Without loss of generality, we assume $p \\left( x \\right)$ to be defined by an energy function $U \\left( x \\right)$ , s.t. $p ( x ) \\propto \\exp ( - U ( { \\bar { x } } ) )$ , and where the state $x \\in \\mathbb { R } ^ { n }$ . HMC extends the state space with an additional momentum vector $v \\in \\mathbb { R } ^ { n }$ , where $v$ is distributed independently from $x$ , as $p ( v ) \\propto \\exp ( - \\frac { 1 } { 2 } v ^ { T } v )$ (i.e., identity-covariance Gaussian). From an augmented state $\\xi \\triangleq ( x , v )$ , HMC produces a proposed state $\\xi ^ { \\prime } = ( \\dot { x } ^ { \\prime } , v ^ { \\prime } )$ by approximately integrating Hamiltonian dynamics jointly on $x$ and $v$ , with $U \\left( x \\right)$ taken to be the potential energy, and $\\scriptstyle { \\frac { 1 } { 2 } } v ^ { \\overline { { T } } } v$ the kinetic energy. Since Hamiltonian dynamics conserve the total energy of a system, their approximate integration moves along approximate iso-probability contours of $p \\bar { ( \\boldsymbol { x } , \\boldsymbol { v } ) } = \\bar { p } ( \\boldsymbol { x } ) p ( \\boldsymbol { v } )$ . ", + "bbox": [ + 173, + 537, + 825, + 655 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The dynamics are typically simulated using the leapfrog integrator (Hairer et al., 2003; Leimkuhler & Reich, 2004), which for a single time step consists of: ", + "bbox": [ + 173, + 660, + 823, + 689 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/129344d4db262ec31f972d8068f8b57408c97a4ca3fe9a09822b67b2b3bcb1d5.jpg", + "text": "$$\n\\begin{array} { r } { v ^ { \\frac { 1 } { 2 } } = v - \\frac { \\epsilon } { 2 } \\partial _ { x } U ( x ) ; \\quad x ^ { \\prime } = x + \\epsilon v ^ { \\frac { 1 } { 2 } } ; \\quad v ^ { \\prime } = v - \\frac { \\epsilon } { 2 } \\partial _ { x } U ( x ^ { \\prime } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 699, + 702, + 719 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Following Sohl-Dickstein et al. (2014), we write the action of the leapfrog integrator in terms of an operator $\\mathbf { L }$ : $\\mathbf { L } \\boldsymbol { \\xi } \\triangleq \\mathbf { L } ( \\boldsymbol { x } , \\boldsymbol { v } ) \\triangleq ( \\boldsymbol { x } ^ { \\prime } , \\boldsymbol { v } ^ { \\prime } )$ , and introduce a momentum flip operator $\\mathbf { F }$ : $\\mathbf { F } ( x , v ) \\triangleq$ $( x , - v )$ . It is important to note two properties of these operators. First, the transformation $\\mathbf { F L }$ is an involution, i.e. $\\mathbf { F L F L } ( x , v ) = \\mathbf { F L } ( x ^ { \\prime } , - v ^ { \\prime } ) = ( x , v )$ . Second, the transformations from $( x , v )$ to $( x , v ^ { \\frac { 1 } { 2 } } )$ , from $( x , v ^ { \\frac { 1 } { 2 } } )$ to $( x ^ { \\prime } , v ^ { \\frac { 1 } { 2 } } )$ , and from $( x ^ { \\prime } , v ^ { \\frac { 1 } { 2 } } )$ to $( x ^ { \\prime } , v ^ { \\prime } )$ are all volume-preserving shear transformations i.e., only one of the variables ( $x$ or $v$ ) changes, by an amount determined by the other one. The determinant of the Jacobian, $\\left. \\frac { \\partial [ \\mathbf { F } \\mathbf { L } \\xi ] } { \\partial \\xi ^ { T } } \\right.$ , is thus easy to compute. For vanilla HMC $\\begin{array} { r } { \\left| \\frac { \\partial [ \\mathbf { F } \\mathbf { L } \\xi ] } { \\partial \\xi ^ { T } } \\right| = 1 } \\end{array}$ , but we will leave it in symbolic form for use in Section 4. The Metropolis-HastingsGreen (Hastings, 1970; Green, 1995) acceptance probability for the HMC proposal is made simple by these two properties, and is ", + "bbox": [ + 173, + 728, + 825, + 892 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/55afa3ae656651f1b5585bdd08c27e684d10f684fe343493a84cd69cf916a2b3.jpg", + "text": "$$\n\\begin{array} { r } { A ( \\mathbf { F } \\mathbf { L } \\xi | \\xi ) = \\operatorname* { m i n } \\left( 1 , \\frac { p ( \\mathbf { F } \\mathbf { L } \\xi ) } { p ( \\xi ) } \\left| \\frac { \\partial [ \\mathbf { F } \\mathbf { L } \\xi ] } { \\partial \\xi ^ { T } } \\right| \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 901, + 632, + 929 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 L2HMC: TRAINING MCMC SAMPLERS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 539, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we describe our proposed method L2HMC (for ‘Learning To Hamiltonian Monte Carlo’). Given access to only an energy function $U$ (and not samples), L2HMC learns a parametric leapfrog operator $\\mathbf { L } _ { \\theta }$ over an augmented state space. We begin by describing what desiderata we have for $\\mathbf { L } _ { \\theta }$ , then go into detail on how we parameterize our sampler. Finally, we conclude this section by describing our training procedure. ", + "bbox": [ + 174, + 132, + 825, + 203 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 AUGMENTING HMC ", + "text_level": 1, + "bbox": [ + 174, + 218, + 356, + 232 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "HMC is a powerful algorithm, but it can still struggle even on very simple problems. For example, a two-dimensional multivariate Gaussian with an ill-conditioned covariance matrix can take arbitrarily long to traverse (even if the covariance is diagonal), whereas it is trivial to sample directly from it. Another problem is that HMC can only move between energy levels via a random walk (Neal, 2011), which leads to slow mixing in some models. Finally, HMC cannot easily traverse low-density zones. For example, given a simple Gaussian mixture model, HMC cannot mix between modes without recourse to additional tricks, as illustrated in Figure 1b. These observations determine the list of desiderata for our learned MCMC kernel: fast mixing, fast burn-in, mixing across energy levels, and mixing between modes. ", + "bbox": [ + 174, + 243, + 825, + 369 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "While pursuing these goals, we must take care to ensure that our proposal operator retains two key features of the leapfrog operator used in HMC: it must be invertible, and the determinant of its Jacobian must be tractable. The leapfrog operator satisfies these properties by ensuring that each sub-update only affects a subset of the variables, and that no sub-update depends nonlinearly on any of the variables being updated. We are free to generalize the leapfrog operator in any way that preserves these properties. In particular, we are free to translate and rescale each sub-update of the leapfrog operator, so long as we are careful to ensure that these translation and scale terms do not depend on the variables being updated. ", + "bbox": [ + 174, + 376, + 825, + 488 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1.1 STATE SPACE ", + "text_level": 1, + "bbox": [ + 174, + 502, + 321, + 516 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As in HMC, we begin by augmenting the current state $x \\in \\mathbb { R } ^ { n }$ with a continuous momentum variable $v \\in \\mathbb { R } ^ { n }$ drawn from a standard normal. We also introduce a binary direction variable $d \\in \\{ - 1 , 1 \\}$ , drawn from a uniform distribution. We will denote the complete augmented state as $\\xi \\triangleq ( x , v , d )$ , with probability density $p ( \\xi ) = p ( x ) p ( v ) p ( d )$ . Finally, to each step $t$ of the operator $\\mathbf { L } _ { \\theta }$ we assign a fixed random binary mask $m ^ { t } \\in \\{ 0 , 1 \\} ^ { n }$ that will determine which variables are affected by each sub-update. We draw $m ^ { t }$ uniformly from the set of binary vectors satisfying $\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } m _ { i } ^ { t } = \\lfloor \\frac { n } { 2 } \\rfloor } \\end{array}$ , that is, half of the entries of $m ^ { t }$ are 0 and half are 1. For convenience, we write $\\bar { m } ^ { t } = 1 - m ^ { t }$ and $x _ { m ^ { t } } = x \\odot m ^ { t }$ $\\odot$ denotes element-wise multiplication, and $\\mathbb { 1 }$ the all ones vector). ", + "bbox": [ + 173, + 525, + 825, + 642 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1.2 UPDATE STEPS ", + "text_level": 1, + "bbox": [ + 174, + 655, + 333, + 670 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now describe the details of our augmented leapfrog integrator $\\mathbf { L } _ { \\theta }$ , for a single time-step $t$ , and for direction $d = 1$ . ", + "bbox": [ + 173, + 679, + 825, + 708 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We first update the momenta $v$ . This update can only depend on a subset $\\zeta _ { 1 } \\triangleq ( x , \\partial _ { x } U ( x ) , t )$ of the full state, which excludes $v$ . It takes the form ", + "bbox": [ + 176, + 714, + 823, + 743 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We have introduced three new functions of $\\zeta _ { 1 } \\colon T _ { v }$ , $Q _ { v }$ , and $S _ { v }$ . $T _ { v }$ is a translation, $\\exp ( Q _ { v } )$ rescales the gradient, and $\\exp ( \\frac { \\epsilon } { 2 } S _ { v } )$ rescales the momentum. The determinant of the Jacobian of this transformation is exp $, \\left( \\frac { \\epsilon } { 2 } \\mathbb { 1 } \\cdot S _ { v } ( \\zeta _ { 1 } ) \\right)$ . Note that if $T _ { v }$ , $Q _ { v }$ , and $S _ { v }$ are all zero, then we recover the standard leapfrog momentum update. ", + "bbox": [ + 174, + 762, + 825, + 821 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now update $x$ . As hinted above, to make our transformation more expressive, we first update a subset of the coordinates of $x$ , followed by the complementary subset. The first update, which yields $x ^ { \\prime }$ and affects only $x _ { m } t$ , depends on the state subset $\\zeta _ { 2 } \\triangleq ( x _ { \\bar { m } ^ { t } } , v , t )$ . Conversely, with $x ^ { \\prime }$ defined below, the second update only affects $\\boldsymbol { x } _ { \\bar { m } ^ { t } } ^ { \\prime }$ and depends only on $\\zeta _ { 3 } \\triangleq ( x _ { m ^ { t } } ^ { \\prime } , v , t )$ : ", + "bbox": [ + 174, + 827, + 823, + 888 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6d17c1fa279df5e04eafd14c968696d59ffdef2d759029cfffe1b6dbb9db5f5e.jpg", + "text": "$$\n\\begin{array} { l } { { x ^ { \\prime } = x _ { \\bar { m } ^ { t } } + m ^ { t } \\odot [ x \\odot \\exp ( \\epsilon S _ { x } ( \\zeta _ { 2 } ) ) + \\epsilon ( v ^ { \\prime } \\odot \\exp ( \\epsilon Q _ { x } ( \\zeta _ { 2 } ) ) + T _ { x } ( \\zeta _ { 2 } ) ) ] } } \\\\ { { x ^ { \\prime \\prime } = x _ { m ^ { t } } ^ { \\prime } + \\bar { m } ^ { t } \\odot [ x ^ { \\prime } \\odot \\exp ( \\epsilon S _ { x } ( \\zeta _ { 3 } ) ) + \\epsilon ( v ^ { \\prime } \\odot \\exp ( \\epsilon Q _ { x } ( \\zeta _ { 3 } ) ) + T _ { x } ( \\zeta _ { 3 } ) ) ] . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 890, + 746, + 930 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Again, $T _ { x }$ is a translation, $\\exp ( Q _ { x } )$ rescales the effect of the momenta, $\\exp ( \\epsilon S _ { x } )$ rescales the positions $x$ , and we recover the original leapfrog position update if $T _ { x } = Q _ { x } = S _ { x } = 0$ . The determinant of the Jacobian of the first transformation is $\\bar { \\exp { ( \\epsilon m ^ { t } \\cdot S _ { x } ( \\zeta _ { 2 } ) ) } }$ , and the determinant of the Jacobian of the second transformation is $\\exp { ( \\epsilon \\bar { m } ^ { t } \\cdot S _ { x } ( \\zeta _ { 3 } ) ) }$ . ", + "bbox": [ + 173, + 103, + 823, + 160 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, we update $v$ again, based on the subset $\\zeta _ { 4 } \\triangleq ( x ^ { \\prime \\prime } , \\partial _ { x } U ( x ^ { \\prime \\prime } ) , t )$ : ", + "bbox": [ + 171, + 167, + 642, + 184 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/98763302a7fd33cd0b70c3d366c06607ba1b58d993c1471f4ae2d2bca64516c5.jpg", + "text": "$$\n\\begin{array} { r } { v ^ { \\prime \\prime } = v ^ { \\prime } \\odot \\exp ( \\frac { \\epsilon } { 2 } S _ { v } ( \\zeta _ { 4 } ) ) - \\frac { \\epsilon } { 2 } ( \\partial _ { x } U ( x ^ { \\prime \\prime } ) \\odot \\exp ( \\epsilon Q _ { v } ( \\zeta _ { 4 } ) ) + T _ { v } ( \\zeta _ { 4 } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 188, + 723, + 207 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This update has the same form as the momentum update in equation 4. ", + "bbox": [ + 173, + 210, + 635, + 224 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To give intuition into these terms, the scaling applied to the momentum can enable, among other things, acceleration in low-density zones, to facilitate mixing between modes. The scaling term applied to the gradient of the energy may allow better conditioning of the energy landscape (e.g., by learning a diagonal inertia tensor), or partial ignoring of the energy gradient for rapidly oscillating energies. ", + "bbox": [ + 174, + 231, + 825, + 301 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The corresponding integrator for $d = - 1$ is given in Appendix A; it essentially just inverts the updates in equations 4, 5 and 6. For all experiments, the functions $Q , S , T$ are implemented using multi-layer perceptrons, with shared weights. We encode the current time step in the MLP input. ", + "bbox": [ + 174, + 308, + 825, + 351 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our leapfrog operator $\\mathbf { L } _ { \\theta }$ corresponds to running $M$ steps of this modified leapfrog, $\\begin{array} { r l } { \\mathbf { L } _ { \\theta } \\boldsymbol { \\xi } } & { { } = } \\end{array}$ ${ \\bf L } _ { \\theta } ( x , v , d ) \\stackrel { } { = } ( x ^ { \\prime \\prime \\times M } , v ^ { \\prime \\prime \\times M } , d )$ , and our flip operator $\\mathbf { F }$ reverses the direction variable $d$ , $\\mathbf { F } \\xi =$ $( x , v , - d )$ . Written in terms of these modified operators, our proposal and acceptance probability are identical to those for standard HMC. Note, however, that this parameterization enables learning non-volume-preserving transformations, as the determinant of the Jacobian is a function of $S _ { x }$ and $S _ { v }$ that does not necessarily evaluate to 1. This quantity is derived in Appendix B. ", + "bbox": [ + 173, + 357, + 825, + 441 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1.3 MCMC TRANSITIONS ", + "text_level": 1, + "bbox": [ + 174, + 455, + 383, + 469 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For convenience, we denote by $\\mathbf { R }$ an operator that re-samples the momentum and direction. I.e., given $\\xi ~ = ~ ( x , v , d )$ , $\\mathbf { R } \\xi = ( x , v ^ { \\prime } , d ^ { \\prime } )$ where $v ^ { \\prime } \\sim \\mathcal { N } ( 0 , I ) , d ^ { \\prime } \\sim \\mathcal { U } \\left( \\{ - 1 , 1 \\} \\right)$ . Sampling thus consists of alternating application of the $\\mathbf { F L } _ { \\theta }$ and $\\mathbf { R }$ , in the following two steps each of which is a Markov transition that satisfies detailed balance with respect to $p$ : ", + "bbox": [ + 174, + 479, + 823, + 536 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. $\\boldsymbol { \\xi } ^ { \\prime } = \\mathbf { F } \\mathbf { L } _ { \\boldsymbol { \\theta } } \\boldsymbol { \\xi }$ with probability $A ( \\mathbf { F L } _ { \\theta } \\xi | \\xi )$ (Equation 3), otherwise $\\xi ^ { \\prime } = \\xi$ \n2. $\\boldsymbol { \\xi } ^ { \\prime } = \\mathbf { R } \\boldsymbol { \\xi }$ ", + "bbox": [ + 210, + 546, + 702, + 580 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This parameterization is effectively a generalization of standard HMC as it is non-volume preserving, with learnable parameters, and easily reduces to standard HMC for $Q , S , T = 0$ . ", + "bbox": [ + 173, + 590, + 823, + 619 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 LOSS AND TRAINING PROCEDURE ", + "text_level": 1, + "bbox": [ + 174, + 636, + 452, + 650 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We need some criterion to train the parameters $\\theta$ that control the functions $Q , S$ , and $T$ . We choose a loss designed to reduce mixing time. Specifically, we aim to minimize lag-one autocorrelation. This is equivalent to maximizing expected squared jumped distance (Pasarica & Gelman, 2010). For $\\xi , \\xi ^ { \\prime }$ in the extended state space, we define $\\delta ( \\xi ^ { \\bar { \\prime } } , \\xi ) \\ = \\ \\delta ( ( x ^ { \\prime } , v ^ { \\prime } , d ^ { \\prime } ) , ( x , v , d ) ) \\ = \\ | | x - x ^ { \\prime } | | _ { 2 } ^ { 2 }$ . Expected squared jumped distance is thus $\\mathbb { E } _ { \\xi \\sim p ( \\xi ) } \\left[ \\delta ( \\mathbf { F L } _ { \\theta } \\xi , \\xi ) A ( \\mathbf { F L } _ { \\theta } \\xi | \\xi ) \\right]$ . However, this loss need not encourage mixing across the entire state space. Indeed, maximizing this objective can lead to regions of state space where almost no mixing occurs, so long as the average squared distance traversed remains high. To optimize both for typical and worst case behavior, we include a reciprocal term in the loss, ", + "bbox": [ + 173, + 661, + 825, + 787 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/9a2e1b2e21533f54c2260ba54c6c204824d0a129dea64628d5765a3ece585936.jpg", + "text": "$$\n\\begin{array} { r } { \\ell _ { \\lambda } ( \\xi , \\xi ^ { \\prime } , A ( \\xi ^ { \\prime } | \\xi ) ) = \\frac { \\lambda ^ { 2 } } { \\delta ( \\xi , \\xi ^ { \\prime } ) A ( \\xi ^ { \\prime } | \\xi ) } - \\frac { \\delta ( \\xi , \\xi ^ { \\prime } ) A ( \\xi ^ { \\prime } | \\xi ) } { \\lambda ^ { 2 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 791, + 661, + 815 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\lambda$ is a scale parameter, capturing the characteristic length scale of the problem. The second term encourages typical moves to be large, while the first term strongly penalizes the sampler if it is ever in a state where it cannot move effectively – $\\delta ( \\xi , \\xi ^ { \\prime } )$ being small resulting in a large loss value. We train our sampler by minimizing this loss over both the target distribution and initialization distribution. Formally, given an initial distribution $\\pi _ { 0 }$ over $\\mathcal { X }$ , we define $q ( \\xi ) = \\pi _ { 0 } ( x ) \\mathcal { N } ( v ; 0 , I ) p ( d )$ , and minimize ", + "bbox": [ + 173, + 819, + 825, + 902 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/059840ffc73a985e769997c8ee82deb69ae6b9c15631887be4f5e0d6bd6ce556.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) \\triangleq \\mathbb { E } _ { p ( \\boldsymbol { \\xi } ) } \\left[ \\ell _ { \\lambda } ( \\boldsymbol { \\xi } , \\mathbf { F } \\mathbf { L } _ { \\boldsymbol { \\theta } } \\boldsymbol { \\xi } , A ( \\mathbf { F } \\mathbf { L } _ { \\boldsymbol { \\theta } } \\boldsymbol { \\xi } | \\boldsymbol { \\xi } ) ) \\right] + \\lambda _ { b } \\mathbb { E } _ { q ( \\boldsymbol { \\xi } ) } \\left[ \\ell _ { \\lambda } ( \\boldsymbol { \\xi } , \\mathbf { F } \\mathbf { L } _ { \\boldsymbol { \\theta } } \\boldsymbol { \\xi } , A ( \\mathbf { F } \\mathbf { L } _ { \\boldsymbol { \\theta } } \\boldsymbol { \\xi } | \\boldsymbol { \\xi } ) ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 906, + 759, + 926 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first term of this loss encourages mixing as it considers our operator applied on draws from the distribution; the second term rewards fast burn-in; $\\lambda _ { b }$ controls the strength of the ‘burn-in’ regularization. Given this loss, we exactly describe our training procedure in Algorithm 1. It is important to note that each training iteration can be done with only one pass through the network and can be efficiently batched. We further emphasize that this training procedure can be applied to any learnable operator whose Jacobian’s determinant is tractable, making it a general framework for training MCMC proposals. ", + "bbox": [ + 173, + 103, + 825, + 200 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1 Training L2HMC ", + "text_level": 1, + "bbox": [ + 174, + 215, + 380, + 231 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Input: Energy function $U : \\mathcal { X } \\mathbb { R }$ and its gradient $\\nabla _ { x } U : x x$ , initial distribution over \nthe augmented state space $q$ , number of iterations $n _ { \\mathrm { i t e r s } }$ , number of leapfrogs $M$ , learning rate \nschedule (αt)t≤n , batch size $N$ , scale parameter $\\lambda$ and regularization strength $\\lambda _ { b }$ . \nInitialize the parameters of the sampler $\\theta$ . \nInitialize $\\{ \\xi _ { p } ^ { ( i ) } \\} _ { i \\le N }$ from $q ( \\xi )$ . \nfor $t = 0$ to $n _ { \\mathrm { i t e r s } } - 1$ do Sample a minibatch $\\{ \\xi _ { q } ^ { ( i ) } \\} _ { i \\leq N }$ from $q ( \\xi )$ . $\\mathcal { L } 0$ for r $\\begin{array} { r l } & { \\xi _ { p } ^ { ( i ) } \\gets \\mathbf { R } \\xi _ { p } ^ { ( i ) } } \\\\ & { \\mathcal { L } \\gets \\mathcal { L } + \\ell _ { \\lambda } \\left( \\xi _ { p } ^ { ( i ) } , \\mathbf { F L } _ { \\theta } \\xi _ { p } ^ { ( i ) } , A ( \\mathbf { F L } _ { \\theta } \\xi _ { p } ^ { ( i ) } | \\xi _ { p } ^ { ( i ) } ) \\right) + \\lambda _ { b } \\ell _ { \\lambda } \\left( \\xi _ { q } ^ { ( i ) } , \\mathbf { F L } _ { \\theta } \\xi _ { q } ^ { ( i ) } , A ( \\mathbf { F L } _ { \\theta } \\xi _ { q } ^ { ( i ) } | \\xi _ { q } ^ { ( i ) } ) \\right) } \\\\ & { } \\end{array}$ $i = 1$ to $N$ do $\\begin{array} { r } { \\xi _ { p } ^ { ( i ) } \\mathbf { F L } _ { \\theta } \\xi _ { p } ^ { ( i ) } } \\end{array}$ with probability $A ( \\mathbf { F L } _ { \\theta } \\xi _ { p } ^ { ( i ) } | \\xi _ { p } ^ { ( i ) } )$ . end for θ θ α t θ \nend for ", + "bbox": [ + 178, + 234, + 823, + 467 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 493, + 326, + 510 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We present an empirical evaluation of our trained sampler on a diverse set of energy functions. We first present results on a collection of toy distributions capturing common pathologies of energy landscapes, followed by results on a task from machine learning: maximum-likelihood training of deep generative models. For each, we compare against HMC with well-tuned step length and show significant gains in mixing time. Code implementing our algorithm is available online1. ", + "bbox": [ + 173, + 525, + 825, + 594 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 VARIED COLLECTION OF ENERGY FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 611, + 537, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate our L2HMC sampler on a diverse collection of energy functions, each posing different challenges for standard HMC. ", + "bbox": [ + 174, + 637, + 821, + 665 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Ill-Conditioned Gaussian (ICG): Gaussian distribution with diagonal covariance spaced loglinearly between $1 0 ^ { - 2 }$ and $1 0 ^ { 2 }$ . This demonstrates that L2HMC can learn a diagonal inertia tensor. ", + "bbox": [ + 173, + 672, + 823, + 700 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Strongly correlated Gaussian (SCG): We rotate a diagonal Gaussian with variances $[ 1 0 ^ { 2 } , 1 0 ^ { - 2 } ]$ by $\\frac { \\pi } { 4 }$ . This is an extreme version of an example from Neal (2011). This problem shows that, although our parametric sampler only applies element-wise transformations, it can adapt to structure which is not axis-aligned. ", + "bbox": [ + 174, + 707, + 825, + 763 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Mixture of Gaussians (MoG): Mixture of two isotropic Gaussians with $\\sigma ^ { 2 } = 0 . 1$ , and centroids separated by distance 4. The means are thus about 12 standard deviations apart, making it almost impossible for HMC to mix between modes. ", + "bbox": [ + 174, + 770, + 825, + 813 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Rough Well: Similar to an example from Sohl-Dickstein et al. (2014), for a given $\\eta > 0 , U ( x ) =$ $\\begin{array} { r } { \\frac { 1 } { 2 } x ^ { T } \\overset { \\smile } { x } + \\eta \\sum _ { i } \\cos ( \\frac { x _ { i } } { \\eta } ) } \\end{array}$ . For small $\\eta$ the energy itself is altered negligibly, but its gradient is perturbed by a high frequency noise oscillating between $- 1$ and 1. In our experiments, we choose $\\eta = 1 0 ^ { - 2 }$ . ", + "bbox": [ + 174, + 819, + 825, + 866 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For each of these distributions, we compare against HMC with the same number of leapfrog steps and a well-tuned step-size. To compare mixing time, we plot auto-correlation for each method and report effective sample size (ESS). We compute those quantities in the same way as Sohl-Dickstein et al. (2014). We observe that samplers trained with L2HMC show greatly improved autocorrelation and ESS on the presented tasks, providing more than $1 0 6 \\times$ improved ESS on the SCG task. In addition, for the MoG, we show that L2HMC can easily mix between modes while standard HMC gets stuck in a mode, unable to traverse the low density zone. Experimental details, as well as a comparison with LAHMC (Sohl-Dickstein et al., 2014), are shown in Appendix C. ", + "bbox": [ + 176, + 872, + 825, + 901 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/e878bce92e8dd48bdb837822419a4ea3132437ffdaebbcd93f07f493cdac4546.jpg", + "image_caption": [ + "Figure 1: L2HMC mixes faster than well-tuned HMC, and than A-NICE-MC, on a collection of toy distributions. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 830, + 491 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 556, + 825, + 641 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison to A-NICE-MC (Song et al., 2017) In addition to the well known challenges associated with adversarial training (Arjovsky et al., 2017), we note that parameterization using a volume-preserving operator can dramatically fail on simple energy landscapes. We build off of the mog2 experiment presented in (Song et al., 2017), which is a 2-d mixture of isotropic Gaussians separated by a distance of 10 with variances 0.5. We consider that setup but increase the ratio of variances: $\\dot { \\sigma } _ { 1 } ^ { 2 } = 3 , \\sigma _ { 2 } ^ { 2 } = 0 . 0 5$ . We show in Figure 1d sample chains trained with L2HMC and A-NICE-MC; A-NICE-MC cannot effectively mix between the two modes as only a fraction of the volume of the large mode can be mapped to the small one, making it highly improbable to traverse. This is also an issue for HMC. On the other hand, L2HMC can both traverse the low-density region between modes, and map a larger volume in the left mode to a smaller volume in the right mode. It is important to note that the distance between both clusters is less than in the mog2 case, and it is thus a good diagnostic of the shortcomings of volume-preserving transformations. ", + "bbox": [ + 173, + 656, + 825, + 821 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 LATENT-VARIABLE GENERATIVE MODEL ", + "text_level": 1, + "bbox": [ + 176, + 840, + 500, + 854 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We apply our learned sampler to the task of training, and sampling from the posterior of, a latentvariable generative model. The model consists of a latent variable $z \\sim p ( z )$ , where we choose $p ( z ) = \\breve { \\mathscr { N } } ( z ; 0 , I )$ , and a conditional distribution $p ( x | z )$ which generates the image $x$ . Given a family of parametric ‘decoders’ $\\{ z \\mapsto p ( x | z ; \\phi ) , \\phi \\in \\Phi \\}$ , and a set of samples $\\mathcal { D } = \\{ x ^ { ( i ) } \\} _ { i \\leq N }$ training involves finding $\\begin{array} { r } { \\phi ^ { * } = \\arg \\operatorname* { m a x } _ { \\phi \\in \\Phi } p ( \\mathcal { D } ; \\phi ) } \\end{array}$ . However, the log-likelihood is intractable as $\\begin{array} { r } { p ( x ; \\boldsymbol { \\phi } ) = \\int p ( x | \\boldsymbol { z } ; \\boldsymbol { \\phi } ) p ( \\boldsymbol { z } ) \\mathrm { d } \\boldsymbol { z } } \\end{array}$ . To remedy that problem, Kingma $\\&$ Welling (2013) proposed jointly training an approximate posterior $q _ { \\psi }$ that maximizes a tractable lower-bound on the log-likelihood: ", + "bbox": [ + 174, + 866, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/250475de256c4eebae15fc4fa637a5547873059d01bec521d1802c331ccf79c1.jpg", + "image_caption": [ + "Figure 2: Training and held-out log-likelihood for models trained with L2HMC, HMC, and the ELBO (VAE). " + ], + "image_footnote": [], + "bbox": [ + 225, + 103, + 769, + 207 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 270, + 825, + 313 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/b24c790c60685bd813e80b83358ec476c2c47c7a4eb3408681732541e70e0073.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { E L B O } } ( x , \\phi , \\psi ) = \\mathbb { E } _ { q _ { \\psi } ( z | x ) } \\left[ p ( x | z ; \\phi ) \\right] - \\mathrm { K L } ( q _ { \\psi } ( z | x ) | | p ( z ) ) \\leq p ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 269, + 328, + 727, + 347 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $q _ { \\psi } ( z | x )$ is a tractable conditional distribution with parameters $\\psi$ , typically parameterized by a neural network. Recently, to improve upon well-known pitfalls like over-pruning (Burda et al., 2015) of the VAE, Hoffman (2017) proposed HMC-DLGM. For a data sample $x ^ { ( i ) }$ , after obtaining a sample from the approximate posterior $q _ { \\psi } ( \\cdot | x ^ { ( i ) } )$ , Hoffman (2017) runs a MCMC algorithm with energy function $U ( z , x ^ { ( i ) } ) = - \\log p ( z ) - \\log p ( x ^ { ( i ) } | z ; \\phi )$ to obtain a more exact posterior sample from $p ( \\boldsymbol { z } | \\boldsymbol { x } ^ { ( i ) } ; \\boldsymbol { \\phi } )$ . Given that better posterior sample $z ^ { \\prime }$ , the algorithm maximizes $\\log p ( x ^ { ( i ) } | z ^ { \\prime } ; \\phi )$ . ", + "bbox": [ + 173, + 361, + 825, + 452 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To show the benefits of L2HMC, we borrow the method from Hoffman (2017), but replace HMC by jointly training an L2HMC sampler to improve the efficiency of the posterior sampling. We call this model L2HMC-DLGM. A diagram of our model and a formal description of our training procedure are presented in Appendix D. We define, for $\\xi = \\{ z , v , d \\} , r ( \\xi | x ; \\psi ) \\triangleq$ $q _ { \\psi } ( z | x ) \\mathcal { N } ( v ; \\bar { 0 , I } ) \\mathcal { U } ( d ; \\{ - 1 , \\bar { 1 } \\} )$ . ", + "bbox": [ + 174, + 458, + 825, + 531 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In the subsequent sections, we compare our method to the standard VAE model from Kingma & Welling (2013) and HMC-DGLM from Hoffman (2017). It is important to note that, since our sampler is trained jointly with $p _ { \\phi }$ and $q _ { \\psi }$ , it performs exactly the same number of gradient computations of the energy function as HMC. We first show that training a latent variable generative model with L2HMC results in better generative models both qualitatively and quantitatively. We then show that our improved sampler enables a more expressive, non-Gaussian, posterior. ", + "bbox": [ + 173, + 536, + 825, + 621 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Implementation details: Our decoder $( p _ { \\phi } )$ is a neural network with 2 fully connected layers, with 1024 units each and softplus non-linearities, and outputs Bernoulli activation probabilities for each pixel. The encoder $( q _ { \\psi } )$ has the same architecture, returning mean and variance for the approximate posterior. Our model was trained for 300 epochs with Adam (Kingma & Ba, 2014) and a learning rate $\\alpha = 1 0 ^ { - 3 }$ . All experiments were done on the dynamically binarized MNIST dataset (LeCun). ", + "bbox": [ + 173, + 627, + 825, + 696 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2.1 SAMPLE QUALITY AND DATA LIKELIHOOD ", + "text_level": 1, + "bbox": [ + 174, + 722, + 526, + 736 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We first present samples from decoders trained with L2HMC, HMC and the ELBO (i.e. vanilla VAE). Although higher log likelihood does not necessarily correspond to better samples (Theis et al., 2015), we can see in Figure 5, shown in the Appendix, that the decoder trained with L2HMC generates sharper samples than the compared methods. ", + "bbox": [ + 174, + 750, + 825, + 805 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now compare our method to HMC in terms of log-likelihood of the data. As we previously stated, the marginal likelihood of a data point $x \\in \\mathcal { X }$ is not tractable as it requires integrating $p ( x , z )$ over a high-dimensional space. However, we can estimate it using annealed importance sampling (AIS; Neal (2001)). Following Wu et al. (2016), we evaluate our generative models on both training and held-out data. In Figure 2, we plot the data’s log-likelihood against the number of gradient computation steps for both HMC-DGLM and L2HMC-DGLM. We can see that for a similar number of gradient computations, L2HMC-DGLM achieves higher likelihood for both training and held-out data. This is a strong indication that L2HMC provides significantly better posterior samples. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/fa42f98b9e8fda6e4c48a55741ecf4bec0e430aa2379cfc10ac8d84350687b92.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 204, + 103, + 637, + 166 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Block Gibbs inpainting of the top half of an MNIST digit, using (top) L2HMC as a posterior sampler, and (bottom) $q _ { \\psi }$ as a posterior sampler. ", + "bbox": [ + 187, + 176, + 643, + 203 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/3b5a148d0cd5e54e8a2a908b2b63c8ebc00143ad9d41f371e71810f20e32da5a.jpg", + "image_caption": [ + "(b) Non-Gaussian posterior ", + "Figure 3: Demonstrations of the value of a more expressive posterior approximation. " + ], + "image_footnote": [], + "bbox": [ + 647, + 104, + 805, + 185 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.2.2 INCREASED EXPRESSIVITY OF THE POSTERIOR ", + "text_level": 1, + "bbox": [ + 174, + 262, + 553, + 276 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In the standard VAE framework, approximate posteriors are often parametrized by a Gaussian, thus making a strong assumption of uni-modality. In this section, we show that using L2HMC to sample from the posterior enables learning of a richer posterior landscape. ", + "bbox": [ + 174, + 292, + 825, + 335 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Block Gibbs Sampling To highlight our ability to capture more expressive posteriors, we in-paint the top of an image using Block Gibbs Sampling using the approximate posterior or L2HMC. Formally, let $x _ { 0 }$ be the starting image. We denote top or bottom-half pixels as $x _ { 0 } ^ { \\mathrm { t o p } }$ and $x _ { 0 } ^ { \\mathrm { b o t t o m } }$ . At each step $t$ , we sample $z ^ { ( t ) } \\sim p ( z | x _ { t } ; \\theta )$ , sample $\\tilde { x } \\sim p ( x | z _ { t } ; \\theta )$ . We then set $x _ { t + 1 } ^ { \\mathrm { t o p } } = \\tilde { x } ^ { \\mathrm { t o p } }$ and $x _ { t + 1 } ^ { \\mathrm { b o t t o m } } = x _ { 0 } ^ { \\mathrm { b o t t o m } }$ . We compare the results obtained by sampling from ior) vs. our trained sampler. The results are reported i $p ( z | x ; \\theta )$ using a. We $q _ { \\psi }$ (i.e. the see that L2HMC easily mixes between modes (3, 5, 8, and plausibly 9 in the figure) while the approximate posterior gets stuck on the same reconstructed digit (3 in the figure). ", + "bbox": [ + 173, + 342, + 825, + 460 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Visualization of the posterior After training a decoder with L2HMC, we randomly choose an element $x _ { 0 } \\in \\mathcal { D }$ and run 512 parallel L2HMC chains for 20, 000 Metropolis-Hastings steps. We then find the direction of highest variance, project the samples along that direction and show a histogram in Figure 3b. This plot shows non-Gaussianity in the latent space for the posterior. Using our improved sampler enables the decoder to make use of a more expressive posterior, and enables the encoder to sample from this non-Gaussian posterior. ", + "bbox": [ + 174, + 467, + 825, + 550 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 FUTURE WORK ", + "text_level": 1, + "bbox": [ + 176, + 588, + 334, + 604 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The loss in Section 4.2 targets lag-one autocorrelation. It should be possible to extend this to also target lag-two and higher autocorrelations. It should also be possible to extend this loss to reward fast decay in the autocorrelation of other statistics of the samples, for instance the sample energy as well as the sample position. These additional statistics could also include learned statistics of the samples, combining benefits of the adversarial approach of (Song et al., 2017) with the current work. ", + "bbox": [ + 174, + 631, + 825, + 714 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our learned generalization of HMC should prove complementary to several other research directions related to HMC. It would be interesting to explore combining our work with the use of HMC in a minibatch setting (Chen et al., 2014); with shadow Hamiltonians (Izaguirre & Hampton, 2004); with gradient pre-conditioning approaches similar to those used in Riemannian HMC (Girolami et al., 2009; Betancourt, 2013); with the use of alternative HMC accept-reject rules (Sohl-Dickstein et al., 2014; Berger et al., 2015); with the use of non-canonical Hamiltonian dynamics (Tripuraneni et al., 2016); with variants of AIS adapted to HMC proposals (Sohl-Dickstein & Culpepper, 2012); with the extension of HMC to discrete state spaces (Zhang et al., 2012); and with the use of alternative Hamiltonian integrators (Creutz & Gocksch, 1989; Chao et al., 2015). ", + "bbox": [ + 173, + 722, + 825, + 847 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, our work is also complementary to other methods not utilizing gradient information. For example, we could incorporate the intuition behind Multiple Try Metropolis schemes (Martino & Read, 2013) by having several parametric operators and training each one when used. In addition, one could draw inspiration from the adaptive literature (Haario et al., 2001; Andrieu & Thoms, 2008) or component-wise strategies (Gilks & Wild, 1992). ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 318, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this work, we presented a general method to train expressive MCMC kernels parameterized with deep neural networks. Given a target distribution $p$ , analytically known up to a constant, our method provides a fast-mixing sampler, able to efficiently explore the state space. Our hope is that our method can be utilized in a “black-box” manner, in domains where sampling constitutes a huge bottleneck such as protein foldings (Schutte et al., 1999) or physics simulations (Olsson, 1995). ¨ ", + "bbox": [ + 174, + 133, + 825, + 203 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 220, + 326, + 234 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Ben Poole, Aditya Grover, David Belanger, and Colin Raffel for insightful comments on the draft, Mohammad Norouzi for support and encouragement launching the project, and Jiaming Song for discussions and help running A-NICE-MC. ", + "bbox": [ + 176, + 246, + 825, + 287 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 308, + 287, + 323 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Christophe Andrieu and Johannes Thoms. 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", + "bbox": [ + 169, + 103, + 828, + 500 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Appendix ", + "text_level": 1, + "bbox": [ + 176, + 99, + 316, + 127 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A REVERSE LEAPFROG OPERATOR", + "text_level": 1, + "bbox": [ + 176, + 148, + 475, + 166 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let $\\xi = \\{ x , v , d \\}$ in the extended state space with $d = - 1$ . Here, we describe the leapfrog updates for a single time step $t$ , this consists of inverting the equations presented in the corresponding section. ", + "bbox": [ + 171, + 180, + 825, + 209 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let $\\zeta _ { 1 } = \\{ x , v , t \\}$ , we have: ", + "bbox": [ + 173, + 214, + 359, + 231 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/ceb2fac295431fe2f99dc3790f1822ea6a7f6b968b6600ea6ead43d231a45ec9.jpg", + "text": "$$\nv ^ { \\prime } = \\left[ v + \\frac { \\epsilon } { 2 } \\left( \\partial _ { x } U ( x ) \\odot \\exp ( \\epsilon Q _ { v } ( \\zeta _ { 1 } ) ) + T _ { v } ( \\zeta _ { 1 } ) \\right) \\right] \\odot \\exp ( - S _ { v } ( \\zeta _ { 1 } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 232, + 727, + 260 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "With the notation from Section 4, let $\\zeta _ { 2 } \\triangleq \\{ x _ { m ^ { t } } , v , t \\}$ ", + "bbox": [ + 173, + 270, + 531, + 286 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/80e54b64007cfd4111de288dbf2b8a2f6a4388b9621004458daca402cc1b7a39.jpg", + "text": "$$\nx ^ { \\prime } = x _ { m ^ { t } } + \\bar { m } ^ { t } \\odot [ ( x - \\epsilon ( \\exp ( \\epsilon Q _ { x } ( \\zeta _ { 2 } ) ) \\odot v ^ { \\prime } + T _ { x } ( \\zeta _ { 2 } ) ) ] \\odot \\exp ( - \\epsilon S _ { v } ( \\zeta _ { 2 } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 287, + 753, + 306 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let us denote $\\zeta _ { 3 } \\triangleq \\{ x _ { \\bar { m } ^ { t } } ^ { \\prime } , v , t \\}$ : ", + "bbox": [ + 174, + 315, + 382, + 332 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/552798bf6c3fc0a56ead396690fcb76efc48a173e27d4055b523a55ea69acdbc.jpg", + "text": "$$\nx ^ { \\prime \\prime } = x _ { \\bar { m } ^ { t } } + m ^ { t } \\odot [ ( x ^ { \\prime } - \\epsilon ( \\exp ( \\epsilon Q _ { x } ( \\zeta _ { 3 } ) ) \\odot v ^ { \\prime } + T _ { x } ( \\zeta _ { 3 } ) ) ] \\odot \\exp ( - \\epsilon S _ { v } ( \\zeta _ { 3 } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 241, + 335, + 756, + 353 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Finally, the last update, with $\\zeta _ { 4 } \\triangleq \\{ x ^ { \\prime \\prime } , \\partial _ { x } U ( x ^ { \\prime \\prime } ) , t \\}$ : ", + "bbox": [ + 174, + 363, + 521, + 381 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/a85ada7ad597b1e2832f12ac22eb8d8920d00ad59045c014d475c6fc5c0e3088.jpg", + "text": "$$\nv ^ { \\prime } = \\left[ v + { \\frac { \\epsilon } { 2 } } \\left( \\partial _ { x } U ( x ^ { \\prime \\prime } ) \\odot \\exp ( \\epsilon Q _ { v } ( \\zeta _ { 4 } ) ) + T _ { v } ( \\zeta _ { 4 } ) \\right) \\right] \\odot \\exp ( - S _ { v } ( \\zeta _ { 4 } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 395, + 732, + 422 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "It is important to note that to invert $\\mathbf { L } _ { \\theta }$ , these steps should be ran for $t$ from $M$ to 1. ", + "bbox": [ + 173, + 429, + 720, + 445 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B DETERMINANT OF THE JACOBIAN ", + "text_level": 1, + "bbox": [ + 173, + 463, + 491, + 481 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Given the derivations (and notations) from Section 4, for the forward operator $\\mathbf { L } _ { \\theta }$ , we can immediately compute the Jacobian: ", + "bbox": [ + 174, + 493, + 823, + 523 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/a484e15c6bb13044942a9af90415d093a5f62b33ce8fb2f4386f3843d29bcf14.jpg", + "text": "$$\n\\log \\left| \\frac { \\partial [ \\mathbf { F } \\mathbf { L } _ { \\theta } \\boldsymbol { \\xi } ] } { \\partial \\boldsymbol { \\xi } ^ { T } } \\right| = d \\sum _ { t \\leq M } \\left[ \\frac { \\epsilon } { 2 } \\mathbf { 1 } \\cdot S _ { v } ( \\boldsymbol { \\zeta } _ { 1 } ^ { t } ) + \\epsilon m ^ { t } \\cdot S _ { x } ( \\boldsymbol { \\zeta } _ { 2 } ^ { t } ) + \\epsilon \\bar { m } ^ { t } \\cdot S _ { x } ( \\boldsymbol { \\zeta } _ { 3 } ^ { t } ) + \\frac { \\epsilon } { 2 } \\mathbf { 1 } \\cdot S _ { v } ( \\boldsymbol { \\zeta } _ { 4 } ^ { t } ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 199, + 542, + 769, + 583 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Where $\\zeta _ { i } ^ { t }$ denotes the intermediary variable $\\zeta _ { i }$ at time step $t$ and $d$ is the direction of $\\xi$ i.e. $\\xi =$ \n$\\{ x , v , d \\}$ . ", + "bbox": [ + 173, + 593, + 825, + 623 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C EXPERIMENTAL DETAILS OF SECTION 5 ", + "text_level": 1, + "bbox": [ + 173, + 642, + 540, + 659 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C.1 IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 672, + 406, + 688 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "First of all, we keep separate parameters for the network responsible for updating $v$ and those updating $x$ . The architectures are the same. Let us take the example of $Q _ { v } , S _ { v } , T _ { v }$ . The time step $t$ is given as input to the MLP, encoded as $\\begin{array} { r } { \\tau ( t ) = ( \\cos ( \\frac { 2 \\pi t } { M } ) , \\sin ( \\frac { 2 \\pi { \\hat { t } } } { M } ) ) } \\end{array}$ . $\\sigma ( \\cdot )$ denotes the ReLU non-linearity. ", + "bbox": [ + 173, + 698, + 826, + 743 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For $n _ { h }$ hidden units per layer: ", + "bbox": [ + 173, + 747, + 370, + 762 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• We first compute $h _ { 1 } = \\sigma ( W _ { 1 } x + W _ { 2 } v + W _ { 3 } \\tau ( t ) + b ) ( h \\in \\mathbb { R } ^ { n _ { h } } ) .$ · $h _ { 2 } = \\sigma ( W _ { 4 } h + b _ { 4 } ) \\in \\mathbb { R } ^ { n _ { h } }$ • $S _ { v } = \\lambda _ { s } \\mathtt { t a n h } ( W _ { s } h _ { 2 } + b _ { s } ) , Q _ { v } = \\lambda _ { q } \\mathtt { t a n h } ( W _ { q } h _ { 2 } + b _ { q } ) , T _ { v } = W _ { t } h _ { 2 } + b _ { t } .$ ", + "bbox": [ + 215, + 771, + 718, + 825 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Section 5.1, the $Q , S , T$ are neural networks with 2 hidden layers with 10 (100 for the 50-d ICG) units and ReLU non-linearities. We train with Adam (Kingma & Ba, 2014) and a learning rate $\\alpha = 1 0 ^ { - 3 }$ . We train for 5, 000 iterations with a batch size of 200. ", + "bbox": [ + 176, + 832, + 820, + 875 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$\\lambda _ { b }$ was set to 0 for ICG and SCG and to 1 for MoG and Rough Well. For the MoG tasks, we train our sampler with a temperature parameter that we continuously anneal; we evaluate the trained sampler without using temperature. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/ce0d3b31ad5969a1d38e2d717519f244a5cf9175ca9b22c578fb2f7627737684.jpg", + "image_caption": [ + "Figure 4: Diagram of our L2HMC-DGLM model. Nodes are functions of their parents. Round nodes are deterministic, diamond nodes are stochastic and the doubly-circled node is observed. " + ], + "image_footnote": [], + "bbox": [ + 210, + 127, + 812, + 189 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.2 AUTO-CORRELATION AND ESS ", + "text_level": 1, + "bbox": [ + 176, + 272, + 434, + 287 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Let $( x _ { \\tau } ) _ { \\tau \\leq T }$ be a set of correlated samples converging to the distribution $p$ with mean $\\mu$ and covariance $\\Sigma$ . We define auto-correlation at time $t$ as: ", + "bbox": [ + 174, + 299, + 823, + 327 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a7ac378a10f2f817f53cb50c754be076297c6173257df1c1149f726855b7df10.jpg", + "text": "$$\n\\rho _ { t } \\triangleq \\frac { 1 } { \\mathrm { T r a c e } ( \\Sigma ) ( T - t ) } \\sum _ { \\tau \\leq T - t - 1 } ( x _ { \\tau } - \\mu ) ^ { T } ( x _ { \\tau + t } - \\mu ) .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 330, + 683, + 368 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We can now define effective sample size (ESS) as: ", + "bbox": [ + 174, + 373, + 504, + 388 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/4d3801f22de10cc50e839d170483171d7f9cd0d0b8536137c71e9799d07059d5.jpg", + "text": "$$\n\\mathrm { E S S } \\left( ( x _ { \\tau } ) _ { \\tau \\leq T } \\right) \\triangleq \\frac { 1 } { 1 + 2 \\sum _ { t } \\rho _ { t } } .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 393, + 606, + 428 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Similar to Hoffman & Gelman (2014), we truncate the sum when the auto-correlation goes below 0.05. ", + "bbox": [ + 176, + 433, + 823, + 460 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.3 COMPARISON WITH LAHMC ", + "text_level": 1, + "bbox": [ + 176, + 477, + 419, + 492 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We compare our trained sampler with LAHMC (Sohl-Dickstein et al., 2014). Results are reported in Table 1. L2HMC largely outperforms LAHMC on all task. LAHMC is also unable to mix between modes for the MoG task. We also note that L2HMC could be easily combined with LAHMC, by replacing the leapfrog integrator of LAHMC with the learned one of L2HMC. ", + "bbox": [ + 174, + 502, + 825, + 559 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/aea1321e1674a8648c0177823d680f4d377937660a6bf2bc646ca7024a3cd04c.jpg", + "table_caption": [ + "Table 1: ESS for a fixed number of gradient evaluations. " + ], + "table_footnote": [], + "table_body": "
DistributionGradient EvaluationsESS-L2HMCESS-LAHMCRatio
50-d ICG2000156.621.47.3
Rough Well20012.58.61.5
2-d SCG500011616.714.9
MoG20,00065.0<0.53> 123.5
", + "bbox": [ + 218, + 571, + 774, + 659 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D L2HMC-DGLM ", + "text_level": 1, + "bbox": [ + 174, + 738, + 351, + 756 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D.1 TRAINING ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 770, + 379, + 785 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we present our training algorithm as well as a diagram explaining L2HMC-DGLM. For conciseness, given our operator $\\mathbf { L } _ { \\theta }$ , we denote by $\\mathbf { K } _ { \\theta } ( \\cdot | x )$ the distribution over next state given sampling of a momentum and direction and the Metropolis-Hastings step. ", + "bbox": [ + 174, + 796, + 825, + 839 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D.2 IMPLEMENTATION DETAILS OF L2HMC-DGLM ", + "text_level": 1, + "bbox": [ + 173, + 854, + 550, + 871 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Similar to our L2HMC training on unconditional sampling, we share weights across $Q , S$ and $T$ . In addition, the auxiliary variable $x$ (here the image from MNIST) is first passed through a 2-layer neural network, with softplus non-linearities and 512 hidden units. This input is given to both ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Algorithm 2 L2HMC for latent variable generative models ", + "text_level": 1, + "bbox": [ + 173, + 103, + 562, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Input: dataset $\\mathcal { D }$ , number of iterations $n _ { \\mathrm { i t e r s } }$ , number of Metropolis-Hastings step $J$ , number of \nleapfrogs M , and learning rate schedule (αt)t≤niters . \nRandomly initialize the decoder’s parameters $\\phi$ and the approximate posterior $\\psi$ . Initialize the \nparameters of the sampler $\\theta$ with $M$ leapfrog steps. \nfor $t = 0$ to $n _ { \\mathrm { i t e r s } } - 1$ do Randomly sample a minibatch $\\boldsymbol { B }$ from the dataset $\\mathcal { D }$ . $\\mathcal { L } _ { \\mathrm { E L B O } } , \\mathcal { L } _ { \\mathrm { S a m p l e r } } , \\mathcal { L } _ { \\mathrm { D e c o d e r } } 0$ for $\\boldsymbol { x } ^ { ( b ) } \\in B \\mathrm { d } \\mathbf { 0 }$ ∈Sample $\\xi _ { 0 } ^ { ( b ) } \\sim r ( \\cdot | x ^ { ( b ) } ; \\psi )$ . $\\mathcal { L } _ { \\mathrm { E L B O } } p ( x ^ { ( b ) } | z _ { 0 } ^ { ( b ) } ; \\phi ) - \\mathrm { K L } ( q _ { \\psi } ( z | x ^ { ( b ) } ) | | p ( z ) )$ . With ξ(b)0 = {z(b)0 , v(b)0 , d(b)0 } Define the energy function $U _ { x ^ { ( b ) } } ( z ) = - \\log p ( x ^ { ( b ) } | z ; \\theta ) - \\log p ( z )$ $\\mathcal { L } _ { \\mathrm { S a m p l e r } } 0$ $\\lambda \\sqrt { \\mathrm { V a r } ( q _ { \\psi } ( z _ { 0 } ^ { ( b ) } | x ^ { ( b ) } ) }$ for $j = 0$ to $J - 1$ do ξ (j −b) ← Rξ(b)j LSampler ← LSampler + \\`λ(ξ(b)j , FLθξ(b)j , A(FLθξ(b)j |ξ(b)j )) Set $\\xi _ { j + 1 } ^ { ( b ) }$ to $\\mathbf { F L } _ { \\theta } \\boldsymbol { \\xi } _ { j } ^ { ( b ) }$ with probability $A ( \\mathbf { F L } _ { \\theta } \\xi _ { j } ^ { ( b ) } | \\xi _ { j } ^ { ( b ) } )$ . $\\begin{array} { r } { \\begin{array} { l l } { \\frac { \\operatorname { c a s s a } ^ { \\ast } \\operatorname { s o r } } { \\mathcal { L } _ { \\mathrm { D e c o d e r } } } \\mathcal { L } _ { \\mathrm { D e c o d e r } } + \\log p ( x ^ { ( b ) } | z _ { J } ^ { ( s ) } ; \\phi ) } & { \\qquad \\mathrm { ~ s ~ W i t h ~ } \\xi _ { J } ^ { ( b ) } = \\{ z _ { J } ^ { ( b ) } , v _ { J } ^ { ( b ) } , d _ { J } ^ { ( b ) } \\} } \\end{array} } \\end{array}$ end for $\\begin{array} { l } { \\phi \\phi + \\alpha _ { t } \\nabla _ { \\phi } \\mathcal { L } _ { \\mathrm { D e c o d e r } } } \\\\ { \\psi \\psi + \\alpha _ { t } \\nabla _ { \\psi } \\mathcal { L } _ { \\mathrm { E L B O } } } \\\\ { \\theta \\theta + \\alpha _ { t } \\nabla _ { \\theta } \\mathcal { L } _ { \\mathrm { S a m p l e r } } } \\end{array}$ \nend for \n$\\begin{array} { r l } { 3 } & { 0 } & { 1 \\leq 9 \\leq 9 \\leq 2 \\leq 9 \\leq 5 } \\\\ { 2 } & { 4 \\times 1 \\leq 2 \\leq 3 \\leq 6 } \\\\ { 6 } & { 5 \\leq 2 \\leq 3 \\leq 5 } \\\\ { 3 } & { 6 \\leq 8 \\leq 7 \\leq 3 \\leq 4 } \\\\ { 4 \\leq 9 \\leq 4 \\leq 9 \\leq 4 } \\\\ { 7 \\geq 5 \\leq 5 \\leq 5 \\leq 2 \\leq 4 \\leq 6 } \\\\ { 0 \\leq 6 \\leq 1 \\leq 2 \\leq 2 \\leq 4 \\leq 6 } \\\\ { 3 \\geq 6 \\leq 8 \\leq 5 \\leq 4 } \\end{array}$ 644又9 7 1 9 9 3 / 2 3 S 6 a a 8 7 2 4 一 6 8 9 9 4 D 4 B 7 5 3 6 9 7 7 9 2 4 6 7 0 8 7 ？6 4S q 1 1 23 3 7 9 1 972 076997 090q499 65『6630295147309 h0？b414593&84076 (a) L2HMC (b) HMC (c) VAE ", + "bbox": [ + 176, + 113, + 826, + 696 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "networks $\\{ \\cdot \\} _ { x }$ and $\\{ \\cdot \\} _ { v }$ . The architecture then consists of 2 hidden layers of 200 units and ReLU non-linearities. For $\\lambda$ (scale parameter of the loss), we use the standard deviation of the approximate posterior. ", + "bbox": [ + 174, + 742, + 825, + 785 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "AIS Evaluation For each data point, we run 20 Markov Chains in parallel, 10, 000 annealing steps with 10 leapfrogs per step and choose the step size for an acceptance rate of 0.65. ", + "bbox": [ + 174, + 800, + 823, + 829 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D.3 MNIST SAMPLES ", + "text_level": 1, + "bbox": [ + 174, + 845, + 343, + 861 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We show in Figure 5 samples from the three evaluated models: VAE (Kingma & Welling, 2013), HMC-DGLM (Hoffman, 2017) and L2HMC-DGLM. 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For instance, they arise in protein folding (Schutte et al., 1999), physics ¨", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "simulations (Olsson, 1995), and machine learning (Andrieu et al., 2003). Sampling from such dis-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "score": 1.0, + "content": "tributions is a critical task for learning and inference (MacKay, 2003), however it is an extremely", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 425, + 207, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 207, + 439 + ], + "score": 1.0, + "content": "hard problem in general.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 380, + 505, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 504, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "Markov Chain Monte Carlo (MCMC) methods promise a solution to this problem. They operate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "by generating a sequence of correlated samples that converge in distribution to the target. This", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 464, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 476 + ], + "score": 1.0, + "content": "convergence is most often guaranteed through detailed balance, a sufficient condition for the chain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "to have the target equilibrium distribution. In practice, for any proposal distribution, one can ensure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 441, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 441, + 501 + ], + "score": 1.0, + "content": "detailed balance through a Metropolis-Hastings (Hastings, 1970) accept/reject step.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 441, + 506, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "Despite theoretical guarantees of eventual convergence, in practice convergence and mixing speed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "depend strongly on choosing a proposal that works well for the task at hand. What’s more, it is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "often more art than science to know when an MCMC chain has converged (“burned-in”), and when", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "the chain has produced a new uncorrelated sample (“mixed”). Additionally, the reliance on detailed", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "balance, which assigns equal probability to the forward and reverse transitions, often encourages", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 459, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 459, + 570 + ], + "score": 1.0, + "content": "random-walk behavior and thus slows exploration of the space (Ichiki & Ohzeki, 2013).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 502, + 506, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "For densities over continuous spaces, Hamiltonian Monte Carlo (HMC; Duane et al., 1987; Neal,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 585, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 599 + ], + "score": 1.0, + "content": "2011) introduces independent, auxiliary momentum variables, and computes a new state by inte-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "score": 1.0, + "content": "grating Hamiltonian dynamics. This method can traverse long distances in state space with a single", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "Metropolis-Hastings test. This is the state-of-the-art method for sampling in many domains. How-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 631 + ], + "score": 1.0, + "content": "ever, HMC can perform poorly in a number of settings. While HMC mixes quickly spatially, it", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "struggles at mixing across energy levels due to its volume-preserving dynamics. HMC also does", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "not work well with multi-modal distributions, as the probability of sampling a large enough mo-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "mentum to traverse a very low-density region is negligibly small. Furthermore, HMC struggles with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 661, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 676 + ], + "score": 1.0, + "content": "ill-conditioned energy landscapes (Girolami & Calderhead, 2011) and deals poorly with rapidly", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 674, + 303, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 303, + 686 + ], + "score": 1.0, + "content": "changing gradients (Sohl-Dickstein et al., 2014).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 575, + 506, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 690, + 503, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "Recently, probabilistic models parameterized by deep neural networks have achieved great success", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 702, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 714 + ], + "score": 1.0, + "content": "at approximately sampling from highly complex, multi-modal empirical distributions (Kingma &", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 106, + 690, + 505, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "Welling, 2013; Rezende et al., 2014; Goodfellow et al., 2014; Bengio et al., 2014; Sohl-Dickstein", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "et al., 2015). Building on these successes, we present a method that, given an analytically described", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "distribution, automatically returns an exact sampler with good convergence and mixing properties,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "from a class of highly expressive parametric models. The proposed family of samplers is a gen-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "eralization of HMC; it transforms the HMC trajectory using parametric functions (deep networks", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "in our experiments), while retaining theoretical guarantees with a tractable Metropolis-Hastings", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "accept/reject step. The sampler is trained to minimize a variation on expected squared jumped dis-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "tance (similar in spirit to Pasarica & Gelman (2010)). Our parameterization reduces easily to stan-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "dard HMC. It is further capable of emulating several common extensions of HMC such as within-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 423, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 423, + 194 + ], + "score": 1.0, + "content": "trajectory tempering (Neal, 1996) and diagonal mass matrices (Bennett, 1975).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 504, + 220 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "We evaluate our method on distributions where HMC usually struggles, as well as on a the real-world", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 306, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 306, + 222 + ], + "score": 1.0, + "content": "task of training latent-variable generative models.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 239, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 240, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 240, + 239 + ], + "score": 1.0, + "content": "Our contributions are as follows:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 127, + 251, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "• We introduce a generic training procedure which takes as input a distribution defined by an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 262, + 375, + 275 + ], + "spans": [ + { + "bbox": [ + 142, + 262, + 375, + 275 + ], + "score": 1.0, + "content": "energy function, and returns a fast-mixing MCMC kernel.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 131, + 283, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 131, + 283, + 505, + 299 + ], + "score": 1.0, + "content": "• We show significant empirical gains on various distributions where HMC performs poorly.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 132, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 132, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "• We finally evaluate our method on the real-world task of training and sampling from a latent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 142, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "variable generative model, where we show improvement in the model’s log-likelihood, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 330, + 379, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 379, + 343 + ], + "score": 1.0, + "content": "greater complexity in the distribution of posterior samples.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 365, + 211, + 378 + ], + "lines": [ + { + "bbox": [ + 104, + 364, + 213, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 213, + 381 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "Adaptively modifying proposal distributions to improve convergence and mixing has been explored", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "in the past (Andrieu & Thoms, 2008). In the case of HMC, prior work has reduced the need to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "score": 1.0, + "content": "choose step size (Neal, 2011) or number of leapfrog steps (Hoffman & Gelman, 2014) by adaptively", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "tuning those parameters. Salimans et al. (2015) proposed an alternate scheme based on variational", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "inference. We adopt the much simpler approach of Pasarica & Gelman (2010), who show that choos-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "ing the hyperparameters of a proposal distribution to maximize expected squared jumped distance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 279, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 279, + 474 + ], + "score": 1.0, + "content": "is both principled and effective in practice.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "Previous work has also explored applying models from machine learning to MCMC tasks. Kernel", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "methods have been used both for learning a proposal distribution (Sejdinovic et al., 2014) and for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "approximating the gradient of the energy (Strathmann et al., 2015). In physics, Restricted and semi-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "Restricted Boltzmann machines have been used both to build approximations of the energy function", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "which allow more rapid sampling (Liu et al., 2017; Huang & Wang, 2017), and to motivate new", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 533, + 267, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 267, + 546 + ], + "score": 1.0, + "content": "hand-designed proposals (Wang, 2017).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "Most similar to our approach is recent work from Song et al. (2017), which uses adversarial training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "of a volume-preserving transformation, which is subsequently used as an MCMC proposal distribu-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "tion. While promising, this technique has several limitations. It does not use gradient information,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "which is often crucial to maintaining high acceptance rates, especially in high dimensions. It also", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "can only indirectly measure the quality of the generated sample using adversarial training, which", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "is notoriously unstable, suffers from “mode collapse” (where only a portion of a target distribution", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "is covered), and often requires objective modification to train in practice (Arjovsky et al., 2017).", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "score": 1.0, + "content": "Finally, since the proposal transformation preserves volume, it can suffer from the same difficulties", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 383, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 383, + 650 + ], + "score": 1.0, + "content": "in mixing across energy levels as HMC, as we illustrate in Section 5.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "To compute the Metropolis-Hastings acceptance probability for a deterministic transition, the oper-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "ator must be invertible and have a tractable Jacobian. Recent work (Dinh et al., 2016), introduces", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "RNVP, an invertible transformation that operates by, at each layer, modifying only a subset of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "variables by a function that depends solely on the remaining variables. This is exactly invertible with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "an efficiently computable Jacobian. Furthermore, by chaining enough of these layers, the model can", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "be made arbitrarily expressive. This parameterization will directly motivate our extension of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 721, + 221, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 221, + 733 + ], + "score": 1.0, + "content": "leapfrog integrator in HMC.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "Welling, 2013; Rezende et al., 2014; Goodfellow et al., 2014; Bengio et al., 2014; Sohl-Dickstein", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "et al., 2015). Building on these successes, we present a method that, given an analytically described", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "distribution, automatically returns an exact sampler with good convergence and mixing properties,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "from a class of highly expressive parametric models. The proposed family of samplers is a gen-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "eralization of HMC; it transforms the HMC trajectory using parametric functions (deep networks", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "in our experiments), while retaining theoretical guarantees with a tractable Metropolis-Hastings", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "accept/reject step. The sampler is trained to minimize a variation on expected squared jumped dis-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "tance (similar in spirit to Pasarica & Gelman (2010)). Our parameterization reduces easily to stan-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "dard HMC. It is further capable of emulating several common extensions of HMC such as within-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 423, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 423, + 194 + ], + "score": 1.0, + "content": "trajectory tempering (Neal, 1996) and diagonal mass matrices (Bennett, 1975).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 83, + 506, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 504, + 220 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "We evaluate our method on distributions where HMC usually struggles, as well as on a the real-world", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 306, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 306, + 222 + ], + "score": 1.0, + "content": "task of training latent-variable generative models.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 106, + 198, + 505, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 239, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 240, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 240, + 239 + ], + "score": 1.0, + "content": "Our contributions are as follows:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 225, + 240, + 239 + ] + }, + { + "type": "list", + "bbox": [ + 127, + 251, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 132, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "• We introduce a generic training procedure which takes as input a distribution defined by an", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 262, + 375, + 275 + ], + "spans": [ + { + "bbox": [ + 142, + 262, + 375, + 275 + ], + "score": 1.0, + "content": "energy function, and returns a fast-mixing MCMC kernel.", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 283, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 131, + 283, + 505, + 299 + ], + "score": 1.0, + "content": "• We show significant empirical gains on various distributions where HMC performs poorly.", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 132, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 132, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "• We finally evaluate our method on the real-world task of training and sampling from a latent", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 142, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "variable generative model, where we show improvement in the model’s log-likelihood, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 330, + 379, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 379, + 343 + ], + "score": 1.0, + "content": "greater complexity in the distribution of posterior samples.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + } + ], + "index": 15.5, + "bbox_fs": [ + 131, + 251, + 506, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 365, + 211, + 378 + ], + "lines": [ + { + "bbox": [ + 104, + 364, + 213, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 213, + 381 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "Adaptively modifying proposal distributions to improve convergence and mixing has been explored", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "in the past (Andrieu & Thoms, 2008). In the case of HMC, prior work has reduced the need to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "score": 1.0, + "content": "choose step size (Neal, 2011) or number of leapfrog steps (Hoffman & Gelman, 2014) by adaptively", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "tuning those parameters. Salimans et al. (2015) proposed an alternate scheme based on variational", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "inference. We adopt the much simpler approach of Pasarica & Gelman (2010), who show that choos-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "ing the hyperparameters of a proposal distribution to maximize expected squared jumped distance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 279, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 279, + 474 + ], + "score": 1.0, + "content": "is both principled and effective in practice.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 395, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "Previous work has also explored applying models from machine learning to MCMC tasks. Kernel", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "methods have been used both for learning a proposal distribution (Sejdinovic et al., 2014) and for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "approximating the gradient of the energy (Strathmann et al., 2015). In physics, Restricted and semi-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "Restricted Boltzmann machines have been used both to build approximations of the energy function", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "which allow more rapid sampling (Liu et al., 2017; Huang & Wang, 2017), and to motivate new", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 533, + 267, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 267, + 546 + ], + "score": 1.0, + "content": "hand-designed proposals (Wang, 2017).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 478, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "Most similar to our approach is recent work from Song et al. (2017), which uses adversarial training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "of a volume-preserving transformation, which is subsequently used as an MCMC proposal distribu-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "tion. While promising, this technique has several limitations. It does not use gradient information,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "which is often crucial to maintaining high acceptance rates, especially in high dimensions. 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This last condition can be expressed as:", + "type": "text" + }, + { + "bbox": [ + 321, + 243, + 429, + 255 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p ( x ^ { \\prime } ) = \\int K ( x ^ { \\prime } | x ) p ( x ) \\mathrm { d } x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 242, + 506, + 256 + ], + "score": 1.0, + "content": ". This condition is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "most often satisfied by satisfying the stronger detailed balance condition, which can be written as:", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 264, + 233, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 228, + 277 + ], + "score": 0.92, + "content": "p ( x ^ { \\prime } ) K ( x | x ^ { \\prime } ) = p ( x ) \\dot { K } ( x ^ { \\prime } | \\dot { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 264, + 233, + 279 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 220, + 506, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 237, + 294 + ], + "score": 1.0, + "content": "Given any proposal distribution", + "type": "text" + }, + { + "bbox": [ + 238, + 284, + 244, + 294 + ], + "score": 0.76, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 281, + 505, + 294 + ], + "score": 1.0, + "content": ", satisfying mild conditions, we can easily construct a transition", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "kernel that respects detailed balance using Metropolis-Hastings (Hastings, 1970) accept/reject rules.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 223, + 317 + ], + "score": 1.0, + "content": "More formally, starting from", + "type": "text" + }, + { + "bbox": [ + 224, + 306, + 258, + 315 + ], + "score": 0.85, + "content": "x _ { 0 } \\sim \\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 303, + 311, + 317 + ], + "score": 1.0, + "content": ", at each step", + "type": "text" + }, + { + "bbox": [ + 311, + 305, + 316, + 314 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 303, + 365, + 317 + ], + "score": 1.0, + "content": ", we sample", + "type": "text" + }, + { + "bbox": [ + 365, + 304, + 417, + 316 + ], + "score": 0.93, + "content": "x ^ { \\prime } \\sim q ( \\cdot | X _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 303, + 505, + 317 + ], + "score": 1.0, + "content": ", and with probability", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 312, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 248, + 335 + ], + "score": 0.93, + "content": "\\begin{array} { r } { A ( x ^ { \\prime } | x _ { t } ) = \\operatorname* { m i n } \\left( 1 , \\frac { p ( x ^ { \\prime } ) q ( x _ { t } | x ^ { \\prime } ) } { p ( x _ { t } ) q ( x ^ { \\prime } | x _ { t } ) } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 312, + 282, + 336 + ], + "score": 1.0, + "content": ", accept", + "type": "text" + }, + { + "bbox": [ + 282, + 318, + 292, + 329 + ], + "score": 0.84, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 312, + 374, + 336 + ], + "score": 1.0, + "content": "as the next sample", + "type": "text" + }, + { + "bbox": [ + 375, + 320, + 395, + 330 + ], + "score": 0.89, + "content": "x _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 312, + 506, + 336 + ], + "score": 1.0, + "content": "in the chain. If we reject", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 116, + 344 + ], + "score": 0.83, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 333, + 280, + 347 + ], + "score": 1.0, + "content": ", then we retain the previous state and", + "type": "text" + }, + { + "bbox": [ + 280, + 336, + 328, + 346 + ], + "score": 0.88, + "content": "x _ { t + 1 } ~ = ~ x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 333, + 505, + 347 + ], + "score": 1.0, + "content": ". For typical proposals this algorithm has", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "strong asymptotic guarantees. But in practice one must often choose between very low acceptance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "probabilities and very cautious proposals, both of which lead to slow mixing. For continuous state", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "spaces, Hamiltonian Monte Carlo (HMC; Neal, 2011) tackles this problem by proposing updates", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 377, + 438, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 428, + 391 + ], + "score": 1.0, + "content": "that move far in state space while staying roughly on iso-probability contours of", + "type": "text" + }, + { + "bbox": [ + 428, + 380, + 434, + 389 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 377, + 438, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 281, + 506, + 391 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 405, + 263, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 264, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 264, + 419 + ], + "score": 1.0, + "content": "3.2 HAMILTONIAN MONTE CARLO", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 275, + 441 + ], + "score": 1.0, + "content": "Without loss of generality, we assume", + "type": "text" + }, + { + "bbox": [ + 275, + 428, + 296, + 439 + ], + "score": 0.9, + "content": "p \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 426, + 460, + 441 + ], + "score": 1.0, + "content": "to be defined by an energy function", + "type": "text" + }, + { + "bbox": [ + 460, + 428, + 484, + 439 + ], + "score": 0.91, + "content": "U \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 426, + 505, + 441 + ], + "score": 1.0, + "content": ", s.t.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 107, + 438, + 191, + 451 + ], + "score": 0.91, + "content": "p ( x ) \\propto \\exp ( - U ( { \\bar { x } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 438, + 275, + 451 + ], + "score": 1.0, + "content": ", and where the state", + "type": "text" + }, + { + "bbox": [ + 275, + 439, + 307, + 449 + ], + "score": 0.9, + "content": "x \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 438, + 506, + 451 + ], + "score": 1.0, + "content": ". 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From an augmented state", + "type": "text" + }, + { + "bbox": [ + 351, + 462, + 393, + 475 + ], + "score": 0.94, + "content": "\\xi \\triangleq ( x , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 462, + 506, + 476 + ], + "score": 1.0, + "content": ", HMC produces a proposed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 127, + 487 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 127, + 474, + 177, + 486 + ], + "score": 0.93, + "content": "\\xi ^ { \\prime } = ( \\dot { x } ^ { \\prime } , v ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 473, + 426, + 487 + ], + "score": 1.0, + "content": "by approximately integrating Hamiltonian dynamics jointly on", + "type": "text" + }, + { + "bbox": [ + 426, + 476, + 433, + 484 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 473, + 450, + 487 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 475, + 456, + 484 + ], + "score": 0.77, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 473, + 479, + 487 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 480, + 474, + 504, + 486 + ], + "score": 0.89, + "content": "U \\left( x \\right)", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 251, + 498 + ], + "score": 1.0, + "content": "taken to be the potential energy, and", + "type": "text" + }, + { + "bbox": [ + 251, + 484, + 275, + 498 + ], + "score": 0.92, + "content": "\\scriptstyle { \\frac { 1 } { 2 } } v ^ { \\overline { { T } } } v", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "the kinetic energy. Since Hamiltonian dynamics conserve", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 496, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 504, + 509 + ], + "score": 1.0, + "content": "the total energy of a system, their approximate integration moves along approximate iso-probability", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 506, + 236, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 154, + 519 + ], + "score": 1.0, + "content": "contours of", + "type": "text" + }, + { + "bbox": [ + 155, + 506, + 233, + 519 + ], + "score": 0.92, + "content": "p \\bar { ( \\boldsymbol { x } , \\boldsymbol { v } ) } = \\bar { p } ( \\boldsymbol { x } ) p ( \\boldsymbol { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 507, + 236, + 519 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 426, + 506, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 504, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "The dynamics are typically simulated using the leapfrog integrator (Hairer et al., 2003; Leimkuhler", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 334, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 334, + 546 + ], + "score": 1.0, + "content": "& Reich, 2004), which for a single time step consists of:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 523, + 505, + 546 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 554, + 430, + 570 + ], + "lines": [ + { + "bbox": [ + 181, + 554, + 430, + 570 + ], + "spans": [ + { + "bbox": [ + 181, + 554, + 430, + 570 + ], + "score": 0.9, + "content": "\\begin{array} { r } { v ^ { \\frac { 1 } { 2 } } = v - \\frac { \\epsilon } { 2 } \\partial _ { x } U ( x ) ; \\quad x ^ { \\prime } = x + \\epsilon v ^ { \\frac { 1 } { 2 } } ; \\quad v ^ { \\prime } = v - \\frac { \\epsilon } { 2 } \\partial _ { x } U ( x ^ { \\prime } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "129344d4db262ec31f972d8068f8b57408c97a4ca3fe9a09822b67b2b3bcb1d5.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 181, + 554, + 430, + 570 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Following Sohl-Dickstein et al. (2014), we write the action of the leapfrog integrator in terms of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 155, + 603 + ], + "score": 1.0, + "content": "an operator", + "type": "text" + }, + { + "bbox": [ + 155, + 590, + 164, + 600 + ], + "score": 0.29, + "content": "\\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 588, + 169, + 603 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 169, + 589, + 273, + 602 + ], + "score": 0.89, + "content": "\\mathbf { L } \\boldsymbol { \\xi } \\triangleq \\mathbf { L } ( \\boldsymbol { x } , \\boldsymbol { v } ) \\triangleq ( \\boldsymbol { x } ^ { \\prime } , \\boldsymbol { v } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 588, + 446, + 603 + ], + "score": 1.0, + "content": ", and introduce a momentum flip operator", + "type": "text" + }, + { + "bbox": [ + 446, + 590, + 454, + 600 + ], + "score": 0.35, + "content": "\\mathbf { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 588, + 460, + 603 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 460, + 589, + 505, + 602 + ], + "score": 0.85, + "content": "\\mathbf { F } ( x , v ) \\triangleq", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 107, + 601, + 138, + 613 + ], + "score": 0.92, + "content": "( x , - v )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 600, + 478, + 613 + ], + "score": 1.0, + "content": ". It is important to note two properties of these operators. First, the transformation", + "type": "text" + }, + { + "bbox": [ + 479, + 602, + 494, + 611 + ], + "score": 0.44, + "content": "\\mathbf { F L }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 181, + 626 + ], + "score": 1.0, + "content": "an involution, i.e.", + "type": "text" + }, + { + "bbox": [ + 182, + 612, + 336, + 624 + ], + "score": 0.89, + "content": "\\mathbf { F L F L } ( x , v ) = \\mathbf { F L } ( x ^ { \\prime } , - v ^ { \\prime } ) = ( x , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 610, + 480, + 626 + ], + "score": 1.0, + "content": ". Second, the transformations from", + "type": "text" + }, + { + "bbox": [ + 480, + 612, + 504, + 624 + ], + "score": 0.9, + "content": "( x , v )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 623, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 118, + 638 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 623, + 147, + 637 + ], + "score": 0.92, + "content": "( x , v ^ { \\frac { 1 } { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 623, + 175, + 638 + ], + "score": 1.0, + "content": ", from", + "type": "text" + }, + { + "bbox": [ + 176, + 623, + 205, + 637 + ], + "score": 0.92, + "content": "( x , v ^ { \\frac { 1 } { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 623, + 218, + 638 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 219, + 623, + 251, + 637 + ], + "score": 0.92, + "content": "( x ^ { \\prime } , v ^ { \\frac { 1 } { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 623, + 296, + 638 + ], + "score": 1.0, + "content": ", and from", + "type": "text" + }, + { + "bbox": [ + 297, + 624, + 329, + 637 + ], + "score": 0.91, + "content": "( x ^ { \\prime } , v ^ { \\frac { 1 } { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 623, + 342, + 638 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 343, + 624, + 372, + 637 + ], + "score": 0.92, + "content": "( x ^ { \\prime } , v ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 623, + 506, + 638 + ], + "score": 1.0, + "content": "are all volume-preserving shear", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 299, + 648 + ], + "score": 1.0, + "content": "transformations i.e., only one of the variables (", + "type": "text" + }, + { + "bbox": [ + 299, + 638, + 306, + 646 + ], + "score": 0.7, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 636, + 319, + 648 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 320, + 638, + 326, + 646 + ], + "score": 0.66, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 636, + 505, + 648 + ], + "score": 1.0, + "content": ") changes, by an amount determined by the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 647, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 289, + 663 + ], + "score": 1.0, + "content": "other one. The determinant of the Jacobian,", + "type": "text" + }, + { + "bbox": [ + 290, + 647, + 324, + 667 + ], + "score": 0.93, + "content": "\\left. \\frac { \\partial [ \\mathbf { F } \\mathbf { L } \\xi ] } { \\partial \\xi ^ { T } } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 649, + 505, + 663 + ], + "score": 1.0, + "content": ", is thus easy to compute. For vanilla HMC", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 108, + 665, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 108, + 665, + 160, + 685 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\left| \\frac { \\partial [ \\mathbf { F } \\mathbf { L } \\xi ] } { \\partial \\xi ^ { T } } \\right| = 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 667, + 505, + 684 + ], + "score": 1.0, + "content": ", but we will leave it in symbolic form for use in Section 4. The Metropolis-Hastings-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "Green (Hastings, 1970; Green, 1995) acceptance probability for the HMC proposal is made simple", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 695, + 230, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 230, + 707 + ], + "score": 1.0, + "content": "by these two properties, and is", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 578, + 506, + 707 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 714, + 387, + 736 + ], + "lines": [ + { + "bbox": [ + 224, + 714, + 387, + 736 + ], + "spans": [ + { + "bbox": [ + 224, + 714, + 387, + 736 + ], + "score": 0.95, + "content": "\\begin{array} { r } { A ( \\mathbf { F } \\mathbf { L } \\xi | \\xi ) = \\operatorname* { m i n } \\left( 1 , \\frac { p ( \\mathbf { F } \\mathbf { L } \\xi ) } { p ( \\xi ) } \\left| \\frac { \\partial [ \\mathbf { F } \\mathbf { L } \\xi ] } { \\partial \\xi ^ { T } } \\right| \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "55afa3ae656651f1b5585bdd08c27e684d10f684fe343493a84cd69cf916a2b3.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 224, + 714, + 387, + 736 + ], + "spans": [], + "index": 44 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 330, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 331, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 331, + 96 + ], + "score": 1.0, + "content": "4 L2HMC: TRAINING MCMC SAMPLERS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "In this section, we describe our proposed method L2HMC (for ‘Learning To Hamiltonian Monte", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 303, + 129 + ], + "score": 1.0, + "content": "Carlo’). Given access to only an energy function", + "type": "text" + }, + { + "bbox": [ + 304, + 117, + 313, + 127 + ], + "score": 0.76, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "(and not samples), L2HMC learns a parametric", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 179, + 140 + ], + "score": 1.0, + "content": "leapfrog operator", + "type": "text" + }, + { + "bbox": [ + 179, + 127, + 192, + 138 + ], + "score": 0.87, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "over an augmented state space. We begin by describing what desiderata we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 143, + 151 + ], + "score": 1.0, + "content": "have for", + "type": "text" + }, + { + "bbox": [ + 144, + 139, + 156, + 149 + ], + "score": 0.87, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 138, + 505, + 151 + ], + "score": 1.0, + "content": ", then go into detail on how we parameterize our sampler. Finally, we conclude this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 287, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 287, + 163 + ], + "score": 1.0, + "content": "section by describing our training procedure.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 173, + 218, + 184 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 219, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 219, + 186 + ], + "score": 1.0, + "content": "4.1 AUGMENTING HMC", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "HMC is a powerful algorithm, but it can still struggle even on very simple problems. For example, a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "two-dimensional multivariate Gaussian with an ill-conditioned covariance matrix can take arbitrarily", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "long to traverse (even if the covariance is diagonal), whereas it is trivial to sample directly from it.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 241 + ], + "score": 1.0, + "content": "Another problem is that HMC can only move between energy levels via a random walk (Neal, 2011),", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "which leads to slow mixing in some models. Finally, HMC cannot easily traverse low-density zones.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "For example, given a simple Gaussian mixture model, HMC cannot mix between modes without", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "recourse to additional tricks, as illustrated in Figure 1b. These observations determine the list of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "desiderata for our learned MCMC kernel: fast mixing, fast burn-in, mixing across energy levels, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 201, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 201, + 294 + ], + "score": 1.0, + "content": "mixing between modes.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "score": 1.0, + "content": "While pursuing these goals, we must take care to ensure that our proposal operator retains two key", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "features of the leapfrog operator used in HMC: it must be invertible, and the determinant of its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "Jacobian must be tractable. The leapfrog operator satisfies these properties by ensuring that each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "sub-update only affects a subset of the variables, and that no sub-update depends nonlinearly on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "any of the variables being updated. We are free to generalize the leapfrog operator in any way that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "preserves these properties. In particular, we are free to translate and rescale each sub-update of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "score": 1.0, + "content": "leapfrog operator, so long as we are careful to ensure that these translation and scale terms do not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 264, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 264, + 388 + ], + "score": 1.0, + "content": "depend on the variables being updated.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 107, + 398, + 197, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 198, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 198, + 411 + ], + "score": 1.0, + "content": "4.1.1 STATE SPACE", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 319, + 430 + ], + "score": 1.0, + "content": "As in HMC, we begin by augmenting the current state", + "type": "text" + }, + { + "bbox": [ + 319, + 418, + 351, + 428 + ], + "score": 0.91, + "content": "x \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "with a continuous momentum variable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 428, + 138, + 438 + ], + "score": 0.9, + "content": "v \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 428, + 450, + 440 + ], + "score": 1.0, + "content": "drawn from a standard normal. We also introduce a binary direction variable", + "type": "text" + }, + { + "bbox": [ + 450, + 428, + 501, + 440 + ], + "score": 0.93, + "content": "d \\in \\{ - 1 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 428, + 505, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 448, + 453 + ], + "score": 1.0, + "content": "drawn from a uniform distribution. We will denote the complete augmented state as", + "type": "text" + }, + { + "bbox": [ + 448, + 441, + 501, + 453 + ], + "score": 0.91, + "content": "\\xi \\triangleq ( x , v , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 440, + 505, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 205, + 464 + ], + "score": 1.0, + "content": "with probability density", + "type": "text" + }, + { + "bbox": [ + 205, + 452, + 292, + 463 + ], + "score": 0.91, + "content": "p ( \\xi ) = p ( x ) p ( v ) p ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 452, + 380, + 464 + ], + "score": 1.0, + "content": ". Finally, to each step", + "type": "text" + }, + { + "bbox": [ + 381, + 453, + 386, + 462 + ], + "score": 0.66, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 452, + 448, + 464 + ], + "score": 1.0, + "content": "of the operator", + "type": "text" + }, + { + "bbox": [ + 449, + 453, + 461, + 463 + ], + "score": 0.84, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "we assign", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 221, + 475 + ], + "score": 1.0, + "content": "a fixed random binary mask", + "type": "text" + }, + { + "bbox": [ + 221, + 463, + 277, + 474 + ], + "score": 0.91, + "content": "m ^ { t } \\in \\{ 0 , 1 \\} ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "that will determine which variables are affected by each", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 470, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 102, + 470, + 194, + 491 + ], + "score": 1.0, + "content": "sub-update. We draw", + "type": "text" + }, + { + "bbox": [ + 195, + 473, + 208, + 484 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 470, + 414, + 491 + ], + "score": 1.0, + "content": "uniformly from the set of binary vectors satisfying", + "type": "text" + }, + { + "bbox": [ + 415, + 473, + 483, + 486 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } m _ { i } ^ { t } = \\lfloor \\frac { n } { 2 } \\rfloor } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 470, + 506, + 491 + ], + "score": 1.0, + "content": ", that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 207, + 498 + ], + "score": 1.0, + "content": "is, half of the entries of", + "type": "text" + }, + { + "bbox": [ + 207, + 486, + 221, + 496 + ], + "score": 0.89, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 485, + 424, + 498 + ], + "score": 1.0, + "content": "are 0 and half are 1. For convenience, we write", + "type": "text" + }, + { + "bbox": [ + 424, + 486, + 486, + 497 + ], + "score": 0.9, + "content": "\\bar { m } ^ { t } = 1 - m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 497, + 438, + 509 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 167, + 509 + ], + "score": 0.91, + "content": "x _ { m ^ { t } } = x \\odot m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 497, + 172, + 509 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 173, + 498, + 182, + 507 + ], + "score": 0.8, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 497, + 349, + 509 + ], + "score": 1.0, + "content": "denotes element-wise multiplication, and", + "type": "text" + }, + { + "bbox": [ + 349, + 497, + 356, + 507 + ], + "score": 0.45, + "content": "\\mathbb { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 497, + 438, + 509 + ], + "score": 1.0, + "content": "the all ones vector).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 519, + 204, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 205, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 205, + 533 + ], + "score": 1.0, + "content": "4.1.2 UPDATE STEPS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 373, + 552 + ], + "score": 1.0, + "content": "We now describe the details of our augmented leapfrog integrator", + "type": "text" + }, + { + "bbox": [ + 374, + 540, + 386, + 550 + ], + "score": 0.88, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 538, + 479, + 552 + ], + "score": 1.0, + "content": ", for a single time-step", + "type": "text" + }, + { + "bbox": [ + 479, + 540, + 484, + 549 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 538, + 506, + 552 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 549, + 188, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 158, + 561 + ], + "score": 1.0, + "content": "for direction", + "type": "text" + }, + { + "bbox": [ + 159, + 550, + 183, + 560 + ], + "score": 0.9, + "content": "d = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 549, + 188, + 561 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 108, + 566, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 223, + 579 + ], + "score": 1.0, + "content": "We first update the momenta", + "type": "text" + }, + { + "bbox": [ + 223, + 569, + 230, + 577 + ], + "score": 0.72, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 565, + 397, + 579 + ], + "score": 1.0, + "content": ". This update can only depend on a subset", + "type": "text" + }, + { + "bbox": [ + 398, + 566, + 478, + 579 + ], + "score": 0.92, + "content": "\\zeta _ { 1 } \\triangleq ( x , \\partial _ { x } U ( x ) , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 578, + 289, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 210, + 589 + ], + "score": 1.0, + "content": "full state, which excludes", + "type": "text" + }, + { + "bbox": [ + 210, + 580, + 216, + 588 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 578, + 289, + 589 + ], + "score": 1.0, + "content": ". It takes the form", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 590, + 439, + 604 + ], + "lines": [], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 171, + 590, + 439, + 604 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 278, + 617 + ], + "score": 1.0, + "content": "We have introduced three new functions of", + "type": "text" + }, + { + "bbox": [ + 278, + 605, + 304, + 616 + ], + "score": 0.65, + "content": "\\zeta _ { 1 } \\colon T _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 603, + 307, + 617 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 308, + 605, + 321, + 616 + ], + "score": 0.71, + "content": "Q _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 603, + 342, + 617 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 342, + 605, + 353, + 615 + ], + "score": 0.84, + "content": "S _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 603, + 358, + 617 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 358, + 605, + 370, + 615 + ], + "score": 0.84, + "content": "T _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 603, + 433, + 617 + ], + "score": 1.0, + "content": "is a translation,", + "type": "text" + }, + { + "bbox": [ + 433, + 604, + 470, + 616 + ], + "score": 0.91, + "content": "\\exp ( Q _ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "rescales", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 176, + 627 + ], + "score": 1.0, + "content": "the gradient, and", + "type": "text" + }, + { + "bbox": [ + 177, + 615, + 217, + 628 + ], + "score": 0.91, + "content": "\\exp ( \\frac { \\epsilon } { 2 } S _ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "rescales the momentum. The determinant of the Jacobian of this trans-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 172, + 642 + ], + "score": 1.0, + "content": "formation is exp", + "type": "text" + }, + { + "bbox": [ + 172, + 628, + 230, + 641 + ], + "score": 0.76, + "content": ", \\left( \\frac { \\epsilon } { 2 } \\mathbb { 1 } \\cdot S _ { v } ( \\zeta _ { 1 } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 627, + 281, + 642 + ], + "score": 1.0, + "content": ". 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Given access to only an energy function", + "type": "text" + }, + { + "bbox": [ + 304, + 117, + 313, + 127 + ], + "score": 0.76, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "(and not samples), L2HMC learns a parametric", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 179, + 140 + ], + "score": 1.0, + "content": "leapfrog operator", + "type": "text" + }, + { + "bbox": [ + 179, + 127, + 192, + 138 + ], + "score": 0.87, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "over an augmented state space. We begin by describing what desiderata we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 143, + 151 + ], + "score": 1.0, + "content": "have for", + "type": "text" + }, + { + "bbox": [ + 144, + 139, + 156, + 149 + ], + "score": 0.87, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 138, + 505, + 151 + ], + "score": 1.0, + "content": ", then go into detail on how we parameterize our sampler. Finally, we conclude this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 287, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 287, + 163 + ], + "score": 1.0, + "content": "section by describing our training procedure.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 105, + 505, + 163 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 173, + 218, + 184 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 219, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 219, + 186 + ], + "score": 1.0, + "content": "4.1 AUGMENTING HMC", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "HMC is a powerful algorithm, but it can still struggle even on very simple problems. For example, a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "two-dimensional multivariate Gaussian with an ill-conditioned covariance matrix can take arbitrarily", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "long to traverse (even if the covariance is diagonal), whereas it is trivial to sample directly from it.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 241 + ], + "score": 1.0, + "content": "Another problem is that HMC can only move between energy levels via a random walk (Neal, 2011),", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "which leads to slow mixing in some models. Finally, HMC cannot easily traverse low-density zones.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "For example, given a simple Gaussian mixture model, HMC cannot mix between modes without", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "recourse to additional tricks, as illustrated in Figure 1b. These observations determine the list of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "desiderata for our learned MCMC kernel: fast mixing, fast burn-in, mixing across energy levels, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 201, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 201, + 294 + ], + "score": 1.0, + "content": "mixing between modes.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 194, + 506, + 294 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "score": 1.0, + "content": "While pursuing these goals, we must take care to ensure that our proposal operator retains two key", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "features of the leapfrog operator used in HMC: it must be invertible, and the determinant of its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "Jacobian must be tractable. The leapfrog operator satisfies these properties by ensuring that each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "sub-update only affects a subset of the variables, and that no sub-update depends nonlinearly on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "any of the variables being updated. We are free to generalize the leapfrog operator in any way that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "preserves these properties. In particular, we are free to translate and rescale each sub-update of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "score": 1.0, + "content": "leapfrog operator, so long as we are careful to ensure that these translation and scale terms do not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 264, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 264, + 388 + ], + "score": 1.0, + "content": "depend on the variables being updated.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 297, + 506, + 388 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 398, + 197, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 198, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 198, + 411 + ], + "score": 1.0, + "content": "4.1.1 STATE SPACE", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 319, + 430 + ], + "score": 1.0, + "content": "As in HMC, we begin by augmenting the current state", + "type": "text" + }, + { + "bbox": [ + 319, + 418, + 351, + 428 + ], + "score": 0.91, + "content": "x \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "with a continuous momentum variable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 428, + 138, + 438 + ], + "score": 0.9, + "content": "v \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 428, + 450, + 440 + ], + "score": 1.0, + "content": "drawn from a standard normal. We also introduce a binary direction variable", + "type": "text" + }, + { + "bbox": [ + 450, + 428, + 501, + 440 + ], + "score": 0.93, + "content": "d \\in \\{ - 1 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 428, + 505, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 448, + 453 + ], + "score": 1.0, + "content": "drawn from a uniform distribution. We will denote the complete augmented state as", + "type": "text" + }, + { + "bbox": [ + 448, + 441, + 501, + 453 + ], + "score": 0.91, + "content": "\\xi \\triangleq ( x , v , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 440, + 505, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 205, + 464 + ], + "score": 1.0, + "content": "with probability density", + "type": "text" + }, + { + "bbox": [ + 205, + 452, + 292, + 463 + ], + "score": 0.91, + "content": "p ( \\xi ) = p ( x ) p ( v ) p ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 452, + 380, + 464 + ], + "score": 1.0, + "content": ". Finally, to each step", + "type": "text" + }, + { + "bbox": [ + 381, + 453, + 386, + 462 + ], + "score": 0.66, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 452, + 448, + 464 + ], + "score": 1.0, + "content": "of the operator", + "type": "text" + }, + { + "bbox": [ + 449, + 453, + 461, + 463 + ], + "score": 0.84, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "we assign", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 221, + 475 + ], + "score": 1.0, + "content": "a fixed random binary mask", + "type": "text" + }, + { + "bbox": [ + 221, + 463, + 277, + 474 + ], + "score": 0.91, + "content": "m ^ { t } \\in \\{ 0 , 1 \\} ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "that will determine which variables are affected by each", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 470, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 102, + 470, + 194, + 491 + ], + "score": 1.0, + "content": "sub-update. We draw", + "type": "text" + }, + { + "bbox": [ + 195, + 473, + 208, + 484 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 470, + 414, + 491 + ], + "score": 1.0, + "content": "uniformly from the set of binary vectors satisfying", + "type": "text" + }, + { + "bbox": [ + 415, + 473, + 483, + 486 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } m _ { i } ^ { t } = \\lfloor \\frac { n } { 2 } \\rfloor } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 470, + 506, + 491 + ], + "score": 1.0, + "content": ", that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 207, + 498 + ], + "score": 1.0, + "content": "is, half of the entries of", + "type": "text" + }, + { + "bbox": [ + 207, + 486, + 221, + 496 + ], + "score": 0.89, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 485, + 424, + 498 + ], + "score": 1.0, + "content": "are 0 and half are 1. For convenience, we write", + "type": "text" + }, + { + "bbox": [ + 424, + 486, + 486, + 497 + ], + "score": 0.9, + "content": "\\bar { m } ^ { t } = 1 - m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 497, + 438, + 509 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 167, + 509 + ], + "score": 0.91, + "content": "x _ { m ^ { t } } = x \\odot m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 497, + 172, + 509 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 173, + 498, + 182, + 507 + ], + "score": 0.8, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 497, + 349, + 509 + ], + "score": 1.0, + "content": "denotes element-wise multiplication, and", + "type": "text" + }, + { + "bbox": [ + 349, + 497, + 356, + 507 + ], + "score": 0.45, + "content": "\\mathbb { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 497, + 438, + 509 + ], + "score": 1.0, + "content": "the all ones vector).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 102, + 417, + 506, + 509 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 519, + 204, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 205, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 205, + 533 + ], + "score": 1.0, + "content": "4.1.2 UPDATE STEPS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 373, + 552 + ], + "score": 1.0, + "content": "We now describe the details of our augmented leapfrog integrator", + "type": "text" + }, + { + "bbox": [ + 374, + 540, + 386, + 550 + ], + "score": 0.88, + "content": "\\mathbf { L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 538, + 479, + 552 + ], + "score": 1.0, + "content": ", for a single time-step", + "type": "text" + }, + { + "bbox": [ + 479, + 540, + 484, + 549 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 538, + 506, + 552 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 549, + 188, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 158, + 561 + ], + "score": 1.0, + "content": "for direction", + "type": "text" + }, + { + "bbox": [ + 159, + 550, + 183, + 560 + ], + "score": 0.9, + "content": "d = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 549, + 188, + 561 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 538, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 566, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 223, + 579 + ], + "score": 1.0, + "content": "We first update the momenta", + "type": "text" + }, + { + "bbox": [ + 223, + 569, + 230, + 577 + ], + "score": 0.72, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 565, + 397, + 579 + ], + "score": 1.0, + "content": ". This update can only depend on a subset", + "type": "text" + }, + { + "bbox": [ + 398, + 566, + 478, + 579 + ], + "score": 0.92, + "content": "\\zeta _ { 1 } \\triangleq ( x , \\partial _ { x } U ( x ) , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 578, + 289, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 210, + 589 + ], + "score": 1.0, + "content": "full state, which excludes", + "type": "text" + }, + { + "bbox": [ + 210, + 580, + 216, + 588 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 578, + 289, + 589 + ], + "score": 1.0, + "content": ". It takes the form", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 565, + 505, + 589 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 590, + 439, + 604 + ], + "lines": [], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 171, + 590, + 439, + 604 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 278, + 617 + ], + "score": 1.0, + "content": "We have introduced three new functions of", + "type": "text" + }, + { + "bbox": [ + 278, + 605, + 304, + 616 + ], + "score": 0.65, + "content": "\\zeta _ { 1 } \\colon T _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 603, + 307, + 617 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 308, + 605, + 321, + 616 + ], + "score": 0.71, + "content": "Q _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 603, + 342, + 617 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 342, + 605, + 353, + 615 + ], + "score": 0.84, + "content": "S _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 603, + 358, + 617 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 358, + 605, + 370, + 615 + ], + "score": 0.84, + "content": "T _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 603, + 433, + 617 + ], + "score": 1.0, + "content": "is a translation,", + "type": "text" + }, + { + "bbox": [ + 433, + 604, + 470, + 616 + ], + "score": 0.91, + "content": "\\exp ( Q _ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "rescales", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 176, + 627 + ], + "score": 1.0, + "content": "the gradient, and", + "type": "text" + }, + { + "bbox": [ + 177, + 615, + 217, + 628 + ], + "score": 0.91, + "content": "\\exp ( \\frac { \\epsilon } { 2 } S _ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "rescales the momentum. The determinant of the Jacobian of this trans-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 172, + 642 + ], + "score": 1.0, + "content": "formation is exp", + "type": "text" + }, + { + "bbox": [ + 172, + 628, + 230, + 641 + ], + "score": 0.76, + "content": ", \\left( \\frac { \\epsilon } { 2 } \\mathbb { 1 } \\cdot S _ { v } ( \\zeta _ { 1 } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 627, + 281, + 642 + ], + "score": 1.0, + "content": ". 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As hinted above, to make our transformation more expressive, we first update a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 217, + 679 + ], + "score": 1.0, + "content": "subset of the coordinates of", + "type": "text" + }, + { + "bbox": [ + 217, + 669, + 224, + 677 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 667, + 505, + 679 + ], + "score": 1.0, + "content": ", followed by the complementary subset. 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For all experiments, the functions", + "type": "text" + }, + { + "bbox": [ + 378, + 255, + 409, + 267 + ], + "score": 0.92, + "content": "Q , S , T", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "are implemented using", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 493, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 493, + 280 + ], + "score": 1.0, + "content": "multi-layer perceptrons, with shared weights. 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This quantity is derived in Appendix B.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 361, + 235, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 236, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 236, + 374 + ], + "score": 1.0, + "content": "4.1.3 MCMC TRANSITIONS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 504, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 236, + 393 + ], + "score": 1.0, + "content": "For convenience, we denote by", + "type": "text" + }, + { + "bbox": [ + 237, + 381, + 247, + 390 + ], + "score": 0.64, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "an operator that re-samples the momentum and direction. I.e.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 132, + 404 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 132, + 391, + 189, + 403 + ], + "score": 0.86, + "content": "\\xi ~ = ~ ( x , v , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 391, + 194, + 404 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 195, + 391, + 266, + 403 + ], + "score": 0.9, + "content": "\\mathbf { R } \\xi = ( x , v ^ { \\prime } , d ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 391, + 297, + 404 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 297, + 391, + 435, + 403 + ], + "score": 0.92, + "content": "v ^ { \\prime } \\sim \\mathcal { N } ( 0 , I ) , d ^ { \\prime } \\sim \\mathcal { U } \\left( \\{ - 1 , 1 \\} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 391, + 506, + 404 + ], + "score": 1.0, + "content": ". Sampling thus", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 270, + 415 + ], + "score": 1.0, + "content": "consists of alternating application of the", + "type": "text" + }, + { + "bbox": [ + 270, + 403, + 290, + 414 + ], + "score": 0.87, + "content": "\\mathbf { F L } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 403, + 308, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 403, + 318, + 413 + ], + "score": 0.59, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 403, + 506, + 415 + ], + "score": 1.0, + "content": ", in the following two steps each of which is a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 412, + 371, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 360, + 427 + ], + "score": 1.0, + "content": "Markov transition that satisfies detailed balance with respect to", + "type": "text" + }, + { + "bbox": [ + 360, + 416, + 366, + 425 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 412, + 371, + 427 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 129, + 433, + 430, + 460 + ], + "lines": [ + { + "bbox": [ + 129, + 432, + 428, + 448 + ], + "spans": [ + { + "bbox": [ + 129, + 432, + 142, + 448 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 434, + 188, + 446 + ], + "score": 0.91, + "content": "\\boldsymbol { \\xi } ^ { \\prime } = \\mathbf { F } \\mathbf { L } _ { \\boldsymbol { \\theta } } \\boldsymbol { \\xi }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 432, + 255, + 448 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 256, + 434, + 303, + 446 + ], + "score": 0.91, + "content": "A ( \\mathbf { F L } _ { \\theta } \\xi | \\xi )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 432, + 400, + 448 + ], + "score": 1.0, + "content": "(Equation 3), otherwise", + "type": "text" + }, + { + "bbox": [ + 401, + 434, + 428, + 446 + ], + "score": 0.9, + "content": "\\xi ^ { \\prime } = \\xi", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 128, + 446, + 178, + 462 + ], + "spans": [ + { + "bbox": [ + 128, + 446, + 142, + 462 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 447, + 178, + 460 + ], + "score": 0.76, + "content": "\\boldsymbol { \\xi } ^ { \\prime } = \\mathbf { R } \\boldsymbol { \\xi }", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 504, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "This parameterization is effectively a generalization of standard HMC as it is non-volume preserv-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 447, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 393, + 492 + ], + "score": 1.0, + "content": "ing, with learnable parameters, and easily reduces to standard HMC for", + "type": "text" + }, + { + "bbox": [ + 394, + 480, + 443, + 491 + ], + "score": 0.92, + "content": "Q , S , T = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 479, + 447, + 492 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 504, + 277, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 278, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 278, + 517 + ], + "score": 1.0, + "content": "4.2 LOSS AND TRAINING PROCEDURE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 295, + 538 + ], + "score": 1.0, + "content": "We need some criterion to train the parameters", + "type": "text" + }, + { + "bbox": [ + 295, + 525, + 302, + 535 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 523, + 405, + 538 + ], + "score": 1.0, + "content": "that control the functions", + "type": "text" + }, + { + "bbox": [ + 405, + 525, + 426, + 536 + ], + "score": 0.61, + "content": "Q , S", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 523, + 446, + 538 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 447, + 525, + 455, + 535 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 523, + 505, + 538 + ], + "score": 1.0, + "content": ". We choose", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "a loss designed to reduce mixing time. Specifically, we aim to minimize lag-one autocorrelation.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "This is equivalent to maximizing expected squared jumped distance (Pasarica & Gelman, 2010).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 123, + 572 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 124, + 558, + 142, + 569 + ], + "score": 0.9, + "content": "\\xi , \\xi ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 556, + 302, + 572 + ], + "score": 1.0, + "content": "in the extended state space, we define", + "type": "text" + }, + { + "bbox": [ + 303, + 558, + 502, + 569 + ], + "score": 0.86, + "content": "\\delta ( \\xi ^ { \\bar { \\prime } } , \\xi ) \\ = \\ \\delta ( ( x ^ { \\prime } , v ^ { \\prime } , d ^ { \\prime } ) , ( x , v , d ) ) \\ = \\ | | x - x ^ { \\prime } | | _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 556, + 506, + 572 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 567, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 274, + 583 + ], + "score": 1.0, + "content": "Expected squared jumped distance is thus", + "type": "text" + }, + { + "bbox": [ + 274, + 569, + 405, + 582 + ], + "score": 0.87, + "content": "\\mathbb { E } _ { \\xi \\sim p ( \\xi ) } \\left[ \\delta ( \\mathbf { F L } _ { \\theta } \\xi , \\xi ) A ( \\mathbf { F L } _ { \\theta } \\xi | \\xi ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 567, + 506, + 583 + ], + "score": 1.0, + "content": ". However, this loss need", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "not encourage mixing across the entire state space. Indeed, maximizing this objective can lead", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "to regions of state space where almost no mixing occurs, so long as the average squared distance", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "traversed remains high. 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The scaling term", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 504, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 504, + 218 + ], + "score": 1.0, + "content": "applied to the gradient of the energy may allow better conditioning of the energy landscape (e.g., by", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 214, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 505, + 231 + ], + "score": 1.0, + "content": "learning a diagonal inertia tensor), or partial ignoring of the energy gradient for rapidly oscillating", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 145, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 145, + 241 + ], + "score": 1.0, + "content": "energies.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 182, + 506, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 244, + 258 + ], + "score": 1.0, + "content": "The corresponding integrator for", + "type": "text" + }, + { + "bbox": [ + 244, + 245, + 281, + 255 + ], + "score": 0.91, + "content": "d = - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "is given in Appendix A; it essentially just inverts the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 378, + 269 + ], + "score": 1.0, + "content": "updates in equations 4, 5 and 6. For all experiments, the functions", + "type": "text" + }, + { + "bbox": [ + 378, + 255, + 409, + 267 + ], + "score": 0.92, + "content": "Q , S , T", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "are implemented using", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 493, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 493, + 280 + ], + "score": 1.0, + "content": "multi-layer perceptrons, with shared weights. 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Written in terms of these modified operators, our proposal and acceptance probability", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 315, + 506, + 330 + ], + "score": 1.0, + "content": "are identical to those for standard HMC. Note, however, that this parameterization enables learning", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 474, + 339 + ], + "score": 1.0, + "content": "non-volume-preserving transformations, as the determinant of the Jacobian is a function of", + "type": "text" + }, + { + "bbox": [ + 474, + 327, + 487, + 338 + ], + "score": 0.92, + "content": "S _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 437, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 118, + 349 + ], + "score": 0.87, + "content": "S _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 338, + 437, + 351 + ], + "score": 1.0, + "content": "that does not necessarily evaluate to 1. 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I.e.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 132, + 404 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 132, + 391, + 189, + 403 + ], + "score": 0.86, + "content": "\\xi ~ = ~ ( x , v , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 391, + 194, + 404 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 195, + 391, + 266, + 403 + ], + "score": 0.9, + "content": "\\mathbf { R } \\xi = ( x , v ^ { \\prime } , d ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 391, + 297, + 404 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 297, + 391, + 435, + 403 + ], + "score": 0.92, + "content": "v ^ { \\prime } \\sim \\mathcal { N } ( 0 , I ) , d ^ { \\prime } \\sim \\mathcal { U } \\left( \\{ - 1 , 1 \\} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 391, + 506, + 404 + ], + "score": 1.0, + "content": ". 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Specifically, we aim to minimize lag-one autocorrelation.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "This is equivalent to maximizing expected squared jumped distance (Pasarica & Gelman, 2010).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 123, + 572 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 124, + 558, + 142, + 569 + ], + "score": 0.9, + "content": "\\xi , \\xi ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 556, + 302, + 572 + ], + "score": 1.0, + "content": "in the extended state space, we define", + "type": "text" + }, + { + "bbox": [ + 303, + 558, + 502, + 569 + ], + "score": 0.86, + "content": "\\delta ( \\xi ^ { \\bar { \\prime } } , \\xi ) \\ = \\ \\delta ( ( x ^ { \\prime } , v ^ { \\prime } , d ^ { \\prime } ) , ( x , v , d ) ) \\ = \\ | | x - x ^ { \\prime } | | _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 556, + 506, + 572 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 567, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 274, + 583 + ], + "score": 1.0, + "content": "Expected squared jumped distance is thus", + "type": "text" + }, + { + "bbox": [ + 274, + 569, + 405, + 582 + ], + "score": 0.87, + "content": "\\mathbb { E } _ { \\xi \\sim p ( \\xi ) } \\left[ \\delta ( \\mathbf { F L } _ { \\theta } \\xi , \\xi ) A ( \\mathbf { F L } _ { \\theta } \\xi | \\xi ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 567, + 506, + 583 + ], + "score": 1.0, + "content": ". 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We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "score": 1.0, + "content": "first present results on a collection of toy distributions capturing common pathologies of energy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "landscapes, followed by results on a task from machine learning: maximum-likelihood training of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "deep generative models. For each, we compare against HMC with well-tuned step length and show", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 460, + 457, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 457, + 473 + ], + "score": 1.0, + "content": "significant gains in mixing time. 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This is an extreme version of an example from Neal (2011). This problem shows that, although", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "our parametric sampler only applies element-wise transformations, it can adapt to structure which is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 175, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 175, + 606 + ], + "score": 1.0, + "content": "not axis-aligned.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 403, + 623 + ], + "score": 1.0, + "content": "Mixture of Gaussians (MoG): Mixture of two isotropic Gaussians with", + "type": "text" + }, + { + "bbox": [ + 403, + 610, + 444, + 621 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 609, + 505, + 623 + ], + "score": 1.0, + "content": ", and centroids", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "separated by distance 4. 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In our experiments, we choose", + "type": "text" + }, + { + "bbox": [ + 459, + 673, + 499, + 686 + ], + "score": 0.92, + "content": "\\eta = 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 673, + 504, + 688 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 691, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "For each of these distributions, we compare against HMC with the same number of leapfrog steps", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 702, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 505, + 714 + ], + "score": 1.0, + "content": "and a well-tuned step-size. 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Given this loss, we exactly describe our training procedure in Algorithm 1. It is important", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 504, + 127 + ], + "score": 1.0, + "content": "to note that each training iteration can be done with only one pass through the network and can be", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "efficiently batched. 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We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "score": 1.0, + "content": "first present results on a collection of toy distributions capturing common pathologies of energy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "landscapes, followed by results on a task from machine learning: maximum-likelihood training of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "deep generative models. For each, we compare against HMC with well-tuned step length and show", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 460, + 457, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 457, + 473 + ], + "score": 1.0, + "content": "significant gains in mixing time. Code implementing our algorithm is available online1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 416, + 506, + 473 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 484, + 329, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 330, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 330, + 497 + ], + "score": 1.0, + "content": "5.1 VARIED COLLECTION OF ENERGY FUNCTIONS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 503, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 504, + 518 + ], + "score": 1.0, + "content": "We evaluate our L2HMC sampler on a diverse collection of energy functions, each posing different", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 515, + 229, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 229, + 528 + ], + "score": 1.0, + "content": "challenges for standard HMC.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 504, + 504, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 504, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 504, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 504, + 547 + ], + "score": 1.0, + "content": "Ill-Conditioned Gaussian (ICG): Gaussian distribution with diagonal covariance spaced log-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 542, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 175, + 558 + ], + "score": 1.0, + "content": "linearly between", + "type": "text" + }, + { + "bbox": [ + 175, + 543, + 197, + 554 + ], + "score": 0.89, + "content": "1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 542, + 215, + 558 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 544, + 231, + 554 + ], + "score": 0.86, + "content": "1 0 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 542, + 504, + 558 + ], + "score": 1.0, + "content": ". This demonstrates that L2HMC can learn a diagonal inertia tensor.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 531, + 504, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 446, + 573 + ], + "score": 1.0, + "content": "Strongly correlated Gaussian (SCG): We rotate a diagonal Gaussian with variances", + "type": "text" + }, + { + "bbox": [ + 446, + 560, + 491, + 573 + ], + "score": 0.92, + "content": "[ 1 0 ^ { 2 } , 1 0 ^ { - 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 567, + 509, + 590 + ], + "spans": [ + { + "bbox": [ + 107, + 572, + 115, + 585 + ], + "score": 0.82, + "content": "\\frac { \\pi } { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 567, + 509, + 590 + ], + "score": 1.0, + "content": ". This is an extreme version of an example from Neal (2011). This problem shows that, although", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "our parametric sampler only applies element-wise transformations, it can adapt to structure which is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 175, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 175, + 606 + ], + "score": 1.0, + "content": "not axis-aligned.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 560, + 509, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 403, + 623 + ], + "score": 1.0, + "content": "Mixture of Gaussians (MoG): Mixture of two isotropic Gaussians with", + "type": "text" + }, + { + "bbox": [ + 403, + 610, + 444, + 621 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 609, + 505, + 623 + ], + "score": 1.0, + "content": ", and centroids", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "separated by distance 4. The means are thus about 12 standard deviations apart, making it almost", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 633, + 286, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 286, + 644 + ], + "score": 1.0, + "content": "impossible for HMC to mix between modes.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 609, + 505, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 441, + 663 + ], + "score": 1.0, + "content": "Rough Well: Similar to an example from Sohl-Dickstein et al. (2014), for a given", + "type": "text" + }, + { + "bbox": [ + 441, + 649, + 505, + 662 + ], + "score": 0.93, + "content": "\\eta > 0 , U ( x ) =", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 657, + 508, + 676 + ], + "spans": [ + { + "bbox": [ + 107, + 660, + 195, + 675 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } x ^ { T } \\overset { \\smile } { x } + \\eta \\sum _ { i } \\cos ( \\frac { x _ { i } } { \\eta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 657, + 239, + 676 + ], + "score": 1.0, + "content": ". For small", + "type": "text" + }, + { + "bbox": [ + 239, + 662, + 246, + 672 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 657, + 508, + 676 + ], + "score": 1.0, + "content": "the energy itself is altered negligibly, but its gradient is perturbed", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 673, + 504, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 291, + 688 + ], + "score": 1.0, + "content": "by a high frequency noise oscillating between", + "type": "text" + }, + { + "bbox": [ + 292, + 675, + 306, + 685 + ], + "score": 0.7, + "content": "- 1", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 673, + 458, + 688 + ], + "score": 1.0, + "content": "and 1. In our experiments, we choose", + "type": "text" + }, + { + "bbox": [ + 459, + 673, + 499, + 686 + ], + "score": 0.92, + "content": "\\eta = 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 673, + 504, + 688 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 647, + 508, + 688 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 691, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "For each of these distributions, we compare against HMC with the same number of leapfrog steps", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 702, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 505, + 714 + ], + "score": 1.0, + "content": "and a well-tuned step-size. To compare mixing time, we plot auto-correlation for each method and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "report effective sample size (ESS). We compute those quantities in the same way as Sohl-Dickstein", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "et al. (2014). We observe that samplers trained with L2HMC show greatly improved autocorrelation", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 329, + 475 + ], + "score": 1.0, + "content": "and ESS on the presented tasks, providing more than", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 329, + 464, + 353, + 474 + ], + "score": 0.87, + "content": "1 0 6 \\times", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 353, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "improved ESS on the SCG task. In", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "score": 1.0, + "content": "addition, for the MoG, we show that L2HMC can easily mix between modes while standard HMC", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 485, + 507, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 507, + 498 + ], + "score": 1.0, + "content": "gets stuck in a mode, unable to traverse the low density zone. Experimental details, as well as a", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 496, + 438, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 438, + 509 + ], + "score": 1.0, + "content": "comparison with LAHMC (Sohl-Dickstein et al., 2014), are shown in Appendix C.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 691, + 505, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 508, + 389 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 508, + 389 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 79, + 508, + 389 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 508, + 389 + ], + "score": 0.87, + "type": "image", + "image_path": "e878bce92e8dd48bdb837822419a4ea3132437ffdaebbcd93f07f493cdac4546.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 508, + 182.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 182.33333333333331, + 508, + 285.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 285.66666666666663, + 508, + 388.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 399, + 504, + 420 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Figure 1: L2HMC mixes faster than well-tuned HMC, and than A-NICE-MC, on a collection of toy distribu-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 408, + 130, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 130, + 421 + ], + "score": 1.0, + "content": "tions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "report effective sample size (ESS). We compute those quantities in the same way as Sohl-Dickstein", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "et al. (2014). We observe that samplers trained with L2HMC show greatly improved autocorrelation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 329, + 475 + ], + "score": 1.0, + "content": "and ESS on the presented tasks, providing more than", + "type": "text" + }, + { + "bbox": [ + 329, + 464, + 353, + 474 + ], + "score": 0.87, + "content": "1 0 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "improved ESS on the SCG task. In", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "score": 1.0, + "content": "addition, for the MoG, we show that L2HMC can easily mix between modes while standard HMC", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 485, + 507, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 507, + 498 + ], + "score": 1.0, + "content": "gets stuck in a mode, unable to traverse the low density zone. Experimental details, as well as a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 496, + 438, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 438, + 509 + ], + "score": 1.0, + "content": "comparison with LAHMC (Sohl-Dickstein et al., 2014), are shown in Appendix C.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "Comparison to A-NICE-MC (Song et al., 2017) In addition to the well known challenges as-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "sociated with adversarial training (Arjovsky et al., 2017), we note that parameterization using a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "volume-preserving operator can dramatically fail on simple energy landscapes. We build off of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "mog2 experiment presented in (Song et al., 2017), which is a 2-d mixture of isotropic Gaussians", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "separated by a distance of 10 with variances 0.5. We consider that setup but increase the ratio of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 151, + 588 + ], + "score": 1.0, + "content": "variances:", + "type": "text" + }, + { + "bbox": [ + 152, + 574, + 235, + 587 + ], + "score": 0.94, + "content": "\\dot { \\sigma } _ { 1 } ^ { 2 } = 3 , \\sigma _ { 2 } ^ { 2 } = 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 575, + 506, + 588 + ], + "score": 1.0, + "content": ". We show in Figure 1d sample chains trained with L2HMC and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "A-NICE-MC; A-NICE-MC cannot effectively mix between the two modes as only a fraction of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "volume of the large mode can be mapped to the small one, making it highly improbable to traverse.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "This is also an issue for HMC. On the other hand, L2HMC can both traverse the low-density region", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "between modes, and map a larger volume in the left mode to a smaller volume in the right mode. It", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 631, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 641 + ], + "score": 1.0, + "content": "is important to note that the distance between both clusters is less than in the mog2 case, and it is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 641, + 436, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 436, + 653 + ], + "score": 1.0, + "content": "thus a good diagnostic of the shortcomings of volume-preserving transformations.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 666, + 306, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 308, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 308, + 678 + ], + "score": 1.0, + "content": "5.2 LATENT-VARIABLE GENERATIVE MODEL", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "score": 1.0, + "content": "We apply our learned sampler to the task of training, and sampling from the posterior of, a latent-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 697, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 384, + 710 + ], + "score": 1.0, + "content": "variable generative model. The model consists of a latent variable", + "type": "text" + }, + { + "bbox": [ + 385, + 697, + 426, + 709 + ], + "score": 0.92, + "content": "z \\sim p ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 697, + 506, + 710 + ], + "score": 1.0, + "content": ", where we choose", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 708, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 708, + 185, + 720 + ], + "score": 0.92, + "content": "p ( z ) = \\breve { \\mathscr { N } } ( z ; 0 , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 708, + 314, + 721 + ], + "score": 1.0, + "content": ", and a conditional distribution", + "type": "text" + }, + { + "bbox": [ + 314, + 708, + 342, + 720 + ], + "score": 0.91, + "content": "p ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 708, + 456, + 721 + ], + "score": 1.0, + "content": "which generates the image", + "type": "text" + }, + { + "bbox": [ + 456, + 710, + 463, + 718 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 708, + 506, + 721 + ], + "score": 1.0, + "content": ". 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We show in Figure 1d sample chains trained with L2HMC and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "A-NICE-MC; A-NICE-MC cannot effectively mix between the two modes as only a fraction of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "volume of the large mode can be mapped to the small one, making it highly improbable to traverse.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "This is also an issue for HMC. 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It", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 631, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 641 + ], + "score": 1.0, + "content": "is important to note that the distance between both clusters is less than in the mog2 case, and it is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 641, + 436, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 436, + 653 + ], + "score": 1.0, + "content": "thus a good diagnostic of the shortcomings of volume-preserving transformations.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 520, + 506, + 653 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 666, + 306, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 308, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 308, + 678 + ], + "score": 1.0, + "content": "5.2 LATENT-VARIABLE GENERATIVE MODEL", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "score": 1.0, + "content": "We apply our learned sampler to the task of training, and sampling from the posterior of, a latent-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 697, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 384, + 710 + ], + "score": 1.0, + "content": "variable generative model. 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Recently, to improve upon well-known pitfalls like over-pruning (Burda et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 308, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 423, + 323 + ], + "score": 1.0, + "content": "2015) of the VAE, Hoffman (2017) proposed HMC-DLGM. For a data sample", + "type": "text" + }, + { + "bbox": [ + 424, + 308, + 439, + 320 + ], + "score": 0.88, + "content": "x ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 308, + 506, + 323 + ], + "score": 1.0, + "content": ", after obtaining", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 271, + 334 + ], + "score": 1.0, + "content": "a sample from the approximate posterior", + "type": "text" + }, + { + "bbox": [ + 272, + 321, + 311, + 334 + ], + "score": 0.91, + "content": "q _ { \\psi } ( \\cdot | x ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 321, + 506, + 334 + ], + "score": 1.0, + "content": ", Hoffman (2017) runs a MCMC algorithm with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 172, + 348 + ], + "score": 1.0, + "content": "energy function", + "type": "text" + }, + { + "bbox": [ + 172, + 333, + 343, + 346 + ], + "score": 0.92, + "content": "U ( z , x ^ { ( i ) } ) = - \\log p ( z ) - \\log p ( x ^ { ( i ) } | z ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "to obtain a more exact posterior sample", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 345, + 502, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 128, + 360 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 345, + 175, + 358 + ], + "score": 0.93, + "content": "p ( \\boldsymbol { z } | \\boldsymbol { x } ^ { ( i ) } ; \\boldsymbol { \\phi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 345, + 317, + 360 + ], + "score": 1.0, + "content": ". Given that better posterior sample", + "type": "text" + }, + { + "bbox": [ + 318, + 347, + 327, + 356 + ], + "score": 0.84, + "content": "z ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 345, + 432, + 360 + ], + "score": 1.0, + "content": ", the algorithm maximizes", + "type": "text" + }, + { + "bbox": [ + 433, + 345, + 497, + 358 + ], + "score": 0.92, + "content": "\\log p ( x ^ { ( i ) } | z ^ { \\prime } ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 345, + 502, + 360 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "To show the benefits of L2HMC, we borrow the method from Hoffman (2017), but replace", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "HMC by jointly training an L2HMC sampler to improve the efficiency of the posterior sam-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "pling. We call this model L2HMC-DLGM. A diagram of our model and a formal description", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 396, + 504, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 395, + 411 + ], + "score": 1.0, + "content": "of our training procedure are presented in Appendix D. We define, for", + "type": "text" + }, + { + "bbox": [ + 395, + 397, + 504, + 410 + ], + "score": 0.85, + "content": "\\xi = \\{ z , v , d \\} , r ( \\xi | x ; \\psi ) \\triangleq", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 407, + 247, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 243, + 421 + ], + "score": 0.9, + "content": "q _ { \\psi } ( z | x ) \\mathcal { N } ( v ; \\bar { 0 , I } ) \\mathcal { U } ( d ; \\{ - 1 , \\bar { 1 } \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 407, + 247, + 422 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "In the subsequent sections, we compare our method to the standard VAE model from Kingma &", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Welling (2013) and HMC-DGLM from Hoffman (2017). It is important to note that, since our sam-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 211, + 461 + ], + "score": 1.0, + "content": "pler is trained jointly with", + "type": "text" + }, + { + "bbox": [ + 212, + 449, + 223, + 460 + ], + "score": 0.86, + "content": "p _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 447, + 240, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 241, + 449, + 252, + 460 + ], + "score": 0.86, + "content": "q _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 447, + 505, + 461 + ], + "score": 1.0, + "content": ", it performs exactly the same number of gradient computations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "of the energy function as HMC. We first show that training a latent variable generative model with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "L2HMC results in better generative models both qualitatively and quantitatively. We then show that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 480, + 405, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 405, + 493 + ], + "score": 1.0, + "content": "our improved sampler enables a more expressive, non-Gaussian, posterior.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 264, + 510 + ], + "score": 1.0, + "content": "Implementation details: Our decoder", + "type": "text" + }, + { + "bbox": [ + 264, + 498, + 282, + 509 + ], + "score": 0.86, + "content": "( p _ { \\phi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "is a neural network with 2 fully connected layers, with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "1024 units each and softplus non-linearities, and outputs Bernoulli activation probabilities for each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 184, + 532 + ], + "score": 1.0, + "content": "pixel. The encoder", + "type": "text" + }, + { + "bbox": [ + 184, + 520, + 202, + 531 + ], + "score": 0.81, + "content": "( q _ { \\psi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "has the same architecture, returning mean and variance for the approximate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "score": 1.0, + "content": "posterior. Our model was trained for 300 epochs with Adam (Kingma & Ba, 2014) and a learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 540, + 501, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 124, + 554 + ], + "score": 1.0, + "content": "rate", + "type": "text" + }, + { + "bbox": [ + 124, + 541, + 165, + 551 + ], + "score": 0.91, + "content": "\\alpha = 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 540, + 501, + 554 + ], + "score": 1.0, + "content": ". All experiments were done on the dynamically binarized MNIST dataset (LeCun).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 572, + 322, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 323, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 323, + 585 + ], + "score": 1.0, + "content": "5.2.1 SAMPLE QUALITY AND DATA LIKELIHOOD", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "We first present samples from decoders trained with L2HMC, HMC and the ELBO (i.e. vanilla", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "VAE). Although higher log likelihood does not necessarily correspond to better samples (Theis", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "et al., 2015), we can see in Figure 5, shown in the Appendix, that the decoder trained with L2HMC", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 628, + 327, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 327, + 639 + ], + "score": 1.0, + "content": "generates sharper samples than the compared methods.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "We now compare our method to HMC in terms of log-likelihood of the data. As we previously", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 288, + 668 + ], + "score": 1.0, + "content": "stated, the marginal likelihood of a data point", + "type": "text" + }, + { + "bbox": [ + 288, + 655, + 315, + 665 + ], + "score": 0.9, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 655, + 474, + 668 + ], + "score": 1.0, + "content": "is not tractable as it requires integrating", + "type": "text" + }, + { + "bbox": [ + 475, + 655, + 504, + 667 + ], + "score": 0.91, + "content": "p ( x , z )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "over a high-dimensional space. However, we can estimate it using annealed importance sampling", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "(AIS; Neal (2001)). Following Wu et al. (2016), we evaluate our generative models on both training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "and held-out data. In Figure 2, we plot the data’s log-likelihood against the number of gradient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "computation steps for both HMC-DGLM and L2HMC-DGLM. We can see that for a similar number", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "of gradient computations, L2HMC-DGLM achieves higher likelihood for both training and held-out", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 478, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 478, + 733 + ], + "score": 1.0, + "content": "data. This is a strong indication that L2HMC provides significantly better posterior samples.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 138, + 82, + 471, + 164 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 82, + 471, + 164 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 82, + 471, + 164 + ], + "spans": [ + { + "bbox": [ + 138, + 82, + 471, + 164 + ], + "score": 0.97, + "type": "image", + "image_path": "250475de256c4eebae15fc4fa637a5547873059d01bec521d1802c331ccf79c1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 138, + 82, + 471, + 109.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 138, + 109.33333333333333, + 471, + 136.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 138, + 136.66666666666666, + 471, + 164.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 176, + 503, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 174, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 504, + 188 + ], + "score": 1.0, + "content": "Figure 2: Training and held-out log-likelihood for models trained with L2HMC, HMC, and the ELBO (VAE).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 248 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 105, + 214, + 505, + 249 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 260, + 445, + 275 + ], + "lines": [ + { + "bbox": [ + 165, + 260, + 445, + 275 + ], + "spans": [ + { + "bbox": [ + 165, + 260, + 445, + 275 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { \\mathrm { E L B O } } ( x , \\phi , \\psi ) = \\mathbb { E } _ { q _ { \\psi } ( z | x ) } \\left[ p ( x | z ; \\phi ) \\right] - \\mathrm { K L } ( q _ { \\psi } ( z | x ) | | p ( z ) ) \\leq p ( x ) ,", + "type": "interline_equation", + "image_path": "b24c790c60685bd813e80b83358ec476c2c47c7a4eb3408681732541e70e0073.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 165, + 260, + 445, + 275 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 133, + 299 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 286, + 166, + 299 + ], + "score": 0.93, + "content": "q _ { \\psi } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 286, + 384, + 299 + ], + "score": 1.0, + "content": "is a tractable conditional distribution with parameters", + "type": "text" + }, + { + "bbox": [ + 384, + 288, + 392, + 298 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 286, + 505, + 299 + ], + "score": 1.0, + "content": ", typically parameterized by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "a neural network. Recently, to improve upon well-known pitfalls like over-pruning (Burda et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 308, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 423, + 323 + ], + "score": 1.0, + "content": "2015) of the VAE, Hoffman (2017) proposed HMC-DLGM. For a data sample", + "type": "text" + }, + { + "bbox": [ + 424, + 308, + 439, + 320 + ], + "score": 0.88, + "content": "x ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 308, + 506, + 323 + ], + "score": 1.0, + "content": ", after obtaining", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 271, + 334 + ], + "score": 1.0, + "content": "a sample from the approximate posterior", + "type": "text" + }, + { + "bbox": [ + 272, + 321, + 311, + 334 + ], + "score": 0.91, + "content": "q _ { \\psi } ( \\cdot | x ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 321, + 506, + 334 + ], + "score": 1.0, + "content": ", Hoffman (2017) runs a MCMC algorithm with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 172, + 348 + ], + "score": 1.0, + "content": "energy function", + "type": "text" + }, + { + "bbox": [ + 172, + 333, + 343, + 346 + ], + "score": 0.92, + "content": "U ( z , x ^ { ( i ) } ) = - \\log p ( z ) - \\log p ( x ^ { ( i ) } | z ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "to obtain a more exact posterior sample", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 345, + 502, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 128, + 360 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 345, + 175, + 358 + ], + "score": 0.93, + "content": "p ( \\boldsymbol { z } | \\boldsymbol { x } ^ { ( i ) } ; \\boldsymbol { \\phi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 345, + 317, + 360 + ], + "score": 1.0, + "content": ". Given that better posterior sample", + "type": "text" + }, + { + "bbox": [ + 318, + 347, + 327, + 356 + ], + "score": 0.84, + "content": "z ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 345, + 432, + 360 + ], + "score": 1.0, + "content": ", the algorithm maximizes", + "type": "text" + }, + { + "bbox": [ + 433, + 345, + 497, + 358 + ], + "score": 0.92, + "content": "\\log p ( x ^ { ( i ) } | z ^ { \\prime } ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 345, + 502, + 360 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 286, + 506, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "To show the benefits of L2HMC, we borrow the method from Hoffman (2017), but replace", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "HMC by jointly training an L2HMC sampler to improve the efficiency of the posterior sam-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "pling. We call this model L2HMC-DLGM. A diagram of our model and a formal description", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 396, + 504, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 395, + 411 + ], + "score": 1.0, + "content": "of our training procedure are presented in Appendix D. We define, for", + "type": "text" + }, + { + "bbox": [ + 395, + 397, + 504, + 410 + ], + "score": 0.85, + "content": "\\xi = \\{ z , v , d \\} , r ( \\xi | x ; \\psi ) \\triangleq", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 407, + 247, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 243, + 421 + ], + "score": 0.9, + "content": "q _ { \\psi } ( z | x ) \\mathcal { N } ( v ; \\bar { 0 , I } ) \\mathcal { U } ( d ; \\{ - 1 , \\bar { 1 } \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 407, + 247, + 422 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 363, + 506, + 422 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "In the subsequent sections, we compare our method to the standard VAE model from Kingma &", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Welling (2013) and HMC-DGLM from Hoffman (2017). It is important to note that, since our sam-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 211, + 461 + ], + "score": 1.0, + "content": "pler is trained jointly with", + "type": "text" + }, + { + "bbox": [ + 212, + 449, + 223, + 460 + ], + "score": 0.86, + "content": "p _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 447, + 240, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 241, + 449, + 252, + 460 + ], + "score": 0.86, + "content": "q _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 447, + 505, + 461 + ], + "score": 1.0, + "content": ", it performs exactly the same number of gradient computations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "of the energy function as HMC. We first show that training a latent variable generative model with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "L2HMC results in better generative models both qualitatively and quantitatively. We then show that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 480, + 405, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 405, + 493 + ], + "score": 1.0, + "content": "our improved sampler enables a more expressive, non-Gaussian, posterior.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 425, + 506, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 264, + 510 + ], + "score": 1.0, + "content": "Implementation details: Our decoder", + "type": "text" + }, + { + "bbox": [ + 264, + 498, + 282, + 509 + ], + "score": 0.86, + "content": "( p _ { \\phi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "is a neural network with 2 fully connected layers, with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "1024 units each and softplus non-linearities, and outputs Bernoulli activation probabilities for each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 184, + 532 + ], + "score": 1.0, + "content": "pixel. The encoder", + "type": "text" + }, + { + "bbox": [ + 184, + 520, + 202, + 531 + ], + "score": 0.81, + "content": "( q _ { \\psi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "has the same architecture, returning mean and variance for the approximate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "score": 1.0, + "content": "posterior. Our model was trained for 300 epochs with Adam (Kingma & Ba, 2014) and a learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 540, + 501, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 124, + 554 + ], + "score": 1.0, + "content": "rate", + "type": "text" + }, + { + "bbox": [ + 124, + 541, + 165, + 551 + ], + "score": 0.91, + "content": "\\alpha = 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 540, + 501, + 554 + ], + "score": 1.0, + "content": ". All experiments were done on the dynamically binarized MNIST dataset (LeCun).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 497, + 506, + 554 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 572, + 322, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 323, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 323, + 585 + ], + "score": 1.0, + "content": "5.2.1 SAMPLE QUALITY AND DATA LIKELIHOOD", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "We first present samples from decoders trained with L2HMC, HMC and the ELBO (i.e. vanilla", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "VAE). Although higher log likelihood does not necessarily correspond to better samples (Theis", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "et al., 2015), we can see in Figure 5, shown in the Appendix, that the decoder trained with L2HMC", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 628, + 327, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 327, + 639 + ], + "score": 1.0, + "content": "generates sharper samples than the compared methods.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 594, + 505, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "We now compare our method to HMC in terms of log-likelihood of the data. As we previously", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 288, + 668 + ], + "score": 1.0, + "content": "stated, the marginal likelihood of a data point", + "type": "text" + }, + { + "bbox": [ + 288, + 655, + 315, + 665 + ], + "score": 0.9, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 655, + 474, + 668 + ], + "score": 1.0, + "content": "is not tractable as it requires integrating", + "type": "text" + }, + { + "bbox": [ + 475, + 655, + 504, + 667 + ], + "score": 0.91, + "content": "p ( x , z )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "over a high-dimensional space. However, we can estimate it using annealed importance sampling", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "(AIS; Neal (2001)). Following Wu et al. (2016), we evaluate our generative models on both training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "and held-out data. In Figure 2, we plot the data’s log-likelihood against the number of gradient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "computation steps for both HMC-DGLM and L2HMC-DGLM. We can see that for a similar number", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "of gradient computations, L2HMC-DGLM achieves higher likelihood for both training and held-out", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 478, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 478, + 733 + ], + "score": 1.0, + "content": "data. This is a strong indication that L2HMC provides significantly better posterior samples.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 104, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 125, + 82, + 390, + 132 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 82, + 390, + 132 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 125, + 82, + 390, + 132 + ], + "spans": [ + { + "bbox": [ + 125, + 82, + 390, + 132 + ], + "score": 0.706, + "type": "image", + "image_path": "fa42f98b9e8fda6e4c48a55741ecf4bec0e430aa2379cfc10ac8d84350687b92.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 125, + 82, + 390, + 98.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 125, + 98.66666666666667, + 390, + 115.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 125, + 115.33333333333334, + 390, + 132.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 115, + 140, + 394, + 161 + ], + "lines": [ + { + "bbox": [ + 114, + 138, + 396, + 153 + ], + "spans": [ + { + "bbox": [ + 114, + 138, + 396, + 153 + ], + "score": 1.0, + "content": "(a) Block Gibbs inpainting of the top half of an MNIST digit, using (top)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 114, + 150, + 374, + 163 + ], + "spans": [ + { + "bbox": [ + 114, + 150, + 278, + 163 + ], + "score": 1.0, + "content": "L2HMC as a posterior sampler, and (bottom)", + "type": "text" + }, + { + "bbox": [ + 279, + 151, + 289, + 161 + ], + "score": 0.86, + "content": "q _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 150, + 374, + 163 + ], + "score": 1.0, + "content": "as a posterior sampler.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "image", + "bbox": [ + 396, + 83, + 493, + 147 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 396, + 83, + 493, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 396, + 83, + 493, + 147 + ], + "spans": [ + { + "bbox": [ + 396, + 83, + 493, + 147 + ], + "score": 0.697, + "type": "image", + "image_path": "3b5a148d0cd5e54e8a2a908b2b63c8ebc00143ad9d41f371e71810f20e32da5a.jpg" + } + ] + } + ], + "index": 3.0, + "virtual_lines": [ + { + "bbox": [ + 396, + 83, + 493, + 115.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 396, + 115.0, + 493, + 147.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 396, + 151, + 495, + 161 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 395, + 150, + 496, + 162 + ], + "spans": [ + { + "bbox": [ + 395, + 150, + 496, + 162 + ], + "score": 1.0, + "content": "(b) Non-Gaussian posterior", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "image_caption", + "bbox": [ + 152, + 163, + 463, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 150, + 161, + 460, + 176 + ], + "spans": [ + { + "bbox": [ + 150, + 161, + 460, + 176 + ], + "score": 1.0, + "content": "Figure 3: Demonstrations of the value of a more expressive posterior approximation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 208, + 339, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 207, + 340, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 340, + 220 + ], + "score": 1.0, + "content": "5.2.2 INCREASED EXPRESSIVITY OF THE POSTERIOR", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 246 + ], + "score": 1.0, + "content": "In the standard VAE framework, approximate posteriors are often parametrized by a Gaussian, thus", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "making a strong assumption of uni-modality. In this section, we show that using L2HMC to sample", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 253, + 374, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 374, + 268 + ], + "score": 1.0, + "content": "from the posterior enables learning of a richer posterior landscape.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 284 + ], + "score": 1.0, + "content": "Block Gibbs Sampling To highlight our ability to capture more expressive posteriors, we in-paint", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "the top of an image using Block Gibbs Sampling using the approximate posterior or L2HMC. For-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 289, + 509, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 147, + 313 + ], + "score": 1.0, + "content": "mally, let", + "type": "text" + }, + { + "bbox": [ + 147, + 296, + 159, + 306 + ], + "score": 0.85, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 289, + 415, + 313 + ], + "score": 1.0, + "content": "be the starting image. 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At", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 302, + 507, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 147, + 324 + ], + "score": 1.0, + "content": "each step", + "type": "text" + }, + { + "bbox": [ + 147, + 308, + 153, + 317 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 302, + 204, + 324 + ], + "score": 1.0, + "content": ", we sample", + "type": "text" + }, + { + "bbox": [ + 204, + 306, + 275, + 319 + ], + "score": 0.93, + "content": "z ^ { ( t ) } \\sim p ( z | x _ { t } ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 302, + 311, + 324 + ], + "score": 1.0, + "content": ", sample", + "type": "text" + }, + { + "bbox": [ + 312, + 307, + 373, + 319 + ], + "score": 0.93, + "content": "\\tilde { x } \\sim p ( x | z _ { t } ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 302, + 432, + 324 + ], + "score": 1.0, + "content": ". We then set", + "type": "text" + }, + { + "bbox": [ + 432, + 307, + 486, + 320 + ], + "score": 0.91, + "content": "x _ { t + 1 } ^ { \\mathrm { t o p } } = \\tilde { x } ^ { \\mathrm { t o p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 302, + 507, + 324 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 313, + 510, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 185, + 333 + ], + "score": 0.93, + "content": "x _ { t + 1 } ^ { \\mathrm { b o t t o m } } = x _ { 0 } ^ { \\mathrm { b o t t o m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 313, + 396, + 344 + ], + "score": 1.0, + "content": ". We compare the results obtained by sampling from ior) vs. our trained sampler. The results are reported i", + "type": "text" + }, + { + "bbox": [ + 396, + 320, + 432, + 332 + ], + "score": 0.92, + "content": "p ( z | x ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 313, + 458, + 344 + ], + "score": 1.0, + "content": "using a. We", + "type": "text" + }, + { + "bbox": [ + 458, + 322, + 470, + 333 + ], + "score": 0.85, + "content": "q _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 313, + 510, + 344 + ], + "score": 1.0, + "content": "(i.e. the see that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "L2HMC easily mixes between modes (3, 5, 8, and plausibly 9 in the figure) while the approximate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 381, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 381, + 366 + ], + "score": 1.0, + "content": "posterior gets stuck on the same reconstructed digit (3 in the figure).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "Visualization of the posterior After training a decoder with L2HMC, we randomly choose an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 141, + 393 + ], + "score": 1.0, + "content": "element", + "type": "text" + }, + { + "bbox": [ + 141, + 381, + 176, + 392 + ], + "score": 0.91, + "content": "x _ { 0 } \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "and run 512 parallel L2HMC chains for 20, 000 Metropolis-Hastings steps. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "then find the direction of highest variance, project the samples along that direction and show a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "score": 1.0, + "content": "histogram in Figure 3b. This plot shows non-Gaussianity in the latent space for the posterior. Using", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "our improved sampler enables the decoder to make use of a more expressive posterior, and enables", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 332, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 332, + 438 + ], + "score": 1.0, + "content": "the encoder to sample from this non-Gaussian posterior.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 205, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 207, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 207, + 481 + ], + "score": 1.0, + "content": "6 FUTURE WORK", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "The loss in Section 4.2 targets lag-one autocorrelation. It should be possible to extend this to also", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "target lag-two and higher autocorrelations. It should also be possible to extend this loss to reward", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "fast decay in the autocorrelation of other statistics of the samples, for instance the sample energy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "as well as the sample position. These additional statistics could also include learned statistics of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "the samples, combining benefits of the adversarial approach of (Song et al., 2017) with the current", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 132, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 132, + 567 + ], + "score": 1.0, + "content": "work.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "Our learned generalization of HMC should prove complementary to several other research directions", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "related to HMC. It would be interesting to explore combining our work with the use of HMC in a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "minibatch setting (Chen et al., 2014); with shadow Hamiltonians (Izaguirre & Hampton, 2004); with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "gradient pre-conditioning approaches similar to those used in Riemannian HMC (Girolami et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "2009; Betancourt, 2013); with the use of alternative HMC accept-reject rules (Sohl-Dickstein et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "2014; Berger et al., 2015); with the use of non-canonical Hamiltonian dynamics (Tripuraneni et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "2016); with variants of AIS adapted to HMC proposals (Sohl-Dickstein & Culpepper, 2012); with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 647, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 662 + ], + "score": 1.0, + "content": "the extension of HMC to discrete state spaces (Zhang et al., 2012); and with the use of alternative", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 388, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 388, + 672 + ], + "score": 1.0, + "content": "Hamiltonian integrators (Creutz & Gocksch, 1989; Chao et al., 2015).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Finally, our work is also complementary to other methods not utilizing gradient information. For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "example, we could incorporate the intuition behind Multiple Try Metropolis schemes (Martino &", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Read, 2013) by having several parametric operators and training each one when used. In addition,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "one could draw inspiration from the adaptive literature (Haario et al., 2001; Andrieu & Thoms, 2008)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 316, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 316, + 733 + ], + "score": 1.0, + "content": "or component-wise strategies (Gilks & Wild, 1992).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + 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For-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 289, + 509, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 147, + 313 + ], + "score": 1.0, + "content": "mally, let", + "type": "text" + }, + { + "bbox": [ + 147, + 296, + 159, + 306 + ], + "score": 0.85, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 289, + 415, + 313 + ], + "score": 1.0, + "content": "be the starting image. 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We then set", + "type": "text" + }, + { + "bbox": [ + 432, + 307, + 486, + 320 + ], + "score": 0.91, + "content": "x _ { t + 1 } ^ { \\mathrm { t o p } } = \\tilde { x } ^ { \\mathrm { t o p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 302, + 507, + 324 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 313, + 510, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 185, + 333 + ], + "score": 0.93, + "content": "x _ { t + 1 } ^ { \\mathrm { b o t t o m } } = x _ { 0 } ^ { \\mathrm { b o t t o m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 313, + 396, + 344 + ], + "score": 1.0, + "content": ". We compare the results obtained by sampling from ior) vs. our trained sampler. The results are reported i", + "type": "text" + }, + { + "bbox": [ + 396, + 320, + 432, + 332 + ], + "score": 0.92, + "content": "p ( z | x ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 313, + 458, + 344 + ], + "score": 1.0, + "content": "using a. We", + "type": "text" + }, + { + "bbox": [ + 458, + 322, + 470, + 333 + ], + "score": 0.85, + "content": "q _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 313, + 510, + 344 + ], + "score": 1.0, + "content": "(i.e. the see that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "L2HMC easily mixes between modes (3, 5, 8, and plausibly 9 in the figure) while the approximate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 381, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 381, + 366 + ], + "score": 1.0, + "content": "posterior gets stuck on the same reconstructed digit (3 in the figure).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 270, + 510, + 366 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "Visualization of the posterior After training a decoder with L2HMC, we randomly choose an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 141, + 393 + ], + "score": 1.0, + "content": "element", + "type": "text" + }, + { + "bbox": [ + 141, + 381, + 176, + 392 + ], + "score": 0.91, + "content": "x _ { 0 } \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "and run 512 parallel L2HMC chains for 20, 000 Metropolis-Hastings steps. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "then find the direction of highest variance, project the samples along that direction and show a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "score": 1.0, + "content": "histogram in Figure 3b. This plot shows non-Gaussianity in the latent space for the posterior. Using", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "our improved sampler enables the decoder to make use of a more expressive posterior, and enables", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 332, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 332, + 438 + ], + "score": 1.0, + "content": "the encoder to sample from this non-Gaussian posterior.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 369, + 506, + 438 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 205, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 207, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 207, + 481 + ], + "score": 1.0, + "content": "6 FUTURE WORK", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "The loss in Section 4.2 targets lag-one autocorrelation. It should be possible to extend this to also", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "target lag-two and higher autocorrelations. It should also be possible to extend this loss to reward", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "fast decay in the autocorrelation of other statistics of the samples, for instance the sample energy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "as well as the sample position. These additional statistics could also include learned statistics of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "the samples, combining benefits of the adversarial approach of (Song et al., 2017) with the current", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 132, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 132, + 567 + ], + "score": 1.0, + "content": "work.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 500, + 506, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "Our learned generalization of HMC should prove complementary to several other research directions", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "related to HMC. It would be interesting to explore combining our work with the use of HMC in a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "minibatch setting (Chen et al., 2014); with shadow Hamiltonians (Izaguirre & Hampton, 2004); with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "gradient pre-conditioning approaches similar to those used in Riemannian HMC (Girolami et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "2009; Betancourt, 2013); with the use of alternative HMC accept-reject rules (Sohl-Dickstein et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "2014; Berger et al., 2015); with the use of non-canonical Hamiltonian dynamics (Tripuraneni et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "2016); with variants of AIS adapted to HMC proposals (Sohl-Dickstein & Culpepper, 2012); with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 647, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 662 + ], + "score": 1.0, + "content": "the extension of HMC to discrete state spaces (Zhang et al., 2012); and with the use of alternative", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 388, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 388, + 672 + ], + "score": 1.0, + "content": "Hamiltonian integrators (Creutz & Gocksch, 1989; Chao et al., 2015).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 571, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Finally, our work is also complementary to other methods not utilizing gradient information. For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "example, we could incorporate the intuition behind Multiple Try Metropolis schemes (Martino &", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Read, 2013) by having several parametric operators and training each one when used. 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The architectures are the same. Let us take the example of", + "type": "text" + }, + { + "bbox": [ + 363, + 565, + 406, + 576 + ], + "score": 0.92, + "content": "Q _ { v } , S _ { v } , T _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 565, + 466, + 577 + ], + "score": 1.0, + "content": ". 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Results are reported in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "Table 1. L2HMC largely outperforms LAHMC on all task. LAHMC is also unable to mix between", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "modes for the MoG task. We also note that L2HMC could be easily combined with LAHMC, by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 420, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 420, + 444 + ], + "score": 1.0, + "content": "replacing the leapfrog integrator of LAHMC with the learned one of L2HMC.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "table", + "bbox": [ + 134, + 453, + 474, + 522 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 134, + 453, + 474, + 522 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 134, + 453, + 474, + 522 + ], + "spans": [ + { + "bbox": [ + 134, + 453, + 474, + 522 + ], + "score": 0.979, + "html": "
DistributionGradient EvaluationsESS-L2HMCESS-LAHMCRatio
50-d ICG2000156.621.47.3
Rough Well20012.58.61.5
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DistributionGradient EvaluationsESS-L2HMCESS-LAHMCRatio
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+ 105, + 645, + 434, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 434, + 657 + ], + "score": 1.0, + "content": "with 10 leapfrogs per step and choose the step size for an acceptance rate of 0.65.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 633, + 505, + 657 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 670, + 210, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 212, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 212, + 683 + ], + "score": 1.0, + "content": "D.3 MNIST SAMPLES", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 690, + 504, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 689, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 704 + ], + "score": 1.0, + "content": "We show in Figure 5 samples from the three evaluated models: VAE (Kingma & Welling, 2013),", + "type": "text" + } + ], + "index": 37 + }, + { + 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Input:Energy function U :X →R and its gradient VxU : X →X, initial distribution over the augmented state space q, number of iterations niters, number of leapfrogs M,learning rate schedule (αt)t≤niters ,batch size N,scale parameter 入and regularization strength 入b. Initialize the parameters of the sampler 0.
Initialize {)≤N from q(E）. fort = 0to niters-1do
Sample a minibatch {ε@)}i≤N from q(ε).
L←0 fori=1toNdo
郎 ←Rp
L←L+ex（),FLθsβ），A（FLθsβ)s{）)）+λblx（s,，FLθs(¹），A(FL)|s() ←FLθ with probabit A(FLθs(0)().
end for θ←θ-αtVθL end for
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0000000000000000000000000000000000000000..9dc43fb180ee09ceb36f7e592efb06c720d9eab5 --- /dev/null +++ b/parse/train/BkbY4psgg/BkbY4psgg.md @@ -0,0 +1,623 @@ +# MAKING NEURAL PROGRAMMING ARCHITECTURES GENERALIZE VIA RECURSION + +Jonathon Cai, Richard Shin, Dawn Song +Department of Computer Science +University of California, Berkeley +Berkeley, CA 94720, USA +{jonathon,ricshin,dawnsong}@cs.berkeley.edu + +# ABSTRACT + +Empirically, neural networks that attempt to learn programs from data have exhibited poor generalizability. Moreover, it has traditionally been difficult to reason about the behavior of these models beyond a certain level of input complexity. In order to address these issues, we propose augmenting neural architectures with a key abstraction: recursion. As an application, we implement recursion in the Neural Programmer-Interpreter framework on four tasks: grade-school addition, bubble sort, topological sort, and quicksort. We demonstrate superior generalizability and interpretability with small amounts of training data. Recursion divides the problem into smaller pieces and drastically reduces the domain of each neural network component, making it tractable to prove guarantees about the overall system’s behavior. Our experience suggests that in order for neural architectures to robustly learn program semantics, it is necessary to incorporate a concept like recursion. + +# 1 INTRODUCTION + +Training neural networks to synthesize robust programs from a small number of examples is a challenging task. The space of possible programs is extremely large, and composing a program that performs robustly on the infinite space of possible inputs is difficult—in part because it is impractical to obtain enough training examples to easily disambiguate amongst all possible programs. Nevertheless, we would like the model to quickly learn to represent the right semantics of the underlying program from a small number of training examples, not an exhaustive number of them. + +Thus far, to evaluate the efficacy of neural models on programming tasks, the only metric that has been used is generalization of expected behavior to inputs of greater complexity (Vinyals et al. (2015), Kaiser & Sutskever (2015), Reed & de Freitas (2016), Graves et al. (2016), Zaremba et al. (2016)). For example, for the addition task, the model is trained on short inputs and then tested on its ability to sum inputs with much longer numbers of digits. Empirically, existing models suffer from a common limitation—generalization becomes poor beyond a threshold level of complexity. Errors arise due to undesirable and uninterpretable dependencies and associations the architecture learns to store in some high-dimensional hidden state. This makes it difficult to reason about what the model will do when given complex inputs. + +One common strategy to improve generalization is to use curriculum learning, where the model is trained on inputs of gradually increasing complexity. However, models that make use of this strategy eventually fail after a certain level of complexity (e.g. the single-digit multiplication task in Zaremba et al. (2016), the bubble sort task in Reed & de Freitas (2016), and the graph tasks in Graves et al. (2016)). In this version of curriculum learning, even though the inputs are gradually becoming more complex, the semantics of the program is succinct and does not change. Although the model is exposed to more and more data, it might learn spurious and overly complex representations of the program, as suggested in Zaremba et al. (2016). That is to say, the network does not learn the true program semantics. + +In this paper, we propose to resolve these issues by explicitly incorporating recursion into neural architectures. Recursion is an important concept in programming languages and a critical tool to reduce the complexity of programs. We find that recursion makes it easier for the network to learn the right program and generalize to unknown situations. Recursion enables provable guarantees on neural programs’ behavior without needing to exhaustively enumerate all possible inputs to the programs. This paper is the first (to our knowledge) to investigate the important problem of provable generalization properties of neural programs. As an application, we incorporate recursion into the Neural Programmer-Interpreter architecture and consider four sample tasks: grade-school addition, bubble sort, topological sort, and quicksort. Empirically, we observe that the learned recursive programs solve all valid inputs with $100 \%$ accuracy after training on a very small number of examples, out-performing previous generalization results. Given verification sets that cover all the base cases and reduction rules, we can provide proofs that these learned programs generalize perfectly. This is the first time one can provide provable guarantees of perfect generalization for neural programs. + +# 2 THE PROBLEM AND OUR APPROACH + +# 2.1 THE PROBLEM OF GENERALIZATION + +When constructing a neural network for the purpose of learning a program, there are two orthogonal aspects to consider. The first is the actual model architecture. Numerous models have been proposed for learning programs; to name a few, this includes the Differentiable Neural Computer (Graves et al., 2016), Neural Turing Machine (Graves et al., 2014), Neural GPU (Kaiser & Sutskever, 2015), Neural Programmer (Neelakantan et al., 2015), Pointer Network (Vinyals et al., 2015), Hierarchical Attentive Memory (Andrychowicz & Kurach, 2016), and Neural Random Access Machine (Kurach et al., 2016). The architecture usually possesses some form of memory, which could be internal (such as the hidden state of a recurrent neural network) or external (such as a discrete “scratch pad” or a memory block with differentiable access). The second is the training procedure, which consists of the form of the training data and the optimization process. Almost all architectures train on program input/output pairs. The only model, to our knowledge, that does not train on input-output pairs is the Neural Programmer-Interpreter (Reed & de Freitas, 2016), which trains on synthetic execution traces. + +To evaluate a neural network that learns a neural program to accomplish a certain task, one common evaluation metric is how well the learned model $M$ generalizes. More specifically, when $M$ is trained on simpler inputs, such as inputs of a small length, the generalization metric evaluates how well $M$ will do on more complex inputs, such as inputs of much longer length. $M$ is considered to have perfect generalization if $M$ can give the right answer for any input, such as inputs of arbitrary length. + +As mentioned in Section 1, all approaches to neural programming today fare poorly on this generalization issue. We hypothesize that the reason for this is that the neural network learns to spuriously depend on specific characteristics of the training examples that are irrelevant to the true program semantics, such as length of the training inputs, and thus fails to generalize to more complex inputs. + +In addition, none of the current approaches to neural programming provide a method or even aim to enable provable guarantees about generalization. The memory updates of these neural programs are so complex and interdependent that it is difficult to reason about the behaviors of the learned neural program under previously unseen situations (such as problems with longer inputs). This is highly undesirable, since being able to provide the correct answer in all possible settings is one of the most important aspects of any learned neural program. + +# 2.2 OUR APPROACH USING RECURSION + +In this paper, we propose that the key abstraction of recursion is necessary for neural programs to generalize. The general notion of recursion has been an important concept in many domains, including mathematics and computer science. In computer science, recursion (as opposed to iteration) involves solving a larger problem by combining solutions to smaller instances of the same problem. Formally, a function exhibits recursive behavior when it possesses two properties: (1) Base cases— terminating scenarios that do not use recursion to produce answers; (2) A set of rules that reduces all other problems toward the base cases. Some functional programming languages go so far as not to define any looping constructs but rely solely on recursion to enable repeated execution of the same code. + +In this paper, we propose that recursion is an important concept for neural programs as well. In fact, we argue that recursion is an essential element for neural programs to generalize, and makes it tractable to prove the generalization of neural programs. Recursion can be implemented differently for different neural programming models. Here as a concrete and general example, we consider a general Neural Programming Architecture (NPA), similar to Neural Programmer-Interpreter (NPI) in Reed & de Freitas (2016). In this architecture, we consider a core controller, e.g., an LSTM in NPI’s case, but possibly other networks in different cases. There is a (changing) list of neural programs used to accomplish a given task. The core controller acts as a dispatcher for the programs. At each time step, the core controller can decide to select one of the programs to call with certain arguments. When the program is called, the current context including the caller’s memory state is stored on a stack; when the program returns, the stored context is popped off the stack to resume execution in the previous caller’s context. + +In this general Neural Programming Architecture, we show it is easy to support recursion. In particular, recursion can be implemented as a program calling itself. Because the context of the caller is stored on a stack when it calls another program and the callee starts in a fresh context, this enables recursion simply by allowing a program to call itself. In practice, we can additionally use tail recursion optimization to avoid problems with the call stack growing too deep. Thus, any general Neural Programming Architecture supporting such a call structure can be made to support recursion. In particular, this condition is satisfied by NPI, and thus the NPI model naturally supports recursion (even though the authors of NPI did not consider this aspect explicitly). + +By nature, recursion reduces the complexity of a problem to simpler instances. Thus, recursion helps decompose a problem and makes it easier to reason about a program’s behavior for previously unseen situations such as longer inputs. In particular, given that a recursion is defined by two properties as mentioned before, the base cases and the set of reduction rules, we can prove a recursive neural program generalizes perfectly if we can prove that (1) it performs correctly on the base cases; (2) it learns the reduction rules correctly. For many problems, the base cases and reduction rules usually consist of a finite (often small) number of cases. For problems where the base cases may be extremely large or infinite, such as certain forms of motor control, recursion can still help reduce the problem of generalization to these two aspects and make the generalization problem significantly simpler to handle and reason about. + +As a concrete instantiation, we show in this paper that we can enable recursive neural programs in the NPI model, and thus enable perfectly generalizable neural programs for tasks such as sorting where the original, non-recursive NPI program fails. As aforementioned, the NPI model naturally supports recursion. However, the authors of NPI did not consider explicitly the notion of recursion and as a consequence, did not learn recursive programs. We show that by modifying the training procedure, we enable the NPI model to learn recursive neural programs. As a consequence, our learned neural programs empirically achieve perfect generalization from a very small number of training examples. Furthermore, given a verification input set that covers all base cases and reduction rules, we can formally prove that the learned neural programs achieve perfect generalization after verifying its behavior on the verification input set. This is the first time one can provide provable guarantees of perfect generalization for neural programs. + +We would also like to point out that in this paper, we provide as an example one way to train a recursive neural program, by providing a certain training execution trace to the NPI model. However, our concept of recursion for neural programs is general. In fact, it is one of our future directions to explore new ways to train a recursive neural program without providing explicit training execution traces or with only partial or non-recursive traces. + +# 3 APPLICATION TO LEARNING RECURSIVE NEURAL PROGRAMS WITH NPI + +# 3.1 BACKGROUND: NPI ARCHITECTURE + +As discussed in Section 2, the Neural Programmer-Interpreter (NPI) is an instance of a Neural Programmer Architecture and hence it naturally supports recursion. In this section, we give a brief review of the NPI architecture from Reed & de Freitas (2016) as background. + +We describe the details of the NPI model relevant to our contributions. We adapt machinery from the original paper slightly to fit our needs. The NPI model has three learnable components: a task-agnostic core, a program-key embedding, and domain-specific encoders that allow the NPI to operate in diverse environments. + +The NPI accesses an external environment, $Q$ , which varies according to the task. The core module of the NPI is an LSTM controller that takes as input a slice of the current external environment, via a set of pointers, and a program and arguments to execute. NPI then outputs the return probability and next program and arguments to execute. Formally, the NPI is represented by the following set of equations: + +$$ +\begin{array} { c } { s _ { t } = f _ { e n c } ( e _ { t } , a _ { t } ) } \\ { h _ { t } = f _ { l s t m } ( s _ { t } , p _ { t } , h _ { t - 1 } ) } \\ { r _ { t } = f _ { e n d } ( h _ { t } ) , p _ { t + 1 } = f _ { p r o g } ( h _ { t } ) , a _ { t + 1 } = f _ { a r g } ( h _ { t } ) } \end{array} +$$ + +$t$ is a subscript denoting the time-step; $f _ { e n c }$ is a domain-specific encoder (to be described later) that takes in the environment slice $e _ { t }$ and arguments $a _ { t }$ ; $f _ { l s t m }$ represents the core module, which takes in the state $s _ { t }$ generated by $f _ { e n c }$ , a program embedding $p _ { t } \in \mathbb { R } ^ { P }$ , and hidden LSTM state $h _ { t }$ ; $f _ { e n d }$ decodes the return probability $r _ { t }$ ; $f _ { p r o g }$ decodes a program key embedding $p _ { t + 1 }$ ;1 and $f _ { a r g }$ decodes arguments $a _ { t + 1 }$ . The outputs $r _ { t } , p _ { t + 1 } , a _ { t + 1 }$ are used to determine the next action, as described in Algorithm 1. If the program is primitive, the next environmental state $e _ { t + 1 }$ will be affected by $p _ { t }$ and $a _ { t }$ , i.e. $e _ { t + 1 } \sim f _ { e n v } ( e _ { t } , p _ { t } , a _ { t } )$ . As with the original NPI architecture, the experiments for this paper always used a 3-tuple of integers $a _ { t } = ( a _ { t } ( 1 ) , \hat { a } _ { t } ( 2 ) , a _ { t } ( 3 ) )$ . + +Algorithm 1 Neural programming inference + +
1:Inputs: Environment observation e, program p,arguments a, stop threshold α
2:function RUN(e,p, a)
3:h↑0,r←0
4:whiler<αdo
5:s ←fenc(e,a),h ←fistm(s,p,h)
6:r ←fend(h),p2 ← fprog(h),a2 ← farg(h)
7:if p is a primitive function then
8:e ← fenv(e,p,a).
9:else
10:function RUN(e,P2, a2)
+ +A description of the inference procedure is given in Algorithm 1. Each step during an execution of the program does one of three things: (1) another subprogram along with associated arguments is called, as in Line 10, (2) the program writes to the environment if it is primitive, as in Line 8, or (3) the loop is terminated if the return probability exceeds a threshold $\alpha$ , after which the stack frame is popped and control is returned to the caller. In all experiments, $\alpha$ is set to 0.5. Each time a subprogram is called, the stack depth increases. + +The training data for the Neural Programmer-Interpreter consists of full execution traces for the program of interest. A single element of an execution trace consists of a step input-step output pair, which can be synthesized from Algorithm 1: this corresponds to, for a given time-step, the step input tuple $( e , p , a )$ and step output tuple $( r , p _ { 2 } , a _ { 2 } )$ . An example of part of an addition task trace, written in shorthand, is given in Figure 1. For example, a step input-step output pair in Lines 2 and 3 of the left-hand side of Figure 1 is (ADD1, WRITE OUT 1). In this pair, the step input runs a subprogram ADD1 that has no arguments, and the step output contains a program WRITE that has arguments of OUT and 1. The environment and return probability are omitted for readability. Indentation indicates the stack is one level deeper than before. + +It is important to emphasize that at inference time in the NPI, the hidden state of the LSTM controller is reset (to zero) at each subprogram call, as in Line 3 of Algorithm 1 $h \mathbf { 0 }$ ). This functionality is critical for implementing recursion, since it permits us to restrict our attention to the currently relevant recursive call, ignoring irrelevant details about other contexts. + +![](images/400738f1bb6ec680e00398081df8b91a36e097048042ec5fdd599e1e2221cb5f.jpg) +Figure 1: Addition Task. The non-recursive trace loops on cycles of ADD1 and LSHIFT, whereas in the recursive version, the ADD function calls itself (bolded). + +# 3.2 RECURSIVE FORMULATIONS FOR NPI PROGRAMS + +We emphasize the overall goal of this work is to enable the learning of a recursive program. The learned recursive program is different from neural programs learned in all previous work in an important aspect: previous approaches do not explicitly incorporate this abstraction, and hence generalize poorly, whereas our learned neural programs incorporate recursion and achieve perfect generalization. + +Since NPI naturally supports the notion of recursion, a key question is how to enable NPI to learn recursive programs. We found that changing the NPI training traces is a simple way to enable this. In particular, we construct new training traces which explicitly contain recursive elements and show that with this type of trace, NPI easily learns recursive programs. In future work, we would like to decrease supervision and construct models that are capable of coming up with recursive abstractions themselves. + +In what follows, we describe the way in which we constructed NPI training traces so as to make them contain recursive elements and thus enable NPI to learn recursive programs. We describe the recursive re-formulation of traces for two tasks from the original NPI paper—grade-school addition and bubble sort. For these programs, we re-use the appropriate program sets (the associated subprograms), and we refer the reader to the appendix of Reed & de Freitas (2016) for further details on the subprograms used in addition and bubble sort. Finally, we implement recursive traces for our own topological sort and quicksort tasks. + +Grade School Addition. For grade-school addition, the domain-specific encoder is + +$$ +\begin{array} { r } { f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , i _ { 4 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } ) , Q ( 4 , i _ { 4 } ) , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \end{array} +$$ + +where the environment $Q \in \mathbb { R } ^ { 4 \times N \times K }$ is a scratch-pad that contains four rows (the first input number, the second input number, the carry bits, and the output) and $N$ columns. $K$ is set to 11, to represent the range of 10 possible digits, along with a token representing the end of input.2 At any given time, the NPI has access to values pointed to by four pointers in each of the four rows, represented by $Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } )$ , and $Q ( 4 , i _ { 4 } )$ . + +The non-recursive trace loops on cycles of ADD1 and LSHIFT. ADD1 is a subprogram that adds the current column (writing the appropriate digit to the output row and carrying a bit to the next column if needed). LSHIFT moves the four pointers to the left, to move to the next column. The program terminates when seeing no numbers in the current column. + +Figure 1 shows examples of non-recursive and recursive addition traces. We make the trace recursive by adding a tail recursive call into the trace for the ADD program after calling ADD1 and LSHIFT, + +# Full Recursive + +![](images/54ac005719045e00f6d938466b5b91cbc2d9b186869b92dd8c3b93eb2423cdb1.jpg) +Figure 2: Bubble Sort Task. The non-recursive trace loops on cycles of BUBBLE and RESET. The difference between the partial recursive and full recursive versions is in the indentation of Lines 10-15 and 20-22 (bolded), since in the full recursive version, BSTEP and LSHIFT are made tail recursive; the final calls to BSTEP and LSHIFT return immediately as they occur after the pointer reaches the end of the array. Also note that COMPSWAP conditionally swaps numbers under the bubble pointers. + +as in Line 13 of the right-hand side of Figure 1. Via the recursive call, we effectively forget that the column just added exists, since the recursive call to ADD starts with a new hidden state for the LSTM controller. Consequently, there is no concept of length relevant to the problem, which has traditionally been an important focus of length-based curriculum learning. + +Bubble Sort. For bubble sort, the domain-specific encoder is + +$$ +f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) , Q ( 1 , i _ { 2 } ) , i _ { 3 } = l e n g t h , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , +$$ + +where the environment $Q \in \mathbb { R } ^ { 1 \times N \times K }$ is a scratch-pad that contains 1 row, to represent the state of the array as sorting proceeds in-place, and $N$ columns. $K$ is set to 11, to denote the range of possible numbers (0 through 9), along with the start/end token (represented with the same encoding) which is observed when a pointer reaches beyond the bounds of the input. At any given time, the NPI has access to the values referred to by two pointers, represented by $Q ( 1 , i _ { 1 } )$ and $Q ( 1 , i _ { 2 } )$ ,. The pointers at index $i _ { 1 }$ and $i _ { 2 }$ are used to compare the pair of numbers considered during the bubble sweep, swapping them if the number at $i _ { 1 }$ is greater than that in $i _ { 2 }$ . These pointers are referred to as bubble pointers. The pointer at index $i _ { 3 }$ represents a counter internal to the environment that is incremented once after each pass of the algorithm (one cycle of BUBBLE and RESET); when incremented a number of times equal to the length of the array, the flag $i _ { 3 } = =$ length becomes true and terminates the entire algorithm . + +The non-recursive trace loops on cycles of BUBBLE and RESET, which logically represents one bubble sweep through the array and reset of the two bubble pointers to the very beginning of the array, respectively. In this version, there is a dependence on length: BSTEP and LSHIFT are called a number of times equivalent to one less than the length of the input array, in BUBBLE and RESET respectively. + +Inside BUBBLE and RESET, there are two operations that can be made recursive. BSTEP, used in BUBBLE, compares pairs of numbers, continuously moving the bubble pointers once to the right each time until reaching the end of the array. LSHIFT, used in RESET, shifts the pointers left until reaching the start token. + +We experiment with two levels of recursion—partial and full. Partial recursion only adds a tail recursive call to BUBBLESORT after BUBBLE and RESET, similar to the tail recursive call described previously for addition. The partial recursion is not enough for perfect generalization, as will be presented later in Section 4. Full recursion, in addition to making the aforementioned tail recursive call, adds two additional recursive calls; BSTEP and LSHIFT are made tail recursive. Figure 2 shows examples of traces for the different versions of bubble sort. Training on the full recursive trace leads to perfect generalization, as shown in Section 4. We performed experiments on the partially recursive version in order to examine what happens when only one recursive call is implemented, when in reality three are required for perfect generalization. + +# Algorithm 2 Depth First Search Topological Sort + +1: Color all vertices white. +2: Initialize an empty stack $S$ and a directed acyclic graph $D A G$ to traverse. 3: Begin traversing from Vertex 1 in the DAG. 4: function TOPOSORT $( D A G )$ 5: while there is still a white vertex $u$ : do 6: color[u] = grey 7: $v _ { a c t i v e } = u$ 8: do +9: if $v _ { a c t i v e }$ has a white child $v$ then +10: $\operatorname { c o l o r } [ v ] = \operatorname { g r e y }$ +11: push $v _ { a c t i v e }$ onto $S$ +12: $v _ { a c t i v e } = v$ +13: else +14: $\mathrm { c o l o r } [ v _ { a c t i v e } ] = \mathrm { b l a c k }$ +15: Write $v _ { a c t i v e }$ to result +16: if $S$ is empty then pass +17: else pop the top vertex off $S$ and set it to $v _ { a c t i v e }$ +18: while $S$ is not empty + +Topological Sort. We choose to implement a topological sort task for graphs. A topological sort is a linear ordering of vertices such that for every directed edge $( u , v )$ from $u$ to $v , u$ comes before $v$ in the ordering. This is possible if and only if the graph has no directed cycles; that is to say, it must be a directed acyclic graph (DAG). In our experiments, we only present DAG’s as inputs and represent the vertices as values ranging from $1 , \ldots , n$ , where the DAG contains $n$ vertices. + +Directed acyclic graphs are structurally more diverse than inputs in the two tasks of grade-school addition and bubble sort. The degree for any vertex in the DAG is variable. Also the DAG can have potentially more than one connected component, meaning it is necessary to transition between these components appropriately. + +Algorithm 2 shows the topological sort task of interest. This algorithm is a variant of depth first search. We created a program set that reflects the semantics of Algorithm 2. For brevity, we refer the reader to the appendix for further details on the program set and non-recursive and recursive trace-generating functions used for topological sort. + +For topological sort, the domain-specific encoder is + +$$ +\begin{array} { r l } & { \ f _ { e n c } ( D A G , Q _ { c o l o r } , p _ { s t a c k } , p _ { s t a r t } , v _ { a c t i v e } , c h i l d L i s t , a _ { t } ) } \\ & { = M L P ( [ Q _ { c o l o r } ( p _ { s t a r t } ) , Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ) , p _ { s t a c k } = = 1 , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) , a _ { t } ) } \end{array} +$$ + +where $Q _ { c o l o r } \in \mathbb { R } ^ { U \times 4 }$ is a scratch-pad that contains $U$ rows, each containing one of four colors (white, gray, black, invalid) with one-hot encoding. $U$ varies with the number of vertices in the graph. We further have $Q _ { r e s u l t } \in \mathbb { N } ^ { U }$ , a scratch-pad which contains the sorted list of vertices at the end of execution, and $Q _ { s t a c k } \in \mathbb { N } ^ { U }$ , which serves the role of the stack $S$ in Algorithm 2. The contents of $Q _ { r e s u l t }$ and $Q _ { s t a c k }$ are not exposed directly through the domain-specific encoder; rather, we define primitive functions which manipulate these scratch-pads. + +The DAG is represented as an adjacency list where $D A G [ i ] [ j ]$ refers to the $j$ -th child of vertex $i$ . There are 3 pointers $( p _ { r e s u l t } , p _ { s t a c k } , p _ { s t a r t } )$ , $p _ { r e s u l t }$ points to the next empty location in $Q _ { r e s u l t }$ $p _ { s t a c k }$ points to the top of the stack in $Q _ { s t a c k }$ , and $p _ { s t a r t }$ points to the candidate starting node for a connected component. There are 2 variables ( $\boldsymbol { v } _ { a c t i v e }$ and $v _ { s a v e . }$ ); $v _ { a c t i v e }$ holds the active vertex (as in Algorithm 2) and $v _ { s a v e }$ holds the value of $v _ { a c t i v e }$ before executing Line 12 of Algorithm 2. $c h i l d L i s t \in \mathbb { N } ^ { U }$ is a vector of pointers, where childList[i] points to the next child under consideration for vertex $i$ . + +The three environment observations aid with control flow in Algorithm 2. $Q _ { c o l o r } ( p _ { s t a r t } )$ contains the color of the current start vertex, used in the evaluation of the condition in the WHILE loop in Line 5 of Algorithm 2. $Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] )$ refers to the color of the next child of $v _ { a c t i v e }$ , used in the evaluation of the condition in the IF branch in Line 9 of Algorithm 2. Finally, the boolean $p _ { s t a c k } = = 1$ is used to check whether the stack is empty in Line 18 of Algorithm 2. + +An alternative way of representing the environment slice is to expose the values of the absolute vertices to the model; however, this makes it difficult to scale the model to larger graphs, since large vertex values are not seen during training time. + +We refer the reader to the appendix for the non-recursive trace generating functions. In the non-recursive trace, there are four functions that can be made recursive—TOPOSORT, CHECK CHILD, EXPLORE, and NEXT START, and we add a tail recursive call to each of these functions in order to make the recursive trace. In particular, in the EXPLORE function, adding a tail recursive call resets and stores the hidden states associated with vertices in a stack-like fashion. This makes it so that we only need to consider the vertices in the subgraph that are currently relevant for computing the sort, allowing simpler reasoning about behavior for large graphs. The sequence of primitive operations (MOVE and WRITE operations) for the non-recursive and recursive versions are exactly the same. + +Quicksort. We implement a quicksort task, in order to demonstrate that recursion helps with learning divide-and-conquer algorithms. We use the Lomuto partition scheme; the logic for the recursive trace is shown in Algorithm 3. For brevity, we refer the reader to the appendix for information about the program set and non-recursive and recursive trace-generating functions for quicksort. The logic for the non-recursive trace is shown in Algorithm 4 in the appendix. + +# Algorithm 3 Recursive Quicksort + +1: Initialize an array $A$ to sort. +2: Initialize $l o$ and $h i$ to be 1 and $n$ , where $n$ is the length of $A$ . +3: +4: function QUICKSORT $( A , l o , h i )$ +5: if $l o < h i$ : then +6: $\mathsf { p } = \mathsf { P A R T I T I O N } ( A , l o , h i )$ +7: $\mathrm { Q U I C K S O R T } ( A , l o , p - 1 )$ +8: $\operatorname { Q U I C K S O R T } ( A , p + 1 , h i )$ +9: +10: function PARTITION $( A , l o , h i )$ +11: $p i v o t = l o$ +12: for $j \in [ l o , h i - 1 ] : \mathbf { d o }$ +13: if $A [ j ] \leq A [ h i ]$ then +14: swap A[pivot] with $A [ j ]$ +15: pivot = pivot + 1 +16: swap A[pivot] with A[hi] +17: return pivot + +For quicksort, the domain-specific encoder is + +$$ +\begin{array} { r l } & { f _ { e n c } ( Q _ { a r r a y } , Q _ { s t a c k L o } , Q _ { s t a c k H i } , p _ { l o } , p _ { h i } , p _ { s t a c k L o } , p _ { s t a c k H i } , p _ { p i v o t } , p _ { j } , a _ { t } ) = } \\ & { \qquad M L P ( [ Q _ { a r r a y } ( p _ { j } ) \leq Q _ { a r r a y } ( p _ { h i } ) , p _ { j } = = p _ { h i } , } \\ & { \qquad Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 ) , p _ { s t a c k L o } = = } \\ & { \qquad \quad M L P ( [ Q _ { a r r a y } ( p _ { j } ) \leq \mathfrak { a } _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \end{array} +$$ + +where $Q _ { a r r a y } \in \mathbb { R } ^ { U \times 1 1 }$ is a scratch-pad that contains $U$ rows, each containing one of 11 values (one of the numbers 0 through 9 or an invalid state). Our implementation uses two stacks $Q _ { s t a c k L o }$ and + +$Q _ { s t a c k H i }$ , each in $\mathbb { R } ^ { U }$ , that store the arguments to the recursive QUICKSORT calls in Algorithm 3; before each recursive call, the appropriate arguments are popped off the stack and written to $p _ { l o }$ and $p _ { h i }$ . + +There are 6 pointers $( p _ { l o } , p _ { h i } , p _ { s t a c k L o } , p _ { s t a c k H i } , p _ { p i v o t } , p _ { j } )$ . $p _ { l o }$ and $p _ { h i }$ point to the lo and hi indices of the array, as in Algorithm 3. $p _ { s t a c k L o }$ and $p _ { s t a c k H i }$ point to the top (empty) positions in $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . $p _ { p i v o t }$ and $p _ { j }$ point to the pivot and $j$ indices of the array, used in the PARTITION function in Algorithm 3. The 4 environment observations aid with control flow; $Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 )$ implements the $l o < h i$ comparison in Line 5 of Algorithm 3, $p _ { s t a c k L o } = = 1$ checks if the stacks are empty in Line 18 of Algorithm 4, and the other observations (all involving $p _ { p i v o t }$ or $p _ { j }$ ) deal with logic in the PARTITION function. + +Note that the recursion for quicksort is not purely tail recursive and therefore represents a more complex kind of recursion that is harder to learn than in the previous tasks. Also, compared to the bubble pointers in bubble sort, the pointers that perform the comparison for quicksort (the COMPSWAP function) are usually not adjacent to each other, making quicksort less local than bubble sort. In order to compensate for this, $p _ { p i v o t }$ and $p _ { j }$ require special functions (MOVE PIVOT LO and MOVE J LO) to properly set them to $l o$ in Lines 11 and 12 of the PARTITION function in Algorithm 3. + +# 3.3 PROVABLY PERFECT GENERALIZATION + +We show that if we incorporate recursion, the learned NPI programs can achieve provably perfect generalization for different tasks. Provably perfect generalization implies the model will behave correctly, given any valid input. In order to claim a proof, we must verify the model produces correct behavior over all base cases and reductions, as described in Section 2. + +We propose and describe our verification procedure. This procedure verifies that all base cases and reductions are handled properly by the model via explicit tests. Note that recursion helps make this process tractable, because we only need to test a finite number of inputs to show that the model will work correctly on inputs of unbounded complexity. This verification phase only needs to be performed once after training. + +Formally, verification consists of proving the following theorem: + +$$ +\forall i \in V , M ( i ) \downdownarrows P ( i ) +$$ + +where $i$ denotes a sequence of step inputs (within one function call), $V$ denotes the set of valid sequences of step inputs, $M$ denotes the neural network model, $P$ denotes the correct program, and $P ( i )$ denotes the next step output from the correct program. The arrow in the theorem refers to evaluation, as in big-step semantics. The theorem states that for the same sequence of step inputs, the model produces the exact same step output as the target program it aims to learn. $M$ , as described in Algorithm 1, processes the sequence of step inputs by using an LSTM. + +Recursion drastically reduces the number of configurations we need to consider during the verification phase and makes the proof tractable, because it introduces structure that eliminates infinitely long sequences of step inputs that would otherwise need to be considered. For instance, for recursive addition, consider the family $F$ of addition problems $a _ { n } a _ { n - 1 } \dots a _ { 1 } a _ { 0 } + b _ { n } b _ { n - 1 } \dots b _ { 1 } b _ { 0 }$ where no CARRY operations occur. We prove every member of $F$ is added properly, given that subproblems $S = \{ a _ { n } a _ { n - 1 } + b _ { n } b _ { n - 1 } , a _ { n - 1 } { \bar { a } } _ { n - 2 } + b _ { n - 1 } b _ { n - 2 } , \dots , a _ { 1 } a _ { 0 } + b _ { 1 } b _ { 0 } \}$ are added properly. + +Without using a recursive program, such a proof is not possible, because the non-recursive program runs on an arbitrarily long addition problem that creates correspondingly long sequences of step inputs; in the non-recursive formulation of addition, ADD calls ADD1 a number of times that is dependent on the length of the input. The core LSTM module’s hidden state is preserved over all these ADD1 calls, and it is difficult to interpret with certainty what happens over longer timesteps without concretely evaluating the LSTM with an input of that length. In contrast, each call to the recursive ADD always runs for a fixed number of steps, even on arbitrarily long problems in $F$ , so we can test that it performs correctly on a small, fixed number of step input sequences. This guarantees that the step input sequences considered during verification contain all step input sequences which arise during execution of an unseen problem in $F$ , leading to generalization to any problem in $F$ . Hence, if all subproblems in $S$ are added correctly, we have proven that any member of $F$ will be added correctly, thus eliminating an infinite family of inputs that need to be tested. + +To perform the verification as described here, it is critical to construct $V$ correctly. If it is too small, then execution of the program on some input might require evaluation of $M ( i )$ on some $i \not \in V$ , and so the behavior of $M ( i )$ might deviate from $P ( i )$ . If it is too large, then the semantics of $P$ might not be well-defined on some elements in $V$ , or the spurious step input sequences may not be reachable from any valid problem input (e.g., an array for quicksort or a DAG for topological sort). + +To construct this set, by using the reference implementation of each subprogram, we construct a mapping between two sets of environment observations: the first set consists of all observations that can occur at the beginning of a particular subprogram’s invocation, and the second set contains the observations at the end of that subprogram. We can obtain this mapping by first considering the possible observations that can arise at the beginning of the entry function (ADD, BUBBLESORT, TOPOSORT, and QUICKSORT) for some valid program input, and iteratively applying the observation-to-observation mapping implied by the reference implementation’s step output at that point in the execution. If the step output specifies a primitive function call, we need to reason about how it can affect the environment so as to change the observation in the next step input. For non-primitive subprograms, we can update the observation-to-observation mapping currently associated with the subprogram and then apply that mapping to the current set. By iterating with this procedure, and then running $P$ on the input observation set that we obtain for the entry point function, we can obtain $V$ precisely. To make an analogy to MDPs, this procedure is analogous to how value iteration obtains the correct value for each state starting from any initialization. + +An alternative method is to run $P$ on many different program inputs and then observe step input sequences which occur, to create $V$ . However, to be sure that the generated $V$ is complete (covers all the cases needed), we need to check all pairs of observations seen in adjacent step inputs (in particular, those before and after a primitive function call), in a similar way as if we were constructing $V$ from scratch. Given a precise definition of $P$ , it may be possible to automate the generation of $V$ from $P$ in future work. + +Note that $V$ should also contain the necessary reductions, which corresponds to making the recursive calls at the correct time, as indicated by $P$ . + +After finding $V$ , we construct a set of problem inputs which, when executed on $P$ , create exactly the step input sequences which make up $V$ . We call this set of inputs the verification set, $S _ { V }$ . + +Given a verification set, we can then run the model on the verification set to check if the produced traces and results are correct. If yes, then this indicates that the learned neural program achieves provably perfect generalization. + +We note that for tasks with very large input domains, such as ones involving MNIST digits or speech samples, the state space of base cases and reduction rules could be prohibitively large, possibly infinite. Consequently, it is infeasible to construct a verification set that covers all cases, and the verification procedure we have described is inadequate. We leave this as future work to devise a verification procedure more appropriate to this setting. + +# 4 EXPERIMENTS + +As there is no public implementation of NPI, we implemented a version of it in Keras that is as faithful to the paper as possible. Our experiments use a small number of training examples. + +Training Setup. The training set for addition contains 200 traces. The maximum problem length in this training set is 3 (e.g., the trace corresponding to the problem $^ { \mathrm { \left. } } 1 0 9 + 1 0 1 ^ { \mathrm { \right. } }$ ). + +The training set for bubble sort contains 100 traces, with maximum problem length of 2 (e.g., the trace corresponding to the array [3,2]). + +The training set for topological sort contains 6 traces, with one synthesized from a graph of size 5 and the rest synthesized from graphs of size 7. + +The training set for quicksort contains 4 traces, synthesized from arrays of length 5. + +The same set of problems was used to generate the training traces for all formulations of the task, for non-recursive and recursive versions. + +Table 1: Accuracy on Randomly Generated Problems for Bubble Sort + +
Length of ArrayNon-RecursivePartiallyRecursiveFullRecursive
２3100%100%100%
6.7%23%100%
410%10%100%
80% 0%0% 0%100% 100%
20100%
900%0%
+ +We train using the Adam optimizer and use a 2-layer LSTM and task-specific state encoders for the external environments, as described in Reed & de Freitas (2016). + +4.1 RESULTS ON GENERALIZATION OF RECURSIVE NEURAL PROGRAMS + +We now report on generalization for the varying tasks. + +Grade-School Addition. Both the non-recursive and recursive learned programs generalize on all input lengths we tried, up to 5000 digits. This agrees with the generalization of non-recursive addition in Reed & de Freitas (2016), where they reported generalization up to 3000 digits. However, note that there is no provable guarantee that the non-recursive learned program will generalize to all inputs, whereas we show later that the recursive learned program has a provable guarantee of perfect generalization. + +In order to demonstrate that recursion can help learn and generalize better, for addition, we trained only on traces for 5 arbitrarily chosen 1-digit addition sum examples. The recursive version can generalize perfectly to long problems constructed from these components (such as the sum $^ { 6 6 } 8 2 2 + 2 3 3 ^ { 3 }$ , where $\because 8 + 2 ^ { , 5 }$ and $\bar { 2 } + 3 \bar { 2 }$ are in the training set), but the non-recursive version fails to sum these long problems properly. + +Bubble Sort. Table 1 presents results on randomly generated arrays of varying length for the learned non-recursive, partially recursive, and full recursive programs. For each length, we test each program on 30 randomly generated problems. Observe that partially recursive does slightly better than non-recursive for the setting in which the length of the array is 3, and that the fully recursive version is able to sort every array given to it. The non-recursive and partially recursive versions are unable to sort long arrays, beyond length 8. + +Topological Sort. Both the non-recursive and recursive learned programs generalize on all graphs we tried, up to 120 vertices. As before, the non-recursive learned program lacks a provable guarantee of generalization, whereas we show later that the recursive learned program has one. + +In order to demonstrate that recursion can help learn and generalize better, we trained a non-recursive and recursive model on just a single execution trace generated from a graph containing 5 nodes3 for the topological sort task. For these models, Table 2 presents results on randomly generated DAGs of varying graph sizes (varying in the number of vertices). For each graph size, we test the learned programs on 30 randomly generated DAGs. The recursive version of topological sort solves all graph instances we tried, from graphs of size 5 through 70. On the other hand, the non-recursive version has low accuracy, beginning from size 5, and fails completely for graphs of size 8 and beyond. + +Quicksort. Table 3 presents results on randomly generated arrays of varying length for the learned non-recursive and recursive programs. For each length, we test each program on 30 randomly generated problems. Observe that the non-recursive program’s correctness degrades for length 11 and beyond, while the recursive program can sort any given array. + +Table 2: Accuracy on Randomly Generated Problems for Topological Sort + +
NumberofVerticesNon-RecursiveRecursive
56.7%100%
66.7%100%
73.3%100%
80%100%
700%100%
+ +Table 3: Accuracy on Randomly Generated Problems for Quicksort + +
LengthofArrayNon-RecursiveRecursive
3100%100%
5100%100%
7100%100%
1173.3%100%
1560%100%
2030%
2220%100%
253.33%100%
303.33%100%
100%
700%100%
+ +As mentioned in Section 2.1, we hypothesize the non-recursive programs do not generalize well because they have learned spurious dependencies specific to the training set, such as length of the input problems. On the other hand, the recursive programs have learned the true program semantics. + +# 4.2 VERIFICATION OF PROVABLY PERFECT GENERALIZATION + +We describe how models trained with recursive traces can be proven to generalize, by using the verification procedure described in Section 3.3. As described in the verification procedure, it is possible to prove our learned recursive program generalizes perfectly by testing on an appropriate set of problem inputs, i.e., the verification set. Recall that this verification procedure cannot be performed for the non-recursive versions, since the propagation of the hidden state in the core LSTM module makes reasoning difficult and so we would need to check an unbounded number of examples. + +We describe the base cases, reduction rules, and the verification set for each task in Appendix A.6. For each task, given the verification set, we check the traces and results of the learned, to-be-verified neural program (described in Section 4.1; and for bubble sort, Appendix A.6) on the verification set, and ensure they match the traces produced by the true program $P$ . Our results show that for all learned, to-be-verified neural programs, they all produced the same traces as those produced by $P$ on the verification set. Thus, we demonstrate that recursion enables provably perfect generalization for different tasks, including addition, topological sort, quicksort, and a variant of bubble sort. + +Note that the training set can often be considerably smaller than the verification set, and despite this, the learned model can still pass the entire verification set. Our result shows that the training procedure and the NPI architecture is capable of generalizing from the step input-output pairs seen in the training data to the unseen ones present in the verification set. + +# 5 CONCLUSION + +We emphasize that the notion of a neural recursive program has not been presented in the literature before: this is our main contribution. Recursion enables provably perfect generalization. To the best of our knowledge, this is the first time verification has been applied to a neural program, providing provable guarantees about its behavior. We instantiated recursion for the Neural ProgrammerInterpreter by changing the training traces. In future work, we seek to enable more tasks with recursive structure. We also hope to decrease supervision, for example by training with only partial or non-recursive traces, and to develop novel Neural Programming Architectures integrated directly with a notion of recursion. + +# ACKNOWLEDGMENTS + +This material is in part based upon work supported by the National Science Foundation under Grant No. TWC-1409915, DARPA under Grant No. FA8750-15-2-0104, and Berkeley Deep Drive. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of National Science Foundation and DARPA. + +# REFERENCES + +Marcin Andrychowicz and Karol Kurach. Learning efficient algorithms with hierarchical attentive memory. CoRR, abs/1602.03218, 2016. URL http://arxiv.org/abs/1602.03218. + +Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. CoRR, abs/1410.5401, 2014. URL http://arxiv.org/abs/1410.5401. + +Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwiska, Sergio Gmez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, Adri Puigdomnech Badia, Karl Moritz Hermann, Yori Zwols, Georg Ostrovski, Adam Cain, Helen King, Christopher Summerfield, Phil Blunsom, Koray Kavukcuoglu, and Demis Hassabis. Hybrid computing using a neural network with dynamic external memory. Nature, 538 (7626):471–476, October 2016. ISSN 0028-0836, 1476-4687. doi: 10.1038/nature20101. URL http://www.nature.com/doifinder/10.1038/nature20101. + +Lukasz Kaiser and Ilya Sutskever. Neural gpus learn algorithms. CoRR, abs/1511.08228, 2015. URL http://arxiv.org/abs/1511.08228. + +Karol Kurach, Marcin Andrychowicz, and Ilya Sutskever. Neural random access machines. ERCIM News, 2016(107), 2016. URL http://ercim-news.ercim.eu/en107/special/ neural-random-access-machines. + +Arvind Neelakantan, Quoc V. Le, and Ilya Sutskever. Neural programmer: Inducing latent programs with gradient descent, 2015. + +Scott Reed and Nando de Freitas. Neural programmer-interpreters. ICLR, 2016. + +Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada, pp. 2692–2700, 2015. URL http://papers.nips.cc/paper/5866-pointer-networks. + +Wojciech Zaremba, Tomas Mikolov, Armand Joulin, and Rob Fergus. Learning simple algorithms from examples. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pp. 421–429, 2016. URL http://jmlr. org/proceedings/papers/v48/zaremba16.html. + +A APPENDIX + +A.1 PROGRAM SET FOR NON-RECURSIVE TOPOLOGICAL SORT + +
ProgramDescriptionsCallsArguments
TOPOSORTPerform topologicalsort on graphTRAVERSE,NEXT_START,WRITE,MOVENONE
TRAVERSETraverse graph untilstack is emptyCHECK_CHILD，EX-PLORENONE
CHECK_CHILDCheck ifawhitechild exists; if so, setchildList[Uactive] topoint to itMOVENONE
EXPLORERepeatedlytraversesubgraphs until stackis emptySTACK,CHECK_CHILD,WRITE,MOVENONE
STACKInteractwith stack,either pushing orpoppingWRITE,MOVEPUSH,POP
NEXT_STARTMove Pstart untilreaching a whitevertex. If a whitevertex is found，setPstart to point to it;this signifies the startof a traversal of anew connected com-ponent.If no whitevertex is found, theentireexecution isterminatedMOVENONE
WRITEWrite a value eitherto environment (e.g.,to color a vertex)or variable (e.g., tochange the value ofUactive)NONEDescribed below
MOVEMove a pointer(e.g, Pstart orchildList[vactive])up or downNONEDescribed below
+ +# Argument Sets for WRITE and MOVE. + +WRITE. The WRITE operation has the following arguments: + +# ARG 1 (Main Action): COLOR CURR, COLOR NEXT, ACTIVE START, ACTIVE NEIGHB, ACTIVE STACK, SAVE, STACK PUSH, STACK POP, RESULT + +COLOR CURR colors $v _ { a c t i v e }$ , COLOR NEXT colors Vertex $D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ,$ , ACTIVE START writes pstart to $v _ { a c t i v e }$ , ACTIVE NEIGHB writes $D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ]$ to $v _ { a c t i v e }$ , ACTIVE STACK writes $Q _ { s t a c k } ( p _ { s t a c k } )$ to $v _ { a c t i v e }$ , SAVE writes $v _ { a c t i v e }$ to $v _ { s a v e }$ , $S T A C K \_ P U S H$ pushes $v _ { a c t i v e }$ to the top of the stack, $S T A C K \_ P O P$ writes a null value to the top of the stack, and $R E S U L T$ writes $v _ { a c t i v e }$ to $Q _ { r e s u l t } ( p _ { r e s u l t } )$ . + +ARG 2 (Auxiliary Variable): COLOR GREY, COLOR BLACK + +COLOR GREY and COLOR BLACK color the given vertex grey and black, respectively. + +MOVE. The MOVE operation has the following arguments: + +ARG 1 (Pointer): $p _ { r e s u l t } , p _ { s t a c k } , p _ { s t a r t } , c h i l d L i s t [ v _ { a c t i v e } ] , c h i l d L i s t [ v _ { s a v e } ]$ + +Note that the argument is the identity of the pointer, not what the pointer points to; in other words, ARG 1 can only take one of 5 values. + +ARG 2 (Increment or Decrement): UP, DOWN + +# A.2 TRACE-GENERATING FUNCTIONS FOR TOPOLOGICAL SORT + +# A.2.1 NON-RECURSIVE TRACE-GENERATING FUNCTIONS + +1 // Top level topological sort call +2 TOPOSORT() { +3 while $( Q _ { c o l o r } \big ( p _ { s t a r t } \big )$ is a valid color): // color invalid when all vertices explored +4 WRITE(ACTIVE_START) +5 WRITE(COLOR_CURR, COLOR_GREY) +6 TRAVERSE() +7 MOVE(pstart, UP) +8 NEXT_START() +9 } +10 +11 TRAVERSE() { +12 CHECK_CHILD() +13 EXPLORE() +14 } +15 +16 CHECK_CHILD() { +17 while $( Q _ { c o l o r } \big ( D A G \big [ v _ { a c t i v e } \big ] \big [ c h i l d L i s t \big [ v _ { a c t i v e } \big ] \big ] \big )$ is not white and is not invalid): // color invalid when all children explored +18 MOVE(childList[vactive], UP) +19 } +20 +21 EXPLORE() { +22 do +23 if (Qcolor(DAG[vactive][childList[vactive]]) is white): +24 WRITE(COLOR_NEXT, COLOR_GREY) +25 STACK(PUSH) +26 WRITE(SAVE) +27 WRITE(ACTIVE_NEIGHB) +28 MOVE(childList[vsave], UP) +29 else: +30 WRITE(COLOR_CURR, COLOR_BLACK) +31 WRITE(RESULT) +32 MOVE(presult, UP) +33 if(pstack == 1): +34 break +35 else: +36 STACK(POP) +37 CHECK_CHILD() +38 while (true) +39 +40 +41 STACK(op) { +42 if (op == PUSH): +43 WRITE(STACK_PUSH) +44 MOVE(pstack, UP) +45 +46 if (op == POP): +47 WRITE(ACTIVE_STACK) +48 WRITE(STACK_POP) +49 MOVE(pstack, DOWN) +50 } +51 +52 NEXT_START() { +53 while(Qcolor(pstart) is not white and is not invalid): // color invalid when all vertices explored +54 MOVE(pstart, UP) +55 } + +# A.2.2 RECURSIVE TRACE-GENERATING FUNCTIONS + +# Altered Recursive Functions + +1 // Top level topological sort call +2 TOPOSORT() { +3 if $\scriptstyle : Q _ { c o l o r }$ $_ { p _ { s t a r t } } ,$ is a valid color): // color invalid when all vertices explored +4 WRITE(ACTIVE_START) +5 WRITE(COLOR_CURR, COLOR_GREY) +6 TRAVERSE() +7 MOVE(pstart, UP) +8 NEXT_START() +9 TOPOSORT() // Recursive Call +10 } +11 +12 CHECK_CHILD() { +13 if $\overline { { ( Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ) } } ) \overline { { { } } }$ is not white and is not invalid): // color invalid when all children explore +14 MOVE(childList[vactive], UP) +15 CHECK_CHILD() // Recursive Call +16 } +17 +18 EXPLORE() { +19 if (Qcolor(DAG[vactive][childList[vactive]]) is white): +20 WRITE(COLOR_NEXT, COLOR_GREY) +21 STACK(PUSH) +22 WRITE(SAVE) +23 WRITE(ACTIVE_NEIGHB) +24 MOVE(childList[vsave], UP) +25 else: +26 WRITE(COLOR_CURR, COLOR_BLACK) +27 WRITE(RESULT) +28 MOVE(presult, UP) +29 if $( p _ { s t a c k } = = 1$ ): +30 return +31 else: +32 STACK(POP) +33 CHECK_CHILD() +34 EXPLORE() // Recursive Call +35 } +36 +37 NEXT_START() { +38 if $( Q _ { c o l o r } \left( p _ { s t a r t } \right)$ is not white and is not invalid): // color invalid when all vertices explored +39 MOVE $( p _ { s t a r t }$ , UP) +40 NEXT_START() // Recursive Call +41 } + +# Algorithm 4 Iterative Quicksort + +1: Initialize an array $A$ to sort and two empty stacks $S _ { l o }$ and $S _ { h i }$ . +2: Initialize lo and $h i$ to be 1 and $n$ , where $n$ is the length of $A$ . +3: +4: function PARTITION $( A , l o , h i )$ +5: $p i v o t = l o$ +6: for $j \in [ l o , h i - 1 ] : \mathbf { d o }$ +7: if $A [ j ] \leq A [ h i ]$ then +8: swap A[pivot] with $A [ j ]$ +9: pivot = pivot + 1 +10: swap A[pivot] with $A [ h i ]$ +11: return pivot +12: +13: function QUICK ${ \mathrm { : } } \operatorname { S o R T } ( A , l o , h i )$ +14: while $S _ { l o }$ and $S _ { h i }$ are not empty: do +15: Pop states off $S _ { l o }$ and $S _ { h i }$ , writing them to $l o$ and $h i$ . +16: $\mathsf { p } = \mathsf { P A R T I T I O N } ( A , l o , h i )$ +17: Push $p + 1$ and $h i$ to $S _ { l o }$ and $S _ { h i }$ . +18: Push lo and $p - 1$ to $S _ { l o }$ and $S _ { h i }$ . + +A.4 PROGRAM SET FOR QUICKSORT + +
ProgramDescriptionsCalls Arguments
QUICKSORTRunthequicksortroutine in place forthearrayA，forindices from lo to hiNon-Recursive: PAR-TITION, STACK,WRITERecursive: same asnon-recursiveversion,alongwithQUICK-SORTImplicitly:arrayA to sort, lo, hi
PARTITIONRunsthepartitionfunction. At end,pointer Ppivot ismoved to the pivotCOMPSWAP_LOOP,MOVE_PIVOT_LO,MOVE_JLO, SWAPNONE
COMPSWAPLOOPRuns the FOR loopinside the partitionfunctionCOMPSWAP, MOVENONE
COMPSWAPComparesA[pivot] ≤ A[j]; ifso,perform a swapand increment PpivotSWAP, MOVENONE
SET_PIVOTLOSets Ppivot to lo in-dexNONENONE
SETJLOSets pj to lo indexNONENONE
SET_J_NULLSets pj to-00NONENONE
STACKPushes lo/hi statesonto stacks Sto andShi according to argument(describedbelow)WRITE, MOVEDescribed below
MOVEMovespointerone unit up or downNONEDescribed below
SWAPSwapselementsatgiven array indicesNONEDescribed below
WRITEWrite a valueeither to stack(e.g QstackLoorQstackHi) or topointer (e.g tochangethe value ofPhi)NONEDescribed below
+ +Argument Sets for STACK, MOVE, SWAP, WRITE. + +STACK. The STACK operation has the following arguments: + +ARG 1 (Operation): STACK PUSH CALL1, STACK PUSH CALL2, STACK POP + +STACK PUSH CALL1 pushes $l o$ and pivot−1 to $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . STACK PUSH CALL2 pushes pivot $+ 1$ and $h i$ to $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . STACK POP pushes $- \infty$ values to $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . + +MOVE. The MOVE operation has the following arguments: + +ARG 1 (Pointer): pstackLo, pstackHi, pj , ppivot + +Note that the argument is the identity of the pointer, not what the pointer points to; in other words, ARG 1 can only take one of 4 values. + +ARG 2 (Increment or Decrement): UP, DOWN + +SWAP. The SWAP operation has the following arguments: + +ARG 1 (Swap Object 1): ppivot + +ARG 2 (Swap Object 2): $p _ { h i } , p _ { j }$ + +WRITE. The WRITE operation has the following arguments: + +ARG 1 (Object to Write): ENV STACK LO, ENV STACK HI, $p _ { h i } , p _ { l o }$ + +ENV STACK LO and ENV STACK HI represent $Q _ { s t a c k L o } ( p _ { s t a c k L o } ) $ and $Q _ { s t a c k H i } ( p _ { s t a c k H i } )$ , re spectively. + +ARG 2 (Object to Copy): ENV STACK LO PEEK, ENV STACK HI PEEK, $p _ { h i } , p _ { l o } , p _ { p i v o t } - 1$ ppivot + 1, RESET + +ENV STACK LO PEEK and ENV STACK HI PEEK represent $Q _ { s t a c k L o } ( p _ { s t a c k L o } \mathrm { ~ - ~ } 1 )$ and $Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 )$ , respectively. RESET represents a $- \infty$ value. + +Note that the argument is the identity of the pointer, not what the pointer points to; in other words, ARG 1 can only take one of 4 values, and ARG 2 can only take one of 7 values. + +# A.5 TRACE-GENERATING FUNCTIONS FOR QUICKSORT + +# A.5.1 NON-RECURSIVE TRACE-GENERATING FUNCTIONS + +1 Initialize $p _ { l o }$ to 1 and $p _ { h i } \ t \circ \ n$ (length of array) +2 Initialize $p _ { j } ~ \mathsf { t o } ~ - \infty$ +3 +4 QUICKSORT() { +5 while $( p _ { s t a c k L o } \neq 1 )$ : +6 if $( \stackrel { } { Q } _ { s t a c k L o } ^ { \prime \prime } ( \stackrel { \prime } { p } _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 ) ) :$ +7 STACK(STACK_POP) +8 else: +9 WRITE $( \boldsymbol { p } _ { h i }$ , ENV_STACK_HI_PEEK) +10 WRITE(plo, ENV_STACK_LO_PEEK) +11 STACK(STACK_POP) +12 PARTITION() +13 STACK(STACK_PUSH_CALL2) +14 STACK(STACK_PUSH_CALL1) +15 } +16 +17 PARTITION() { +18 SET_PIVOT_LO() +19 SET_J_LO() +20 COMPSWAP_LOOP() +24 SWAP(ppivot, phi) +SET_J_NULL() +} +COMPSWAP_LOOP() { +while $( p _ { j } \neq p _ { h i } )$ : +COMPSWAP() +MOVE $( p _ { j }$ , UP) +} +30 +31 COMPSWAP() { +32 if (A[pj ] ≤ A[phi]): +33 SWAP(ppivot, pj) +34 MOVE(ppivot, UP) +35 } +36 +37 STACK(op) { +38 if (op == STACK_PUSH_CALL1): +39 WRITE(ENV_STACK_LO, plo) +40 WRITE(ENV_STACK_HI, ppivot − 1) +41 MOVE(pstackLo, UP) +42 MOVE(pstackHi, UP) +43 +44 if (op $= =$ STACK_PUSH_CALL2): +45 WRITE(ENV_STACK_LO, ppivot $^ { + 1 1 }$ ) +46 WRITE(ENV_STACK_HI, $p _ { h i }$ ) +47 MOVE(pstackLo, UP) +48 MOVE(pstackHi, UP) +49 +50 if (op $= =$ STACK_POP): +51 WRITE(ENV_STACK_LO, RESET) +52 WRITE(ENV_STACK_HI, RESET) +53 MOVE(pstackLo, DOWN) +54 MOVE(pstackHi, DOWN) +55 } + +# A.5.2 RECURSIVE TRACE-GENERATING FUNCTIONS + +# Altered Recursive Functions + +Initialize to 1 and $p _ { h i } \ t \circ \ n$ (length of array) + +1 $p _ { l o }$ +2 Initialize $p _ { j }$ to −∞ +3 +4 QUIC $\begin{array} { r l } & { \mathrm { \sf { A S O R T } \left( \tau \right) } \quad \mathrm { \sf { \{ } } } \\ & { \mathrm { \sf { ( } } Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 ) ) : } \\ & { \mathrm { \sf { 2 } } \mathrm { \sf { A R T I T I O N } \left( \tau \right) } } \end{array}$ +5 if +6 +7 STACK(STACK_PUSH_CALL2) +8 STACK(STACK_PUSH_CALL1) +9 WRITE $( p _ { h i }$ , ENV_STACK_HI_PEEK) +10 WRITE $( p _ { l o }$ , ENV_STACK_LO_PEEK) +11 QUICKSORT() // Recursive Call +12 STACK(STACK_POP) +13 WRITE $( p _ { h i }$ , ENV_STACK_HI_PEEK) +14 WRITE $( p _ { l o }$ , ENV_STACK_LO_PEEK) +15 QUICKSORT() // Recursive Call +16 STACK(STACK_POP) +17 } +18 +19 COMPSWAP_LOOP() { +20 if $( p _ { j } \neq p _ { h i } )$ : +21 COMPSWAP() +22 MOVE $( p _ { j }$ , UP) +23 COMPSWAP_LOOP() // Recursive Call +24 } + +A.6 BASE CASES, REDUCTION RULES, AND VERIFICATION SETS + +In this section, we describe the space of base cases and reduction rules that must be covered for each of the four sample tasks, in order to create the verification set. + +For addition, we analytically determine the verification set. For tasks other than addition, it is difficult to analytically determine the verification set, so instead, we randomly generate input candidates until they completely cover the base cases and reduction rules. + +Base Cases and Reduction Rules for Addition. For the recursive formulation of addition, we analytically construct the set of input problems that cover all base cases and reduction rules. We outline how to construct this set. + +It is sufficient to construct problems where every transition between two adjacent columns is covered. The ADD reduction rule ensures that each call to ADD only covers two adjacent columns, and so the LSTM only ever runs for a fixed number of steps necessary to process these two columns. + +We construct input problems by splitting into two cases: one case in which the left column contains a null value and another in which the left column does not contain any null values. We then construct problem configurations that span all possible valid environment states (for instance, in order to force the carry bit in a column to be 1, one can add the sum $\cdot _ { 1 + 9 } ,$ in the column to the right). + +The operations we need to be concerned most about are CARRY and LSHIFT, which induce partial environment states spanning two columns. It is straightforward to deal with all other operations, which do not induce partial environment states. + +Under the assumption that there are no leading 0’s (except in the case of single digits) and the two numbers to be added have the same number of digits, the verification set for addition contains 20,181 input problems. The assumption of leading 0’s can be easily removed, at the cost of slightly increasing the size of the verification set. We made the assumption of equivalent lengths in order to parametrize the input format with respect to length, but this assumption can be removed as well. + +Base Cases and Reduction Rules for Bubble Sort. The original version of the bubblesort implementation exposes the values within the array. While this matches the description from Reed & de Freitas (2016), we found that this causes an unnecessary blowup in the size of $V$ and makes it much more difficult to construct the verification set. For purposes of verification, we replace the domain-specific encoder with the following: + +$$ +\begin{array} { r l } & { f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) \leq Q ( 1 , i _ { 2 } ) , 1 \leq i _ { 1 } \leq l e n g t h , 1 \leq i _ { 2 } \leq l e n g t h , } \\ & { ~ i _ { 3 } = = l e n g t h , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \end{array} +$$ + +Table 4: Accuracy on Randomly Generated Problems for Variant of Bubble Sort + +
Length of ArrayNon-RecursiveRecursive
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+ +which directly exposes which of the two values pointed to is larger. This modification also enables us to sort arrays containing arbitrary comparable elements. + +By reasoning about the possible set of environment observations created by all valid inputs, we construct $V$ using the procedure described in Section 3.3. Using this modification, we constructed a verification set consisting of one array of size 10. + +We also report on generalization results for the non-recursive and recursive versions of this variant of bubble sort. Table 4 demonstrates that the accuracy of the non-recursive program degrades sharply when moving from arrays of length 7 to arrays of length 8. This is due to the properties of the training set – we trained on 2 traces synthesized from arrays of length 7 and 1 trace synthesized from an array of length 6. Table 4 also demonstrates that the (verified) recursive program generalizes perfectly. + +Base Cases and Reduction Rules for Topological Sort. For each function we use to implement the recursive version of topological sort, we need to consider the set of possible environment observation sequences we can create from all valid inputs and test that the learned program produces the correct behavior on each of these inputs. We have three observations: the color of the start node, the color of the active node’s next child to be considered, and whether the stack is empty. Na¨ıvely, we might expect to synthesize and test an input for any sequence created by combining the four possible colors in two variables and another boolean variable for whether the stack is empty (so 32 possible observations at any point), but for various reasons, most of these combinations are impossible to occur at any given point in the execution trace. + +Through careful reasoning about the possible set of environment observations created by all valid inputs, and how each of the operations in the execution trace affects the environment, we can construct $V$ using the procedure described in Section 3.3. We then construct a verification set of size 73 by ensuring that randomly generated graphs cover the analytically derived $V$ . The model described in the training setup of Section 4 (trained on 6 traces) was verified to be correct via the matching procedure described in Section 4.2. + +Base Cases and Reduction Rules for Quicksort. As with the others, we apply the procedure described in Section 3.3 to construct $V$ and then empirically create a verification set which covers $V$ . The verification set can be very small, as we found a 10-element array ([8,2,1,2,0,8,5,8,3,7]) is sufficient to cover all of $V$ . We note that an earlier version of quicksort we tried lacked primitive operations to directly move a pointer to another, and therefore needed more functions and observations. As this complexity interfered with determining the base cases and reductions, we changed the algorithm to its current form. Even though the earlier version also generalized just as well in practice, relatively small differences in the formulation of the traces and the environment observations can drastically change the difficulty of verification. \ No newline at end of file diff --git a/parse/train/BkbY4psgg/BkbY4psgg_content_list.json b/parse/train/BkbY4psgg/BkbY4psgg_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..82f7adf2fbfe4a72e4090ac3c08447a752ddb192 --- /dev/null +++ b/parse/train/BkbY4psgg/BkbY4psgg_content_list.json @@ -0,0 +1,2200 @@ +[ + { + "type": "text", + "text": "MAKING NEURAL PROGRAMMING ARCHITECTURES GENERALIZE VIA RECURSION ", + "text_level": 1, + "bbox": [ + 174, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jonathon Cai, Richard Shin, Dawn Song \nDepartment of Computer Science \nUniversity of California, Berkeley \nBerkeley, CA 94720, USA \n{jonathon,ricshin,dawnsong}@cs.berkeley.edu ", + "bbox": [ + 184, + 170, + 604, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Empirically, neural networks that attempt to learn programs from data have exhibited poor generalizability. Moreover, it has traditionally been difficult to reason about the behavior of these models beyond a certain level of input complexity. In order to address these issues, we propose augmenting neural architectures with a key abstraction: recursion. As an application, we implement recursion in the Neural Programmer-Interpreter framework on four tasks: grade-school addition, bubble sort, topological sort, and quicksort. We demonstrate superior generalizability and interpretability with small amounts of training data. Recursion divides the problem into smaller pieces and drastically reduces the domain of each neural network component, making it tractable to prove guarantees about the overall system’s behavior. Our experience suggests that in order for neural architectures to robustly learn program semantics, it is necessary to incorporate a concept like recursion. ", + "bbox": [ + 233, + 306, + 764, + 486 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 510, + 336, + 526 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Training neural networks to synthesize robust programs from a small number of examples is a challenging task. The space of possible programs is extremely large, and composing a program that performs robustly on the infinite space of possible inputs is difficult—in part because it is impractical to obtain enough training examples to easily disambiguate amongst all possible programs. Nevertheless, we would like the model to quickly learn to represent the right semantics of the underlying program from a small number of training examples, not an exhaustive number of them. ", + "bbox": [ + 174, + 541, + 823, + 625 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Thus far, to evaluate the efficacy of neural models on programming tasks, the only metric that has been used is generalization of expected behavior to inputs of greater complexity (Vinyals et al. (2015), Kaiser & Sutskever (2015), Reed & de Freitas (2016), Graves et al. (2016), Zaremba et al. (2016)). For example, for the addition task, the model is trained on short inputs and then tested on its ability to sum inputs with much longer numbers of digits. Empirically, existing models suffer from a common limitation—generalization becomes poor beyond a threshold level of complexity. Errors arise due to undesirable and uninterpretable dependencies and associations the architecture learns to store in some high-dimensional hidden state. This makes it difficult to reason about what the model will do when given complex inputs. ", + "bbox": [ + 174, + 631, + 825, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One common strategy to improve generalization is to use curriculum learning, where the model is trained on inputs of gradually increasing complexity. However, models that make use of this strategy eventually fail after a certain level of complexity (e.g. the single-digit multiplication task in Zaremba et al. (2016), the bubble sort task in Reed & de Freitas (2016), and the graph tasks in Graves et al. (2016)). In this version of curriculum learning, even though the inputs are gradually becoming more complex, the semantics of the program is succinct and does not change. Although the model is exposed to more and more data, it might learn spurious and overly complex representations of the program, as suggested in Zaremba et al. (2016). That is to say, the network does not learn the true program semantics. ", + "bbox": [ + 174, + 763, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we propose to resolve these issues by explicitly incorporating recursion into neural architectures. Recursion is an important concept in programming languages and a critical tool to reduce the complexity of programs. We find that recursion makes it easier for the network to learn the right program and generalize to unknown situations. Recursion enables provable guarantees on neural programs’ behavior without needing to exhaustively enumerate all possible inputs to the programs. This paper is the first (to our knowledge) to investigate the important problem of provable generalization properties of neural programs. As an application, we incorporate recursion into the Neural Programmer-Interpreter architecture and consider four sample tasks: grade-school addition, bubble sort, topological sort, and quicksort. Empirically, we observe that the learned recursive programs solve all valid inputs with $100 \\%$ accuracy after training on a very small number of examples, out-performing previous generalization results. Given verification sets that cover all the base cases and reduction rules, we can provide proofs that these learned programs generalize perfectly. This is the first time one can provide provable guarantees of perfect generalization for neural programs. ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 256 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 THE PROBLEM AND OUR APPROACH", + "text_level": 1, + "bbox": [ + 174, + 275, + 511, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 THE PROBLEM OF GENERALIZATION ", + "text_level": 1, + "bbox": [ + 176, + 306, + 467, + 320 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "When constructing a neural network for the purpose of learning a program, there are two orthogonal aspects to consider. The first is the actual model architecture. Numerous models have been proposed for learning programs; to name a few, this includes the Differentiable Neural Computer (Graves et al., 2016), Neural Turing Machine (Graves et al., 2014), Neural GPU (Kaiser & Sutskever, 2015), Neural Programmer (Neelakantan et al., 2015), Pointer Network (Vinyals et al., 2015), Hierarchical Attentive Memory (Andrychowicz & Kurach, 2016), and Neural Random Access Machine (Kurach et al., 2016). The architecture usually possesses some form of memory, which could be internal (such as the hidden state of a recurrent neural network) or external (such as a discrete “scratch pad” or a memory block with differentiable access). The second is the training procedure, which consists of the form of the training data and the optimization process. Almost all architectures train on program input/output pairs. The only model, to our knowledge, that does not train on input-output pairs is the Neural Programmer-Interpreter (Reed & de Freitas, 2016), which trains on synthetic execution traces. ", + "bbox": [ + 174, + 332, + 825, + 512 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To evaluate a neural network that learns a neural program to accomplish a certain task, one common evaluation metric is how well the learned model $M$ generalizes. More specifically, when $M$ is trained on simpler inputs, such as inputs of a small length, the generalization metric evaluates how well $M$ will do on more complex inputs, such as inputs of much longer length. $M$ is considered to have perfect generalization if $M$ can give the right answer for any input, such as inputs of arbitrary length. ", + "bbox": [ + 174, + 520, + 825, + 603 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As mentioned in Section 1, all approaches to neural programming today fare poorly on this generalization issue. We hypothesize that the reason for this is that the neural network learns to spuriously depend on specific characteristics of the training examples that are irrelevant to the true program semantics, such as length of the training inputs, and thus fails to generalize to more complex inputs. ", + "bbox": [ + 176, + 609, + 823, + 666 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In addition, none of the current approaches to neural programming provide a method or even aim to enable provable guarantees about generalization. The memory updates of these neural programs are so complex and interdependent that it is difficult to reason about the behaviors of the learned neural program under previously unseen situations (such as problems with longer inputs). This is highly undesirable, since being able to provide the correct answer in all possible settings is one of the most important aspects of any learned neural program. ", + "bbox": [ + 174, + 672, + 823, + 757 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 OUR APPROACH USING RECURSION", + "text_level": 1, + "bbox": [ + 176, + 772, + 465, + 786 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we propose that the key abstraction of recursion is necessary for neural programs to generalize. The general notion of recursion has been an important concept in many domains, including mathematics and computer science. In computer science, recursion (as opposed to iteration) involves solving a larger problem by combining solutions to smaller instances of the same problem. Formally, a function exhibits recursive behavior when it possesses two properties: (1) Base cases— terminating scenarios that do not use recursion to produce answers; (2) A set of rules that reduces all other problems toward the base cases. Some functional programming languages go so far as not to define any looping constructs but rely solely on recursion to enable repeated execution of the same code. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we propose that recursion is an important concept for neural programs as well. In fact, we argue that recursion is an essential element for neural programs to generalize, and makes it tractable to prove the generalization of neural programs. Recursion can be implemented differently for different neural programming models. Here as a concrete and general example, we consider a general Neural Programming Architecture (NPA), similar to Neural Programmer-Interpreter (NPI) in Reed & de Freitas (2016). In this architecture, we consider a core controller, e.g., an LSTM in NPI’s case, but possibly other networks in different cases. There is a (changing) list of neural programs used to accomplish a given task. The core controller acts as a dispatcher for the programs. At each time step, the core controller can decide to select one of the programs to call with certain arguments. When the program is called, the current context including the caller’s memory state is stored on a stack; when the program returns, the stored context is popped off the stack to resume execution in the previous caller’s context. ", + "bbox": [ + 174, + 103, + 825, + 270 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this general Neural Programming Architecture, we show it is easy to support recursion. In particular, recursion can be implemented as a program calling itself. Because the context of the caller is stored on a stack when it calls another program and the callee starts in a fresh context, this enables recursion simply by allowing a program to call itself. In practice, we can additionally use tail recursion optimization to avoid problems with the call stack growing too deep. Thus, any general Neural Programming Architecture supporting such a call structure can be made to support recursion. In particular, this condition is satisfied by NPI, and thus the NPI model naturally supports recursion (even though the authors of NPI did not consider this aspect explicitly). ", + "bbox": [ + 174, + 277, + 825, + 388 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "By nature, recursion reduces the complexity of a problem to simpler instances. Thus, recursion helps decompose a problem and makes it easier to reason about a program’s behavior for previously unseen situations such as longer inputs. In particular, given that a recursion is defined by two properties as mentioned before, the base cases and the set of reduction rules, we can prove a recursive neural program generalizes perfectly if we can prove that (1) it performs correctly on the base cases; (2) it learns the reduction rules correctly. For many problems, the base cases and reduction rules usually consist of a finite (often small) number of cases. For problems where the base cases may be extremely large or infinite, such as certain forms of motor control, recursion can still help reduce the problem of generalization to these two aspects and make the generalization problem significantly simpler to handle and reason about. ", + "bbox": [ + 174, + 395, + 825, + 534 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As a concrete instantiation, we show in this paper that we can enable recursive neural programs in the NPI model, and thus enable perfectly generalizable neural programs for tasks such as sorting where the original, non-recursive NPI program fails. As aforementioned, the NPI model naturally supports recursion. However, the authors of NPI did not consider explicitly the notion of recursion and as a consequence, did not learn recursive programs. We show that by modifying the training procedure, we enable the NPI model to learn recursive neural programs. As a consequence, our learned neural programs empirically achieve perfect generalization from a very small number of training examples. Furthermore, given a verification input set that covers all base cases and reduction rules, we can formally prove that the learned neural programs achieve perfect generalization after verifying its behavior on the verification input set. This is the first time one can provide provable guarantees of perfect generalization for neural programs. ", + "bbox": [ + 174, + 541, + 825, + 694 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We would also like to point out that in this paper, we provide as an example one way to train a recursive neural program, by providing a certain training execution trace to the NPI model. However, our concept of recursion for neural programs is general. In fact, it is one of our future directions to explore new ways to train a recursive neural program without providing explicit training execution traces or with only partial or non-recursive traces. ", + "bbox": [ + 174, + 702, + 823, + 770 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 APPLICATION TO LEARNING RECURSIVE NEURAL PROGRAMS WITH NPI ", + "text_level": 1, + "bbox": [ + 174, + 808, + 810, + 824 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 BACKGROUND: NPI ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 848, + 470, + 863 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As discussed in Section 2, the Neural Programmer-Interpreter (NPI) is an instance of a Neural Programmer Architecture and hence it naturally supports recursion. In this section, we give a brief review of the NPI architecture from Reed & de Freitas (2016) as background. ", + "bbox": [ + 176, + 882, + 823, + 922 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We describe the details of the NPI model relevant to our contributions. We adapt machinery from the original paper slightly to fit our needs. The NPI model has three learnable components: a task-agnostic core, a program-key embedding, and domain-specific encoders that allow the NPI to operate in diverse environments. ", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The NPI accesses an external environment, $Q$ , which varies according to the task. The core module of the NPI is an LSTM controller that takes as input a slice of the current external environment, via a set of pointers, and a program and arguments to execute. NPI then outputs the return probability and next program and arguments to execute. Formally, the NPI is represented by the following set of equations: ", + "bbox": [ + 173, + 166, + 825, + 236 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/bfd328979f3862d272f2a19a564f1f89362dc096d4ed4d62f9e093d1d99b6bb8.jpg", + "text": "$$\n\\begin{array} { c } { s _ { t } = f _ { e n c } ( e _ { t } , a _ { t } ) } \\\\ { h _ { t } = f _ { l s t m } ( s _ { t } , p _ { t } , h _ { t - 1 } ) } \\\\ { r _ { t } = f _ { e n d } ( h _ { t } ) , p _ { t + 1 } = f _ { p r o g } ( h _ { t } ) , a _ { t + 1 } = f _ { a r g } ( h _ { t } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 330, + 248, + 668, + 313 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$t$ is a subscript denoting the time-step; $f _ { e n c }$ is a domain-specific encoder (to be described later) that takes in the environment slice $e _ { t }$ and arguments $a _ { t }$ ; $f _ { l s t m }$ represents the core module, which takes in the state $s _ { t }$ generated by $f _ { e n c }$ , a program embedding $p _ { t } \\in \\mathbb { R } ^ { P }$ , and hidden LSTM state $h _ { t }$ ; $f _ { e n d }$ decodes the return probability $r _ { t }$ ; $f _ { p r o g }$ decodes a program key embedding $p _ { t + 1 }$ ;1 and $f _ { a r g }$ decodes arguments $a _ { t + 1 }$ . The outputs $r _ { t } , p _ { t + 1 } , a _ { t + 1 }$ are used to determine the next action, as described in Algorithm 1. If the program is primitive, the next environmental state $e _ { t + 1 }$ will be affected by $p _ { t }$ and $a _ { t }$ , i.e. $e _ { t + 1 } \\sim f _ { e n v } ( e _ { t } , p _ { t } , a _ { t } )$ . As with the original NPI architecture, the experiments for this paper always used a 3-tuple of integers $a _ { t } = ( a _ { t } ( 1 ) , \\hat { a } _ { t } ( 2 ) , a _ { t } ( 3 ) )$ . ", + "bbox": [ + 173, + 323, + 825, + 435 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/e6560f320f41f348fdd426619c47efa3064b98bfb7c7dc35cb65540f5dbaec9a.jpg", + "table_caption": [ + "Algorithm 1 Neural programming inference " + ], + "table_footnote": [], + "table_body": "
1:Inputs: Environment observation e, program p,arguments a, stop threshold α
2:function RUN(e,p, a)
3:h↑0,r←0
4:whiler<αdo
5:s ←fenc(e,a),h ←fistm(s,p,h)
6:r ←fend(h),p2 ← fprog(h),a2 ← farg(h)
7:if p is a primitive function then
8:e ← fenv(e,p,a).
9:else
10:function RUN(e,P2, a2)
", + "bbox": [ + 179, + 470, + 714, + 614 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A description of the inference procedure is given in Algorithm 1. Each step during an execution of the program does one of three things: (1) another subprogram along with associated arguments is called, as in Line 10, (2) the program writes to the environment if it is primitive, as in Line 8, or (3) the loop is terminated if the return probability exceeds a threshold $\\alpha$ , after which the stack frame is popped and control is returned to the caller. In all experiments, $\\alpha$ is set to 0.5. Each time a subprogram is called, the stack depth increases. ", + "bbox": [ + 173, + 631, + 825, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The training data for the Neural Programmer-Interpreter consists of full execution traces for the program of interest. A single element of an execution trace consists of a step input-step output pair, which can be synthesized from Algorithm 1: this corresponds to, for a given time-step, the step input tuple $( e , p , a )$ and step output tuple $( r , p _ { 2 } , a _ { 2 } )$ . An example of part of an addition task trace, written in shorthand, is given in Figure 1. For example, a step input-step output pair in Lines 2 and 3 of the left-hand side of Figure 1 is (ADD1, WRITE OUT 1). In this pair, the step input runs a subprogram ADD1 that has no arguments, and the step output contains a program WRITE that has arguments of OUT and 1. The environment and return probability are omitted for readability. Indentation indicates the stack is one level deeper than before. ", + "bbox": [ + 173, + 722, + 825, + 847 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is important to emphasize that at inference time in the NPI, the hidden state of the LSTM controller is reset (to zero) at each subprogram call, as in Line 3 of Algorithm 1 $h \\mathbf { 0 }$ ). This functionality is critical for implementing recursion, since it permits us to restrict our attention to the currently relevant recursive call, ignoring irrelevant details about other contexts. ", + "bbox": [ + 174, + 853, + 823, + 882 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/400738f1bb6ec680e00398081df8b91a36e097048042ec5fdd599e1e2221cb5f.jpg", + "image_caption": [ + "Figure 1: Addition Task. The non-recursive trace loops on cycles of ADD1 and LSHIFT, whereas in the recursive version, the ADD function calls itself (bolded). " + ], + "image_footnote": [], + "bbox": [ + 295, + 98, + 666, + 246 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 318, + 823, + 345 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 RECURSIVE FORMULATIONS FOR NPI PROGRAMS ", + "text_level": 1, + "bbox": [ + 174, + 363, + 558, + 377 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We emphasize the overall goal of this work is to enable the learning of a recursive program. The learned recursive program is different from neural programs learned in all previous work in an important aspect: previous approaches do not explicitly incorporate this abstraction, and hence generalize poorly, whereas our learned neural programs incorporate recursion and achieve perfect generalization. ", + "bbox": [ + 174, + 390, + 825, + 459 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since NPI naturally supports the notion of recursion, a key question is how to enable NPI to learn recursive programs. We found that changing the NPI training traces is a simple way to enable this. In particular, we construct new training traces which explicitly contain recursive elements and show that with this type of trace, NPI easily learns recursive programs. In future work, we would like to decrease supervision and construct models that are capable of coming up with recursive abstractions themselves. ", + "bbox": [ + 173, + 467, + 825, + 550 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In what follows, we describe the way in which we constructed NPI training traces so as to make them contain recursive elements and thus enable NPI to learn recursive programs. We describe the recursive re-formulation of traces for two tasks from the original NPI paper—grade-school addition and bubble sort. For these programs, we re-use the appropriate program sets (the associated subprograms), and we refer the reader to the appendix of Reed & de Freitas (2016) for further details on the subprograms used in addition and bubble sort. Finally, we implement recursive traces for our own topological sort and quicksort tasks. ", + "bbox": [ + 173, + 556, + 825, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Grade School Addition. For grade-school addition, the domain-specific encoder is ", + "bbox": [ + 178, + 670, + 732, + 685 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6c1b6d2e5e1d9059fe38faa7c40cbd3c86e9a2bd9bba26354ec307dad09105fd.jpg", + "text": "$$\n\\begin{array} { r } { f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , i _ { 4 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } ) , Q ( 4 , i _ { 4 } ) , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 693, + 805, + 710 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the environment $Q \\in \\mathbb { R } ^ { 4 \\times N \\times K }$ is a scratch-pad that contains four rows (the first input number, the second input number, the carry bits, and the output) and $N$ columns. $K$ is set to 11, to represent the range of 10 possible digits, along with a token representing the end of input.2 At any given time, the NPI has access to values pointed to by four pointers in each of the four rows, represented by $Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } )$ , and $Q ( 4 , i _ { 4 } )$ . ", + "bbox": [ + 173, + 717, + 825, + 789 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The non-recursive trace loops on cycles of ADD1 and LSHIFT. ADD1 is a subprogram that adds the current column (writing the appropriate digit to the output row and carrying a bit to the next column if needed). LSHIFT moves the four pointers to the left, to move to the next column. The program terminates when seeing no numbers in the current column. ", + "bbox": [ + 173, + 794, + 825, + 851 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 1 shows examples of non-recursive and recursive addition traces. We make the trace recursive by adding a tail recursive call into the trace for the ADD program after calling ADD1 and LSHIFT, ", + "bbox": [ + 174, + 857, + 821, + 886 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Full Recursive ", + "text_level": 1, + "bbox": [ + 643, + 99, + 740, + 113 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/54ac005719045e00f6d938466b5b91cbc2d9b186869b92dd8c3b93eb2423cdb1.jpg", + "image_caption": [ + "Figure 2: Bubble Sort Task. The non-recursive trace loops on cycles of BUBBLE and RESET. The difference between the partial recursive and full recursive versions is in the indentation of Lines 10-15 and 20-22 (bolded), since in the full recursive version, BSTEP and LSHIFT are made tail recursive; the final calls to BSTEP and LSHIFT return immediately as they occur after the pointer reaches the end of the array. Also note that COMPSWAP conditionally swaps numbers under the bubble pointers. " + ], + "image_footnote": [], + "bbox": [ + 192, + 104, + 776, + 369 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "as in Line 13 of the right-hand side of Figure 1. Via the recursive call, we effectively forget that the column just added exists, since the recursive call to ADD starts with a new hidden state for the LSTM controller. Consequently, there is no concept of length relevant to the problem, which has traditionally been an important focus of length-based curriculum learning. ", + "bbox": [ + 174, + 500, + 825, + 556 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Bubble Sort. For bubble sort, the domain-specific encoder is ", + "bbox": [ + 173, + 577, + 586, + 592 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c97b7e8ea72d9311b1673fe553df45d3a94bc2d0065e66963871c32bbca0af23.jpg", + "text": "$$\nf _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) , Q ( 1 , i _ { 2 } ) , i _ { 3 } = l e n g t h , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) ,\n$$", + "text_format": "latex", + "bbox": [ + 212, + 602, + 784, + 621 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where the environment $Q \\in \\mathbb { R } ^ { 1 \\times N \\times K }$ is a scratch-pad that contains 1 row, to represent the state of the array as sorting proceeds in-place, and $N$ columns. $K$ is set to 11, to denote the range of possible numbers (0 through 9), along with the start/end token (represented with the same encoding) which is observed when a pointer reaches beyond the bounds of the input. At any given time, the NPI has access to the values referred to by two pointers, represented by $Q ( 1 , i _ { 1 } )$ and $Q ( 1 , i _ { 2 } )$ ,. The pointers at index $i _ { 1 }$ and $i _ { 2 }$ are used to compare the pair of numbers considered during the bubble sweep, swapping them if the number at $i _ { 1 }$ is greater than that in $i _ { 2 }$ . These pointers are referred to as bubble pointers. The pointer at index $i _ { 3 }$ represents a counter internal to the environment that is incremented once after each pass of the algorithm (one cycle of BUBBLE and RESET); when incremented a number of times equal to the length of the array, the flag $i _ { 3 } = =$ length becomes true and terminates the entire algorithm . ", + "bbox": [ + 174, + 630, + 825, + 784 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The non-recursive trace loops on cycles of BUBBLE and RESET, which logically represents one bubble sweep through the array and reset of the two bubble pointers to the very beginning of the array, respectively. In this version, there is a dependence on length: BSTEP and LSHIFT are called a number of times equivalent to one less than the length of the input array, in BUBBLE and RESET respectively. ", + "bbox": [ + 174, + 790, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Inside BUBBLE and RESET, there are two operations that can be made recursive. BSTEP, used in BUBBLE, compares pairs of numbers, continuously moving the bubble pointers once to the right each time until reaching the end of the array. LSHIFT, used in RESET, shifts the pointers left until reaching the start token. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We experiment with two levels of recursion—partial and full. Partial recursion only adds a tail recursive call to BUBBLESORT after BUBBLE and RESET, similar to the tail recursive call described previously for addition. The partial recursion is not enough for perfect generalization, as will be presented later in Section 4. Full recursion, in addition to making the aforementioned tail recursive call, adds two additional recursive calls; BSTEP and LSHIFT are made tail recursive. Figure 2 shows examples of traces for the different versions of bubble sort. Training on the full recursive trace leads to perfect generalization, as shown in Section 4. We performed experiments on the partially recursive version in order to examine what happens when only one recursive call is implemented, when in reality three are required for perfect generalization. ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Algorithm 2 Depth First Search Topological Sort ", + "text_level": 1, + "bbox": [ + 176, + 243, + 500, + 258 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1: Color all vertices white. \n2: Initialize an empty stack $S$ and a directed acyclic graph $D A G$ to traverse. 3: Begin traversing from Vertex 1 in the DAG. 4: function TOPOSORT $( D A G )$ 5: while there is still a white vertex $u$ : do 6: color[u] = grey 7: $v _ { a c t i v e } = u$ 8: do \n9: if $v _ { a c t i v e }$ has a white child $v$ then \n10: $\\operatorname { c o l o r } [ v ] = \\operatorname { g r e y }$ \n11: push $v _ { a c t i v e }$ onto $S$ \n12: $v _ { a c t i v e } = v$ \n13: else \n14: $\\mathrm { c o l o r } [ v _ { a c t i v e } ] = \\mathrm { b l a c k }$ \n15: Write $v _ { a c t i v e }$ to result \n16: if $S$ is empty then pass \n17: else pop the top vertex off $S$ and set it to $v _ { a c t i v e }$ \n18: while $S$ is not empty ", + "bbox": [ + 176, + 260, + 689, + 517 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Topological Sort. We choose to implement a topological sort task for graphs. A topological sort is a linear ordering of vertices such that for every directed edge $( u , v )$ from $u$ to $v , u$ comes before $v$ in the ordering. This is possible if and only if the graph has no directed cycles; that is to say, it must be a directed acyclic graph (DAG). In our experiments, we only present DAG’s as inputs and represent the vertices as values ranging from $1 , \\ldots , n$ , where the DAG contains $n$ vertices. ", + "bbox": [ + 174, + 539, + 825, + 609 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Directed acyclic graphs are structurally more diverse than inputs in the two tasks of grade-school addition and bubble sort. The degree for any vertex in the DAG is variable. Also the DAG can have potentially more than one connected component, meaning it is necessary to transition between these components appropriately. ", + "bbox": [ + 174, + 616, + 825, + 672 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Algorithm 2 shows the topological sort task of interest. This algorithm is a variant of depth first search. We created a program set that reflects the semantics of Algorithm 2. For brevity, we refer the reader to the appendix for further details on the program set and non-recursive and recursive trace-generating functions used for topological sort. ", + "bbox": [ + 174, + 679, + 825, + 736 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For topological sort, the domain-specific encoder is ", + "bbox": [ + 176, + 742, + 511, + 756 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/a592788c363fd85d2c133b3b98a5611788d42e231bad314abd3d9ec7325ac3e1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\ f _ { e n c } ( D A G , Q _ { c o l o r } , p _ { s t a c k } , p _ { s t a r t } , v _ { a c t i v e } , c h i l d L i s t , a _ { t } ) } \\\\ & { = M L P ( [ Q _ { c o l o r } ( p _ { s t a r t } ) , Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ) , p _ { s t a c k } = = 1 , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) , a _ { t } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 763, + 839, + 800 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $Q _ { c o l o r } \\in \\mathbb { R } ^ { U \\times 4 }$ is a scratch-pad that contains $U$ rows, each containing one of four colors (white, gray, black, invalid) with one-hot encoding. $U$ varies with the number of vertices in the graph. We further have $Q _ { r e s u l t } \\in \\mathbb { N } ^ { U }$ , a scratch-pad which contains the sorted list of vertices at the end of execution, and $Q _ { s t a c k } \\in \\mathbb { N } ^ { U }$ , which serves the role of the stack $S$ in Algorithm 2. The contents of $Q _ { r e s u l t }$ and $Q _ { s t a c k }$ are not exposed directly through the domain-specific encoder; rather, we define primitive functions which manipulate these scratch-pads. ", + "bbox": [ + 176, + 804, + 826, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The DAG is represented as an adjacency list where $D A G [ i ] [ j ]$ refers to the $j$ -th child of vertex $i$ . There are 3 pointers $( p _ { r e s u l t } , p _ { s t a c k } , p _ { s t a r t } )$ , $p _ { r e s u l t }$ points to the next empty location in $Q _ { r e s u l t }$ $p _ { s t a c k }$ points to the top of the stack in $Q _ { s t a c k }$ , and $p _ { s t a r t }$ points to the candidate starting node for a connected component. There are 2 variables ( $\\boldsymbol { v } _ { a c t i v e }$ and $v _ { s a v e . }$ ); $v _ { a c t i v e }$ holds the active vertex (as in Algorithm 2) and $v _ { s a v e }$ holds the value of $v _ { a c t i v e }$ before executing Line 12 of Algorithm 2. $c h i l d L i s t \\in \\mathbb { N } ^ { U }$ is a vector of pointers, where childList[i] points to the next child under consideration for vertex $i$ . ", + "bbox": [ + 174, + 895, + 823, + 925 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 172 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The three environment observations aid with control flow in Algorithm 2. $Q _ { c o l o r } ( p _ { s t a r t } )$ contains the color of the current start vertex, used in the evaluation of the condition in the WHILE loop in Line 5 of Algorithm 2. $Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] )$ refers to the color of the next child of $v _ { a c t i v e }$ , used in the evaluation of the condition in the IF branch in Line 9 of Algorithm 2. Finally, the boolean $p _ { s t a c k } = = 1$ is used to check whether the stack is empty in Line 18 of Algorithm 2. ", + "bbox": [ + 174, + 180, + 825, + 251 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "An alternative way of representing the environment slice is to expose the values of the absolute vertices to the model; however, this makes it difficult to scale the model to larger graphs, since large vertex values are not seen during training time. ", + "bbox": [ + 176, + 257, + 825, + 299 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We refer the reader to the appendix for the non-recursive trace generating functions. In the non-recursive trace, there are four functions that can be made recursive—TOPOSORT, CHECK CHILD, EXPLORE, and NEXT START, and we add a tail recursive call to each of these functions in order to make the recursive trace. In particular, in the EXPLORE function, adding a tail recursive call resets and stores the hidden states associated with vertices in a stack-like fashion. This makes it so that we only need to consider the vertices in the subgraph that are currently relevant for computing the sort, allowing simpler reasoning about behavior for large graphs. The sequence of primitive operations (MOVE and WRITE operations) for the non-recursive and recursive versions are exactly the same. ", + "bbox": [ + 174, + 306, + 825, + 431 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Quicksort. We implement a quicksort task, in order to demonstrate that recursion helps with learning divide-and-conquer algorithms. We use the Lomuto partition scheme; the logic for the recursive trace is shown in Algorithm 3. For brevity, we refer the reader to the appendix for information about the program set and non-recursive and recursive trace-generating functions for quicksort. The logic for the non-recursive trace is shown in Algorithm 4 in the appendix. ", + "bbox": [ + 174, + 446, + 825, + 517 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Algorithm 3 Recursive Quicksort ", + "text_level": 1, + "bbox": [ + 176, + 532, + 398, + 546 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "1: Initialize an array $A$ to sort. \n2: Initialize $l o$ and $h i$ to be 1 and $n$ , where $n$ is the length of $A$ . \n3: \n4: function QUICKSORT $( A , l o , h i )$ \n5: if $l o < h i$ : then \n6: $\\mathsf { p } = \\mathsf { P A R T I T I O N } ( A , l o , h i )$ \n7: $\\mathrm { Q U I C K S O R T } ( A , l o , p - 1 )$ \n8: $\\operatorname { Q U I C K S O R T } ( A , p + 1 , h i )$ \n9: \n10: function PARTITION $( A , l o , h i )$ \n11: $p i v o t = l o$ \n12: for $j \\in [ l o , h i - 1 ] : \\mathbf { d o }$ \n13: if $A [ j ] \\leq A [ h i ]$ then \n14: swap A[pivot] with $A [ j ]$ \n15: pivot = pivot + 1 \n16: swap A[pivot] with A[hi] \n17: return pivot ", + "bbox": [ + 178, + 550, + 602, + 794 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For quicksort, the domain-specific encoder is ", + "bbox": [ + 173, + 809, + 470, + 824 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/1e687a901424b80f7b8998994be9caf74d69a7b05809a2c4c258213aa1c3636c.jpg", + "text": "$$\n\\begin{array} { r l } & { f _ { e n c } ( Q _ { a r r a y } , Q _ { s t a c k L o } , Q _ { s t a c k H i } , p _ { l o } , p _ { h i } , p _ { s t a c k L o } , p _ { s t a c k H i } , p _ { p i v o t } , p _ { j } , a _ { t } ) = } \\\\ & { \\qquad M L P ( [ Q _ { a r r a y } ( p _ { j } ) \\leq Q _ { a r r a y } ( p _ { h i } ) , p _ { j } = = p _ { h i } , } \\\\ & { \\qquad Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 ) , p _ { s t a c k L o } = = } \\\\ & { \\qquad \\quad M L P ( [ Q _ { a r r a y } ( p _ { j } ) \\leq \\mathfrak { a } _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 834, + 807, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $Q _ { a r r a y } \\in \\mathbb { R } ^ { U \\times 1 1 }$ is a scratch-pad that contains $U$ rows, each containing one of 11 values (one of the numbers 0 through 9 or an invalid state). Our implementation uses two stacks $Q _ { s t a c k L o }$ and ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$Q _ { s t a c k H i }$ , each in $\\mathbb { R } ^ { U }$ , that store the arguments to the recursive QUICKSORT calls in Algorithm 3; before each recursive call, the appropriate arguments are popped off the stack and written to $p _ { l o }$ and $p _ { h i }$ . ", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "There are 6 pointers $( p _ { l o } , p _ { h i } , p _ { s t a c k L o } , p _ { s t a c k H i } , p _ { p i v o t } , p _ { j } )$ . $p _ { l o }$ and $p _ { h i }$ point to the lo and hi indices of the array, as in Algorithm 3. $p _ { s t a c k L o }$ and $p _ { s t a c k H i }$ point to the top (empty) positions in $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . $p _ { p i v o t }$ and $p _ { j }$ point to the pivot and $j$ indices of the array, used in the PARTITION function in Algorithm 3. The 4 environment observations aid with control flow; $Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 )$ implements the $l o < h i$ comparison in Line 5 of Algorithm 3, $p _ { s t a c k L o } = = 1$ checks if the stacks are empty in Line 18 of Algorithm 4, and the other observations (all involving $p _ { p i v o t }$ or $p _ { j }$ ) deal with logic in the PARTITION function. ", + "bbox": [ + 174, + 154, + 825, + 251 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Note that the recursion for quicksort is not purely tail recursive and therefore represents a more complex kind of recursion that is harder to learn than in the previous tasks. Also, compared to the bubble pointers in bubble sort, the pointers that perform the comparison for quicksort (the COMPSWAP function) are usually not adjacent to each other, making quicksort less local than bubble sort. In order to compensate for this, $p _ { p i v o t }$ and $p _ { j }$ require special functions (MOVE PIVOT LO and MOVE J LO) to properly set them to $l o$ in Lines 11 and 12 of the PARTITION function in Algorithm 3. ", + "bbox": [ + 174, + 257, + 825, + 354 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.3 PROVABLY PERFECT GENERALIZATION ", + "text_level": 1, + "bbox": [ + 178, + 371, + 486, + 385 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We show that if we incorporate recursion, the learned NPI programs can achieve provably perfect generalization for different tasks. Provably perfect generalization implies the model will behave correctly, given any valid input. In order to claim a proof, we must verify the model produces correct behavior over all base cases and reductions, as described in Section 2. ", + "bbox": [ + 174, + 397, + 825, + 453 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose and describe our verification procedure. This procedure verifies that all base cases and reductions are handled properly by the model via explicit tests. Note that recursion helps make this process tractable, because we only need to test a finite number of inputs to show that the model will work correctly on inputs of unbounded complexity. This verification phase only needs to be performed once after training. ", + "bbox": [ + 174, + 459, + 825, + 530 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Formally, verification consists of proving the following theorem: ", + "bbox": [ + 174, + 536, + 598, + 551 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/c8b90ef3f1f4a594a65d98322c24c8c407c2f5d51069a21a999c875e2f9ac463.jpg", + "text": "$$\n\\forall i \\in V , M ( i ) \\downdownarrows P ( i )\n$$", + "text_format": "latex", + "bbox": [ + 426, + 555, + 570, + 573 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $i$ denotes a sequence of step inputs (within one function call), $V$ denotes the set of valid sequences of step inputs, $M$ denotes the neural network model, $P$ denotes the correct program, and $P ( i )$ denotes the next step output from the correct program. The arrow in the theorem refers to evaluation, as in big-step semantics. The theorem states that for the same sequence of step inputs, the model produces the exact same step output as the target program it aims to learn. $M$ , as described in Algorithm 1, processes the sequence of step inputs by using an LSTM. ", + "bbox": [ + 174, + 575, + 825, + 660 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Recursion drastically reduces the number of configurations we need to consider during the verification phase and makes the proof tractable, because it introduces structure that eliminates infinitely long sequences of step inputs that would otherwise need to be considered. For instance, for recursive addition, consider the family $F$ of addition problems $a _ { n } a _ { n - 1 } \\dots a _ { 1 } a _ { 0 } + b _ { n } b _ { n - 1 } \\dots b _ { 1 } b _ { 0 }$ where no CARRY operations occur. We prove every member of $F$ is added properly, given that subproblems $S = \\{ a _ { n } a _ { n - 1 } + b _ { n } b _ { n - 1 } , a _ { n - 1 } { \\bar { a } } _ { n - 2 } + b _ { n - 1 } b _ { n - 2 } , \\dots , a _ { 1 } a _ { 0 } + b _ { 1 } b _ { 0 } \\}$ are added properly. ", + "bbox": [ + 174, + 666, + 825, + 751 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Without using a recursive program, such a proof is not possible, because the non-recursive program runs on an arbitrarily long addition problem that creates correspondingly long sequences of step inputs; in the non-recursive formulation of addition, ADD calls ADD1 a number of times that is dependent on the length of the input. The core LSTM module’s hidden state is preserved over all these ADD1 calls, and it is difficult to interpret with certainty what happens over longer timesteps without concretely evaluating the LSTM with an input of that length. In contrast, each call to the recursive ADD always runs for a fixed number of steps, even on arbitrarily long problems in $F$ , so we can test that it performs correctly on a small, fixed number of step input sequences. This guarantees that the step input sequences considered during verification contain all step input sequences which arise during execution of an unseen problem in $F$ , leading to generalization to any problem in $F$ . Hence, if all subproblems in $S$ are added correctly, we have proven that any member of $F$ will be added correctly, thus eliminating an infinite family of inputs that need to be tested. ", + "bbox": [ + 173, + 757, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To perform the verification as described here, it is critical to construct $V$ correctly. If it is too small, then execution of the program on some input might require evaluation of $M ( i )$ on some $i \\not \\in V$ , and so the behavior of $M ( i )$ might deviate from $P ( i )$ . If it is too large, then the semantics of $P$ might not be well-defined on some elements in $V$ , or the spurious step input sequences may not be reachable from any valid problem input (e.g., an array for quicksort or a DAG for topological sort). ", + "bbox": [ + 174, + 103, + 823, + 174 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "To construct this set, by using the reference implementation of each subprogram, we construct a mapping between two sets of environment observations: the first set consists of all observations that can occur at the beginning of a particular subprogram’s invocation, and the second set contains the observations at the end of that subprogram. We can obtain this mapping by first considering the possible observations that can arise at the beginning of the entry function (ADD, BUBBLESORT, TOPOSORT, and QUICKSORT) for some valid program input, and iteratively applying the observation-to-observation mapping implied by the reference implementation’s step output at that point in the execution. If the step output specifies a primitive function call, we need to reason about how it can affect the environment so as to change the observation in the next step input. For non-primitive subprograms, we can update the observation-to-observation mapping currently associated with the subprogram and then apply that mapping to the current set. By iterating with this procedure, and then running $P$ on the input observation set that we obtain for the entry point function, we can obtain $V$ precisely. To make an analogy to MDPs, this procedure is analogous to how value iteration obtains the correct value for each state starting from any initialization. ", + "bbox": [ + 174, + 180, + 825, + 375 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "An alternative method is to run $P$ on many different program inputs and then observe step input sequences which occur, to create $V$ . However, to be sure that the generated $V$ is complete (covers all the cases needed), we need to check all pairs of observations seen in adjacent step inputs (in particular, those before and after a primitive function call), in a similar way as if we were constructing $V$ from scratch. Given a precise definition of $P$ , it may be possible to automate the generation of $V$ from $P$ in future work. ", + "bbox": [ + 174, + 382, + 823, + 465 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Note that $V$ should also contain the necessary reductions, which corresponds to making the recursive calls at the correct time, as indicated by $P$ . ", + "bbox": [ + 176, + 472, + 821, + 501 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "After finding $V$ , we construct a set of problem inputs which, when executed on $P$ , create exactly the step input sequences which make up $V$ . We call this set of inputs the verification set, $S _ { V }$ . ", + "bbox": [ + 176, + 507, + 820, + 536 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Given a verification set, we can then run the model on the verification set to check if the produced traces and results are correct. If yes, then this indicates that the learned neural program achieves provably perfect generalization. ", + "bbox": [ + 176, + 542, + 821, + 584 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We note that for tasks with very large input domains, such as ones involving MNIST digits or speech samples, the state space of base cases and reduction rules could be prohibitively large, possibly infinite. Consequently, it is infeasible to construct a verification set that covers all cases, and the verification procedure we have described is inadequate. We leave this as future work to devise a verification procedure more appropriate to this setting. ", + "bbox": [ + 176, + 592, + 825, + 661 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 685, + 326, + 702 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "As there is no public implementation of NPI, we implemented a version of it in Keras that is as faithful to the paper as possible. Our experiments use a small number of training examples. ", + "bbox": [ + 174, + 719, + 823, + 748 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Training Setup. The training set for addition contains 200 traces. The maximum problem length in this training set is 3 (e.g., the trace corresponding to the problem $^ { \\mathrm { \\left. } } 1 0 9 + 1 0 1 ^ { \\mathrm { \\right. } }$ ). ", + "bbox": [ + 173, + 768, + 823, + 796 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The training set for bubble sort contains 100 traces, with maximum problem length of 2 (e.g., the trace corresponding to the array [3,2]). ", + "bbox": [ + 174, + 803, + 823, + 832 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The training set for topological sort contains 6 traces, with one synthesized from a graph of size 5 and the rest synthesized from graphs of size 7. ", + "bbox": [ + 173, + 838, + 823, + 867 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The training set for quicksort contains 4 traces, synthesized from arrays of length 5. ", + "bbox": [ + 174, + 875, + 722, + 888 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The same set of problems was used to generate the training traces for all formulations of the task, for non-recursive and recursive versions. ", + "bbox": [ + 173, + 895, + 823, + 922 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/677499324ed68bf480a7e15b02f721a5e60409de047d5e783482d2582f0c0582.jpg", + "table_caption": [ + "Table 1: Accuracy on Randomly Generated Problems for Bubble Sort " + ], + "table_footnote": [], + "table_body": "
Length of ArrayNon-RecursivePartiallyRecursiveFullRecursive
２3100%100%100%
6.7%23%100%
410%10%100%
80% 0%0% 0%100% 100%
20100%
900%0%
", + "bbox": [ + 233, + 131, + 764, + 248 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We train using the Adam optimizer and use a 2-layer LSTM and task-specific state encoders for the external environments, as described in Reed & de Freitas (2016). ", + "bbox": [ + 173, + 281, + 825, + 309 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "4.1 RESULTS ON GENERALIZATION OF RECURSIVE NEURAL PROGRAMS ", + "bbox": [ + 176, + 329, + 687, + 344 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We now report on generalization for the varying tasks. ", + "bbox": [ + 176, + 357, + 529, + 371 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Grade-School Addition. Both the non-recursive and recursive learned programs generalize on all input lengths we tried, up to 5000 digits. This agrees with the generalization of non-recursive addition in Reed & de Freitas (2016), where they reported generalization up to 3000 digits. However, note that there is no provable guarantee that the non-recursive learned program will generalize to all inputs, whereas we show later that the recursive learned program has a provable guarantee of perfect generalization. ", + "bbox": [ + 174, + 390, + 825, + 473 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In order to demonstrate that recursion can help learn and generalize better, for addition, we trained only on traces for 5 arbitrarily chosen 1-digit addition sum examples. The recursive version can generalize perfectly to long problems constructed from these components (such as the sum $^ { 6 6 } 8 2 2 + 2 3 3 ^ { 3 }$ , where $\\because 8 + 2 ^ { , 5 }$ and $\\bar { 2 } + 3 \\bar { 2 }$ are in the training set), but the non-recursive version fails to sum these long problems properly. ", + "bbox": [ + 174, + 481, + 825, + 551 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Bubble Sort. Table 1 presents results on randomly generated arrays of varying length for the learned non-recursive, partially recursive, and full recursive programs. For each length, we test each program on 30 randomly generated problems. Observe that partially recursive does slightly better than non-recursive for the setting in which the length of the array is 3, and that the fully recursive version is able to sort every array given to it. The non-recursive and partially recursive versions are unable to sort long arrays, beyond length 8. ", + "bbox": [ + 174, + 569, + 825, + 654 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Topological Sort. Both the non-recursive and recursive learned programs generalize on all graphs we tried, up to 120 vertices. As before, the non-recursive learned program lacks a provable guarantee of generalization, whereas we show later that the recursive learned program has one. ", + "bbox": [ + 174, + 672, + 825, + 714 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In order to demonstrate that recursion can help learn and generalize better, we trained a non-recursive and recursive model on just a single execution trace generated from a graph containing 5 nodes3 for the topological sort task. For these models, Table 2 presents results on randomly generated DAGs of varying graph sizes (varying in the number of vertices). For each graph size, we test the learned programs on 30 randomly generated DAGs. The recursive version of topological sort solves all graph instances we tried, from graphs of size 5 through 70. On the other hand, the non-recursive version has low accuracy, beginning from size 5, and fails completely for graphs of size 8 and beyond. ", + "bbox": [ + 174, + 720, + 825, + 819 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Quicksort. Table 3 presents results on randomly generated arrays of varying length for the learned non-recursive and recursive programs. For each length, we test each program on 30 randomly generated problems. Observe that the non-recursive program’s correctness degrades for length 11 and beyond, while the recursive program can sort any given array. ", + "bbox": [ + 174, + 838, + 825, + 893 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/951009b59193f83793a8ee61d1018dd22ca8ecf19a9b7466ad10ba0e9fc84226.jpg", + "table_caption": [ + "Table 2: Accuracy on Randomly Generated Problems for Topological Sort " + ], + "table_footnote": [], + "table_body": "
NumberofVerticesNon-RecursiveRecursive
56.7%100%
66.7%100%
73.3%100%
80%100%
700%100%
", + "bbox": [ + 315, + 131, + 683, + 231 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/9811922eec9e8a30d6382d695ea75057133fee7a76233a946edf07d7de5ba752.jpg", + "table_caption": [ + "Table 3: Accuracy on Randomly Generated Problems for Quicksort " + ], + "table_footnote": [], + "table_body": "
LengthofArrayNon-RecursiveRecursive
3100%100%
5100%100%
7100%100%
1173.3%100%
1560%100%
2030%
2220%100%
253.33%100%
303.33%100%
100%
700%100%
", + "bbox": [ + 325, + 284, + 674, + 453 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "As mentioned in Section 2.1, we hypothesize the non-recursive programs do not generalize well because they have learned spurious dependencies specific to the training set, such as length of the input problems. On the other hand, the recursive programs have learned the true program semantics. ", + "bbox": [ + 176, + 487, + 823, + 530 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "4.2 VERIFICATION OF PROVABLY PERFECT GENERALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 547, + 612, + 563 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We describe how models trained with recursive traces can be proven to generalize, by using the verification procedure described in Section 3.3. As described in the verification procedure, it is possible to prove our learned recursive program generalizes perfectly by testing on an appropriate set of problem inputs, i.e., the verification set. Recall that this verification procedure cannot be performed for the non-recursive versions, since the propagation of the hidden state in the core LSTM module makes reasoning difficult and so we would need to check an unbounded number of examples. ", + "bbox": [ + 174, + 575, + 825, + 659 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We describe the base cases, reduction rules, and the verification set for each task in Appendix A.6. For each task, given the verification set, we check the traces and results of the learned, to-be-verified neural program (described in Section 4.1; and for bubble sort, Appendix A.6) on the verification set, and ensure they match the traces produced by the true program $P$ . Our results show that for all learned, to-be-verified neural programs, they all produced the same traces as those produced by $P$ on the verification set. Thus, we demonstrate that recursion enables provably perfect generalization for different tasks, including addition, topological sort, quicksort, and a variant of bubble sort. ", + "bbox": [ + 174, + 666, + 825, + 763 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Note that the training set can often be considerably smaller than the verification set, and despite this, the learned model can still pass the entire verification set. Our result shows that the training procedure and the NPI architecture is capable of generalizing from the step input-output pairs seen in the training data to the unseen ones present in the verification set. ", + "bbox": [ + 174, + 770, + 823, + 827 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 848, + 318, + 864 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We emphasize that the notion of a neural recursive program has not been presented in the literature before: this is our main contribution. Recursion enables provably perfect generalization. To the best of our knowledge, this is the first time verification has been applied to a neural program, providing provable guarantees about its behavior. We instantiated recursion for the Neural ProgrammerInterpreter by changing the training traces. In future work, we seek to enable more tasks with recursive structure. We also hope to decrease supervision, for example by training with only partial or non-recursive traces, and to develop novel Neural Programming Architectures integrated directly with a notion of recursion. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 172 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 190, + 326, + 203 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This material is in part based upon work supported by the National Science Foundation under Grant No. TWC-1409915, DARPA under Grant No. FA8750-15-2-0104, and Berkeley Deep Drive. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of National Science Foundation and DARPA. ", + "bbox": [ + 176, + 213, + 825, + 268 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 290, + 285, + 305 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Marcin Andrychowicz and Karol Kurach. Learning efficient algorithms with hierarchical attentive memory. CoRR, abs/1602.03218, 2016. URL http://arxiv.org/abs/1602.03218. ", + "bbox": [ + 173, + 314, + 825, + 342 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. CoRR, abs/1410.5401, 2014. URL http://arxiv.org/abs/1410.5401. ", + "bbox": [ + 169, + 351, + 823, + 380 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwiska, Sergio Gmez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, Adri Puigdomnech Badia, Karl Moritz Hermann, Yori Zwols, Georg Ostrovski, Adam Cain, Helen King, Christopher Summerfield, Phil Blunsom, Koray Kavukcuoglu, and Demis Hassabis. Hybrid computing using a neural network with dynamic external memory. Nature, 538 (7626):471–476, October 2016. ISSN 0028-0836, 1476-4687. doi: 10.1038/nature20101. URL http://www.nature.com/doifinder/10.1038/nature20101. ", + "bbox": [ + 176, + 388, + 825, + 486 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lukasz Kaiser and Ilya Sutskever. Neural gpus learn algorithms. CoRR, abs/1511.08228, 2015. URL http://arxiv.org/abs/1511.08228. ", + "bbox": [ + 174, + 496, + 820, + 525 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Karol Kurach, Marcin Andrychowicz, and Ilya Sutskever. Neural random access machines. ERCIM News, 2016(107), 2016. URL http://ercim-news.ercim.eu/en107/special/ neural-random-access-machines. ", + "bbox": [ + 174, + 534, + 825, + 575 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Arvind Neelakantan, Quoc V. Le, and Ilya Sutskever. Neural programmer: Inducing latent programs with gradient descent, 2015. ", + "bbox": [ + 174, + 584, + 821, + 613 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Scott Reed and Nando de Freitas. Neural programmer-interpreters. ICLR, 2016. ", + "bbox": [ + 174, + 622, + 699, + 637 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada, pp. 2692–2700, 2015. URL http://papers.nips.cc/paper/5866-pointer-networks. ", + "bbox": [ + 174, + 646, + 825, + 703 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Wojciech Zaremba, Tomas Mikolov, Armand Joulin, and Rob Fergus. Learning simple algorithms from examples. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pp. 421–429, 2016. URL http://jmlr. org/proceedings/papers/v48/zaremba16.html. ", + "bbox": [ + 174, + 712, + 823, + 768 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A APPENDIX ", + "bbox": [ + 176, + 103, + 295, + 117 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/1ac6208d018efec0f8dd33d75e116555f134a4699946cf4c6542ccbd5ac94e7f.jpg", + "table_caption": [ + "A.1 PROGRAM SET FOR NON-RECURSIVE TOPOLOGICAL SORT " + ], + "table_footnote": [], + "table_body": "
ProgramDescriptionsCallsArguments
TOPOSORTPerform topologicalsort on graphTRAVERSE,NEXT_START,WRITE,MOVENONE
TRAVERSETraverse graph untilstack is emptyCHECK_CHILD，EX-PLORENONE
CHECK_CHILDCheck ifawhitechild exists; if so, setchildList[Uactive] topoint to itMOVENONE
EXPLORERepeatedlytraversesubgraphs until stackis emptySTACK,CHECK_CHILD,WRITE,MOVENONE
STACKInteractwith stack,either pushing orpoppingWRITE,MOVEPUSH,POP
NEXT_STARTMove Pstart untilreaching a whitevertex. If a whitevertex is found，setPstart to point to it;this signifies the startof a traversal of anew connected com-ponent.If no whitevertex is found, theentireexecution isterminatedMOVENONE
WRITEWrite a value eitherto environment (e.g.,to color a vertex)or variable (e.g., tochange the value ofUactive)NONEDescribed below
MOVEMove a pointer(e.g, Pstart orchildList[vactive])up or downNONEDescribed below
", + "bbox": [ + 200, + 155, + 797, + 694 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Argument Sets for WRITE and MOVE. ", + "text_level": 1, + "bbox": [ + 174, + 708, + 452, + 724 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "WRITE. The WRITE operation has the following arguments: ", + "bbox": [ + 173, + 741, + 593, + 756 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "ARG 1 (Main Action): COLOR CURR, COLOR NEXT, ACTIVE START, ACTIVE NEIGHB, ACTIVE STACK, SAVE, STACK PUSH, STACK POP, RESULT ", + "text_level": 1, + "bbox": [ + 174, + 761, + 821, + 790 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "COLOR CURR colors $v _ { a c t i v e }$ , COLOR NEXT colors Vertex $D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ,$ , ACTIVE START writes pstart to $v _ { a c t i v e }$ , ACTIVE NEIGHB writes $D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ]$ to $v _ { a c t i v e }$ , ACTIVE STACK writes $Q _ { s t a c k } ( p _ { s t a c k } )$ to $v _ { a c t i v e }$ , SAVE writes $v _ { a c t i v e }$ to $v _ { s a v e }$ , $S T A C K \\_ P U S H$ pushes $v _ { a c t i v e }$ to the top of the stack, $S T A C K \\_ P O P$ writes a null value to the top of the stack, and $R E S U L T$ writes $v _ { a c t i v e }$ to $Q _ { r e s u l t } ( p _ { r e s u l t } )$ . ", + "bbox": [ + 173, + 796, + 825, + 882 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "ARG 2 (Auxiliary Variable): COLOR GREY, COLOR BLACK ", + "bbox": [ + 174, + 887, + 596, + 902 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "COLOR GREY and COLOR BLACK color the given vertex grey and black, respectively. ", + "bbox": [ + 171, + 909, + 763, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "MOVE. The MOVE operation has the following arguments: ", + "bbox": [ + 173, + 103, + 581, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "ARG 1 (Pointer): $p _ { r e s u l t } , p _ { s t a c k } , p _ { s t a r t } , c h i l d L i s t [ v _ { a c t i v e } ] , c h i l d L i s t [ v _ { s a v e } ]$ ", + "bbox": [ + 174, + 123, + 679, + 141 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Note that the argument is the identity of the pointer, not what the pointer points to; in other words, ARG 1 can only take one of 5 values. ", + "bbox": [ + 174, + 146, + 825, + 174 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "ARG 2 (Increment or Decrement): UP, DOWN ", + "bbox": [ + 174, + 181, + 482, + 196 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 TRACE-GENERATING FUNCTIONS FOR TOPOLOGICAL SORT ", + "text_level": 1, + "bbox": [ + 176, + 213, + 632, + 228 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2.1 NON-RECURSIVE TRACE-GENERATING FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 238, + 593, + 253 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1 // Top level topological sort call \n2 TOPOSORT() { \n3 while $( Q _ { c o l o r } \\big ( p _ { s t a r t } \\big )$ is a valid color): // color invalid when all vertices explored \n4 WRITE(ACTIVE_START) \n5 WRITE(COLOR_CURR, COLOR_GREY) \n6 TRAVERSE() \n7 MOVE(pstart, UP) \n8 NEXT_START() \n9 } \n10 \n11 TRAVERSE() { \n12 CHECK_CHILD() \n13 EXPLORE() \n14 } \n15 \n16 CHECK_CHILD() { \n17 while $( Q _ { c o l o r } \\big ( D A G \\big [ v _ { a c t i v e } \\big ] \\big [ c h i l d L i s t \\big [ v _ { a c t i v e } \\big ] \\big ] \\big )$ is not white and is not invalid): // color invalid when all children explored \n18 MOVE(childList[vactive], UP) \n19 } \n20 \n21 EXPLORE() { \n22 do \n23 if (Qcolor(DAG[vactive][childList[vactive]]) is white): \n24 WRITE(COLOR_NEXT, COLOR_GREY) \n25 STACK(PUSH) \n26 WRITE(SAVE) \n27 WRITE(ACTIVE_NEIGHB) \n28 MOVE(childList[vsave], UP) \n29 else: \n30 WRITE(COLOR_CURR, COLOR_BLACK) \n31 WRITE(RESULT) \n32 MOVE(presult, UP) \n33 if(pstack == 1): \n34 break \n35 else: \n36 STACK(POP) \n37 CHECK_CHILD() \n38 while (true) \n39 \n40 \n41 STACK(op) { \n42 if (op == PUSH): \n43 WRITE(STACK_PUSH) \n44 MOVE(pstack, UP) \n45 \n46 if (op == POP): \n47 WRITE(ACTIVE_STACK) \n48 WRITE(STACK_POP) \n49 MOVE(pstack, DOWN) \n50 } \n51 \n52 NEXT_START() { \n53 while(Qcolor(pstart) is not white and is not invalid): // color invalid when all vertices explored \n54 MOVE(pstart, UP) \n55 } ", + "bbox": [ + 145, + 261, + 910, + 753 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2.2 RECURSIVE TRACE-GENERATING FUNCTIONS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 550, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Altered Recursive Functions ", + "text_level": 1, + "bbox": [ + 176, + 138, + 361, + 152 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "1 // Top level topological sort call \n2 TOPOSORT() { \n3 if $\\scriptstyle : Q _ { c o l o r }$ $_ { p _ { s t a r t } } ,$ is a valid color): // color invalid when all vertices explored \n4 WRITE(ACTIVE_START) \n5 WRITE(COLOR_CURR, COLOR_GREY) \n6 TRAVERSE() \n7 MOVE(pstart, UP) \n8 NEXT_START() \n9 TOPOSORT() // Recursive Call \n10 } \n11 \n12 CHECK_CHILD() { \n13 if $\\overline { { ( Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ) } } ) \\overline { { { } } }$ is not white and is not invalid): // color invalid when all children explore \n14 MOVE(childList[vactive], UP) \n15 CHECK_CHILD() // Recursive Call \n16 } \n17 \n18 EXPLORE() { \n19 if (Qcolor(DAG[vactive][childList[vactive]]) is white): \n20 WRITE(COLOR_NEXT, COLOR_GREY) \n21 STACK(PUSH) \n22 WRITE(SAVE) \n23 WRITE(ACTIVE_NEIGHB) \n24 MOVE(childList[vsave], UP) \n25 else: \n26 WRITE(COLOR_CURR, COLOR_BLACK) \n27 WRITE(RESULT) \n28 MOVE(presult, UP) \n29 if $( p _ { s t a c k } = = 1$ ): \n30 return \n31 else: \n32 STACK(POP) \n33 CHECK_CHILD() \n34 EXPLORE() // Recursive Call \n35 } \n36 \n37 NEXT_START() { \n38 if $( Q _ { c o l o r } \\left( p _ { s t a r t } \\right)$ is not white and is not invalid): // color invalid when all vertices explored \n39 MOVE $( p _ { s t a r t }$ , UP) \n40 NEXT_START() // Recursive Call \n41 } ", + "bbox": [ + 147, + 170, + 880, + 532 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Algorithm 4 Iterative Quicksort ", + "text_level": 1, + "bbox": [ + 174, + 627, + 387, + 641 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "1: Initialize an array $A$ to sort and two empty stacks $S _ { l o }$ and $S _ { h i }$ . \n2: Initialize lo and $h i$ to be 1 and $n$ , where $n$ is the length of $A$ . \n3: \n4: function PARTITION $( A , l o , h i )$ \n5: $p i v o t = l o$ \n6: for $j \\in [ l o , h i - 1 ] : \\mathbf { d o }$ \n7: if $A [ j ] \\leq A [ h i ]$ then \n8: swap A[pivot] with $A [ j ]$ \n9: pivot = pivot + 1 \n10: swap A[pivot] with $A [ h i ]$ \n11: return pivot \n12: \n13: function QUICK ${ \\mathrm { : } } \\operatorname { S o R T } ( A , l o , h i )$ \n14: while $S _ { l o }$ and $S _ { h i }$ are not empty: do \n15: Pop states off $S _ { l o }$ and $S _ { h i }$ , writing them to $l o$ and $h i$ . \n16: $\\mathsf { p } = \\mathsf { P A R T I T I O N } ( A , l o , h i )$ \n17: Push $p + 1$ and $h i$ to $S _ { l o }$ and $S _ { h i }$ . \n18: Push lo and $p - 1$ to $S _ { l o }$ and $S _ { h i }$ . ", + "bbox": [ + 176, + 643, + 611, + 902 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/19091c5a8769f57c237066c44de4aee890ae4f19a5be4e2f6be1e61fe6dee7d8.jpg", + "table_caption": [ + "A.4 PROGRAM SET FOR QUICKSORT " + ], + "table_footnote": [], + "table_body": "
ProgramDescriptionsCalls Arguments
QUICKSORTRunthequicksortroutine in place forthearrayA，forindices from lo to hiNon-Recursive: PAR-TITION, STACK,WRITERecursive: same asnon-recursiveversion,alongwithQUICK-SORTImplicitly:arrayA to sort, lo, hi
PARTITIONRunsthepartitionfunction. At end,pointer Ppivot ismoved to the pivotCOMPSWAP_LOOP,MOVE_PIVOT_LO,MOVE_JLO, SWAPNONE
COMPSWAPLOOPRuns the FOR loopinside the partitionfunctionCOMPSWAP, MOVENONE
COMPSWAPComparesA[pivot] ≤ A[j]; ifso,perform a swapand increment PpivotSWAP, MOVENONE
SET_PIVOTLOSets Ppivot to lo in-dexNONENONE
SETJLOSets pj to lo indexNONENONE
SET_J_NULLSets pj to-00NONENONE
STACKPushes lo/hi statesonto stacks Sto andShi according to argument(describedbelow)WRITE, MOVEDescribed below
MOVEMovespointerone unit up or downNONEDescribed below
SWAPSwapselementsatgiven array indicesNONEDescribed below
WRITEWrite a valueeither to stack(e.g QstackLoorQstackHi) or topointer (e.g tochangethe value ofPhi)NONEDescribed below
", + "bbox": [ + 186, + 126, + 810, + 680 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Argument Sets for STACK, MOVE, SWAP, WRITE. ", + "bbox": [ + 173, + 695, + 540, + 712 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "STACK. The STACK operation has the following arguments: ", + "bbox": [ + 173, + 728, + 591, + 743 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ARG 1 (Operation): STACK PUSH CALL1, STACK PUSH CALL2, STACK POP ", + "bbox": [ + 174, + 750, + 725, + 765 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "STACK PUSH CALL1 pushes $l o$ and pivot−1 to $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . STACK PUSH CALL2 pushes pivot $+ 1$ and $h i$ to $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . STACK POP pushes $- \\infty$ values to $Q _ { s t a c k L o }$ and $Q _ { s t a c k H i }$ . ", + "bbox": [ + 174, + 771, + 823, + 815 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "MOVE. The MOVE operation has the following arguments: ", + "bbox": [ + 173, + 830, + 581, + 847 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ARG 1 (Pointer): pstackLo, pstackHi, pj , ppivot ", + "bbox": [ + 174, + 853, + 482, + 869 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Note that the argument is the identity of the pointer, not what the pointer points to; in other words, ARG 1 can only take one of 4 values. ", + "bbox": [ + 176, + 875, + 825, + 902 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ARG 2 (Increment or Decrement): UP, DOWN ", + "bbox": [ + 173, + 909, + 482, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "SWAP. The SWAP operation has the following arguments: ", + "bbox": [ + 173, + 103, + 571, + 118 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ARG 1 (Swap Object 1): ppivot ", + "bbox": [ + 173, + 125, + 380, + 140 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ARG 2 (Swap Object 2): $p _ { h i } , p _ { j }$ ", + "bbox": [ + 174, + 146, + 385, + 161 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "WRITE. The WRITE operation has the following arguments: ", + "bbox": [ + 173, + 178, + 593, + 193 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ARG 1 (Object to Write): ENV STACK LO, ENV STACK HI, $p _ { h i } , p _ { l o }$ ", + "bbox": [ + 174, + 198, + 645, + 213 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ENV STACK LO and ENV STACK HI represent $Q _ { s t a c k L o } ( p _ { s t a c k L o } ) $ and $Q _ { s t a c k H i } ( p _ { s t a c k H i } )$ , re spectively. ", + "bbox": [ + 174, + 219, + 818, + 250 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ARG 2 (Object to Copy): ENV STACK LO PEEK, ENV STACK HI PEEK, $p _ { h i } , p _ { l o } , p _ { p i v o t } - 1$ ppivot + 1, RESET ", + "bbox": [ + 171, + 255, + 823, + 284 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ENV STACK LO PEEK and ENV STACK HI PEEK represent $Q _ { s t a c k L o } ( p _ { s t a c k L o } \\mathrm { ~ - ~ } 1 )$ and $Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 )$ , respectively. RESET represents a $- \\infty$ value. ", + "bbox": [ + 174, + 290, + 823, + 319 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Note that the argument is the identity of the pointer, not what the pointer points to; in other words, ARG 1 can only take one of 4 values, and ARG 2 can only take one of 7 values. ", + "bbox": [ + 173, + 327, + 823, + 354 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.5 TRACE-GENERATING FUNCTIONS FOR QUICKSORT ", + "text_level": 1, + "bbox": [ + 174, + 373, + 573, + 387 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.5.1 NON-RECURSIVE TRACE-GENERATING FUNCTIONS ", + "text_level": 1, + "bbox": [ + 176, + 398, + 591, + 414 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "1 Initialize $p _ { l o }$ to 1 and $p _ { h i } \\ t \\circ \\ n$ (length of array) \n2 Initialize $p _ { j } ~ \\mathsf { t o } ~ - \\infty$ \n3 \n4 QUICKSORT() { \n5 while $( p _ { s t a c k L o } \\neq 1 )$ : \n6 if $( \\stackrel { } { Q } _ { s t a c k L o } ^ { \\prime \\prime } ( \\stackrel { \\prime } { p } _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 ) ) :$ \n7 STACK(STACK_POP) \n8 else: \n9 WRITE $( \\boldsymbol { p } _ { h i }$ , ENV_STACK_HI_PEEK) \n10 WRITE(plo, ENV_STACK_LO_PEEK) \n11 STACK(STACK_POP) \n12 PARTITION() \n13 STACK(STACK_PUSH_CALL2) \n14 STACK(STACK_PUSH_CALL1) \n15 } \n16 \n17 PARTITION() { \n18 SET_PIVOT_LO() \n19 SET_J_LO() \n20 COMPSWAP_LOOP() \n24 SWAP(ppivot, phi) \nSET_J_NULL() \n} \nCOMPSWAP_LOOP() { \nwhile $( p _ { j } \\neq p _ { h i } )$ : \nCOMPSWAP() \nMOVE $( p _ { j }$ , UP) \n} \n30 \n31 COMPSWAP() { \n32 if (A[pj ] ≤ A[phi]): \n33 SWAP(ppivot, pj) \n34 MOVE(ppivot, UP) \n35 } \n36 \n37 STACK(op) { \n38 if (op == STACK_PUSH_CALL1): \n39 WRITE(ENV_STACK_LO, plo) \n40 WRITE(ENV_STACK_HI, ppivot − 1) \n41 MOVE(pstackLo, UP) \n42 MOVE(pstackHi, UP) \n43 \n44 if (op $= =$ STACK_PUSH_CALL2): \n45 WRITE(ENV_STACK_LO, ppivot $^ { + 1 1 }$ ) \n46 WRITE(ENV_STACK_HI, $p _ { h i }$ ) \n47 MOVE(pstackLo, UP) \n48 MOVE(pstackHi, UP) \n49 \n50 if (op $= =$ STACK_POP): \n51 WRITE(ENV_STACK_LO, RESET) \n52 WRITE(ENV_STACK_HI, RESET) \n53 MOVE(pstackLo, DOWN) \n54 MOVE(pstackHi, DOWN) \n55 } ", + "bbox": [ + 147, + 425, + 521, + 922 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.5.2 RECURSIVE TRACE-GENERATING FUNCTIONS ", + "text_level": 1, + "bbox": [ + 178, + 103, + 550, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Altered Recursive Functions ", + "text_level": 1, + "bbox": [ + 176, + 133, + 362, + 148 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Initialize to 1 and $p _ { h i } \\ t \\circ \\ n$ (length of array) ", + "bbox": [ + 166, + 154, + 468, + 172 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "1 $p _ { l o }$ \n2 Initialize $p _ { j }$ to −∞ \n3 \n4 QUIC $\\begin{array} { r l } & { \\mathrm { \\sf { A S O R T } \\left( \\tau \\right) } \\quad \\mathrm { \\sf { \\{ } } } \\\\ & { \\mathrm { \\sf { ( } } Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 ) ) : } \\\\ & { \\mathrm { \\sf { 2 } } \\mathrm { \\sf { A R T I T I O N } \\left( \\tau \\right) } } \\end{array}$ \n5 if \n6 \n7 STACK(STACK_PUSH_CALL2) \n8 STACK(STACK_PUSH_CALL1) \n9 WRITE $( p _ { h i }$ , ENV_STACK_HI_PEEK) \n10 WRITE $( p _ { l o }$ , ENV_STACK_LO_PEEK) \n11 QUICKSORT() // Recursive Call \n12 STACK(STACK_POP) \n13 WRITE $( p _ { h i }$ , ENV_STACK_HI_PEEK) \n14 WRITE $( p _ { l o }$ , ENV_STACK_LO_PEEK) \n15 QUICKSORT() // Recursive Call \n16 STACK(STACK_POP) \n17 } \n18 \n19 COMPSWAP_LOOP() { \n20 if $( p _ { j } \\neq p _ { h i } )$ : \n21 COMPSWAP() \n22 MOVE $( p _ { j }$ , UP) \n23 COMPSWAP_LOOP() // Recursive Call \n24 } ", + "bbox": [ + 148, + 159, + 501, + 369 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.6 BASE CASES, REDUCTION RULES, AND VERIFICATION SETS ", + "bbox": [ + 174, + 396, + 625, + 409 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this section, we describe the space of base cases and reduction rules that must be covered for each of the four sample tasks, in order to create the verification set. ", + "bbox": [ + 173, + 420, + 820, + 448 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For addition, we analytically determine the verification set. For tasks other than addition, it is difficult to analytically determine the verification set, so instead, we randomly generate input candidates until they completely cover the base cases and reduction rules. ", + "bbox": [ + 174, + 455, + 821, + 497 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Base Cases and Reduction Rules for Addition. For the recursive formulation of addition, we analytically construct the set of input problems that cover all base cases and reduction rules. We outline how to construct this set. ", + "bbox": [ + 174, + 512, + 825, + 554 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "It is sufficient to construct problems where every transition between two adjacent columns is covered. The ADD reduction rule ensures that each call to ADD only covers two adjacent columns, and so the LSTM only ever runs for a fixed number of steps necessary to process these two columns. ", + "bbox": [ + 174, + 561, + 825, + 603 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We construct input problems by splitting into two cases: one case in which the left column contains a null value and another in which the left column does not contain any null values. We then construct problem configurations that span all possible valid environment states (for instance, in order to force the carry bit in a column to be 1, one can add the sum $\\cdot _ { 1 + 9 } ,$ in the column to the right). ", + "bbox": [ + 176, + 611, + 825, + 666 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The operations we need to be concerned most about are CARRY and LSHIFT, which induce partial environment states spanning two columns. It is straightforward to deal with all other operations, which do not induce partial environment states. ", + "bbox": [ + 174, + 672, + 825, + 715 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Under the assumption that there are no leading 0’s (except in the case of single digits) and the two numbers to be added have the same number of digits, the verification set for addition contains 20,181 input problems. The assumption of leading 0’s can be easily removed, at the cost of slightly increasing the size of the verification set. We made the assumption of equivalent lengths in order to parametrize the input format with respect to length, but this assumption can be removed as well. ", + "bbox": [ + 174, + 723, + 825, + 792 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Base Cases and Reduction Rules for Bubble Sort. The original version of the bubblesort implementation exposes the values within the array. While this matches the description from Reed & de Freitas (2016), we found that this causes an unnecessary blowup in the size of $V$ and makes it much more difficult to construct the verification set. For purposes of verification, we replace the domain-specific encoder with the following: ", + "bbox": [ + 174, + 808, + 823, + 878 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/d6404bbac9ba6a2fa37db9e83467937b3e09f0d707243a22dbb09b327365ec60.jpg", + "text": "$$\n\\begin{array} { r l } & { f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) \\leq Q ( 1 , i _ { 2 } ) , 1 \\leq i _ { 1 } \\leq l e n g t h , 1 \\leq i _ { 2 } \\leq l e n g t h , } \\\\ & { ~ i _ { 3 } = = l e n g t h , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 886, + 808, + 922 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/cece422b89b73ac0c163839ed153b4f1e0f3158314f89a10519ab896503c1b32.jpg", + "table_caption": [ + "Table 4: Accuracy on Randomly Generated Problems for Variant of Bubble Sort " + ], + "table_footnote": [], + "table_body": "
Length of ArrayNon-RecursiveRecursive
100%100%
２3100%100%
4100%100%
5100%100%
690%100%
786.7%100%
86.7%100%
90%100%
100%
120%100%
150%100%
100%
700%100%
", + "bbox": [ + 325, + 131, + 674, + 329 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "which directly exposes which of the two values pointed to is larger. This modification also enables us to sort arrays containing arbitrary comparable elements. ", + "bbox": [ + 174, + 361, + 823, + 388 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "By reasoning about the possible set of environment observations created by all valid inputs, we construct $V$ using the procedure described in Section 3.3. Using this modification, we constructed a verification set consisting of one array of size 10. ", + "bbox": [ + 174, + 396, + 825, + 438 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We also report on generalization results for the non-recursive and recursive versions of this variant of bubble sort. Table 4 demonstrates that the accuracy of the non-recursive program degrades sharply when moving from arrays of length 7 to arrays of length 8. This is due to the properties of the training set – we trained on 2 traces synthesized from arrays of length 7 and 1 trace synthesized from an array of length 6. Table 4 also demonstrates that the (verified) recursive program generalizes perfectly. ", + "bbox": [ + 174, + 445, + 825, + 515 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Base Cases and Reduction Rules for Topological Sort. For each function we use to implement the recursive version of topological sort, we need to consider the set of possible environment observation sequences we can create from all valid inputs and test that the learned program produces the correct behavior on each of these inputs. We have three observations: the color of the start node, the color of the active node’s next child to be considered, and whether the stack is empty. Na¨ıvely, we might expect to synthesize and test an input for any sequence created by combining the four possible colors in two variables and another boolean variable for whether the stack is empty (so 32 possible observations at any point), but for various reasons, most of these combinations are impossible to occur at any given point in the execution trace. ", + "bbox": [ + 173, + 530, + 825, + 655 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Through careful reasoning about the possible set of environment observations created by all valid inputs, and how each of the operations in the execution trace affects the environment, we can construct $V$ using the procedure described in Section 3.3. We then construct a verification set of size 73 by ensuring that randomly generated graphs cover the analytically derived $V$ . The model described in the training setup of Section 4 (trained on 6 traces) was verified to be correct via the matching procedure described in Section 4.2. ", + "bbox": [ + 174, + 662, + 825, + 746 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Base Cases and Reduction Rules for Quicksort. As with the others, we apply the procedure described in Section 3.3 to construct $V$ and then empirically create a verification set which covers $V$ . The verification set can be very small, as we found a 10-element array ([8,2,1,2,0,8,5,8,3,7]) is sufficient to cover all of $V$ . We note that an earlier version of quicksort we tried lacked primitive operations to directly move a pointer to another, and therefore needed more functions and observations. As this complexity interfered with determining the base cases and reductions, we changed the algorithm to its current form. Even though the earlier version also generalized just as well in practice, relatively small differences in the formulation of the traces and the environment observations can drastically change the difficulty of verification. 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The space of possible programs is extremely large, and composing a program that per-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "forms robustly on the infinite space of possible inputs is difficult—in part because it is impractical", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "to obtain enough training examples to easily disambiguate amongst all possible programs. Never-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "score": 1.0, + "content": "theless, we would like the model to quickly learn to represent the right semantics of the underlying", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 455, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 455, + 496 + ], + "score": 1.0, + "content": "program from a small number of training examples, not an exhaustive number of them.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "Thus far, to evaluate the efficacy of neural models on programming tasks, the only metric that has", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "been used is generalization of expected behavior to inputs of greater complexity (Vinyals et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "(2015), Kaiser & Sutskever (2015), Reed & de Freitas (2016), Graves et al. 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Empirically, existing models suffer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "score": 1.0, + "content": "from a common limitation—generalization becomes poor beyond a threshold level of complexity.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Errors arise due to undesirable and uninterpretable dependencies and associations the architecture", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "learns to store in some high-dimensional hidden state. This makes it difficult to reason about what", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 293, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 293, + 600 + ], + "score": 1.0, + "content": "the model will do when given complex inputs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "One common strategy to improve generalization is to use curriculum learning, where the model is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "score": 1.0, + "content": "trained on inputs of gradually increasing complexity. 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That is to say, the network does not learn the true", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 694, + 187, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 187, + 705 + ], + "score": 1.0, + "content": "program semantics.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In this paper, we propose to resolve these issues by explicitly incorporating recursion into neural", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "architectures. 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Moreover, it has traditionally been difficult to reason", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 264, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 264, + 469, + 277 + ], + "score": 1.0, + "content": "about the behavior of these models beyond a certain level of input complexity. In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 275, + 469, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 469, + 288 + ], + "score": 1.0, + "content": "order to address these issues, we propose augmenting neural architectures with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "score": 1.0, + "content": "a key abstraction: recursion. As an application, we implement recursion in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "score": 1.0, + "content": "Neural Programmer-Interpreter framework on four tasks: grade-school addition,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "bubble sort, topological sort, and quicksort. We demonstrate superior generaliz-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 319, + 470, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 470, + 332 + ], + "score": 1.0, + "content": "ability and interpretability with small amounts of training data. Recursion divides", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "the problem into smaller pieces and drastically reduces the domain of each neu-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 341, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 341, + 470, + 354 + ], + "score": 1.0, + "content": "ral network component, making it tractable to prove guarantees about the overall", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "system’s behavior. Our experience suggests that in order for neural architectures", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "score": 1.0, + "content": "to robustly learn program semantics, it is necessary to incorporate a concept like", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 376, + 184, + 385 + ], + "spans": [ + { + "bbox": [ + 142, + 376, + 184, + 385 + ], + "score": 1.0, + "content": "recursion.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 243, + 470, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 404, + 206, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 208, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 208, + 420 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 504, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "Training neural networks to synthesize robust programs from a small number of examples is a chal-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "lenging task. The space of possible programs is extremely large, and composing a program that per-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "forms robustly on the infinite space of possible inputs is difficult—in part because it is impractical", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "to obtain enough training examples to easily disambiguate amongst all possible programs. Never-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "score": 1.0, + "content": "theless, we would like the model to quickly learn to represent the right semantics of the underlying", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 455, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 455, + 496 + ], + "score": 1.0, + "content": "program from a small number of training examples, not an exhaustive number of them.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 428, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "Thus far, to evaluate the efficacy of neural models on programming tasks, the only metric that has", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "been used is generalization of expected behavior to inputs of greater complexity (Vinyals et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "(2015), Kaiser & Sutskever (2015), Reed & de Freitas (2016), Graves et al. 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Empirically, existing models suffer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "score": 1.0, + "content": "from a common limitation—generalization becomes poor beyond a threshold level of complexity.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Errors arise due to undesirable and uninterpretable dependencies and associations the architecture", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "learns to store in some high-dimensional hidden state. This makes it difficult to reason about what", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 293, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 293, + 600 + ], + "score": 1.0, + "content": "the model will do when given complex inputs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 500, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "One common strategy to improve generalization is to use curriculum learning, where the model is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "score": 1.0, + "content": "trained on inputs of gradually increasing complexity. However, models that make use of this strategy", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "eventually fail after a certain level of complexity (e.g. the single-digit multiplication task in Zaremba", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "et al. (2016), the bubble sort task in Reed & de Freitas (2016), and the graph tasks in Graves et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "(2016)). In this version of curriculum learning, even though the inputs are gradually becoming more", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "complex, the semantics of the program is succinct and does not change. Although the model is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "exposed to more and more data, it might learn spurious and overly complex representations of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "program, as suggested in Zaremba et al. (2016). That is to say, the network does not learn the true", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 694, + 187, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 187, + 705 + ], + "score": 1.0, + "content": "program semantics.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 605, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In this paper, we propose to resolve these issues by explicitly incorporating recursion into neural", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "architectures. Recursion is an important concept in programming languages and a critical tool to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "reduce the complexity of programs. We find that recursion makes it easier for the network to learn", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "the right program and generalize to unknown situations. Recursion enables provable guarantees on", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "neural programs’ behavior without needing to exhaustively enumerate all possible inputs to the pro-", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "grams. This paper is the first (to our knowledge) to investigate the important problem of provable", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "generalization properties of neural programs. As an application, we incorporate recursion into the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Neural Programmer-Interpreter architecture and consider four sample tasks: grade-school addition,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "bubble sort, topological sort, and quicksort. Empirically, we observe that the learned recursive pro-", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 238, + 172 + ], + "score": 1.0, + "content": "grams solve all valid inputs with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 239, + 159, + 263, + 170 + ], + "score": 0.92, + "content": "100 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 263, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "accuracy after training on a very small number of examples,", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "out-performing previous generalization results. Given verification sets that cover all the base cases", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "and reduction rules, we can provide proofs that these learned programs generalize perfectly. This is", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 492, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 492, + 205 + ], + "score": 1.0, + "content": "the first time one can provide provable guarantees of perfect generalization for neural programs.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "reduce the complexity of programs. We find that recursion makes it easier for the network to learn", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "the right program and generalize to unknown situations. Recursion enables provable guarantees on", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "neural programs’ behavior without needing to exhaustively enumerate all possible inputs to the pro-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "grams. This paper is the first (to our knowledge) to investigate the important problem of provable", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "generalization properties of neural programs. As an application, we incorporate recursion into the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Neural Programmer-Interpreter architecture and consider four sample tasks: grade-school addition,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "bubble sort, topological sort, and quicksort. Empirically, we observe that the learned recursive pro-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 238, + 172 + ], + "score": 1.0, + "content": "grams solve all valid inputs with", + "type": "text" + }, + { + "bbox": [ + 239, + 159, + 263, + 170 + ], + "score": 0.92, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "accuracy after training on a very small number of examples,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "out-performing previous generalization results. Given verification sets that cover all the base cases", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "and reduction rules, we can provide proofs that these learned programs generalize perfectly. This is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 492, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 492, + 205 + ], + "score": 1.0, + "content": "the first time one can provide provable guarantees of perfect generalization for neural programs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 218, + 313, + 231 + ], + "lines": [ + { + "bbox": [ + 104, + 217, + 315, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 315, + 233 + ], + "score": 1.0, + "content": "2 THE PROBLEM AND OUR APPROACH", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 243, + 286, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 288, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 288, + 255 + ], + "score": 1.0, + "content": "2.1 THE PROBLEM OF GENERALIZATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "When constructing a neural network for the purpose of learning a program, there are two orthogonal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "aspects to consider. The first is the actual model architecture. Numerous models have been proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "for learning programs; to name a few, this includes the Differentiable Neural Computer (Graves", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "et al., 2016), Neural Turing Machine (Graves et al., 2014), Neural GPU (Kaiser & Sutskever, 2015),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "Neural Programmer (Neelakantan et al., 2015), Pointer Network (Vinyals et al., 2015), Hierarchical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "Attentive Memory (Andrychowicz & Kurach, 2016), and Neural Random Access Machine (Kurach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "et al., 2016). The architecture usually possesses some form of memory, which could be internal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "(such as the hidden state of a recurrent neural network) or external (such as a discrete “scratch pad”", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "or a memory block with differentiable access). The second is the training procedure, which consists", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "of the form of the training data and the optimization process. Almost all architectures train on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "program input/output pairs. The only model, to our knowledge, that does not train on input-output", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "pairs is the Neural Programmer-Interpreter (Reed & de Freitas, 2016), which trains on synthetic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 397, + 175, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 175, + 407 + ], + "score": 1.0, + "content": "execution traces.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "To evaluate a neural network that learns a neural program to accomplish a certain task, one common", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 310, + 434 + ], + "score": 1.0, + "content": "evaluation metric is how well the learned model", + "type": "text" + }, + { + "bbox": [ + 311, + 423, + 323, + 433 + ], + "score": 0.72, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 423, + 481, + 434 + ], + "score": 1.0, + "content": "generalizes. More specifically, when", + "type": "text" + }, + { + "bbox": [ + 482, + 423, + 493, + 433 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "trained on simpler inputs, such as inputs of a small length, the generalization metric evaluates how", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 445, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 126, + 456 + ], + "score": 1.0, + "content": "well", + "type": "text" + }, + { + "bbox": [ + 127, + 445, + 138, + 455 + ], + "score": 0.66, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 445, + 426, + 456 + ], + "score": 1.0, + "content": "will do on more complex inputs, such as inputs of much longer length.", + "type": "text" + }, + { + "bbox": [ + 426, + 445, + 438, + 455 + ], + "score": 0.66, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 445, + 505, + 456 + ], + "score": 1.0, + "content": "is considered to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 226, + 470 + ], + "score": 1.0, + "content": "have perfect generalization if", + "type": "text" + }, + { + "bbox": [ + 226, + 456, + 238, + 466 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "can give the right answer for any input, such as inputs of arbitrary", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 466, + 137, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 137, + 480 + ], + "score": 1.0, + "content": "length.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 108, + 483, + 504, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "As mentioned in Section 1, all approaches to neural programming today fare poorly on this general-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "ization issue. We hypothesize that the reason for this is that the neural network learns to spuriously", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "depend on specific characteristics of the training examples that are irrelevant to the true program", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "semantics, such as length of the training inputs, and thus fails to generalize to more complex inputs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 504, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "In addition, none of the current approaches to neural programming provide a method or even aim to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "enable provable guarantees about generalization. The memory updates of these neural programs are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "so complex and interdependent that it is difficult to reason about the behaviors of the learned neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "program under previously unseen situations (such as problems with longer inputs). This is highly", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "undesirable, since being able to provide the correct answer in all possible settings is one of the most", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 589, + 304, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 304, + 601 + ], + "score": 1.0, + "content": "important aspects of any learned neural program.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 108, + 612, + 285, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 286, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 286, + 624 + ], + "score": 1.0, + "content": "2.2 OUR APPROACH USING RECURSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In this paper, we propose that the key abstraction of recursion is necessary for neural programs to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "generalize. The general notion of recursion has been an important concept in many domains, in-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "cluding mathematics and computer science. In computer science, recursion (as opposed to iteration)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "score": 1.0, + "content": "involves solving a larger problem by combining solutions to smaller instances of the same problem.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "Formally, a function exhibits recursive behavior when it possesses two properties: (1) Base cases—", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "terminating scenarios that do not use recursion to produce answers; (2) A set of rules that reduces all", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "other problems toward the base cases. Some functional programming languages go so far as not to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "define any looping constructs but rely solely on recursion to enable repeated execution of the same", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 131, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 131, + 733 + ], + "score": 1.0, + "content": "code.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 105, + 83, + 505, + 205 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 218, + 313, + 231 + ], + "lines": [ + { + "bbox": [ + 104, + 217, + 315, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 315, + 233 + ], + "score": 1.0, + "content": "2 THE PROBLEM AND OUR APPROACH", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 243, + 286, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 288, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 288, + 255 + ], + "score": 1.0, + "content": "2.1 THE PROBLEM OF GENERALIZATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "When constructing a neural network for the purpose of learning a program, there are two orthogonal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "aspects to consider. The first is the actual model architecture. Numerous models have been proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "for learning programs; to name a few, this includes the Differentiable Neural Computer (Graves", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "et al., 2016), Neural Turing Machine (Graves et al., 2014), Neural GPU (Kaiser & Sutskever, 2015),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "Neural Programmer (Neelakantan et al., 2015), Pointer Network (Vinyals et al., 2015), Hierarchical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "Attentive Memory (Andrychowicz & Kurach, 2016), and Neural Random Access Machine (Kurach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "et al., 2016). The architecture usually possesses some form of memory, which could be internal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "(such as the hidden state of a recurrent neural network) or external (such as a discrete “scratch pad”", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "or a memory block with differentiable access). The second is the training procedure, which consists", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "of the form of the training data and the optimization process. Almost all architectures train on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "program input/output pairs. The only model, to our knowledge, that does not train on input-output", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "pairs is the Neural Programmer-Interpreter (Reed & de Freitas, 2016), which trains on synthetic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 397, + 175, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 175, + 407 + ], + "score": 1.0, + "content": "execution traces.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 263, + 506, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "To evaluate a neural network that learns a neural program to accomplish a certain task, one common", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 310, + 434 + ], + "score": 1.0, + "content": "evaluation metric is how well the learned model", + "type": "text" + }, + { + "bbox": [ + 311, + 423, + 323, + 433 + ], + "score": 0.72, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 423, + 481, + 434 + ], + "score": 1.0, + "content": "generalizes. More specifically, when", + "type": "text" + }, + { + "bbox": [ + 482, + 423, + 493, + 433 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "trained on simpler inputs, such as inputs of a small length, the generalization metric evaluates how", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 445, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 126, + 456 + ], + "score": 1.0, + "content": "well", + "type": "text" + }, + { + "bbox": [ + 127, + 445, + 138, + 455 + ], + "score": 0.66, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 445, + 426, + 456 + ], + "score": 1.0, + "content": "will do on more complex inputs, such as inputs of much longer length.", + "type": "text" + }, + { + "bbox": [ + 426, + 445, + 438, + 455 + ], + "score": 0.66, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 445, + 505, + 456 + ], + "score": 1.0, + "content": "is considered to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 226, + 470 + ], + "score": 1.0, + "content": "have perfect generalization if", + "type": "text" + }, + { + "bbox": [ + 226, + 456, + 238, + 466 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "can give the right answer for any input, such as inputs of arbitrary", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 466, + 137, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 137, + 480 + ], + "score": 1.0, + "content": "length.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 411, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 483, + 504, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "As mentioned in Section 1, all approaches to neural programming today fare poorly on this general-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "ization issue. We hypothesize that the reason for this is that the neural network learns to spuriously", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "depend on specific characteristics of the training examples that are irrelevant to the true program", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "semantics, such as length of the training inputs, and thus fails to generalize to more complex inputs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 483, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 504, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "In addition, none of the current approaches to neural programming provide a method or even aim to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "enable provable guarantees about generalization. The memory updates of these neural programs are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "so complex and interdependent that it is difficult to reason about the behaviors of the learned neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "program under previously unseen situations (such as problems with longer inputs). This is highly", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "undesirable, since being able to provide the correct answer in all possible settings is one of the most", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 589, + 304, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 304, + 601 + ], + "score": 1.0, + "content": "important aspects of any learned neural program.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 533, + 506, + 601 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 612, + 285, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 286, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 286, + 624 + ], + "score": 1.0, + "content": "2.2 OUR APPROACH USING RECURSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In this paper, we propose that the key abstraction of recursion is necessary for neural programs to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "generalize. The general notion of recursion has been an important concept in many domains, in-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "cluding mathematics and computer science. In computer science, recursion (as opposed to iteration)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "score": 1.0, + "content": "involves solving a larger problem by combining solutions to smaller instances of the same problem.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "Formally, a function exhibits recursive behavior when it possesses two properties: (1) Base cases—", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "terminating scenarios that do not use recursion to produce answers; (2) A set of rules that reduces all", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "other problems toward the base cases. Some functional programming languages go so far as not to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "define any looping constructs but rely solely on recursion to enable repeated execution of the same", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 131, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 131, + 733 + ], + "score": 1.0, + "content": "code.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47, + "bbox_fs": [ + 104, + 633, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "In this paper, we propose that recursion is an important concept for neural programs as well. In", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "fact, we argue that recursion is an essential element for neural programs to generalize, and makes it", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 504, + 117 + ], + "score": 1.0, + "content": "tractable to prove the generalization of neural programs. Recursion can be implemented differently", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "for different neural programming models. Here as a concrete and general example, we consider a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "general Neural Programming Architecture (NPA), similar to Neural Programmer-Interpreter (NPI)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "in Reed & de Freitas (2016). In this architecture, we consider a core controller, e.g., an LSTM", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "in NPI’s case, but possibly other networks in different cases. There is a (changing) list of neural", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "programs used to accomplish a given task. The core controller acts as a dispatcher for the programs.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "At each time step, the core controller can decide to select one of the programs to call with certain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "arguments. When the program is called, the current context including the caller’s memory state is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "stored on a stack; when the program returns, the stored context is popped off the stack to resume", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 274, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 274, + 215 + ], + "score": 1.0, + "content": "execution in the previous caller’s context.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 220, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "In this general Neural Programming Architecture, we show it is easy to support recursion. In par-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "ticular, recursion can be implemented as a program calling itself. Because the context of the caller", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "is stored on a stack when it calls another program and the callee starts in a fresh context, this en-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "ables recursion simply by allowing a program to call itself. In practice, we can additionally use tail", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "recursion optimization to avoid problems with the call stack growing too deep. Thus, any general", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "Neural Programming Architecture supporting such a call structure can be made to support recursion.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "In particular, this condition is satisfied by NPI, and thus the NPI model naturally supports recursion", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 393, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 393, + 310 + ], + "score": 1.0, + "content": "(even though the authors of NPI did not consider this aspect explicitly).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "By nature, recursion reduces the complexity of a problem to simpler instances. Thus, recursion", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "helps decompose a problem and makes it easier to reason about a program’s behavior for previously", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "unseen situations such as longer inputs. In particular, given that a recursion is defined by two", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "properties as mentioned before, the base cases and the set of reduction rules, we can prove a recursive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "neural program generalizes perfectly if we can prove that (1) it performs correctly on the base cases;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "(2) it learns the reduction rules correctly. For many problems, the base cases and reduction rules", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "usually consist of a finite (often small) number of cases. For problems where the base cases may be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "extremely large or infinite, such as certain forms of motor control, recursion can still help reduce the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "problem of generalization to these two aspects and make the generalization problem significantly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 413, + 250, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 250, + 424 + ], + "score": 1.0, + "content": "simpler to handle and reason about.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "As a concrete instantiation, we show in this paper that we can enable recursive neural programs in the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "NPI model, and thus enable perfectly generalizable neural programs for tasks such as sorting where", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "the original, non-recursive NPI program fails. As aforementioned, the NPI model naturally supports", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "recursion. However, the authors of NPI did not consider explicitly the notion of recursion and as a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "score": 1.0, + "content": "consequence, did not learn recursive programs. We show that by modifying the training procedure,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "we enable the NPI model to learn recursive neural programs. As a consequence, our learned neural", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "programs empirically achieve perfect generalization from a very small number of training examples.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "Furthermore, given a verification input set that covers all base cases and reduction rules, we can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "formally prove that the learned neural programs achieve perfect generalization after verifying its", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "behavior on the verification input set. This is the first time one can provide provable guarantees of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 540, + 279, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 279, + 551 + ], + "score": 1.0, + "content": "perfect generalization for neural programs.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "We would also like to point out that in this paper, we provide as an example one way to train a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "recursive neural program, by providing a certain training execution trace to the NPI model. However,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "our concept of recursion for neural programs is general. In fact, it is one of our future directions to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "explore new ways to train a recursive neural program without providing explicit training execution", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 600, + 307, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 307, + 612 + ], + "score": 1.0, + "content": "traces or with only partial or non-recursive traces.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 496, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 498, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 498, + 655 + ], + "score": 1.0, + "content": "3 APPLICATION TO LEARNING RECURSIVE NEURAL PROGRAMS WITH NPI", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "title", + "bbox": [ + 108, + 672, + 288, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 289, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 289, + 685 + ], + "score": 1.0, + "content": "3.1 BACKGROUND: NPI ARCHITECTURE", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "As discussed in Section 2, the Neural Programmer-Interpreter (NPI) is an instance of a Neural", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "Programmer Architecture and hence it naturally supports recursion. In this section, we give a brief", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 720, + 417, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 417, + 732 + ], + "score": 1.0, + "content": "review of the NPI architecture from Reed & de Freitas (2016) as background.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "In this paper, we propose that recursion is an important concept for neural programs as well. In", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "fact, we argue that recursion is an essential element for neural programs to generalize, and makes it", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 504, + 117 + ], + "score": 1.0, + "content": "tractable to prove the generalization of neural programs. Recursion can be implemented differently", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "for different neural programming models. Here as a concrete and general example, we consider a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "general Neural Programming Architecture (NPA), similar to Neural Programmer-Interpreter (NPI)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "in Reed & de Freitas (2016). In this architecture, we consider a core controller, e.g., an LSTM", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "in NPI’s case, but possibly other networks in different cases. There is a (changing) list of neural", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "programs used to accomplish a given task. The core controller acts as a dispatcher for the programs.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "At each time step, the core controller can decide to select one of the programs to call with certain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "arguments. When the program is called, the current context including the caller’s memory state is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "stored on a stack; when the program returns, the stored context is popped off the stack to resume", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 274, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 274, + 215 + ], + "score": 1.0, + "content": "execution in the previous caller’s context.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 81, + 506, + 215 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 220, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "In this general Neural Programming Architecture, we show it is easy to support recursion. In par-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "ticular, recursion can be implemented as a program calling itself. Because the context of the caller", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "is stored on a stack when it calls another program and the callee starts in a fresh context, this en-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "ables recursion simply by allowing a program to call itself. In practice, we can additionally use tail", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "recursion optimization to avoid problems with the call stack growing too deep. Thus, any general", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "Neural Programming Architecture supporting such a call structure can be made to support recursion.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "In particular, this condition is satisfied by NPI, and thus the NPI model naturally supports recursion", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 393, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 393, + 310 + ], + "score": 1.0, + "content": "(even though the authors of NPI did not consider this aspect explicitly).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 219, + 505, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "By nature, recursion reduces the complexity of a problem to simpler instances. Thus, recursion", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "helps decompose a problem and makes it easier to reason about a program’s behavior for previously", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "unseen situations such as longer inputs. 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As aforementioned, the NPI model naturally supports", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "recursion. However, the authors of NPI did not consider explicitly the notion of recursion and as a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "score": 1.0, + "content": "consequence, did not learn recursive programs. We show that by modifying the training procedure,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "we enable the NPI model to learn recursive neural programs. As a consequence, our learned neural", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "programs empirically achieve perfect generalization from a very small number of training examples.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "Furthermore, given a verification input set that covers all base cases and reduction rules, we can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "formally prove that the learned neural programs achieve perfect generalization after verifying its", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "behavior on the verification input set. This is the first time one can provide provable guarantees of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 540, + 279, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 279, + 551 + ], + "score": 1.0, + "content": "perfect generalization for neural programs.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 429, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "We would also like to point out that in this paper, we provide as an example one way to train a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "recursive neural program, by providing a certain training execution trace to the NPI model. However,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "our concept of recursion for neural programs is general. In fact, it is one of our future directions to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "explore new ways to train a recursive neural program without providing explicit training execution", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 600, + 307, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 307, + 612 + ], + "score": 1.0, + "content": "traces or with only partial or non-recursive traces.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 556, + 506, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 496, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 498, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 498, + 655 + ], + "score": 1.0, + "content": "3 APPLICATION TO LEARNING RECURSIVE NEURAL PROGRAMS WITH NPI", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "title", + "bbox": [ + 108, + 672, + 288, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 289, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 289, + 685 + ], + "score": 1.0, + "content": "3.1 BACKGROUND: NPI ARCHITECTURE", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "As discussed in Section 2, the Neural Programmer-Interpreter (NPI) is an instance of a Neural", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "Programmer Architecture and hence it naturally supports recursion. 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The core module", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "of the NPI is an LSTM controller that takes as input a slice of the current external environment, via", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "a set of pointers, and a program and arguments to execute. NPI then outputs the return probability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "and next program and arguments to execute. 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The outputs", + "type": "text" + }, + { + "bbox": [ + 229, + 303, + 284, + 312 + ], + "score": 0.81, + "content": "r _ { t } , p _ { t + 1 } , a _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "are used to determine the next action, as described in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 393, + 326 + ], + "score": 1.0, + "content": "Algorithm 1. If the program is primitive, the next environmental state", + "type": "text" + }, + { + "bbox": [ + 394, + 313, + 413, + 324 + ], + "score": 0.89, + "content": "e _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 310, + 493, + 326 + ], + "score": 1.0, + "content": "will be affected by", + "type": "text" + }, + { + "bbox": [ + 494, + 314, + 504, + 324 + ], + "score": 0.82, + "content": "p _ { t }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 123, + 335 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 324, + 133, + 334 + ], + "score": 0.85, + "content": "a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 322, + 154, + 335 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 154, + 323, + 248, + 334 + ], + "score": 0.91, + "content": "e _ { t + 1 } \\sim f _ { e n v } ( e _ { t } , p _ { t } , a _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 322, + 506, + 335 + ], + "score": 1.0, + "content": ". As with the original NPI architecture, the experiments for this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 372, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 264, + 347 + ], + "score": 1.0, + "content": "paper always used a 3-tuple of integers", + "type": "text" + }, + { + "bbox": [ + 264, + 333, + 368, + 345 + ], + "score": 0.93, + "content": "a _ { t } = ( a _ { t } ( 1 ) , \\hat { a } _ { t } ( 2 ) , a _ { t } ( 3 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 333, + 372, + 347 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 110, + 373, + 437, + 487 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 360, + 285, + 371 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 358, + 287, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 287, + 374 + ], + "score": 1.0, + "content": "Algorithm 1 Neural programming inference", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 110, + 373, + 437, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 373, + 437, + 487 + ], + "spans": [ + { + "bbox": [ + 110, + 373, + 437, + 487 + ], + "score": 0.768, + "html": "
1:Inputs: Environment observation e, program p,arguments a, stop threshold α
2:function RUN(e,p, a)
3:h↑0,r←0
4:whiler<αdo
5:s ←fenc(e,a),h ←fistm(s,p,h)
6:r ←fend(h),p2 ← fprog(h),a2 ← farg(h)
7:if p is a primitive function then
8:e ← fenv(e,p,a).
9:else
10:function RUN(e,P2, a2)
", + "type": "table", + "image_path": "e6560f320f41f348fdd426619c47efa3064b98bfb7c7dc35cb65540f5dbaec9a.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 110, + 373, + 437, + 411.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 110, + 411.0, + 437, + 449.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 110, + 449.0, + 437, + 487.0 + ], + "spans": [], + "index": 23 + } + ] + } + ], + "index": 21.0 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "A description of the inference procedure is given in Algorithm 1. Each step during an execution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "of the program does one of three things: (1) another subprogram along with associated arguments", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "is called, as in Line 10, (2) the program writes to the environment if it is primitive, as in Line 8,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 405, + 545 + ], + "score": 1.0, + "content": "or (3) the loop is terminated if the return probability exceeds a threshold", + "type": "text" + }, + { + "bbox": [ + 406, + 535, + 413, + 543 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 534, + 505, + 545 + ], + "score": 1.0, + "content": ", after which the stack", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 395, + 556 + ], + "score": 1.0, + "content": "frame is popped and control is returned to the caller. In all experiments,", + "type": "text" + }, + { + "bbox": [ + 395, + 546, + 403, + 554 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "is set to 0.5. Each time a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 555, + 298, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 298, + 568 + ], + "score": 1.0, + "content": "subprogram is called, the stack depth increases.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "The training data for the Neural Programmer-Interpreter consists of full execution traces for the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "program of interest. A single element of an execution trace consists of a step input-step output pair,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 592, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 609 + ], + "score": 1.0, + "content": "which can be synthesized from Algorithm 1: this corresponds to, for a given time-step, the step", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 153, + 619 + ], + "score": 1.0, + "content": "input tuple", + "type": "text" + }, + { + "bbox": [ + 153, + 605, + 185, + 617 + ], + "score": 0.93, + "content": "( e , p , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 604, + 275, + 619 + ], + "score": 1.0, + "content": "and step output tuple", + "type": "text" + }, + { + "bbox": [ + 275, + 605, + 316, + 617 + ], + "score": 0.92, + "content": "( r , p _ { 2 } , a _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 604, + 506, + 619 + ], + "score": 1.0, + "content": ". An example of part of an addition task trace,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "written in shorthand, is given in Figure 1. For example, a step input-step output pair in Lines 2 and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "3 of the left-hand side of Figure 1 is (ADD1, WRITE OUT 1). In this pair, the step input runs", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "a subprogram ADD1 that has no arguments, and the step output contains a program WRITE that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "has arguments of OUT and 1. The environment and return probability are omitted for readability.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 658, + 356, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 356, + 672 + ], + "score": 1.0, + "content": "Indentation indicates the stack is one level deeper than before.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "It is important to emphasize that at inference time in the NPI, the hidden state of the LSTM controller", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 504, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 393, + 701 + ], + "score": 1.0, + "content": "is reset (to zero) at each subprogram call, as in Line 3 of Algorithm 1", + "type": "text" + }, + { + "bbox": [ + 393, + 688, + 423, + 698 + ], + "score": 0.82, + "content": "h \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 687, + 504, + 701 + ], + "score": 1.0, + "content": "). This functionality", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 353, + 724 + ], + "score": 1.0, + "content": "1The original NPI paper decodes to a program key embedding", + "type": "text" + }, + { + "bbox": [ + 353, + 710, + 390, + 721 + ], + "score": 0.92, + "content": "\\boldsymbol { k } _ { t } ~ \\in \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "and then computes a program", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 424, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 148, + 733 + ], + "score": 1.0, + "content": "embedding", + "type": "text" + }, + { + "bbox": [ + 148, + 723, + 167, + 732 + ], + "score": 0.88, + "content": "p _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 721, + 424, + 733 + ], + "score": 1.0, + "content": ", which we also did in our implementation, but we omit this for brevity.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "We describe the details of the NPI model relevant to our contributions. We adapt machinery from", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "the original paper slightly to fit our needs. The NPI model has three learnable components: a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "task-agnostic core, a program-key embedding, and domain-specific encoders that allow the NPI to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 238, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 238, + 128 + ], + "score": 1.0, + "content": "operate in diverse environments.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 81, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 280, + 145 + ], + "score": 1.0, + "content": "The NPI accesses an external environment,", + "type": "text" + }, + { + "bbox": [ + 281, + 132, + 290, + 144 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 131, + 506, + 145 + ], + "score": 1.0, + "content": ", which varies according to the task. The core module", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "of the NPI is an LSTM controller that takes as input a slice of the current external environment, via", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "a set of pointers, and a program and arguments to execute. NPI then outputs the return probability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "and next program and arguments to execute. Formally, the NPI is represented by the following set", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 161, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 161, + 190 + ], + "score": 1.0, + "content": "of equations:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 131, + 506, + 190 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 197, + 409, + 248 + ], + "lines": [ + { + "bbox": [ + 202, + 197, + 409, + 248 + ], + "spans": [ + { + "bbox": [ + 202, + 197, + 409, + 248 + ], + "score": 0.89, + "content": "\\begin{array} { c } { s _ { t } = f _ { e n c } ( e _ { t } , a _ { t } ) } \\\\ { h _ { t } = f _ { l s t m } ( s _ { t } , p _ { t } , h _ { t - 1 } ) } \\\\ { r _ { t } = f _ { e n d } ( h _ { t } ) , p _ { t + 1 } = f _ { p r o g } ( h _ { t } ) , a _ { t + 1 } = f _ { a r g } ( h _ { t } ) } \\end{array}", + "type": "interline_equation", + "image_path": "bfd328979f3862d272f2a19a564f1f89362dc096d4ed4d62f9e093d1d99b6bb8.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 202, + 197, + 409, + 214.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 202, + 214.0, + 409, + 231.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 202, + 231.0, + 409, + 248.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 111, + 267 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 256, + 261, + 270 + ], + "score": 1.0, + "content": "is a subscript denoting the time-step;", + "type": "text" + }, + { + "bbox": [ + 262, + 257, + 281, + 268 + ], + "score": 0.91, + "content": "f _ { e n c }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "is a domain-specific encoder (to be described later) that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 229, + 281 + ], + "score": 1.0, + "content": "takes in the environment slice", + "type": "text" + }, + { + "bbox": [ + 229, + 270, + 239, + 279 + ], + "score": 0.83, + "content": "e _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 267, + 302, + 281 + ], + "score": 1.0, + "content": "and arguments", + "type": "text" + }, + { + "bbox": [ + 302, + 269, + 312, + 279 + ], + "score": 0.74, + "content": "a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 267, + 316, + 281 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 317, + 268, + 339, + 279 + ], + "score": 0.78, + "content": "f _ { l s t m }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "represents the core module, which takes", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 276, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 153, + 293 + ], + "score": 1.0, + "content": "in the state", + "type": "text" + }, + { + "bbox": [ + 153, + 280, + 162, + 290 + ], + "score": 0.84, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 276, + 218, + 293 + ], + "score": 1.0, + "content": "generated by", + "type": "text" + }, + { + "bbox": [ + 218, + 279, + 236, + 290 + ], + "score": 0.89, + "content": "f _ { e n c }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 276, + 331, + 293 + ], + "score": 1.0, + "content": ", a program embedding", + "type": "text" + }, + { + "bbox": [ + 331, + 278, + 368, + 290 + ], + "score": 0.92, + "content": "p _ { t } \\in \\mathbb { R } ^ { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 276, + 469, + 293 + ], + "score": 1.0, + "content": ", and hidden LSTM state", + "type": "text" + }, + { + "bbox": [ + 469, + 279, + 480, + 290 + ], + "score": 0.82, + "content": "h _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 276, + 484, + 293 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 484, + 279, + 504, + 290 + ], + "score": 0.52, + "content": "f _ { e n d }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 227, + 304 + ], + "score": 1.0, + "content": "decodes the return probability", + "type": "text" + }, + { + "bbox": [ + 228, + 291, + 237, + 301 + ], + "score": 0.65, + "content": "r _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 289, + 241, + 304 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 241, + 290, + 264, + 302 + ], + "score": 0.57, + "content": "f _ { p r o g }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 289, + 405, + 304 + ], + "score": 1.0, + "content": "decodes a program key embedding", + "type": "text" + }, + { + "bbox": [ + 406, + 290, + 425, + 302 + ], + "score": 0.88, + "content": "p _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 289, + 450, + 304 + ], + "score": 1.0, + "content": ";1 and", + "type": "text" + }, + { + "bbox": [ + 451, + 290, + 469, + 302 + ], + "score": 0.91, + "content": "f _ { a r g }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "decodes", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 151, + 314 + ], + "score": 1.0, + "content": "arguments", + "type": "text" + }, + { + "bbox": [ + 151, + 303, + 171, + 312 + ], + "score": 0.88, + "content": "a _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 300, + 228, + 314 + ], + "score": 1.0, + "content": ". The outputs", + "type": "text" + }, + { + "bbox": [ + 229, + 303, + 284, + 312 + ], + "score": 0.81, + "content": "r _ { t } , p _ { t + 1 } , a _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "are used to determine the next action, as described in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 393, + 326 + ], + "score": 1.0, + "content": "Algorithm 1. If the program is primitive, the next environmental state", + "type": "text" + }, + { + "bbox": [ + 394, + 313, + 413, + 324 + ], + "score": 0.89, + "content": "e _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 310, + 493, + 326 + ], + "score": 1.0, + "content": "will be affected by", + "type": "text" + }, + { + "bbox": [ + 494, + 314, + 504, + 324 + ], + "score": 0.82, + "content": "p _ { t }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 123, + 335 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 324, + 133, + 334 + ], + "score": 0.85, + "content": "a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 322, + 154, + 335 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 154, + 323, + 248, + 334 + ], + "score": 0.91, + "content": "e _ { t + 1 } \\sim f _ { e n v } ( e _ { t } , p _ { t } , a _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 322, + 506, + 335 + ], + "score": 1.0, + "content": ". As with the original NPI architecture, the experiments for this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 372, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 264, + 347 + ], + "score": 1.0, + "content": "paper always used a 3-tuple of integers", + "type": "text" + }, + { + "bbox": [ + 264, + 333, + 368, + 345 + ], + "score": 0.93, + "content": "a _ { t } = ( a _ { t } ( 1 ) , \\hat { a } _ { t } ( 2 ) , a _ { t } ( 3 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 333, + 372, + 347 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 256, + 506, + 347 + ] + }, + { + "type": "table", + "bbox": [ + 110, + 373, + 437, + 487 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 360, + 285, + 371 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 358, + 287, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 287, + 374 + ], + "score": 1.0, + "content": "Algorithm 1 Neural programming inference", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 110, + 373, + 437, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 373, + 437, + 487 + ], + "spans": [ + { + "bbox": [ + 110, + 373, + 437, + 487 + ], + "score": 0.768, + "html": "
1:Inputs: Environment observation e, program p,arguments a, stop threshold α
2:function RUN(e,p, a)
3:h↑0,r←0
4:whiler<αdo
5:s ←fenc(e,a),h ←fistm(s,p,h)
6:r ←fend(h),p2 ← fprog(h),a2 ← farg(h)
7:if p is a primitive function then
8:e ← fenv(e,p,a).
9:else
10:function RUN(e,P2, a2)
", + "type": "table", + "image_path": "e6560f320f41f348fdd426619c47efa3064b98bfb7c7dc35cb65540f5dbaec9a.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 110, + 373, + 437, + 411.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 110, + 411.0, + 437, + 449.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 110, + 449.0, + 437, + 487.0 + ], + "spans": [], + "index": 23 + } + ] + } + ], + "index": 21.0 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "A description of the inference procedure is given in Algorithm 1. Each step during an execution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "of the program does one of three things: (1) another subprogram along with associated arguments", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "is called, as in Line 10, (2) the program writes to the environment if it is primitive, as in Line 8,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 405, + 545 + ], + "score": 1.0, + "content": "or (3) the loop is terminated if the return probability exceeds a threshold", + "type": "text" + }, + { + "bbox": [ + 406, + 535, + 413, + 543 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 534, + 505, + 545 + ], + "score": 1.0, + "content": ", after which the stack", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 395, + 556 + ], + "score": 1.0, + "content": "frame is popped and control is returned to the caller. In all experiments,", + "type": "text" + }, + { + "bbox": [ + 395, + 546, + 403, + 554 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "is set to 0.5. Each time a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 555, + 298, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 298, + 568 + ], + "score": 1.0, + "content": "subprogram is called, the stack depth increases.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 501, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "The training data for the Neural Programmer-Interpreter consists of full execution traces for the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "program of interest. A single element of an execution trace consists of a step input-step output pair,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 592, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 609 + ], + "score": 1.0, + "content": "which can be synthesized from Algorithm 1: this corresponds to, for a given time-step, the step", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 153, + 619 + ], + "score": 1.0, + "content": "input tuple", + "type": "text" + }, + { + "bbox": [ + 153, + 605, + 185, + 617 + ], + "score": 0.93, + "content": "( e , p , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 604, + 275, + 619 + ], + "score": 1.0, + "content": "and step output tuple", + "type": "text" + }, + { + "bbox": [ + 275, + 605, + 316, + 617 + ], + "score": 0.92, + "content": "( r , p _ { 2 } , a _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 604, + 506, + 619 + ], + "score": 1.0, + "content": ". An example of part of an addition task trace,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "written in shorthand, is given in Figure 1. For example, a step input-step output pair in Lines 2 and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "3 of the left-hand side of Figure 1 is (ADD1, WRITE OUT 1). In this pair, the step input runs", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "a subprogram ADD1 that has no arguments, and the step output contains a program WRITE that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "has arguments of OUT and 1. The environment and return probability are omitted for readability.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 658, + 356, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 356, + 672 + ], + "score": 1.0, + "content": "Indentation indicates the stack is one level deeper than before.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 572, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "It is important to emphasize that at inference time in the NPI, the hidden state of the LSTM controller", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 504, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 393, + 701 + ], + "score": 1.0, + "content": "is reset (to zero) at each subprogram call, as in Line 3 of Algorithm 1", + "type": "text" + }, + { + "bbox": [ + 393, + 688, + 423, + 698 + ], + "score": 0.82, + "content": "h \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 687, + 504, + 701 + ], + "score": 1.0, + "content": "). This functionality", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "is critical for implementing recursion, since it permits us to restrict our attention to the currently", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 263, + 389, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 389, + 275 + ], + "score": 1.0, + "content": "relevant recursive call, ignoring irrelevant details about other contexts.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 676, + 505, + 701 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 181, + 78, + 408, + 195 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 181, + 78, + 408, + 195 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 181, + 78, + 408, + 195 + ], + "spans": [ + { + "bbox": [ + 181, + 78, + 408, + 195 + ], + "score": 0.831, + "type": "image", + "image_path": "400738f1bb6ec680e00398081df8b91a36e097048042ec5fdd599e1e2221cb5f.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 181, + 78, + 408, + 91.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 181, + 91.0, + 408, + 104.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 181, + 104.0, + 408, + 117.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 181, + 117.0, + 408, + 130.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 181, + 130.0, + 408, + 143.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 181, + 143.0, + 408, + 156.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 181, + 156.0, + 408, + 169.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 181, + 169.0, + 408, + 182.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 181, + 182.0, + 408, + 195.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 209, + 503, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "Figure 1: Addition Task. The non-recursive trace loops on cycles of ADD1 and LSHIFT, whereas", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 220, + 361, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 361, + 231 + ], + "score": 1.0, + "content": "in the recursive version, the ADD function calls itself (bolded).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 105, + 252, + 504, + 274 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "is critical for implementing recursion, since it permits us to restrict our attention to the currently", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 263, + 389, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 389, + 275 + ], + "score": 1.0, + "content": "relevant recursive call, ignoring irrelevant details about other contexts.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 288, + 342, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 344, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 344, + 301 + ], + "score": 1.0, + "content": "3.2 RECURSIVE FORMULATIONS FOR NPI PROGRAMS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "We emphasize the overall goal of this work is to enable the learning of a recursive program. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 321, + 504, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 504, + 333 + ], + "score": 1.0, + "content": "learned recursive program is different from neural programs learned in all previous work in an impor-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "tant aspect: previous approaches do not explicitly incorporate this abstraction, and hence generalize", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 341, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 341, + 505, + 356 + ], + "score": 1.0, + "content": "poorly, whereas our learned neural programs incorporate recursion and achieve perfect generaliza-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 353, + 127, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 127, + 366 + ], + "score": 1.0, + "content": "tion.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "Since NPI naturally supports the notion of recursion, a key question is how to enable NPI to learn", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "recursive programs. We found that changing the NPI training traces is a simple way to enable this.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "In particular, we construct new training traces which explicitly contain recursive elements and show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "that with this type of trace, NPI easily learns recursive programs. In future work, we would like to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "decrease supervision and construct models that are capable of coming up with recursive abstractions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 425, + 155, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 155, + 436 + ], + "score": 1.0, + "content": "themselves.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "In what follows, we describe the way in which we constructed NPI training traces so as to make", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "them contain recursive elements and thus enable NPI to learn recursive programs. We describe the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "recursive re-formulation of traces for two tasks from the original NPI paper—grade-school addition", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 475, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 488 + ], + "score": 1.0, + "content": "and bubble sort. For these programs, we re-use the appropriate program sets (the associated subpro-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "grams), and we refer the reader to the appendix of Reed & de Freitas (2016) for further details on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "the subprograms used in addition and bubble sort. Finally, we implement recursive traces for our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 508, + 272, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 272, + 520 + ], + "score": 1.0, + "content": "own topological sort and quicksort tasks.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 109, + 531, + 448, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 449, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 449, + 546 + ], + "score": 1.0, + "content": "Grade School Addition. For grade-school addition, the domain-specific encoder is", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 549, + 493, + 563 + ], + "lines": [ + { + "bbox": [ + 118, + 549, + 493, + 563 + ], + "spans": [ + { + "bbox": [ + 118, + 549, + 493, + 563 + ], + "score": 0.89, + "content": "\\begin{array} { r } { f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , i _ { 4 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } ) , Q ( 4 , i _ { 4 } ) , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \\end{array}", + "type": "interline_equation", + "image_path": "6c1b6d2e5e1d9059fe38faa7c40cbd3c86e9a2bd9bba26354ec307dad09105fd.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 118, + 549, + 493, + 563 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 201, + 582 + ], + "score": 1.0, + "content": "where the environment", + "type": "text" + }, + { + "bbox": [ + 201, + 568, + 261, + 580 + ], + "score": 0.93, + "content": "Q \\in \\mathbb { R } ^ { 4 \\times N \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 566, + 506, + 582 + ], + "score": 1.0, + "content": "is a scratch-pad that contains four rows (the first input num-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 375, + 592 + ], + "score": 1.0, + "content": "ber, the second input number, the carry bits, and the output) and", + "type": "text" + }, + { + "bbox": [ + 375, + 580, + 386, + 590 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 580, + 430, + 592 + ], + "score": 1.0, + "content": "columns.", + "type": "text" + }, + { + "bbox": [ + 430, + 580, + 441, + 590 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "is set to 11, to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "represent the range of 10 possible digits, along with a token representing the end of input.2 At", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "any given time, the NPI has access to values pointed to by four pointers in each of the four rows,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 613, + 335, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 167, + 626 + ], + "score": 1.0, + "content": "represented by", + "type": "text" + }, + { + "bbox": [ + 168, + 613, + 276, + 625 + ], + "score": 0.92, + "content": "Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 613, + 297, + 626 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 297, + 613, + 331, + 625 + ], + "score": 0.93, + "content": "Q ( 4 , i _ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 613, + 335, + 626 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "The non-recursive trace loops on cycles of ADD1 and LSHIFT. ADD1 is a subprogram that adds", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "score": 1.0, + "content": "the current column (writing the appropriate digit to the output row and carrying a bit to the next", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 652, + 504, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 504, + 664 + ], + "score": 1.0, + "content": "column if needed). LSHIFT moves the four pointers to the left, to move to the next column. The", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 663, + 379, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 379, + 675 + ], + "score": 1.0, + "content": "program terminates when seeing no numbers in the current column.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 679, + 503, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "Figure 1 shows examples of non-recursive and recursive addition traces. We make the trace recursive", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "by adding a tail recursive call into the trace for the ADD program after calling ADD1 and LSHIFT,", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 117, + 709, + 211, + 724 + ], + "score": 1.0, + "content": "2The original paper uses", + "type": "text" + }, + { + "bbox": [ + 211, + 712, + 244, + 721 + ], + "score": 0.83, + "content": "K = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 709, + 506, + 724 + ], + "score": 1.0, + "content": ", but we found it necessary to augment the range with an end token, in", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 208, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 208, + 734 + ], + "score": 1.0, + "content": "order to terminate properly.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 181, + 78, + 408, + 195 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 181, + 78, + 408, + 195 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 181, + 78, + 408, + 195 + ], + "spans": [ + { + "bbox": [ + 181, + 78, + 408, + 195 + ], + "score": 0.831, + "type": "image", + "image_path": "400738f1bb6ec680e00398081df8b91a36e097048042ec5fdd599e1e2221cb5f.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 181, + 78, + 408, + 91.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 181, + 91.0, + 408, + 104.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 181, + 104.0, + 408, + 117.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 181, + 117.0, + 408, + 130.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 181, + 130.0, + 408, + 143.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 181, + 143.0, + 408, + 156.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 181, + 156.0, + 408, + 169.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 181, + 169.0, + 408, + 182.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 181, + 182.0, + 408, + 195.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 209, + 503, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "Figure 1: Addition Task. The non-recursive trace loops on cycles of ADD1 and LSHIFT, whereas", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 220, + 361, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 361, + 231 + ], + "score": 1.0, + "content": "in the recursive version, the ADD function calls itself (bolded).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 105, + 252, + 504, + 274 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 106, + 253, + 504, + 275 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 288, + 342, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 344, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 344, + 301 + ], + "score": 1.0, + "content": "3.2 RECURSIVE FORMULATIONS FOR NPI PROGRAMS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "We emphasize the overall goal of this work is to enable the learning of a recursive program. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 321, + 504, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 504, + 333 + ], + "score": 1.0, + "content": "learned recursive program is different from neural programs learned in all previous work in an impor-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "tant aspect: previous approaches do not explicitly incorporate this abstraction, and hence generalize", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 341, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 341, + 505, + 356 + ], + "score": 1.0, + "content": "poorly, whereas our learned neural programs incorporate recursion and achieve perfect generaliza-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 353, + 127, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 127, + 366 + ], + "score": 1.0, + "content": "tion.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 309, + 505, + 366 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "Since NPI naturally supports the notion of recursion, a key question is how to enable NPI to learn", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "recursive programs. We found that changing the NPI training traces is a simple way to enable this.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "In particular, we construct new training traces which explicitly contain recursive elements and show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "that with this type of trace, NPI easily learns recursive programs. In future work, we would like to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "decrease supervision and construct models that are capable of coming up with recursive abstractions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 425, + 155, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 155, + 436 + ], + "score": 1.0, + "content": "themselves.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 370, + 506, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "In what follows, we describe the way in which we constructed NPI training traces so as to make", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "them contain recursive elements and thus enable NPI to learn recursive programs. We describe the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "recursive re-formulation of traces for two tasks from the original NPI paper—grade-school addition", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 475, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 488 + ], + "score": 1.0, + "content": "and bubble sort. For these programs, we re-use the appropriate program sets (the associated subpro-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "grams), and we refer the reader to the appendix of Reed & de Freitas (2016) for further details on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "the subprograms used in addition and bubble sort. Finally, we implement recursive traces for our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 508, + 272, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 272, + 520 + ], + "score": 1.0, + "content": "own topological sort and quicksort tasks.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 441, + 505, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 531, + 448, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 449, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 449, + 546 + ], + "score": 1.0, + "content": "Grade School Addition. For grade-school addition, the domain-specific encoder is", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 529, + 449, + 546 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 549, + 493, + 563 + ], + "lines": [ + { + "bbox": [ + 118, + 549, + 493, + 563 + ], + "spans": [ + { + "bbox": [ + 118, + 549, + 493, + 563 + ], + "score": 0.89, + "content": "\\begin{array} { r } { f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , i _ { 4 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } ) , Q ( 4 , i _ { 4 } ) , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \\end{array}", + "type": "interline_equation", + "image_path": "6c1b6d2e5e1d9059fe38faa7c40cbd3c86e9a2bd9bba26354ec307dad09105fd.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 118, + 549, + 493, + 563 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 201, + 582 + ], + "score": 1.0, + "content": "where the environment", + "type": "text" + }, + { + "bbox": [ + 201, + 568, + 261, + 580 + ], + "score": 0.93, + "content": "Q \\in \\mathbb { R } ^ { 4 \\times N \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 566, + 506, + 582 + ], + "score": 1.0, + "content": "is a scratch-pad that contains four rows (the first input num-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 375, + 592 + ], + "score": 1.0, + "content": "ber, the second input number, the carry bits, and the output) and", + "type": "text" + }, + { + "bbox": [ + 375, + 580, + 386, + 590 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 580, + 430, + 592 + ], + "score": 1.0, + "content": "columns.", + "type": "text" + }, + { + "bbox": [ + 430, + 580, + 441, + 590 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "is set to 11, to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "represent the range of 10 possible digits, along with a token representing the end of input.2 At", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "any given time, the NPI has access to values pointed to by four pointers in each of the four rows,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 613, + 335, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 167, + 626 + ], + "score": 1.0, + "content": "represented by", + "type": "text" + }, + { + "bbox": [ + 168, + 613, + 276, + 625 + ], + "score": 0.92, + "content": "Q ( 1 , i _ { 1 } ) , Q ( 2 , i _ { 2 } ) , Q ( 3 , i _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 613, + 297, + 626 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 297, + 613, + 331, + 625 + ], + "score": 0.93, + "content": "Q ( 4 , i _ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 613, + 335, + 626 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 566, + 506, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "The non-recursive trace loops on cycles of ADD1 and LSHIFT. ADD1 is a subprogram that adds", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "score": 1.0, + "content": "the current column (writing the appropriate digit to the output row and carrying a bit to the next", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 652, + 504, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 504, + 664 + ], + "score": 1.0, + "content": "column if needed). LSHIFT moves the four pointers to the left, to move to the next column. The", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 663, + 379, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 379, + 675 + ], + "score": 1.0, + "content": "program terminates when seeing no numbers in the current column.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 629, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 679, + 503, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "Figure 1 shows examples of non-recursive and recursive addition traces. We make the trace recursive", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "by adding a tail recursive call into the trace for the ADD program after calling ADD1 and LSHIFT,", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 679, + 505, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 394, + 79, + 453, + 90 + ], + "lines": [ + { + "bbox": [ + 393, + 78, + 455, + 91 + ], + "spans": [ + { + "bbox": [ + 393, + 78, + 455, + 91 + ], + "score": 1.0, + "content": "Full Recursive", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image", + "bbox": [ + 118, + 83, + 475, + 293 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 118, + 83, + 475, + 293 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 83, + 475, + 293 + ], + "spans": [ + { + "bbox": [ + 118, + 83, + 475, + 293 + ], + "score": 0.791, + "type": "image", + "image_path": "54ac005719045e00f6d938466b5b91cbc2d9b186869b92dd8c3b93eb2423cdb1.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 118, + 83, + 475, + 153.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 118, + 153.0, + 475, + 223.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 118, + 223.0, + 475, + 293.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 306, + 505, + 372 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "Figure 2: Bubble Sort Task. The non-recursive trace loops on cycles of BUBBLE and RESET.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "The difference between the partial recursive and full recursive versions is in the indentation of Lines", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "10-15 and 20-22 (bolded), since in the full recursive version, BSTEP and LSHIFT are made tail", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "recursive; the final calls to BSTEP and LSHIFT return immediately as they occur after the pointer", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "reaches the end of the array. Also note that COMPSWAP conditionally swaps numbers under the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 360, + 173, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 173, + 375 + ], + "score": 1.0, + "content": "bubble pointers.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + } + ], + "index": 4.25 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 504, + 408 + ], + "score": 1.0, + "content": "as in Line 13 of the right-hand side of Figure 1. Via the recursive call, we effectively forget that the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "column just added exists, since the recursive call to ADD starts with a new hidden state for the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "LSTM controller. 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A topological sort", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 363, + 451 + ], + "score": 1.0, + "content": "is a linear ordering of vertices such that for every directed edge", + "type": "text" + }, + { + "bbox": [ + 364, + 439, + 387, + 451 + ], + "score": 0.94, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 438, + 411, + 451 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 411, + 441, + 419, + 448 + ], + "score": 0.66, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 438, + 430, + 451 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 430, + 440, + 448, + 449 + ], + "score": 0.28, + "content": "v , u", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "comes before", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 113, + 459 + ], + "score": 0.58, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "in the ordering. This is possible if and only if the graph has no directed cycles; that is to say, it", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "must be a directed acyclic graph (DAG). In our experiments, we only present DAG’s as inputs and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 472, + 471, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 287, + 484 + ], + "score": 1.0, + "content": "represent the vertices as values ranging from", + "type": "text" + }, + { + "bbox": [ + 287, + 472, + 322, + 483 + ], + "score": 0.9, + "content": "1 , \\ldots , n", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 472, + 428, + 484 + ], + "score": 1.0, + "content": ", where the DAG contains", + "type": "text" + }, + { + "bbox": [ + 428, + 474, + 435, + 481 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 472, + 471, + 484 + ], + "score": 1.0, + "content": "vertices.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "Directed acyclic graphs are structurally more diverse than inputs in the two tasks of grade-school", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "addition and bubble sort. The degree for any vertex in the DAG is variable. Also the DAG can have", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "potentially more than one connected component, meaning it is necessary to transition between these", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 214, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 214, + 534 + ], + "score": 1.0, + "content": "components appropriately.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "Algorithm 2 shows the topological sort task of interest. This algorithm is a variant of depth first", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "search. We created a program set that reflects the semantics of Algorithm 2. For brevity, we refer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "score": 1.0, + "content": "the reader to the appendix for further details on the program set and non-recursive and recursive", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 316, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 316, + 583 + ], + "score": 1.0, + "content": "trace-generating functions used for topological sort.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 108, + 588, + 313, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 314, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 314, + 601 + ], + "score": 1.0, + "content": "For topological sort, the domain-specific encoder is", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 605, + 514, + 634 + ], + "lines": [ + { + "bbox": [ + 111, + 605, + 514, + 634 + ], + "spans": [ + { + "bbox": [ + 111, + 605, + 514, + 634 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\ f _ { e n c } ( D A G , Q _ { c o l o r } , p _ { s t a c k } , p _ { s t a r t } , v _ { a c t i v e } , c h i l d L i s t , a _ { t } ) } \\\\ & { = M L P ( [ Q _ { c o l o r } ( p _ { s t a r t } ) , Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ) , p _ { s t a c k } = = 1 , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) , a _ { t } ) } \\end{array}", + "type": "interline_equation", + "image_path": "a592788c363fd85d2c133b3b98a5611788d42e231bad314abd3d9ec7325ac3e1.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 111, + 605, + 514, + 614.6666666666666 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 111, + 614.6666666666666, + 514, + 624.3333333333333 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 111, + 624.3333333333333, + 514, + 633.9999999999999 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 506, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 134, + 651 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 637, + 201, + 649 + ], + "score": 0.92, + "content": "Q _ { c o l o r } \\in \\mathbb { R } ^ { U \\times 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 635, + 325, + 651 + ], + "score": 1.0, + "content": "is a scratch-pad that contains", + "type": "text" + }, + { + "bbox": [ + 325, + 639, + 334, + 648 + ], + "score": 0.84, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 635, + 506, + 651 + ], + "score": 1.0, + "content": "rows, each containing one of four colors", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 323, + 662 + ], + "score": 1.0, + "content": "(white, gray, black, invalid) with one-hot encoding.", + "type": "text" + }, + { + "bbox": [ + 324, + 649, + 333, + 659 + ], + "score": 0.79, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "varies with the number of vertices in the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 657, + 507, + 674 + ], + "spans": [ + { + "bbox": [ + 104, + 657, + 205, + 674 + ], + "score": 1.0, + "content": "graph. 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The", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 153, + 695 + ], + "score": 1.0, + "content": "contents of", + "type": "text" + }, + { + "bbox": [ + 153, + 682, + 183, + 693 + ], + "score": 0.92, + "content": "Q _ { r e s u l t }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 681, + 201, + 695 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 682, + 230, + 693 + ], + "score": 0.92, + "content": "Q _ { s t a c k }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "are not exposed directly through the domain-specific encoder; rather,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 693, + 377, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 377, + 705 + ], + "score": 1.0, + "content": "we define primitive functions which manipulate these scratch-pads.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 317, + 723 + ], + "score": 1.0, + "content": "The DAG is represented as an adjacency list where", + "type": "text" + }, + { + "bbox": [ + 318, + 709, + 362, + 722 + ], + "score": 0.92, + "content": "D A G [ i ] [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 708, + 415, + 723 + ], + "score": 1.0, + "content": "refers to the", + "type": "text" + }, + { + "bbox": [ + 416, + 711, + 421, + 721 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 708, + 496, + 723 + ], + "score": 1.0, + "content": "-th child of vertex", + "type": "text" + }, + { + "bbox": [ + 497, + 711, + 501, + 720 + ], + "score": 0.59, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 708, + 506, + 723 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 718, + 501, + 736 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 193, + 736 + ], + "score": 1.0, + "content": "There are 3 pointers", + "type": "text" + }, + { + "bbox": [ + 193, + 722, + 284, + 732 + ], + "score": 0.81, + "content": "( p _ { r e s u l t } , p _ { s t a c k } , p _ { s t a r t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 718, + 289, + 736 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 290, + 722, + 318, + 732 + ], + "score": 0.47, + "content": "p _ { r e s u l t }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 718, + 470, + 736 + ], + "score": 1.0, + "content": "points to the next empty location in", + "type": "text" + }, + { + "bbox": [ + 470, + 721, + 501, + 732 + ], + "score": 0.9, + "content": "Q _ { r e s u l t }", + "type": "inline_equation" + } + ], + "index": 52 + } + ], + "index": 51.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "We experiment with two levels of recursion—partial and full. 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We performed experiments", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "on the partially recursive version in order to examine what happens when only one recursive call is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 404, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 404, + 183 + ], + "score": 1.0, + "content": "implemented, when in reality three are required for perfect generalization.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 83, + 506, + 183 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 193, + 306, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 191, + 308, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 308, + 208 + ], + "score": 1.0, + "content": "Algorithm 2 Depth First Search Topological Sort", + "type": "text" + } + ], + "index": 9 + } + 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This algorithm is a variant of depth first", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "search. We created a program set that reflects the semantics of Algorithm 2. For brevity, we refer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "score": 1.0, + "content": "the reader to the appendix for further details on the program set and non-recursive and recursive", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 316, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 316, + 583 + ], + "score": 1.0, + "content": "trace-generating functions used for topological sort.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 538, + 505, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 588, + 313, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 314, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 314, + 601 + ], + "score": 1.0, + "content": "For topological sort, the domain-specific encoder is", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 587, + 314, + 601 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 605, + 514, + 634 + ], + "lines": [ + { + "bbox": [ + 111, + 605, + 514, + 634 + ], + "spans": [ + { + "bbox": [ + 111, + 605, + 514, + 634 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\ f _ { e n c } ( D A G , Q _ { c o l o r } , p _ { s t a c k } , p _ { s t a r t } , v _ { a c t i v e } , c h i l d L i s t , a _ { t } ) } \\\\ & { = M L P ( [ Q _ { c o l o r } ( p _ { s t a r t } ) , Q _ { c o l o r } ( D A G [ v _ { a c t i v e } ] [ c h i l d L i s t [ v _ { a c t i v e } ] ] ) , p _ { s t a c k } = = 1 , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) , a _ { t } ) } \\end{array}", + "type": "interline_equation", + "image_path": "a592788c363fd85d2c133b3b98a5611788d42e231bad314abd3d9ec7325ac3e1.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 111, + 605, + 514, + 614.6666666666666 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 111, + 614.6666666666666, + 514, + 624.3333333333333 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 111, + 624.3333333333333, + 514, + 633.9999999999999 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 506, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 134, + 651 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 637, + 201, + 649 + ], + "score": 0.92, + "content": "Q _ { c o l o r } \\in \\mathbb { R } ^ { U \\times 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 635, + 325, + 651 + ], + "score": 1.0, + "content": "is a scratch-pad that contains", + "type": "text" + }, + { + "bbox": [ + 325, + 639, + 334, + 648 + ], + "score": 0.84, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 635, + 506, + 651 + ], + "score": 1.0, + "content": "rows, each containing one of four colors", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 323, + 662 + ], + "score": 1.0, + "content": "(white, gray, black, invalid) with one-hot encoding.", + "type": "text" + }, + { + "bbox": [ + 324, + 649, + 333, + 659 + ], + "score": 0.79, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "varies with the number of vertices in the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 657, + 507, + 674 + ], + "spans": [ + { + "bbox": [ + 104, + 657, + 205, + 674 + ], + "score": 1.0, + "content": "graph. 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The", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 153, + 695 + ], + "score": 1.0, + "content": "contents of", + "type": "text" + }, + { + "bbox": [ + 153, + 682, + 183, + 693 + ], + "score": 0.92, + "content": "Q _ { r e s u l t }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 681, + 201, + 695 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 682, + 230, + 693 + ], + "score": 0.92, + "content": "Q _ { s t a c k }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "are not exposed directly through the domain-specific encoder; rather,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 693, + 377, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 377, + 705 + ], + "score": 1.0, + "content": "we define primitive functions which manipulate these scratch-pads.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 104, + 635, + 507, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 317, + 723 + ], + "score": 1.0, + "content": "The DAG is represented as an adjacency list where", + "type": "text" + }, + { + "bbox": [ + 318, + 709, + 362, + 722 + ], + "score": 0.92, + "content": "D A G [ i ] [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 708, + 415, + 723 + ], + "score": 1.0, + "content": "refers to the", + "type": "text" + }, + { + "bbox": [ + 416, + 711, + 421, + 721 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 708, + 496, + 723 + ], + "score": 1.0, + "content": "-th child of vertex", + "type": "text" + }, + { + "bbox": [ + 497, + 711, + 501, + 720 + ], + "score": 0.59, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 708, + 506, + 723 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 718, + 501, + 736 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 193, + 736 + ], + "score": 1.0, + "content": "There are 3 pointers", + "type": "text" + }, + { + "bbox": [ + 193, + 722, + 284, + 732 + ], + "score": 0.81, + "content": "( p _ { r e s u l t } , p _ { s t a c k } , p _ { s t a r t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 718, + 289, + 736 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 290, + 722, + 318, + 732 + ], + "score": 0.47, + "content": "p _ { r e s u l t }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 718, + 470, + 736 + ], + "score": 1.0, + "content": "points to the next empty location in", + "type": "text" + }, + { + "bbox": [ + 470, + 721, + 501, + 732 + ], + "score": 0.9, + "content": "Q _ { r e s u l t }", + "type": "inline_equation" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 132, + 94 + ], + "score": 0.41, + "content": "p _ { s t a c k }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 133, + 82, + 264, + 95 + ], + "score": 1.0, + "content": "points to the top of the stack in", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 264, + 83, + 292, + 94 + ], + "score": 0.92, + "content": "Q _ { s t a c k }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 293, + 82, + 314, + 95 + ], + "score": 1.0, + "content": ", and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 315, + 84, + 339, + 94 + ], + "score": 0.83, + "content": "p _ { s t a r t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 340, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "points to the candidate starting node for", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 302, + 107 + ], + "score": 1.0, + "content": "a connected component. 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Finally,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 491, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 155, + 200 + ], + "score": 1.0, + "content": "the boolean", + "type": "text" + }, + { + "bbox": [ + 155, + 188, + 207, + 199 + ], + "score": 0.91, + "content": "p _ { s t a c k } = = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 186, + 491, + 200 + ], + "score": 1.0, + "content": "is used to check whether the stack is empty in Line 18 of Algorithm 2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 204, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "An alternative way of representing the environment slice is to expose the values of the absolute", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "vertices to the model; however, this makes it difficult to scale the model to larger graphs, since large", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 296, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 296, + 239 + ], + "score": 1.0, + "content": "vertex values are not seen during training time.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "We refer the reader to the appendix for the non-recursive trace generating functions. 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The logic", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 397, + 379, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 379, + 411 + ], + "score": 1.0, + "content": "for the non-recursive trace is shown in Algorithm 4 in the appendix.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 422, + 244, + 433 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 245, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 245, + 435 + ], + "score": 1.0, + "content": "Algorithm 3 Recursive Quicksort", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 109, + 436, + 369, + 629 + ], + "lines": [ + { + "bbox": [ + 111, + 435, + 236, + 446 + ], + "spans": [ + { + "bbox": [ + 111, + 435, + 196, + 446 + ], + "score": 1.0, + "content": "1: Initialize an array", + "type": "text" + }, + { + "bbox": [ + 196, + 435, + 205, + 445 + ], + "score": 0.67, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 435, + 236, + 446 + ], + "score": 1.0, + "content": "to sort.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 111, + 446, + 367, + 457 + ], + "spans": [ + { + "bbox": [ + 111, + 446, + 161, + 457 + ], + "score": 1.0, + "content": "2: Initialize", + "type": "text" + }, + { + "bbox": [ + 162, + 446, + 171, + 456 + ], + "score": 0.27, + "content": "l o", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 446, + 189, + 457 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 189, + 446, + 199, + 456 + ], + "score": 0.74, + "content": "h i", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 446, + 246, + 457 + ], + "score": 1.0, + "content": "to be 1 and", + "type": "text" + }, + { + "bbox": [ + 247, + 448, + 254, + 456 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 446, + 284, + 457 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 285, + 448, + 292, + 456 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 446, + 355, + 457 + ], + "score": 1.0, + "content": "is the length of", + "type": "text" + }, + { + "bbox": [ + 355, + 446, + 363, + 456 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 446, + 367, + 457 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 110, + 457, + 121, + 468 + ], + "spans": [ + { + "bbox": [ + 110, + 457, + 121, + 468 + ], + "score": 1.0, + "content": "3:", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 110, + 466, + 255, + 480 + ], + "spans": [ + { + "bbox": [ + 110, + 468, + 121, + 478 + ], + "score": 1.0, + "content": "4:", + "type": "text" + }, + { + "bbox": [ + 122, + 466, + 214, + 480 + ], + "score": 1.0, + "content": "function QUICKSORT", + "type": "text" + }, + { + "bbox": [ + 214, + 468, + 255, + 479 + ], + "score": 0.46, + "content": "( A , l o , h i )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 110, + 478, + 205, + 491 + ], + "spans": [ + { + "bbox": [ + 110, + 479, + 121, + 490 + ], + "score": 1.0, + "content": "5:", + "type": "text" + }, + { + "bbox": [ + 137, + 478, + 147, + 491 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 147, + 479, + 178, + 489 + ], + "score": 0.88, + "content": "l o < h i", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 478, + 205, + 491 + ], + "score": 1.0, + "content": ": then", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 110, + 490, + 257, + 501 + ], + "spans": [ + { + "bbox": [ + 110, + 490, + 121, + 501 + ], + "score": 1.0, + "content": "6:", + "type": "text" + }, + { + "bbox": [ + 153, + 491, + 257, + 501 + ], + "score": 0.41, + "content": "\\mathsf { p } = \\mathsf { P A R T I T I O N } ( A , l o , h i )", + "type": "inline_equation", + "image_path": "61ad663f7bc2eff26a6f07baef4103e489e83e804f7b93f24b74c528d142e09c.jpg" + } + ], + "index": 33 + }, + { + "bbox": [ + 110, + 501, + 258, + 512 + ], + "spans": [ + { + "bbox": [ + 110, + 501, + 121, + 512 + ], + "score": 1.0, + "content": "7:", + "type": "text" + }, + { + "bbox": [ + 153, + 502, + 258, + 512 + ], + "score": 0.62, + "content": "\\mathrm { Q U I C K S O R T } ( A , l o , p - 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Finally,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 491, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 155, + 200 + ], + "score": 1.0, + "content": "the boolean", + "type": "text" + }, + { + "bbox": [ + 155, + 188, + 207, + 199 + ], + "score": 0.91, + "content": "p _ { s t a c k } = = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 186, + 491, + 200 + ], + "score": 1.0, + "content": "is used to check whether the stack is empty in Line 18 of Algorithm 2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 141, + 507, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 204, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "An alternative way of representing the environment slice is to expose the values of the absolute", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "vertices to the model; however, this makes it difficult to scale the model to larger graphs, since large", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 296, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 296, + 239 + ], + "score": 1.0, + "content": "vertex values are not seen during training time.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 204, + 505, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "We refer the reader to the appendix for the non-recursive trace generating functions. In", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 504, + 265 + ], + "score": 1.0, + "content": "the non-recursive trace, there are four functions that can be made recursive—TOPOSORT,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 276 + ], + "score": 1.0, + "content": "CHECK CHILD, EXPLORE, and NEXT START, and we add a tail recursive call to each of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "these functions in order to make the recursive trace. In particular, in the EXPLORE function,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "adding a tail recursive call resets and stores the hidden states associated with vertices in a stack-like", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "fashion. This makes it so that we only need to consider the vertices in the subgraph that are cur-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "rently relevant for computing the sort, allowing simpler reasoning about behavior for large graphs.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "The sequence of primitive operations (MOVE and WRITE operations) for the non-recursive and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 331, + 267, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 267, + 343 + ], + "score": 1.0, + "content": "recursive versions are exactly the same.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 242, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "Quicksort. 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The 4 environment observations aid with control flow;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 311, + 177 + ], + "score": 0.88, + "content": "Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 165, + 374, + 178 + ], + "score": 1.0, + "content": "implements the", + "type": "text" + }, + { + "bbox": [ + 375, + 165, + 406, + 176 + ], + "score": 0.89, + "content": "l o < h i", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "comparison in Line 5 of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 160, + 189 + ], + "score": 1.0, + "content": "Algorithm 3,", + "type": "text" + }, + { + "bbox": [ + 161, + 178, + 221, + 187 + ], + "score": 0.88, + "content": "p _ { s t a c k L o } = = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "checks if the stacks are empty in Line 18 of Algorithm 4, and the other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 439, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 214, + 200 + ], + "score": 1.0, + "content": "observations (all involving", + "type": "text" + }, + { + "bbox": [ + 215, + 189, + 240, + 200 + ], + "score": 0.59, + "content": "p _ { p i v o t }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 186, + 251, + 200 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 252, + 189, + 262, + 200 + ], + "score": 0.78, + "content": "p _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 186, + 439, + 200 + ], + "score": 1.0, + "content": ") deal with logic in the PARTITION function.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "Note that the recursion for quicksort is not purely tail recursive and therefore represents a more", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "complex kind of recursion that is harder to learn than in the previous tasks. Also, compared to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "bubble pointers in bubble sort, the pointers that perform the comparison for quicksort (the COMP-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "SWAP function) are usually not adjacent to each other, making quicksort less local than bubble", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 258, + 262 + ], + "score": 1.0, + "content": "sort. In order to compensate for this,", + "type": "text" + }, + { + "bbox": [ + 258, + 249, + 283, + 260 + ], + "score": 0.88, + "content": "p _ { p i v o t }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 246, + 302, + 262 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 249, + 312, + 260 + ], + "score": 0.84, + "content": "p _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 246, + 505, + 262 + ], + "score": 1.0, + "content": "require special functions (MOVE PIVOT LO", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 286, + 271 + ], + "score": 1.0, + "content": "and MOVE J LO) to properly set them to", + "type": "text" + }, + { + "bbox": [ + 287, + 259, + 297, + 269 + ], + "score": 0.74, + "content": "l o", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "in Lines 11 and 12 of the PARTITION function in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 162, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 162, + 282 + ], + "score": 1.0, + "content": "Algorithm 3.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 109, + 294, + 298, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 299, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 299, + 307 + ], + "score": 1.0, + "content": "3.3 PROVABLY PERFECT GENERALIZATION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "We show that if we incorporate recursion, the learned NPI programs can achieve provably perfect", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "score": 1.0, + "content": "generalization for different tasks. Provably perfect generalization implies the model will behave", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "correctly, given any valid input. In order to claim a proof, we must verify the model produces", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 418, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 418, + 359 + ], + "score": 1.0, + "content": "correct behavior over all base cases and reductions, as described in Section 2.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "We propose and describe our verification procedure. This procedure verifies that all base cases and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "reductions are handled properly by the model via explicit tests. Note that recursion helps make this", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "process tractable, because we only need to test a finite number of inputs to show that the model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "score": 1.0, + "content": "will work correctly on inputs of unbounded complexity. This verification phase only needs to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 228, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 228, + 423 + ], + "score": 1.0, + "content": "performed once after training.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 366, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 367, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 367, + 439 + ], + "score": 1.0, + "content": "Formally, verification consists of proving the following theorem:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 440, + 349, + 454 + ], + "lines": [ + { + "bbox": [ + 261, + 440, + 349, + 454 + ], + "spans": [ + { + "bbox": [ + 261, + 440, + 349, + 454 + ], + "score": 0.92, + "content": "\\forall i \\in V , M ( i ) \\downdownarrows P ( i )", + "type": "interline_equation", + "image_path": "c8b90ef3f1f4a594a65d98322c24c8c407c2f5d51069a21a999c875e2f9ac463.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 261, + 440, + 349, + 454 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 134, + 469 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 457, + 140, + 466 + ], + "score": 0.64, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 456, + 394, + 469 + ], + "score": 1.0, + "content": "denotes a sequence of step inputs (within one function call),", + "type": "text" + }, + { + "bbox": [ + 394, + 457, + 404, + 467 + ], + "score": 0.74, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "denotes the set of valid", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 208, + 479 + ], + "score": 1.0, + "content": "sequences of step inputs,", + "type": "text" + }, + { + "bbox": [ + 208, + 468, + 220, + 478 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 467, + 361, + 479 + ], + "score": 1.0, + "content": "denotes the neural network model,", + "type": "text" + }, + { + "bbox": [ + 361, + 468, + 371, + 477 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "denotes the correct program, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 127, + 491 + ], + "score": 0.91, + "content": "P ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "denotes the next step output from the correct program. The arrow in the theorem refers to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "evaluation, as in big-step semantics. The theorem states that for the same sequence of step inputs,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 438, + 513 + ], + "score": 1.0, + "content": "the model produces the exact same step output as the target program it aims to learn.", + "type": "text" + }, + { + "bbox": [ + 439, + 501, + 451, + 510 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 500, + 505, + 513 + ], + "score": 1.0, + "content": ", as described", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 402, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 402, + 524 + ], + "score": 1.0, + "content": "in Algorithm 1, processes the sequence of step inputs by using an LSTM.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "Recursion drastically reduces the number of configurations we need to consider during the verifi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 538, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 504, + 551 + ], + "score": 1.0, + "content": "cation phase and makes the proof tractable, because it introduces structure that eliminates infinitely", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "long sequences of step inputs that would otherwise need to be considered. For instance, for recursive", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 225, + 576 + ], + "score": 1.0, + "content": "addition, consider the family", + "type": "text" + }, + { + "bbox": [ + 225, + 562, + 234, + 571 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 560, + 321, + 576 + ], + "score": 1.0, + "content": "of addition problems", + "type": "text" + }, + { + "bbox": [ + 322, + 561, + 464, + 573 + ], + "score": 0.92, + "content": "a _ { n } a _ { n - 1 } \\dots a _ { 1 } a _ { 0 } + b _ { n } b _ { n - 1 } \\dots b _ { 1 } b _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 560, + 506, + 576 + ], + "score": 1.0, + "content": "where no", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 326, + 586 + ], + "score": 1.0, + "content": "CARRY operations occur. We prove every member of", + "type": "text" + }, + { + "bbox": [ + 327, + 573, + 335, + 582 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "is added properly, given that subproblems", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 581, + 462, + 597 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 383, + 596 + ], + "score": 0.65, + "content": "S = \\{ a _ { n } a _ { n - 1 } + b _ { n } b _ { n - 1 } , a _ { n - 1 } { \\bar { a } } _ { n - 2 } + b _ { n - 1 } b _ { n - 2 } , \\dots , a _ { 1 } a _ { 0 } + b _ { 1 } b _ { 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 581, + 462, + 597 + ], + "score": 1.0, + "content": "are added properly.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "Without using a recursive program, such a proof is not possible, because the non-recursive program", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "runs on an arbitrarily long addition problem that creates correspondingly long sequences of step", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "inputs; in the non-recursive formulation of addition, ADD calls ADD1 a number of times that is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "dependent on the length of the input. The core LSTM module’s hidden state is preserved over all", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "these ADD1 calls, and it is difficult to interpret with certainty what happens over longer timesteps", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "without concretely evaluating the LSTM with an input of that length. In contrast, each call to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the recursive ADD always runs for a fixed number of steps, even on arbitrarily long problems", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 117, + 690 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 118, + 677, + 127, + 687 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 677, + 505, + 690 + ], + "score": 1.0, + "content": ", so we can test that it performs correctly on a small, fixed number of step input sequences.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "This guarantees that the step input sequences considered during verification contain all step input", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 365, + 712 + ], + "score": 1.0, + "content": "sequences which arise during execution of an unseen problem in", + "type": "text" + }, + { + "bbox": [ + 365, + 700, + 373, + 708 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ", leading to generalization to any", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 152, + 722 + ], + "score": 1.0, + "content": "problem in", + "type": "text" + }, + { + "bbox": [ + 153, + 710, + 161, + 720 + ], + "score": 0.77, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 710, + 281, + 722 + ], + "score": 1.0, + "content": ". 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The 4 environment observations aid with control flow;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 311, + 177 + ], + "score": 0.88, + "content": "Q _ { s t a c k L o } ( p _ { s t a c k L o } - 1 ) < Q _ { s t a c k H i } ( p _ { s t a c k H i } - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 165, + 374, + 178 + ], + "score": 1.0, + "content": "implements the", + "type": "text" + }, + { + "bbox": [ + 375, + 165, + 406, + 176 + ], + "score": 0.89, + "content": "l o < h i", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "comparison in Line 5 of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 160, + 189 + ], + "score": 1.0, + "content": "Algorithm 3,", + "type": "text" + }, + { + "bbox": [ + 161, + 178, + 221, + 187 + ], + "score": 0.88, + "content": "p _ { s t a c k L o } = = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "checks if the stacks are empty in Line 18 of Algorithm 4, and the other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 439, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 214, + 200 + ], + "score": 1.0, + "content": "observations (all involving", + "type": "text" + }, + { + "bbox": [ + 215, + 189, + 240, + 200 + ], + "score": 0.59, + "content": "p _ { p i v o t }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 186, + 251, + 200 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 252, + 189, + 262, + 200 + ], + "score": 0.78, + "content": "p _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 186, + 439, + 200 + ], + "score": 1.0, + "content": ") deal with logic in the PARTITION function.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 120, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "Note that the recursion for quicksort is not purely tail recursive and therefore represents a more", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "complex kind of recursion that is harder to learn than in the previous tasks. 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Provably perfect generalization implies the model will behave", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "correctly, given any valid input. In order to claim a proof, we must verify the model produces", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 418, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 418, + 359 + ], + "score": 1.0, + "content": "correct behavior over all base cases and reductions, as described in Section 2.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 315, + 506, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "We propose and describe our verification procedure. 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This verification phase only needs to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 228, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 228, + 423 + ], + "score": 1.0, + "content": "performed once after training.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 365, + 505, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 366, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 367, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 367, + 439 + ], + "score": 1.0, + "content": "Formally, verification consists of proving the following theorem:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 424, + 367, + 439 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 440, + 349, + 454 + ], + "lines": [ + { + "bbox": [ + 261, + 440, + 349, + 454 + ], + "spans": [ + { + "bbox": [ + 261, + 440, + 349, + 454 + ], + "score": 0.92, + "content": "\\forall i \\in V , M ( i ) \\downdownarrows P ( i )", + "type": "interline_equation", + "image_path": "c8b90ef3f1f4a594a65d98322c24c8c407c2f5d51069a21a999c875e2f9ac463.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 261, + 440, + 349, + 454 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 134, + 469 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 457, + 140, + 466 + ], + "score": 0.64, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 456, + 394, + 469 + ], + "score": 1.0, + "content": "denotes a sequence of step inputs (within one function call),", + "type": "text" + }, + { + "bbox": [ + 394, + 457, + 404, + 467 + ], + "score": 0.74, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "denotes the set of valid", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 208, + 479 + ], + "score": 1.0, + "content": "sequences of step inputs,", + "type": "text" + }, + { + "bbox": [ + 208, + 468, + 220, + 478 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 467, + 361, + 479 + ], + "score": 1.0, + "content": "denotes the neural network model,", + "type": "text" + }, + { + "bbox": [ + 361, + 468, + 371, + 477 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "denotes the correct program, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 127, + 491 + ], + "score": 0.91, + "content": "P ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "denotes the next step output from the correct program. The arrow in the theorem refers to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "evaluation, as in big-step semantics. The theorem states that for the same sequence of step inputs,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 438, + 513 + ], + "score": 1.0, + "content": "the model produces the exact same step output as the target program it aims to learn.", + "type": "text" + }, + { + "bbox": [ + 439, + 501, + 451, + 510 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 500, + 505, + 513 + ], + "score": 1.0, + "content": ", as described", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 402, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 402, + 524 + ], + "score": 1.0, + "content": "in Algorithm 1, processes the sequence of step inputs by using an LSTM.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 456, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "Recursion drastically reduces the number of configurations we need to consider during the verifi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 538, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 504, + 551 + ], + "score": 1.0, + "content": "cation phase and makes the proof tractable, because it introduces structure that eliminates infinitely", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "long sequences of step inputs that would otherwise need to be considered. For instance, for recursive", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 225, + 576 + ], + "score": 1.0, + "content": "addition, consider the family", + "type": "text" + }, + { + "bbox": [ + 225, + 562, + 234, + 571 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 560, + 321, + 576 + ], + "score": 1.0, + "content": "of addition problems", + "type": "text" + }, + { + "bbox": [ + 322, + 561, + 464, + 573 + ], + "score": 0.92, + "content": "a _ { n } a _ { n - 1 } \\dots a _ { 1 } a _ { 0 } + b _ { n } b _ { n - 1 } \\dots b _ { 1 } b _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 560, + 506, + 576 + ], + "score": 1.0, + "content": "where no", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 326, + 586 + ], + "score": 1.0, + "content": "CARRY operations occur. We prove every member of", + "type": "text" + }, + { + "bbox": [ + 327, + 573, + 335, + 582 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "is added properly, given that subproblems", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 581, + 462, + 597 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 383, + 596 + ], + "score": 0.65, + "content": "S = \\{ a _ { n } a _ { n - 1 } + b _ { n } b _ { n - 1 } , a _ { n - 1 } { \\bar { a } } _ { n - 2 } + b _ { n - 1 } b _ { n - 2 } , \\dots , a _ { 1 } a _ { 0 } + b _ { 1 } b _ { 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 581, + 462, + 597 + ], + "score": 1.0, + "content": "are added properly.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 528, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "Without using a recursive program, such a proof is not possible, because the non-recursive program", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "runs on an arbitrarily long addition problem that creates correspondingly long sequences of step", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "inputs; in the non-recursive formulation of addition, ADD calls ADD1 a number of times that is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "dependent on the length of the input. The core LSTM module’s hidden state is preserved over all", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "these ADD1 calls, and it is difficult to interpret with certainty what happens over longer timesteps", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "without concretely evaluating the LSTM with an input of that length. In contrast, each call to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the recursive ADD always runs for a fixed number of steps, even on arbitrarily long problems", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 117, + 690 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 118, + 677, + 127, + 687 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 677, + 505, + 690 + ], + "score": 1.0, + "content": ", so we can test that it performs correctly on a small, fixed number of step input sequences.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "This guarantees that the step input sequences considered during verification contain all step input", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 365, + 712 + ], + "score": 1.0, + "content": "sequences which arise during execution of an unseen problem in", + "type": "text" + }, + { + "bbox": [ + 365, + 700, + 373, + 708 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ", leading to generalization to any", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 152, + 722 + ], + "score": 1.0, + "content": "problem in", + "type": "text" + }, + { + "bbox": [ + 153, + 710, + 161, + 720 + ], + "score": 0.77, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 710, + 281, + 722 + ], + "score": 1.0, + "content": ". 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If it is too large, then the semantics of", + "type": "text" + }, + { + "bbox": [ + 455, + 105, + 464, + 114 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "might not", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 256, + 128 + ], + "score": 1.0, + "content": "be well-defined on some elements in", + "type": "text" + }, + { + "bbox": [ + 256, + 116, + 265, + 125 + ], + "score": 0.76, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 115, + 506, + 128 + ], + "score": 1.0, + "content": ", or the spurious step input sequences may not be reachable", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 462, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 462, + 140 + ], + "score": 1.0, + "content": "from any valid problem input (e.g., an array for quicksort or a DAG for topological sort).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "To construct this set, by using the reference implementation of each subprogram, we construct a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "mapping between two sets of environment observations: the first set consists of all observations", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "that can occur at the beginning of a particular subprogram’s invocation, and the second set contains", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "score": 1.0, + "content": "the observations at the end of that subprogram. We can obtain this mapping by first considering", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "the possible observations that can arise at the beginning of the entry function (ADD, BUBBLE-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 506, + 212 + ], + "score": 1.0, + "content": "SORT, TOPOSORT, and QUICKSORT) for some valid program input, and iteratively applying", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "the observation-to-observation mapping implied by the reference implementation’s step output at", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "that point in the execution. If the step output specifies a primitive function call, we need to reason", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "about how it can affect the environment so as to change the observation in the next step input. For", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "non-primitive subprograms, we can update the observation-to-observation mapping currently asso-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "ciated with the subprogram and then apply that mapping to the current set. By iterating with this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 222, + 277 + ], + "score": 1.0, + "content": "procedure, and then running", + "type": "text" + }, + { + "bbox": [ + 222, + 264, + 231, + 274 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "on the input observation set that we obtain for the entry point func-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 185, + 288 + ], + "score": 1.0, + "content": "tion, we can obtain", + "type": "text" + }, + { + "bbox": [ + 186, + 275, + 195, + 285 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "precisely. To make an analogy to MDPs, this procedure is analogous to how", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 448, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 448, + 299 + ], + "score": 1.0, + "content": "value iteration obtains the correct value for each state starting from any initialization.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 504, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 238, + 315 + ], + "score": 1.0, + "content": "An alternative method is to run", + "type": "text" + }, + { + "bbox": [ + 238, + 303, + 247, + 313 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "on many different program inputs and then observe step input", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 241, + 326 + ], + "score": 1.0, + "content": "sequences which occur, to create", + "type": "text" + }, + { + "bbox": [ + 241, + 314, + 250, + 324 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 313, + 414, + 326 + ], + "score": 1.0, + "content": ". However, to be sure that the generated", + "type": "text" + }, + { + "bbox": [ + 414, + 314, + 423, + 324 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "is complete (covers", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "all the cases needed), we need to check all pairs of observations seen in adjacent step inputs (in par-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "score": 1.0, + "content": "ticular, those before and after a primitive function call), in a similar way as if we were constructing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 346, + 504, + 359 + ], + "spans": [ + { + "bbox": [ + 107, + 347, + 116, + 356 + ], + "score": 0.76, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 346, + 288, + 359 + ], + "score": 1.0, + "content": "from scratch. Given a precise definition of", + "type": "text" + }, + { + "bbox": [ + 288, + 347, + 297, + 356 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 346, + 495, + 359 + ], + "score": 1.0, + "content": ", it may be possible to automate the generation of", + "type": "text" + }, + { + "bbox": [ + 495, + 347, + 504, + 356 + ], + "score": 0.81, + "content": "V", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 358, + 200, + 368 + ], + "spans": [ + { + "bbox": [ + 107, + 358, + 128, + 368 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 358, + 137, + 367 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 358, + 200, + 368 + ], + "score": 1.0, + "content": "in future work.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 108, + 374, + 503, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 145, + 388 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 145, + 375, + 154, + 385 + ], + "score": 0.64, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 373, + 505, + 388 + ], + "score": 1.0, + "content": "should also contain the necessary reductions, which corresponds to making the recursive", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 279, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 266, + 398 + ], + "score": 1.0, + "content": "calls at the correct time, as indicated by", + "type": "text" + }, + { + "bbox": [ + 267, + 386, + 276, + 395 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 385, + 279, + 398 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 402, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 160, + 415 + ], + "score": 1.0, + "content": "After finding", + "type": "text" + }, + { + "bbox": [ + 160, + 403, + 169, + 412 + ], + "score": 0.68, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 402, + 421, + 415 + ], + "score": 1.0, + "content": ", we construct a set of problem inputs which, when executed on", + "type": "text" + }, + { + "bbox": [ + 421, + 403, + 430, + 412 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 402, + 504, + 415 + ], + "score": 1.0, + "content": ", create exactly the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 412, + 465, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 254, + 428 + ], + "score": 1.0, + "content": "step input sequences which make up", + "type": "text" + }, + { + "bbox": [ + 254, + 414, + 263, + 423 + ], + "score": 0.61, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 412, + 447, + 428 + ], + "score": 1.0, + "content": ". We call this set of inputs the verification set,", + "type": "text" + }, + { + "bbox": [ + 447, + 414, + 461, + 425 + ], + "score": 0.89, + "content": "S _ { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 412, + 465, + 428 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 108, + 430, + 503, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "Given a verification set, we can then run the model on the verification set to check if the produced", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "traces and results are correct. If yes, then this indicates that the learned neural program achieves", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 453, + 235, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 235, + 464 + ], + "score": 1.0, + "content": "provably perfect generalization.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 469, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "We note that for tasks with very large input domains, such as ones involving MNIST digits or speech", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "samples, the state space of base cases and reduction rules could be prohibitively large, possibly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "infinite. Consequently, it is infeasible to construct a verification set that covers all cases, and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 107, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "verification procedure we have described is inadequate. We leave this as future work to devise a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 513, + 326, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 326, + 527 + ], + "score": 1.0, + "content": "verification procedure more appropriate to this setting.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 200, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 201, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 201, + 558 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 504, + 593 + ], + "lines": [ + { + "bbox": [ + 107, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 107, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "As there is no public implementation of NPI, we implemented a version of it in Keras that is as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 582, + 472, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 472, + 595 + ], + "score": 1.0, + "content": "faithful to the paper as possible. Our experiments use a small number of training examples.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 504, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "Training Setup. The training set for addition contains 200 traces. The maximum problem length", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 619, + 434, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 377, + 632 + ], + "score": 1.0, + "content": "in this training set is 3 (e.g., the trace corresponding to the problem", + "type": "text" + }, + { + "bbox": [ + 377, + 620, + 428, + 631 + ], + "score": 0.85, + "content": "^ { \\mathrm { \\left. } } 1 0 9 + 1 0 1 ^ { \\mathrm { \\right. } }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 619, + 434, + 632 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 650 + ], + "score": 1.0, + "content": "The training set for bubble sort contains 100 traces, with maximum problem length of 2 (e.g., the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 648, + 263, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 263, + 660 + ], + "score": 1.0, + "content": "trace corresponding to the array [3,2]).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "The training set for topological sort contains 6 traces, with one synthesized from a graph of size 5", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 675, + 294, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 294, + 688 + ], + "score": 1.0, + "content": "and the rest synthesized from graphs of size 7.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 693, + 442, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 691, + 443, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 443, + 707 + ], + "score": 1.0, + "content": "The training set for quicksort contains 4 traces, synthesized from arrays of length 5.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "The same set of problems was used to generate the training traces for all formulations of the task,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 271, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 271, + 732 + ], + "score": 1.0, + "content": "for non-recursive and recursive versions.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 386, + 95 + ], + "score": 1.0, + "content": "To perform the verification as described here, it is critical to construct", + "type": "text" + }, + { + "bbox": [ + 386, + 83, + 396, + 92 + ], + "score": 0.68, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "correctly. 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If it is too large, then the semantics of", + "type": "text" + }, + { + "bbox": [ + 455, + 105, + 464, + 114 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "might not", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 256, + 128 + ], + "score": 1.0, + "content": "be well-defined on some elements in", + "type": "text" + }, + { + "bbox": [ + 256, + 116, + 265, + 125 + ], + "score": 0.76, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 115, + 506, + 128 + ], + "score": 1.0, + "content": ", or the spurious step input sequences may not be reachable", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 462, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 462, + 140 + ], + "score": 1.0, + "content": "from any valid problem input (e.g., an array for quicksort or a DAG for topological sort).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "To construct this set, by using the reference implementation of each subprogram, we construct a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "mapping between two sets of environment observations: the first set consists of all observations", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "that can occur at the beginning of a particular subprogram’s invocation, and the second set contains", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "score": 1.0, + "content": "the observations at the end of that subprogram. We can obtain this mapping by first considering", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "the possible observations that can arise at the beginning of the entry function (ADD, BUBBLE-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 506, + 212 + ], + "score": 1.0, + "content": "SORT, TOPOSORT, and QUICKSORT) for some valid program input, and iteratively applying", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "the observation-to-observation mapping implied by the reference implementation’s step output at", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "that point in the execution. If the step output specifies a primitive function call, we need to reason", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "about how it can affect the environment so as to change the observation in the next step input. For", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "non-primitive subprograms, we can update the observation-to-observation mapping currently asso-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "ciated with the subprogram and then apply that mapping to the current set. By iterating with this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 222, + 277 + ], + "score": 1.0, + "content": "procedure, and then running", + "type": "text" + }, + { + "bbox": [ + 222, + 264, + 231, + 274 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "on the input observation set that we obtain for the entry point func-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 185, + 288 + ], + "score": 1.0, + "content": "tion, we can obtain", + "type": "text" + }, + { + "bbox": [ + 186, + 275, + 195, + 285 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "precisely. To make an analogy to MDPs, this procedure is analogous to how", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 448, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 448, + 299 + ], + "score": 1.0, + "content": "value iteration obtains the correct value for each state starting from any initialization.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 143, + 506, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 504, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 238, + 315 + ], + "score": 1.0, + "content": "An alternative method is to run", + "type": "text" + }, + { + "bbox": [ + 238, + 303, + 247, + 313 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "on many different program inputs and then observe step input", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 241, + 326 + ], + "score": 1.0, + "content": "sequences which occur, to create", + "type": "text" + }, + { + "bbox": [ + 241, + 314, + 250, + 324 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 313, + 414, + 326 + ], + "score": 1.0, + "content": ". However, to be sure that the generated", + "type": "text" + }, + { + "bbox": [ + 414, + 314, + 423, + 324 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "is complete (covers", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "all the cases needed), we need to check all pairs of observations seen in adjacent step inputs (in par-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "score": 1.0, + "content": "ticular, those before and after a primitive function call), in a similar way as if we were constructing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 346, + 504, + 359 + ], + "spans": [ + { + "bbox": [ + 107, + 347, + 116, + 356 + ], + "score": 0.76, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 346, + 288, + 359 + ], + "score": 1.0, + "content": "from scratch. Given a precise definition of", + "type": "text" + }, + { + "bbox": [ + 288, + 347, + 297, + 356 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 346, + 495, + 359 + ], + "score": 1.0, + "content": ", it may be possible to automate the generation of", + "type": "text" + }, + { + "bbox": [ + 495, + 347, + 504, + 356 + ], + "score": 0.81, + "content": "V", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 358, + 200, + 368 + ], + "spans": [ + { + "bbox": [ + 107, + 358, + 128, + 368 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 358, + 137, + 367 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 358, + 200, + 368 + ], + "score": 1.0, + "content": "in future work.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 302, + 506, + 368 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 374, + 503, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 145, + 388 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 145, + 375, + 154, + 385 + ], + "score": 0.64, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 373, + 505, + 388 + ], + "score": 1.0, + "content": "should also contain the necessary reductions, which corresponds to making the recursive", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 279, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 266, + 398 + ], + "score": 1.0, + "content": "calls at the correct time, as indicated by", + "type": "text" + }, + { + "bbox": [ + 267, + 386, + 276, + 395 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 385, + 279, + 398 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 373, + 505, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 402, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 160, + 415 + ], + "score": 1.0, + "content": "After finding", + "type": "text" + }, + { + "bbox": [ + 160, + 403, + 169, + 412 + ], + "score": 0.68, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 402, + 421, + 415 + ], + "score": 1.0, + "content": ", we construct a set of problem inputs which, when executed on", + "type": "text" + }, + { + "bbox": [ + 421, + 403, + 430, + 412 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 402, + 504, + 415 + ], + "score": 1.0, + "content": ", create exactly the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 412, + 465, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 254, + 428 + ], + "score": 1.0, + "content": "step input sequences which make up", + "type": "text" + }, + { + "bbox": [ + 254, + 414, + 263, + 423 + ], + "score": 0.61, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 412, + 447, + 428 + ], + "score": 1.0, + "content": ". We call this set of inputs the verification set,", + "type": "text" + }, + { + "bbox": [ + 447, + 414, + 461, + 425 + ], + "score": 0.89, + "content": "S _ { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 412, + 465, + 428 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 402, + 504, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 430, + 503, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "Given a verification set, we can then run the model on the verification set to check if the produced", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "traces and results are correct. If yes, then this indicates that the learned neural program achieves", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 453, + 235, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 235, + 464 + ], + "score": 1.0, + "content": "provably perfect generalization.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 430, + 505, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 469, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "We note that for tasks with very large input domains, such as ones involving MNIST digits or speech", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "samples, the state space of base cases and reduction rules could be prohibitively large, possibly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "infinite. Consequently, it is infeasible to construct a verification set that covers all cases, and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 107, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "verification procedure we have described is inadequate. We leave this as future work to devise a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 513, + 326, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 326, + 527 + ], + "score": 1.0, + "content": "verification procedure more appropriate to this setting.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 469, + 506, + 527 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 200, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 201, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 201, + 558 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 504, + 593 + ], + "lines": [ + { + "bbox": [ + 107, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 107, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "As there is no public implementation of NPI, we implemented a version of it in Keras that is as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 582, + 472, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 472, + 595 + ], + "score": 1.0, + "content": "faithful to the paper as possible. Our experiments use a small number of training examples.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 106, + 571, + 505, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 504, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "Training Setup. The training set for addition contains 200 traces. The maximum problem length", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 619, + 434, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 377, + 632 + ], + "score": 1.0, + "content": "in this training set is 3 (e.g., the trace corresponding to the problem", + "type": "text" + }, + { + "bbox": [ + 377, + 620, + 428, + 631 + ], + "score": 0.85, + "content": "^ { \\mathrm { \\left. } } 1 0 9 + 1 0 1 ^ { \\mathrm { \\right. } }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 619, + 434, + 632 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 106, + 608, + 505, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 650 + ], + "score": 1.0, + "content": "The training set for bubble sort contains 100 traces, with maximum problem length of 2 (e.g., the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 648, + 263, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 263, + 660 + ], + "score": 1.0, + "content": "trace corresponding to the array [3,2]).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 635, + 505, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "The training set for topological sort contains 6 traces, with one synthesized from a graph of size 5", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 675, + 294, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 294, + 688 + ], + "score": 1.0, + "content": "and the rest synthesized from graphs of size 7.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 664, + 505, + 688 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 693, + 442, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 691, + 443, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 443, + 707 + ], + "score": 1.0, + "content": "The training set for quicksort contains 4 traces, synthesized from arrays of length 5.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46, + "bbox_fs": [ + 106, + 691, + 443, + 707 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "The same set of problems was used to generate the training traces for all formulations of the task,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 271, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 271, + 732 + ], + "score": 1.0, + "content": "for non-recursive and recursive versions.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 143, + 104, + 468, + 197 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 164, + 89, + 446, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 88, + 446, + 102 + ], + "spans": [ + { + "bbox": [ + 165, + 88, + 446, + 102 + ], + "score": 1.0, + "content": "Table 1: Accuracy on Randomly Generated Problems for Bubble Sort", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 143, + 104, + 468, + 197 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 104, + 468, + 197 + ], + "spans": [ + { + "bbox": [ + 143, + 104, + 468, + 197 + ], + "score": 0.976, + "html": "
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Both the non-recursive and recursive learned programs generalize on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "all input lengths we tried, up to 5000 digits. This agrees with the generalization of non-recursive", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 330, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 345 + ], + "score": 1.0, + "content": "addition in Reed & de Freitas (2016), where they reported generalization up to 3000 digits. However,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "score": 1.0, + "content": "note that there is no provable guarantee that the non-recursive learned program will generalize to all", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "inputs, whereas we show later that the recursive learned program has a provable guarantee of perfect", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 365, + 168, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 168, + 376 + ], + "score": 1.0, + "content": "generalization.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "In order to demonstrate that recursion can help learn and generalize better, for addition, we trained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "only on traces for 5 arbitrarily chosen 1-digit addition sum examples. The recursive version can gen-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 456, + 416 + ], + "score": 1.0, + "content": "eralize perfectly to long problems constructed from these components (such as the sum", + "type": "text" + }, + { + "bbox": [ + 456, + 403, + 501, + 414 + ], + "score": 0.85, + "content": "^ { 6 6 } 8 2 2 + 2 3 3 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 402, + 505, + 416 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 133, + 428 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 414, + 159, + 425 + ], + "score": 0.79, + "content": "\\because 8 + 2 ^ { , 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 412, + 178, + 428 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 414, + 203, + 425 + ], + "score": 0.79, + "content": "\\bar { 2 } + 3 \\bar { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 412, + 506, + 428 + ], + "score": 1.0, + "content": "are in the training set), but the non-recursive version fails to sum these long", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 425, + 184, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 184, + 439 + ], + "score": 1.0, + "content": "problems properly.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Bubble Sort. Table 1 presents results on randomly generated arrays of varying length for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "learned non-recursive, partially recursive, and full recursive programs. For each length, we test each", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "program on 30 randomly generated problems. Observe that partially recursive does slightly better", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "than non-recursive for the setting in which the length of the array is 3, and that the fully recursive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "version is able to sort every array given to it. The non-recursive and partially recursive versions are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 506, + 282, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 282, + 519 + ], + "score": 1.0, + "content": "unable to sort long arrays, beyond length 8.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "Topological Sort. Both the non-recursive and recursive learned programs generalize on all graphs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "we tried, up to 120 vertices. As before, the non-recursive learned program lacks a provable guarantee", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 554, + 446, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 446, + 568 + ], + "score": 1.0, + "content": "of generalization, whereas we show later that the recursive learned program has one.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "In order to demonstrate that recursion can help learn and generalize better, we trained a non-recursive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "and recursive model on just a single execution trace generated from a graph containing 5 nodes3 for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "the topological sort task. For these models, Table 2 presents results on randomly generated DAGs", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "of varying graph sizes (varying in the number of vertices). For each graph size, we test the learned", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "programs on 30 randomly generated DAGs. The recursive version of topological sort solves all graph", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "score": 1.0, + "content": "instances we tried, from graphs of size 5 through 70. On the other hand, the non-recursive version", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 484, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 484, + 650 + ], + "score": 1.0, + "content": "has low accuracy, beginning from size 5, and fails completely for graphs of size 8 and beyond.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "Quicksort. Table 3 presents results on randomly generated arrays of varying length for the learned", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 675, + 504, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 504, + 687 + ], + "score": 1.0, + "content": "non-recursive and recursive programs. For each length, we test each program on 30 randomly gen-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "erated problems. 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Both the non-recursive and recursive learned programs generalize on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "all input lengths we tried, up to 5000 digits. This agrees with the generalization of non-recursive", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 330, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 345 + ], + "score": 1.0, + "content": "addition in Reed & de Freitas (2016), where they reported generalization up to 3000 digits. However,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "score": 1.0, + "content": "note that there is no provable guarantee that the non-recursive learned program will generalize to all", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "inputs, whereas we show later that the recursive learned program has a provable guarantee of perfect", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 365, + 168, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 168, + 376 + ], + "score": 1.0, + "content": "generalization.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 309, + 506, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "In order to demonstrate that recursion can help learn and generalize better, for addition, we trained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "only on traces for 5 arbitrarily chosen 1-digit addition sum examples. The recursive version can gen-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 456, + 416 + ], + "score": 1.0, + "content": "eralize perfectly to long problems constructed from these components (such as the sum", + "type": "text" + }, + { + "bbox": [ + 456, + 403, + 501, + 414 + ], + "score": 0.85, + "content": "^ { 6 6 } 8 2 2 + 2 3 3 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 402, + 505, + 416 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 133, + 428 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 414, + 159, + 425 + ], + "score": 0.79, + "content": "\\because 8 + 2 ^ { , 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 412, + 178, + 428 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 414, + 203, + 425 + ], + "score": 0.79, + "content": "\\bar { 2 } + 3 \\bar { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 412, + 506, + 428 + ], + "score": 1.0, + "content": "are in the training set), but the non-recursive version fails to sum these long", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 425, + 184, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 184, + 439 + ], + "score": 1.0, + "content": "problems properly.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 381, + 506, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Bubble Sort. Table 1 presents results on randomly generated arrays of varying length for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "learned non-recursive, partially recursive, and full recursive programs. For each length, we test each", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "program on 30 randomly generated problems. Observe that partially recursive does slightly better", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "than non-recursive for the setting in which the length of the array is 3, and that the fully recursive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "version is able to sort every array given to it. The non-recursive and partially recursive versions are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 506, + 282, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 282, + 519 + ], + "score": 1.0, + "content": "unable to sort long arrays, beyond length 8.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 451, + 506, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "Topological Sort. Both the non-recursive and recursive learned programs generalize on all graphs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "we tried, up to 120 vertices. As before, the non-recursive learned program lacks a provable guarantee", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 554, + 446, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 446, + 568 + ], + "score": 1.0, + "content": "of generalization, whereas we show later that the recursive learned program has one.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 532, + 505, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "In order to demonstrate that recursion can help learn and generalize better, we trained a non-recursive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "and recursive model on just a single execution trace generated from a graph containing 5 nodes3 for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "the topological sort task. For these models, Table 2 presents results on randomly generated DAGs", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "of varying graph sizes (varying in the number of vertices). For each graph size, we test the learned", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "programs on 30 randomly generated DAGs. The recursive version of topological sort solves all graph", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "score": 1.0, + "content": "instances we tried, from graphs of size 5 through 70. 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For each length, we test each program on 30 randomly gen-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "erated problems. Observe that the non-recursive program’s correctness degrades for length 11 and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 697, + 354, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 354, + 710 + ], + "score": 1.0, + "content": "beyond, while the recursive program can sort any given array.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 664, + 505, + 710 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 193, + 104, + 418, + 183 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 156, + 89, + 455, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 155, + 88, + 456, + 103 + ], + "spans": [ + { + "bbox": [ + 155, + 88, + 456, + 103 + ], + "score": 1.0, + "content": "Table 2: Accuracy on Randomly Generated Problems for Topological Sort", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 193, + 104, + 418, + 183 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 193, + 104, + 418, + 183 + ], + "spans": [ + { + "bbox": [ + 193, + 104, + 418, + 183 + ], + "score": 0.971, + "html": "
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On the other hand, the recursive programs have learned the true program semantics.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 375, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 377, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 377, + 447 + ], + "score": 1.0, + "content": "4.2 VERIFICATION OF PROVABLY PERFECT GENERALIZATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "We describe how models trained with recursive traces can be proven to generalize, by using the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "verification procedure described in Section 3.3. As described in the verification procedure, it is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "possible to prove our learned recursive program generalizes perfectly by testing on an appropriate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "set of problem inputs, i.e., the verification set. Recall that this verification procedure cannot be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "performed for the non-recursive versions, since the propagation of the hidden state in the core LSTM", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "module makes reasoning difficult and so we would need to check an unbounded number of examples.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "We describe the base cases, reduction rules, and the verification set for each task in Appendix A.6.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "For each task, given the verification set, we check the traces and results of the learned, to-be-verified", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "neural program (described in Section 4.1; and for bubble sort, Appendix A.6) on the verification", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 377, + 573 + ], + "score": 1.0, + "content": "set, and ensure they match the traces produced by the true program", + "type": "text" + }, + { + "bbox": [ + 377, + 561, + 386, + 570 + ], + "score": 0.77, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 561, + 505, + 573 + ], + "score": 1.0, + "content": ". Our results show that for all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 495, + 584 + ], + "score": 1.0, + "content": "learned, to-be-verified neural programs, they all produced the same traces as those produced by", + "type": "text" + }, + { + "bbox": [ + 495, + 572, + 504, + 582 + ], + "score": 0.79, + "content": "P", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "on the verification set. Thus, we demonstrate that recursion enables provably perfect generalization", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 594, + 482, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 482, + 606 + ], + "score": 1.0, + "content": "for different tasks, including addition, topological sort, quicksort, and a variant of bubble sort.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 504, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Note that the training set can often be considerably smaller than the verification set, and despite", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "this, the learned model can still pass the entire verification set. Our result shows that the training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "procedure and the NPI architecture is capable of generalizing from the step input-output pairs seen", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 380, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 380, + 655 + ], + "score": 1.0, + "content": "in the training data to the unseen ones present in the verification set.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 108, + 672, + 195, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 197, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 197, + 688 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "We emphasize that the notion of a neural recursive program has not been presented in the literature", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "before: this is our main contribution. Recursion enables provably perfect generalization. To the best", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "score": 1.0, + "content": "of our knowledge, this is the first time verification has been applied to a neural program, provid-", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 193, + 104, + 418, + 183 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 156, + 89, + 455, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 155, + 88, + 456, + 103 + ], + "spans": [ + { + "bbox": [ + 155, + 88, + 456, + 103 + ], + "score": 1.0, + "content": "Table 2: Accuracy on Randomly Generated Problems for Topological Sort", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 193, + 104, + 418, + 183 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 193, + 104, + 418, + 183 + ], + "spans": [ + { + "bbox": [ + 193, + 104, + 418, + 183 + ], + "score": 0.971, + "html": "
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On the other hand, the recursive programs have learned the true program semantics.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 386, + 506, + 421 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 375, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 377, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 377, + 447 + ], + "score": 1.0, + "content": "4.2 VERIFICATION OF PROVABLY PERFECT GENERALIZATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "We describe how models trained with recursive traces can be proven to generalize, by using the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "verification procedure described in Section 3.3. As described in the verification procedure, it is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "possible to prove our learned recursive program generalizes perfectly by testing on an appropriate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "set of problem inputs, i.e., the verification set. Recall that this verification procedure cannot be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "performed for the non-recursive versions, since the propagation of the hidden state in the core LSTM", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "module makes reasoning difficult and so we would need to check an unbounded number of examples.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 455, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "We describe the base cases, reduction rules, and the verification set for each task in Appendix A.6.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "For each task, given the verification set, we check the traces and results of the learned, to-be-verified", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "neural program (described in Section 4.1; and for bubble sort, Appendix A.6) on the verification", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 377, + 573 + ], + "score": 1.0, + "content": "set, and ensure they match the traces produced by the true program", + "type": "text" + }, + { + "bbox": [ + 377, + 561, + 386, + 570 + ], + "score": 0.77, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 561, + 505, + 573 + ], + "score": 1.0, + "content": ". Our results show that for all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 495, + 584 + ], + "score": 1.0, + "content": "learned, to-be-verified neural programs, they all produced the same traces as those produced by", + "type": "text" + }, + { + "bbox": [ + 495, + 572, + 504, + 582 + ], + "score": 0.79, + "content": "P", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "on the verification set. Thus, we demonstrate that recursion enables provably perfect generalization", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 594, + 482, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 482, + 606 + ], + "score": 1.0, + "content": "for different tasks, including addition, topological sort, quicksort, and a variant of bubble sort.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 527, + 505, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 504, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Note that the training set can often be considerably smaller than the verification set, and despite", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "this, the learned model can still pass the entire verification set. Our result shows that the training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "procedure and the NPI architecture is capable of generalizing from the step input-output pairs seen", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 380, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 380, + 655 + ], + "score": 1.0, + "content": "in the training data to the unseen ones present in the verification set.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 611, + 506, + 655 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 672, + 195, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 197, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 197, + 688 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "We emphasize that the notion of a neural recursive program has not been presented in the literature", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "before: this is our main contribution. Recursion enables provably perfect generalization. To the best", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "score": 1.0, + "content": "of our knowledge, this is the first time verification has been applied to a neural program, provid-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "ing provable guarantees about its behavior. We instantiated recursion for the Neural Programmer-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "Interpreter by changing the training traces. 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Any", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "opinions, findings, and conclusions or recommendations expressed in this material are those of the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 202, + 487, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 487, + 213 + ], + "score": 1.0, + "content": "author(s) and do not necessarily reflect the views of National Science Foundation and DARPA.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 230, + 175, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 176, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 176, + 243 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "Marcin Andrychowicz and Karol Kurach. 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ProgramDescriptionsCallsArguments
TOPOSORTPerform topologicalsort on graphTRAVERSE,NEXT_START,WRITE,MOVENONE
TRAVERSETraverse graph untilstack is emptyCHECK_CHILD，EX-PLORENONE
CHECK_CHILDCheck ifawhitechild exists; if so, setchildList[Uactive] topoint to itMOVENONE
EXPLORERepeatedlytraversesubgraphs until stackis emptySTACK,CHECK_CHILD,WRITE,MOVENONE
STACKInteractwith stack,either pushing orpoppingWRITE,MOVEPUSH,POP
NEXT_STARTMove Pstart untilreaching a whitevertex. If a whitevertex is found，setPstart to point to it;this signifies the startof a traversal of anew connected com-ponent.If no whitevertex is found, theentireexecution isterminatedMOVENONE
WRITEWrite a value eitherto environment (e.g.,to color a vertex)or variable (e.g., tochange the value ofUactive)NONEDescribed below
MOVEMove a pointer(e.g, Pstart orchildList[vactive])up or downNONEDescribed below
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ProgramDescriptionsCallsArguments
TOPOSORTPerform topologicalsort on graphTRAVERSE,NEXT_START,WRITE,MOVENONE
TRAVERSETraverse graph untilstack is emptyCHECK_CHILD，EX-PLORENONE
CHECK_CHILDCheck ifawhitechild exists; if so, setchildList[Uactive] topoint to itMOVENONE
EXPLORERepeatedlytraversesubgraphs until stackis emptySTACK,CHECK_CHILD,WRITE,MOVENONE
STACKInteractwith stack,either pushing orpoppingWRITE,MOVEPUSH,POP
NEXT_STARTMove Pstart untilreaching a whitevertex. If a whitevertex is found，setPstart to point to it;this signifies the startof a traversal of anew connected com-ponent.If no whitevertex is found, theentireexecution isterminatedMOVENONE
WRITEWrite a value eitherto environment (e.g.,to color a vertex)or variable (e.g., tochange the value ofUactive)NONEDescribed below
MOVEMove a pointer(e.g, Pstart orchildList[vactive])up or downNONEDescribed below
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ProgramDescriptionsCalls Arguments
QUICKSORTRunthequicksortroutine in place forthearrayA，forindices from lo to hiNon-Recursive: PAR-TITION, STACK,WRITERecursive: same asnon-recursiveversion,alongwithQUICK-SORTImplicitly:arrayA to sort, lo, hi
PARTITIONRunsthepartitionfunction. At end,pointer Ppivot ismoved to the pivotCOMPSWAP_LOOP,MOVE_PIVOT_LO,MOVE_JLO, SWAPNONE
COMPSWAPLOOPRuns the FOR loopinside the partitionfunctionCOMPSWAP, MOVENONE
COMPSWAPComparesA[pivot] ≤ A[j]; ifso,perform a swapand increment PpivotSWAP, MOVENONE
SET_PIVOTLOSets Ppivot to lo in-dexNONENONE
SETJLOSets pj to lo indexNONENONE
SET_J_NULLSets pj to-00NONENONE
STACKPushes lo/hi statesonto stacks Sto andShi according to argument(describedbelow)WRITE, MOVEDescribed below
MOVEMovespointerone unit up or downNONEDescribed below
SWAPSwapselementsatgiven array indicesNONEDescribed below
WRITEWrite a valueeither to stack(e.g QstackLoorQstackHi) or topointer (e.g tochangethe value ofPhi)NONEDescribed below
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ProgramDescriptionsCalls Arguments
QUICKSORTRunthequicksortroutine in place forthearrayA，forindices from lo to hiNon-Recursive: PAR-TITION, STACK,WRITERecursive: same asnon-recursiveversion,alongwithQUICK-SORTImplicitly:arrayA to sort, lo, hi
PARTITIONRunsthepartitionfunction. At end,pointer Ppivot ismoved to the pivotCOMPSWAP_LOOP,MOVE_PIVOT_LO,MOVE_JLO, SWAPNONE
COMPSWAPLOOPRuns the FOR loopinside the partitionfunctionCOMPSWAP, MOVENONE
COMPSWAPComparesA[pivot] ≤ A[j]; ifso,perform a swapand increment PpivotSWAP, MOVENONE
SET_PIVOTLOSets Ppivot to lo in-dexNONENONE
SETJLOSets pj to lo indexNONENONE
SET_J_NULLSets pj to-00NONENONE
STACKPushes lo/hi statesonto stacks Sto andShi according to argument(describedbelow)WRITE, MOVEDescribed below
MOVEMovespointerone unit up or downNONEDescribed below
SWAPSwapselementsatgiven array indicesNONEDescribed below
WRITEWrite a valueeither to stack(e.g QstackLoorQstackHi) or topointer (e.g tochangethe value ofPhi)NONEDescribed below
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For tasks other than addition, it is diffi-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "cult to analytically determine the verification set, so instead, we randomly generate input candidates", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 384, + 357, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 357, + 395 + ], + "score": 1.0, + "content": "until they completely cover the base cases and reduction rules.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "Base Cases and Reduction Rules for Addition. For the recursive formulation of addition, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "analytically construct the set of input problems that cover all base cases and reduction rules. 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The ADD reduction rule ensures that each call to ADD only covers two adjacent columns,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "and so the LSTM only ever runs for a fixed number of steps necessary to process these two columns.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 108, + 484, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "We construct input problems by splitting into two cases: one case in which the left column contains", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "a null value and another in which the left column does not contain any null values. 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It is straightforward to deal with all other operations,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 556, + 298, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 298, + 568 + ], + "score": 1.0, + "content": "which do not induce partial environment states.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "Under the assumption that there are no leading 0’s (except in the case of single digits) and the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "two numbers to be added have the same number of digits, the verification set for addition contains", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "20,181 input problems. The assumption of leading 0’s can be easily removed, at the cost of slightly", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "increasing the size of the verification set. We made the assumption of equivalent lengths in order to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 617, + 492, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 492, + 630 + ], + "score": 1.0, + "content": "parametrize the input format with respect to length, but this assumption can be removed as well.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 504, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 504, + 651 + ], + "score": 1.0, + "content": "Base Cases and Reduction Rules for Bubble Sort. 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It is straightforward to deal with all other operations,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 556, + 298, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 298, + 568 + ], + "score": 1.0, + "content": "which do not induce partial environment states.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 534, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "Under the assumption that there are no leading 0’s (except in the case of single digits) and the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "two numbers to be added have the same number of digits, the verification set for addition contains", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "20,181 input problems. The assumption of leading 0’s can be easily removed, at the cost of slightly", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "increasing the size of the verification set. We made the assumption of equivalent lengths in order to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 617, + 492, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 492, + 630 + ], + "score": 1.0, + "content": "parametrize the input format with respect to length, but this assumption can be removed as well.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 573, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 504, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 504, + 651 + ], + "score": 1.0, + "content": "Base Cases and Reduction Rules for Bubble Sort. The original version of the bubblesort im-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "plementation exposes the values within the array. While this matches the description from Reed &", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 440, + 675 + ], + "score": 1.0, + "content": "de Freitas (2016), we found that this causes an unnecessary blowup in the size of", + "type": "text" + }, + { + "bbox": [ + 440, + 662, + 450, + 672 + ], + "score": 0.72, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 662, + 506, + 675 + ], + "score": 1.0, + "content": "and makes it", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "much more difficult to construct the verification set. For purposes of verification, we replace the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 683, + 285, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 285, + 698 + ], + "score": 1.0, + "content": "domain-specific encoder with the following:", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53, + "bbox_fs": [ + 105, + 641, + 506, + 698 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 702, + 495, + 731 + ], + "lines": [ + { + "bbox": [ + 116, + 702, + 495, + 731 + ], + "spans": [ + { + "bbox": [ + 116, + 702, + 495, + 731 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { f _ { e n c } ( Q , i _ { 1 } , i _ { 2 } , i _ { 3 } , a _ { t } ) = M L P ( [ Q ( 1 , i _ { 1 } ) \\leq Q ( 1 , i _ { 2 } ) , 1 \\leq i _ { 1 } \\leq l e n g t h , 1 \\leq i _ { 2 } \\leq l e n g t h , } \\\\ & { ~ i _ { 3 } = = l e n g t h , a _ { t } ( 1 ) , a _ { t } ( 2 ) , a _ { t } ( 3 ) ] ) , } \\end{array}", + "type": "interline_equation", + "image_path": "d6404bbac9ba6a2fa37db9e83467937b3e09f0d707243a22dbb09b327365ec60.jpg" + } + ] + } + ], + "index": 57, + "virtual_lines": [ + { + "bbox": [ + 116, + 702, + 495, + 711.6666666666666 + ], + "spans": [], + "index": 56 + }, + { + "bbox": [ + 116, + 711.6666666666666, + 495, + 721.3333333333333 + ], + "spans": [], + "index": 57 + }, + { + "bbox": [ + 116, + 721.3333333333333, + 495, + 730.9999999999999 + ], + "spans": [], + "index": 58 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 199, + 104, + 413, + 261 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 144, + 89, + 466, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 88, + 467, + 101 + ], + "spans": [ + { + "bbox": [ + 144, + 88, + 467, + 101 + ], + "score": 1.0, + "content": "Table 4: Accuracy on Randomly Generated Problems for Variant of Bubble Sort", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 199, + 104, + 413, + 261 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 199, + 104, + 413, + 261 + ], + "spans": [ + { + "bbox": [ + 199, + 104, + 413, + 261 + ], + "score": 0.97, + "html": "
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This modification also enables", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 297, + 343, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 343, + 310 + ], + "score": 1.0, + "content": "us to sort arrays containing arbitrary comparable elements.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "By reasoning about the possible set of environment observations created by all valid inputs, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 145, + 337 + ], + "score": 1.0, + "content": "construct", + "type": "text" + }, + { + "bbox": [ + 145, + 326, + 155, + 335 + ], + "score": 0.64, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "using the procedure described in Section 3.3. Using this modification, we constructed a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 336, + 305, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 305, + 348 + ], + "score": 1.0, + "content": "verification set consisting of one array of size 10.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "We also report on generalization results for the non-recursive and recursive versions of this variant of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "bubble sort. Table 4 demonstrates that the accuracy of the non-recursive program degrades sharply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 389 + ], + "score": 1.0, + "content": "when moving from arrays of length 7 to arrays of length 8. This is due to the properties of the training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 400 + ], + "score": 1.0, + "content": "set – we trained on 2 traces synthesized from arrays of length 7 and 1 trace synthesized from an array", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 396, + 494, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 494, + 410 + ], + "score": 1.0, + "content": "of length 6. Table 4 also demonstrates that the (verified) recursive program generalizes perfectly.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Base Cases and Reduction Rules for Topological Sort. For each function we use to implement", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "score": 1.0, + "content": "the recursive version of topological sort, we need to consider the set of possible environment obser-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "vation sequences we can create from all valid inputs and test that the learned program produces the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "correct behavior on each of these inputs. We have three observations: the color of the start node, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "color of the active node’s next child to be considered, and whether the stack is empty. Na¨ıvely, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "might expect to synthesize and test an input for any sequence created by combining the four possible", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "colors in two variables and another boolean variable for whether the stack is empty (so 32 possible", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "observations at any point), but for various reasons, most of these combinations are impossible to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 294, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 294, + 521 + ], + "score": 1.0, + "content": "occur at any given point in the execution trace.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "Through careful reasoning about the possible set of environment observations created by all valid", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "score": 1.0, + "content": "inputs, and how each of the operations in the execution trace affects the environment, we can con-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 131, + 559 + ], + "score": 1.0, + "content": "struct", + "type": "text" + }, + { + "bbox": [ + 131, + 547, + 140, + 557 + ], + "score": 0.72, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "using the procedure described in Section 3.3. We then construct a verification set of size 73", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 405, + 570 + ], + "score": 1.0, + "content": "by ensuring that randomly generated graphs cover the analytically derived", + "type": "text" + }, + { + "bbox": [ + 405, + 558, + 414, + 568 + ], + "score": 0.54, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 557, + 505, + 570 + ], + "score": 1.0, + "content": ". The model described", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "in the training setup of Section 4 (trained on 6 traces) was verified to be correct via the matching", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 580, + 250, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 250, + 591 + ], + "score": 1.0, + "content": "procedure described in Section 4.2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "score": 1.0, + "content": "Base Cases and Reduction Rules for Quicksort. As with the others, we apply the procedure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 256, + 627 + ], + "score": 1.0, + "content": "described in Section 3.3 to construct", + "type": "text" + }, + { + "bbox": [ + 257, + 615, + 266, + 624 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "and then empirically create a verification set which covers", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 107, + 626, + 115, + 635 + ], + "score": 0.7, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 624, + 506, + 638 + ], + "score": 1.0, + "content": ". 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Using this modification, we constructed a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 336, + 305, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 305, + 348 + ], + "score": 1.0, + "content": "verification set consisting of one array of size 10.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 313, + 506, + 348 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "We also report on generalization results for the non-recursive and recursive versions of this variant of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "bubble sort. Table 4 demonstrates that the accuracy of the non-recursive program degrades sharply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 389 + ], + "score": 1.0, + "content": "when moving from arrays of length 7 to arrays of length 8. This is due to the properties of the training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 400 + ], + "score": 1.0, + "content": "set – we trained on 2 traces synthesized from arrays of length 7 and 1 trace synthesized from an array", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 396, + 494, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 494, + 410 + ], + "score": 1.0, + "content": "of length 6. Table 4 also demonstrates that the (verified) recursive program generalizes perfectly.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 352, + 505, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Base Cases and Reduction Rules for Topological Sort. For each function we use to implement", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 504, + 443 + ], + "score": 1.0, + "content": "the recursive version of topological sort, we need to consider the set of possible environment obser-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "vation sequences we can create from all valid inputs and test that the learned program produces the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "correct behavior on each of these inputs. We have three observations: the color of the start node, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "color of the active node’s next child to be considered, and whether the stack is empty. Na¨ıvely, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "might expect to synthesize and test an input for any sequence created by combining the four possible", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "colors in two variables and another boolean variable for whether the stack is empty (so 32 possible", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "observations at any point), but for various reasons, most of these combinations are impossible to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 294, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 294, + 521 + ], + "score": 1.0, + "content": "occur at any given point in the execution trace.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 420, + 506, + 521 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "Through careful reasoning about the possible set of environment observations created by all valid", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "score": 1.0, + "content": "inputs, and how each of the operations in the execution trace affects the environment, we can con-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 131, + 559 + ], + "score": 1.0, + "content": "struct", + "type": "text" + }, + { + "bbox": [ + 131, + 547, + 140, + 557 + ], + "score": 0.72, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "using the procedure described in Section 3.3. We then construct a verification set of size 73", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 405, + 570 + ], + "score": 1.0, + "content": "by ensuring that randomly generated graphs cover the analytically derived", + "type": "text" + }, + { + "bbox": [ + 405, + 558, + 414, + 568 + ], + "score": 0.54, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 557, + 505, + 570 + ], + "score": 1.0, + "content": ". The model described", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "in the training setup of Section 4 (trained on 6 traces) was verified to be correct via the matching", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 580, + 250, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 250, + 591 + ], + "score": 1.0, + "content": "procedure described in Section 4.2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 524, + 505, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "score": 1.0, + "content": "Base Cases and Reduction Rules for Quicksort. As with the others, we apply the procedure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 256, + 627 + ], + "score": 1.0, + "content": "described in Section 3.3 to construct", + "type": "text" + }, + { + "bbox": [ + 257, + 615, + 266, + 624 + ], + "score": 0.69, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "and then empirically create a verification set which covers", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 107, + 626, + 115, + 635 + ], + "score": 0.7, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 624, + 506, + 638 + ], + "score": 1.0, + "content": ". The verification set can be very small, as we found a 10-element array ([8,2,1,2,0,8,5,8,3,7]) is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 207, + 649 + ], + "score": 1.0, + "content": "sufficient to cover all of", + "type": "text" + }, + { + "bbox": [ + 207, + 637, + 216, + 646 + ], + "score": 0.68, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 635, + 506, + 649 + ], + "score": 1.0, + "content": ". We note that an earlier version of quicksort we tried lacked primitive", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 659 + ], + "score": 1.0, + "content": "operations to directly move a pointer to another, and therefore needed more functions and observa-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 658, + 504, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 504, + 670 + ], + "score": 1.0, + "content": "tions. As this complexity interfered with determining the base cases and reductions, we changed the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "algorithm to its current form. 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1:Inputs: Environment observation e, program p,arguments a, stop threshold α
2:function RUN(e,p, a)
3:h↑0,r←0
4:whiler<αdo
5:s ←fenc(e,a),h ←fistm(s,p,h)
6:r ←fend(h),p2 ← fprog(h),a2 ← farg(h)
7:if p is a primitive function then
8:e ← fenv(e,p,a).
9:else
10:function RUN(e,P2, a2)
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1: Colorall verticeswhite.
2: Initialize an empty stack S and a directed acyclic graph DAG to traverse. 3: Begin traversing from Vertex1in the DAG.
4: function TOPOSORT(DAG)
5: while there is still a white vertex u: do
6: color[u] = grey
7: Vactive=u
8: do
9: if Uactive has a white child v then
10: color[v]= grey
11: push Uactive onto S
12: Vactive = U
13: else
14: color[Uactive] = black
15: Write Uactive to result
16: if S is empty then pass
17: else pop the top vertex off S and set it to Uactive
18: while S is not empty
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ProgramDescriptionsCallsArguments
TOPOSORTPerform topologicalsort on graphTRAVERSE,NEXT_START,WRITE,MOVENONE
TRAVERSETraverse graph untilstack is emptyCHECK_CHILD，EX-PLORENONE
CHECK_CHILDCheck ifawhitechild exists; if so, setchildList[Uactive] topoint to itMOVENONE
EXPLORERepeatedlytraversesubgraphs until stackis emptySTACK,CHECK_CHILD,WRITE,MOVENONE
STACKInteractwith stack,either pushing orpoppingWRITE,MOVEPUSH,POP
NEXT_STARTMove Pstart untilreaching a whitevertex. If a whitevertex is found，setPstart to point to it;this signifies the startof a traversal of anew connected com-ponent.If no whitevertex is found, theentireexecution isterminatedMOVENONE
WRITEWrite a value eitherto environment (e.g.,to color a vertex)or variable (e.g., tochange the value ofUactive)NONEDescribed below
MOVEMove a pointer(e.g, Pstart orchildList[vactive])up or downNONEDescribed below
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ProgramDescriptionsCalls Arguments
QUICKSORTRunthequicksortroutine in place forthearrayA，forindices from lo to hiNon-Recursive: PAR-TITION, STACK,WRITERecursive: same asnon-recursiveversion,alongwithQUICK-SORTImplicitly:arrayA to sort, lo, hi
PARTITIONRunsthepartitionfunction. At end,pointer Ppivot ismoved to the pivotCOMPSWAP_LOOP,MOVE_PIVOT_LO,MOVE_JLO, SWAPNONE
COMPSWAPLOOPRuns the FOR loopinside the partitionfunctionCOMPSWAP, MOVENONE
COMPSWAPComparesA[pivot] ≤ A[j]; ifso,perform a swapand increment PpivotSWAP, MOVENONE
SET_PIVOTLOSets Ppivot to lo in-dexNONENONE
SETJLOSets pj to lo indexNONENONE
SET_J_NULLSets pj to-00NONENONE
STACKPushes lo/hi statesonto stacks Sto andShi according to argument(describedbelow)WRITE, MOVEDescribed below
MOVEMovespointerone unit up or downNONEDescribed below
SWAPSwapselementsatgiven array indicesNONEDescribed below
WRITEWrite a valueeither to stack(e.g QstackLoorQstackHi) or topointer (e.g tochangethe value ofPhi)NONEDescribed below
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a/parse/train/BygZK2VYvB/BygZK2VYvB.md b/parse/train/BygZK2VYvB/BygZK2VYvB.md new file mode 100644 index 0000000000000000000000000000000000000000..c5476c2faed55cfcaf88ac870b7a2c35ea8c727c --- /dev/null +++ b/parse/train/BygZK2VYvB/BygZK2VYvB.md @@ -0,0 +1,316 @@ +# Utilizing Edge Features in Graph Neural Networks via Variational Information Maximization + +Anonymous authors Paper under double-blind review + +# Abstract + +Graph Neural Networks (GNNs) broadly follow the scheme that the representation vector of each node is updated recursively using the message from neighbor nodes, where the message of a neighbor is usually pre-processed with a parameterized transform matrix. To make better use of edge features, we propose the Edge Information maximized Graph Neural Network (EIGNN) that maximizes the Mutual Information (MI) between edge features and message passing channels. The MI is reformulated as a differentiable objective via a variational approach. We theoretically show that the newly introduced objective enables the model to preserve edge information, and empirically corroborate the enhanced performance of MI-maximized models across a broad range of learning tasks including regression on molecular graphs and relation prediction in knowledge graphs. + +# 1 Introduction + +Many real-world datasets naturally come in the form of graphs, such as citation networks (Kipf & Welling, 2017), social networks Hamilton et al. (2017), knowledge graphs (Schlichtkrull et al., 2018), molecular graphs (Scarselli et al., 2009; Duvenaud et al., 2015) etc., all of which consist of a number of nodes and edges equipped with their inherent features. Recently, impressive performance has been achieved in graph learning tasks with various forms of Graph Neural Networks (GNNs) (Zhou et al., 2018). Compared to prior works, such as node2vec (Grover & Leskovec, 2016), GNNs learn the state of a node by recursively aggregating messages from its neighbors: combining the graph structure with node features. Intuitively, edge features should play an important role in graph learning tasks. For example, chemical bonds in a molecule have a high impact on chemical properties of molecules, and edge features in knowledge graphs encode important relations between concepts, data, and entities. Our proposed method focuses on improving the usage of edge features in GNNs. + +The expressive power of GNNs largely depends on how the message is passed between nodes. A widely adopted scheme is multiplying neighbor node states with a parameterized transform matrix before aggregation (Gilmer et al., 2017; Xu et al., 2019). Despite tremendous success of GNNs, existing models do not exhaustively exploit the full potentials of edge features on graphs. For example, many GNNs such as GCN (Kipf & Welling, 2017), ChebyNet (Defferrard et al., 2016) and GAT (Veličković et al., 2018) do not even consider categorized edge types. To utilize edge features in multi-relational graphs, RGCN (Schlichtkrull et al., 2018) proposes to learn a different transform matrix for each edge type, respectively. However, it does not generalize to edge features in continuous space. MPNN (Gilmer et al., 2017) introduces an edge network that takes edge feature vectors as input and outputs transform matrices, which are used to transform states of neighbor nodes. In principle, the MPNN framework can handle complex edge features. Yet, the lack of maximization of MI between edge and message channels implies that the MPNN may give an edge-independent transform matrix. + +In this work, we aim to more efficiently exploit the full potentials of edge features from the perspective of training. We propose the Edge Information maximized Graph Neural Network (EIGNN) that maximizes the Mutual Information (MI) between edge features and the message passing channel which is parameterized as the transform matrix in the widely-accepted message passing framework (transformation and aggregation) (Gilmer et al., 2017; Xu et al., 2019). Considering the challenge of computing the MI, we adopt a variational approach to reformulate it as an differentiable objective, which can be easily applied as a regularization term. We theoretically show that EIGNN can reduce information loss of edge features. Apart from demonstrating the impressive performance of EIGNN on extensive benchmarks of molecular graphs and knowledge graphs, we also analyze and attribute the enhanced effectiveness of EIGNN to the exploitation of edge features instead of the regularization effects. Notably, attribution analysis on molecular graphs show that EIGNN can capture domain knowledge without human interference. + +Preliminaries Let $G = ( \nu , \mathcal { E } )$ be a graph with node feature vectors $x _ { v } \in \mathbb { R } ^ { d }$ for node $v \in \nu$ and edge feature vectors $e _ { v w } \in \mathcal { E }$ for the edge connecting node $v$ and $w$ . In GNNs, the state of each node is updated recursively using neighbor nodes. Let $\mathcal { N } _ { v }$ be the set of neighbor nodes of $v$ and $h _ { v } ^ { ( l ) } \in \mathbb { R } ^ { d _ { l } }$ be the hidden state of $v$ at $\it l$ -th layer, where $d _ { l }$ is the dimension of the hidden layer. For simplicity of notation, we use a single $d$ to denote the dimension such that $h _ { v } ^ { ( l ) } \in \mathbb { R } ^ { d }$ . We also have $h _ { v } ^ { ( 0 ) } = x _ { v }$ at the input layer. + +# 2 Related works + +# 2.1 Relational Modeling in Graph Neural Networks + +Single-relational modeling. Many variants such as GCN (Kipf & Welling, 2017), GAT (Veličković et al., 2018), ChebyNet (Defferrard et al., 2016), GraphSAGE (Hamilton et al., 2017) focus on learning node states. These models can assign weight to neighbors, but they can not handle various edge features. A typical neighborhood aggregation scheme is + +$$ +h _ { v } ^ { ( l + 1 ) } = \sigma \left( \sum _ { w \in \mathcal { N } _ { v } } \alpha _ { v w } W _ { 1 } ^ { ( l ) } h _ { w } ^ { ( l ) } + W _ { 0 } ^ { ( l ) } h _ { v } ^ { ( l ) } \right) , +$$ + +where $\sigma$ denotes an activation function, $\alpha _ { v w }$ can be a normalization constant or a learned attention coefficient (Veličković et al., 2018). States of all neighbors are multiplied by the same trainable transform matrix $W _ { 1 } ^ { ( l ) }$ . Sometimes the self-connection is also treated in the same way, s.t., W (l)0 = W (l)1 . + +Multi-relational modeling. A simple strategy to handle multi-relational graphs is assigning each edge type with a separate transform matrix as presented in RGCN (Schlichtkrull et al., 2018) and adopted by GGNN (Li et al., 2016) and LNet (Liao et al., 2019). RGCN updates node states according to the following scheme + +$$ +\begin{array} { r } { h _ { v } ^ { ( l + 1 ) } = \sigma \left( \sum _ { r \in \mathcal { R } } \sum _ { w \in \mathcal { N } _ { v } ^ { r } } \alpha _ { v w , r } W _ { r } ^ { ( l ) } h _ { w } ^ { ( l ) } + W _ { 0 } ^ { ( l ) } h _ { v } ^ { ( l ) } \right) , } \end{array} +$$ + +where $\mathcal { N } _ { v } ^ { r }$ is the collection of neighboring nodes of $\boldsymbol { v }$ with relation $r \in \mathcal { R }$ and $\alpha _ { v w , r }$ is a normalization constant similar as $\alpha _ { v w }$ in Eq. (1). Such a scheme faces challenge in handling edge features of continuous space. GGNN and LNet do not focus on the improvement of edge expressibility. GGNN introduces Gated Recurrent Unit (GRU) (Cho et al., 2014) and LNet focus on handling multi-scale connections. + +Complex-relational modeling. The relation in a graph can be quite complex, expressed as a general feature vector $e$ . MPNN (Gilmer et al., 2017) introduces an edge network which takes edge feature vectors as input and outputs transform matrices. A single edge network is shared in a MPNN model. The forward propagation is formalized as + +$$ +m _ { v } ^ { ( l + 1 ) } = \sigma \left( \sum _ { w \in \mathcal { N } _ { v } } f ( e _ { v w } ) h _ { w } ^ { ( l ) } + W _ { 0 } ^ { ( l ) } h _ { v } ^ { ( l ) } \right) , \quad h _ { v } ^ { ( l + 1 ) } = \mathrm { G R U } ( h _ { v } ^ { ( l ) } , m _ { v } ^ { ( l + 1 ) } ) , +$$ + +where $f : e W$ denotes the edge network. Recently, some research works treat a multirelational problem as the complex-relational one by introducing a continuous edge embedding vector for each edge type, so as to handle increasing number of relations (Nathani et al., 2019). Although the MPNN architecture allows the usage of arbitrary edge features, this advantage is not utilized in practice. MPNN can actually learn an edge-independent transform matrix. A GNN model that efficiently utilizes edge features is yet to emerge. + +# 2.2 Readout functions + +After several forward propagations, GNNs yield final states of all nodes, which are suitable for node/edge classification or regression. For graph classification or regression tasks, we can apply a readout function (Ying et al., 2018; Vinyals et al., 2015) such that + +$$ +y = R ( \{ h _ { v } ^ { L } | v \in G \} ) , +$$ + +where $h _ { v } ^ { L }$ is the state of $v$ at the last layer, $R$ is the readout function that outputs a graph-level representation $y$ , e.g., summing up the final node states, applying hierarchical pooling (Ying et al., 2018) or using the set2set model (Vinyals et al., 2015). + +# 3 Our method + +# 3.1 The Usage of Mutual Information + +In probability theory and information theory, MI is a measure of mutual dependence between two random variables. Our method proposes to preserve edge information in GNNs, which is important in many real-world graph structures such as molecules - apart from node (atom) features, attributes of edges (bonds) are equally important for predicting properties of molecules. To this end, we maximize $I ( e ; W )$ - the MI between the edge feature vector $e$ and the message passing channel, i.e., the transform matrix $W$ which is used to transform neighbor node states in the forward propagation. Our method can be easily generalized to directly maximize the MI between edge features and the message itself in methods that do not explicitly have the transform matrix, e.g., the message from node $w$ to node $v$ can be expressed as $f ( h _ { v } , e _ { v w } , h _ { w } )$ rather than $f ( e _ { v w } ) h _ { w }$ , which is shown in Section 4.3. + +An general principle of maximum MI is described for unsupervised learning task by (Linsker, 1988) and MI inspired objective functions have long been adopted in unsupervised learning (Bridle et al., 1992; Barber & Agakov, 2006; Veličković et al., 2019; Hjelm et al., 2019), semi-supervised learning (Krause et al., 2010) and generative adversarial networks (Chen et al., 2016). Specifically, DGI (Veličković et al., 2019) also applies MI to GNNs. DGI proposes to learn node-wise representations in an unsupervised manner by maximizing the MI between node representations and corresponding high-level summaries of graphs, using adversarial learning and negative sampling. The node representations may then be retrieved and used for downstream tasks, such as node classification. DGI can be used to pre-train GNNs, as demonstrated in Hu et al. (2019). Our EIGNN also proposes information maximization but targets a completely different objective and adopts a quite different approach. + +# 3.2 A Variational Approach to Maximize Mutual Information + +Computing $I ( e ; W )$ itself is intractable in practice, needless to say that training a model requires the derivative. Thus, we adopt a variational approach (Agakov, 2004) to reformulate $I ( e ; W )$ as a differentiable objective. We show that our objective is an approximated lower bound of $I ( e ; W )$ and notably, optimizing our objective does lead to maximizing $I ( e ; W )$ . Following MPNN (Gilmer et al., 2017), we use an edge network to parameterize the transform matrix $W$ and relate it to edge features. Therefore, the prior $p ( W | e )$ is + +$$ +p ( W | e ) = \delta ( W - f ( e ) ) , +$$ + +where $\delta ( \cdot )$ is the Dirac delta function. The posterior $p ( e | W )$ is intractable, so we define a variational distribution $q ( e | W )$ , which can be obtained by defining a neural network $g : W \to e$ Specifically, $q ( e | W )$ substitutes to some distribution (such as Gaussian distribution) with parameter $g ( W )$ . In this way, $f$ and $g$ are similar to the probabilistic encoder and decoder in the Variational Auto-Encoder (VAE) (Kingma $\&$ Welling, 2013). Then we can approximate $I ( e ; W )$ with a differentiable objective $L _ { I } ( f , g ; e )$ as follows. + +Theorem 1. Let e be the edge feature vector, $W$ be the transform matrix with conditional distribution $p ( W | e )$ specified by the probabilistic encoder $f$ as shown in Eq. (5) and $q ( e | W )$ be the variational distribution specified by the probabilistic decoder $g$ , then we have + +$$ +I ( e ; W ) \geq H ( e ) + \mathbb { E } _ { e \sim p ( e ) } [ \mathcal { L } _ { I } ( f , g ; e ) ] , +$$ + +where $\mathcal { L } _ { I } ( f , g ; e ) = \log q ( e | f ( e ) )$ and $H ( \cdot )$ denotes the entropy. + +Proof. Let $D _ { K L } ( \cdot \parallel \cdot )$ denote the KL-divergence, which should be nonnegative, then we have + +$$ +\begin{array} { r l } { I ( e ; W ) = H ( e ) - H ( e | W ) } & { } \\ { \ } & { = H ( e ) + \mathbb { E } _ { W \sim p ( W ) } [ \mathbb { E } _ { e \sim p ( e | W ) } [ \log p ( e | W ) ] ] } \\ { \ } & { = H ( e ) + \mathbb { E } _ { W \sim p ( W ) } [ \mathbb { E } _ { e \sim p ( e | W ) } [ \log p ( e | W ) - \log q ( e | W ) + \log q ( e | W ) ] ] } \\ { \ } & { = H ( e ) + \mathbb { E } _ { W \sim p ( W ) } [ D _ { K L } ( p ( e | W ) \| q ( e | W ) ) + \mathbb { E } _ { e \sim p ( e | W ) } [ \log q ( e | W ) ] ] } \\ { \ } & { \geq H ( e ) + \mathbb { E } _ { W \sim p ( W ) } [ \mathbb { E } _ { e \sim p ( e | W ) } [ \log q ( e | W ) ] ] } \\ { \ } & { = H ( e ) + \mathbb { E } _ { e \sim p ( e ) , W \sim p ( W | e ) } [ \log q ( e | W ) ] } \\ { \ } & { \stackrel { ( a ) } { = } H ( e ) + \mathbb { E } _ { e \sim p ( e ) } [ \log q ( e | f ( e ) ) ] } \end{array} +$$ + +where the equality ( $a$ ) follows from Eq. (5). + +According to Theorem 1, we can maximize the variational lower bound for $I ( e ; W )$ . The bound becomes tight when the variational distribution $q ( e | W )$ approaches the true posterior $p ( e | W )$ . Moreover, $H ( e )$ is a constant because the distribution of edge feature vector $e$ is fixed for given graphs, hence we can equivalently maximize $\mathcal { L } _ { I } ( f , g ; e )$ . We choose the widely accepted Gaussian distribution as the prior distribution for the probabilistic decoder $g$ , + +$$ +q ( e | W ) = \mathcal { N } ( e ; g ( W ) , \sigma ^ { 2 } I ) . +$$ + +Then we have + +$$ +\mathcal { L } _ { I } ( f , g ; e ) = \log q ( e | f ( e ) ) = \log \mathcal { N } ( e ; g ( f ( e ) ) , \sigma ^ { 2 } I ) = - \lambda \| e - g ( f ( e ) ) \| _ { 2 } ^ { 2 } +$$ + +where $\lambda > 0$ is a constant determined by $\sigma$ and the dimension of $e$ , taken as a tunable parameter. The following Theorem 2 shows that maximizing the objective in Eq. (8) does lead to the maximization of $I ( e ; W )$ , hence enables the model to preserve edge information. + +Theorem 2. Assume the optimal solution of maximizing $\mathcal { L } _ { I } ( f , g ; e )$ is $f ^ { \star }$ and $g ^ { \star }$ , then $f ^ { \star }$ also maximizes $I ( e ; W )$ . + +Proof. Note that $H ( e )$ is a constant when the graphs are given. In information theory, we have + +$$ +H ( g ( f ( e ) ) ) \leq H ( f ( e ) ) \leq H ( e ) . +$$ + +$I ( e ; W )$ is upper bounded by $H ( e )$ , + +$$ +I ( e ; W ) = I ( e ; f ( e ) ) = H ( f ( e ) ) - H ( f ( e ) | e ) = H ( f ( e ) ) \leq H ( e ) . +$$ + +Since $f ^ { \star }$ and $g ^ { \star }$ is the optimal solution of maximizing $\mathcal { L } _ { I } ( f , g ; e )$ presented in Eq. (8), it is not difficult to see that $e = g ^ { \star } ( f ^ { \star } ( e ) ) , \forall e \in \mathcal { E }$ . In this case, the inequalities in Eq. 9 become equalities, i.e., + +$$ +H ( g ^ { \star } ( f ^ { \star } ( e ) ) ) = H ( f ^ { \star } ( e ) ) = H ( e ) . +$$ + +Therefore, we have $I ( e ; W ) = H ( e )$ , i.e., the maximum is attained in this case. + +3.3 Edge Information Maximized Graph Neural Networks + +Our EIGNN is derived by implementing our MI objective in GNNs. As a concrete example, the forward propagation of our model follows the formulation in Eq. (3), where the dege network $f : e W$ is expressed as a multi-layer perceptron (MLP). According to theoretical analysis presented in Sec. 3.2 , we introduce another MLP $g : W \to e$ as the decoder. + +For graph regression or classification tasks, the model outputs a prediction $y$ for each graph $G$ , which has label $\hat { y }$ . Without MI maximization, we denote the vanilla loss as $\mathcal { L } _ { 0 } ( \hat { y } , y ; G )$ . Common choice of $\mathcal { L } _ { 0 }$ includes Mean Square Error (MSE), Mean Absolute Error (MAE) and Cross Entropy (CE). For a graph $G = ( \nu , \mathcal { E } )$ , EIGNN maximizes $\mathcal { L } _ { I } ( f , g ; e )$ and minimizes $\mathcal { L } _ { 0 } ( \hat { y } , y ; G )$ using the following loss function + +$$ +\mathcal { L } _ { E I G N N } ( G ) = \mathcal { L } _ { 0 } ( \hat { y } , y ; G ) - \lambda \mathbb { E } _ { e \in \mathcal { E } } [ \mathcal { L } _ { I } ( f , g ; e ) ] , +$$ + +where $\mathbb { E } _ { e \in \mathcal { E } } [ \cdot ]$ denotes taking the mean over all edges in $G = ( \nu , \mathcal { E } )$ and $\lambda$ is the regularization parameter. When EIGNN is trained using mini-batches, $\mathcal { L } _ { 0 } ( \hat { y } , y ; G )$ is averaged over all graphs in the batch while $\mathcal { L } _ { I } ( f , g ; e )$ is averaged over all edges of all graphs in the batch. + +Similarly, for relational prediction tasks in knowledge graphs, EIGNN directly yields nodelevel representations $h _ { v }$ for each node $v \in \mathcal V$ and edge-level representations $e$ for each relationship. The objective function of EIGNN can be derived from the translational scoring function Bordes et al. (2013), which learns embedding such that for a given valid triple $t _ { v w } = ( h _ { v } , e _ { v w } , h _ { w } )$ from the valid set $S$ , the condition $d _ { t _ { v w } } = h _ { v } + e _ { v w } - h _ { w } \approx 0$ holds. Let $\mathcal { L } _ { 0 } = \mathbb { E } _ { t _ { v w } \in S } \mathbb { E } _ { t _ { v w } ^ { \prime } \in S ^ { \prime } } \operatorname* { m a x } \{ d _ { t _ { v w } ^ { \prime } } - d _ { t _ { v w } } + \gamma , 0 \}$ , where $S ^ { \prime }$ denotes a set of invalid triples and $\gamma$ is a margin hyper-parameter, EIGNN can be trained by minimizing the following loss, + +$$ +\mathcal { L } _ { E I G N N } ( G ) = \mathcal { L } _ { 0 } - \lambda \mathbb { E } _ { e \in \mathcal { E } } [ \mathcal { L } _ { I } ( f , g ; e ) ] . +$$ + +# 4 Experiments + +In this section, we first conduct experiments on a large quantum chemistry benchmark QM9, which is challenging for most baselines. Then we evaluate EIGNN on several useful molecule benchmarks and use attribution analysis to show that EIGNN increases the impact of edges and captures domain knowledge without human interference. Finally, we adopt our method to large-scale knowledge graphs and evaluate the performance on challenging relation prediction tasks using a wide variety of real-world datasets. All experimental results demonstrate a clear and substantial improvement of EIGNN over the state-of-the-art methods. + +# 4.1 Quantum Chemistry + +QM9 (Ramakrishnan et al., 2014) is a large benchmark containing 134k molecules with 12 quantum chemistry regression properties, which have been show to be quite challenging for many GNNs (Gilmer et al., 2017). Feature engineering of nodes and edges exactly follows (Gilmer et al., 2017) such that molecules are preprocessed as graphs according to atom features and bond features. We compare our EIGNN with nine state-of-the-art baselines which can be categorized into three groups according to the ability of handling edge features: i) GCN, ChebyNet, GAT and GIN (Xu et al., 2019) which simply use binary edge features to indicate the existence of a bond without any other edge features; ii) RGCN, GGNN, LNet and simplified MPNN (sMPNN) which consider bond types (no bond, single, double, triple, or aromatic); iii) MPNN and our EIGNN which use edge feature vectors to indicate both edge types and pairwise distance between atoms. + +For a fair comparison, we repeat all experiments 3 times with different random seeds while during each run, all methods share the same random seed. We randomly choose 10k molecules for validation, 10k molecules for testing, and keep the rest for training. Each target property is normalized to zero mean and unit variance for training. Each model is trained to predict the 12 target properties simultaneously. $\lambda$ is naively set to 1 for EIGNN. We use mean square error (MSE) loss to train the models for at most 300 epochs till convergence, and the performance is measured by mean absolute error (MAE). For LNet and GGNN, implementation of the readout function follows the original paper. While for all other models, we use the same set2set (Vinyals et al., 2015) readout, which has been demonstrated to work well in (Gilmer et al., 2017). + +Table 1: Quantum property regressions for 12 targets and overall performance (top two raws) on QM9. We repeat all experiments 3 times with different random seeds and report the average performance. Full results with standard deviation are presented in Appendix A, e.g., for MPNN and EIGNN, we have Avg. nMAE $0 . 0 3 9 8 \pm 0 . 0 0 0 2$ and 0.0357 ± 0.0005. + +
MethodGCNChebyNetGATGINRGCNGGNNLNetsMPNNMPNNEIGNN
Avg.nMAE0.1350.1210.1370.1000.1020.0990.0990.0890.0400.036
Avg.MAE5.3064.3035.4703.4803.8173.6613.6533.1610.6930.633
mu0.5680.5180.5670.4780.5060.5180.4720.4720.1100.097
alpha0.8810.7930.8910.6210.6320.6080.6230.5280.3320.294
HOMO(10-3)5.4514.7755.4294.1834.4534.4833.8893.8542.4812.230
LUMO(10-3)6.4005.6746.3314.7965.1385.1534.1944.5492.8622.593
gap(10-3)8.2017.0978.1936.0966.5006.6025.8135.6343.6203.275
R253.5641.9554.5234.6540.1039.6835.2733.496.0645.646
ZPVE(10-3)2.5332.5272.2711.7441.4771.2921.4381.3450.6790.612
UO2.0421.9842.2901.4221.0590.6971.8060.7910.4160.357
U2.0421.9842.2901.4221.0590.6971.7550.7910.4160.357
H2.0421.9842.2901.4221.0590.6971.7960.7910.4160.357
G2.0421.9842.2901.4221.0590.6961.7780.7910.4160.357
Cv0.4730.4200.4790.3090.3170.3150.3120.2620.1340.121
+ +![](images/b51754269df1a1a81766b31ec23b13e16833a1c14fe21641771ca4dfac60a05d.jpg) +Figure 1: Ablation study. Training and validation error on QM9. The shadow area indicates $m e a n \pm s t d$ over 3 runs. + +In Table 1, we list regression results for all methods. We report individual MAE for each target in their original scale, averaged MAE (Avg. MAE) over 12 properties, and averaged normalized MAE (Avg. nMAE; averaged over normalized target properties since different targets have different units and ranges). Our EIGNN achieves the best performance for each metric and each target. Now we are ready to answer the following research questions. i) Are edge features important? Yes. The error has a trend of decreasing with increasing edge features. The comparison between sMPNN (using edge types) and MPNN (using edge types and distance) directly verifies the importance of edge features. It is also consistent with the expert knowledge that distances between pairwise atoms are closely related to quantum properties. For example, the smaller the distance between the two atoms, the stronger the bond is, and consequently a higher bond energy is associated with this atom pair. ii) Does the EIGNN work? Yes. EIGNN achieves the best performance on each target, outperforming the strong baseline MPNN. Moreover, the advantage of EIGNN over MPNN is consistent over 3 runs and the standard deviation on this task is quite small. Detailed results are shown in Table 4 of Appendix A. iii) How does the EIGNN work? Our MI objective is easily implemented on top of vanilla loss function. We have shown that our objective enables preserving of edge information. Fig. 1 demonstrates that regularization such as $L _ { 2 }$ weight decay can increase training error while our objective does not. Moreover, the validation performance verifies that regularization itself does not reduce the validation error. Thus, the effectiveness of EIGNN is due to exploiting edge features rather than the regularization effect. We further run an ablation study where we concatenate edge features to node representations (i.e., MPNN+concat in Fig. 1) in message passing. Concatenation is unable to identify correlations between edges and nodes (Gilmer et al., 2017) and our results show that it slightly reduces the mean validation error but increases the variance. + +# 4.2 More Molecule Benchmarks with Potential Applications + +We further evaluate EIGNN on three molecule benchmarks: Lipophilicity (Wu et al., 2018), ESOL (Delaney, 2004) and FreeSolv (Mobley & Guthrie, 2014). These datasets contain fewer molecules, and have potential usages in applications such as chemistry, drug discovery, and materials science. For example, the property lipophilicity is an important feature of drug molecules that affects both membrane permeability and solubility. The dataset Lipophilicity contains 4200 compounds. ESOL provides water solubility data for 1128 compounds. FreeSolv contains hydration free energy of 642 small molecules in water. We conduct graph regression experiments on these benchmarks. All datasets are split into training, validation and test according to a proportion of 0.8/0.1/0.1. MPNN and our EIGNN share the same architecture with 3 layers of message passing and 3 steps of set2set. We repeat each experiment 3 times with different random seeds. Results of testing root mean square error (RMSE) in Table 2 verify the effectiveness of our method. Our EIGNN outperforms MPNN on each dataset and each run. Detailed results for each run are presented in Appendix B. + +Table 2: Testing RMSE on Lipophilicity, ESOL and FreeSolv. + +
DatasetLipophilicityESOLFreeSolv
MethodMPNNEIGNNMPNNEIGNNMPNNEIGNN
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
+ +![](images/93a30757aa01764b32a397063bc6a0e3a215232f566f99e62524e437f83f0e93.jpg) +Figure 2: Attribution analysis. The color indicates the impact of an edge/atom on the output, i.e., the regression result. EIGNN i) increases the edge attribution, ii) reduces the prediction error and iii) can learn domain knowledge without human interference. + +Attribution analysis. To understand how our EIGNN reduces the regression error, we conduct attribution analysis, i.e., attributing the prediction of a deep network to its input features, which usually builds up on the standard gradient operator (Sundararajan et al., features 2017). For an output $e$ as $\begin{array} { r } { S _ { e } = | \frac { \partial y } { \partial e } | } \end{array}$ $y$ , i.e., the prediction of a GNN, we define its sensitivity to an edge with , where $\left. \cdot \right.$ denotes the $L _ { 1 }$ norm. Similarly, the sensitivity to an atom with features $x$ is $\begin{array} { r } { S _ { x } = | \frac { \partial y } { \partial x } | } \end{array}$ . Then $S _ { e }$ and $S _ { x }$ are used as the metrics of attribution in our experiments. As an example, we show in Fig. 2 the attribution for two molecules in Lipophilicity: (a) $N c I n o n c I C ( = N O ) N c \mathcal { Q } c c c ( F ) c ( C l ) c \mathcal { Q }$ and (b) Oc1c2ncc3ccccc3c2nn1c4ccccc4 . Molecule in (a) has the potential to be used to treat, prevent and/or diagnose cancer Prinz et al. (2019). Compared with MPNN, we can observe an increasing of overall edge attribution under our EIGNN and a decreasing of prediction error in both cases. Interestingly, the attribution under EIGNN is similar to the expert knowledge of chemists: halogen atoms such as {Cl, Br, I} and their bond with the carbon atom greatly effect the lipophilicity of a molecule Wilcken et al. (2013), while atoms {O, N} also have high impact on the lipophilicity but usually in a negative way (Augustijns & Brewster, 2007). In Fig. 2 (a), the attribution of the halogen bond C-Cl and the pair {O, N} under our EIGNN is much higher than the one under MPNN, which is consistent with the expert knowledge. In Fig. 2 (b), similarly, the attribution of atoms {O, N} and the bond C-O under EIGNN is much higher. More examples are presented in Appendix C. + +# 4.3 Predicting Relations in Knowledge Graphs + +In this subsection, we adopt EIGNN to tackle the problem of relation prediction in knowledge graphs (KGs), which entails predicting whether a given triple is valid or not. For example, a triple (London, capital of, United Kingdom) should be classified as valid or London should be predicted as the capital of United Kingdom. KGs represent human knowledge as a directed graph, and have been widely used in practical applications, such as semantic search, dialogue generation, question answering etc. Recovering missing relations in KGs have been a major task for practical usages of KGs. We evaluate our methods on three benchmark datasets, WN18RR (Dettmers et al., 2018), FB15k-237 (Toutanova et al., 2015) and NELL-995 (Xiong et al., 2017). Without the reversible relation problem (Dettmers et al., 2018), WN18RR includes 11 relations scraped from WordNet for 40, 943 synsets. FB15k-237 is a subset of Freebase, and contains 14, 541 entities associated with 237 types of edge. NELL-995 is constructed from the $9 9 5 ^ { t h }$ iteration of NELL system, containing 75, 492 entities and 200 types of edge. + +Table 3: Experimental results on WN18RR, FB15K-237 and NELL-995 test sets. Hits@N values are in percentage. The best score is in bold and second best score is underlined. + +
DatasetWN18RRFB15K-237NELL-995
MRRHit@N%MRRHit@N %
MRR@1@3@10@1@3@10@1@3@10
DistMult0.44441.24750.40.28119.930.144.60.48540.152.461
ComplEx0.44940.946.9530.27819.429.7450.48239.952.860.6
ConvE0.45641.94753.10.31222.534.149.70.49140.353.161.3
TransE0.24342.744.153.20.27919.837.644.10.40134.447.250.1
ConvKB0.26558.244.555.80.28919.832.447.10.4337.04754.5
R-GCN0.123813.720.70.1641018.1300.128.212.618.8
KBGAT0.43635.848.157.80.43136.145.856.90.51442.955.367.8
EIGNN0.43835.748.858.10.45137.448.260.50.52343.856.168.3
+ +A critical issue of applying Eq. (3) to KGs is that high-dimensional embedding vectors are required to distinguish massive amount of entities and relations, leading to a rapid growth in number of parameters in EIGNN. To address this issue, we adopt the architecture of (Nathani et al., 2019) and learn graph attention based embeddings that target relation prediction on KGs as follows, + +$$ +\begin{array} { r } { m _ { v w } = f \big ( h _ { v } ^ { ( l ) } , e _ { v w } ^ { l } , h _ { w } ^ { ( l ) } \big ) , \alpha _ { v w } ^ { l } = \mathrm { s o f t m a x } \big ( a ^ { l } m _ { v w } ^ { l } \big ) , h _ { v } ^ { ( l + 1 ) } = \sigma \left( \sum _ { w \in \mathcal { N } _ { v } } \alpha _ { v w } ^ { l } m _ { v w } ^ { l } \right) . } \end{array} +$$ + +Compared with Eq. (3), where $m _ { v w } = f ( e _ { v w } ) h _ { w }$ , the above equation absorbs the transform matrix $f ( e _ { v w } )$ into $f ( h _ { v } ^ { ( l ) } , e _ { v w } , h _ { w } ^ { ( l ) } )$ and reduces the model parameters. In the following experiments, we implement $f$ using a MLP as in previous experiments, and maximizes the MI between $e _ { v w }$ and $m _ { v w }$ by introducing another MLP with $\lambda = 0 . 0 1$ . Multi-head attention is further introduced to stabilize the learning process and encapsulate more information about neighbors according to (Veličković et al., 2018). After training EIGNN, ConvKB (Nguyen et al., 2018) is adopted as a regression function for a given triple by analyzing the global embedding properties across each dimension. + +In the relation prediction task, the aim is to predict a triple $( v , e _ { v w } , w )$ with $\boldsymbol { v }$ or $w$ missing. We can generate a set of candidate triples for each missing entity $v$ by randomly replacing it with an arbitrary one. Scores can be calculated by ConvKB for all triples, and we find the rank of a correct triple by sorting all scores in ascending order. Thus, the performance of relation prediction task can be evaluated by mean reciprocal rank (MRR) and the proportion of correct entities in the top $N$ ranks (Hits@N) for $N = 1 , 3$ , and 10 (Bordes et al., 2013). We compare our EIGNN with seven state-of-the-art baselines focusing on this task: DistMult (Yang et al., 2014), ComplEx (Trouillon et al., 2016), ConvE (Dettmers et al., 2018), TransE (Bordes et al., 2013), ConvKB (Nguyen et al., 2018), RGCN (Schlichtkrull et al., 2018) and KBGAT (Nathani et al., 2019). As shown in Table 3, our EIGNN achieves the best performance for each metric on FB15K-237 and NELL-995, and achieves the best performance on WN18RR with Hit $^ \mathrm { ( a 3 }$ and 10 metrics. The results of KBGAT are reproduced following the official implementation1, and the results of other methods can be found in the previous peer-reviewed publications, i.e. (Nathani et al., 2019). + +# 5 Conclusions + +In this work, to make better use of edge features in GNNs, we proposed the edge information maximized graph neural network (EIGNN) that maximizes the mutual information between edge feature vectors and message passing channels. We reformulated the mutual information as a differentiable objective by adopting a variational approach. We have theoretically proved that our proposed objective enables EIGNN to preserve edge information and empirically evaluated EIGNN’s performance on a variety of benchmarks incorporating an array of challenging molecular datasets and knowledge graphs. These results clearly manifested a substantial improvement of EIGNN over the prior state-of-the-art methods. 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Embedding entities and relations for learning and inference in knowledge bases. ICLR, 2014. +Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In NeurIPS, pp. 4800–4810, 2018. +Jie Zhou, Ganqu Cui, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, and Maosong Sun. Graph neural networks: A review of methods and applications. arXiv preprint arXiv:1812.08434, 2018. + +# A Full Results on QM9 + +Table 4: Full results of quantum property regressions for 12 targets and overall performance (nMAE and MAE in top two raws) on QM9. We repeat all experiments 3 times with different random seeds and report the average performance and standard deviation. This is a supplement for Table 1. in the main text. + +
MethodGCNChebyNetGATGIN
Avg.nMAE0.1350±0.00460.1206±0.00840.1367±0.00500.1001±0.0007
Avg.MAE5.3063±0.19644.3032±0.48145.4698±0.20403.4799±0.0402
mu0.5679±0.00780.5180±0.01310.5670±0.01020.4783±0.0041
alpha0.8811±0.03080.7932±0.07780.8913±0.03140.6209±0.0028
HOMO(10-3)5.4510±0.07904.7750±0.20405.4290±0.16604.1830±0.0370
LUMO(10-3)6.4000±0.12305.6740±0.26706.3310±0.23904.7960±0.0520
gap(10-3)8.2010±0.21507.0970±0.36208.1930±0.29106.0960±0.0420
R253.563±1.031941.950±4.828954.519±1.599234.647±0.2167
ZPVE(10-3)2.5330±0.10702.5270±0.35602.2710±0.14501.7440±0.0100
UO2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
U2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
H2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
G2.0423±0.32811.9842±0.20362.2899±0.20001.4215±0.0857
Cv0.4730±0.02090.4199±0.04990.4787±0.01640.3093±0.0035
MethodRGCNGGNNLNetsMPNN
Avg.nMAE0.1021±0.00160.0992±0.00130.0992±0.00610.0888±0.0014
Avg.MAE3.8175±0.06053.6608±0.07233.6527±0.34173.1610±0.0697
mu0.5056±0.00480.5179±0.00760.4717±0.00630.4718±0.0096
alpha0.6321±0.01450.6077±0.00920.6225±0.05080.5278±0.0106
HOMO(10-3)4.4530±0.12904.4830±0.06503.8889±0.16173.8540±0.0410
LUMO(10-3)5.1380±0.12105.1530±0.08904.1935±0.22054.5490±0.0810
gap(10-3)6.5000±0.13906.6020±0.14005.8132±0.64565.6340±0.0570
R240.102±0.742839.685±0.821235.275±3.053133.489±0.6562
ZPVE(10-3)1.4770±0.00901.2920±0.03401.4376±0.07691.3450±0.0260
UO1.0589±0.02310.6969±0.04131.8058±0.25330.7914±0.0446
U1.0589±0.02310.6966±0.04181.7555±0.21960.7914±0.0446
H1.0589±0.02310.6974±0.04081.7964±0.24280.7914±0.0446
G1.0589±0.02310.6961±0.04211.7780±0.24580.7914±0.0446
Cv0.3170±0.01520.3146±0.01250.3124±0.03030.2625±0.0040
MethodMPNNEIGNN
Avg.nMAE0.0398±0.00020.0357±0.0005
Avg.MAE0.6929±0.02120.6331±0.0298
mu0.1095±0.00140.0974±0.0026
alpha0.3318±0.00260.2939±0.0054
HOMO(10-3)2.4810±0.02002.2300±0.0310
LUMO(10-3)2.8620±0.03702.5930±0.0440
gap(10-3)3.6200±0.01803.2750±0.0520
R26.0637±0.25115.6464±0.3098
ZPVE(10-3)0.6790±0.01400.6120±0.0170
UO0.4164±0.02250.3574±0.0100
U0.4164±0.02250.3575±0.0100
H0.4164±0.02250.3574±0.0100
G0.4164±0.02250.3575±0.0101
Cv0.1339±0.00130.1208±0.0027
+ +We present full results of quantum property regressions on QM9 with average performance and standard deviation in Table 4. The $( 1 0 ^ { - 3 }$ ) in the parentheses indicates that the values in the corresponding raw of the table should multiply by $1 0 ^ { - 3 }$ . This is simply for clear presentation of the values. We also present a detailed descriptions on the target properties in Table 5 for your reference. + +In our results, we directly report MAE instead of Error Ratio [(MAE)/(Chemical Accuracy) (Gilmer et al., 2017)], because it is common to report MAE in terms of chemical unit. This practice has been widely adopted not only in computational chemistry but also in the machine learning community working on molecular graphs (Wu et al., 2018; Morris et al., 2019). Still, we add Table 6 which contains the Error Ratio for MPNN and our EIGNN. In this comparison, we follow (Gilmer et al., 2017) and train models separately to predict each target. Our EIGNN consistently outperforms MPNN. + +Table 5: Regression targets on QM9. + +
Target propertyDescriptionUnit
muDipole momentD
alphaIsotropic polarizabilitya
HOMOHighest occupied molecular orbital energyEh
LUMOLowest unoccupied molecular orbital energyEh
gapGap between HOMO and LUMOEh
R2Electronic spatial extent品
ZPVEZero point vibrational energyEh
UOInternal energy at OKEh
UInternal energy at 298.15KEh
HEnthalpy at 298.15KEh
GFree energy at 298.15KEh
CvHeat capavity at 298.15Kcal molK
+ +Table 6: Error Ratio [(MAE)/(Chemical Accuracy) (Gilmer et al., 2017)] on QM9. Note that the energy values of $\{ \mathrm { U 0 , ~ U , ~ H , ~ G } \}$ are per molecule rather than per atom. Following (Gilmer et al., 2017), models are separately trained on each target. + +
TargetMPNNEIGNNImprovement (%)
mu0.870.808.24
alpha2.642.447.57
HOMO1.541.3910.2
LUMO1.311.282.13
gap2.201.9710.4
R21.000.7228.4
ZPVE5.314.4316.5
UO64.335.844.4
U56.623.658.3
H77.435.254.5
G46.634.426.1
Cv1.691.79-6.22
+ +B Full Results on Lipophilicity, ESOL and FreeSolv + +In this section, we present experimental results on Lipophilicity, ESOL and FreeSolv with detailed results for each run. The results in Table 7 verify that our EIGNN consistently outperforms MPNN. + +# C More Examples of Attribution + +In this section, we present more examples of attribution analysis on Lipophilicity. As a supplement for Fig. 2 in the main text, the observation here is similar. Compared with MPNN, we can observe an increasing of overall edge attribution under our EIGNN and a decreasing of prediction error in both cases. Notably, our EIGNN is able to capture the expert knowledge. (a) The molecule is $C N [ C @ \mathbb { Q } H ] ( C ) C ( = O ) N [ C @ \mathbb { Q } H ] ( C 1 C C C C T 1 ) C ( =$ $O ) N [ C @ H ] 2 C C C N ( C C c ( F ) c c 3 ) C 2$ . The attribution of $\{ \mathrm { O } , \mathrm { N } \}$ and the halogen atom $\mathrm { F }$ is higher under EIGNN. (b) The molecule is $C C ( C ) N 1 C C N [ C @ H ] ( C 1 ) C ( =$ $O ) N 2 C C N ( C C 2 ) C ( = O ) N c 3 c c c ( C l ) c ( C l ) c 3$ . Our EIGNN successfully captures the importance of two critical halogen atoms Cl and several atoms N. (c) The molecule is $C O c 1 c c ( c c 1 ) C ( = \ O ) N 2 C C C C 2 \ = \ O$ . The attribution of two atoms $\mathrm { \ o }$ at the top is + +Table 7: Testing RMSE on Lipophilicity, ESOL and FreeSolv. This is a supplement for Table 2 in the main text. + +
DatasetLipophilicityESOLFreeSolv
SeedMPNNEIGNNMPNNEIGNNMPNNEIGNN
00.7180.6760.7700.7181.3961.109
10.6960.6640.7500.7331.2991.265
20.6200.6190.8940.8761.4991.443
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
+ +much higher under EIGNN. (d) The molecule is $C c 1 c c 2 N C ( = O ) C ( = C C ( = O ) c 2 c c 1 C ) O$ . +The attribution of atoms O is much higher under EIGNN. + +![](images/d35b4c5026ccc9e9cd8864632869ef2ae4dd487409a61f9850c2293c8640530b.jpg) +Figure 3: More examples of attribution analysis. The color indicates the impact of an edge/atom on the output, i.e., the regression result. EIGNN i) increases the edge attribution, ii) reduces the prediction error and iii) can learn domain knowledge without human interference. \ No newline at end of file diff --git a/parse/train/BygZK2VYvB/BygZK2VYvB_content_list.json b/parse/train/BygZK2VYvB/BygZK2VYvB_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..46d818e55649853c618d755a88c3d17780f7df42 --- /dev/null +++ b/parse/train/BygZK2VYvB/BygZK2VYvB_content_list.json @@ -0,0 +1,1581 @@ +[ + { + "type": "text", + "text": "Utilizing Edge Features in Graph Neural Networks via Variational Information Maximization ", + "text_level": 1, + "bbox": [ + 176, + 98, + 826, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 184, + 195, + 419, + 223 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 452, + 260, + 545, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph Neural Networks (GNNs) broadly follow the scheme that the representation vector of each node is updated recursively using the message from neighbor nodes, where the message of a neighbor is usually pre-processed with a parameterized transform matrix. To make better use of edge features, we propose the Edge Information maximized Graph Neural Network (EIGNN) that maximizes the Mutual Information (MI) between edge features and message passing channels. The MI is reformulated as a differentiable objective via a variational approach. We theoretically show that the newly introduced objective enables the model to preserve edge information, and empirically corroborate the enhanced performance of MI-maximized models across a broad range of learning tasks including regression on molecular graphs and relation prediction in knowledge graphs. ", + "bbox": [ + 233, + 291, + 764, + 459 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 176, + 488, + 339, + 503 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many real-world datasets naturally come in the form of graphs, such as citation networks (Kipf & Welling, 2017), social networks Hamilton et al. (2017), knowledge graphs (Schlichtkrull et al., 2018), molecular graphs (Scarselli et al., 2009; Duvenaud et al., 2015) etc., all of which consist of a number of nodes and edges equipped with their inherent features. Recently, impressive performance has been achieved in graph learning tasks with various forms of Graph Neural Networks (GNNs) (Zhou et al., 2018). Compared to prior works, such as node2vec (Grover & Leskovec, 2016), GNNs learn the state of a node by recursively aggregating messages from its neighbors: combining the graph structure with node features. Intuitively, edge features should play an important role in graph learning tasks. For example, chemical bonds in a molecule have a high impact on chemical properties of molecules, and edge features in knowledge graphs encode important relations between concepts, data, and entities. Our proposed method focuses on improving the usage of edge features in GNNs. ", + "bbox": [ + 174, + 520, + 825, + 686 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The expressive power of GNNs largely depends on how the message is passed between nodes. A widely adopted scheme is multiplying neighbor node states with a parameterized transform matrix before aggregation (Gilmer et al., 2017; Xu et al., 2019). Despite tremendous success of GNNs, existing models do not exhaustively exploit the full potentials of edge features on graphs. For example, many GNNs such as GCN (Kipf & Welling, 2017), ChebyNet (Defferrard et al., 2016) and GAT (Veličković et al., 2018) do not even consider categorized edge types. To utilize edge features in multi-relational graphs, RGCN (Schlichtkrull et al., 2018) proposes to learn a different transform matrix for each edge type, respectively. However, it does not generalize to edge features in continuous space. MPNN (Gilmer et al., 2017) introduces an edge network that takes edge feature vectors as input and outputs transform matrices, which are used to transform states of neighbor nodes. In principle, the MPNN framework can handle complex edge features. Yet, the lack of maximization of MI between edge and message channels implies that the MPNN may give an edge-independent transform matrix. ", + "bbox": [ + 174, + 694, + 825, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we aim to more efficiently exploit the full potentials of edge features from the perspective of training. We propose the Edge Information maximized Graph Neural Network (EIGNN) that maximizes the Mutual Information (MI) between edge features and the message passing channel which is parameterized as the transform matrix in the widely-accepted message passing framework (transformation and aggregation) (Gilmer et al., 2017; Xu et al., 2019). Considering the challenge of computing the MI, we adopt a variational approach to reformulate it as an differentiable objective, which can be easily applied as a regularization term. We theoretically show that EIGNN can reduce information loss of edge features. Apart from demonstrating the impressive performance of EIGNN on extensive benchmarks of molecular graphs and knowledge graphs, we also analyze and attribute the enhanced effectiveness of EIGNN to the exploitation of edge features instead of the regularization effects. Notably, attribution analysis on molecular graphs show that EIGNN can capture domain knowledge without human interference. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 243 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Preliminaries Let $G = ( \\nu , \\mathcal { E } )$ be a graph with node feature vectors $x _ { v } \\in \\mathbb { R } ^ { d }$ for node $v \\in \\nu$ and edge feature vectors $e _ { v w } \\in \\mathcal { E }$ for the edge connecting node $v$ and $w$ . In GNNs, the state of each node is updated recursively using neighbor nodes. Let $\\mathcal { N } _ { v }$ be the set of neighbor nodes of $v$ and $h _ { v } ^ { ( l ) } \\in \\mathbb { R } ^ { d _ { l } }$ be the hidden state of $v$ at $\\it l$ -th layer, where $d _ { l }$ is the dimension of the hidden layer. For simplicity of notation, we use a single $d$ to denote the dimension such that $h _ { v } ^ { ( l ) } \\in \\mathbb { R } ^ { d }$ . We also have $h _ { v } ^ { ( 0 ) } = x _ { v }$ at the input layer. ", + "bbox": [ + 173, + 256, + 825, + 347 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related works ", + "text_level": 1, + "bbox": [ + 176, + 364, + 357, + 381 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 Relational Modeling in Graph Neural Networks ", + "text_level": 1, + "bbox": [ + 174, + 393, + 630, + 410 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Single-relational modeling. Many variants such as GCN (Kipf & Welling, 2017), GAT (Veličković et al., 2018), ChebyNet (Defferrard et al., 2016), GraphSAGE (Hamilton et al., 2017) focus on learning node states. These models can assign weight to neighbors, but they can not handle various edge features. A typical neighborhood aggregation scheme is ", + "bbox": [ + 173, + 420, + 825, + 477 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/897a3f0394352afdd981afc2c5beb5e43696de6f029404d486b83b2d43b34699.jpg", + "text": "$$\nh _ { v } ^ { ( l + 1 ) } = \\sigma \\left( \\sum _ { w \\in \\mathcal { N } _ { v } } \\alpha _ { v w } W _ { 1 } ^ { ( l ) } h _ { w } ^ { ( l ) } + W _ { 0 } ^ { ( l ) } h _ { v } ^ { ( l ) } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 334, + 478, + 663, + 503 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\sigma$ denotes an activation function, $\\alpha _ { v w }$ can be a normalization constant or a learned attention coefficient (Veličković et al., 2018). States of all neighbors are multiplied by the same trainable transform matrix $W _ { 1 } ^ { ( l ) }$ . Sometimes the self-connection is also treated in the same way, s.t., W (l)0 = W (l)1 . ", + "bbox": [ + 174, + 505, + 825, + 569 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Multi-relational modeling. A simple strategy to handle multi-relational graphs is assigning each edge type with a separate transform matrix as presented in RGCN (Schlichtkrull et al., 2018) and adopted by GGNN (Li et al., 2016) and LNet (Liao et al., 2019). RGCN updates node states according to the following scheme ", + "bbox": [ + 174, + 582, + 825, + 637 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/2ed04d513db0882ec07a019e0ecc97ba5d85793352a6b0896db29b31bf4e31a2.jpg", + "text": "$$\n\\begin{array} { r } { h _ { v } ^ { ( l + 1 ) } = \\sigma \\left( \\sum _ { r \\in \\mathcal { R } } \\sum _ { w \\in \\mathcal { N } _ { v } ^ { r } } \\alpha _ { v w , r } W _ { r } ^ { ( l ) } h _ { w } ^ { ( l ) } + W _ { 0 } ^ { ( l ) } h _ { v } ^ { ( l ) } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 638, + 697, + 672 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathcal { N } _ { v } ^ { r }$ is the collection of neighboring nodes of $\\boldsymbol { v }$ with relation $r \\in \\mathcal { R }$ and $\\alpha _ { v w , r }$ is a normalization constant similar as $\\alpha _ { v w }$ in Eq. (1). Such a scheme faces challenge in handling edge features of continuous space. GGNN and LNet do not focus on the improvement of edge expressibility. GGNN introduces Gated Recurrent Unit (GRU) (Cho et al., 2014) and LNet focus on handling multi-scale connections. ", + "bbox": [ + 174, + 672, + 825, + 743 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Complex-relational modeling. The relation in a graph can be quite complex, expressed as a general feature vector $e$ . MPNN (Gilmer et al., 2017) introduces an edge network which takes edge feature vectors as input and outputs transform matrices. A single edge network is shared in a MPNN model. The forward propagation is formalized as ", + "bbox": [ + 174, + 756, + 823, + 811 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/fa5600ecca6e4db258cd8b03cebc33d5e5982081f6001312ac655c155dc4299d.jpg", + "text": "$$\nm _ { v } ^ { ( l + 1 ) } = \\sigma \\left( \\sum _ { w \\in \\mathcal { N } _ { v } } f ( e _ { v w } ) h _ { w } ^ { ( l ) } + W _ { 0 } ^ { ( l ) } h _ { v } ^ { ( l ) } \\right) , \\quad h _ { v } ^ { ( l + 1 ) } = \\mathrm { G R U } ( h _ { v } ^ { ( l ) } , m _ { v } ^ { ( l + 1 ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 232, + 813, + 764, + 839 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $f : e W$ denotes the edge network. Recently, some research works treat a multirelational problem as the complex-relational one by introducing a continuous edge embedding vector for each edge type, so as to handle increasing number of relations (Nathani et al., 2019). Although the MPNN architecture allows the usage of arbitrary edge features, this advantage is not utilized in practice. MPNN can actually learn an edge-independent transform matrix. A GNN model that efficiently utilizes edge features is yet to emerge. ", + "bbox": [ + 173, + 840, + 826, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 Readout functions ", + "text_level": 1, + "bbox": [ + 174, + 103, + 377, + 117 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "After several forward propagations, GNNs yield final states of all nodes, which are suitable for node/edge classification or regression. For graph classification or regression tasks, we can apply a readout function (Ying et al., 2018; Vinyals et al., 2015) such that ", + "bbox": [ + 174, + 128, + 825, + 171 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6b3aeb614b48bbadc542b9ff62623957cbe52cd0d89c9bf49da41a0802158ccd.jpg", + "text": "$$\ny = R ( \\{ h _ { v } ^ { L } | v \\in G \\} ) ,\n$$", + "text_format": "latex", + "bbox": [ + 428, + 175, + 570, + 193 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $h _ { v } ^ { L }$ is the state of $v$ at the last layer, $R$ is the readout function that outputs a graph-level representation $y$ , e.g., summing up the final node states, applying hierarchical pooling (Ying et al., 2018) or using the set2set model (Vinyals et al., 2015). ", + "bbox": [ + 174, + 195, + 825, + 238 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Our method ", + "text_level": 1, + "bbox": [ + 176, + 256, + 328, + 272 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 The Usage of Mutual Information ", + "text_level": 1, + "bbox": [ + 173, + 285, + 508, + 300 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In probability theory and information theory, MI is a measure of mutual dependence between two random variables. Our method proposes to preserve edge information in GNNs, which is important in many real-world graph structures such as molecules - apart from node (atom) features, attributes of edges (bonds) are equally important for predicting properties of molecules. To this end, we maximize $I ( e ; W )$ - the MI between the edge feature vector $e$ and the message passing channel, i.e., the transform matrix $W$ which is used to transform neighbor node states in the forward propagation. Our method can be easily generalized to directly maximize the MI between edge features and the message itself in methods that do not explicitly have the transform matrix, e.g., the message from node $w$ to node $v$ can be expressed as $f ( h _ { v } , e _ { v w } , h _ { w } )$ rather than $f ( e _ { v w } ) h _ { w }$ , which is shown in Section 4.3. ", + "bbox": [ + 173, + 311, + 825, + 452 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "An general principle of maximum MI is described for unsupervised learning task by (Linsker, 1988) and MI inspired objective functions have long been adopted in unsupervised learning (Bridle et al., 1992; Barber & Agakov, 2006; Veličković et al., 2019; Hjelm et al., 2019), semi-supervised learning (Krause et al., 2010) and generative adversarial networks (Chen et al., 2016). Specifically, DGI (Veličković et al., 2019) also applies MI to GNNs. DGI proposes to learn node-wise representations in an unsupervised manner by maximizing the MI between node representations and corresponding high-level summaries of graphs, using adversarial learning and negative sampling. The node representations may then be retrieved and used for downstream tasks, such as node classification. DGI can be used to pre-train GNNs, as demonstrated in Hu et al. (2019). Our EIGNN also proposes information maximization but targets a completely different objective and adopts a quite different approach. ", + "bbox": [ + 173, + 457, + 825, + 611 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 A Variational Approach to Maximize Mutual Information ", + "text_level": 1, + "bbox": [ + 174, + 626, + 704, + 641 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Computing $I ( e ; W )$ itself is intractable in practice, needless to say that training a model requires the derivative. Thus, we adopt a variational approach (Agakov, 2004) to reformulate $I ( e ; W )$ as a differentiable objective. We show that our objective is an approximated lower bound of $I ( e ; W )$ and notably, optimizing our objective does lead to maximizing $I ( e ; W )$ . Following MPNN (Gilmer et al., 2017), we use an edge network to parameterize the transform matrix $W$ and relate it to edge features. Therefore, the prior $p ( W | e )$ is ", + "bbox": [ + 174, + 651, + 825, + 737 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/064575c5b1ecce9a7f7dbda9406409bc548a24e03885ab642abea6e3861b477f.jpg", + "text": "$$\np ( W | e ) = \\delta ( W - f ( e ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 739, + 581, + 756 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\delta ( \\cdot )$ is the Dirac delta function. The posterior $p ( e | W )$ is intractable, so we define a variational distribution $q ( e | W )$ , which can be obtained by defining a neural network $g : W \\to e$ Specifically, $q ( e | W )$ substitutes to some distribution (such as Gaussian distribution) with parameter $g ( W )$ . In this way, $f$ and $g$ are similar to the probabilistic encoder and decoder in the Variational Auto-Encoder (VAE) (Kingma $\\&$ Welling, 2013). Then we can approximate $I ( e ; W )$ with a differentiable objective $L _ { I } ( f , g ; e )$ as follows. ", + "bbox": [ + 174, + 757, + 825, + 843 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 1. Let e be the edge feature vector, $W$ be the transform matrix with conditional distribution $p ( W | e )$ specified by the probabilistic encoder $f$ as shown in Eq. (5) and $q ( e | W )$ be the variational distribution specified by the probabilistic decoder $g$ , then we have ", + "bbox": [ + 174, + 844, + 825, + 886 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1ddb6427dd9ce6d7604222cb7abb1323181e83e653a98be12246cba2bb6046d8.jpg", + "text": "$$\nI ( e ; W ) \\geq H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) } [ \\mathcal { L } _ { I } ( f , g ; e ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 890, + 632, + 906 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { L } _ { I } ( f , g ; e ) = \\log q ( e | f ( e ) )$ and $H ( \\cdot )$ denotes the entropy. ", + "bbox": [ + 173, + 909, + 617, + 925 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proof. Let $D _ { K L } ( \\cdot \\parallel \\cdot )$ denote the KL-divergence, which should be nonnegative, then we have ", + "bbox": [ + 173, + 102, + 823, + 119 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d1f9d2080ad50a36557e6e9a92ee313414557012e82f952539a8c31f6ce4b4bc.jpg", + "text": "$$\n\\begin{array} { r l } { I ( e ; W ) = H ( e ) - H ( e | W ) } & { } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log p ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log p ( e | W ) - \\log q ( e | W ) + \\log q ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ D _ { K L } ( p ( e | W ) \\| q ( e | W ) ) + \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log q ( e | W ) ] ] } \\\\ { \\ } & { \\geq H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log q ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) , W \\sim p ( W | e ) } [ \\log q ( e | W ) ] } \\\\ { \\ } & { \\stackrel { ( a ) } { = } H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) } [ \\log q ( e | f ( e ) ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 222, + 123, + 774, + 262 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the equality ( $a$ ) follows from Eq. (5). ", + "bbox": [ + 173, + 265, + 485, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "According to Theorem 1, we can maximize the variational lower bound for $I ( e ; W )$ . The bound becomes tight when the variational distribution $q ( e | W )$ approaches the true posterior $p ( e | W )$ . Moreover, $H ( e )$ is a constant because the distribution of edge feature vector $e$ is fixed for given graphs, hence we can equivalently maximize $\\mathcal { L } _ { I } ( f , g ; e )$ . We choose the widely accepted Gaussian distribution as the prior distribution for the probabilistic decoder $g$ , ", + "bbox": [ + 173, + 295, + 825, + 366 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f8a4483a3180430a353a9084a23a720e3b8ae44919cce3c219973627c201c39d.jpg", + "text": "$$\nq ( e | W ) = \\mathcal { N } ( e ; g ( W ) , \\sigma ^ { 2 } I ) .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 369, + 596, + 387 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then we have ", + "bbox": [ + 174, + 391, + 274, + 405 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f9aff49d6e2e211ee2f081b682d6476c98870ba18a9bff3393ceb13b8bcdb085.jpg", + "text": "$$\n\\mathcal { L } _ { I } ( f , g ; e ) = \\log q ( e | f ( e ) ) = \\log \\mathcal { N } ( e ; g ( f ( e ) ) , \\sigma ^ { 2 } I ) = - \\lambda \\| e - g ( f ( e ) ) \\| _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 250, + 410, + 750, + 428 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\lambda > 0$ is a constant determined by $\\sigma$ and the dimension of $e$ , taken as a tunable parameter. The following Theorem 2 shows that maximizing the objective in Eq. (8) does lead to the maximization of $I ( e ; W )$ , hence enables the model to preserve edge information. ", + "bbox": [ + 174, + 433, + 825, + 474 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 2. Assume the optimal solution of maximizing $\\mathcal { L } _ { I } ( f , g ; e )$ is $f ^ { \\star }$ and $g ^ { \\star }$ , then $f ^ { \\star }$ also maximizes $I ( e ; W )$ . ", + "bbox": [ + 173, + 477, + 821, + 507 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proof. Note that $H ( e )$ is a constant when the graphs are given. In information theory, we have ", + "bbox": [ + 169, + 521, + 821, + 547 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2a778d2e9fad80245dd43235b7a2babb5068136039b0572e8acfda673570fcb7.jpg", + "text": "$$\nH ( g ( f ( e ) ) ) \\leq H ( f ( e ) ) \\leq H ( e ) .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 549, + 611, + 564 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$I ( e ; W )$ is upper bounded by $H ( e )$ , ", + "bbox": [ + 176, + 566, + 428, + 582 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a305eb6f7f8a26722c22a657ceb5288b473051298c22bbc93edee7a5d363b310.jpg", + "text": "$$\nI ( e ; W ) = I ( e ; f ( e ) ) = H ( f ( e ) ) - H ( f ( e ) | e ) = H ( f ( e ) ) \\leq H ( e ) .\n$$", + "text_format": "latex", + "bbox": [ + 272, + 588, + 723, + 603 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Since $f ^ { \\star }$ and $g ^ { \\star }$ is the optimal solution of maximizing $\\mathcal { L } _ { I } ( f , g ; e )$ presented in Eq. (8), it is not difficult to see that $e = g ^ { \\star } ( f ^ { \\star } ( e ) ) , \\forall e \\in \\mathcal { E }$ . In this case, the inequalities in Eq. 9 become equalities, i.e., ", + "bbox": [ + 174, + 607, + 825, + 648 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/809dfe6268777d2ba705ed212906089485ef5b8dcf78fafc8f140c20b3eecaed.jpg", + "text": "$$\nH ( g ^ { \\star } ( f ^ { \\star } ( e ) ) ) = H ( f ^ { \\star } ( e ) ) = H ( e ) .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 650, + 617, + 665 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Therefore, we have $I ( e ; W ) = H ( e )$ , i.e., the maximum is attained in this case. ", + "bbox": [ + 174, + 666, + 735, + 681 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 Edge Information Maximized Graph Neural Networks ", + "bbox": [ + 176, + 695, + 673, + 712 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our EIGNN is derived by implementing our MI objective in GNNs. As a concrete example, the forward propagation of our model follows the formulation in Eq. (3), where the dege network $f : e W$ is expressed as a multi-layer perceptron (MLP). According to theoretical analysis presented in Sec. 3.2 , we introduce another MLP $g : W \\to e$ as the decoder. ", + "bbox": [ + 173, + 722, + 825, + 779 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For graph regression or classification tasks, the model outputs a prediction $y$ for each graph $G$ , which has label $\\hat { y }$ . Without MI maximization, we denote the vanilla loss as $\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )$ . Common choice of $\\mathcal { L } _ { 0 }$ includes Mean Square Error (MSE), Mean Absolute Error (MAE) and Cross Entropy (CE). For a graph $G = ( \\nu , \\mathcal { E } )$ , EIGNN maximizes $\\mathcal { L } _ { I } ( f , g ; e )$ and minimizes $\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )$ using the following loss function ", + "bbox": [ + 173, + 785, + 825, + 856 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e9ffed3d9ec064286879e4ca1479706a35dcaf6e9fa768bf67f604003845c77f.jpg", + "text": "$$\n\\mathcal { L } _ { E I G N N } ( G ) = \\mathcal { L } _ { 0 } ( \\hat { y } , y ; G ) - \\lambda \\mathbb { E } _ { e \\in \\mathcal { E } } [ \\mathcal { L } _ { I } ( f , g ; e ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 331, + 861, + 666, + 877 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbb { E } _ { e \\in \\mathcal { E } } [ \\cdot ]$ denotes taking the mean over all edges in $G = ( \\nu , \\mathcal { E } )$ and $\\lambda$ is the regularization parameter. When EIGNN is trained using mini-batches, $\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )$ is averaged over all graphs in the batch while $\\mathcal { L } _ { I } ( f , g ; e )$ is averaged over all edges of all graphs in the batch. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Similarly, for relational prediction tasks in knowledge graphs, EIGNN directly yields nodelevel representations $h _ { v }$ for each node $v \\in \\mathcal V$ and edge-level representations $e$ for each relationship. The objective function of EIGNN can be derived from the translational scoring function Bordes et al. (2013), which learns embedding such that for a given valid triple $t _ { v w } = ( h _ { v } , e _ { v w } , h _ { w } )$ from the valid set $S$ , the condition $d _ { t _ { v w } } = h _ { v } + e _ { v w } - h _ { w } \\approx 0$ holds. Let $\\mathcal { L } _ { 0 } = \\mathbb { E } _ { t _ { v w } \\in S } \\mathbb { E } _ { t _ { v w } ^ { \\prime } \\in S ^ { \\prime } } \\operatorname* { m a x } \\{ d _ { t _ { v w } ^ { \\prime } } - d _ { t _ { v w } } + \\gamma , 0 \\}$ , where $S ^ { \\prime }$ denotes a set of invalid triples and $\\gamma$ is a margin hyper-parameter, EIGNN can be trained by minimizing the following loss, ", + "bbox": [ + 173, + 102, + 825, + 202 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6b4942e4c4e30ad4a1ef853dea11681cee7cc1ec7a668a18ce808e38004cbe7e.jpg", + "text": "$$\n\\mathcal { L } _ { E I G N N } ( G ) = \\mathcal { L } _ { 0 } - \\lambda \\mathbb { E } _ { e \\in \\mathcal { E } } [ \\mathcal { L } _ { I } ( f , g ; e ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 209, + 638, + 226 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 176, + 244, + 333, + 261 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we first conduct experiments on a large quantum chemistry benchmark QM9, which is challenging for most baselines. Then we evaluate EIGNN on several useful molecule benchmarks and use attribution analysis to show that EIGNN increases the impact of edges and captures domain knowledge without human interference. Finally, we adopt our method to large-scale knowledge graphs and evaluate the performance on challenging relation prediction tasks using a wide variety of real-world datasets. All experimental results demonstrate a clear and substantial improvement of EIGNN over the state-of-the-art methods. ", + "bbox": [ + 173, + 276, + 825, + 375 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 Quantum Chemistry ", + "text_level": 1, + "bbox": [ + 176, + 391, + 382, + 406 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "QM9 (Ramakrishnan et al., 2014) is a large benchmark containing 134k molecules with 12 quantum chemistry regression properties, which have been show to be quite challenging for many GNNs (Gilmer et al., 2017). Feature engineering of nodes and edges exactly follows (Gilmer et al., 2017) such that molecules are preprocessed as graphs according to atom features and bond features. We compare our EIGNN with nine state-of-the-art baselines which can be categorized into three groups according to the ability of handling edge features: i) GCN, ChebyNet, GAT and GIN (Xu et al., 2019) which simply use binary edge features to indicate the existence of a bond without any other edge features; ii) RGCN, GGNN, LNet and simplified MPNN (sMPNN) which consider bond types (no bond, single, double, triple, or aromatic); iii) MPNN and our EIGNN which use edge feature vectors to indicate both edge types and pairwise distance between atoms. ", + "bbox": [ + 174, + 417, + 825, + 570 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For a fair comparison, we repeat all experiments 3 times with different random seeds while during each run, all methods share the same random seed. We randomly choose 10k molecules for validation, 10k molecules for testing, and keep the rest for training. Each target property is normalized to zero mean and unit variance for training. Each model is trained to predict the 12 target properties simultaneously. $\\lambda$ is naively set to 1 for EIGNN. We use mean square error (MSE) loss to train the models for at most 300 epochs till convergence, and the performance is measured by mean absolute error (MAE). For LNet and GGNN, implementation of the readout function follows the original paper. While for all other models, we use the same set2set (Vinyals et al., 2015) readout, which has been demonstrated to work well in (Gilmer et al., 2017). ", + "bbox": [ + 174, + 577, + 825, + 689 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/e2363b0c98ec786aa722ef1c4ce1f710a68fcc7b20e4906ed858c3a4646bd1e1.jpg", + "table_caption": [ + "Table 1: Quantum property regressions for 12 targets and overall performance (top two raws) on QM9. We repeat all experiments 3 times with different random seeds and report the average performance. Full results with standard deviation are presented in Appendix A, e.g., for MPNN and EIGNN, we have Avg. nMAE $0 . 0 3 9 8 \\pm 0 . 0 0 0 2$ and 0.0357 ± 0.0005. " + ], + "table_footnote": [], + "table_body": "
MethodGCNChebyNetGATGINRGCNGGNNLNetsMPNNMPNNEIGNN
Avg.nMAE0.1350.1210.1370.1000.1020.0990.0990.0890.0400.036
Avg.MAE5.3064.3035.4703.4803.8173.6613.6533.1610.6930.633
mu0.5680.5180.5670.4780.5060.5180.4720.4720.1100.097
alpha0.8810.7930.8910.6210.6320.6080.6230.5280.3320.294
HOMO(10-3)5.4514.7755.4294.1834.4534.4833.8893.8542.4812.230
LUMO(10-3)6.4005.6746.3314.7965.1385.1534.1944.5492.8622.593
gap(10-3)8.2017.0978.1936.0966.5006.6025.8135.6343.6203.275
R253.5641.9554.5234.6540.1039.6835.2733.496.0645.646
ZPVE(10-3)2.5332.5272.2711.7441.4771.2921.4381.3450.6790.612
UO2.0421.9842.2901.4221.0590.6971.8060.7910.4160.357
U2.0421.9842.2901.4221.0590.6971.7550.7910.4160.357
H2.0421.9842.2901.4221.0590.6971.7960.7910.4160.357
G2.0421.9842.2901.4221.0590.6961.7780.7910.4160.357
Cv0.4730.4200.4790.3090.3170.3150.3120.2620.1340.121
", + "bbox": [ + 176, + 762, + 820, + 920 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/b51754269df1a1a81766b31ec23b13e16833a1c14fe21641771ca4dfac60a05d.jpg", + "image_caption": [ + "Figure 1: Ablation study. Training and validation error on QM9. The shadow area indicates $m e a n \\pm s t d$ over 3 runs. " + ], + "image_footnote": [], + "bbox": [ + 238, + 99, + 758, + 246 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 287, + 821, + 316 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Table 1, we list regression results for all methods. We report individual MAE for each target in their original scale, averaged MAE (Avg. MAE) over 12 properties, and averaged normalized MAE (Avg. nMAE; averaged over normalized target properties since different targets have different units and ranges). Our EIGNN achieves the best performance for each metric and each target. Now we are ready to answer the following research questions. i) Are edge features important? Yes. The error has a trend of decreasing with increasing edge features. The comparison between sMPNN (using edge types) and MPNN (using edge types and distance) directly verifies the importance of edge features. It is also consistent with the expert knowledge that distances between pairwise atoms are closely related to quantum properties. For example, the smaller the distance between the two atoms, the stronger the bond is, and consequently a higher bond energy is associated with this atom pair. ii) Does the EIGNN work? Yes. EIGNN achieves the best performance on each target, outperforming the strong baseline MPNN. Moreover, the advantage of EIGNN over MPNN is consistent over 3 runs and the standard deviation on this task is quite small. Detailed results are shown in Table 4 of Appendix A. iii) How does the EIGNN work? Our MI objective is easily implemented on top of vanilla loss function. We have shown that our objective enables preserving of edge information. Fig. 1 demonstrates that regularization such as $L _ { 2 }$ weight decay can increase training error while our objective does not. Moreover, the validation performance verifies that regularization itself does not reduce the validation error. Thus, the effectiveness of EIGNN is due to exploiting edge features rather than the regularization effect. We further run an ablation study where we concatenate edge features to node representations (i.e., MPNN+concat in Fig. 1) in message passing. Concatenation is unable to identify correlations between edges and nodes (Gilmer et al., 2017) and our results show that it slightly reduces the mean validation error but increases the variance. ", + "bbox": [ + 173, + 324, + 825, + 655 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 More Molecule Benchmarks with Potential Applications ", + "text_level": 1, + "bbox": [ + 176, + 676, + 700, + 690 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We further evaluate EIGNN on three molecule benchmarks: Lipophilicity (Wu et al., 2018), ESOL (Delaney, 2004) and FreeSolv (Mobley & Guthrie, 2014). These datasets contain fewer molecules, and have potential usages in applications such as chemistry, drug discovery, and materials science. For example, the property lipophilicity is an important feature of drug molecules that affects both membrane permeability and solubility. The dataset Lipophilicity contains 4200 compounds. ESOL provides water solubility data for 1128 compounds. FreeSolv contains hydration free energy of 642 small molecules in water. We conduct graph regression experiments on these benchmarks. All datasets are split into training, validation and test according to a proportion of 0.8/0.1/0.1. MPNN and our EIGNN share the same architecture with 3 layers of message passing and 3 steps of set2set. We repeat each experiment 3 times with different random seeds. Results of testing root mean square error (RMSE) in Table 2 verify the effectiveness of our method. Our EIGNN outperforms MPNN on each dataset and each run. Detailed results for each run are presented in Appendix B. ", + "bbox": [ + 174, + 703, + 825, + 842 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/7582535dc47cc56dc3b59505e6ee0ff5b99c6069343599af156f76f4881db6c9.jpg", + "table_caption": [ + "Table 2: Testing RMSE on Lipophilicity, ESOL and FreeSolv. " + ], + "table_footnote": [], + "table_body": "
DatasetLipophilicityESOLFreeSolv
MethodMPNNEIGNNMPNNEIGNNMPNNEIGNN
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
", + "bbox": [ + 178, + 876, + 820, + 921 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/93a30757aa01764b32a397063bc6a0e3a215232f566f99e62524e437f83f0e93.jpg", + "image_caption": [ + "Figure 2: Attribution analysis. The color indicates the impact of an edge/atom on the output, i.e., the regression result. EIGNN i) increases the edge attribution, ii) reduces the prediction error and iii) can learn domain knowledge without human interference. " + ], + "image_footnote": [], + "bbox": [ + 246, + 99, + 750, + 311 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 369, + 825, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Attribution analysis. To understand how our EIGNN reduces the regression error, we conduct attribution analysis, i.e., attributing the prediction of a deep network to its input features, which usually builds up on the standard gradient operator (Sundararajan et al., features 2017). For an output $e$ as $\\begin{array} { r } { S _ { e } = | \\frac { \\partial y } { \\partial e } | } \\end{array}$ $y$ , i.e., the prediction of a GNN, we define its sensitivity to an edge with , where $\\left. \\cdot \\right.$ denotes the $L _ { 1 }$ norm. Similarly, the sensitivity to an atom with features $x$ is $\\begin{array} { r } { S _ { x } = | \\frac { \\partial y } { \\partial x } | } \\end{array}$ . Then $S _ { e }$ and $S _ { x }$ are used as the metrics of attribution in our experiments. As an example, we show in Fig. 2 the attribution for two molecules in Lipophilicity: (a) $N c I n o n c I C ( = N O ) N c \\mathcal { Q } c c c ( F ) c ( C l ) c \\mathcal { Q }$ and (b) Oc1c2ncc3ccccc3c2nn1c4ccccc4 . Molecule in (a) has the potential to be used to treat, prevent and/or diagnose cancer Prinz et al. (2019). Compared with MPNN, we can observe an increasing of overall edge attribution under our EIGNN and a decreasing of prediction error in both cases. Interestingly, the attribution under EIGNN is similar to the expert knowledge of chemists: halogen atoms such as {Cl, Br, I} and their bond with the carbon atom greatly effect the lipophilicity of a molecule Wilcken et al. (2013), while atoms {O, N} also have high impact on the lipophilicity but usually in a negative way (Augustijns & Brewster, 2007). In Fig. 2 (a), the attribution of the halogen bond C-Cl and the pair {O, N} under our EIGNN is much higher than the one under MPNN, which is consistent with the expert knowledge. In Fig. 2 (b), similarly, the attribution of atoms {O, N} and the bond C-O under EIGNN is much higher. More examples are presented in Appendix C. ", + "bbox": [ + 174, + 428, + 825, + 698 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Predicting Relations in Knowledge Graphs ", + "text_level": 1, + "bbox": [ + 174, + 717, + 581, + 731 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this subsection, we adopt EIGNN to tackle the problem of relation prediction in knowledge graphs (KGs), which entails predicting whether a given triple is valid or not. For example, a triple (London, capital of, United Kingdom) should be classified as valid or London should be predicted as the capital of United Kingdom. KGs represent human knowledge as a directed graph, and have been widely used in practical applications, such as semantic search, dialogue generation, question answering etc. Recovering missing relations in KGs have been a major task for practical usages of KGs. We evaluate our methods on three benchmark datasets, WN18RR (Dettmers et al., 2018), FB15k-237 (Toutanova et al., 2015) and NELL-995 (Xiong et al., 2017). Without the reversible relation problem (Dettmers et al., 2018), WN18RR includes 11 relations scraped from WordNet for 40, 943 synsets. FB15k-237 is a subset of Freebase, and contains 14, 541 entities associated with 237 types of edge. NELL-995 is constructed from the $9 9 5 ^ { t h }$ iteration of NELL system, containing 75, 492 entities and 200 types of edge. ", + "bbox": [ + 174, + 743, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/b9f6d05fff4b5da5ff97fa116ab1faa3e0ecaf14f59bf84883052dee805e45d1.jpg", + "table_caption": [ + "Table 3: Experimental results on WN18RR, FB15K-237 and NELL-995 test sets. Hits@N values are in percentage. The best score is in bold and second best score is underlined. " + ], + "table_footnote": [], + "table_body": "
DatasetWN18RRFB15K-237NELL-995
MRRHit@N%MRRHit@N %
MRR@1@3@10@1@3@10@1@3@10
DistMult0.44441.24750.40.28119.930.144.60.48540.152.461
ComplEx0.44940.946.9530.27819.429.7450.48239.952.860.6
ConvE0.45641.94753.10.31222.534.149.70.49140.353.161.3
TransE0.24342.744.153.20.27919.837.644.10.40134.447.250.1
ConvKB0.26558.244.555.80.28919.832.447.10.4337.04754.5
R-GCN0.123813.720.70.1641018.1300.128.212.618.8
KBGAT0.43635.848.157.80.43136.145.856.90.51442.955.367.8
EIGNN0.43835.748.858.10.45137.448.260.50.52343.856.168.3
", + "bbox": [ + 186, + 122, + 810, + 241 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A critical issue of applying Eq. (3) to KGs is that high-dimensional embedding vectors are required to distinguish massive amount of entities and relations, leading to a rapid growth in number of parameters in EIGNN. To address this issue, we adopt the architecture of (Nathani et al., 2019) and learn graph attention based embeddings that target relation prediction on KGs as follows, ", + "bbox": [ + 174, + 263, + 823, + 335 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/56b7e18c1c9472053b2ca7278088a13ab3c9e8c0952468df13b763434be1907f.jpg", + "text": "$$\n\\begin{array} { r } { m _ { v w } = f \\big ( h _ { v } ^ { ( l ) } , e _ { v w } ^ { l } , h _ { w } ^ { ( l ) } \\big ) , \\alpha _ { v w } ^ { l } = \\mathrm { s o f t m a x } \\big ( a ^ { l } m _ { v w } ^ { l } \\big ) , h _ { v } ^ { ( l + 1 ) } = \\sigma \\left( \\sum _ { w \\in \\mathcal { N } _ { v } } \\alpha _ { v w } ^ { l } m _ { v w } ^ { l } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 197, + 338, + 772, + 366 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Compared with Eq. (3), where $m _ { v w } = f ( e _ { v w } ) h _ { w }$ , the above equation absorbs the transform matrix $f ( e _ { v w } )$ into $f ( h _ { v } ^ { ( l ) } , e _ { v w } , h _ { w } ^ { ( l ) } )$ and reduces the model parameters. In the following experiments, we implement $f$ using a MLP as in previous experiments, and maximizes the MI between $e _ { v w }$ and $m _ { v w }$ by introducing another MLP with $\\lambda = 0 . 0 1$ . Multi-head attention is further introduced to stabilize the learning process and encapsulate more information about neighbors according to (Veličković et al., 2018). After training EIGNN, ConvKB (Nguyen et al., 2018) is adopted as a regression function for a given triple by analyzing the global embedding properties across each dimension. ", + "bbox": [ + 174, + 369, + 825, + 486 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In the relation prediction task, the aim is to predict a triple $( v , e _ { v w } , w )$ with $\\boldsymbol { v }$ or $w$ missing. We can generate a set of candidate triples for each missing entity $v$ by randomly replacing it with an arbitrary one. Scores can be calculated by ConvKB for all triples, and we find the rank of a correct triple by sorting all scores in ascending order. Thus, the performance of relation prediction task can be evaluated by mean reciprocal rank (MRR) and the proportion of correct entities in the top $N$ ranks (Hits@N) for $N = 1 , 3$ , and 10 (Bordes et al., 2013). We compare our EIGNN with seven state-of-the-art baselines focusing on this task: DistMult (Yang et al., 2014), ComplEx (Trouillon et al., 2016), ConvE (Dettmers et al., 2018), TransE (Bordes et al., 2013), ConvKB (Nguyen et al., 2018), RGCN (Schlichtkrull et al., 2018) and KBGAT (Nathani et al., 2019). As shown in Table 3, our EIGNN achieves the best performance for each metric on FB15K-237 and NELL-995, and achieves the best performance on WN18RR with Hit $^ \\mathrm { ( a 3 }$ and 10 metrics. The results of KBGAT are reproduced following the official implementation1, and the results of other methods can be found in the previous peer-reviewed publications, i.e. (Nathani et al., 2019). ", + "bbox": [ + 173, + 492, + 825, + 686 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 Conclusions ", + "text_level": 1, + "bbox": [ + 176, + 705, + 328, + 722 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this work, to make better use of edge features in GNNs, we proposed the edge information maximized graph neural network (EIGNN) that maximizes the mutual information between edge feature vectors and message passing channels. We reformulated the mutual information as a differentiable objective by adopting a variational approach. We have theoretically proved that our proposed objective enables EIGNN to preserve edge information and empirically evaluated EIGNN’s performance on a variety of benchmarks incorporating an array of challenging molecular datasets and knowledge graphs. These results clearly manifested a substantial improvement of EIGNN over the prior state-of-the-art methods. Apart from demonstrating the impressive performance of EIGNN, we also showed that its effectiveness is due to exploitation of edge features instead of the regularization effect. Notably, attribution analysis on molecular graphs show that EIGNN can capture domain knowledge in an end-to-end fashion. ", + "bbox": [ + 174, + 737, + 825, + 904 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 176, + 102, + 290, + 118 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "David Barber Felix Agakov. The im algorithm: a variational approach to information maximization. NeurIPS, 2004. ", + "bbox": [ + 174, + 126, + 823, + 155 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Patrick Augustijns and Marcus E Brewster. Solvent systems and their selection in pharmaceutics and biopharmaceutics, volume 190. Springer, 2007. 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", + "bbox": [ + 171, + 102, + 826, + 377 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A Full Results on QM9 ", + "text_level": 1, + "bbox": [ + 176, + 101, + 426, + 118 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/a301b4667b007d0ca0d55c084b4ad788389f2f177ec6b78e47c696d5409f9f41.jpg", + "table_caption": [ + "Table 4: Full results of quantum property regressions for 12 targets and overall performance (nMAE and MAE in top two raws) on QM9. We repeat all experiments 3 times with different random seeds and report the average performance and standard deviation. This is a supplement for Table 1. in the main text. " + ], + "table_footnote": [], + "table_body": "
MethodGCNChebyNetGATGIN
Avg.nMAE0.1350±0.00460.1206±0.00840.1367±0.00500.1001±0.0007
Avg.MAE5.3063±0.19644.3032±0.48145.4698±0.20403.4799±0.0402
mu0.5679±0.00780.5180±0.01310.5670±0.01020.4783±0.0041
alpha0.8811±0.03080.7932±0.07780.8913±0.03140.6209±0.0028
HOMO(10-3)5.4510±0.07904.7750±0.20405.4290±0.16604.1830±0.0370
LUMO(10-3)6.4000±0.12305.6740±0.26706.3310±0.23904.7960±0.0520
gap(10-3)8.2010±0.21507.0970±0.36208.1930±0.29106.0960±0.0420
R253.563±1.031941.950±4.828954.519±1.599234.647±0.2167
ZPVE(10-3)2.5330±0.10702.5270±0.35602.2710±0.14501.7440±0.0100
UO2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
U2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
H2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
G2.0423±0.32811.9842±0.20362.2899±0.20001.4215±0.0857
Cv0.4730±0.02090.4199±0.04990.4787±0.01640.3093±0.0035
MethodRGCNGGNNLNetsMPNN
Avg.nMAE0.1021±0.00160.0992±0.00130.0992±0.00610.0888±0.0014
Avg.MAE3.8175±0.06053.6608±0.07233.6527±0.34173.1610±0.0697
mu0.5056±0.00480.5179±0.00760.4717±0.00630.4718±0.0096
alpha0.6321±0.01450.6077±0.00920.6225±0.05080.5278±0.0106
HOMO(10-3)4.4530±0.12904.4830±0.06503.8889±0.16173.8540±0.0410
LUMO(10-3)5.1380±0.12105.1530±0.08904.1935±0.22054.5490±0.0810
gap(10-3)6.5000±0.13906.6020±0.14005.8132±0.64565.6340±0.0570
R240.102±0.742839.685±0.821235.275±3.053133.489±0.6562
ZPVE(10-3)1.4770±0.00901.2920±0.03401.4376±0.07691.3450±0.0260
UO1.0589±0.02310.6969±0.04131.8058±0.25330.7914±0.0446
U1.0589±0.02310.6966±0.04181.7555±0.21960.7914±0.0446
H1.0589±0.02310.6974±0.04081.7964±0.24280.7914±0.0446
G1.0589±0.02310.6961±0.04211.7780±0.24580.7914±0.0446
Cv0.3170±0.01520.3146±0.01250.3124±0.03030.2625±0.0040
MethodMPNNEIGNN
Avg.nMAE0.0398±0.00020.0357±0.0005
Avg.MAE0.6929±0.02120.6331±0.0298
mu0.1095±0.00140.0974±0.0026
alpha0.3318±0.00260.2939±0.0054
HOMO(10-3)2.4810±0.02002.2300±0.0310
LUMO(10-3)2.8620±0.03702.5930±0.0440
gap(10-3)3.6200±0.01803.2750±0.0520
R26.0637±0.25115.6464±0.3098
ZPVE(10-3)0.6790±0.01400.6120±0.0170
UO0.4164±0.02250.3574±0.0100
U0.4164±0.02250.3575±0.0100
H0.4164±0.02250.3574±0.0100
G0.4164±0.02250.3575±0.0101
Cv0.1339±0.00130.1208±0.0027
", + "bbox": [ + 243, + 202, + 753, + 738 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We present full results of quantum property regressions on QM9 with average performance and standard deviation in Table 4. The $( 1 0 ^ { - 3 }$ ) in the parentheses indicates that the values in the corresponding raw of the table should multiply by $1 0 ^ { - 3 }$ . This is simply for clear presentation of the values. We also present a detailed descriptions on the target properties in Table 5 for your reference. ", + "bbox": [ + 173, + 750, + 825, + 819 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In our results, we directly report MAE instead of Error Ratio [(MAE)/(Chemical Accuracy) (Gilmer et al., 2017)], because it is common to report MAE in terms of chemical unit. This practice has been widely adopted not only in computational chemistry but also in the machine learning community working on molecular graphs (Wu et al., 2018; Morris et al., 2019). Still, we add Table 6 which contains the Error Ratio for MPNN and our EIGNN. In this comparison, we follow (Gilmer et al., 2017) and train models separately to predict each target. Our EIGNN consistently outperforms MPNN. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/91a3cbe5a783bcdbe8a1afa2b6c12f24ae9c429e52bd0844c20c200534394b6c.jpg", + "table_caption": [ + "Table 5: Regression targets on QM9. " + ], + "table_footnote": [], + "table_body": "
Target propertyDescriptionUnit
muDipole momentD
alphaIsotropic polarizabilitya
HOMOHighest occupied molecular orbital energyEh
LUMOLowest unoccupied molecular orbital energyEh
gapGap between HOMO and LUMOEh
R2Electronic spatial extent品
ZPVEZero point vibrational energyEh
UOInternal energy at OKEh
UInternal energy at 298.15KEh
HEnthalpy at 298.15KEh
GFree energy at 298.15KEh
CvHeat capavity at 298.15Kcal molK
", + "bbox": [ + 281, + 127, + 717, + 275 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/2c21c7621d41da0431991f555ba9609c7ac7f048f42d5cf034e99ce8a99c7454.jpg", + "table_caption": [ + "Table 6: Error Ratio [(MAE)/(Chemical Accuracy) (Gilmer et al., 2017)] on QM9. Note that the energy values of $\\{ \\mathrm { U 0 , ~ U , ~ H , ~ G } \\}$ are per molecule rather than per atom. Following (Gilmer et al., 2017), models are separately trained on each target. " + ], + "table_footnote": [], + "table_body": "
TargetMPNNEIGNNImprovement (%)
mu0.870.808.24
alpha2.642.447.57
HOMO1.541.3910.2
LUMO1.311.282.13
gap2.201.9710.4
R21.000.7228.4
ZPVE5.314.4316.5
UO64.335.844.4
U56.623.658.3
H77.435.254.5
G46.634.426.1
Cv1.691.79-6.22
", + "bbox": [ + 346, + 345, + 650, + 491 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B Full Results on Lipophilicity, ESOL and FreeSolv ", + "bbox": [ + 173, + 515, + 717, + 534 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section, we present experimental results on Lipophilicity, ESOL and FreeSolv with detailed results for each run. The results in Table 7 verify that our EIGNN consistently outperforms MPNN. ", + "bbox": [ + 174, + 549, + 823, + 590 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C More Examples of Attribution ", + "text_level": 1, + "bbox": [ + 174, + 611, + 524, + 627 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section, we present more examples of attribution analysis on Lipophilicity. As a supplement for Fig. 2 in the main text, the observation here is similar. Compared with MPNN, we can observe an increasing of overall edge attribution under our EIGNN and a decreasing of prediction error in both cases. Notably, our EIGNN is able to capture the expert knowledge. (a) The molecule is $C N [ C @ \\mathbb { Q } H ] ( C ) C ( = O ) N [ C @ \\mathbb { Q } H ] ( C 1 C C C C T 1 ) C ( =$ $O ) N [ C @ H ] 2 C C C N ( C C c ( F ) c c 3 ) C 2$ . The attribution of $\\{ \\mathrm { O } , \\mathrm { N } \\}$ and the halogen atom $\\mathrm { F }$ is higher under EIGNN. (b) The molecule is $C C ( C ) N 1 C C N [ C @ H ] ( C 1 ) C ( =$ $O ) N 2 C C N ( C C 2 ) C ( = O ) N c 3 c c c ( C l ) c ( C l ) c 3$ . Our EIGNN successfully captures the importance of two critical halogen atoms Cl and several atoms N. (c) The molecule is $C O c 1 c c ( c c 1 ) C ( = \\ O ) N 2 C C C C 2 \\ = \\ O$ . The attribution of two atoms $\\mathrm { \\ o }$ at the top is ", + "bbox": [ + 173, + 643, + 825, + 784 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/901ad24ad330961b104145c59ad458a6aba349587ab22792703374b07939c3f2.jpg", + "table_caption": [ + "Table 7: Testing RMSE on Lipophilicity, ESOL and FreeSolv. This is a supplement for Table 2 in the main text. " + ], + "table_footnote": [], + "table_body": "
DatasetLipophilicityESOLFreeSolv
SeedMPNNEIGNNMPNNEIGNNMPNNEIGNN
00.7180.6760.7700.7181.3961.109
10.6960.6640.7500.7331.2991.265
20.6200.6190.8940.8761.4991.443
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
", + "bbox": [ + 178, + 845, + 818, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "much higher under EIGNN. (d) The molecule is $C c 1 c c 2 N C ( = O ) C ( = C C ( = O ) c 2 c c 1 C ) O$ . \nThe attribution of atoms O is much higher under EIGNN. ", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/d35b4c5026ccc9e9cd8864632869ef2ae4dd487409a61f9850c2293c8640530b.jpg", + "image_caption": [ + "Figure 3: More examples of attribution analysis. The color indicates the impact of an edge/atom on the output, i.e., the regression result. EIGNN i) increases the edge attribution, ii) reduces the prediction error and iii) can learn domain knowledge without human interference. 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We theoretically show that the newly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "score": 1.0, + "content": "introduced objective enables the model to preserve edge information, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 331, + 469, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 469, + 343 + ], + "score": 1.0, + "content": "empirically corroborate the enhanced performance of MI-maximized models", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 342, + 469, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 355 + ], + "score": 1.0, + "content": "across a broad range of learning tasks including regression on molecular", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 352, + 371, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 371, + 366 + ], + "score": 1.0, + "content": "graphs and relation prediction in knowledge graphs.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5, + "bbox_fs": [ + 141, + 231, + 470, + 366 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 387, + 208, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 210, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 210, + 402 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "Many real-world datasets naturally come in the form of graphs, such as citation networks (Kipf", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "& Welling, 2017), social networks Hamilton et al. 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Recently,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "impressive performance has been achieved in graph learning tasks with various forms of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "score": 1.0, + "content": "Graph Neural Networks (GNNs) (Zhou et al., 2018). Compared to prior works, such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "as node2vec (Grover & Leskovec, 2016), GNNs learn the state of a node by recursively", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "aggregating messages from its neighbors: combining the graph structure with node features.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "Intuitively, edge features should play an important role in graph learning tasks. For example,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "chemical bonds in a molecule have a high impact on chemical properties of molecules, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "edge features in knowledge graphs encode important relations between concepts, data, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 498, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 498, + 546 + ], + "score": 1.0, + "content": "entities. Our proposed method focuses on improving the usage of edge features in GNNs.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 412, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "The expressive power of GNNs largely depends on how the message is passed between nodes.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 562, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 504, + 573 + ], + "score": 1.0, + "content": "A widely adopted scheme is multiplying neighbor node states with a parameterized transform", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 570, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 505, + 586 + ], + "score": 1.0, + "content": "matrix before aggregation (Gilmer et al., 2017; Xu et al., 2019). Despite tremendous success", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "of GNNs, existing models do not exhaustively exploit the full potentials of edge features on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "graphs. For example, many GNNs such as GCN (Kipf & Welling, 2017), ChebyNet (Defferrard", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "et al., 2016) and GAT (Veličković et al., 2018) do not even consider categorized edge types.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "score": 1.0, + "content": "To utilize edge features in multi-relational graphs, RGCN (Schlichtkrull et al., 2018) proposes", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "to learn a different transform matrix for each edge type, respectively. However, it does not", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "generalize to edge features in continuous space. MPNN (Gilmer et al., 2017) introduces", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "an edge network that takes edge feature vectors as input and outputs transform matrices,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "which are used to transform states of neighbor nodes. In principle, the MPNN framework", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "can handle complex edge features. Yet, the lack of maximization of MI between edge and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "message channels implies that the MPNN may give an edge-independent transform matrix.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 550, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 504, + 711 + ], + "score": 1.0, + "content": "In this work, we aim to more efficiently exploit the full potentials of edge features from", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "the perspective of training. We propose the Edge Information maximized Graph Neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "Network (EIGNN) that maximizes the Mutual Information (MI) between edge features", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "and the message passing channel which is parameterized as the transform matrix in the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "widely-accepted message passing framework (transformation and aggregation) (Gilmer et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "2017; Xu et al., 2019). Considering the challenge of computing the MI, we adopt a variational", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "approach to reformulate it as an differentiable objective, which can be easily applied as a", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "regularization term. We theoretically show that EIGNN can reduce information loss of edge", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "features. Apart from demonstrating the impressive performance of EIGNN on extensive", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "benchmarks of molecular graphs and knowledge graphs, we also analyze and attribute", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "the enhanced effectiveness of EIGNN to the exploitation of edge features instead of the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "regularization effects. Notably, attribution analysis on molecular graphs show that EIGNN", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 369, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 369, + 193 + ], + "score": 1.0, + "content": "can capture domain knowledge without human interference.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 698, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "and the message passing channel which is parameterized as the transform matrix in the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "widely-accepted message passing framework (transformation and aggregation) (Gilmer et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "2017; Xu et al., 2019). Considering the challenge of computing the MI, we adopt a variational", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "approach to reformulate it as an differentiable objective, which can be easily applied as a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "regularization term. We theoretically show that EIGNN can reduce information loss of edge", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "features. Apart from demonstrating the impressive performance of EIGNN on extensive", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "benchmarks of molecular graphs and knowledge graphs, we also analyze and attribute", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "the enhanced effectiveness of EIGNN to the exploitation of edge features instead of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "regularization effects. 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Let", + "type": "text" + }, + { + "bbox": [ + 431, + 227, + 444, + 237 + ], + "score": 0.92, + "content": "\\mathcal { N } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 225, + 506, + 238 + ], + "score": 1.0, + "content": "be the set of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 234, + 507, + 253 + ], + "spans": [ + { + "bbox": [ + 103, + 234, + 190, + 253 + ], + "score": 1.0, + "content": "neighbor nodes of", + "type": "text" + }, + { + "bbox": [ + 190, + 244, + 195, + 249 + ], + "score": 0.89, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 234, + 218, + 253 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 237, + 262, + 250 + ], + "score": 0.93, + "content": "h _ { v } ^ { ( l ) } \\in \\mathbb { R } ^ { d _ { l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 234, + 368, + 253 + ], + "score": 1.0, + "content": "be the hidden state of", + "type": "text" + }, + { + "bbox": [ + 368, + 244, + 373, + 249 + ], + "score": 0.89, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 234, + 389, + 253 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 389, + 241, + 393, + 249 + ], + "score": 0.86, + "content": "\\it l", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 234, + 466, + 253 + ], + "score": 1.0, + "content": "-th layer, where", + "type": "text" + }, + { + "bbox": [ + 466, + 241, + 475, + 250 + ], + "score": 0.91, + "content": "d _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 234, + 507, + 253 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 435, + 263 + ], + "score": 1.0, + "content": "dimension of the hidden layer. 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We also have", + "type": "text" + }, + { + "bbox": [ + 305, + 262, + 345, + 274 + ], + "score": 0.93, + "content": "h _ { v } ^ { ( 0 ) } = x _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 256, + 431, + 279 + ], + "score": 1.0, + "content": "at the input layer.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 289, + 219, + 302 + ], + "lines": [ + { + "bbox": [ + 104, + 288, + 221, + 304 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 221, + 304 + ], + "score": 1.0, + "content": "2 Related works", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 312, + 386, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 387, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 387, + 326 + ], + "score": 1.0, + "content": "2.1 Relational Modeling in Graph Neural Networks", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "Single-relational modeling. Many variants such as GCN (Kipf & Welling, 2017),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "GAT (Veličković et al., 2018), ChebyNet (Defferrard et al., 2016), GraphSAGE (Hamilton", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "et al., 2017) focus on learning node states. These models can assign weight to neighbors, but", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 498, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 498, + 379 + ], + "score": 1.0, + "content": "they can not handle various edge features. A typical neighborhood aggregation scheme is", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 379, + 406, + 399 + ], + "lines": [ + { + "bbox": [ + 205, + 379, + 406, + 399 + ], + "spans": [ + { + "bbox": [ + 205, + 379, + 406, + 399 + ], + "score": 0.91, + "content": "h _ { v } ^ { ( l + 1 ) } = \\sigma \\left( \\sum _ { w \\in \\mathcal { N } _ { v } } \\alpha _ { v w } W _ { 1 } ^ { ( l ) } h _ { w } ^ { ( l ) } + W _ { 0 } ^ { ( l ) } h _ { v } ^ { ( l ) } \\right) ,", + "type": "interline_equation", + "image_path": "897a3f0394352afdd981afc2c5beb5e43696de6f029404d486b83b2d43b34699.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 205, + 379, + 406, + 399 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 136, + 412 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 405, + 142, + 410 + ], + "score": 0.9, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 399, + 285, + 412 + ], + "score": 1.0, + "content": "denotes an activation function,", + "type": "text" + }, + { + "bbox": [ + 285, + 405, + 302, + 411 + ], + "score": 0.88, + "content": "\\alpha _ { v w }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "can be a normalization constant or a learned", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "attention coefficient (Veličković et al., 2018). 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Many variants such as GCN (Kipf & Welling, 2017),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "GAT (Veličković et al., 2018), ChebyNet (Defferrard et al., 2016), GraphSAGE (Hamilton", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "et al., 2017) focus on learning node states. These models can assign weight to neighbors, but", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 498, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 498, + 379 + ], + "score": 1.0, + "content": "they can not handle various edge features. 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MPNN can actually learn an edge-independent transform matrix.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 720, + 407, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 407, + 734 + ], + "score": 1.0, + "content": "A GNN model that efficiently utilizes edge features is yet to emerge.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 665, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 231, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 232, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 232, + 95 + ], + "score": 1.0, + "content": "2.2 Readout functions", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "After several forward propagations, GNNs yield final states of all nodes, which are suitable", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "for node/edge classification or regression. 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Our method proposes to preserve edge information in GNNs, which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 270, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 504, + 281 + ], + "score": 1.0, + "content": "is important in many real-world graph structures such as molecules - apart from node", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "(atom) features, attributes of edges (bonds) are equally important for predicting properties", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 291, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 281, + 304 + ], + "score": 1.0, + "content": "of molecules. To this end, we maximize", + "type": "text" + }, + { + "bbox": [ + 281, + 293, + 315, + 304 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 291, + 498, + 304 + ], + "score": 1.0, + "content": "- the MI between the edge feature vector", + "type": "text" + }, + { + "bbox": [ + 498, + 297, + 504, + 301 + ], + "score": 0.88, + "content": "e", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 372, + 315 + ], + "score": 1.0, + "content": "and the message passing channel, i.e., the transform matrix", + "type": "text" + }, + { + "bbox": [ + 372, + 305, + 383, + 312 + ], + "score": 0.78, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "which is used to transform", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 313, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 504, + 326 + ], + "score": 1.0, + "content": "neighbor node states in the forward propagation. Our method can be easily generalized to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 325, + 504, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 504, + 336 + ], + "score": 1.0, + "content": "directly maximize the MI between edge features and the message itself in methods that do", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 418, + 348 + ], + "score": 1.0, + "content": "not explicitly have the transform matrix, e.g., the message from node", + "type": "text" + }, + { + "bbox": [ + 418, + 340, + 426, + 345 + ], + "score": 0.88, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 335, + 465, + 348 + ], + "score": 1.0, + "content": "to node", + "type": "text" + }, + { + "bbox": [ + 466, + 340, + 471, + 345 + ], + "score": 0.88, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 346, + 461, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 164, + 359 + ], + "score": 1.0, + "content": "expressed as", + "type": "text" + }, + { + "bbox": [ + 164, + 348, + 225, + 358 + ], + "score": 0.94, + "content": "f ( h _ { v } , e _ { v w } , h _ { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 346, + 281, + 359 + ], + "score": 1.0, + "content": "rather than", + "type": "text" + }, + { + "bbox": [ + 281, + 348, + 322, + 358 + ], + "score": 0.94, + "content": "f ( e _ { v w } ) h _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 346, + 461, + 359 + ], + "score": 1.0, + "content": ", which is shown in Section 4.3.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 377 + ], + "score": 1.0, + "content": "An general principle of maximum MI is described for unsupervised learning task by (Linsker,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 374, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 506, + 386 + ], + "score": 1.0, + "content": "1988) and MI inspired objective functions have long been adopted in unsupervised learn-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "ing (Bridle et al., 1992; Barber & Agakov, 2006; Veličković et al., 2019; Hjelm et al., 2019),", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "semi-supervised learning (Krause et al., 2010) and generative adversarial networks (Chen", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "et al., 2016). Specifically, DGI (Veličković et al., 2019) also applies MI to GNNs. DGI", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "proposes to learn node-wise representations in an unsupervised manner by maximizing the MI", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "between node representations and corresponding high-level summaries of graphs, using adver-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "sarial learning and negative sampling. The node representations may then be retrieved and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "used for downstream tasks, such as node classification. DGI can be used to pre-train GNNs,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "as demonstrated in Hu et al. (2019). Our EIGNN also proposes information maximization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 473, + 466, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 466, + 484 + ], + "score": 1.0, + "content": "but targets a completely different objective and adopts a quite different approach.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 496, + 431, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 433, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 433, + 509 + ], + "score": 1.0, + "content": "3.2 A Variational Approach to Maximize Mutual Information", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 160, + 530 + ], + "score": 1.0, + "content": "Computing", + "type": "text" + }, + { + "bbox": [ + 160, + 519, + 194, + 529 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 517, + 504, + 530 + ], + "score": 1.0, + "content": "itself is intractable in practice, needless to say that training a model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "score": 1.0, + "content": "requires the derivative. 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Then we can approximate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 655, + 369, + 668 + ], + "spans": [ + { + "bbox": [ + 107, + 657, + 140, + 668 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 655, + 276, + 668 + ], + "score": 1.0, + "content": "with a differentiable objective", + "type": "text" + }, + { + "bbox": [ + 276, + 657, + 320, + 668 + ], + "score": 0.94, + "content": "L _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 655, + 369, + 668 + ], + "score": 1.0, + "content": "as follows.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 311, + 681 + ], + "score": 1.0, + "content": "Theorem 1. 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Specifically, DGI (Veličković et al., 2019) also applies MI to GNNs. DGI", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "proposes to learn node-wise representations in an unsupervised manner by maximizing the MI", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "between node representations and corresponding high-level summaries of graphs, using adver-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "sarial learning and negative sampling. The node representations may then be retrieved and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "used for downstream tasks, such as node classification. DGI can be used to pre-train GNNs,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "as demonstrated in Hu et al. (2019). Our EIGNN also proposes information maximization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 473, + 466, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 466, + 484 + ], + "score": 1.0, + "content": "but targets a completely different objective and adopts a quite different approach.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 362, + 506, + 484 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 496, + 431, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 433, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 433, + 509 + ], + "score": 1.0, + "content": "3.2 A Variational Approach to Maximize Mutual Information", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 160, + 530 + ], + "score": 1.0, + "content": "Computing", + "type": "text" + }, + { + "bbox": [ + 160, + 519, + 194, + 529 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 517, + 504, + 530 + ], + "score": 1.0, + "content": "itself is intractable in practice, needless to say that training a model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "score": 1.0, + "content": "requires the derivative. Thus, we adopt a variational approach (Agakov, 2004) to reformulate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 107, + 541, + 140, + 551 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "as a differentiable objective. We show that our objective is an approximated lower", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 150, + 564 + ], + "score": 1.0, + "content": "bound of", + "type": "text" + }, + { + "bbox": [ + 150, + 552, + 183, + 562 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 549, + 469, + 564 + ], + "score": 1.0, + "content": "and notably, optimizing our objective does lead to maximizing", + "type": "text" + }, + { + "bbox": [ + 469, + 552, + 502, + 562 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 549, + 506, + 564 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "Following MPNN (Gilmer et al., 2017), we use an edge network to parameterize the transform", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 420, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 139, + 585 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 140, + 574, + 150, + 582 + ], + "score": 0.89, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 572, + 376, + 585 + ], + "score": 1.0, + "content": "and relate it to edge features. Therefore, the prior", + "type": "text" + }, + { + "bbox": [ + 376, + 574, + 407, + 584 + ], + "score": 0.94, + "content": "p ( W | e )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 572, + 420, + 585 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 517, + 506, + 585 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 586, + 356, + 599 + ], + "lines": [ + { + "bbox": [ + 254, + 586, + 356, + 599 + ], + "spans": [ + { + "bbox": [ + 254, + 586, + 356, + 599 + ], + "score": 0.93, + "content": "p ( W | e ) = \\delta ( W - f ( e ) ) ,", + "type": "interline_equation", + "image_path": "064575c5b1ecce9a7f7dbda9406409bc548a24e03885ab642abea6e3861b477f.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 254, + 586, + 356, + 599 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 136, + 613 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 602, + 152, + 613 + ], + "score": 0.93, + "content": "\\delta ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 599, + 342, + 613 + ], + "score": 1.0, + "content": "is the Dirac delta function. The posterior", + "type": "text" + }, + { + "bbox": [ + 343, + 602, + 374, + 613 + ], + "score": 0.94, + "content": "p ( e | W )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "is intractable, so we define a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 503, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 207, + 624 + ], + "score": 1.0, + "content": "variational distribution", + "type": "text" + }, + { + "bbox": [ + 207, + 613, + 237, + 624 + ], + "score": 0.94, + "content": "q ( e | W )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 612, + 458, + 624 + ], + "score": 1.0, + "content": ", which can be obtained by defining a neural network", + "type": "text" + }, + { + "bbox": [ + 459, + 614, + 503, + 623 + ], + "score": 0.92, + "content": "g : W \\to e", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 162, + 635 + ], + "score": 1.0, + "content": "Specifically,", + "type": "text" + }, + { + "bbox": [ + 162, + 624, + 193, + 635 + ], + "score": 0.94, + "content": "q ( e | W )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "substitutes to some distribution (such as Gaussian distribution) with", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 153, + 646 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 154, + 635, + 177, + 646 + ], + "score": 0.93, + "content": "g ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 633, + 236, + 646 + ], + "score": 1.0, + "content": ". In this way,", + "type": "text" + }, + { + "bbox": [ + 236, + 636, + 243, + 645 + ], + "score": 0.91, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 633, + 264, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 264, + 639, + 270, + 645 + ], + "score": 0.89, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "are similar to the probabilistic encoder and decoder in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 311, + 657 + ], + "score": 1.0, + "content": "the Variational Auto-Encoder (VAE) (Kingma", + "type": "text" + }, + { + "bbox": [ + 311, + 646, + 319, + 654 + ], + "score": 0.64, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "Welling, 2013). Then we can approximate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 655, + 369, + 668 + ], + "spans": [ + { + "bbox": [ + 107, + 657, + 140, + 668 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 655, + 276, + 668 + ], + "score": 1.0, + "content": "with a differentiable objective", + "type": "text" + }, + { + "bbox": [ + 276, + 657, + 320, + 668 + ], + "score": 0.94, + "content": "L _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 655, + 369, + 668 + ], + "score": 1.0, + "content": "as follows.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 599, + 506, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 311, + 681 + ], + "score": 1.0, + "content": "Theorem 1. Let e be the edge feature vector,", + "type": "text" + }, + { + "bbox": [ + 312, + 671, + 322, + 678 + ], + "score": 0.86, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "be the transform matrix with conditional", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 160, + 693 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 160, + 682, + 191, + 692 + ], + "score": 0.94, + "content": "p ( W | e )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 680, + 354, + 693 + ], + "score": 1.0, + "content": "specified by the probabilistic encoder", + "type": "text" + }, + { + "bbox": [ + 354, + 682, + 361, + 692 + ], + "score": 0.89, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 680, + 473, + 693 + ], + "score": 1.0, + "content": "as shown in Eq. (5) and", + "type": "text" + }, + { + "bbox": [ + 474, + 682, + 505, + 692 + ], + "score": 0.93, + "content": "q ( e | W )", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 690, + 468, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 397, + 703 + ], + "score": 1.0, + "content": "be the variational distribution specified by the probabilistic decoder", + "type": "text" + }, + { + "bbox": [ + 397, + 696, + 402, + 703 + ], + "score": 0.88, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 690, + 468, + 703 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 668, + 505, + 703 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 705, + 387, + 718 + ], + "lines": [ + { + "bbox": [ + 223, + 705, + 387, + 718 + ], + "spans": [ + { + "bbox": [ + 223, + 705, + 387, + 718 + ], + "score": 0.93, + "content": "I ( e ; W ) \\geq H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) } [ \\mathcal { L } _ { I } ( f , g ; e ) ] ,", + "type": "interline_equation", + "image_path": "1ddb6427dd9ce6d7604222cb7abb1323181e83e653a98be12246cba2bb6046d8.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 223, + 705, + 387, + 718 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 378, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 717, + 380, + 736 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 134, + 736 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 722, + 245, + 732 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { I } ( f , g ; e ) = \\log q ( e | f ( e ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 717, + 266, + 736 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 267, + 722, + 286, + 733 + ], + "score": 0.93, + "content": "H ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 717, + 380, + 736 + ], + "score": 1.0, + "content": "denotes the entropy.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 717, + 380, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 155, + 96 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 155, + 84, + 201, + 95 + ], + "score": 0.94, + "content": "D _ { K L } ( \\cdot \\parallel \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 80, + 506, + 96 + ], + "score": 1.0, + "content": "denote the KL-divergence, which should be nonnegative, then we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 98, + 474, + 208 + ], + "lines": [ + { + "bbox": [ + 136, + 98, + 474, + 208 + ], + "spans": [ + { + "bbox": [ + 136, + 98, + 474, + 208 + ], + "score": 0.96, + "content": "\\begin{array} { r l } { I ( e ; W ) = H ( e ) - H ( e | W ) } & { } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log p ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log p ( e | W ) - \\log q ( e | W ) + \\log q ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ D _ { K L } ( p ( e | W ) \\| q ( e | W ) ) + \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log q ( e | W ) ] ] } \\\\ { \\ } & { \\geq H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log q ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) , W \\sim p ( W | e ) } [ \\log q ( e | W ) ] } \\\\ { \\ } & { \\stackrel { ( a ) } { = } H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) } [ \\log q ( e | f ( e ) ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "d1f9d2080ad50a36557e6e9a92ee313414557012e82f952539a8c31f6ce4b4bc.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 136, + 98, + 474, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 136, + 134.66666666666666, + 474, + 171.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 136, + 171.33333333333331, + 474, + 207.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 297, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 299, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 194, + 226 + ], + "score": 1.0, + "content": "where the equality (", + "type": "text" + }, + { + "bbox": [ + 194, + 215, + 200, + 221 + ], + "score": 0.4, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 208, + 299, + 226 + ], + "score": 1.0, + "content": ") follows from Eq. (5).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 444, + 248 + ], + "score": 1.0, + "content": "According to Theorem 1, we can maximize the variational lower bound for", + "type": "text" + }, + { + "bbox": [ + 444, + 236, + 478, + 247 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 234, + 506, + 248 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 344, + 258 + ], + "score": 1.0, + "content": "bound becomes tight when the variational distribution", + "type": "text" + }, + { + "bbox": [ + 344, + 247, + 375, + 258 + ], + "score": 0.94, + "content": "q ( e | W )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "approaches the true posterior", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 107, + 258, + 138, + 269 + ], + "score": 0.93, + "content": "p ( e | W )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 256, + 193, + 270 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 193, + 258, + 215, + 269 + ], + "score": 0.94, + "content": "H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 256, + 488, + 270 + ], + "score": 1.0, + "content": "is a constant because the distribution of edge feature vector", + "type": "text" + }, + { + "bbox": [ + 488, + 262, + 493, + 266 + ], + "score": 0.88, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 361, + 280 + ], + "score": 1.0, + "content": "fixed for given graphs, hence we can equivalently maximize", + "type": "text" + }, + { + "bbox": [ + 362, + 269, + 405, + 280 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 267, + 505, + 280 + ], + "score": 1.0, + "content": ". We choose the widely", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 277, + 490, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 277, + 479, + 293 + ], + "score": 1.0, + "content": "accepted Gaussian distribution as the prior distribution for the probabilistic decoder", + "type": "text" + }, + { + "bbox": [ + 479, + 283, + 485, + 290 + ], + "score": 0.89, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 277, + 490, + 293 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 293, + 365, + 307 + ], + "lines": [ + { + "bbox": [ + 246, + 293, + 365, + 307 + ], + "spans": [ + { + "bbox": [ + 246, + 293, + 365, + 307 + ], + "score": 0.91, + "content": "q ( e | W ) = \\mathcal { N } ( e ; g ( W ) , \\sigma ^ { 2 } I ) .", + "type": "interline_equation", + "image_path": "f8a4483a3180430a353a9084a23a720e3b8ae44919cce3c219973627c201c39d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 246, + 293, + 365, + 307 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 168, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 170, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 170, + 323 + ], + "score": 1.0, + "content": "Then we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 325, + 459, + 339 + ], + "lines": [ + { + "bbox": [ + 153, + 325, + 459, + 339 + ], + "spans": [ + { + "bbox": [ + 153, + 325, + 459, + 339 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { I } ( f , g ; e ) = \\log q ( e | f ( e ) ) = \\log \\mathcal { N } ( e ; g ( f ( e ) ) , \\sigma ^ { 2 } I ) = - \\lambda \\| e - g ( f ( e ) ) \\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "f9aff49d6e2e211ee2f081b682d6476c98870ba18a9bff3393ceb13b8bcdb085.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 153, + 325, + 459, + 339 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 136, + 355 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 137, + 345, + 164, + 353 + ], + "score": 0.92, + "content": "\\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 343, + 297, + 355 + ], + "score": 1.0, + "content": "is a constant determined by", + "type": "text" + }, + { + "bbox": [ + 298, + 348, + 304, + 352 + ], + "score": 0.9, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 343, + 407, + 355 + ], + "score": 1.0, + "content": "and the dimension of", + "type": "text" + }, + { + "bbox": [ + 407, + 348, + 412, + 352 + ], + "score": 0.86, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 343, + 505, + 355 + ], + "score": 1.0, + "content": ", taken as a tunable", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "parameter. The following Theorem 2 shows that maximizing the objective in Eq. (8) does", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 230, + 378 + ], + "score": 1.0, + "content": "lead to the maximization of", + "type": "text" + }, + { + "bbox": [ + 231, + 366, + 264, + 377 + ], + "score": 0.95, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 363, + 506, + 378 + ], + "score": 1.0, + "content": ", hence enables the model to preserve edge information.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 503, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 503, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 362, + 392 + ], + "score": 1.0, + "content": "Theorem 2. Assume the optimal solution of maximizing", + "type": "text" + }, + { + "bbox": [ + 363, + 380, + 406, + 391 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 376, + 420, + 392 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 420, + 381, + 431, + 390 + ], + "score": 0.9, + "content": "f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 376, + 454, + 392 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 454, + 381, + 464, + 390 + ], + "score": 0.89, + "content": "g ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 376, + 492, + 392 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 493, + 381, + 503, + 390 + ], + "score": 0.9, + "content": "f ^ { \\star }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 389, + 212, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 174, + 402 + ], + "score": 1.0, + "content": "also maximizes", + "type": "text" + }, + { + "bbox": [ + 175, + 392, + 208, + 402 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 389, + 212, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 104, + 413, + 503, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 184, + 428 + ], + "score": 1.0, + "content": "Proof. Note that", + "type": "text" + }, + { + "bbox": [ + 185, + 415, + 206, + 426 + ], + "score": 0.94, + "content": "H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 411, + 505, + 428 + ], + "score": 1.0, + "content": "is a constant when the graphs are given. In information theory, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 423, + 131, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 131, + 438 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 435, + 374, + 447 + ], + "lines": [ + { + "bbox": [ + 237, + 435, + 374, + 447 + ], + "spans": [ + { + "bbox": [ + 237, + 435, + 374, + 447 + ], + "score": 0.9, + "content": "H ( g ( f ( e ) ) ) \\leq H ( f ( e ) ) \\leq H ( e ) .", + "type": "interline_equation", + "image_path": "2a778d2e9fad80245dd43235b7a2babb5068136039b0572e8acfda673570fcb7.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 237, + 435, + 374, + 447 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 449, + 262, + 461 + ], + "lines": [ + { + "bbox": [ + 107, + 447, + 263, + 464 + ], + "spans": [ + { + "bbox": [ + 107, + 451, + 140, + 461 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 447, + 236, + 464 + ], + "score": 1.0, + "content": "is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 236, + 451, + 258, + 461 + ], + "score": 0.93, + "content": "H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 447, + 263, + 464 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 466, + 443, + 478 + ], + "lines": [ + { + "bbox": [ + 167, + 466, + 443, + 478 + ], + "spans": [ + { + "bbox": [ + 167, + 466, + 443, + 478 + ], + "score": 0.87, + "content": "I ( e ; W ) = I ( e ; f ( e ) ) = H ( f ( e ) ) - H ( f ( e ) | e ) = H ( f ( e ) ) \\leq H ( e ) .", + "type": "interline_equation", + "image_path": "a305eb6f7f8a26722c22a657ceb5288b473051298c22bbc93edee7a5d363b310.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 167, + 466, + 443, + 478 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 133, + 494 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 133, + 483, + 144, + 493 + ], + "score": 0.91, + "content": "f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 480, + 166, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 167, + 484, + 176, + 493 + ], + "score": 0.92, + "content": "g ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 480, + 346, + 494 + ], + "score": 1.0, + "content": "is the optimal solution of maximizing", + "type": "text" + }, + { + "bbox": [ + 346, + 483, + 390, + 493 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "presented in Eq. (8), it is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 210, + 505 + ], + "score": 1.0, + "content": "not difficult to see that", + "type": "text" + }, + { + "bbox": [ + 210, + 494, + 302, + 505 + ], + "score": 0.95, + "content": "e = g ^ { \\star } ( f ^ { \\star } ( e ) ) , \\forall e \\in \\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 492, + 505, + 505 + ], + "score": 1.0, + "content": ". In this case, the inequalities in Eq. 9 become", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 503, + 172, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 172, + 517 + ], + "score": 1.0, + "content": "equalities, i.e.,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 515, + 378, + 527 + ], + "lines": [ + { + "bbox": [ + 231, + 515, + 378, + 527 + ], + "spans": [ + { + "bbox": [ + 231, + 515, + 378, + 527 + ], + "score": 0.89, + "content": "H ( g ^ { \\star } ( f ^ { \\star } ( e ) ) ) = H ( f ^ { \\star } ( e ) ) = H ( e ) .", + "type": "interline_equation", + "image_path": "809dfe6268777d2ba705ed212906089485ef5b8dcf78fafc8f140c20b3eecaed.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 231, + 515, + 378, + 527 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 450, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 451, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 192, + 541 + ], + "score": 1.0, + "content": "Therefore, we have", + "type": "text" + }, + { + "bbox": [ + 192, + 530, + 260, + 540 + ], + "score": 0.94, + "content": "I ( e ; W ) = H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 527, + 451, + 541 + ], + "score": 1.0, + "content": ", i.e., the maximum is attained in this case.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 551, + 412, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 413, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 413, + 566 + ], + "score": 1.0, + "content": "3.3 Edge Information Maximized Graph Neural Networks", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "Our EIGNN is derived by implementing our MI objective in GNNs. As a concrete example,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "the forward propagation of our model follows the formulation in Eq. (3), where the dege", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 595, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 144, + 607 + ], + "score": 1.0, + "content": "network", + "type": "text" + }, + { + "bbox": [ + 144, + 597, + 189, + 606 + ], + "score": 0.92, + "content": "f : e W", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 595, + 504, + 607 + ], + "score": 1.0, + "content": "is expressed as a multi-layer perceptron (MLP). According to theoretical", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 606, + 479, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 364, + 618 + ], + "score": 1.0, + "content": "analysis presented in Sec. 3.2 , we introduce another MLP", + "type": "text" + }, + { + "bbox": [ + 364, + 608, + 409, + 617 + ], + "score": 0.93, + "content": "g : W \\to e", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 606, + 479, + 618 + ], + "score": 1.0, + "content": "as the decoder.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 504, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 432, + 635 + ], + "score": 1.0, + "content": "For graph regression or classification tasks, the model outputs a prediction", + "type": "text" + }, + { + "bbox": [ + 433, + 627, + 438, + 634 + ], + "score": 0.89, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 622, + 504, + 635 + ], + "score": 1.0, + "content": "for each graph", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 636, + 115, + 643 + ], + "score": 0.88, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 632, + 192, + 646 + ], + "score": 1.0, + "content": ", which has label", + "type": "text" + }, + { + "bbox": [ + 192, + 636, + 198, + 645 + ], + "score": 0.89, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 632, + 455, + 646 + ], + "score": 1.0, + "content": ". Without MI maximization, we denote the vanilla loss as", + "type": "text" + }, + { + "bbox": [ + 456, + 635, + 502, + 646 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 632, + 506, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 188, + 657 + ], + "score": 1.0, + "content": "Common choice of", + "type": "text" + }, + { + "bbox": [ + 189, + 647, + 200, + 656 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "includes Mean Square Error (MSE), Mean Absolute Error (MAE) and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 255, + 669 + ], + "score": 1.0, + "content": "Cross Entropy (CE). For a graph", + "type": "text" + }, + { + "bbox": [ + 256, + 658, + 302, + 668 + ], + "score": 0.94, + "content": "G = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 654, + 393, + 669 + ], + "score": 1.0, + "content": ", EIGNN maximizes", + "type": "text" + }, + { + "bbox": [ + 394, + 657, + 437, + 668 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "and minimizes", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 667, + 298, + 679 + ], + "spans": [ + { + "bbox": [ + 107, + 668, + 154, + 678 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 667, + 298, + 679 + ], + "score": 1.0, + "content": "using the following loss function", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 682, + 408, + 695 + ], + "lines": [ + { + "bbox": [ + 203, + 682, + 408, + 695 + ], + "spans": [ + { + "bbox": [ + 203, + 682, + 408, + 695 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { E I G N N } ( G ) = \\mathcal { L } _ { 0 } ( \\hat { y } , y ; G ) - \\lambda \\mathbb { E } _ { e \\in \\mathcal { E } } [ \\mathcal { L } _ { I } ( f , g ; e ) ] ,", + "type": "interline_equation", + "image_path": "e9ffed3d9ec064286879e4ca1479706a35dcaf6e9fa768bf67f604003845c77f.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 203, + 682, + 408, + 695 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 134, + 712 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 700, + 164, + 711 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { e \\in \\mathcal { E } } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 698, + 343, + 712 + ], + "score": 1.0, + "content": "denotes taking the mean over all edges in", + "type": "text" + }, + { + "bbox": [ + 343, + 700, + 389, + 711 + ], + "score": 0.93, + "content": "G = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 698, + 410, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 410, + 701, + 416, + 708 + ], + "score": 0.89, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "is the regularization", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 366, + 722 + ], + "score": 1.0, + "content": "parameter. When EIGNN is trained using mini-batches,", + "type": "text" + }, + { + "bbox": [ + 366, + 712, + 413, + 722 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is averaged over all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 495, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 221, + 733 + ], + "score": 1.0, + "content": "graphs in the batch while", + "type": "text" + }, + { + "bbox": [ + 221, + 722, + 265, + 732 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 720, + 495, + 733 + ], + "score": 1.0, + "content": "is averaged over all edges of all graphs in the batch.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 327, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 327, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 327, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 528, + 504, + 539 + ], + "lines": [ + { + "bbox": [ + 496, + 530, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 496, + 530, + 504, + 540 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 211, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 495, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 495, + 212, + 505, + 223 + ], + "score": 0.996, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 155, + 96 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 155, + 84, + 201, + 95 + ], + "score": 0.94, + "content": "D _ { K L } ( \\cdot \\parallel \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 80, + 506, + 96 + ], + "score": 1.0, + "content": "denote the KL-divergence, which should be nonnegative, then we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 80, + 506, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 98, + 474, + 208 + ], + "lines": [ + { + "bbox": [ + 136, + 98, + 474, + 208 + ], + "spans": [ + { + "bbox": [ + 136, + 98, + 474, + 208 + ], + "score": 0.96, + "content": "\\begin{array} { r l } { I ( e ; W ) = H ( e ) - H ( e | W ) } & { } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log p ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log p ( e | W ) - \\log q ( e | W ) + \\log q ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ D _ { K L } ( p ( e | W ) \\| q ( e | W ) ) + \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log q ( e | W ) ] ] } \\\\ { \\ } & { \\geq H ( e ) + \\mathbb { E } _ { W \\sim p ( W ) } [ \\mathbb { E } _ { e \\sim p ( e | W ) } [ \\log q ( e | W ) ] ] } \\\\ { \\ } & { = H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) , W \\sim p ( W | e ) } [ \\log q ( e | W ) ] } \\\\ { \\ } & { \\stackrel { ( a ) } { = } H ( e ) + \\mathbb { E } _ { e \\sim p ( e ) } [ \\log q ( e | f ( e ) ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "d1f9d2080ad50a36557e6e9a92ee313414557012e82f952539a8c31f6ce4b4bc.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 136, + 98, + 474, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 136, + 134.66666666666666, + 474, + 171.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 136, + 171.33333333333331, + 474, + 207.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 297, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 299, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 194, + 226 + ], + "score": 1.0, + "content": "where the equality (", + "type": "text" + }, + { + "bbox": [ + 194, + 215, + 200, + 221 + ], + "score": 0.4, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 208, + 299, + 226 + ], + "score": 1.0, + "content": ") follows from Eq. (5).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 208, + 299, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 444, + 248 + ], + "score": 1.0, + "content": "According to Theorem 1, we can maximize the variational lower bound for", + "type": "text" + }, + { + "bbox": [ + 444, + 236, + 478, + 247 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 234, + 506, + 248 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 344, + 258 + ], + "score": 1.0, + "content": "bound becomes tight when the variational distribution", + "type": "text" + }, + { + "bbox": [ + 344, + 247, + 375, + 258 + ], + "score": 0.94, + "content": "q ( e | W )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "approaches the true posterior", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 107, + 258, + 138, + 269 + ], + "score": 0.93, + "content": "p ( e | W )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 256, + 193, + 270 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 193, + 258, + 215, + 269 + ], + "score": 0.94, + "content": "H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 256, + 488, + 270 + ], + "score": 1.0, + "content": "is a constant because the distribution of edge feature vector", + "type": "text" + }, + { + "bbox": [ + 488, + 262, + 493, + 266 + ], + "score": 0.88, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 361, + 280 + ], + "score": 1.0, + "content": "fixed for given graphs, hence we can equivalently maximize", + "type": "text" + }, + { + "bbox": [ + 362, + 269, + 405, + 280 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 267, + 505, + 280 + ], + "score": 1.0, + "content": ". We choose the widely", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 277, + 490, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 277, + 479, + 293 + ], + "score": 1.0, + "content": "accepted Gaussian distribution as the prior distribution for the probabilistic decoder", + "type": "text" + }, + { + "bbox": [ + 479, + 283, + 485, + 290 + ], + "score": 0.89, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 277, + 490, + 293 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 234, + 506, + 293 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 293, + 365, + 307 + ], + "lines": [ + { + "bbox": [ + 246, + 293, + 365, + 307 + ], + "spans": [ + { + "bbox": [ + 246, + 293, + 365, + 307 + ], + "score": 0.91, + "content": "q ( e | W ) = \\mathcal { N } ( e ; g ( W ) , \\sigma ^ { 2 } I ) .", + "type": "interline_equation", + "image_path": "f8a4483a3180430a353a9084a23a720e3b8ae44919cce3c219973627c201c39d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 246, + 293, + 365, + 307 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 168, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 170, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 170, + 323 + ], + "score": 1.0, + "content": "Then we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 309, + 170, + 323 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 325, + 459, + 339 + ], + "lines": [ + { + "bbox": [ + 153, + 325, + 459, + 339 + ], + "spans": [ + { + "bbox": [ + 153, + 325, + 459, + 339 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { I } ( f , g ; e ) = \\log q ( e | f ( e ) ) = \\log \\mathcal { N } ( e ; g ( f ( e ) ) , \\sigma ^ { 2 } I ) = - \\lambda \\| e - g ( f ( e ) ) \\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "f9aff49d6e2e211ee2f081b682d6476c98870ba18a9bff3393ceb13b8bcdb085.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 153, + 325, + 459, + 339 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 136, + 355 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 137, + 345, + 164, + 353 + ], + "score": 0.92, + "content": "\\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 343, + 297, + 355 + ], + "score": 1.0, + "content": "is a constant determined by", + "type": "text" + }, + { + "bbox": [ + 298, + 348, + 304, + 352 + ], + "score": 0.9, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 343, + 407, + 355 + ], + "score": 1.0, + "content": "and the dimension of", + "type": "text" + }, + { + "bbox": [ + 407, + 348, + 412, + 352 + ], + "score": 0.86, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 343, + 505, + 355 + ], + "score": 1.0, + "content": ", taken as a tunable", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "parameter. The following Theorem 2 shows that maximizing the objective in Eq. (8) does", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 230, + 378 + ], + "score": 1.0, + "content": "lead to the maximization of", + "type": "text" + }, + { + "bbox": [ + 231, + 366, + 264, + 377 + ], + "score": 0.95, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 363, + 506, + 378 + ], + "score": 1.0, + "content": ", hence enables the model to preserve edge information.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 343, + 506, + 378 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 503, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 503, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 362, + 392 + ], + "score": 1.0, + "content": "Theorem 2. Assume the optimal solution of maximizing", + "type": "text" + }, + { + "bbox": [ + 363, + 380, + 406, + 391 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 376, + 420, + 392 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 420, + 381, + 431, + 390 + ], + "score": 0.9, + "content": "f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 376, + 454, + 392 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 454, + 381, + 464, + 390 + ], + "score": 0.89, + "content": "g ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 376, + 492, + 392 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 493, + 381, + 503, + 390 + ], + "score": 0.9, + "content": "f ^ { \\star }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 389, + 212, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 174, + 402 + ], + "score": 1.0, + "content": "also maximizes", + "type": "text" + }, + { + "bbox": [ + 175, + 392, + 208, + 402 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 389, + 212, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 376, + 503, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 413, + 503, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 184, + 428 + ], + "score": 1.0, + "content": "Proof. Note that", + "type": "text" + }, + { + "bbox": [ + 185, + 415, + 206, + 426 + ], + "score": 0.94, + "content": "H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 411, + 505, + 428 + ], + "score": 1.0, + "content": "is a constant when the graphs are given. In information theory, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 423, + 131, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 131, + 438 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 411, + 505, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 435, + 374, + 447 + ], + "lines": [ + { + "bbox": [ + 237, + 435, + 374, + 447 + ], + "spans": [ + { + "bbox": [ + 237, + 435, + 374, + 447 + ], + "score": 0.9, + "content": "H ( g ( f ( e ) ) ) \\leq H ( f ( e ) ) \\leq H ( e ) .", + "type": "interline_equation", + "image_path": "2a778d2e9fad80245dd43235b7a2babb5068136039b0572e8acfda673570fcb7.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 237, + 435, + 374, + 447 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 449, + 262, + 461 + ], + "lines": [ + { + "bbox": [ + 107, + 447, + 263, + 464 + ], + "spans": [ + { + "bbox": [ + 107, + 451, + 140, + 461 + ], + "score": 0.94, + "content": "I ( e ; W )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 447, + 236, + 464 + ], + "score": 1.0, + "content": "is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 236, + 451, + 258, + 461 + ], + "score": 0.93, + "content": "H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 447, + 263, + 464 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 107, + 447, + 263, + 464 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 466, + 443, + 478 + ], + "lines": [ + { + "bbox": [ + 167, + 466, + 443, + 478 + ], + "spans": [ + { + "bbox": [ + 167, + 466, + 443, + 478 + ], + "score": 0.87, + "content": "I ( e ; W ) = I ( e ; f ( e ) ) = H ( f ( e ) ) - H ( f ( e ) | e ) = H ( f ( e ) ) \\leq H ( e ) .", + "type": "interline_equation", + "image_path": "a305eb6f7f8a26722c22a657ceb5288b473051298c22bbc93edee7a5d363b310.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 167, + 466, + 443, + 478 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 133, + 494 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 133, + 483, + 144, + 493 + ], + "score": 0.91, + "content": "f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 480, + 166, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 167, + 484, + 176, + 493 + ], + "score": 0.92, + "content": "g ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 480, + 346, + 494 + ], + "score": 1.0, + "content": "is the optimal solution of maximizing", + "type": "text" + }, + { + "bbox": [ + 346, + 483, + 390, + 493 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "presented in Eq. (8), it is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 210, + 505 + ], + "score": 1.0, + "content": "not difficult to see that", + "type": "text" + }, + { + "bbox": [ + 210, + 494, + 302, + 505 + ], + "score": 0.95, + "content": "e = g ^ { \\star } ( f ^ { \\star } ( e ) ) , \\forall e \\in \\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 492, + 505, + 505 + ], + "score": 1.0, + "content": ". In this case, the inequalities in Eq. 9 become", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 503, + 172, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 172, + 517 + ], + "score": 1.0, + "content": "equalities, i.e.,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 480, + 506, + 517 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 515, + 378, + 527 + ], + "lines": [ + { + "bbox": [ + 231, + 515, + 378, + 527 + ], + "spans": [ + { + "bbox": [ + 231, + 515, + 378, + 527 + ], + "score": 0.89, + "content": "H ( g ^ { \\star } ( f ^ { \\star } ( e ) ) ) = H ( f ^ { \\star } ( e ) ) = H ( e ) .", + "type": "interline_equation", + "image_path": "809dfe6268777d2ba705ed212906089485ef5b8dcf78fafc8f140c20b3eecaed.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 231, + 515, + 378, + 527 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 450, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 451, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 192, + 541 + ], + "score": 1.0, + "content": "Therefore, we have", + "type": "text" + }, + { + "bbox": [ + 192, + 530, + 260, + 540 + ], + "score": 0.94, + "content": "I ( e ; W ) = H ( e )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 527, + 451, + 541 + ], + "score": 1.0, + "content": ", i.e., the maximum is attained in this case.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 527, + 451, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 551, + 412, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 413, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 413, + 566 + ], + "score": 1.0, + "content": "3.3 Edge Information Maximized Graph Neural Networks", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 551, + 413, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "Our EIGNN is derived by implementing our MI objective in GNNs. As a concrete example,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "the forward propagation of our model follows the formulation in Eq. (3), where the dege", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 595, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 144, + 607 + ], + "score": 1.0, + "content": "network", + "type": "text" + }, + { + "bbox": [ + 144, + 597, + 189, + 606 + ], + "score": 0.92, + "content": "f : e W", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 595, + 504, + 607 + ], + "score": 1.0, + "content": "is expressed as a multi-layer perceptron (MLP). According to theoretical", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 606, + 479, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 364, + 618 + ], + "score": 1.0, + "content": "analysis presented in Sec. 3.2 , we introduce another MLP", + "type": "text" + }, + { + "bbox": [ + 364, + 608, + 409, + 617 + ], + "score": 0.93, + "content": "g : W \\to e", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 606, + 479, + 618 + ], + "score": 1.0, + "content": "as the decoder.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 572, + 506, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 504, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 432, + 635 + ], + "score": 1.0, + "content": "For graph regression or classification tasks, the model outputs a prediction", + "type": "text" + }, + { + "bbox": [ + 433, + 627, + 438, + 634 + ], + "score": 0.89, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 622, + 504, + 635 + ], + "score": 1.0, + "content": "for each graph", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 636, + 115, + 643 + ], + "score": 0.88, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 632, + 192, + 646 + ], + "score": 1.0, + "content": ", which has label", + "type": "text" + }, + { + "bbox": [ + 192, + 636, + 198, + 645 + ], + "score": 0.89, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 632, + 455, + 646 + ], + "score": 1.0, + "content": ". Without MI maximization, we denote the vanilla loss as", + "type": "text" + }, + { + "bbox": [ + 456, + 635, + 502, + 646 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 632, + 506, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 188, + 657 + ], + "score": 1.0, + "content": "Common choice of", + "type": "text" + }, + { + "bbox": [ + 189, + 647, + 200, + 656 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "includes Mean Square Error (MSE), Mean Absolute Error (MAE) and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 255, + 669 + ], + "score": 1.0, + "content": "Cross Entropy (CE). For a graph", + "type": "text" + }, + { + "bbox": [ + 256, + 658, + 302, + 668 + ], + "score": 0.94, + "content": "G = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 654, + 393, + 669 + ], + "score": 1.0, + "content": ", EIGNN maximizes", + "type": "text" + }, + { + "bbox": [ + 394, + 657, + 437, + 668 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { I } ( f , g ; e )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "and minimizes", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 667, + 298, + 679 + ], + "spans": [ + { + "bbox": [ + 107, + 668, + 154, + 678 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { 0 } ( \\hat { y } , y ; G )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 667, + 298, + 679 + ], + "score": 1.0, + "content": "using the following loss function", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 622, + 506, + 679 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 682, + 408, + 695 + ], + "lines": [ + { + "bbox": [ + 203, + 682, + 408, + 695 + ], + "spans": [ + { + "bbox": [ + 203, + 682, + 408, + 695 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { E I G N N } ( G ) = \\mathcal { L } _ { 0 } ( \\hat { y } , y ; G ) - \\lambda \\mathbb { E } _ { e \\in \\mathcal { E } } [ \\mathcal { L } _ { I } ( f , g ; e ) ] ,", + "type": "interline_equation", + "image_path": "e9ffed3d9ec064286879e4ca1479706a35dcaf6e9fa768bf67f604003845c77f.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 203, + 682, + 408, + 695 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 134, + 712 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 700, + 164, + 711 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { e \\in \\mathcal { E } } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 698, + 343, + 712 + ], + "score": 1.0, + "content": "denotes taking the mean over all edges in", + "type": "text" + }, + { + "bbox": [ + 343, + 700, + 389, + 711 + ], + "score": 0.93, + "content": "G = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 698, + 410, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 410, + 701, + 416, + 708 + ], + "score": 0.89, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "is the regularization", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 366, + 722 + ], + "score": 1.0, + "content": "parameter. 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The objective function of EIGNN can be derived from the translational scoring", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "function Bordes et al. (2013), which learns embedding such that for a given valid triple", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 107, + 128, + 190, + 138 + ], + "score": 0.91, + "content": "t _ { v w } = ( h _ { v } , e _ { v w } , h _ { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 126, + 271, + 140 + ], + "score": 1.0, + "content": "from the valid set", + "type": "text" + }, + { + "bbox": [ + 272, + 129, + 279, + 136 + ], + "score": 0.89, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 126, + 344, + 140 + ], + "score": 1.0, + "content": ", the condition", + "type": "text" + }, + { + "bbox": [ + 345, + 129, + 457, + 138 + ], + "score": 0.92, + "content": "d _ { t _ { v w } } = h _ { v } + e _ { v w } - h _ { w } \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "holds. Let", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 136, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 301, + 150 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { 0 } = \\mathbb { E } _ { t _ { v w } \\in S } \\mathbb { E } _ { t _ { v w } ^ { \\prime } \\in S ^ { \\prime } } \\operatorname* { m a x } \\{ d _ { t _ { v w } ^ { \\prime } } - d _ { t _ { v w } } + \\gamma , 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 136, + 334, + 151 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 334, + 139, + 344, + 147 + ], + "score": 0.91, + "content": "S ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 136, + 497, + 151 + ], + "score": 1.0, + "content": "denotes a set of invalid triples and", + "type": "text" + }, + { + "bbox": [ + 497, + 142, + 504, + 149 + ], + "score": 0.89, + "content": "\\gamma", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 484, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 484, + 161 + ], + "score": 1.0, + "content": "is a margin hyper-parameter, EIGNN can be trained by minimizing the following loss,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 166, + 391, + 179 + ], + "lines": [ + { + "bbox": [ + 220, + 166, + 391, + 179 + ], + "spans": [ + { + "bbox": [ + 220, + 166, + 391, + 179 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { E I G N N } ( G ) = \\mathcal { L } _ { 0 } - \\lambda \\mathbb { E } _ { e \\in \\mathcal { E } } [ \\mathcal { L } _ { I } ( f , g ; e ) ] .", + "type": "interline_equation", + "image_path": "6b4942e4c4e30ad4a1ef853dea11681cee7cc1ec7a668a18ce808e38004cbe7e.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 220, + 166, + 391, + 179 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 194, + 204, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 205, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 205, + 209 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "In this section, we first conduct experiments on a large quantum chemistry benchmark QM9,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "which is challenging for most baselines. Then we evaluate EIGNN on several useful molecule", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "benchmarks and use attribution analysis to show that EIGNN increases the impact of edges", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "and captures domain knowledge without human interference. Finally, we adopt our method to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 263, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 504, + 276 + ], + "score": 1.0, + "content": "large-scale knowledge graphs and evaluate the performance on challenging relation prediction", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "tasks using a wide variety of real-world datasets. All experimental results demonstrate a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 286, + 457, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 457, + 297 + ], + "score": 1.0, + "content": "clear and substantial improvement of EIGNN over the state-of-the-art methods.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 310, + 234, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 236, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 236, + 324 + ], + "score": 1.0, + "content": "4.1 Quantum Chemistry", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "QM9 (Ramakrishnan et al., 2014) is a large benchmark containing 134k molecules with 12", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "quantum chemistry regression properties, which have been show to be quite challenging", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 504, + 366 + ], + "score": 1.0, + "content": "for many GNNs (Gilmer et al., 2017). Feature engineering of nodes and edges exactly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "follows (Gilmer et al., 2017) such that molecules are preprocessed as graphs according to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "atom features and bond features. We compare our EIGNN with nine state-of-the-art baselines", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "which can be categorized into three groups according to the ability of handling edge features:", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "i) GCN, ChebyNet, GAT and GIN (Xu et al., 2019) which simply use binary edge features", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "to indicate the existence of a bond without any other edge features; ii) RGCN, GGNN, LNet", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "and simplified MPNN (sMPNN) which consider bond types (no bond, single, double, triple,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "or aromatic); iii) MPNN and our EIGNN which use edge feature vectors to indicate both", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 441, + 322, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 322, + 453 + ], + "score": 1.0, + "content": "edge types and pairwise distance between atoms.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "For a fair comparison, we repeat all experiments 3 times with different random seeds while", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "during each run, all methods share the same random seed. We randomly choose 10k", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "molecules for validation, 10k molecules for testing, and keep the rest for training. Each target", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "property is normalized to zero mean and unit variance for training. Each model is trained to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 324, + 515 + ], + "score": 1.0, + "content": "predict the 12 target properties simultaneously.", + "type": "text" + }, + { + "bbox": [ + 324, + 504, + 330, + 511 + ], + "score": 0.88, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "is naively set to 1 for EIGNN. We use", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "score": 1.0, + "content": "mean square error (MSE) loss to train the models for at most 300 epochs till convergence,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "and the performance is measured by mean absolute error (MAE). For LNet and GGNN,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "implementation of the readout function follows the original paper. While for all other models,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "table", + "bbox": [ + 108, + 604, + 502, + 729 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 560, + 505, + 604 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "Table 1: Quantum property regressions for 12 targets and overall performance (top two raws)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "on QM9. We repeat all experiments 3 times with different random seeds and report the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 581, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 507, + 596 + ], + "score": 1.0, + "content": "average performance. 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MethodGCNChebyNetGATGINRGCNGGNNLNetsMPNNMPNNEIGNN
Avg.nMAE0.1350.1210.1370.1000.1020.0990.0990.0890.0400.036
Avg.MAE5.3064.3035.4703.4803.8173.6613.6533.1610.6930.633
mu0.5680.5180.5670.4780.5060.5180.4720.4720.1100.097
alpha0.8810.7930.8910.6210.6320.6080.6230.5280.3320.294
HOMO(10-3)5.4514.7755.4294.1834.4534.4833.8893.8542.4812.230
LUMO(10-3)6.4005.6746.3314.7965.1385.1534.1944.5492.8622.593
gap(10-3)8.2017.0978.1936.0966.5006.6025.8135.6343.6203.275
R253.5641.9554.5234.6540.1039.6835.2733.496.0645.646
ZPVE(10-3)2.5332.5272.2711.7441.4771.2921.4381.3450.6790.612
UO2.0421.9842.2901.4221.0590.6971.8060.7910.4160.357
U2.0421.9842.2901.4221.0590.6971.7550.7910.4160.357
H2.0421.9842.2901.4221.0590.6971.7960.7910.4160.357
G2.0421.9842.2901.4221.0590.6961.7780.7910.4160.357
Cv0.4730.4200.4790.3090.3170.3150.3120.2620.1340.121
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(2013), which learns embedding such that for a given valid triple", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 107, + 128, + 190, + 138 + ], + "score": 0.91, + "content": "t _ { v w } = ( h _ { v } , e _ { v w } , h _ { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 126, + 271, + 140 + ], + "score": 1.0, + "content": "from the valid set", + "type": "text" + }, + { + "bbox": [ + 272, + 129, + 279, + 136 + ], + "score": 0.89, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 126, + 344, + 140 + ], + "score": 1.0, + "content": ", the condition", + "type": "text" + }, + { + "bbox": [ + 345, + 129, + 457, + 138 + ], + "score": 0.92, + "content": "d _ { t _ { v w } } = h _ { v } + e _ { v w } - h _ { w } \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "holds. Let", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 136, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 301, + 150 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { 0 } = \\mathbb { E } _ { t _ { v w } \\in S } \\mathbb { E } _ { t _ { v w } ^ { \\prime } \\in S ^ { \\prime } } \\operatorname* { m a x } \\{ d _ { t _ { v w } ^ { \\prime } } - d _ { t _ { v w } } + \\gamma , 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 136, + 334, + 151 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 334, + 139, + 344, + 147 + ], + "score": 0.91, + "content": "S ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 136, + 497, + 151 + ], + "score": 1.0, + "content": "denotes a set of invalid triples and", + "type": "text" + }, + { + "bbox": [ + 497, + 142, + 504, + 149 + ], + "score": 0.89, + "content": "\\gamma", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 484, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 484, + 161 + ], + "score": 1.0, + "content": "is a margin hyper-parameter, EIGNN can be trained by minimizing the following loss,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 81, + 506, + 161 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 166, + 391, + 179 + ], + "lines": [ + { + "bbox": [ + 220, + 166, + 391, + 179 + ], + "spans": [ + { + "bbox": [ + 220, + 166, + 391, + 179 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { E I G N N } ( G ) = \\mathcal { L } _ { 0 } - \\lambda \\mathbb { E } _ { e \\in \\mathcal { E } } [ \\mathcal { L } _ { I } ( f , g ; e ) ] .", + "type": "interline_equation", + "image_path": "6b4942e4c4e30ad4a1ef853dea11681cee7cc1ec7a668a18ce808e38004cbe7e.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 220, + 166, + 391, + 179 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 194, + 204, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 205, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 205, + 209 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "In this section, we first conduct experiments on a large quantum chemistry benchmark QM9,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "which is challenging for most baselines. Then we evaluate EIGNN on several useful molecule", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "benchmarks and use attribution analysis to show that EIGNN increases the impact of edges", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "and captures domain knowledge without human interference. Finally, we adopt our method to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 263, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 504, + 276 + ], + "score": 1.0, + "content": "large-scale knowledge graphs and evaluate the performance on challenging relation prediction", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "tasks using a wide variety of real-world datasets. All experimental results demonstrate a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 286, + 457, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 457, + 297 + ], + "score": 1.0, + "content": "clear and substantial improvement of EIGNN over the state-of-the-art methods.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 219, + 506, + 297 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 310, + 234, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 236, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 236, + 324 + ], + "score": 1.0, + "content": "4.1 Quantum Chemistry", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "QM9 (Ramakrishnan et al., 2014) is a large benchmark containing 134k molecules with 12", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "quantum chemistry regression properties, which have been show to be quite challenging", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 504, + 366 + ], + "score": 1.0, + "content": "for many GNNs (Gilmer et al., 2017). Feature engineering of nodes and edges exactly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "follows (Gilmer et al., 2017) such that molecules are preprocessed as graphs according to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "atom features and bond features. We compare our EIGNN with nine state-of-the-art baselines", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "which can be categorized into three groups according to the ability of handling edge features:", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "i) GCN, ChebyNet, GAT and GIN (Xu et al., 2019) which simply use binary edge features", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "to indicate the existence of a bond without any other edge features; ii) RGCN, GGNN, LNet", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "and simplified MPNN (sMPNN) which consider bond types (no bond, single, double, triple,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "or aromatic); iii) MPNN and our EIGNN which use edge feature vectors to indicate both", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 441, + 322, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 322, + 453 + ], + "score": 1.0, + "content": "edge types and pairwise distance between atoms.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 331, + 506, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "For a fair comparison, we repeat all experiments 3 times with different random seeds while", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "during each run, all methods share the same random seed. We randomly choose 10k", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "molecules for validation, 10k molecules for testing, and keep the rest for training. Each target", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "property is normalized to zero mean and unit variance for training. Each model is trained to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 324, + 515 + ], + "score": 1.0, + "content": "predict the 12 target properties simultaneously.", + "type": "text" + }, + { + "bbox": [ + 324, + 504, + 330, + 511 + ], + "score": 0.88, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "is naively set to 1 for EIGNN. We use", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "score": 1.0, + "content": "mean square error (MSE) loss to train the models for at most 300 epochs till convergence,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "and the performance is measured by mean absolute error (MAE). For LNet and GGNN,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "implementation of the readout function follows the original paper. While for all other models,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "score": 1.0, + "content": "we use the same set2set (Vinyals et al., 2015) readout, which has been demonstrated to work", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 239, + 232, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 232, + 252 + ], + "score": 1.0, + "content": "well in (Gilmer et al., 2017).", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 457, + 506, + 547 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 604, + 502, + 729 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 560, + 505, + 604 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "Table 1: Quantum property regressions for 12 targets and overall performance (top two raws)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "on QM9. We repeat all experiments 3 times with different random seeds and report the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 581, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 507, + 596 + ], + "score": 1.0, + "content": "average performance. 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MethodGCNChebyNetGATGINRGCNGGNNLNetsMPNNMPNNEIGNN
Avg.nMAE0.1350.1210.1370.1000.1020.0990.0990.0890.0400.036
Avg.MAE5.3064.3035.4703.4803.8173.6613.6533.1610.6930.633
mu0.5680.5180.5670.4780.5060.5180.4720.4720.1100.097
alpha0.8810.7930.8910.6210.6320.6080.6230.5280.3320.294
HOMO(10-3)5.4514.7755.4294.1834.4534.4833.8893.8542.4812.230
LUMO(10-3)6.4005.6746.3314.7965.1385.1534.1944.5492.8622.593
gap(10-3)8.2017.0978.1936.0966.5006.6025.8135.6343.6203.275
R253.5641.9554.5234.6540.1039.6835.2733.496.0645.646
ZPVE(10-3)2.5332.5272.2711.7441.4771.2921.4381.3450.6790.612
UO2.0421.9842.2901.4221.0590.6971.8060.7910.4160.357
U2.0421.9842.2901.4221.0590.6971.7550.7910.4160.357
H2.0421.9842.2901.4221.0590.6971.7960.7910.4160.357
G2.0421.9842.2901.4221.0590.6961.7780.7910.4160.357
Cv0.4730.4200.4790.3090.3170.3150.3120.2620.1340.121
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Training and validation error on QM9. The shadow area indicates", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 209, + 213, + 221 + ], + "spans": [ + { + "bbox": [ + 107, + 211, + 158, + 220 + ], + "score": 0.84, + "content": "m e a n \\pm s t d", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 209, + 213, + 221 + ], + "score": 1.0, + "content": "over 3 runs.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 503, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "score": 1.0, + "content": "we use the same set2set (Vinyals et al., 2015) readout, which has been demonstrated to work", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 239, + 232, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 232, + 252 + ], + "score": 1.0, + "content": "well in (Gilmer et al., 2017).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 256, + 504, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 504, + 268 + ], + "score": 1.0, + "content": "In Table 1, we list regression results for all methods. We report individual MAE for each", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "target in their original scale, averaged MAE (Avg. MAE) over 12 properties, and averaged", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "normalized MAE (Avg. nMAE; averaged over normalized target properties since different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "targets have different units and ranges). Our EIGNN achieves the best performance for each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "metric and each target. Now we are ready to answer the following research questions. i) Are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 310, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 325 + ], + "score": 1.0, + "content": "edge features important? Yes. The error has a trend of decreasing with increasing edge", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "score": 1.0, + "content": "features. The comparison between sMPNN (using edge types) and MPNN (using edge types", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 333, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 504, + 345 + ], + "score": 1.0, + "content": "and distance) directly verifies the importance of edge features. It is also consistent with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 344, + 504, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 504, + 356 + ], + "score": 1.0, + "content": "the expert knowledge that distances between pairwise atoms are closely related to quantum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "properties. For example, the smaller the distance between the two atoms, the stronger", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "the bond is, and consequently a higher bond energy is associated with this atom pair. ii)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 507, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 507, + 390 + ], + "score": 1.0, + "content": "Does the EIGNN work? Yes. EIGNN achieves the best performance on each target,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "outperforming the strong baseline MPNN. Moreover, the advantage of EIGNN over MPNN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "is consistent over 3 runs and the standard deviation on this task is quite small. Detailed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "results are shown in Table 4 of Appendix A. iii) How does the EIGNN work? Our MI", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "score": 1.0, + "content": "objective is easily implemented on top of vanilla loss function. We have shown that our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "objective enables preserving of edge information. Fig. 1 demonstrates that regularization", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 441, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 141, + 456 + ], + "score": 1.0, + "content": "such as", + "type": "text" + }, + { + "bbox": [ + 142, + 445, + 153, + 453 + ], + "score": 0.92, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 441, + 507, + 456 + ], + "score": 1.0, + "content": "weight decay can increase training error while our objective does not. Moreover,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "the validation performance verifies that regularization itself does not reduce the validation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "error. Thus, the effectiveness of EIGNN is due to exploiting edge features rather than the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "regularization effect. We further run an ablation study where we concatenate edge features", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "to node representations (i.e., MPNN+concat in Fig. 1) in message passing. Concatenation is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "unable to identify correlations between edges and nodes (Gilmer et al., 2017) and our results", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 508, + 465, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 465, + 521 + ], + "score": 1.0, + "content": "show that it slightly reduces the mean validation error but increases the variance.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 536, + 429, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 432, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 432, + 548 + ], + "score": 1.0, + "content": "4.2 More Molecule Benchmarks with Potential Applications", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "We further evaluate EIGNN on three molecule benchmarks: Lipophilicity (Wu et al., 2018),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "ESOL (Delaney, 2004) and FreeSolv (Mobley & Guthrie, 2014). These datasets contain fewer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "molecules, and have potential usages in applications such as chemistry, drug discovery, and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 604 + ], + "score": 1.0, + "content": "materials science. For example, the property lipophilicity is an important feature of drug", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 600, + 504, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 504, + 613 + ], + "score": 1.0, + "content": "molecules that affects both membrane permeability and solubility. The dataset Lipophilicity", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "contains 4200 compounds. ESOL provides water solubility data for 1128 compounds. FreeSolv", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "contains hydration free energy of 642 small molecules in water. We conduct graph regression", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "experiments on these benchmarks. All datasets are split into training, validation and test", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "according to a proportion of 0.8/0.1/0.1. MPNN and our EIGNN share the same architecture", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "with 3 layers of message passing and 3 steps of set2set. We repeat each experiment 3 times", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + }, + { + "type": "table", + "bbox": [ + 109, + 694, + 502, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 169, + 682, + 441, + 694 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 169, + 682, + 441, + 696 + ], + "spans": [ + { + "bbox": [ + 169, + 682, + 441, + 696 + ], + "score": 1.0, + "content": "Table 2: Testing RMSE on Lipophilicity, ESOL and FreeSolv.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "table_body", + "bbox": [ + 109, + 694, + 502, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 694, + 502, + 730 + ], + "spans": [ + { + "bbox": [ + 109, + 694, + 502, + 730 + ], + "score": 0.806, + "html": "
DatasetLipophilicityESOLFreeSolv
MethodMPNNEIGNNMPNNEIGNNMPNNEIGNN
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
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Training and validation error on QM9. The shadow area indicates", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 209, + 213, + 221 + ], + "spans": [ + { + "bbox": [ + 107, + 211, + 158, + 220 + ], + "score": 0.84, + "content": "m e a n \\pm s t d", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 209, + 213, + 221 + ], + "score": 1.0, + "content": "over 3 runs.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 503, + 251 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 105, + 227, + 505, + 252 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 256, + 504, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 504, + 268 + ], + "score": 1.0, + "content": "In Table 1, we list regression results for all methods. We report individual MAE for each", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "target in their original scale, averaged MAE (Avg. MAE) over 12 properties, and averaged", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "normalized MAE (Avg. nMAE; averaged over normalized target properties since different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "targets have different units and ranges). Our EIGNN achieves the best performance for each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "metric and each target. Now we are ready to answer the following research questions. i) Are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 310, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 325 + ], + "score": 1.0, + "content": "edge features important? Yes. The error has a trend of decreasing with increasing edge", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "score": 1.0, + "content": "features. The comparison between sMPNN (using edge types) and MPNN (using edge types", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 333, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 504, + 345 + ], + "score": 1.0, + "content": "and distance) directly verifies the importance of edge features. It is also consistent with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 344, + 504, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 504, + 356 + ], + "score": 1.0, + "content": "the expert knowledge that distances between pairwise atoms are closely related to quantum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "properties. For example, the smaller the distance between the two atoms, the stronger", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "the bond is, and consequently a higher bond energy is associated with this atom pair. ii)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 507, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 507, + 390 + ], + "score": 1.0, + "content": "Does the EIGNN work? Yes. EIGNN achieves the best performance on each target,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "outperforming the strong baseline MPNN. Moreover, the advantage of EIGNN over MPNN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "is consistent over 3 runs and the standard deviation on this task is quite small. Detailed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "results are shown in Table 4 of Appendix A. iii) How does the EIGNN work? Our MI", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "score": 1.0, + "content": "objective is easily implemented on top of vanilla loss function. We have shown that our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "objective enables preserving of edge information. Fig. 1 demonstrates that regularization", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 441, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 141, + 456 + ], + "score": 1.0, + "content": "such as", + "type": "text" + }, + { + "bbox": [ + 142, + 445, + 153, + 453 + ], + "score": 0.92, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 441, + 507, + 456 + ], + "score": 1.0, + "content": "weight decay can increase training error while our objective does not. Moreover,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "the validation performance verifies that regularization itself does not reduce the validation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "error. Thus, the effectiveness of EIGNN is due to exploiting edge features rather than the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "regularization effect. We further run an ablation study where we concatenate edge features", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "to node representations (i.e., MPNN+concat in Fig. 1) in message passing. Concatenation is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "unable to identify correlations between edges and nodes (Gilmer et al., 2017) and our results", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 508, + 465, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 465, + 521 + ], + "score": 1.0, + "content": "show that it slightly reduces the mean validation error but increases the variance.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 256, + 507, + 521 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 536, + 429, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 432, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 432, + 548 + ], + "score": 1.0, + "content": "4.2 More Molecule Benchmarks with Potential Applications", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "We further evaluate EIGNN on three molecule benchmarks: Lipophilicity (Wu et al., 2018),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "ESOL (Delaney, 2004) and FreeSolv (Mobley & Guthrie, 2014). These datasets contain fewer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "molecules, and have potential usages in applications such as chemistry, drug discovery, and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 604 + ], + "score": 1.0, + "content": "materials science. For example, the property lipophilicity is an important feature of drug", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 600, + 504, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 504, + 613 + ], + "score": 1.0, + "content": "molecules that affects both membrane permeability and solubility. The dataset Lipophilicity", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "contains 4200 compounds. ESOL provides water solubility data for 1128 compounds. FreeSolv", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "contains hydration free energy of 642 small molecules in water. We conduct graph regression", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "experiments on these benchmarks. All datasets are split into training, validation and test", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "according to a proportion of 0.8/0.1/0.1. MPNN and our EIGNN share the same architecture", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "with 3 layers of message passing and 3 steps of set2set. We repeat each experiment 3 times", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "with different random seeds. Results of testing root mean square error (RMSE) in Table 2", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "verify the effectiveness of our method. Our EIGNN outperforms MPNN on each dataset and", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 315, + 408, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 408, + 327 + ], + "score": 1.0, + "content": "each run. Detailed results for each run are presented in Appendix B.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 556, + 506, + 668 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 694, + 502, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 169, + 682, + 441, + 694 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 169, + 682, + 441, + 696 + ], + "spans": [ + { + "bbox": [ + 169, + 682, + 441, + 696 + ], + "score": 1.0, + "content": "Table 2: Testing RMSE on Lipophilicity, ESOL and FreeSolv.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "table_body", + "bbox": [ + 109, + 694, + 502, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 694, + 502, + 730 + ], + "spans": [ + { + "bbox": [ + 109, + 694, + 502, + 730 + ], + "score": 0.806, + "html": "
DatasetLipophilicityESOLFreeSolv
MethodMPNNEIGNNMPNNEIGNNMPNNEIGNN
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
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The color indicates the impact of an edge/atom on the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 262, + 504, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 504, + 274 + ], + "score": 1.0, + "content": "output, i.e., the regression result. EIGNN i) increases the edge attribution, ii) reduces the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 273, + 466, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 466, + 286 + ], + "score": 1.0, + "content": "prediction error and iii) can learn domain knowledge without human interference.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "with different random seeds. Results of testing root mean square error (RMSE) in Table 2", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "verify the effectiveness of our method. Our EIGNN outperforms MPNN on each dataset and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 315, + 408, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 408, + 327 + ], + "score": 1.0, + "content": "each run. Detailed results for each run are presented in Appendix B.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "Attribution analysis. To understand how our EIGNN reduces the regression error, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "conduct attribution analysis, i.e., attributing the prediction of a deep network to its input", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "features, which usually builds up on the standard gradient operator (Sundararajan et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 372, + 509, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 144, + 404 + ], + "score": 1.0, + "content": "features", + "type": "text" + }, + { + "bbox": [ + 106, + 372, + 199, + 385 + ], + "score": 1.0, + "content": "2017). For an output", + "type": "text" + }, + { + "bbox": [ + 144, + 390, + 149, + 395 + ], + "score": 0.88, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 379, + 164, + 404 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 164, + 385, + 205, + 398 + ], + "score": 0.92, + "content": "\\begin{array} { r } { S _ { e } = | \\frac { \\partial y } { \\partial e } | } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 378, + 204, + 384 + ], + "score": 0.87, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 372, + 505, + 385 + ], + "score": 1.0, + "content": ", i.e., the prediction of a GNN, we define its sensitivity to an edge with", + "type": "text" + }, + { + "bbox": [ + 205, + 379, + 239, + 404 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 239, + 387, + 248, + 397 + ], + "score": 0.89, + "content": "\\left. \\cdot \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 379, + 303, + 404 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 304, + 388, + 315, + 396 + ], + "score": 0.92, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 379, + 509, + 404 + ], + "score": 1.0, + "content": "norm. Similarly, the sensitivity to an atom", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 394, + 507, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 165, + 415 + ], + "score": 1.0, + "content": "with features", + "type": "text" + }, + { + "bbox": [ + 165, + 404, + 171, + 408 + ], + "score": 0.88, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 394, + 183, + 415 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 183, + 399, + 225, + 412 + ], + "score": 0.93, + "content": "\\begin{array} { r } { S _ { x } = | \\frac { \\partial y } { \\partial x } | } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 394, + 257, + 415 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 257, + 401, + 267, + 410 + ], + "score": 0.93, + "content": "S _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 394, + 289, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 401, + 300, + 410 + ], + "score": 0.92, + "content": "S _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 394, + 507, + 415 + ], + "score": 1.0, + "content": "are used as the metrics of attribution in our ex-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "periments. As an example, we show in Fig. 2 the attribution for two molecules in Lipophilicity:", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 122, + 434 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 122, + 423, + 280, + 433 + ], + "score": 0.86, + "content": "N c I n o n c I C ( = N O ) N c \\mathcal { Q } c c c ( F ) c ( C l ) c \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "and (b) Oc1c2ncc3ccccc3c2nn1c4ccccc4 . Molecule", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 431, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 445 + ], + "score": 1.0, + "content": "in (a) has the potential to be used to treat, prevent and/or diagnose cancer Prinz et al.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "score": 1.0, + "content": "(2019). Compared with MPNN, we can observe an increasing of overall edge attribution", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "under our EIGNN and a decreasing of prediction error in both cases. Interestingly, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "score": 1.0, + "content": "attribution under EIGNN is similar to the expert knowledge of chemists: halogen atoms", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "such as {Cl, Br, I} and their bond with the carbon atom greatly effect the lipophilicity of a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "molecule Wilcken et al. (2013), while atoms {O, N} also have high impact on the lipophilicity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "but usually in a negative way (Augustijns & Brewster, 2007). In Fig. 2 (a), the attribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "of the halogen bond C-Cl and the pair {O, N} under our EIGNN is much higher than the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "one under MPNN, which is consistent with the expert knowledge. In Fig. 2 (b), similarly,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "the attribution of atoms {O, N} and the bond C-O under EIGNN is much higher. More", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 541, + 280, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 280, + 554 + ], + "score": 1.0, + "content": "examples are presented in Appendix C.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 568, + 356, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 358, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 358, + 581 + ], + "score": 1.0, + "content": "4.3 Predicting Relations in Knowledge Graphs", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "In this subsection, we adopt EIGNN to tackle the problem of relation prediction in knowledge", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "score": 1.0, + "content": "graphs (KGs), which entails predicting whether a given triple is valid or not. For example, a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "triple (London, capital of, United Kingdom) should be classified as valid or London should be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "predicted as the capital of United Kingdom. KGs represent human knowledge as a directed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "graph, and have been widely used in practical applications, such as semantic search, dialogue", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "generation, question answering etc. Recovering missing relations in KGs have been a major", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "task for practical usages of KGs. We evaluate our methods on three benchmark datasets,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "WN18RR (Dettmers et al., 2018), FB15k-237 (Toutanova et al., 2015) and NELL-995 (Xiong", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "et al., 2017). Without the reversible relation problem (Dettmers et al., 2018), WN18RR", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "includes 11 relations scraped from WordNet for 40, 943 synsets. FB15k-237 is a subset of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Freebase, and contains 14, 541 entities associated with 237 types of edge. 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The color indicates the impact of an edge/atom on the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 262, + 504, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 504, + 274 + ], + "score": 1.0, + "content": "output, i.e., the regression result. EIGNN i) increases the edge attribution, ii) reduces the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 273, + 466, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 466, + 286 + ], + "score": 1.0, + "content": "prediction error and iii) can learn domain knowledge without human interference.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 326 + ], + "lines": [], + "index": 7, + "bbox_fs": [ + 105, + 292, + 505, + 327 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "Attribution analysis. To understand how our EIGNN reduces the regression error, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "conduct attribution analysis, i.e., attributing the prediction of a deep network to its input", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "features, which usually builds up on the standard gradient operator (Sundararajan et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 372, + 509, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 144, + 404 + ], + "score": 1.0, + "content": "features", + "type": "text" + }, + { + "bbox": [ + 106, + 372, + 199, + 385 + ], + "score": 1.0, + "content": "2017). For an output", + "type": "text" + }, + { + "bbox": [ + 144, + 390, + 149, + 395 + ], + "score": 0.88, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 379, + 164, + 404 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 164, + 385, + 205, + 398 + ], + "score": 0.92, + "content": "\\begin{array} { r } { S _ { e } = | \\frac { \\partial y } { \\partial e } | } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 378, + 204, + 384 + ], + "score": 0.87, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 372, + 505, + 385 + ], + "score": 1.0, + "content": ", i.e., the prediction of a GNN, we define its sensitivity to an edge with", + "type": "text" + }, + { + "bbox": [ + 205, + 379, + 239, + 404 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 239, + 387, + 248, + 397 + ], + "score": 0.89, + "content": "\\left. \\cdot \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 379, + 303, + 404 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 304, + 388, + 315, + 396 + ], + "score": 0.92, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 379, + 509, + 404 + ], + "score": 1.0, + "content": "norm. 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Then", + "type": "text" + }, + { + "bbox": [ + 257, + 401, + 267, + 410 + ], + "score": 0.93, + "content": "S _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 394, + 289, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 401, + 300, + 410 + ], + "score": 0.92, + "content": "S _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 394, + 507, + 415 + ], + "score": 1.0, + "content": "are used as the metrics of attribution in our ex-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "periments. As an example, we show in Fig. 2 the attribution for two molecules in Lipophilicity:", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 122, + 434 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 122, + 423, + 280, + 433 + ], + "score": 0.86, + "content": "N c I n o n c I C ( = N O ) N c \\mathcal { Q } c c c ( F ) c ( C l ) c \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "and (b) Oc1c2ncc3ccccc3c2nn1c4ccccc4 . Molecule", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 431, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 445 + ], + "score": 1.0, + "content": "in (a) has the potential to be used to treat, prevent and/or diagnose cancer Prinz et al.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "score": 1.0, + "content": "(2019). Compared with MPNN, we can observe an increasing of overall edge attribution", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "under our EIGNN and a decreasing of prediction error in both cases. Interestingly, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "score": 1.0, + "content": "attribution under EIGNN is similar to the expert knowledge of chemists: halogen atoms", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "such as {Cl, Br, I} and their bond with the carbon atom greatly effect the lipophilicity of a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "molecule Wilcken et al. (2013), while atoms {O, N} also have high impact on the lipophilicity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "but usually in a negative way (Augustijns & Brewster, 2007). In Fig. 2 (a), the attribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "of the halogen bond C-Cl and the pair {O, N} under our EIGNN is much higher than the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "one under MPNN, which is consistent with the expert knowledge. In Fig. 2 (b), similarly,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "the attribution of atoms {O, N} and the bond C-O under EIGNN is much higher. More", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 541, + 280, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 280, + 554 + ], + "score": 1.0, + "content": "examples are presented in Appendix C.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 338, + 509, + 554 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 568, + 356, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 358, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 358, + 581 + ], + "score": 1.0, + "content": "4.3 Predicting Relations in Knowledge Graphs", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "In this subsection, we adopt EIGNN to tackle the problem of relation prediction in knowledge", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "score": 1.0, + "content": "graphs (KGs), which entails predicting whether a given triple is valid or not. For example, a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "triple (London, capital of, United Kingdom) should be classified as valid or London should be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "predicted as the capital of United Kingdom. KGs represent human knowledge as a directed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "graph, and have been widely used in practical applications, such as semantic search, dialogue", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "generation, question answering etc. Recovering missing relations in KGs have been a major", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "task for practical usages of KGs. We evaluate our methods on three benchmark datasets,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "WN18RR (Dettmers et al., 2018), FB15k-237 (Toutanova et al., 2015) and NELL-995 (Xiong", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "et al., 2017). Without the reversible relation problem (Dettmers et al., 2018), WN18RR", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "includes 11 relations scraped from WordNet for 40, 943 synsets. FB15k-237 is a subset of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Freebase, and contains 14, 541 entities associated with 237 types of edge. 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Hits@N", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 86, + 487, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 86, + 487, + 98 + ], + "score": 1.0, + "content": "values are in percentage. The best score is in bold and second best score is underlined.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 114, + 97, + 496, + 191 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 97, + 496, + 191 + ], + "spans": [ + { + "bbox": [ + 114, + 97, + 496, + 191 + ], + "score": 0.982, + "html": "
DatasetWN18RRFB15K-237NELL-995
MRRHit@N%MRRHit@N %
MRR@1@3@10@1@3@10@1@3@10
DistMult0.44441.24750.40.28119.930.144.60.48540.152.461
ComplEx0.44940.946.9530.27819.429.7450.48239.952.860.6
ConvE0.45641.94753.10.31222.534.149.70.49140.353.161.3
TransE0.24342.744.153.20.27919.837.644.10.40134.447.250.1
ConvKB0.26558.244.555.80.28919.832.447.10.4337.04754.5
R-GCN0.123813.720.70.1641018.1300.128.212.618.8
KBGAT0.43635.848.157.80.43136.145.856.90.51442.955.367.8
EIGNN0.43835.748.858.10.45137.448.260.50.52343.856.168.3
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(3) to KGs is that high-dimensional embedding vectors are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "required to distinguish massive amount of entities and relations, leading to a rapid growth in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "number of parameters in EIGNN. 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After training EIGNN, ConvKB (Nguyen", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "et al., 2018) is adopted as a regression function for a given triple by analyzing the global", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 374, + 305, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 305, + 386 + ], + "score": 1.0, + "content": "embedding properties across each dimension.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 507, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 369, + 404 + ], + "score": 1.0, + "content": "In the relation prediction task, the aim is to predict a triple", + "type": "text" + }, + { + "bbox": [ + 369, + 392, + 414, + 402 + ], + "score": 0.93, + "content": "( v , e _ { v w } , w )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 389, + 439, + 404 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 439, + 395, + 444, + 400 + ], + "score": 0.9, + "content": "\\boldsymbol { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 389, + 460, + 404 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 460, + 395, + 468, + 400 + ], + "score": 0.89, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 389, + 507, + 404 + ], + "score": 1.0, + "content": "missing.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 400, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 396, + 415 + ], + "score": 1.0, + "content": "We can generate a set of candidate triples for each missing entity", + "type": "text" + }, + { + "bbox": [ + 396, + 406, + 402, + 411 + ], + "score": 0.86, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 400, + 506, + 415 + ], + "score": 1.0, + "content": "by randomly replacing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 412, + 504, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 424 + ], + "score": 1.0, + "content": "it with an arbitrary one. Scores can be calculated by ConvKB for all triples, and we find", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "the rank of a correct triple by sorting all scores in ascending order. 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The results of KBGAT are reproduced", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "following the official implementation1, and the results of other methods can be found in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 383, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 383, + 545 + ], + "score": 1.0, + "content": "previous peer-reviewed publications, i.e. (Nathani et al., 2019).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 559, + 201, + 572 + ], + "lines": [ + { + "bbox": [ + 104, + 557, + 204, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 557, + 204, + 575 + ], + "score": 1.0, + "content": "5 Conclusions", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "In this work, to make better use of edge features in GNNs, we proposed the edge information", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "maximized graph neural network (EIGNN) that maximizes the mutual information between", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "edge feature vectors and message passing channels. 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These results clearly manifested a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "substantial improvement of EIGNN over the prior state-of-the-art methods. Apart from", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "demonstrating the impressive performance of EIGNN, we also showed that its effectiveness is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "due to exploitation of edge features instead of the regularization effect. Notably, attribution", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 694, + 505, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 505, + 706 + ], + "score": 1.0, + "content": "analysis on molecular graphs show that EIGNN can capture domain knowledge in an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 704, + 192, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 192, + 717 + ], + "score": 1.0, + "content": "end-to-end fashion.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 325, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 326, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 326, + 733 + ], + "score": 1.0, + "content": "1https://github.com/deepakn97/relationPrediction", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 326, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 327, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 327, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 114, + 97, + 496, + 191 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 74, + 502, + 97 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 74, + 504, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 74, + 504, + 87 + ], + "score": 1.0, + "content": "Table 3: Experimental results on WN18RR, FB15K-237 and NELL-995 test sets. 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DatasetWN18RRFB15K-237NELL-995
MRRHit@N%MRRHit@N %
MRR@1@3@10@1@3@10@1@3@10
DistMult0.44441.24750.40.28119.930.144.60.48540.152.461
ComplEx0.44940.946.9530.27819.429.7450.48239.952.860.6
ConvE0.45641.94753.10.31222.534.149.70.49140.353.161.3
TransE0.24342.744.153.20.27919.837.644.10.40134.447.250.1
ConvKB0.26558.244.555.80.28919.832.447.10.4337.04754.5
R-GCN0.123813.720.70.1641018.1300.128.212.618.8
KBGAT0.43635.848.157.80.43136.145.856.90.51442.955.367.8
EIGNN0.43835.748.858.10.45137.448.260.50.52343.856.168.3
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Multi-head attention is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "further introduced to stabilize the learning process and encapsulate more information about", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "neighbors according to (Veličković et al., 2018). 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Scores can be calculated by ConvKB for all triples, and we find", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "the rank of a correct triple by sorting all scores in ascending order. 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We compare our EIGNN with seven state-of-the-art baselines focusing on this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "task: DistMult (Yang et al., 2014), ComplEx (Trouillon et al., 2016), ConvE (Dettmers et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "2018), TransE (Bordes et al., 2013), ConvKB (Nguyen et al., 2018), RGCN (Schlichtkrull", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "et al., 2018) and KBGAT (Nathani et al., 2019). 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The results of KBGAT are reproduced", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "following the official implementation1, and the results of other methods can be found in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 383, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 383, + 545 + ], + "score": 1.0, + "content": "previous peer-reviewed publications, i.e. (Nathani et al., 2019).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 389, + 507, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 559, + 201, + 572 + ], + "lines": [ + { + "bbox": [ + 104, + 557, + 204, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 557, + 204, + 575 + ], + "score": 1.0, + "content": "5 Conclusions", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "In this work, to make better use of edge features in GNNs, we proposed the edge information", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "maximized graph neural network (EIGNN) that maximizes the mutual information between", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "edge feature vectors and message passing channels. 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These results clearly manifested a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "substantial improvement of EIGNN over the prior state-of-the-art methods. Apart from", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "demonstrating the impressive performance of EIGNN, we also showed that its effectiveness is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "due to exploitation of edge features instead of the regularization effect. 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MethodGCNChebyNetGATGIN
Avg.nMAE0.1350±0.00460.1206±0.00840.1367±0.00500.1001±0.0007
Avg.MAE5.3063±0.19644.3032±0.48145.4698±0.20403.4799±0.0402
mu0.5679±0.00780.5180±0.01310.5670±0.01020.4783±0.0041
alpha0.8811±0.03080.7932±0.07780.8913±0.03140.6209±0.0028
HOMO(10-3)5.4510±0.07904.7750±0.20405.4290±0.16604.1830±0.0370
LUMO(10-3)6.4000±0.12305.6740±0.26706.3310±0.23904.7960±0.0520
gap(10-3)8.2010±0.21507.0970±0.36208.1930±0.29106.0960±0.0420
R253.563±1.031941.950±4.828954.519±1.599234.647±0.2167
ZPVE(10-3)2.5330±0.10702.5270±0.35602.2710±0.14501.7440±0.0100
UO2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
U2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
H2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
G2.0423±0.32811.9842±0.20362.2899±0.20001.4215±0.0857
Cv0.4730±0.02090.4199±0.04990.4787±0.01640.3093±0.0035
MethodRGCNGGNNLNetsMPNN
Avg.nMAE0.1021±0.00160.0992±0.00130.0992±0.00610.0888±0.0014
Avg.MAE3.8175±0.06053.6608±0.07233.6527±0.34173.1610±0.0697
mu0.5056±0.00480.5179±0.00760.4717±0.00630.4718±0.0096
alpha0.6321±0.01450.6077±0.00920.6225±0.05080.5278±0.0106
HOMO(10-3)4.4530±0.12904.4830±0.06503.8889±0.16173.8540±0.0410
LUMO(10-3)5.1380±0.12105.1530±0.08904.1935±0.22054.5490±0.0810
gap(10-3)6.5000±0.13906.6020±0.14005.8132±0.64565.6340±0.0570
R240.102±0.742839.685±0.821235.275±3.053133.489±0.6562
ZPVE(10-3)1.4770±0.00901.2920±0.03401.4376±0.07691.3450±0.0260
UO1.0589±0.02310.6969±0.04131.8058±0.25330.7914±0.0446
U1.0589±0.02310.6966±0.04181.7555±0.21960.7914±0.0446
H1.0589±0.02310.6974±0.04081.7964±0.24280.7914±0.0446
G1.0589±0.02310.6961±0.04211.7780±0.24580.7914±0.0446
Cv0.3170±0.01520.3146±0.01250.3124±0.03030.2625±0.0040
MethodMPNNEIGNN
Avg.nMAE0.0398±0.00020.0357±0.0005
Avg.MAE0.6929±0.02120.6331±0.0298
mu0.1095±0.00140.0974±0.0026
alpha0.3318±0.00260.2939±0.0054
HOMO(10-3)2.4810±0.02002.2300±0.0310
LUMO(10-3)2.8620±0.03702.5930±0.0440
gap(10-3)3.6200±0.01803.2750±0.0520
R26.0637±0.25115.6464±0.3098
ZPVE(10-3)0.6790±0.01400.6120±0.0170
UO0.4164±0.02250.3574±0.0100
U0.4164±0.02250.3575±0.0100
H0.4164±0.02250.3574±0.0100
G0.4164±0.02250.3575±0.0101
Cv0.1339±0.00130.1208±0.0027
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The", + "type": "text" + }, + { + "bbox": [ + 283, + 606, + 309, + 617 + ], + "score": 0.72, + "content": "( 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 603, + 505, + 618 + ], + "score": 1.0, + "content": ") in the parentheses indicates that the values", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 368, + 629 + ], + "score": 1.0, + "content": "in the corresponding raw of the table should multiply by", + "type": "text" + }, + { + "bbox": [ + 369, + 617, + 389, + 626 + ], + "score": 0.89, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 614, + 506, + 629 + ], + "score": 1.0, + "content": ". This is simply for clear", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "presentation of the values. We also present a detailed descriptions on the target properties", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 637, + 235, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 235, + 650 + ], + "score": 1.0, + "content": "in Table 5 for your reference.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "In our results, we directly report MAE instead of Error Ratio [(MAE)/(Chemical Accu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "racy) (Gilmer et al., 2017)], because it is common to report MAE in terms of chemical unit.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "This practice has been widely adopted not only in computational chemistry but also in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "machine learning community working on molecular graphs (Wu et al., 2018; Morris et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "2019). Still, we add Table 6 which contains the Error Ratio for MPNN and our EIGNN. In", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "this comparison, we follow (Gilmer et al., 2017) and train models separately to predict each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 721, + 343, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 343, + 733 + ], + "score": 1.0, + "content": "target. 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MethodGCNChebyNetGATGIN
Avg.nMAE0.1350±0.00460.1206±0.00840.1367±0.00500.1001±0.0007
Avg.MAE5.3063±0.19644.3032±0.48145.4698±0.20403.4799±0.0402
mu0.5679±0.00780.5180±0.01310.5670±0.01020.4783±0.0041
alpha0.8811±0.03080.7932±0.07780.8913±0.03140.6209±0.0028
HOMO(10-3)5.4510±0.07904.7750±0.20405.4290±0.16604.1830±0.0370
LUMO(10-3)6.4000±0.12305.6740±0.26706.3310±0.23904.7960±0.0520
gap(10-3)8.2010±0.21507.0970±0.36208.1930±0.29106.0960±0.0420
R253.563±1.031941.950±4.828954.519±1.599234.647±0.2167
ZPVE(10-3)2.5330±0.10702.5270±0.35602.2710±0.14501.7440±0.0100
UO2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
U2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
H2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
G2.0423±0.32811.9842±0.20362.2899±0.20001.4215±0.0857
Cv0.4730±0.02090.4199±0.04990.4787±0.01640.3093±0.0035
MethodRGCNGGNNLNetsMPNN
Avg.nMAE0.1021±0.00160.0992±0.00130.0992±0.00610.0888±0.0014
Avg.MAE3.8175±0.06053.6608±0.07233.6527±0.34173.1610±0.0697
mu0.5056±0.00480.5179±0.00760.4717±0.00630.4718±0.0096
alpha0.6321±0.01450.6077±0.00920.6225±0.05080.5278±0.0106
HOMO(10-3)4.4530±0.12904.4830±0.06503.8889±0.16173.8540±0.0410
LUMO(10-3)5.1380±0.12105.1530±0.08904.1935±0.22054.5490±0.0810
gap(10-3)6.5000±0.13906.6020±0.14005.8132±0.64565.6340±0.0570
R240.102±0.742839.685±0.821235.275±3.053133.489±0.6562
ZPVE(10-3)1.4770±0.00901.2920±0.03401.4376±0.07691.3450±0.0260
UO1.0589±0.02310.6969±0.04131.8058±0.25330.7914±0.0446
U1.0589±0.02310.6966±0.04181.7555±0.21960.7914±0.0446
H1.0589±0.02310.6974±0.04081.7964±0.24280.7914±0.0446
G1.0589±0.02310.6961±0.04211.7780±0.24580.7914±0.0446
Cv0.3170±0.01520.3146±0.01250.3124±0.03030.2625±0.0040
MethodMPNNEIGNN
Avg.nMAE0.0398±0.00020.0357±0.0005
Avg.MAE0.6929±0.02120.6331±0.0298
mu0.1095±0.00140.0974±0.0026
alpha0.3318±0.00260.2939±0.0054
HOMO(10-3)2.4810±0.02002.2300±0.0310
LUMO(10-3)2.8620±0.03702.5930±0.0440
gap(10-3)3.6200±0.01803.2750±0.0520
R26.0637±0.25115.6464±0.3098
ZPVE(10-3)0.6790±0.01400.6120±0.0170
UO0.4164±0.02250.3574±0.0100
U0.4164±0.02250.3575±0.0100
H0.4164±0.02250.3574±0.0100
G0.4164±0.02250.3575±0.0101
Cv0.1339±0.00130.1208±0.0027
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The", + "type": "text" + }, + { + "bbox": [ + 283, + 606, + 309, + 617 + ], + "score": 0.72, + "content": "( 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 603, + 505, + 618 + ], + "score": 1.0, + "content": ") in the parentheses indicates that the values", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 368, + 629 + ], + "score": 1.0, + "content": "in the corresponding raw of the table should multiply by", + "type": "text" + }, + { + "bbox": [ + 369, + 617, + 389, + 626 + ], + "score": 0.89, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 614, + 506, + 629 + ], + "score": 1.0, + "content": ". This is simply for clear", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "presentation of the values. We also present a detailed descriptions on the target properties", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 637, + 235, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 235, + 650 + ], + "score": 1.0, + "content": "in Table 5 for your reference.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 594, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "In our results, we directly report MAE instead of Error Ratio [(MAE)/(Chemical Accu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "racy) (Gilmer et al., 2017)], because it is common to report MAE in terms of chemical unit.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "This practice has been widely adopted not only in computational chemistry but also in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "machine learning community working on molecular graphs (Wu et al., 2018; Morris et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "2019). Still, we add Table 6 which contains the Error Ratio for MPNN and our EIGNN. In", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "this comparison, we follow (Gilmer et al., 2017) and train models separately to predict each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 721, + 343, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 343, + 733 + ], + "score": 1.0, + "content": "target. Our EIGNN consistently outperforms MPNN.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 653, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 172, + 101, + 439, + 218 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 224, + 90, + 385, + 100 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 223, + 88, + 387, + 102 + ], + "spans": [ + { + "bbox": [ + 223, + 88, + 387, + 102 + ], + "score": 1.0, + "content": "Table 5: Regression targets on QM9.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 172, + 101, + 439, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 172, + 101, + 439, + 218 + ], + "spans": [ + { + "bbox": [ + 172, + 101, + 439, + 218 + ], + "score": 0.98, + "html": "
Target propertyDescriptionUnit
muDipole momentD
alphaIsotropic polarizabilitya
HOMOHighest occupied molecular orbital energyEh
LUMOLowest unoccupied molecular orbital energyEh
gapGap between HOMO and LUMOEh
R2Electronic spatial extent品
ZPVEZero point vibrational energyEh
UOInternal energy at OKEh
UInternal energy at 298.15KEh
HEnthalpy at 298.15KEh
GFree energy at 298.15KEh
CvHeat capavity at 298.15Kcal molK
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TargetMPNNEIGNNImprovement (%)
mu0.870.808.24
alpha2.642.447.57
HOMO1.541.3910.2
LUMO1.311.282.13
gap2.201.9710.4
R21.000.7228.4
ZPVE5.314.4316.5
UO64.335.844.4
U56.623.658.3
H77.435.254.5
G46.634.426.1
Cv1.691.79-6.22
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The results in Table 7 verify that our EIGNN consistently", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 198, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 198, + 468 + ], + "score": 1.0, + "content": "outperforms MPNN.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 484, + 321, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 324, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 324, + 500 + ], + "score": 1.0, + "content": "C More Examples of Attribution", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 104, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 506, + 524 + ], + "score": 1.0, + "content": "In this section, we present more examples of attribution analysis on Lipophilicity. As a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "supplement for Fig. 2 in the main text, the observation here is similar. Compared with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 531, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 546 + ], + "score": 1.0, + "content": "MPNN, we can observe an increasing of overall edge attribution under our EIGNN and a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "decreasing of prediction error in both cases. Notably, our EIGNN is able to capture the ex-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 268, + 567 + ], + "score": 1.0, + "content": "pert knowledge. (a) The molecule is", + "type": "text" + }, + { + "bbox": [ + 268, + 556, + 506, + 567 + ], + "score": 0.81, + "content": "C N [ C @ \\mathbb { Q } H ] ( C ) C ( = O ) N [ C @ \\mathbb { Q } H ] ( C 1 C C C C T 1 ) C ( =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 107, + 567, + 284, + 577 + ], + "score": 0.91, + "content": "O ) N [ C @ H ] 2 C C C N ( C C c ( F ) c c 3 ) C 2", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 564, + 388, + 579 + ], + "score": 1.0, + "content": ". The attribution of", + "type": "text" + }, + { + "bbox": [ + 388, + 567, + 423, + 577 + ], + "score": 0.68, + "content": "\\{ \\mathrm { O } , \\mathrm { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "and the halogen", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 574, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 134, + 591 + ], + "score": 1.0, + "content": "atom", + "type": "text" + }, + { + "bbox": [ + 135, + 578, + 142, + 586 + ], + "score": 0.43, + "content": "\\mathrm { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 574, + 361, + 591 + ], + "score": 1.0, + "content": "is higher under EIGNN. (b) The molecule is", + "type": "text" + }, + { + "bbox": [ + 362, + 578, + 506, + 588 + ], + "score": 0.89, + "content": "C C ( C ) N 1 C C N [ C @ H ] ( C 1 ) C ( =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 587, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 108, + 589, + 306, + 599 + ], + "score": 0.88, + "content": "O ) N 2 C C N ( C C 2 ) C ( = O ) N c 3 c c c ( C l ) c ( C l ) c 3", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 587, + 507, + 601 + ], + "score": 1.0, + "content": ". Our EIGNN successfully captures the im-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "portance of two critical halogen atoms Cl and several atoms N. (c) The molecule is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 107, + 611, + 280, + 621 + ], + "score": 0.91, + "content": "C O c 1 c c ( c c 1 ) C ( = \\ O ) N 2 C C C C 2 \\ = \\ O", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 609, + 431, + 622 + ], + "score": 1.0, + "content": ". The attribution of two atoms", + "type": "text" + }, + { + "bbox": [ + 432, + 612, + 440, + 619 + ], + "score": 0.61, + "content": "\\mathrm { \\ o }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "at the top is", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "table", + "bbox": [ + 109, + 670, + 501, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 649, + 502, + 670 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 105, + 649, + 504, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 504, + 663 + ], + "score": 1.0, + "content": "Table 7: Testing RMSE on Lipophilicity, ESOL and FreeSolv. 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DatasetLipophilicityESOLFreeSolv
SeedMPNNEIGNNMPNNEIGNNMPNNEIGNN
00.7180.6760.7700.7181.3961.109
10.6960.6640.7500.7331.2991.265
20.6200.6190.8940.8761.4991.443
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
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Target propertyDescriptionUnit
muDipole momentD
alphaIsotropic polarizabilitya
HOMOHighest occupied molecular orbital energyEh
LUMOLowest unoccupied molecular orbital energyEh
gapGap between HOMO and LUMOEh
R2Electronic spatial extent品
ZPVEZero point vibrational energyEh
UOInternal energy at OKEh
UInternal energy at 298.15KEh
HEnthalpy at 298.15KEh
GFree energy at 298.15KEh
CvHeat capavity at 298.15Kcal molK
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TargetMPNNEIGNNImprovement (%)
mu0.870.808.24
alpha2.642.447.57
HOMO1.541.3910.2
LUMO1.311.282.13
gap2.201.9710.4
R21.000.7228.4
ZPVE5.314.4316.5
UO64.335.844.4
U56.623.658.3
H77.435.254.5
G46.634.426.1
Cv1.691.79-6.22
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The results in Table 7 verify that our EIGNN consistently", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 198, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 198, + 468 + ], + "score": 1.0, + "content": "outperforms MPNN.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 435, + 505, + 468 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 484, + 321, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 324, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 324, + 500 + ], + "score": 1.0, + "content": "C More Examples of Attribution", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 104, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 506, + 524 + ], + "score": 1.0, + "content": "In this section, we present more examples of attribution analysis on Lipophilicity. As a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "supplement for Fig. 2 in the main text, the observation here is similar. Compared with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 531, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 546 + ], + "score": 1.0, + "content": "MPNN, we can observe an increasing of overall edge attribution under our EIGNN and a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "decreasing of prediction error in both cases. 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The attribution of", + "type": "text" + }, + { + "bbox": [ + 388, + 567, + 423, + 577 + ], + "score": 0.68, + "content": "\\{ \\mathrm { O } , \\mathrm { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "and the halogen", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 574, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 134, + 591 + ], + "score": 1.0, + "content": "atom", + "type": "text" + }, + { + "bbox": [ + 135, + 578, + 142, + 586 + ], + "score": 0.43, + "content": "\\mathrm { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 574, + 361, + 591 + ], + "score": 1.0, + "content": "is higher under EIGNN. (b) The molecule is", + "type": "text" + }, + { + "bbox": [ + 362, + 578, + 506, + 588 + ], + "score": 0.89, + "content": "C C ( C ) N 1 C C N [ C @ H ] ( C 1 ) C ( =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 587, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 108, + 589, + 306, + 599 + ], + "score": 0.88, + "content": "O ) N 2 C C N ( C C 2 ) C ( = O ) N c 3 c c c ( C l ) c ( C l ) c 3", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 587, + 507, + 601 + ], + "score": 1.0, + "content": ". Our EIGNN successfully captures the im-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "portance of two critical halogen atoms Cl and several atoms N. (c) The molecule is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 107, + 611, + 280, + 621 + ], + "score": 0.91, + "content": "C O c 1 c c ( c c 1 ) C ( = \\ O ) N 2 C C C C 2 \\ = \\ O", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 609, + 431, + 622 + ], + "score": 1.0, + "content": ". The attribution of two atoms", + "type": "text" + }, + { + "bbox": [ + 432, + 612, + 440, + 619 + ], + "score": 0.61, + "content": "\\mathrm { \\ o }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "at the top is", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 509, + 507, + 622 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 670, + 501, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 649, + 502, + 670 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 105, + 649, + 504, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 504, + 663 + ], + "score": 1.0, + "content": "Table 7: Testing RMSE on Lipophilicity, ESOL and FreeSolv. 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DatasetLipophilicityESOLFreeSolv
SeedMPNNEIGNNMPNNEIGNNMPNNEIGNN
00.7180.6760.7700.7181.3961.109
10.6960.6640.7500.7331.2991.265
20.6200.6190.8940.8761.4991.443
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
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MethodGCNChebyNetGATGINRGCNGGNNLNetsMPNNMPNNEIGNN
Avg.nMAE0.1350.1210.1370.1000.1020.0990.0990.0890.0400.036
Avg.MAE5.3064.3035.4703.4803.8173.6613.6533.1610.6930.633
mu0.5680.5180.5670.4780.5060.5180.4720.4720.1100.097
alpha0.8810.7930.8910.6210.6320.6080.6230.5280.3320.294
HOMO(10-3)5.4514.7755.4294.1834.4534.4833.8893.8542.4812.230
LUMO(10-3)6.4005.6746.3314.7965.1385.1534.1944.5492.8622.593
gap(10-3)8.2017.0978.1936.0966.5006.6025.8135.6343.6203.275
R253.5641.9554.5234.6540.1039.6835.2733.496.0645.646
ZPVE(10-3)2.5332.5272.2711.7441.4771.2921.4381.3450.6790.612
UO2.0421.9842.2901.4221.0590.6971.8060.7910.4160.357
U2.0421.9842.2901.4221.0590.6971.7550.7910.4160.357
H2.0421.9842.2901.4221.0590.6971.7960.7910.4160.357
G2.0421.9842.2901.4221.0590.6961.7780.7910.4160.357
Cv0.4730.4200.4790.3090.3170.3150.3120.2620.1340.121
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DatasetLipophilicityESOLFreeSolv
MethodMPNNEIGNNMPNNEIGNNMPNNEIGNN
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
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DatasetWN18RRFB15K-237NELL-995
MRRHit@N%MRRHit@N %
MRR@1@3@10@1@3@10@1@3@10
DistMult0.44441.24750.40.28119.930.144.60.48540.152.461
ComplEx0.44940.946.9530.27819.429.7450.48239.952.860.6
ConvE0.45641.94753.10.31222.534.149.70.49140.353.161.3
TransE0.24342.744.153.20.27919.837.644.10.40134.447.250.1
ConvKB0.26558.244.555.80.28919.832.447.10.4337.04754.5
R-GCN0.123813.720.70.1641018.1300.128.212.618.8
KBGAT0.43635.848.157.80.43136.145.856.90.51442.955.367.8
EIGNN0.43835.748.858.10.45137.448.260.50.52343.856.168.3
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MethodGCNChebyNetGATGIN
Avg.nMAE0.1350±0.00460.1206±0.00840.1367±0.00500.1001±0.0007
Avg.MAE5.3063±0.19644.3032±0.48145.4698±0.20403.4799±0.0402
mu0.5679±0.00780.5180±0.01310.5670±0.01020.4783±0.0041
alpha0.8811±0.03080.7932±0.07780.8913±0.03140.6209±0.0028
HOMO(10-3)5.4510±0.07904.7750±0.20405.4290±0.16604.1830±0.0370
LUMO(10-3)6.4000±0.12305.6740±0.26706.3310±0.23904.7960±0.0520
gap(10-3)8.2010±0.21507.0970±0.36208.1930±0.29106.0960±0.0420
R253.563±1.031941.950±4.828954.519±1.599234.647±0.2167
ZPVE(10-3)2.5330±0.10702.5270±0.35602.2710±0.14501.7440±0.0100
UO2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
U2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
H2.0422±0.32811.9842±0.20352.2899±0.20001.4215±0.0857
G2.0423±0.32811.9842±0.20362.2899±0.20001.4215±0.0857
Cv0.4730±0.02090.4199±0.04990.4787±0.01640.3093±0.0035
MethodRGCNGGNNLNetsMPNN
Avg.nMAE0.1021±0.00160.0992±0.00130.0992±0.00610.0888±0.0014
Avg.MAE3.8175±0.06053.6608±0.07233.6527±0.34173.1610±0.0697
mu0.5056±0.00480.5179±0.00760.4717±0.00630.4718±0.0096
alpha0.6321±0.01450.6077±0.00920.6225±0.05080.5278±0.0106
HOMO(10-3)4.4530±0.12904.4830±0.06503.8889±0.16173.8540±0.0410
LUMO(10-3)5.1380±0.12105.1530±0.08904.1935±0.22054.5490±0.0810
gap(10-3)6.5000±0.13906.6020±0.14005.8132±0.64565.6340±0.0570
R240.102±0.742839.685±0.821235.275±3.053133.489±0.6562
ZPVE(10-3)1.4770±0.00901.2920±0.03401.4376±0.07691.3450±0.0260
UO1.0589±0.02310.6969±0.04131.8058±0.25330.7914±0.0446
U1.0589±0.02310.6966±0.04181.7555±0.21960.7914±0.0446
H1.0589±0.02310.6974±0.04081.7964±0.24280.7914±0.0446
G1.0589±0.02310.6961±0.04211.7780±0.24580.7914±0.0446
Cv0.3170±0.01520.3146±0.01250.3124±0.03030.2625±0.0040
MethodMPNNEIGNN
Avg.nMAE0.0398±0.00020.0357±0.0005
Avg.MAE0.6929±0.02120.6331±0.0298
mu0.1095±0.00140.0974±0.0026
alpha0.3318±0.00260.2939±0.0054
HOMO(10-3)2.4810±0.02002.2300±0.0310
LUMO(10-3)2.8620±0.03702.5930±0.0440
gap(10-3)3.6200±0.01803.2750±0.0520
R26.0637±0.25115.6464±0.3098
ZPVE(10-3)0.6790±0.01400.6120±0.0170
UO0.4164±0.02250.3574±0.0100
U0.4164±0.02250.3575±0.0100
H0.4164±0.02250.3574±0.0100
G0.4164±0.02250.3575±0.0101
Cv0.1339±0.00130.1208±0.0027
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Target propertyDescriptionUnit
muDipole momentD
alphaIsotropic polarizabilitya
HOMOHighest occupied molecular orbital energyEh
LUMOLowest unoccupied molecular orbital energyEh
gapGap between HOMO and LUMOEh
R2Electronic spatial extent品
ZPVEZero point vibrational energyEh
UOInternal energy at OKEh
UInternal energy at 298.15KEh
HEnthalpy at 298.15KEh
GFree energy at 298.15KEh
CvHeat capavity at 298.15Kcal molK
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TargetMPNNEIGNNImprovement (%)
mu0.870.808.24
alpha2.642.447.57
HOMO1.541.3910.2
LUMO1.311.282.13
gap2.201.9710.4
R21.000.7228.4
ZPVE5.314.4316.5
UO64.335.844.4
U56.623.658.3
H77.435.254.5
G46.634.426.1
Cv1.691.79-6.22
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DatasetLipophilicityESOLFreeSolv
SeedMPNNEIGNNMPNNEIGNNMPNNEIGNN
00.7180.6760.7700.7181.3961.109
10.6960.6640.7500.7331.2991.265
20.6200.6190.8940.8761.4991.443
mean±std0.678±0.0420.653±0.0250.805±0.0640.776±0.0711.398±0.0811.273±0.137
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Zhong +Huawei +zorro.zhongzhao@huawei.com + +# ABSTRACT + +Data augmentation (DA) has been widely utilized to improve generalization in training deep neural networks. Recently, human-designed data augmentation has been gradually replaced by automatically learned augmentation policy. Through finding the best policy in well-designed search space of data augmentation, AutoAugment (Cubuk et al., 2019) can significantly improve validation accuracy on image classification tasks. However, this approach is not computationally practical for large-scale problems. In this paper, we develop an adversarial method to arrive at a computationally-affordable solution called Adversarial AutoAugment, which can simultaneously optimize target related object and augmentation policy search loss. The augmentation policy network attempts to increase the training loss of a target network through generating adversarial augmentation policies, while the target network can learn more robust features from harder examples to improve the generalization. In contrast to prior work, we reuse the computation in target network training for policy evaluation, and dispense with the retraining of the target network. Compared to AutoAugment, this leads to about $1 2 \times$ reduction in computing cost and $1 1 \times$ shortening in time overhead on ImageNet. We show experimental results of our approach on CIFAR-10/CIFAR-100, ImageNet, and demonstrate significant performance improvements over state-of-the-art. On CIFAR-10, we achieve a top-1 test error of $1 . 3 6 \%$ , which is the currently best performing single model. On ImageNet, we achieve a leading performance of top-1 accuracy $7 9 . 4 0 \%$ on ResNet-50 and $8 0 . 0 0 \%$ on ResNet-50-D without extra data. + +# 1 INTRODUCTION + +Massive amount of data have promoted the great success of deep learning in academia and industry. The performance of deep neural networks (DNNs) would be improved substantially when more supervised data is available or better data augmentation method is adapted. Data augmentation such as rotation, flipping, cropping, etc., is a powerful technique to increase the amount and diversity of data. Experiments show that the generalization of a neural network can be efficiently improved through manually designing data augmentation policies. However, this needs lots of knowledge of human expert, and sometimes shows the weak transferability across different tasks and datasets in practical applications. Inspired by neural architecture search (NAS)(Zoph & Le, 2016; Zoph et al., 2017; Zhong et al., 2018a;b; Guo et al., 2018), a reinforcement learning (RL) (Williams, 1992) method called AutoAugment is proposed by Cubuk et al. (2019), which can automatically learn the augmentation policy from data and provide an exciting performance improvement on image classification tasks. However, the computing cost is huge for training and evaluating thousands of sampled policies in the search process. Although proxy tasks, i.e., smaller models and reduced datasets, are taken to accelerate the searching process, tens of thousands of GPU-hours of consumption are still required. In addition, these data augmentation policies optimized on proxy tasks are not guaranteed to be optimal on the target task, and the fixed augmentation policy is also sub-optimal for the whole training process. + +![](images/30303714f451b21eaee42acd79567fecfdfac2cdfb67ff8adc6f2486bb4f5ec4.jpg) +Figure 1: The overview of our proposed method. We formulate it as a Min-Max game. The data of each batch is augmented by multiple pre-processing components with sampled policies $\{ \tau _ { 1 } , \tau _ { 2 } , \cdots , \tau _ { M } \}$ , respectively. Then, a target network is trained to minimize the loss of a large batch, which is formed by multiple augmented instances of the input batch. We extract the training losses of a target network corresponding to different augmentation policies as the reward signal. Finally, the augmentation policy network is trained with the guideline of the processed reward signal, and aims to maximize the training loss of the target network through generating adversarial policies. + +In this paper, we propose an efficient data augmentation method to address the problems mentioned above, which can directly search the best augmentation policy on the full dataset during training a target network, as shown in Figure 1. We first organize the network training and augmentation policy search in an adversarial and online manner. The augmentation policy is dynamically changed along with the training state of the target network, rather than fixed throughout the whole training process like normal AutoAugment (Cubuk et al., 2019). Due to reusing the computation in policy evaluation and dispensing with the retraining of the target network, the computing cost and time overhead are extremely reduced. Then, the augmentation policy network is taken as an adversary to explore the weakness of the target network. We augment the data of each min-batch with various adversarial policies in parallel, rather than the same data augmentation taken in batch augmentation (BA) (Hoffer et al., 2019). Then, several augmented instances of each mini-batch are formed into a large batch for target network learning. As an indicator of the hardness of augmentation policies, the training losses of the target network are used to guide the policy network to generate more aggressive and efficient policies based on REINFORCE algorithm (Williams, 1992). Through adversarial learning, we can train the target network more efficiently and robustly. + +The contributions can be summarized as follows: + +• Our method can directly learn augmentation policies on target tasks, i.e., target networks and full datasets, with a quite low computing cost and time overhead. The direct policy search avoids the performance degradation caused by the policy transfer from proxy tasks to target tasks. +• We propose an adversarial framework to jointly optimize target network training and augmentation policy search. The harder samples augmented by adversarial policies are constantly fed into the target network to promote robust feature learning. Hence, the generalization of the target network can be significantly improved. +• The experiment results show that our proposed method outperforms previous augmentation methods. For instance, we achieve a top-1 test error of $1 . 3 6 \%$ with PyramidNet+ShakeDrop (Yamada et al., 2018) on CIFAR-10, which is the state-of-the-art performance. On ImageNet, we improve the top-1 accuracy of ResNet-50 (He et al., 2016) from $7 6 . 3 \%$ to $7 9 . 4 \%$ without extra data, which is even $1 . 7 7 \%$ better than AutoAugment (Cubuk et al., 2019). + +# 2 RELATED WORK + +Common data augmentation, which can generate extra samples by some label-preserved transformations, is usually used to increase the size of datasets and improve the generalization of networks, such as on MINST, CIFAR-10 and ImageNet (Krizhevsky et al., 2012; Wan et al., 2013; Szegedy et al., 2015). However, human-designed augmentation policies are specified for different datasets. For example, flipping, the widely used transformation on CIFAR-10/CIFAR-100 and ImageNet, is not suitable for MINST, which will destroy the property of original samples. + +Hence, several works (Lemley et al., 2017; Cubuk et al., 2019; Lin et al., 2019; Ho et al., 2019) have attempted to automatically learn data augmentation policies. Lemley et al. (2017) propose a method called Smart Augmentation, which merges two or more samples of a class to improve the generalization of a target network. The result also indicates that an augmentation network can be learned when a target network is being training. Through well designing the search space of data augmentation policies, AutoAugment (Cubuk et al., 2019) takes a recurrent neural network (RNN) as a sample controller to find the best data augmentation policy for a selected dataset. To reduce the computing cost, the augmentation policy search is performed on proxy tasks. Population based augmentation (PBA) (Ho et al., 2019) replaces the fixed augmentation policy with a dynamic schedule of augmentation policy along with the training process, which is mostly related to our work. Inspired by population based training (PBT) (Jaderberg et al., 2017), the augmentation policy search problem in PBA is modeled as a process of hyperparameter schedule learning. However, the augmentation schedule learning is still performed on proxy tasks. The learned policy schedule should be manually adjusted when the training process of a target network is non-matched with proxy tasks. + +Another related topic is Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which has recently attracted lots of research attention due to its fascinating performance, and also been used to enlarge datasets through directly synthesizing new images (Tran et al., 2017; Perez & Wang, 2017; Antoniou et al., 2017; Gurumurthy et al., 2017; Frid-Adar et al., 2018). Although we formulate our proposed method as a Min-Max game, there exists an obvious difference with traditional GANs. We want to find the best augmentation policy to perform image transformation along with the training process, rather than synthesize new images. Peng et al. (2018) also take such an idea to optimize the training process of a target network in human pose estimation. + +# 3 METHOD + +In this section, we present the implementation of Adversarial AutoAugment. First, the motivation for the adversarial relation between network learning and augmentation policy is discussed. Then, we introduce the search space with the dynamic augmentation policy. Finally, the joint framework for network training and augmentation policy search is presented in detail. + +# 3.1 MOTIVATIONS + +Although some human-designed data augmentations have been used in the training of DNNs, such as randomly cropping and horizontally flipping on CIFAR-10/CIFAR-100 and ImageNet, limited randomness will make it very difficult to generate effective samples at the tail end of the training. To struggle with the problem, more randomness about image transformation is introduced into the search space of AutoAugment (Cubuk et al., 2019) (described in Section 3.2). However, the learned policy is fixed for the entire training process. All of possible instances of each example will be send to the target network repeatedly, which still results in an inevitable overfitting in a long-epoch training. This phenomenon indicates that the learned policy is not adaptive to the training process of a target network, especially found on proxy tasks. Hence, the dynamic and adversarial augmentation policy with the training process is considered as the crucial feature in our search space. + +Another consideration is how to improve the efficiency of the policy search. In AutoAugment (Cubuk et al., 2019), to evaluate the performance of augmentation policies, a lot of child models should be trained from scratch nearly to convergence. The computation in training and evaluating the performance of different sampled policies can not be reused, which leads to huge waste of computation resources. In this paper, we propose a computing-efficient policy search framework through reusing prior computation in policy evaluation. Only one target network is used to evaluate the performance of different policies with the help of the training losses of corresponding augmented instances. The augmentation policy network is learned from the intermediate state of the target network, which makes generated augmentation policies more aggressive and adaptive. On the contrary, to combat harder examples augmented by adversarial policies, the target network has to learn more robust features, which makes the training more efficiently. + +![](images/cf1281f8bae28f7b31d6aab6d29d8044913d73f1d9a58a90985849e432656219.jpg) +Figure 2: An example of dynamic augmentation policies learned with ResNet-50 on ImageNet. With the training process of the target network, harder augmentation policies are sampled to combat overfitting. Intuitively, more geometric transformations, such as TranslateX, ShearY and Rotate, are picked in our sampled policies, which is obviously different from AutoAugment (Cubuk et al., 2019) concentrating on color-based transformations. + +# 3.2 SEARCH SPACE + +In this paper, the basic structure of the search space of AutoAugment (Cubuk et al., 2019) is reserved. An augmentation policy is defined as that it is composed by 5 sub-policies, each sub-policy contains two image operations to be applied orderly, each operation has two corresponding parameters, i.e., the probability and magnitude of the operation. Finally, the 5 best policies are concatenated to form a single policy with 25 sub-policies. For each image in a mini-batch, only one sub-policy will be randomly selected to be applied. To compare with AutoAugment (Cubuk et al., 2019) conveniently, we just slightly modify the search space with removing the probability of each operation. This is because that we think the stochasticity of an operation with a probability requires a certain epochs to take effect, which will detain the feedback of the intermediate state of the target network. There are totally 16 image operations in our search space, including ShearX/Y, TranslateX/Y, Rotate, AutoContrast, Invert, Equalize, Solarize, Posterize, Contrast, Color, Brightness, Sharpness, Cutout (Devries & Taylor, 2017) and Sample Pairing (Inoue, 2018). The range of the magnitude is also discretized uniformly into 10 values. To guarantee the convergence during adversarial learning, the magnitude of all the operations are set in a moderate range.1 Besides, the randomness during the training process is introduced into our search space. Hence, the search space of the policy in each epoch has $| S | = ( 1 6 \times 1 0 ) ^ { 1 0 } \approx 1 . 1 \times 1 0 ^ { 2 2 }$ possibilities. Considering the dynamic policy, the number of possible policies with the whole training process can be expressed as $| S | ^ { \# e p o c h s }$ . An example of dynamically learning the augmentation policy along with the training process is shown in Figure 2. We observe that the magnitude (an indication of difficulty) gradually increases with the training process. + +# 3.3 ADVERSARIAL LEARNING + +In this section, the adversarial framework of jointly optimizing network training and augmentation policy search is presented in detail. We use the augmentation policy network $\boldsymbol { \mathcal { A } } ( \cdot , \pmb { \theta } )$ as an adversary, which attempts to increase the training loss of the target network $\mathcal { F } ( \cdot , w )$ through adversarial learning. The target network is trained by a large batch formed by multiple augmented instances of each batch to promote invariant learning (Salazar et al., 2018), and the losses of different augmentation policies applied on the same data are used to train the augmentation policy network by RL algorithm. + +Considering the target network $\mathcal { F } ( \cdot , w )$ with a loss function $\mathcal { L } [ \mathcal { F } ( \pmb { x } , \pmb { w } ) , \pmb { y } ]$ , where each example is transformed by some random data augmentation $o ( \cdot )$ , the learning process of the target network can be defined as the following minimization problem + +$$ +\pmb { w } ^ { * } = \underset { \pmb { w } } { \arg \operatorname* { m i n } } \ \underset { \pmb { x } \sim \Omega } { \mathbb { E } } \mathcal { L } [ \mathcal { F } ( o ( \pmb { x } ) , \pmb { w } ) , \pmb { y } ] , +$$ + +where $\Omega$ is the training set, $_ { \textbf { \em x } }$ and $\textbf { { y } }$ are the input image and the corresponding label, respectively. The problem is usually solved by vanilla SGD with a learning rate $\eta$ and batch size $N$ , and the training procedure for each batch can be expressed as + +$$ +\pmb { w } _ { t + 1 } = \pmb { w } _ { t } - \eta \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \nabla _ { \pmb { w } } \mathcal { L } [ \mathcal { F } ( o ( x _ { n } ) , \pmb { w } , y _ { n } ] . +$$ + +To improve the convergence performance of DNNs, more random and efficient data augmentation is performed under the help of the augmentation policy network. Hence, the minimization problem should be slightly modified as + +$$ +\pmb { w } ^ { * } = \underset { \pmb { w } } { \arg \operatorname* { m i n } } \ \underset { \pmb { x } \sim \Omega } { \mathbb { E } } \ \underset { \pmb { \mathcal { A } } ( \cdot , \pmb { \theta } ) } { \mathbb { E } } \mathcal { L } [ \mathcal { F } ( \tau ( \pmb { x } ) , \pmb { w } ) , \pmb { y } ] , +$$ + +where $\tau ( \cdot )$ represents the augmentation policy generated by the network $\boldsymbol { \mathcal { A } } ( \cdot , \pmb { \theta } )$ . Accordingly, the training rule can be rewritten as + +$$ +\pmb { w } _ { t + 1 } = \pmb { w } _ { t } - \eta \frac { 1 } { M \cdot N } \sum _ { m = 1 } ^ { M } \sum _ { n = 1 } ^ { N } \nabla _ { \pmb { w } } \mathcal { L } [ \mathcal { F } ( \tau _ { m } ( x _ { n } ) , \pmb { w } ) , y _ { n } ] , +$$ + +where we introduce $M$ different instances of each input example augmented by adversarial policies $\{ \tau _ { 1 } , \tau _ { 2 } , \cdots , \tau _ { M } \}$ . For convenience, we denote the training loss of a mini-batch corresponding to the augmentation policy $\tau _ { m }$ as + +$$ +\mathcal { L } _ { m } = \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \mathcal { L } [ \mathcal { F } ( \tau _ { m } ( x _ { n } ) , \pmb { w } ) , y _ { n } ] . +$$ + +Hence, we have an equivalent form of Equation 4 + +$$ +\pmb { w } _ { t + 1 } = \pmb { w } _ { t } - \eta \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \nabla _ { \pmb { w } } \mathcal { L } _ { m } . +$$ + +Note that the training procedure can be regarded as a larger $N \cdot M$ batch training or an average over $M$ instances of gradient computation without changing the learning rate, which will lead to a reduction of gradient variance and a faster convergence of the target network Hoffer et al. (2019). However, overfitting will also come. To overcome the problem, the augmentation policy network is designed to increase the training loss of the target network with harder augmentation policies. Therefore, we can mathematically express the object as the following maximization problem + +$$ +\begin{array} { c } { \pmb { \theta } ^ { * } = \underset { \pmb { \theta } } { \arg \operatorname* { m a x } } J ( \pmb { \theta } ) , } \\ { \mathrm { w h e r e } ~ J ( \pmb { \theta } ) = \underset { \pmb { x } \sim \Omega } { \mathbb { E } } ~ \underset { \pmb { \mathbb { E } } ( \cdot , \pmb { \theta } ) } { \mathbb { E } } \mathcal { L } [ \mathcal { F } ( \tau ( \pmb { x } ) , \pmb { w } ) , \pmb { y } ] . } \end{array} +$$ + +Similar to AutoAugment (Cubuk et al., 2019), the augmentation policy network is also implemented as a RNN shown in Figure 3. At each time step of the RNN controller, the softmax layer will predict an action corresponding to a discrete parameter of a sub-policy, and then an embedding of the predicted action will be fed into the next time step. In our experiments, the RNN controller will predict 20 discrete parameters to form a whole policy. + +![](images/d2f4b35084c6e611d6139d1fbef94757f5d6b6d525d6edbd65d47563cffa3f4c.jpg) +Figure 3: The basic architecture of the controller for generating a sub-policy, which consists of two operations with corresponding parameters, the type and magnitude of each operation. When a policy contains $Q$ sub-policies, the basic architecture will be repeated $Q$ times. Following the setting of AutoAugment (Cubuk et al., 2019), the number of sub-policies $Q$ is set to 5 in this paper. + +However, there has a severe problem in jointly optimizing target network training and augmentation policy search. This is because that non-differentiable augmentation operations break gradient flow from the target network $\mathcal { F }$ to the augmentation policy network $\mathcal { A }$ (Wang et al., 2017; Peng et al., 2018). As an alternative approach, REINFORCE algorithm (Williams, 1992) is applied to optimize the augmentation policy network as + +$$ +\begin{array} { r l } & { \nabla _ { \theta } J ( \theta ) = \nabla _ { \theta } \underset { x \sim \Omega \tau \sim A ( \cdot , \theta ) } { \mathbb { E } } ~ \underset { { A } ^ { ( \cdot ) } } { \mathbb { E } } ~ \mathcal { L } [ \mathcal { F } ( \tau ( x ) , w ) , y ] } \\ & { \approx \displaystyle \sum _ { m } \mathcal { L } _ { m } \nabla _ { \theta } p _ { m } = \sum _ { m } \mathcal { L } _ { m } p _ { m } \nabla _ { \theta } \log p _ { m } } \\ & { ~ = \underset { { \tau \sim A ( \cdot , \theta ) } } { \mathbb { E } } ~ \mathcal { L } _ { m } \nabla _ { \theta } \log p _ { m } } \\ & { ~ \approx \frac { 1 } { M } \displaystyle \sum _ { m = 1 } ^ { M } \mathcal { L } _ { m } \nabla _ { \theta } \log p _ { m } , } \end{array} +$$ + +where $p _ { m }$ represents the probability of the policy $\tau _ { m }$ . To reduce the variance of gradient $\nabla _ { \pmb { \theta } } J ( \pmb { \theta } )$ , we replace the training loss of a mini-batch ${ \mathcal { L } } _ { m }$ with ${ \widehat { \mathcal { L } } } _ { m }$ a moving average over a certain minibatches2, and then normalize it among $M$ instances as ${ \widetilde { \mathcal { L } } } _ { m }$ . Hence, the training procedure of the augmentation policy network can be expressed as + +$$ +\begin{array} { r l } & { \nabla _ { \pmb { \theta } } J ( \pmb { \theta } ) \approx \cfrac { 1 } { M } \displaystyle \sum _ { m = 1 } ^ { M } \widetilde { \mathcal { L } } _ { m } \nabla _ { \pmb { \theta } } \log p _ { m } , } \\ & { \theta _ { e + 1 } = \theta _ { e } + \beta \displaystyle \frac { 1 } { M } \displaystyle \sum _ { m = 1 } ^ { M } \widetilde { \mathcal { L } } _ { m } \nabla _ { \pmb { \theta } } \log p _ { m } , } \end{array} +$$ + +The adversarial learning of target network training and augmentation policy search is summarized as Algorithm 1. + +# 4 EXPERIMENTS AND ANALYSIS + +In this section, we first reveal the details of experiment settings. Then, we evaluate our proposed method on CIFAR-10/CIFAR-100, ImageNet, and compare it with previous methods. Results in Figure 4 show our method achieves the state-of-the-art performance with higher computing and time efficiency3. + +Algorithm 1 Joint Training of Target Network and Augmentation Policy Network + +Initialization: target network $\mathcal { F } ( \cdot , w )$ , augmentation policy network $\boldsymbol { \mathcal { A } } ( \cdot , \pmb { \theta } )$ Input: input examples $_ { \textbf { \em x } }$ , corresponding labels $\textbf { { y } }$ + +1: for $1 \leq e \leq$ epochs do +2: Initialize $\widehat { \mathcal { L } } _ { m } = 0 , \forall m \in \{ 1 , 2 , \cdots , M \}$ ; +3: Generate $M$ policies with the probabilities $\{ p _ { 1 } , p _ { 2 } , \cdots , p _ { M } \}$ ; +4: for $1 \leq t \leq T$ do +5: Augment each batch data with $M$ generated policies, respectively; +6: Update $w _ { e , t + 1 }$ according to Equation 4; +7: Update ${ \widehat { \mathcal { L } } } _ { m }$ through moving average, $\forall m \in \{ 1 , 2 , \cdot \cdot \cdot , M \}$ ; +8: Collect $\{ \widehat { \mathcal { L } } _ { 1 } , \widehat { \mathcal { L } } _ { 2 } , \cdots , \widehat { \mathcal { L } } _ { M } \}$ ; +9: Normalize ${ \widehat { \mathcal { L } } } _ { m }$ among $M$ instances as $\widetilde { \mathcal { L } } _ { m } , \forall m \in \{ 1 , 2 , \cdots , M \}$ ; +10: Update $\pmb { \theta } _ { e + 1 }$ via Equation 9; +11: Output $w ^ { \ast } , \theta ^ { \ast }$ + +# 4.1 EXPERIMENT SETTINGS + +The RNN controller is implemented as a one-layer LSTM (Hochreiter & Schmidhuber, 1997). We set the hidden size to 100, and the embedding size to 32. We use Adam optimizer (Kingma & Ba, 2015) with a initial learning rate 0.00035 to train the controller. To avoid unexpected rapid convergence, an entropy penalty of a weight of 0.00001 is applied. All the reported results are the mean of five runs with different initializations. + +# 4.2 EXPERIMENTS ON CIFAR-10 AND CIFAR-100 + +CIFAR-10 dataset (Krizhevsky & Hinton, 2009) has totally 60000 images. The training and test sets have 50000 and 10000 images, respectively. Each image in size of $3 2 \times 3 2$ belongs to one of 10 classes. We evaluate our proposed method with the following models: Wide-ResNet-28- 10 (Zagoruyko & Komodakis, 2016), Shake-Shake $( 2 6 ~ 2 \mathrm { x } 3 2 \mathrm { d } )$ (Gastaldi, 2017), Shake-Shake (26 $2 \mathrm { x } 9 6 \mathrm { d } )$ (Gastaldi, 2017), Shake-Shake $( 2 6 2 \mathrm { x } 1 1 2 \mathrm { d } )$ (Gastaldi, 2017), PyramidNet+ShakeDrop (Han et al., 2017; Yamada et al., 2018). All the models are trained on the full training set. + +Training details: The Baseline is trained with the standard data augmentation, namely, randomly cropping a part of $3 2 \times 3 2$ from the padded image and horizontally flipping it with a probability of 0.5. The Cutout (Devries & Taylor, 2017) randomly select a $1 6 \times 1 6$ patch of each image, and then set the pixels of the selected patch to zeros. For our method, the searched policy is applied in addition to standard data augmentation and Cutout. For each image in the training process, standard data augmentation, the searched policy and Cutout are applied in sequence. For Wide-ResNet-28- 10, the step learning rate (LR) schedule is adopted. The cosine LR schedule is adopted for the other models. More details about model hyperparameters are supplied in A.1. + +Choice of $M$ : To choose the optimal $M$ , we select Wide-ResNet-28-10 as a target network, and evaluate the performance of our proposed method verse different $M$ , where $M \in \{ 2 , 4 , 8 , 1 6 , 3 2 \}$ . From Figure 5, we can observe that the test accuracy of the model improves rapidly with the increase of $M$ up to 8. The further increase of $M$ does not bring a significant improvement. Therefore, to balance the performance and the computing cost, $M$ is set to 8 in all the following experiments. + +CIFAR-10 results: In Table 1, we report the test error of these models on CIFAR-10. For all of these models, our proposed method can achieve better performance compared to previous methods. We achieve $0 . 7 8 \%$ and $0 . 6 8 \%$ improvement on Wide-ResNet-28-10 compared to AutoAugment and PBA, respectively. We achieve a top-1 test error of $1 . 3 6 \%$ with PyramidNet $^ +$ ShakeDrop, which is $0 . 1 \%$ better than the current state-of-the-art reported in Ho et al. (2019). As shown in Figure 6(a) and 6(b),we further visualize the probability distribution of the parameters of the augmentation policies learned with PyramidNet+ShakeDrop on CIFAR-10 over time. From Figure 6(a), we can find that the percentages of some operations, such as TranslateY, Rotate, Posterize, and SampleParing, gradually increase along with the training process. Meanwhile, more geometric transformations, such as TranslateX, TranslateY, and Rotate, are picked in the sampled augmentation policies, which is different from color-focused AutoAugment (Cubuk et al., 2019) on CIFAR-10. Figure 6(b) shows that large magnitudes gain higher percentages during training. However, at the tail of training, low magnitudes remain considerable percentages. This indicates that our method does not simply learn the transformations with the extremes of the allowed magnitudes to spoil the target network. + +![](images/1d9bd1c440b329d4cde0cd4933e8cd51662c730a2c81bacfeed27c2444ff1516.jpg) +Figure 4: The Comparison of normalized performance between AutoAugment and our method. Please refer to the following tables for more details. + +![](images/8f0622955a06ab6ba860d80c3c02ae266fb49a0c8467d420de114247a0cfe754.jpg) +Figure 5: The Top-1 test accuracy of WideResNet-28-10 on CIFAR-10 verse different $M$ , where $M \in \{ 2 , 4 , 8 , 1 6 , 3 2 \}$ . + +![](images/969c5da3cca5ba7fe0f9f8865ef328f9131f97876658c78dfb77dd5e89fdd3d5.jpg) +Figure 6: Probability distribution of the parameters in the learned augmentation policies on CIFAR10 over time. The number in (b) represents the magnitude of one operation. Larger number stands for more dramatic image transformations. The probability distribution of each parameter is the mean of each five epochs. + +CIFAR-100 results: We also evaluate our proposed method on CIFAR-100, as shown in Table 2. +As we can observe from the table, we also achieve the state-of-the-art performance on this dataset. + +Table 1: Top-1 test error $( \% )$ on CIFAR-10. We replicate the results of Baseline, Cutout and AutoAugment methods from Cubuk et al. (2019), and the results of PBA from Ho et al. (2019) in all of our experiments. + +
ModelBaselineCutoutAutoAugmentPBAOur Method
Wide-ResNet-28-103.873.082.682.581.90±0.15
Shake-Shake (26 2x32d)3.553.022.472.542.36±0.10
Shake-Shake (26 2x96d)2.862.561.992.031.85±0.12
Shake-Shake (26 2x112d)2.822.571.892.031.78±0.05
PyramidNet+ShakeDrop2.672.311.481.461.36±0.06
+ +# 4.3 EXPERIMENTS ON IMAGENET + +As a great challenge in image recognition, ImageNet dataset (Deng et al., 2009) has about 1.2 million training images and 50000 validation images with 1000 classes. In this section, we directly search the augmentation policy on the full training set and train ResNet-50 (He et al., 2016), ResNet-50-D (He et al., 2018) and ResNet-200 (He et al., 2016) from scratch. + +Table 2: Top-1 test error $( \% )$ on CIFAR-100. + +
ModelBaselineCutout AutoAugmentPBAOur Method
Wide-ResNet-28-1018.8018.4117.0916.7315.49±0.18
Shake-Shake (26 2x96d)17.0516.0014.2815.3114.10±0.15
PyramidNet+ShakeDrop13.9912.1910.6710.9410.42±0.20
+ +Training details: For the baseline augmentation, we randomly resize and crop each input image to a size of $2 2 4 \times 2 2 4$ , and then horizontally flip it with a probability of 0.5. For AutoAugment (Cubuk et al., 2019) and our method, the baseline augmentation and the augmentation policy are both used for each image. The cosine LR schedule is adopted in the training process. The model hyperparameters on ImageNet is also detailed in A.1. + +ImageNet results: The performance of our proposed method on ImageNet is presented in Table 3. It can be observed that we achieve a top-1 accuracy $7 9 . 4 0 \%$ on ResNet-50 without extra data. To the best of our knowledge, this is the highest top-1 accuracy for ResNet-50 learned on ImageNet. Besides, we only replace the ResNet-50 architecture with ResNet-50-D, and achieve a consistent improvement with a top-1 accuracy of $8 0 . 0 0 \%$ . + +Table 3: Top-1 / Top-5 test error $( \% )$ on ImageNet. Note that the result of ResNet-50-D is achieved only through substituting the architecture. + +
ModelBaselineAutoAugmentPBAOur Method
ResNet-5023.69 / 6.9222.37 /6.18120.60±0.15 /5.53±0.05
ResNet-50-D22.84 / 6.48120.00±0.12/5.25±0.03
ResNet-20021.52 / 5.8520.00 /4.90118.68±0.18 /4.70±0.05
+ +# 4.4 ABLATION STUDY + +To check the effect of each component in our proposed method, we report the test error of ResNet-50 on ImageNet the following augmentation methods in Table 4. + +• Baseline: Training regularly with the standard data augmentation and step LR schedule. +• Fixed: Augmenting all the instances of each batch with the standard data augmentation fixed throughout the entire training process. Random: Augmenting all the instances of each batch with randomly and dynamically generated policies. Ours: Augmenting all the instances of each batch with adversarial policies sampled by the policy network along with the training process. + +From the table, we can find that Fixed can achieve $0 . 9 9 \%$ error reduction compared to Baseline. This shows that a large-batch training with multiple augmented instances of each mini-batch can indeed improve the generalization of the model, which is consistent with the conclusion presented in Hoffer et al. (2019). In addition, the test error of Random is $1 . 0 2 \%$ better than Fixed. This indicates that augmenting batch with randomly generated policies can reduce overfitting in a certain extent. Furthermore, our method achieves the best test error of $2 0 . 6 0 \%$ through augmenting samples with adversarial policies. From the result, we can conclude that these policies generated by the policy network are more adaptive to the training process, and make the target network have to learn more robust features. + +# 4.5 COMPUTING COST AND TIME OVERHEAD + +Computing Cost: The computation in target network training is reused for policy evaluation. This makes the computing cost in policy search become negligible. Although there exists an increase of computing cost in target network training, the total computing cost in training one target network with augmentation policies is quite small compared to prior work. + +Time Overhead: Since we just train one target network with a large batch distributedly and simultaneously, the time overhead of the large-batch training is equal to the regular training. Meanwhile, the joint optimization of target network training and augmentation policy search dispenses with the process of offline policy search and the retraining of a target network, which leads to a extreme time overhead reduction. + +Table 4: Top-1 test error $( \% )$ of ResNet-50 with different augmentation methods on ImageNet. + +
MethodAug. PolicyEnlarge BatchLR ScheduleTest Error
BaselinestandardM=1step23.69
FixedstandardM=8cosine22.70
RandomrandomM=8cosine21.68
OursadversarialM=8cosine20.60
+ +In Table 5, we take the training of ResNet-50 on ImageNet as an example to compare the computing cost and time overhead of our method and AutoAugment. From the table, we can find that our method is $1 2 \times$ less computing cost and $1 1 \times$ shorter time overhead than AutoAugment. + +Table 5: The comparison of computing cost (GPU hours) and time overhead (days) in training ResNet-50 on ImageNet between AutoAugment and our method. The computing cost and time overhead are estimated on 64 NVIDIA Tesla V100s. + +
MethodComputing CostTime Overhead
SearchingTrainingTotalSearchingTrainingTotal
AutoAugment150001601516010111
Our Method~012801280~011
+ +# 4.6 TRANSFERABILITY ACROSS DATASETS AND ARCHITECTURES + +To further show the higher efficiency of our method, the transferability of the learned augmentation policies is evaluated in this section. We first take a snapshot of the adversarial training process of ResNet-50 on ImageNet, and then directly use the learned dynamic augmentation policies to regularly train the following models: Wide-ResNet-28-10 on CIFAR-10/100, ResNet-50-D on ImageNet and ResNet200 on ImageNet. Table 6 presents the experimental results of the transferability. From the table, we can find that a competitive performance can be still achieved through direct policy transfer. This indicates that the learned augmentation policies transfer well across datasets and architectures. However, compared to the proposed method, the policy transfer results in an obvious performance degradation, especially the transfer across datasets. + +Table 6: Top-1 test error $( \% )$ of the transfer of the augmentation policies learned with ResNet-50 on ImageNet. + +
MethodDatasetAutoAugmentOur MethodPolicy Transfer
Wide-ResNet-28-10CIFAR-102.681.902.45±0.13
Wide-ResNet-28-10CIFAR-10017.0915.4916.48±0.15
ResNet-50-DImageNet120.0020.20±0.05
ResNet-200ImageNet20.0018.6819.05±0.10
+ +# 5 CONCLUSION + +In this paper, we introduce the idea of adversarial learning into automatic data augmentation. The policy network tries to combat the overfitting of the target network through generating adversarial policies with the training process. To oppose this, robust features are learned in the target network, which leads to a significant performance improvement. Meanwhile, the augmentation policy search is performed along with the training of a target network, and the computation in network training is reused for policy evaluation, which can extremely reduce the search cost and make our method more computing-efficient. + +# REFERENCES + +Antreas Antoniou, Amos J. Storkey, and Harrison Edwards. Data augmentation generative adversarial networks. ICLR, 2017. + +Ekin D. Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V. Le. Autoaugment: ´ Learning augmentation policies from data. 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A bayesian data augmentation approach for learning deep models. NIPS, 2017. + +Li Wan, Matthew Zeiler, Sixin Zhang, Yann LeCun, and Rob Fergus. Regularization of neural networks using dropconnect. ICML, 2013. + +Xiaolong Wang, Abhinav Shrivastava, and Abhinav Gupta. A-fast-rcnn: Hard positive generation via adversary for object detection. CVPR, 2017. + +Ronald J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 1992. + +Yoshihiro Yamada, Masakazu Iwamura, and Koichi Kise. Shakedrop regularization. CoRR, abs/1802.02375, 2018. + +Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. British Machine Vision Conference, 2016. + +Zhao Zhong, Junjie Yan, and Cheng-Lin Liu. Practical network blocks design with Q-learning. CVPR, 2018a. + +Zhao Zhong, Zichen Yang, Boyang Deng, Junjie Yan, Wei Wu, Jing Shao, and Cheng-Lin Liu. BlockQNN: Efficient block-wise neural network architecture generation. CoRR, abs/1808.05584, 2018b. + +Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. ICLR, 2016. + +Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. CVPR, 2017. + +A APPENDIX + +# A.1 HYPERPARAMETERS + +We detail the model hyperparameters on CIFAR-10/CIFAR-100 and ImageNet in Table 7. + +Table 7: Model hyperparameters on CIFAR-10/CIFAR-100 and ImageNet. LR represents learning rate, and WD represents weight decay. We do not specifically tune these hyperparameters, and all of these are consistent with previous works, expect for the number of epochs. + +
DatasetModelBatch Size (N · M)LRWDEpoch
CIFAR-10Wide-ResNet-28-10128·80.15e-4200
CIFAR-10Shake-Shake (26 2x32d)128·80.21e-4600
CIFAR-10Shake-Shake ( (262x96d)128·80.21e-4600
CIFAR-10Shake-Shake (26 2x112d)128·80.21e-4600
CIFAR-10PyramidNet+ShakeDrop128·80.11e-4600
CIFAR-100Wide-ResNet-28-10128·80.15e-4200
CIFAR-100Shake-Shake (26 2x96d)128·80.15e-41200
CIFAR-100PyramidNet+ShakeDrop128·80.51e-41200
ImageNetResNet-502048·80.81e-4120
ImageNetResNet-50-D2048·80.81e-4120
ImageNetResNet-2002048·80.81e-4120
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Recently, human-designed data augmentation has been gradually replaced by automatically learned augmentation policy. Through finding the best policy in well-designed search space of data augmentation, AutoAugment (Cubuk et al., 2019) can significantly improve validation accuracy on image classification tasks. However, this approach is not computationally practical for large-scale problems. In this paper, we develop an adversarial method to arrive at a computationally-affordable solution called Adversarial AutoAugment, which can simultaneously optimize target related object and augmentation policy search loss. The augmentation policy network attempts to increase the training loss of a target network through generating adversarial augmentation policies, while the target network can learn more robust features from harder examples to improve the generalization. In contrast to prior work, we reuse the computation in target network training for policy evaluation, and dispense with the retraining of the target network. Compared to AutoAugment, this leads to about $1 2 \\times$ reduction in computing cost and $1 1 \\times$ shortening in time overhead on ImageNet. We show experimental results of our approach on CIFAR-10/CIFAR-100, ImageNet, and demonstrate significant performance improvements over state-of-the-art. On CIFAR-10, we achieve a top-1 test error of $1 . 3 6 \\%$ , which is the currently best performing single model. On ImageNet, we achieve a leading performance of top-1 accuracy $7 9 . 4 0 \\%$ on ResNet-50 and $8 0 . 0 0 \\%$ on ResNet-50-D without extra data. ", + "bbox": [ + 233, + 335, + 764, + 627 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 655, + 334, + 671 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Massive amount of data have promoted the great success of deep learning in academia and industry. The performance of deep neural networks (DNNs) would be improved substantially when more supervised data is available or better data augmentation method is adapted. Data augmentation such as rotation, flipping, cropping, etc., is a powerful technique to increase the amount and diversity of data. Experiments show that the generalization of a neural network can be efficiently improved through manually designing data augmentation policies. However, this needs lots of knowledge of human expert, and sometimes shows the weak transferability across different tasks and datasets in practical applications. Inspired by neural architecture search (NAS)(Zoph & Le, 2016; Zoph et al., 2017; Zhong et al., 2018a;b; Guo et al., 2018), a reinforcement learning (RL) (Williams, 1992) method called AutoAugment is proposed by Cubuk et al. (2019), which can automatically learn the augmentation policy from data and provide an exciting performance improvement on image classification tasks. However, the computing cost is huge for training and evaluating thousands of sampled policies in the search process. Although proxy tasks, i.e., smaller models and reduced datasets, are taken to accelerate the searching process, tens of thousands of GPU-hours of consumption are still required. In addition, these data augmentation policies optimized on proxy tasks are not guaranteed to be optimal on the target task, and the fixed augmentation policy is also sub-optimal for the whole training process. ", + "bbox": [ + 174, + 688, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/30303714f451b21eaee42acd79567fecfdfac2cdfb67ff8adc6f2486bb4f5ec4.jpg", + "image_caption": [ + "Figure 1: The overview of our proposed method. We formulate it as a Min-Max game. The data of each batch is augmented by multiple pre-processing components with sampled policies $\\{ \\tau _ { 1 } , \\tau _ { 2 } , \\cdots , \\tau _ { M } \\}$ , respectively. Then, a target network is trained to minimize the loss of a large batch, which is formed by multiple augmented instances of the input batch. We extract the training losses of a target network corresponding to different augmentation policies as the reward signal. Finally, the augmentation policy network is trained with the guideline of the processed reward signal, and aims to maximize the training loss of the target network through generating adversarial policies. " + ], + "image_footnote": [], + "bbox": [ + 200, + 102, + 730, + 369 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we propose an efficient data augmentation method to address the problems mentioned above, which can directly search the best augmentation policy on the full dataset during training a target network, as shown in Figure 1. We first organize the network training and augmentation policy search in an adversarial and online manner. The augmentation policy is dynamically changed along with the training state of the target network, rather than fixed throughout the whole training process like normal AutoAugment (Cubuk et al., 2019). Due to reusing the computation in policy evaluation and dispensing with the retraining of the target network, the computing cost and time overhead are extremely reduced. Then, the augmentation policy network is taken as an adversary to explore the weakness of the target network. We augment the data of each min-batch with various adversarial policies in parallel, rather than the same data augmentation taken in batch augmentation (BA) (Hoffer et al., 2019). Then, several augmented instances of each mini-batch are formed into a large batch for target network learning. As an indicator of the hardness of augmentation policies, the training losses of the target network are used to guide the policy network to generate more aggressive and efficient policies based on REINFORCE algorithm (Williams, 1992). Through adversarial learning, we can train the target network more efficiently and robustly. ", + "bbox": [ + 173, + 486, + 825, + 694 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The contributions can be summarized as follows: ", + "bbox": [ + 176, + 700, + 495, + 714 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Our method can directly learn augmentation policies on target tasks, i.e., target networks and full datasets, with a quite low computing cost and time overhead. The direct policy search avoids the performance degradation caused by the policy transfer from proxy tasks to target tasks. \n• We propose an adversarial framework to jointly optimize target network training and augmentation policy search. The harder samples augmented by adversarial policies are constantly fed into the target network to promote robust feature learning. Hence, the generalization of the target network can be significantly improved. \n• The experiment results show that our proposed method outperforms previous augmentation methods. For instance, we achieve a top-1 test error of $1 . 3 6 \\%$ with PyramidNet+ShakeDrop (Yamada et al., 2018) on CIFAR-10, which is the state-of-the-art performance. On ImageNet, we improve the top-1 accuracy of ResNet-50 (He et al., 2016) from $7 6 . 3 \\%$ to $7 9 . 4 \\%$ without extra data, which is even $1 . 7 7 \\%$ better than AutoAugment (Cubuk et al., 2019). ", + "bbox": [ + 215, + 720, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 102, + 344, + 117 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Common data augmentation, which can generate extra samples by some label-preserved transformations, is usually used to increase the size of datasets and improve the generalization of networks, such as on MINST, CIFAR-10 and ImageNet (Krizhevsky et al., 2012; Wan et al., 2013; Szegedy et al., 2015). However, human-designed augmentation policies are specified for different datasets. For example, flipping, the widely used transformation on CIFAR-10/CIFAR-100 and ImageNet, is not suitable for MINST, which will destroy the property of original samples. ", + "bbox": [ + 174, + 132, + 825, + 217 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hence, several works (Lemley et al., 2017; Cubuk et al., 2019; Lin et al., 2019; Ho et al., 2019) have attempted to automatically learn data augmentation policies. Lemley et al. (2017) propose a method called Smart Augmentation, which merges two or more samples of a class to improve the generalization of a target network. The result also indicates that an augmentation network can be learned when a target network is being training. Through well designing the search space of data augmentation policies, AutoAugment (Cubuk et al., 2019) takes a recurrent neural network (RNN) as a sample controller to find the best data augmentation policy for a selected dataset. To reduce the computing cost, the augmentation policy search is performed on proxy tasks. Population based augmentation (PBA) (Ho et al., 2019) replaces the fixed augmentation policy with a dynamic schedule of augmentation policy along with the training process, which is mostly related to our work. Inspired by population based training (PBT) (Jaderberg et al., 2017), the augmentation policy search problem in PBA is modeled as a process of hyperparameter schedule learning. However, the augmentation schedule learning is still performed on proxy tasks. The learned policy schedule should be manually adjusted when the training process of a target network is non-matched with proxy tasks. ", + "bbox": [ + 174, + 223, + 825, + 431 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Another related topic is Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which has recently attracted lots of research attention due to its fascinating performance, and also been used to enlarge datasets through directly synthesizing new images (Tran et al., 2017; Perez & Wang, 2017; Antoniou et al., 2017; Gurumurthy et al., 2017; Frid-Adar et al., 2018). Although we formulate our proposed method as a Min-Max game, there exists an obvious difference with traditional GANs. We want to find the best augmentation policy to perform image transformation along with the training process, rather than synthesize new images. Peng et al. (2018) also take such an idea to optimize the training process of a target network in human pose estimation. ", + "bbox": [ + 174, + 438, + 825, + 550 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 564, + 282, + 580 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we present the implementation of Adversarial AutoAugment. First, the motivation for the adversarial relation between network learning and augmentation policy is discussed. Then, we introduce the search space with the dynamic augmentation policy. Finally, the joint framework for network training and augmentation policy search is presented in detail. ", + "bbox": [ + 176, + 589, + 823, + 645 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 MOTIVATIONS ", + "text_level": 1, + "bbox": [ + 174, + 654, + 313, + 669 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Although some human-designed data augmentations have been used in the training of DNNs, such as randomly cropping and horizontally flipping on CIFAR-10/CIFAR-100 and ImageNet, limited randomness will make it very difficult to generate effective samples at the tail end of the training. To struggle with the problem, more randomness about image transformation is introduced into the search space of AutoAugment (Cubuk et al., 2019) (described in Section 3.2). However, the learned policy is fixed for the entire training process. All of possible instances of each example will be send to the target network repeatedly, which still results in an inevitable overfitting in a long-epoch training. This phenomenon indicates that the learned policy is not adaptive to the training process of a target network, especially found on proxy tasks. Hence, the dynamic and adversarial augmentation policy with the training process is considered as the crucial feature in our search space. ", + "bbox": [ + 174, + 680, + 825, + 819 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Another consideration is how to improve the efficiency of the policy search. In AutoAugment (Cubuk et al., 2019), to evaluate the performance of augmentation policies, a lot of child models should be trained from scratch nearly to convergence. The computation in training and evaluating the performance of different sampled policies can not be reused, which leads to huge waste of computation resources. In this paper, we propose a computing-efficient policy search framework through reusing prior computation in policy evaluation. Only one target network is used to evaluate the performance of different policies with the help of the training losses of corresponding augmented instances. The augmentation policy network is learned from the intermediate state of the target network, which makes generated augmentation policies more aggressive and adaptive. On the contrary, to combat harder examples augmented by adversarial policies, the target network has to learn more robust features, which makes the training more efficiently. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/cf1281f8bae28f7b31d6aab6d29d8044913d73f1d9a58a90985849e432656219.jpg", + "image_caption": [ + "Figure 2: An example of dynamic augmentation policies learned with ResNet-50 on ImageNet. With the training process of the target network, harder augmentation policies are sampled to combat overfitting. Intuitively, more geometric transformations, such as TranslateX, ShearY and Rotate, are picked in our sampled policies, which is obviously different from AutoAugment (Cubuk et al., 2019) concentrating on color-based transformations. " + ], + "image_footnote": [], + "bbox": [ + 269, + 103, + 728, + 426 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 512, + 825, + 569 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 SEARCH SPACE ", + "text_level": 1, + "bbox": [ + 176, + 587, + 321, + 601 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this paper, the basic structure of the search space of AutoAugment (Cubuk et al., 2019) is reserved. An augmentation policy is defined as that it is composed by 5 sub-policies, each sub-policy contains two image operations to be applied orderly, each operation has two corresponding parameters, i.e., the probability and magnitude of the operation. Finally, the 5 best policies are concatenated to form a single policy with 25 sub-policies. For each image in a mini-batch, only one sub-policy will be randomly selected to be applied. To compare with AutoAugment (Cubuk et al., 2019) conveniently, we just slightly modify the search space with removing the probability of each operation. This is because that we think the stochasticity of an operation with a probability requires a certain epochs to take effect, which will detain the feedback of the intermediate state of the target network. There are totally 16 image operations in our search space, including ShearX/Y, TranslateX/Y, Rotate, AutoContrast, Invert, Equalize, Solarize, Posterize, Contrast, Color, Brightness, Sharpness, Cutout (Devries & Taylor, 2017) and Sample Pairing (Inoue, 2018). The range of the magnitude is also discretized uniformly into 10 values. To guarantee the convergence during adversarial learning, the magnitude of all the operations are set in a moderate range.1 Besides, the randomness during the training process is introduced into our search space. Hence, the search space of the policy in each epoch has $| S | = ( 1 6 \\times 1 0 ) ^ { 1 0 } \\approx 1 . 1 \\times 1 0 ^ { 2 2 }$ possibilities. Considering the dynamic policy, the number of possible policies with the whole training process can be expressed as $| S | ^ { \\# e p o c h s }$ . An example of dynamically learning the augmentation policy along with the training process is shown in Figure 2. We observe that the magnitude (an indication of difficulty) gradually increases with the training process. ", + "bbox": [ + 173, + 613, + 825, + 891 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 ADVERSARIAL LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 103, + 397, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, the adversarial framework of jointly optimizing network training and augmentation policy search is presented in detail. We use the augmentation policy network $\\boldsymbol { \\mathcal { A } } ( \\cdot , \\pmb { \\theta } )$ as an adversary, which attempts to increase the training loss of the target network $\\mathcal { F } ( \\cdot , w )$ through adversarial learning. The target network is trained by a large batch formed by multiple augmented instances of each batch to promote invariant learning (Salazar et al., 2018), and the losses of different augmentation policies applied on the same data are used to train the augmentation policy network by RL algorithm. ", + "bbox": [ + 173, + 128, + 825, + 227 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Considering the target network $\\mathcal { F } ( \\cdot , w )$ with a loss function $\\mathcal { L } [ \\mathcal { F } ( \\pmb { x } , \\pmb { w } ) , \\pmb { y } ]$ , where each example is transformed by some random data augmentation $o ( \\cdot )$ , the learning process of the target network can be defined as the following minimization problem ", + "bbox": [ + 173, + 233, + 825, + 276 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/357437841a8baa4fa9afe57c2203e6f99e1e22cebf9010ee0941a751ac3842af.jpg", + "text": "$$\n\\pmb { w } ^ { * } = \\underset { \\pmb { w } } { \\arg \\operatorname* { m i n } } \\ \\underset { \\pmb { x } \\sim \\Omega } { \\mathbb { E } } \\mathcal { L } [ \\mathcal { F } ( o ( \\pmb { x } ) , \\pmb { w } ) , \\pmb { y } ] ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 282, + 625, + 308 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\Omega$ is the training set, $_ { \\textbf { \\em x } }$ and $\\textbf { { y } }$ are the input image and the corresponding label, respectively. The problem is usually solved by vanilla SGD with a learning rate $\\eta$ and batch size $N$ , and the training procedure for each batch can be expressed as ", + "bbox": [ + 174, + 313, + 825, + 356 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7faff2f8aad97c6f5a263204249682b43d9c780cd18be5222f7b9a8d73afeb5d.jpg", + "text": "$$\n\\pmb { w } _ { t + 1 } = \\pmb { w } _ { t } - \\eta \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\nabla _ { \\pmb { w } } \\mathcal { L } [ \\mathcal { F } ( o ( x _ { n } ) , \\pmb { w } , y _ { n } ] .\n$$", + "text_format": "latex", + "bbox": [ + 343, + 362, + 655, + 406 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To improve the convergence performance of DNNs, more random and efficient data augmentation is performed under the help of the augmentation policy network. Hence, the minimization problem should be slightly modified as ", + "bbox": [ + 173, + 411, + 825, + 454 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8079bec44737be130c48f9d62df760c081ece4d44dde85d73f5880512a4aa3a1.jpg", + "text": "$$\n\\pmb { w } ^ { * } = \\underset { \\pmb { w } } { \\arg \\operatorname* { m i n } } \\ \\underset { \\pmb { x } \\sim \\Omega } { \\mathbb { E } } \\ \\underset { \\pmb { \\mathcal { A } } ( \\cdot , \\pmb { \\theta } ) } { \\mathbb { E } } \\mathcal { L } [ \\mathcal { F } ( \\tau ( \\pmb { x } ) , \\pmb { w } ) , \\pmb { y } ] ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 459, + 653, + 487 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\tau ( \\cdot )$ represents the augmentation policy generated by the network $\\boldsymbol { \\mathcal { A } } ( \\cdot , \\pmb { \\theta } )$ . Accordingly, the training rule can be rewritten as ", + "bbox": [ + 171, + 493, + 825, + 522 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3f6c2b66da389f91cb63fc352188ac2f1e21fad261eccee4d496f485d8497008.jpg", + "text": "$$\n\\pmb { w } _ { t + 1 } = \\pmb { w } _ { t } - \\eta \\frac { 1 } { M \\cdot N } \\sum _ { m = 1 } ^ { M } \\sum _ { n = 1 } ^ { N } \\nabla _ { \\pmb { w } } \\mathcal { L } [ \\mathcal { F } ( \\tau _ { m } ( x _ { n } ) , \\pmb { w } ) , y _ { n } ] ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 527, + 694, + 573 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where we introduce $M$ different instances of each input example augmented by adversarial policies $\\{ \\tau _ { 1 } , \\tau _ { 2 } , \\cdots , \\tau _ { M } \\}$ . For convenience, we denote the training loss of a mini-batch corresponding to the augmentation policy $\\tau _ { m }$ as ", + "bbox": [ + 173, + 578, + 823, + 621 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ad5ce4403f7216948077f7576c678356d0e67d7debf75ab5897c632920a41053.jpg", + "text": "$$\n\\mathcal { L } _ { m } = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathcal { L } [ \\mathcal { F } ( \\tau _ { m } ( x _ { n } ) , \\pmb { w } ) , y _ { n } ] .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 627, + 620, + 670 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Hence, we have an equivalent form of Equation 4 ", + "bbox": [ + 173, + 675, + 500, + 690 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/bb4c92c686635270ad2964c689792a678bc42b36ac081a645445e349d60251b0.jpg", + "text": "$$\n\\pmb { w } _ { t + 1 } = \\pmb { w } _ { t } - \\eta \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\nabla _ { \\pmb { w } } \\mathcal { L } _ { m } .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 696, + 607, + 741 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that the training procedure can be regarded as a larger $N \\cdot M$ batch training or an average over $M$ instances of gradient computation without changing the learning rate, which will lead to a reduction of gradient variance and a faster convergence of the target network Hoffer et al. (2019). However, overfitting will also come. To overcome the problem, the augmentation policy network is designed to increase the training loss of the target network with harder augmentation policies. Therefore, we can mathematically express the object as the following maximization problem ", + "bbox": [ + 173, + 744, + 826, + 829 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/107b151f7b522f76c993d3bbdfae27d7ce8897abeaefdc3b4bea230fc92817c0.jpg", + "text": "$$\n\\begin{array} { c } { \\pmb { \\theta } ^ { * } = \\underset { \\pmb { \\theta } } { \\arg \\operatorname* { m a x } } J ( \\pmb { \\theta } ) , } \\\\ { \\mathrm { w h e r e } ~ J ( \\pmb { \\theta } ) = \\underset { \\pmb { x } \\sim \\Omega } { \\mathbb { E } } ~ \\underset { \\pmb { \\mathbb { E } } ( \\cdot , \\pmb { \\theta } ) } { \\mathbb { E } } \\mathcal { L } [ \\mathcal { F } ( \\tau ( \\pmb { x } ) , \\pmb { w } ) , \\pmb { y } ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 834, + 655, + 890 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Similar to AutoAugment (Cubuk et al., 2019), the augmentation policy network is also implemented as a RNN shown in Figure 3. At each time step of the RNN controller, the softmax layer will predict an action corresponding to a discrete parameter of a sub-policy, and then an embedding of the predicted action will be fed into the next time step. In our experiments, the RNN controller will predict 20 discrete parameters to form a whole policy. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/d2f4b35084c6e611d6139d1fbef94757f5d6b6d525d6edbd65d47563cffa3f4c.jpg", + "image_caption": [ + "Figure 3: The basic architecture of the controller for generating a sub-policy, which consists of two operations with corresponding parameters, the type and magnitude of each operation. When a policy contains $Q$ sub-policies, the basic architecture will be repeated $Q$ times. Following the setting of AutoAugment (Cubuk et al., 2019), the number of sub-policies $Q$ is set to 5 in this paper. " + ], + "image_footnote": [], + "bbox": [ + 240, + 102, + 759, + 234 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 306, + 825, + 349 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "However, there has a severe problem in jointly optimizing target network training and augmentation policy search. This is because that non-differentiable augmentation operations break gradient flow from the target network $\\mathcal { F }$ to the augmentation policy network $\\mathcal { A }$ (Wang et al., 2017; Peng et al., 2018). As an alternative approach, REINFORCE algorithm (Williams, 1992) is applied to optimize the augmentation policy network as ", + "bbox": [ + 174, + 356, + 823, + 426 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0cd49c12d4aebfe70e605f93dd6deb419619fe0bd1e60dbd2c8f5cc69d4e9f2e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla _ { \\theta } J ( \\theta ) = \\nabla _ { \\theta } \\underset { x \\sim \\Omega \\tau \\sim A ( \\cdot , \\theta ) } { \\mathbb { E } } ~ \\underset { { A } ^ { ( \\cdot ) } } { \\mathbb { E } } ~ \\mathcal { L } [ \\mathcal { F } ( \\tau ( x ) , w ) , y ] } \\\\ & { \\approx \\displaystyle \\sum _ { m } \\mathcal { L } _ { m } \\nabla _ { \\theta } p _ { m } = \\sum _ { m } \\mathcal { L } _ { m } p _ { m } \\nabla _ { \\theta } \\log p _ { m } } \\\\ & { ~ = \\underset { { \\tau \\sim A ( \\cdot , \\theta ) } } { \\mathbb { E } } ~ \\mathcal { L } _ { m } \\nabla _ { \\theta } \\log p _ { m } } \\\\ & { ~ \\approx \\frac { 1 } { M } \\displaystyle \\sum _ { m = 1 } ^ { M } \\mathcal { L } _ { m } \\nabla _ { \\theta } \\log p _ { m } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 433, + 665, + 569 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $p _ { m }$ represents the probability of the policy $\\tau _ { m }$ . To reduce the variance of gradient $\\nabla _ { \\pmb { \\theta } } J ( \\pmb { \\theta } )$ , we replace the training loss of a mini-batch ${ \\mathcal { L } } _ { m }$ with ${ \\widehat { \\mathcal { L } } } _ { m }$ a moving average over a certain minibatches2, and then normalize it among $M$ instances as ${ \\widetilde { \\mathcal { L } } } _ { m }$ . Hence, the training procedure of the augmentation policy network can be expressed as ", + "bbox": [ + 173, + 575, + 826, + 637 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/95ede88d418cd5eb2dfdb765c787784d9b7309625eafd669d96747e2ab95096d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla _ { \\pmb { \\theta } } J ( \\pmb { \\theta } ) \\approx \\cfrac { 1 } { M } \\displaystyle \\sum _ { m = 1 } ^ { M } \\widetilde { \\mathcal { L } } _ { m } \\nabla _ { \\pmb { \\theta } } \\log p _ { m } , } \\\\ & { \\theta _ { e + 1 } = \\theta _ { e } + \\beta \\displaystyle \\frac { 1 } { M } \\displaystyle \\sum _ { m = 1 } ^ { M } \\widetilde { \\mathcal { L } } _ { m } \\nabla _ { \\pmb { \\theta } } \\log p _ { m } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 645, + 625, + 734 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The adversarial learning of target network training and augmentation policy search is summarized as Algorithm 1. ", + "bbox": [ + 173, + 741, + 825, + 770 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 787, + 459, + 804 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we first reveal the details of experiment settings. Then, we evaluate our proposed method on CIFAR-10/CIFAR-100, ImageNet, and compare it with previous methods. Results in Figure 4 show our method achieves the state-of-the-art performance with higher computing and time efficiency3. ", + "bbox": [ + 173, + 813, + 825, + 869 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1 Joint Training of Target Network and Augmentation Policy Network ", + "bbox": [ + 173, + 102, + 710, + 118 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Initialization: target network $\\mathcal { F } ( \\cdot , w )$ , augmentation policy network $\\boldsymbol { \\mathcal { A } } ( \\cdot , \\pmb { \\theta } )$ Input: input examples $_ { \\textbf { \\em x } }$ , corresponding labels $\\textbf { { y } }$ ", + "bbox": [ + 173, + 121, + 676, + 150 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1: for $1 \\leq e \\leq$ epochs do \n2: Initialize $\\widehat { \\mathcal { L } } _ { m } = 0 , \\forall m \\in \\{ 1 , 2 , \\cdots , M \\}$ ; \n3: Generate $M$ policies with the probabilities $\\{ p _ { 1 } , p _ { 2 } , \\cdots , p _ { M } \\}$ ; \n4: for $1 \\leq t \\leq T$ do \n5: Augment each batch data with $M$ generated policies, respectively; \n6: Update $w _ { e , t + 1 }$ according to Equation 4; \n7: Update ${ \\widehat { \\mathcal { L } } } _ { m }$ through moving average, $\\forall m \\in \\{ 1 , 2 , \\cdot \\cdot \\cdot , M \\}$ ; \n8: Collect $\\{ \\widehat { \\mathcal { L } } _ { 1 } , \\widehat { \\mathcal { L } } _ { 2 } , \\cdots , \\widehat { \\mathcal { L } } _ { M } \\}$ ; \n9: Normalize ${ \\widehat { \\mathcal { L } } } _ { m }$ among $M$ instances as $\\widetilde { \\mathcal { L } } _ { m } , \\forall m \\in \\{ 1 , 2 , \\cdots , M \\}$ ; \n10: Update $\\pmb { \\theta } _ { e + 1 }$ via Equation 9; \n11: Output $w ^ { \\ast } , \\theta ^ { \\ast }$ ", + "bbox": [ + 176, + 151, + 684, + 321 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 EXPERIMENT SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 352, + 382, + 367 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The RNN controller is implemented as a one-layer LSTM (Hochreiter & Schmidhuber, 1997). We set the hidden size to 100, and the embedding size to 32. We use Adam optimizer (Kingma & Ba, 2015) with a initial learning rate 0.00035 to train the controller. To avoid unexpected rapid convergence, an entropy penalty of a weight of 0.00001 is applied. All the reported results are the mean of five runs with different initializations. ", + "bbox": [ + 173, + 378, + 825, + 449 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 EXPERIMENTS ON CIFAR-10 AND CIFAR-100 ", + "text_level": 1, + "bbox": [ + 174, + 462, + 542, + 476 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "CIFAR-10 dataset (Krizhevsky & Hinton, 2009) has totally 60000 images. The training and test sets have 50000 and 10000 images, respectively. Each image in size of $3 2 \\times 3 2$ belongs to one of 10 classes. We evaluate our proposed method with the following models: Wide-ResNet-28- 10 (Zagoruyko & Komodakis, 2016), Shake-Shake $( 2 6 ~ 2 \\mathrm { x } 3 2 \\mathrm { d } )$ (Gastaldi, 2017), Shake-Shake (26 $2 \\mathrm { x } 9 6 \\mathrm { d } )$ (Gastaldi, 2017), Shake-Shake $( 2 6 2 \\mathrm { x } 1 1 2 \\mathrm { d } )$ (Gastaldi, 2017), PyramidNet+ShakeDrop (Han et al., 2017; Yamada et al., 2018). All the models are trained on the full training set. ", + "bbox": [ + 174, + 484, + 825, + 569 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training details: The Baseline is trained with the standard data augmentation, namely, randomly cropping a part of $3 2 \\times 3 2$ from the padded image and horizontally flipping it with a probability of 0.5. The Cutout (Devries & Taylor, 2017) randomly select a $1 6 \\times 1 6$ patch of each image, and then set the pixels of the selected patch to zeros. For our method, the searched policy is applied in addition to standard data augmentation and Cutout. For each image in the training process, standard data augmentation, the searched policy and Cutout are applied in sequence. For Wide-ResNet-28- 10, the step learning rate (LR) schedule is adopted. The cosine LR schedule is adopted for the other models. More details about model hyperparameters are supplied in A.1. ", + "bbox": [ + 174, + 575, + 825, + 688 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Choice of $M$ : To choose the optimal $M$ , we select Wide-ResNet-28-10 as a target network, and evaluate the performance of our proposed method verse different $M$ , where $M \\in \\{ 2 , 4 , 8 , 1 6 , 3 2 \\}$ . From Figure 5, we can observe that the test accuracy of the model improves rapidly with the increase of $M$ up to 8. The further increase of $M$ does not bring a significant improvement. Therefore, to balance the performance and the computing cost, $M$ is set to 8 in all the following experiments. ", + "bbox": [ + 174, + 694, + 825, + 763 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "CIFAR-10 results: In Table 1, we report the test error of these models on CIFAR-10. For all of these models, our proposed method can achieve better performance compared to previous methods. We achieve $0 . 7 8 \\%$ and $0 . 6 8 \\%$ improvement on Wide-ResNet-28-10 compared to AutoAugment and PBA, respectively. We achieve a top-1 test error of $1 . 3 6 \\%$ with PyramidNet $^ +$ ShakeDrop, which is $0 . 1 \\%$ better than the current state-of-the-art reported in Ho et al. (2019). As shown in Figure 6(a) and 6(b),we further visualize the probability distribution of the parameters of the augmentation policies learned with PyramidNet+ShakeDrop on CIFAR-10 over time. From Figure 6(a), we can find that the percentages of some operations, such as TranslateY, Rotate, Posterize, and SampleParing, gradually increase along with the training process. Meanwhile, more geometric transformations, such as TranslateX, TranslateY, and Rotate, are picked in the sampled augmentation policies, which is different from color-focused AutoAugment (Cubuk et al., 2019) on CIFAR-10. Figure 6(b) shows that large magnitudes gain higher percentages during training. However, at the tail of training, low magnitudes remain considerable percentages. This indicates that our method does not simply learn the transformations with the extremes of the allowed magnitudes to spoil the target network. ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1d9bd1c440b329d4cde0cd4933e8cd51662c730a2c81bacfeed27c2444ff1516.jpg", + "image_caption": [ + "Figure 4: The Comparison of normalized performance between AutoAugment and our method. Please refer to the following tables for more details. " + ], + "image_footnote": [], + "bbox": [ + 191, + 104, + 473, + 272 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/8f0622955a06ab6ba860d80c3c02ae266fb49a0c8467d420de114247a0cfe754.jpg", + "image_caption": [ + "Figure 5: The Top-1 test accuracy of WideResNet-28-10 on CIFAR-10 verse different $M$ , where $M \\in \\{ 2 , 4 , 8 , 1 6 , 3 2 \\}$ . " + ], + "image_footnote": [], + "bbox": [ + 542, + 112, + 794, + 273 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/969c5da3cca5ba7fe0f9f8865ef328f9131f97876658c78dfb77dd5e89fdd3d5.jpg", + "image_caption": [ + "Figure 6: Probability distribution of the parameters in the learned augmentation policies on CIFAR10 over time. The number in (b) represents the magnitude of one operation. Larger number stands for more dramatic image transformations. The probability distribution of each parameter is the mean of each five epochs. " + ], + "image_footnote": [], + "bbox": [ + 191, + 347, + 813, + 530 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 595, + 825, + 637 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "CIFAR-100 results: We also evaluate our proposed method on CIFAR-100, as shown in Table 2. \nAs we can observe from the table, we also achieve the state-of-the-art performance on this dataset. ", + "bbox": [ + 176, + 645, + 820, + 674 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/d929706459bceab76309d542582ff87eb9b290a004d6adec648ac5702809b1fa.jpg", + "table_caption": [ + "Table 1: Top-1 test error $( \\% )$ on CIFAR-10. We replicate the results of Baseline, Cutout and AutoAugment methods from Cubuk et al. (2019), and the results of PBA from Ho et al. (2019) in all of our experiments. " + ], + "table_footnote": [], + "table_body": "
ModelBaselineCutoutAutoAugmentPBAOur Method
Wide-ResNet-28-103.873.082.682.581.90±0.15
Shake-Shake (26 2x32d)3.553.022.472.542.36±0.10
Shake-Shake (26 2x96d)2.862.561.992.031.85±0.12
Shake-Shake (26 2x112d)2.822.571.892.031.78±0.05
PyramidNet+ShakeDrop2.672.311.481.461.36±0.06
", + "bbox": [ + 199, + 731, + 795, + 832 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 EXPERIMENTS ON IMAGENET ", + "text_level": 1, + "bbox": [ + 174, + 847, + 421, + 862 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As a great challenge in image recognition, ImageNet dataset (Deng et al., 2009) has about 1.2 million training images and 50000 validation images with 1000 classes. In this section, we directly search the augmentation policy on the full training set and train ResNet-50 (He et al., 2016), ResNet-50-D (He et al., 2018) and ResNet-200 (He et al., 2016) from scratch. ", + "bbox": [ + 174, + 868, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/1c93b5f77c1ed4a44c3eae869260059d2b4c4df4c9ddaf09ea86ef7706a1d1fc.jpg", + "table_caption": [ + "Table 2: Top-1 test error $( \\% )$ on CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
ModelBaselineCutout AutoAugmentPBAOur Method
Wide-ResNet-28-1018.8018.4117.0916.7315.49±0.18
Shake-Shake (26 2x96d)17.0516.0014.2815.3114.10±0.15
PyramidNet+ShakeDrop13.9912.1910.6710.9410.42±0.20
", + "bbox": [ + 199, + 116, + 794, + 189 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Training details: For the baseline augmentation, we randomly resize and crop each input image to a size of $2 2 4 \\times 2 2 4$ , and then horizontally flip it with a probability of 0.5. For AutoAugment (Cubuk et al., 2019) and our method, the baseline augmentation and the augmentation policy are both used for each image. The cosine LR schedule is adopted in the training process. The model hyperparameters on ImageNet is also detailed in A.1. ", + "bbox": [ + 174, + 199, + 825, + 268 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ImageNet results: The performance of our proposed method on ImageNet is presented in Table 3. It can be observed that we achieve a top-1 accuracy $7 9 . 4 0 \\%$ on ResNet-50 without extra data. To the best of our knowledge, this is the highest top-1 accuracy for ResNet-50 learned on ImageNet. Besides, we only replace the ResNet-50 architecture with ResNet-50-D, and achieve a consistent improvement with a top-1 accuracy of $8 0 . 0 0 \\%$ . ", + "bbox": [ + 174, + 275, + 825, + 344 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/c11394c053c797ac41deef027c8e6c75b7d63680580b038230cdbafd75d497b2.jpg", + "table_caption": [ + "Table 3: Top-1 / Top-5 test error $( \\% )$ on ImageNet. Note that the result of ResNet-50-D is achieved only through substituting the architecture. " + ], + "table_footnote": [], + "table_body": "
ModelBaselineAutoAugmentPBAOur Method
ResNet-5023.69 / 6.9222.37 /6.18120.60±0.15 /5.53±0.05
ResNet-50-D22.84 / 6.48120.00±0.12/5.25±0.03
ResNet-20021.52 / 5.8520.00 /4.90118.68±0.18 /4.70±0.05
", + "bbox": [ + 220, + 386, + 772, + 459 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 174, + 472, + 341, + 486 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To check the effect of each component in our proposed method, we report the test error of ResNet-50 on ImageNet the following augmentation methods in Table 4. ", + "bbox": [ + 174, + 494, + 823, + 523 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• Baseline: Training regularly with the standard data augmentation and step LR schedule. \n• Fixed: Augmenting all the instances of each batch with the standard data augmentation fixed throughout the entire training process. Random: Augmenting all the instances of each batch with randomly and dynamically generated policies. Ours: Augmenting all the instances of each batch with adversarial policies sampled by the policy network along with the training process. ", + "bbox": [ + 215, + 535, + 825, + 647 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "From the table, we can find that Fixed can achieve $0 . 9 9 \\%$ error reduction compared to Baseline. This shows that a large-batch training with multiple augmented instances of each mini-batch can indeed improve the generalization of the model, which is consistent with the conclusion presented in Hoffer et al. (2019). In addition, the test error of Random is $1 . 0 2 \\%$ better than Fixed. This indicates that augmenting batch with randomly generated policies can reduce overfitting in a certain extent. Furthermore, our method achieves the best test error of $2 0 . 6 0 \\%$ through augmenting samples with adversarial policies. From the result, we can conclude that these policies generated by the policy network are more adaptive to the training process, and make the target network have to learn more robust features. ", + "bbox": [ + 173, + 657, + 825, + 782 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 COMPUTING COST AND TIME OVERHEAD ", + "text_level": 1, + "bbox": [ + 174, + 796, + 504, + 810 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Computing Cost: The computation in target network training is reused for policy evaluation. This makes the computing cost in policy search become negligible. Although there exists an increase of computing cost in target network training, the total computing cost in training one target network with augmentation policies is quite small compared to prior work. ", + "bbox": [ + 174, + 819, + 825, + 875 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Time Overhead: Since we just train one target network with a large batch distributedly and simultaneously, the time overhead of the large-batch training is equal to the regular training. Meanwhile, the joint optimization of target network training and augmentation policy search dispenses with the process of offline policy search and the retraining of a target network, which leads to a extreme time overhead reduction. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/0d33c9b0d3bb4d596dbaf5db23898beb2fc80c5c532a9c04e77237c677eaa18c.jpg", + "table_caption": [ + "Table 4: Top-1 test error $( \\% )$ of ResNet-50 with different augmentation methods on ImageNet. " + ], + "table_footnote": [], + "table_body": "
MethodAug. PolicyEnlarge BatchLR ScheduleTest Error
BaselinestandardM=1step23.69
FixedstandardM=8cosine22.70
RandomrandomM=8cosine21.68
OursadversarialM=8cosine20.60
", + "bbox": [ + 256, + 116, + 736, + 203 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 213, + 825, + 241 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In Table 5, we take the training of ResNet-50 on ImageNet as an example to compare the computing cost and time overhead of our method and AutoAugment. From the table, we can find that our method is $1 2 \\times$ less computing cost and $1 1 \\times$ shorter time overhead than AutoAugment. ", + "bbox": [ + 174, + 248, + 825, + 290 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/f3bf8b2d6f177971a2bdad2dbb26e8734c9a3d3754c0e1aeba92d92343c857b2.jpg", + "table_caption": [ + "Table 5: The comparison of computing cost (GPU hours) and time overhead (days) in training ResNet-50 on ImageNet between AutoAugment and our method. The computing cost and time overhead are estimated on 64 NVIDIA Tesla V100s. " + ], + "table_footnote": [], + "table_body": "
MethodComputing CostTime Overhead
SearchingTrainingTotalSearchingTrainingTotal
AutoAugment150001601516010111
Our Method~012801280~011
", + "bbox": [ + 222, + 345, + 772, + 424 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.6 TRANSFERABILITY ACROSS DATASETS AND ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 174, + 438, + 640, + 452 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "To further show the higher efficiency of our method, the transferability of the learned augmentation policies is evaluated in this section. We first take a snapshot of the adversarial training process of ResNet-50 on ImageNet, and then directly use the learned dynamic augmentation policies to regularly train the following models: Wide-ResNet-28-10 on CIFAR-10/100, ResNet-50-D on ImageNet and ResNet200 on ImageNet. Table 6 presents the experimental results of the transferability. From the table, we can find that a competitive performance can be still achieved through direct policy transfer. This indicates that the learned augmentation policies transfer well across datasets and architectures. However, compared to the proposed method, the policy transfer results in an obvious performance degradation, especially the transfer across datasets. ", + "bbox": [ + 173, + 459, + 825, + 585 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/e885dd982a0940ae52993dc12f9f60d83acd25879acc9f8565db376eeebf1417.jpg", + "table_caption": [ + "Table 6: Top-1 test error $( \\% )$ of the transfer of the augmentation policies learned with ResNet-50 on ImageNet. " + ], + "table_footnote": [], + "table_body": "
MethodDatasetAutoAugmentOur MethodPolicy Transfer
Wide-ResNet-28-10CIFAR-102.681.902.45±0.13
Wide-ResNet-28-10CIFAR-10017.0915.4916.48±0.15
ResNet-50-DImageNet120.0020.20±0.05
ResNet-200ImageNet20.0018.6819.05±0.10
", + "bbox": [ + 204, + 626, + 789, + 713 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 729, + 320, + 746 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we introduce the idea of adversarial learning into automatic data augmentation. The policy network tries to combat the overfitting of the target network through generating adversarial policies with the training process. To oppose this, robust features are learned in the target network, which leads to a significant performance improvement. Meanwhile, the augmentation policy search is performed along with the training of a target network, and the computation in network training is reused for policy evaluation, which can extremely reduce the search cost and make our method more computing-efficient. 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", + "bbox": [ + 171, + 669, + 761, + 684 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/805358f66a6525e1f2d65ec0f4b18937182bf4e5a10355c998155b8c28719ccb.jpg", + "table_caption": [ + "Table 7: Model hyperparameters on CIFAR-10/CIFAR-100 and ImageNet. LR represents learning rate, and WD represents weight decay. We do not specifically tune these hyperparameters, and all of these are consistent with previous works, expect for the number of epochs. " + ], + "table_footnote": [], + "table_body": "
DatasetModelBatch Size (N · M)LRWDEpoch
CIFAR-10Wide-ResNet-28-10128·80.15e-4200
CIFAR-10Shake-Shake (26 2x32d)128·80.21e-4600
CIFAR-10Shake-Shake ( (262x96d)128·80.21e-4600
CIFAR-10Shake-Shake (26 2x112d)128·80.21e-4600
CIFAR-10PyramidNet+ShakeDrop128·80.11e-4600
CIFAR-100Wide-ResNet-28-10128·80.15e-4200
CIFAR-100Shake-Shake (26 2x96d)128·80.15e-41200
CIFAR-100PyramidNet+ShakeDrop128·80.51e-41200
ImageNetResNet-502048·80.81e-4120
ImageNetResNet-50-D2048·80.81e-4120
ImageNetResNet-2002048·80.81e-4120
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Recently, human-designed data augmentation has", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "score": 1.0, + "content": "been gradually replaced by automatically learned augmentation policy. Through", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 299, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 312 + ], + "score": 1.0, + "content": "finding the best policy in well-designed search space of data augmentation, Au-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 311, + 470, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 324 + ], + "score": 1.0, + "content": "toAugment (Cubuk et al., 2019) can significantly improve validation accuracy on", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "score": 1.0, + "content": "image classification tasks. 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The augmentation policy network attempts to increase the train-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 376, + 470, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 470, + 389 + ], + "score": 1.0, + "content": "ing loss of a target network through generating adversarial augmentation policies,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "score": 1.0, + "content": "while the target network can learn more robust features from harder examples to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 398, + 470, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 470, + 411 + ], + "score": 1.0, + "content": "improve the generalization. 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We", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 442, + 470, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 470, + 455 + ], + "score": 1.0, + "content": "show experimental results of our approach on CIFAR-10/CIFAR-100, ImageNet,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 453, + 470, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 470, + 466 + ], + "score": 1.0, + "content": "and demonstrate significant performance improvements over state-of-the-art. 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Data augmentation such", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "as rotation, flipping, cropping, etc., is a powerful technique to increase the amount and diversity", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "of data. Experiments show that the generalization of a neural network can be efficiently improved", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "through manually designing data augmentation policies. However, this needs lots of knowledge of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "human expert, and sometimes shows the weak transferability across different tasks and datasets in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "practical applications. Inspired by neural architecture search (NAS)(Zoph & Le, 2016; Zoph et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "2017; Zhong et al., 2018a;b; Guo et al., 2018), a reinforcement learning (RL) (Williams, 1992)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "method called AutoAugment is proposed by Cubuk et al. (2019), which can automatically learn the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "augmentation policy from data and provide an exciting performance improvement on image classifi-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "cation tasks. However, the computing cost is huge for training and evaluating thousands of sampled", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "policies in the search process. Although proxy tasks, i.e., smaller models and reduced datasets, are", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "taken to accelerate the searching process, tens of thousands of GPU-hours of consumption are still", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "required. In addition, these data augmentation policies optimized on proxy tasks are not guaranteed", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "to be optimal on the target task, and the fixed augmentation policy is also sub-optimal for the whole", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 175, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 175, + 735 + ], + "score": 1.0, + "content": "training process.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 545, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 123, + 81, + 447, + 293 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 123, + 81, + 447, + 293 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 81, + 447, + 293 + ], + "spans": [ + { + "bbox": [ + 123, + 81, + 447, + 293 + ], + "score": 0.971, + "type": "image", + "image_path": "30303714f451b21eaee42acd79567fecfdfac2cdfb67ff8adc6f2486bb4f5ec4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 123, + 81, + 447, + 151.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 123, + 151.66666666666669, + 447, + 222.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 123, + 222.33333333333337, + 447, + 293.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 295, + 505, + 373 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "score": 1.0, + "content": "Figure 1: The overview of our proposed method. We formulate it as a Min-Max game. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "data of each batch is augmented by multiple pre-processing components with sampled policies", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 316, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 317, + 176, + 329 + ], + "score": 0.92, + "content": "\\{ \\tau _ { 1 } , \\tau _ { 2 } , \\cdots , \\tau _ { M } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 316, + 506, + 331 + ], + "score": 1.0, + "content": ", respectively. Then, a target network is trained to minimize the loss of a large", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "batch, which is formed by multiple augmented instances of the input batch. We extract the training", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "losses of a target network corresponding to different augmentation policies as the reward signal. Fi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "nally, the augmentation policy network is trained with the guideline of the processed reward signal,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "and aims to maximize the training loss of the target network through generating adversarial policies.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "In this paper, we propose an efficient data augmentation method to address the problems mentioned", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 394, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 409 + ], + "score": 1.0, + "content": "above, which can directly search the best augmentation policy on the full dataset during training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "a target network, as shown in Figure 1. We first organize the network training and augmentation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "policy search in an adversarial and online manner. The augmentation policy is dynamically changed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 427, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 443 + ], + "score": 1.0, + "content": "along with the training state of the target network, rather than fixed throughout the whole training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "process like normal AutoAugment (Cubuk et al., 2019). Due to reusing the computation in policy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "evaluation and dispensing with the retraining of the target network, the computing cost and time", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "overhead are extremely reduced. Then, the augmentation policy network is taken as an adversary to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "explore the weakness of the target network. We augment the data of each min-batch with various", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "adversarial policies in parallel, rather than the same data augmentation taken in batch augmentation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "(BA) (Hoffer et al., 2019). Then, several augmented instances of each mini-batch are formed into a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "large batch for target network learning. As an indicator of the hardness of augmentation policies, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "training losses of the target network are used to guide the policy network to generate more aggres-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "sive and efficient policies based on REINFORCE algorithm (Williams, 1992). Through adversarial", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 538, + 389, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 389, + 551 + ], + "score": 1.0, + "content": "learning, we can train the target network more efficiently and robustly.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 555, + 303, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 304, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 304, + 567 + ], + "score": 1.0, + "content": "The contributions can be summarized as follows:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 132, + 571, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 133, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "• Our method can directly learn augmentation policies on target tasks, i.e., target networks", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 141, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "and full datasets, with a quite low computing cost and time overhead. The direct policy", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 142, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "search avoids the performance degradation caused by the policy transfer from proxy tasks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 605, + 201, + 616 + ], + "spans": [ + { + "bbox": [ + 141, + 605, + 201, + 616 + ], + "score": 1.0, + "content": "to target tasks.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 132, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 132, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "• We propose an adversarial framework to jointly optimize target network training and aug-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 141, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "mentation policy search. The harder samples augmented by adversarial policies are con-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 142, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "stantly fed into the target network to promote robust feature learning. Hence, the general-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 657, + 380, + 669 + ], + "spans": [ + { + "bbox": [ + 142, + 657, + 380, + 669 + ], + "score": 1.0, + "content": "ization of the target network can be significantly improved.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 135, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 135, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "• The experiment results show that our proposed method outperforms previous augmentation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 141, + 686, + 357, + 702 + ], + "score": 1.0, + "content": "methods. For instance, we achieve a top-1 test error of", + "type": "text" + }, + { + "bbox": [ + 357, + 688, + 384, + 698 + ], + "score": 0.88, + "content": "1 . 3 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "with PyramidNet+ShakeDrop", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 142, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "(Yamada et al., 2018) on CIFAR-10, which is the state-of-the-art performance. On Ima-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 710, + 439, + 722 + ], + "score": 1.0, + "content": "geNet, we improve the top-1 accuracy of ResNet-50 (He et al., 2016) from", + "type": "text" + }, + { + "bbox": [ + 439, + 710, + 466, + 720 + ], + "score": 0.88, + "content": "7 6 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 710, + 477, + 722 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 477, + 710, + 504, + 720 + ], + "score": 0.87, + "content": "7 9 . 4 \\%", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 720, + 493, + 733 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 276, + 733 + ], + "score": 1.0, + "content": "without extra data, which is even", + "type": "text" + }, + { + "bbox": [ + 276, + 721, + 303, + 731 + ], + "score": 0.86, + "content": "1 . 7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 720, + 493, + 733 + ], + "score": 1.0, + "content": "better than AutoAugment (Cubuk et al., 2019).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 123, + 81, + 447, + 293 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 123, + 81, + 447, + 293 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 81, + 447, + 293 + ], + "spans": [ + { + "bbox": [ + 123, + 81, + 447, + 293 + ], + "score": 0.971, + "type": "image", + "image_path": "30303714f451b21eaee42acd79567fecfdfac2cdfb67ff8adc6f2486bb4f5ec4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 123, + 81, + 447, + 151.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 123, + 151.66666666666669, + 447, + 222.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 123, + 222.33333333333337, + 447, + 293.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 295, + 505, + 373 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "score": 1.0, + "content": "Figure 1: The overview of our proposed method. We formulate it as a Min-Max game. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "data of each batch is augmented by multiple pre-processing components with sampled policies", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 316, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 317, + 176, + 329 + ], + "score": 0.92, + "content": "\\{ \\tau _ { 1 } , \\tau _ { 2 } , \\cdots , \\tau _ { M } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 316, + 506, + 331 + ], + "score": 1.0, + "content": ", respectively. Then, a target network is trained to minimize the loss of a large", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "batch, which is formed by multiple augmented instances of the input batch. We extract the training", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "losses of a target network corresponding to different augmentation policies as the reward signal. Fi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "nally, the augmentation policy network is trained with the guideline of the processed reward signal,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "and aims to maximize the training loss of the target network through generating adversarial policies.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "In this paper, we propose an efficient data augmentation method to address the problems mentioned", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 394, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 409 + ], + "score": 1.0, + "content": "above, which can directly search the best augmentation policy on the full dataset during training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "a target network, as shown in Figure 1. We first organize the network training and augmentation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "policy search in an adversarial and online manner. The augmentation policy is dynamically changed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 427, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 443 + ], + "score": 1.0, + "content": "along with the training state of the target network, rather than fixed throughout the whole training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "process like normal AutoAugment (Cubuk et al., 2019). Due to reusing the computation in policy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "evaluation and dispensing with the retraining of the target network, the computing cost and time", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "overhead are extremely reduced. Then, the augmentation policy network is taken as an adversary to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "explore the weakness of the target network. We augment the data of each min-batch with various", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "adversarial policies in parallel, rather than the same data augmentation taken in batch augmentation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "(BA) (Hoffer et al., 2019). Then, several augmented instances of each mini-batch are formed into a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "large batch for target network learning. As an indicator of the hardness of augmentation policies, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "training losses of the target network are used to guide the policy network to generate more aggres-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "sive and efficient policies based on REINFORCE algorithm (Williams, 1992). Through adversarial", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 538, + 389, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 389, + 551 + ], + "score": 1.0, + "content": "learning, we can train the target network more efficiently and robustly.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 385, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 555, + 303, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 304, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 304, + 567 + ], + "score": 1.0, + "content": "The contributions can be summarized as follows:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 554, + 304, + 567 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 571, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 133, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "• Our method can directly learn augmentation policies on target tasks, i.e., target networks", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 141, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "and full datasets, with a quite low computing cost and time overhead. The direct policy", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 142, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "search avoids the performance degradation caused by the policy transfer from proxy tasks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 605, + 201, + 616 + ], + "spans": [ + { + "bbox": [ + 141, + 605, + 201, + 616 + ], + "score": 1.0, + "content": "to target tasks.", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 132, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "• We propose an adversarial framework to jointly optimize target network training and aug-", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 141, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "mentation policy search. The harder samples augmented by adversarial policies are con-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 142, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "stantly fed into the target network to promote robust feature learning. Hence, the general-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 657, + 380, + 669 + ], + "spans": [ + { + "bbox": [ + 142, + 657, + 380, + 669 + ], + "score": 1.0, + "content": "ization of the target network can be significantly improved.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + }, + { + "bbox": [ + 135, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 135, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "• The experiment results show that our proposed method outperforms previous augmentation", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 141, + 686, + 357, + 702 + ], + "score": 1.0, + "content": "methods. For instance, we achieve a top-1 test error of", + "type": "text" + }, + { + "bbox": [ + 357, + 688, + 384, + 698 + ], + "score": 0.88, + "content": "1 . 3 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "with PyramidNet+ShakeDrop", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 142, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "(Yamada et al., 2018) on CIFAR-10, which is the state-of-the-art performance. On Ima-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 710, + 439, + 722 + ], + "score": 1.0, + "content": "geNet, we improve the top-1 accuracy of ResNet-50 (He et al., 2016) from", + "type": "text" + }, + { + "bbox": [ + 439, + 710, + 466, + 720 + ], + "score": 0.88, + "content": "7 6 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 710, + 477, + 722 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 477, + 710, + 504, + 720 + ], + "score": 0.87, + "content": "7 9 . 4 \\%", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 720, + 493, + 733 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 276, + 733 + ], + "score": 1.0, + "content": "without extra data, which is even", + "type": "text" + }, + { + "bbox": [ + 276, + 721, + 303, + 731 + ], + "score": 0.86, + "content": "1 . 7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 720, + 493, + 733 + ], + "score": 1.0, + "content": "better than AutoAugment (Cubuk et al., 2019).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32, + "bbox_fs": [ + 132, + 571, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 211, + 93 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 213, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 213, + 96 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "Common data augmentation, which can generate extra samples by some label-preserved transfor-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "mations, is usually used to increase the size of datasets and improve the generalization of networks,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "score": 1.0, + "content": "such as on MINST, CIFAR-10 and ImageNet (Krizhevsky et al., 2012; Wan et al., 2013; Szegedy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "et al., 2015). However, human-designed augmentation policies are specified for different datasets.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "For example, flipping, the widely used transformation on CIFAR-10/CIFAR-100 and ImageNet, is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 413, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 413, + 173 + ], + "score": 1.0, + "content": "not suitable for MINST, which will destroy the property of original samples.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "Hence, several works (Lemley et al., 2017; Cubuk et al., 2019; Lin et al., 2019; Ho et al., 2019)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "have attempted to automatically learn data augmentation policies. Lemley et al. (2017) propose", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "score": 1.0, + "content": "a method called Smart Augmentation, which merges two or more samples of a class to improve", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "the generalization of a target network. The result also indicates that an augmentation network can", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "be learned when a target network is being training. Through well designing the search space of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "data augmentation policies, AutoAugment (Cubuk et al., 2019) takes a recurrent neural network", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "(RNN) as a sample controller to find the best data augmentation policy for a selected dataset. To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "reduce the computing cost, the augmentation policy search is performed on proxy tasks. Population", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "based augmentation (PBA) (Ho et al., 2019) replaces the fixed augmentation policy with a dynamic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "schedule of augmentation policy along with the training process, which is mostly related to our", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 301 + ], + "score": 1.0, + "content": "work. Inspired by population based training (PBT) (Jaderberg et al., 2017), the augmentation policy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "score": 1.0, + "content": "search problem in PBA is modeled as a process of hyperparameter schedule learning. However,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "the augmentation schedule learning is still performed on proxy tasks. The learned policy schedule", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "should be manually adjusted when the training process of a target network is non-matched with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 157, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 157, + 343 + ], + "score": 1.0, + "content": "proxy tasks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "Another related topic is Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "has recently attracted lots of research attention due to its fascinating performance, and also been used", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "to enlarge datasets through directly synthesizing new images (Tran et al., 2017; Perez & Wang, 2017;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Antoniou et al., 2017; Gurumurthy et al., 2017; Frid-Adar et al., 2018). Although we formulate our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "proposed method as a Min-Max game, there exists an obvious difference with traditional GANs. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "want to find the best augmentation policy to perform image transformation along with the training", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "process, rather than synthesize new images. Peng et al. (2018) also take such an idea to optimize the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 425, + 357, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 357, + 437 + ], + "score": 1.0, + "content": "training process of a target network in human pose estimation.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 447, + 173, + 460 + ], + "lines": [ + { + "bbox": [ + 104, + 444, + 174, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 174, + 462 + ], + "score": 1.0, + "content": "3 METHOD", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 467, + 504, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "In this section, we present the implementation of Adversarial AutoAugment. First, the motivation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "for the adversarial relation between network learning and augmentation policy is discussed. Then,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "we introduce the search space with the dynamic augmentation policy. Finally, the joint framework", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 405, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 405, + 512 + ], + "score": 1.0, + "content": "for network training and augmentation policy search is presented in detail.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 518, + 192, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 194, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 194, + 532 + ], + "score": 1.0, + "content": "3.1 MOTIVATIONS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "Although some human-designed data augmentations have been used in the training of DNNs, such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "as randomly cropping and horizontally flipping on CIFAR-10/CIFAR-100 and ImageNet, limited", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "randomness will make it very difficult to generate effective samples at the tail end of the training.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "To struggle with the problem, more randomness about image transformation is introduced into the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "search space of AutoAugment (Cubuk et al., 2019) (described in Section 3.2). However, the learned", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "policy is fixed for the entire training process. All of possible instances of each example will be", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "send to the target network repeatedly, which still results in an inevitable overfitting in a long-epoch", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "training. This phenomenon indicates that the learned policy is not adaptive to the training process of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "a target network, especially found on proxy tasks. Hence, the dynamic and adversarial augmentation", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 638, + 456, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 456, + 651 + ], + "score": 1.0, + "content": "policy with the training process is considered as the crucial feature in our search space.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "Another consideration is how to improve the efficiency of the policy search. In AutoAugment", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "(Cubuk et al., 2019), to evaluate the performance of augmentation policies, a lot of child models", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "should be trained from scratch nearly to convergence. The computation in training and evaluat-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "ing the performance of different sampled policies can not be reused, which leads to huge waste of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "computation resources. In this paper, we propose a computing-efficient policy search framework", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "through reusing prior computation in policy evaluation. Only one target network is used to evaluate", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "the performance of different policies with the help of the training losses of corresponding augmented", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 211, + 93 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 213, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 213, + 96 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "Common data augmentation, which can generate extra samples by some label-preserved transfor-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "mations, is usually used to increase the size of datasets and improve the generalization of networks,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "score": 1.0, + "content": "such as on MINST, CIFAR-10 and ImageNet (Krizhevsky et al., 2012; Wan et al., 2013; Szegedy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "et al., 2015). However, human-designed augmentation policies are specified for different datasets.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "For example, flipping, the widely used transformation on CIFAR-10/CIFAR-100 and ImageNet, is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 413, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 413, + 173 + ], + "score": 1.0, + "content": "not suitable for MINST, which will destroy the property of original samples.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 505, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "Hence, several works (Lemley et al., 2017; Cubuk et al., 2019; Lin et al., 2019; Ho et al., 2019)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "have attempted to automatically learn data augmentation policies. Lemley et al. (2017) propose", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "score": 1.0, + "content": "a method called Smart Augmentation, which merges two or more samples of a class to improve", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "the generalization of a target network. The result also indicates that an augmentation network can", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "be learned when a target network is being training. Through well designing the search space of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "data augmentation policies, AutoAugment (Cubuk et al., 2019) takes a recurrent neural network", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "(RNN) as a sample controller to find the best data augmentation policy for a selected dataset. To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "reduce the computing cost, the augmentation policy search is performed on proxy tasks. Population", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "based augmentation (PBA) (Ho et al., 2019) replaces the fixed augmentation policy with a dynamic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "schedule of augmentation policy along with the training process, which is mostly related to our", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 301 + ], + "score": 1.0, + "content": "work. Inspired by population based training (PBT) (Jaderberg et al., 2017), the augmentation policy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "score": 1.0, + "content": "search problem in PBA is modeled as a process of hyperparameter schedule learning. However,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "the augmentation schedule learning is still performed on proxy tasks. The learned policy schedule", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "should be manually adjusted when the training process of a target network is non-matched with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 157, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 157, + 343 + ], + "score": 1.0, + "content": "proxy tasks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 177, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "Another related topic is Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "has recently attracted lots of research attention due to its fascinating performance, and also been used", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "to enlarge datasets through directly synthesizing new images (Tran et al., 2017; Perez & Wang, 2017;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Antoniou et al., 2017; Gurumurthy et al., 2017; Frid-Adar et al., 2018). Although we formulate our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "proposed method as a Min-Max game, there exists an obvious difference with traditional GANs. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "want to find the best augmentation policy to perform image transformation along with the training", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "process, rather than synthesize new images. Peng et al. (2018) also take such an idea to optimize the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 425, + 357, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 357, + 437 + ], + "score": 1.0, + "content": "training process of a target network in human pose estimation.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 347, + 506, + 437 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 447, + 173, + 460 + ], + "lines": [ + { + "bbox": [ + 104, + 444, + 174, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 174, + 462 + ], + "score": 1.0, + "content": "3 METHOD", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 467, + 504, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "In this section, we present the implementation of Adversarial AutoAugment. First, the motivation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "for the adversarial relation between network learning and augmentation policy is discussed. Then,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "we introduce the search space with the dynamic augmentation policy. Finally, the joint framework", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 405, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 405, + 512 + ], + "score": 1.0, + "content": "for network training and augmentation policy search is presented in detail.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 467, + 506, + 512 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 518, + 192, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 194, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 194, + 532 + ], + "score": 1.0, + "content": "3.1 MOTIVATIONS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "Although some human-designed data augmentations have been used in the training of DNNs, such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "as randomly cropping and horizontally flipping on CIFAR-10/CIFAR-100 and ImageNet, limited", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "randomness will make it very difficult to generate effective samples at the tail end of the training.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "To struggle with the problem, more randomness about image transformation is introduced into the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "search space of AutoAugment (Cubuk et al., 2019) (described in Section 3.2). However, the learned", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "policy is fixed for the entire training process. All of possible instances of each example will be", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "send to the target network repeatedly, which still results in an inevitable overfitting in a long-epoch", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "training. This phenomenon indicates that the learned policy is not adaptive to the training process of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "a target network, especially found on proxy tasks. Hence, the dynamic and adversarial augmentation", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 638, + 456, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 456, + 651 + ], + "score": 1.0, + "content": "policy with the training process is considered as the crucial feature in our search space.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 539, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "Another consideration is how to improve the efficiency of the policy search. In AutoAugment", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "(Cubuk et al., 2019), to evaluate the performance of augmentation policies, a lot of child models", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "should be trained from scratch nearly to convergence. The computation in training and evaluat-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "ing the performance of different sampled policies can not be reused, which leads to huge waste of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "computation resources. In this paper, we propose a computing-efficient policy search framework", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "through reusing prior computation in policy evaluation. Only one target network is used to evaluate", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "the performance of different policies with the help of the training losses of corresponding augmented", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "instances. The augmentation policy network is learned from the intermediate state of the target net-", + "type": "text", + "cross_page": true + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "score": 1.0, + "content": "work, which makes generated augmentation policies more aggressive and adaptive. On the contrary,", + "type": "text", + "cross_page": true + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "to combat harder examples augmented by adversarial policies, the target network has to learn more", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 340, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 340, + 453 + ], + "score": 1.0, + "content": "robust features, which makes the training more efficiently.", + "type": "text", + "cross_page": true + } + ], + "index": 27 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 654, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 165, + 82, + 446, + 338 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 165, + 82, + 446, + 338 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 82, + 446, + 338 + ], + "spans": [ + { + "bbox": [ + 165, + 82, + 446, + 338 + ], + "score": 0.972, + "type": "image", + "image_path": "cf1281f8bae28f7b31d6aab6d29d8044913d73f1d9a58a90985849e432656219.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 165, + 82, + 446, + 95.47368421052632 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 165, + 95.47368421052632, + 446, + 108.94736842105263 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 165, + 108.94736842105263, + 446, + 122.42105263157895 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 165, + 122.42105263157895, + 446, + 135.89473684210526 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 165, + 135.89473684210526, + 446, + 149.36842105263156 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 165, + 149.36842105263156, + 446, + 162.84210526315786 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 165, + 162.84210526315786, + 446, + 176.31578947368416 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 165, + 176.31578947368416, + 446, + 189.78947368421046 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 165, + 189.78947368421046, + 446, + 203.26315789473676 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 165, + 203.26315789473676, + 446, + 216.73684210526307 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 165, + 216.73684210526307, + 446, + 230.21052631578937 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 165, + 230.21052631578937, + 446, + 243.68421052631567 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 165, + 243.68421052631567, + 446, + 257.15789473684197 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 165, + 257.15789473684197, + 446, + 270.63157894736827 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 165, + 270.63157894736827, + 446, + 284.10526315789457 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 165, + 284.10526315789457, + 446, + 297.57894736842087 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 165, + 297.57894736842087, + 446, + 311.05263157894717 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 165, + 311.05263157894717, + 446, + 324.5263157894735 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 165, + 324.5263157894735, + 446, + 337.9999999999998 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 340, + 505, + 396 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Figure 2: An example of dynamic augmentation policies learned with ResNet-50 on ImageNet.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "With the training process of the target network, harder augmentation policies are sampled to combat", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 361, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 376 + ], + "score": 1.0, + "content": "overfitting. Intuitively, more geometric transformations, such as TranslateX, ShearY and Rotate,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "are picked in our sampled policies, which is obviously different from AutoAugment (Cubuk et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 383, + 318, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 318, + 398 + ], + "score": 1.0, + "content": "2019) concentrating on color-based transformations.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + } + ], + "index": 15.0 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "instances. The augmentation policy network is learned from the intermediate state of the target net-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "score": 1.0, + "content": "work, which makes generated augmentation policies more aggressive and adaptive. On the contrary,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "to combat harder examples augmented by adversarial policies, the target network has to learn more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 340, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 340, + 453 + ], + "score": 1.0, + "content": "robust features, which makes the training more efficiently.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 465, + 197, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 199, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 199, + 478 + ], + "score": 1.0, + "content": "3.2 SEARCH SPACE", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "In this paper, the basic structure of the search space of AutoAugment (Cubuk et al., 2019) is re-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 504, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 504, + 511 + ], + "score": 1.0, + "content": "served. An augmentation policy is defined as that it is composed by 5 sub-policies, each sub-policy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "contains two image operations to be applied orderly, each operation has two corresponding parame-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "ters, i.e., the probability and magnitude of the operation. Finally, the 5 best policies are concatenated", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "to form a single policy with 25 sub-policies. For each image in a mini-batch, only one sub-policy", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "will be randomly selected to be applied. To compare with AutoAugment (Cubuk et al., 2019) con-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "veniently, we just slightly modify the search space with removing the probability of each operation.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "score": 1.0, + "content": "This is because that we think the stochasticity of an operation with a probability requires a certain", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "epochs to take effect, which will detain the feedback of the intermediate state of the target network.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "There are totally 16 image operations in our search space, including ShearX/Y, TranslateX/Y, Rotate,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "AutoContrast, Invert, Equalize, Solarize, Posterize, Contrast, Color, Brightness, Sharpness, Cutout", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "(Devries & Taylor, 2017) and Sample Pairing (Inoue, 2018). The range of the magnitude is also", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "discretized uniformly into 10 values. To guarantee the convergence during adversarial learning, the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "magnitude of all the operations are set in a moderate range.1 Besides, the randomness during the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "training process is introduced into our search space. Hence, the search space of the policy in each", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 147, + 665 + ], + "score": 1.0, + "content": "epoch has", + "type": "text" + }, + { + "bbox": [ + 148, + 650, + 273, + 663 + ], + "score": 0.92, + "content": "| S | = ( 1 6 \\times 1 0 ) ^ { 1 0 } \\approx 1 . 1 \\times 1 0 ^ { 2 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 650, + 506, + 665 + ], + "score": 1.0, + "content": "possibilities. Considering the dynamic policy, the number", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 104, + 661, + 403, + 675 + ], + "score": 1.0, + "content": "of possible policies with the whole training process can be expressed as", + "type": "text" + }, + { + "bbox": [ + 403, + 662, + 447, + 675 + ], + "score": 0.88, + "content": "| S | ^ { \\# e p o c h s }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 661, + 506, + 675 + ], + "score": 1.0, + "content": ". An example", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "score": 1.0, + "content": "of dynamically learning the augmentation policy along with the training process is shown in Figure", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 683, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 104, + 683, + 506, + 698 + ], + "score": 1.0, + "content": "2. 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Results in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "Figure 4 show our method achieves the state-of-the-art performance with higher computing and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 677, + 176, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 176, + 690 + ], + "score": 1.0, + "content": "time efficiency3.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 700, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 697, + 390, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 697, + 390, + 714 + ], + "score": 1.0, + "content": "2The length of the moving average is fixed to an epoch in our experiments.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 506, + 725 + ], + "score": 1.0, + "content": "3To clearly present the advantage of our proposed method, we normalize the performance of our method in", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 720, + 375, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 375, + 734 + ], + "score": 1.0, + "content": "the Figure 4, and the performance of AutoAugment is plotted accordingly.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 147, + 81, + 465, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 81, + 465, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 81, + 465, + 186 + ], + "spans": [ + { + "bbox": [ + 147, + 81, + 465, + 186 + ], + "score": 0.962, + "type": "image", + "image_path": "d2f4b35084c6e611d6139d1fbef94757f5d6b6d525d6edbd65d47563cffa3f4c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 147, + 81, + 465, + 116.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 147, + 116.0, + 465, + 151.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 147, + 151.0, + 465, + 186.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 187, + 505, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "Figure 3: The basic architecture of the controller for generating a sub-policy, which consists of two", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "operations with corresponding parameters, the type and magnitude of each operation. 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We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "set the hidden size to 100, and the embedding size to 32. We use Adam optimizer (Kingma &", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "Ba, 2015) with a initial learning rate 0.00035 to train the controller. To avoid unexpected rapid", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "convergence, an entropy penalty of a weight of 0.00001 is applied. All the reported results are the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 344, + 294, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 294, + 356 + ], + "score": 1.0, + "content": "mean of five runs with different initializations.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 366, + 332, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 333, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 333, + 379 + ], + "score": 1.0, + "content": "4.2 EXPERIMENTS ON CIFAR-10 AND CIFAR-100", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "CIFAR-10 dataset (Krizhevsky & Hinton, 2009) has totally 60000 images. The training and test", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 405, + 408 + ], + "score": 1.0, + "content": "sets have 50000 and 10000 images, respectively. Each image in size of", + "type": "text" + }, + { + "bbox": [ + 405, + 396, + 441, + 406 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "belongs to one", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "of 10 classes. We evaluate our proposed method with the following models: Wide-ResNet-28-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 317, + 430 + ], + "score": 1.0, + "content": "10 (Zagoruyko & Komodakis, 2016), Shake-Shake", + "type": "text" + }, + { + "bbox": [ + 318, + 418, + 360, + 429 + ], + "score": 0.44, + "content": "( 2 6 ~ 2 \\mathrm { x } 3 2 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "(Gastaldi, 2017), Shake-Shake (26", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 135, + 440 + ], + "score": 0.51, + "content": "2 \\mathrm { x } 9 6 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 429, + 263, + 442 + ], + "score": 1.0, + "content": "(Gastaldi, 2017), Shake-Shake", + "type": "text" + }, + { + "bbox": [ + 263, + 429, + 311, + 440 + ], + "score": 0.52, + "content": "( 2 6 2 \\mathrm { x } 1 1 2 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "(Gastaldi, 2017), PyramidNet+ShakeDrop (Han", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 440, + 443, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 443, + 452 + ], + "score": 1.0, + "content": "et al., 2017; Yamada et al., 2018). All the models are trained on the full training set.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "Training details: The Baseline is trained with the standard data augmentation, namely, randomly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 183, + 480 + ], + "score": 1.0, + "content": "cropping a part of", + "type": "text" + }, + { + "bbox": [ + 183, + 468, + 218, + 478 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "from the padded image and horizontally flipping it with a probability", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 365, + 491 + ], + "score": 1.0, + "content": "of 0.5. The Cutout (Devries & Taylor, 2017) randomly select a", + "type": "text" + }, + { + "bbox": [ + 365, + 479, + 400, + 489 + ], + "score": 0.9, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "patch of each image, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "then set the pixels of the selected patch to zeros. For our method, the searched policy is applied in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "addition to standard data augmentation and Cutout. For each image in the training process, standard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "data augmentation, the searched policy and Cutout are applied in sequence. For Wide-ResNet-28-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "10, the step learning rate (LR) schedule is adopted. The cosine LR schedule is adopted for the other", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 396, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 396, + 546 + ], + "score": 1.0, + "content": "models. More details about model hyperparameters are supplied in A.1.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 150, + 562 + ], + "score": 1.0, + "content": "Choice of", + "type": "text" + }, + { + "bbox": [ + 151, + 550, + 163, + 560 + ], + "score": 0.51, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 550, + 263, + 562 + ], + "score": 1.0, + "content": ": To choose the optimal", + "type": "text" + }, + { + "bbox": [ + 263, + 550, + 275, + 560 + ], + "score": 0.76, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 550, + 504, + 562 + ], + "score": 1.0, + "content": ", we select Wide-ResNet-28-10 as a target network, and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 370, + 575 + ], + "score": 1.0, + "content": "evaluate the performance of our proposed method verse different", + "type": "text" + }, + { + "bbox": [ + 370, + 562, + 382, + 572 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 559, + 413, + 575 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 414, + 561, + 501, + 573 + ], + "score": 0.92, + "content": "M \\in \\{ 2 , 4 , 8 , 1 6 , 3 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 559, + 505, + 575 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "From Figure 5, we can observe that the test accuracy of the model improves rapidly with the increase", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 581, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 118, + 597 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 583, + 130, + 593 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 581, + 263, + 597 + ], + "score": 1.0, + "content": "up to 8. The further increase of", + "type": "text" + }, + { + "bbox": [ + 263, + 583, + 275, + 593 + ], + "score": 0.74, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 581, + 506, + 597 + ], + "score": 1.0, + "content": "does not bring a significant improvement. Therefore, to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 593, + 491, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 304, + 608 + ], + "score": 1.0, + "content": "balance the performance and the computing cost,", + "type": "text" + }, + { + "bbox": [ + 305, + 594, + 317, + 604 + ], + "score": 0.7, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 593, + 491, + 608 + ], + "score": 1.0, + "content": "is set to 8 in all the following experiments.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "CIFAR-10 results: In Table 1, we report the test error of these models on CIFAR-10. For all of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "these models, our proposed method can achieve better performance compared to previous methods.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 154, + 645 + ], + "score": 1.0, + "content": "We achieve", + "type": "text" + }, + { + "bbox": [ + 154, + 633, + 182, + 644 + ], + "score": 0.89, + "content": "0 . 7 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 632, + 199, + 645 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 633, + 226, + 644 + ], + "score": 0.89, + "content": "0 . 6 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "improvement on Wide-ResNet-28-10 compared to AutoAugment and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 315, + 656 + ], + "score": 1.0, + "content": "PBA, respectively. We achieve a top-1 test error of", + "type": "text" + }, + { + "bbox": [ + 315, + 644, + 343, + 655 + ], + "score": 0.89, + "content": "1 . 3 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 644, + 413, + 656 + ], + "score": 1.0, + "content": "with PyramidNet", + "type": "text" + }, + { + "bbox": [ + 413, + 645, + 420, + 653 + ], + "score": 0.36, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "ShakeDrop, which is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 653, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 129, + 666 + ], + "score": 0.87, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 653, + 505, + 668 + ], + "score": 1.0, + "content": "better than the current state-of-the-art reported in Ho et al. (2019). As shown in Figure 6(a)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "and 6(b),we further visualize the probability distribution of the parameters of the augmentation poli-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "cies learned with PyramidNet+ShakeDrop on CIFAR-10 over time. From Figure 6(a), we can find", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "that the percentages of some operations, such as TranslateY, Rotate, Posterize, and SampleParing,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "gradually increase along with the training process. 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We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "set the hidden size to 100, and the embedding size to 32. We use Adam optimizer (Kingma &", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "Ba, 2015) with a initial learning rate 0.00035 to train the controller. To avoid unexpected rapid", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "convergence, an entropy penalty of a weight of 0.00001 is applied. All the reported results are the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 344, + 294, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 294, + 356 + ], + "score": 1.0, + "content": "mean of five runs with different initializations.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 300, + 505, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 366, + 332, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 333, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 333, + 379 + ], + "score": 1.0, + "content": "4.2 EXPERIMENTS ON CIFAR-10 AND CIFAR-100", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "CIFAR-10 dataset (Krizhevsky & Hinton, 2009) has totally 60000 images. The training and test", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 405, + 408 + ], + "score": 1.0, + "content": "sets have 50000 and 10000 images, respectively. Each image in size of", + "type": "text" + }, + { + "bbox": [ + 405, + 396, + 441, + 406 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "belongs to one", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "of 10 classes. We evaluate our proposed method with the following models: Wide-ResNet-28-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 317, + 430 + ], + "score": 1.0, + "content": "10 (Zagoruyko & Komodakis, 2016), Shake-Shake", + "type": "text" + }, + { + "bbox": [ + 318, + 418, + 360, + 429 + ], + "score": 0.44, + "content": "( 2 6 ~ 2 \\mathrm { x } 3 2 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "(Gastaldi, 2017), Shake-Shake (26", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 135, + 440 + ], + "score": 0.51, + "content": "2 \\mathrm { x } 9 6 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 429, + 263, + 442 + ], + "score": 1.0, + "content": "(Gastaldi, 2017), Shake-Shake", + "type": "text" + }, + { + "bbox": [ + 263, + 429, + 311, + 440 + ], + "score": 0.52, + "content": "( 2 6 2 \\mathrm { x } 1 1 2 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "(Gastaldi, 2017), PyramidNet+ShakeDrop (Han", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 440, + 443, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 443, + 452 + ], + "score": 1.0, + "content": "et al., 2017; Yamada et al., 2018). All the models are trained on the full training set.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 385, + 505, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "Training details: The Baseline is trained with the standard data augmentation, namely, randomly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 183, + 480 + ], + "score": 1.0, + "content": "cropping a part of", + "type": "text" + }, + { + "bbox": [ + 183, + 468, + 218, + 478 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "from the padded image and horizontally flipping it with a probability", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 365, + 491 + ], + "score": 1.0, + "content": "of 0.5. The Cutout (Devries & Taylor, 2017) randomly select a", + "type": "text" + }, + { + "bbox": [ + 365, + 479, + 400, + 489 + ], + "score": 0.9, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "patch of each image, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "then set the pixels of the selected patch to zeros. For our method, the searched policy is applied in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "addition to standard data augmentation and Cutout. For each image in the training process, standard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "data augmentation, the searched policy and Cutout are applied in sequence. For Wide-ResNet-28-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "10, the step learning rate (LR) schedule is adopted. The cosine LR schedule is adopted for the other", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 396, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 396, + 546 + ], + "score": 1.0, + "content": "models. More details about model hyperparameters are supplied in A.1.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 457, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 150, + 562 + ], + "score": 1.0, + "content": "Choice of", + "type": "text" + }, + { + "bbox": [ + 151, + 550, + 163, + 560 + ], + "score": 0.51, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 550, + 263, + 562 + ], + "score": 1.0, + "content": ": To choose the optimal", + "type": "text" + }, + { + "bbox": [ + 263, + 550, + 275, + 560 + ], + "score": 0.76, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 550, + 504, + 562 + ], + "score": 1.0, + "content": ", we select Wide-ResNet-28-10 as a target network, and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 370, + 575 + ], + "score": 1.0, + "content": "evaluate the performance of our proposed method verse different", + "type": "text" + }, + { + "bbox": [ + 370, + 562, + 382, + 572 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 559, + 413, + 575 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 414, + 561, + 501, + 573 + ], + "score": 0.92, + "content": "M \\in \\{ 2 , 4 , 8 , 1 6 , 3 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 559, + 505, + 575 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "From Figure 5, we can observe that the test accuracy of the model improves rapidly with the increase", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 581, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 118, + 597 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 583, + 130, + 593 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 581, + 263, + 597 + ], + "score": 1.0, + "content": "up to 8. The further increase of", + "type": "text" + }, + { + "bbox": [ + 263, + 583, + 275, + 593 + ], + "score": 0.74, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 581, + 506, + 597 + ], + "score": 1.0, + "content": "does not bring a significant improvement. Therefore, to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 593, + 491, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 304, + 608 + ], + "score": 1.0, + "content": "balance the performance and the computing cost,", + "type": "text" + }, + { + "bbox": [ + 305, + 594, + 317, + 604 + ], + "score": 0.7, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 593, + 491, + 608 + ], + "score": 1.0, + "content": "is set to 8 in all the following experiments.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 550, + 506, + 608 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "CIFAR-10 results: In Table 1, we report the test error of these models on CIFAR-10. For all of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "these models, our proposed method can achieve better performance compared to previous methods.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 154, + 645 + ], + "score": 1.0, + "content": "We achieve", + "type": "text" + }, + { + "bbox": [ + 154, + 633, + 182, + 644 + ], + "score": 0.89, + "content": "0 . 7 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 632, + 199, + 645 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 633, + 226, + 644 + ], + "score": 0.89, + "content": "0 . 6 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "improvement on Wide-ResNet-28-10 compared to AutoAugment and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 315, + 656 + ], + "score": 1.0, + "content": "PBA, respectively. We achieve a top-1 test error of", + "type": "text" + }, + { + "bbox": [ + 315, + 644, + 343, + 655 + ], + "score": 0.89, + "content": "1 . 3 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 644, + 413, + 656 + ], + "score": 1.0, + "content": "with PyramidNet", + "type": "text" + }, + { + "bbox": [ + 413, + 645, + 420, + 653 + ], + "score": 0.36, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "ShakeDrop, which is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 653, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 129, + 666 + ], + "score": 0.87, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 653, + 505, + 668 + ], + "score": 1.0, + "content": "better than the current state-of-the-art reported in Ho et al. (2019). As shown in Figure 6(a)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "and 6(b),we further visualize the probability distribution of the parameters of the augmentation poli-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "cies learned with PyramidNet+ShakeDrop on CIFAR-10 over time. 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ModelBaselineCutoutAutoAugmentPBAOur Method
Wide-ResNet-28-103.873.082.682.581.90±0.15
Shake-Shake (26 2x32d)3.553.022.472.542.36±0.10
Shake-Shake (26 2x96d)2.862.561.992.031.85±0.12
Shake-Shake (26 2x112d)2.822.571.892.031.78±0.05
PyramidNet+ShakeDrop2.672.311.481.461.36±0.06
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We replicate the results of Baseline, Cutout and Au-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "toAugment methods from Cubuk et al. (2019), and the results of PBA from Ho et al. (2019) in all of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 568, + 176, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 176, + 581 + ], + "score": 1.0, + "content": "our experiments.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "table_body", + "bbox": [ + 122, + 579, + 487, + 659 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 579, + 487, + 659 + ], + "spans": [ + { + "bbox": [ + 122, + 579, + 487, + 659 + ], + "score": 0.983, + "html": "
ModelBaselineCutoutAutoAugmentPBAOur Method
Wide-ResNet-28-103.873.082.682.581.90±0.15
Shake-Shake (26 2x32d)3.553.022.472.542.36±0.10
Shake-Shake (26 2x96d)2.862.561.992.031.85±0.12
Shake-Shake (26 2x112d)2.822.571.892.031.78±0.05
PyramidNet+ShakeDrop2.672.311.481.461.36±0.06
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ModelBaselineCutout AutoAugmentPBAOur Method
Wide-ResNet-28-1018.8018.4117.0916.7315.49±0.18
Shake-Shake (26 2x96d)17.0516.0014.2815.3114.10±0.15
PyramidNet+ShakeDrop13.9912.1910.6710.9410.42±0.20
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For AutoAugment", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "(Cubuk et al., 2019) and our method, the baseline augmentation and the augmentation policy are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 189, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 204 + ], + "score": 1.0, + "content": "both used for each image. The cosine LR schedule is adopted in the training process. The model", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 202, + 321, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 321, + 214 + ], + "score": 1.0, + "content": "hyperparameters on ImageNet is also detailed in A.1.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "ImageNet results: The performance of our proposed method on ImageNet is presented in Table 3.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 319, + 241 + ], + "score": 1.0, + "content": "It can be observed that we achieve a top-1 accuracy", + "type": "text" + }, + { + "bbox": [ + 320, + 229, + 352, + 240 + ], + "score": 0.89, + "content": "7 9 . 4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "on ResNet-50 without extra data. To", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "the best of our knowledge, this is the highest top-1 accuracy for ResNet-50 learned on ImageNet.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "Besides, we only replace the ResNet-50 architecture with ResNet-50-D, and achieve a consistent", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 297, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 261, + 275 + ], + "score": 1.0, + "content": "improvement with a top-1 accuracy of", + "type": "text" + }, + { + "bbox": [ + 261, + 262, + 293, + 273 + ], + "score": 0.89, + "content": "8 0 . 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 262, + 297, + 275 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "table", + "bbox": [ + 135, + 306, + 473, + 364 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 284, + 504, + 305 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 238, + 297 + ], + "score": 1.0, + "content": "Table 3: Top-1 / Top-5 test error", + "type": "text" + }, + { + "bbox": [ + 238, + 284, + 254, + 295 + ], + "score": 0.72, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 283, + 506, + 297 + ], + "score": 1.0, + "content": "on ImageNet. Note that the result of ResNet-50-D is achieved", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 294, + 276, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 276, + 307 + ], + "score": 1.0, + "content": "only through substituting the architecture.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "table_body", + "bbox": [ + 135, + 306, + 473, + 364 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 135, + 306, + 473, + 364 + ], + "spans": [ + { + "bbox": [ + 135, + 306, + 473, + 364 + ], + "score": 0.978, + "html": "
ModelBaselineAutoAugmentPBAOur Method
ResNet-5023.69 / 6.9222.37 /6.18120.60±0.15 /5.53±0.05
ResNet-50-D22.84 / 6.48120.00±0.12/5.25±0.03
ResNet-20021.52 / 5.8520.00 /4.90118.68±0.18 /4.70±0.05
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(2019). In addition, the test error of Random is", + "type": "text" + }, + { + "bbox": [ + 346, + 554, + 373, + 565 + ], + "score": 0.89, + "content": "1 . 0 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "better than Fixed. This indicates", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "that augmenting batch with randomly generated policies can reduce overfitting in a certain extent.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 332, + 589 + ], + "score": 1.0, + "content": "Furthermore, our method achieves the best test error of", + "type": "text" + }, + { + "bbox": [ + 332, + 576, + 365, + 587 + ], + "score": 0.9, + "content": "2 0 . 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "through augmenting samples with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "score": 1.0, + "content": "adversarial policies. From the result, we can conclude that these policies generated by the policy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "network are more adaptive to the training process, and make the target network have to learn more", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 609, + 169, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 169, + 621 + ], + "score": 1.0, + "content": "robust features.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 631, + 309, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 310, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 310, + 644 + ], + "score": 1.0, + "content": "4.5 COMPUTING COST AND TIME OVERHEAD", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "Computing Cost: The computation in target network training is reused for policy evaluation. This", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "makes the computing cost in policy search become negligible. Although there exists an increase of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "computing cost in target network training, the total computing cost in training one target network", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 682, + 371, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 371, + 694 + ], + "score": 1.0, + "content": "with augmentation policies is quite small compared to prior work.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Time Overhead: Since we just train one target network with a large batch distributedly and simul-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "taneously, the time overhead of the large-batch training is equal to the regular training. 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ModelBaselineCutout AutoAugmentPBAOur Method
Wide-ResNet-28-1018.8018.4117.0916.7315.49±0.18
Shake-Shake (26 2x96d)17.0516.0014.2815.3114.10±0.15
PyramidNet+ShakeDrop13.9912.1910.6710.9410.42±0.20
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For AutoAugment", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "(Cubuk et al., 2019) and our method, the baseline augmentation and the augmentation policy are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 189, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 204 + ], + "score": 1.0, + "content": "both used for each image. The cosine LR schedule is adopted in the training process. The model", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 202, + 321, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 321, + 214 + ], + "score": 1.0, + "content": "hyperparameters on ImageNet is also detailed in A.1.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 156, + 506, + 214 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "ImageNet results: The performance of our proposed method on ImageNet is presented in Table 3.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 319, + 241 + ], + "score": 1.0, + "content": "It can be observed that we achieve a top-1 accuracy", + "type": "text" + }, + { + "bbox": [ + 320, + 229, + 352, + 240 + ], + "score": 0.89, + "content": "7 9 . 4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "on ResNet-50 without extra data. To", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "the best of our knowledge, this is the highest top-1 accuracy for ResNet-50 learned on ImageNet.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "Besides, we only replace the ResNet-50 architecture with ResNet-50-D, and achieve a consistent", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 297, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 261, + 275 + ], + "score": 1.0, + "content": "improvement with a top-1 accuracy of", + "type": "text" + }, + { + "bbox": [ + 261, + 262, + 293, + 273 + ], + "score": 0.89, + "content": "8 0 . 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 262, + 297, + 275 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 218, + 506, + 275 + ] + }, + { + "type": "table", + "bbox": [ + 135, + 306, + 473, + 364 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 284, + 504, + 305 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 238, + 297 + ], + "score": 1.0, + "content": "Table 3: Top-1 / Top-5 test error", + "type": "text" + }, + { + "bbox": [ + 238, + 284, + 254, + 295 + ], + "score": 0.72, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 283, + 506, + 297 + ], + "score": 1.0, + "content": "on ImageNet. Note that the result of ResNet-50-D is achieved", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 294, + 276, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 276, + 307 + ], + "score": 1.0, + "content": "only through substituting the architecture.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "table_body", + "bbox": [ + 135, + 306, + 473, + 364 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 135, + 306, + 473, + 364 + ], + "spans": [ + { + "bbox": [ + 135, + 306, + 473, + 364 + ], + "score": 0.978, + "html": "
ModelBaselineAutoAugmentPBAOur Method
ResNet-5023.69 / 6.9222.37 /6.18120.60±0.15 /5.53±0.05
ResNet-50-D22.84 / 6.48120.00±0.12/5.25±0.03
ResNet-20021.52 / 5.8520.00 /4.90118.68±0.18 /4.70±0.05
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This", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "makes the computing cost in policy search become negligible. 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MethodAug. PolicyEnlarge BatchLR ScheduleTest Error
BaselinestandardM=1step23.69
FixedstandardM=8cosine22.70
RandomrandomM=8cosine21.68
OursadversarialM=8cosine20.60
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SearchingTrainingTotalSearchingTrainingTotal
AutoAugment150001601516010111
Our Method~012801280~011
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MethodDatasetAutoAugmentOur MethodPolicy Transfer
Wide-ResNet-28-10CIFAR-102.681.902.45±0.13
Wide-ResNet-28-10CIFAR-10017.0915.4916.48±0.15
ResNet-50-DImageNet120.0020.20±0.05
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MethodAug. PolicyEnlarge BatchLR ScheduleTest Error
BaselinestandardM=1step23.69
FixedstandardM=8cosine22.70
RandomrandomM=8cosine21.68
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SearchingTrainingTotalSearchingTrainingTotal
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DatasetModelBatch Size (N · M)LRWDEpoch
CIFAR-10Wide-ResNet-28-10128·80.15e-4200
CIFAR-10Shake-Shake (26 2x32d)128·80.21e-4600
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CIFAR-10PyramidNet+ShakeDrop128·80.11e-4600
CIFAR-100Wide-ResNet-28-10128·80.15e-4200
CIFAR-100Shake-Shake (26 2x96d)128·80.15e-41200
CIFAR-100PyramidNet+ShakeDrop128·80.51e-41200
ImageNetResNet-502048·80.81e-4120
ImageNetResNet-50-D2048·80.81e-4120
ImageNetResNet-2002048·80.81e-4120
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DatasetModelBatch Size (N · M)LRWDEpoch
CIFAR-10Wide-ResNet-28-10128·80.15e-4200
CIFAR-10Shake-Shake (26 2x32d)128·80.21e-4600
CIFAR-10Shake-Shake ( (262x96d)128·80.21e-4600
CIFAR-10Shake-Shake (26 2x112d)128·80.21e-4600
CIFAR-10PyramidNet+ShakeDrop128·80.11e-4600
CIFAR-100Wide-ResNet-28-10128·80.15e-4200
CIFAR-100Shake-Shake (26 2x96d)128·80.15e-41200
CIFAR-100PyramidNet+ShakeDrop128·80.51e-41200
ImageNetResNet-502048·80.81e-4120
ImageNetResNet-50-D2048·80.81e-4120
ImageNetResNet-2002048·80.81e-4120
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ModelBaselineCutoutAutoAugmentPBAOur Method
Wide-ResNet-28-103.873.082.682.581.90±0.15
Shake-Shake (26 2x32d)3.553.022.472.542.36±0.10
Shake-Shake (26 2x96d)2.862.561.992.031.85±0.12
Shake-Shake (26 2x112d)2.822.571.892.031.78±0.05
PyramidNet+ShakeDrop2.672.311.481.461.36±0.06
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ModelBaselineCutout AutoAugmentPBAOur Method
Wide-ResNet-28-1018.8018.4117.0916.7315.49±0.18
Shake-Shake (26 2x96d)17.0516.0014.2815.3114.10±0.15
PyramidNet+ShakeDrop13.9912.1910.6710.9410.42±0.20
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ModelBaselineAutoAugmentPBAOur Method
ResNet-5023.69 / 6.9222.37 /6.18120.60±0.15 /5.53±0.05
ResNet-50-D22.84 / 6.48120.00±0.12/5.25±0.03
ResNet-20021.52 / 5.8520.00 /4.90118.68±0.18 /4.70±0.05
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MethodDatasetAutoAugmentOur MethodPolicy Transfer
Wide-ResNet-28-10CIFAR-102.681.902.45±0.13
Wide-ResNet-28-10CIFAR-10017.0915.4916.48±0.15
ResNet-50-DImageNet120.0020.20±0.05
ResNet-200ImageNet20.0018.6819.05±0.10
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MethodAug. PolicyEnlarge BatchLR ScheduleTest Error
BaselinestandardM=1step23.69
FixedstandardM=8cosine22.70
RandomrandomM=8cosine21.68
OursadversarialM=8cosine20.60
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MethodComputing CostTime Overhead
SearchingTrainingTotalSearchingTrainingTotal
AutoAugment150001601516010111
Our Method~012801280~011
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DatasetModelBatch Size (N · M)LRWDEpoch
CIFAR-10Wide-ResNet-28-10128·80.15e-4200
CIFAR-10Shake-Shake (26 2x32d)128·80.21e-4600
CIFAR-10Shake-Shake ( (262x96d)128·80.21e-4600
CIFAR-10Shake-Shake (26 2x112d)128·80.21e-4600
CIFAR-10PyramidNet+ShakeDrop128·80.11e-4600
CIFAR-100Wide-ResNet-28-10128·80.15e-4200
CIFAR-100Shake-Shake (26 2x96d)128·80.15e-41200
CIFAR-100PyramidNet+ShakeDrop128·80.51e-41200
ImageNetResNet-502048·80.81e-4120
ImageNetResNet-50-D2048·80.81e-4120
ImageNetResNet-2002048·80.81e-4120
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However, most recent work focuses on discriminative classifiers, which only model the conditional distribution of the labels given the inputs. In this paper, we propose and investigate the deep Bayes classifier, which improves classical naive Bayes with conditional deep generative models. We further develop detection methods for adversarial examples, which reject inputs with low likelihood under the generative model. Experimental results suggest that deep Bayes classifiers are more robust than deep discriminative classifiers, and that the proposed detection methods are effective against many recently proposed attacks. + +# 1 INTRODUCTION + +Deep neural networks have been shown to be vulnerable to adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2014). The latest attack techniques can easily fool a deep neural network with imperceptible perturbations (Goodfellow et al., 2014; Papernot et al., 2016b; Carlini & Wagner, 2017a; Kurakin et al., 2016; Madry et al., 2018; Chen et al., 2017a), even in the black-box case, where the attacker does not have access to the network’s weights (Papernot et al., 2017b; Chen et al., 2017b; Alzantot et al., 2018a). Adversarial attacks are serious security threats to machine learning systems, threatening applications beyond image classification (Carlini & Wagner, 2018; Alzantot et al., 2018b). + +To address this outstanding security issue, researchers have proposed defence mechanisms against adversarial attacks. Adversarial training, which augments the training data with adversarially perturbed inputs, has shown moderate success at defending against recently proposed attack techniques (Szegedy et al., 2013; Goodfellow et al., 2014; Tramer et al., 2017; Madry et al., 2018). In addition, \` recent advances in Bayesian neural networks have demonstrated that uncertainty estimates can be used to detect adversarial attacks (Li & Gal, 2017; Feinman et al., 2017; Louizos & Welling, 2017; Smith & Gal, 2018). Another notable category of defence techniques involves the usage of generative models. For example, Gu & Rigazio (2014) used an auto-encoder to denoise the inputs before feeding them to the classifier. This denoising approach has been extensively investigated, and the “denoisers” in usage include generative adversarial networks (Samangouei et al., 2018), PixelCNNs (Song et al., 2018) and denoising auto-encoders (Kurakin et al., 2018). These developments rely on the “off-manifold” conjecture, that is, that adversarial examples are far away from the data manifold, although Gilmer et al. (2018) has challenged this observation with a synthetic example. + +Surprisingly, much less recent work has investigated the robustness of generative classifiers (Ng & Jordan, 2002) against adversarial attacks for multi-class classification, where such classifiers explicitly model the conditional distribution of the inputs given labels. Typically, a generative classifier produces predictions by comparing between the likelihood of the labels for a given input, which is closely related to the “distance” of the input to the data manifold associated with a class. Therefore, generative classifiers should be robust to many recently proposed adversarial attacks if the “off-manifold” conjecture holds for many real-world applications. Unfortunately, many generative classifiers in popular use, including naive Bayes and linear discriminant analysis (Fisher, 1936), perform poorly on natural image classification tasks, making it difficult to verify the “off-manifold” conjecture and the robustness of generative classifiers with these tools. In recent work, $\mathbf { k }$ -nearest neighbors (Cover & Hart, 1967), a method which shares many similarities with generative classifiers, has been significantly improved in handling natural images by leveraging deep feature representations (Papernot & McDaniel, 2018). To the best of our knowledge, an approach which targets a similar contribution has not yet been proposed for generative classifiers. + +Are generative classifiers more robust to recently proposed adversarial attack techniques? To answer this, we improve the naive Bayes algorithm by using conditional deep generative models, and evaluate the conjecture on the proposed generative classifier. In summary, our contributions include: + +• We propose deep Bayes as an extension of naive Bayes, in which the conditional distribution of an input, given a label, is parameterised by a deep latent variable model (LVM). We learn the LVM with the variational auto-encoder algorithm (Kingma & Welling, 2013; Rezende et al., 2014), and approximate Bayes’ rule using importance sampling. We propose three detection methods for adversarial perturbations. The first two use the learned generative model as a proxy of the data manifold, and reject inputs that are far away from it. The third computes statistics for the classifier’s output probability vector on training data, and rejects inputs that lead to under-confident predictions. We evaluate the robustness of the proposed generative classifier on the MNIST multi-class and CIFAR binary classification tasks. We also improve the robustness of deep neural networks on CIFAR-10 multi-class classification, by fusing discriminatively learned visual feature representations with the proposed generative classifiers. We further compare the generative classifiers with a number of discriminative classifiers, including Bayesian neural networks and discriminative latent variable models. + +# 2 DEEP BAYES: CONDITIONAL DEEP LVM AS A GENERATIVE CLASSIFIER + +Denote $p _ { \mathcal { D } } ( \pmb { x } , \pmb { y } )$ the data distribution for the input $\pmb { x } \in \mathbb { R } ^ { D }$ and label $\pmb { y } \in \{ \pmb { y } _ { c } | c = 1 , . . . , C \}$ , where $\mathbf { \nabla } \mathbf { \textbf { { y } } } _ { c }$ denotes the one-hot encoding vector for class $c$ . For a given $\pmb { x } \in \mathbb { R } ^ { D }$ we can define the ground-truth label by + +$$ +\pmb { y } \sim p _ { \mathcal { D } } ( \pmb { y } | \pmb { x } ) \quad \mathrm { ~ i f ~ } \pmb { x } \in \mathrm { s u p p } ( p _ { \mathcal { D } } ( \pmb { x } ) ) . +$$ + +We assume the data distribution $p _ { \mathcal { D } } ( \pmb { x } , \pmb { y } )$ follows the manifold assumption: for every class $c$ , the conditional distribution $p _ { \mathcal { D } } ( \pmb { x } | \pmb { y } _ { c } )$ has a low-dimensional manifold support $\mathcal { M } _ { c } = \operatorname { s u p p } ( p _ { \mathcal { D } } ( \pmb { x } | \pmb { y } _ { c } ) )$ . Therefore the training dataset $\mathcal { D } = \{ ( \pmb { x } ^ { ( n ) } , \pmb { y } ^ { ( n ) } ) \} _ { n = 1 } ^ { N }$ is generated as the following: + +$$ +( { \pmb x } ^ { ( n ) } , { \pmb y } ^ { ( n ) } ) \sim p _ { \mathcal { D } } ( { \pmb x } , { \pmb y } ) \Leftrightarrow { \pmb y } ^ { ( n ) } \sim p _ { \mathcal { D } } ( { \pmb y } ) , { \pmb x } ^ { ( n ) } \sim p _ { \mathcal { D } } ( { \pmb x } | { \pmb y } ) . +$$ + +A (Bayesian) generative classifier first builds a generative model $p ( { \pmb x } , { \pmb y } ) = p ( { \pmb x } | { \pmb y } ) p ( { \pmb y } )$ , and then, in prediction time, predicts the label $\boldsymbol { y } ^ { * }$ of a test input $\pmb { x } ^ { * }$ using Bayes’ rule, + +$$ +p ( { \pmb y } ^ { * } | { \pmb x } ^ { * } ) = \frac { p ( { \pmb x } ^ { * } | { \pmb y } ^ { * } ) p ( { \pmb y } ^ { * } ) } { p ( { \pmb x } ^ { * } ) } = \mathrm { s o f t m a x } _ { c = 1 } ^ { C } \left[ \log p ( { \pmb x } ^ { * } , { \pmb y } _ { c } ) \right] , +$$ + +where $\mathrm { s o f t m a x } _ { c = 1 } ^ { C }$ denotes the softmax operator over the $c$ axis. Therefore, the output probability vector is computed analogously to many deep discriminative classifiers which use softmax activation in the output layer, so many existing attacks can be tested directly. However, unlike discriminative classifiers, the “logit” values prior to softmax activation have a clear meaning here, which is the (approximated) log joint distribution $\log p ( { \pmb x } ^ { * } , { \pmb y } _ { c } )$ of input $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } } \ast \mathbf { \ v { x } }$ conditioned on a given label $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } _ { c } }$ . Therefore, one can also analyse the logit values to determine whether the unseen pair $( { \boldsymbol { \it x } } ^ { * } , { \boldsymbol { \it y } } ^ { * } )$ is legitimate, a utility which will be discussed further in later sections. + +Naive Bayes is perhaps the most well-known generative classifier; it assumes a factorised distribution for the conditional generator, i.e. $\begin{array} { r } { p ( \pmb { x } | \pmb { y } ) = \prod _ { d = 1 } ^ { D } p ( x _ { d } | \pmb { y } ) } \end{array}$ . However, this factorisation assumption is inappropriate for e.g. image and speech data. Fortunately, we can leverage the recent advances in generative modelling and apply a deep generative model for the joint distribution $p ( { \pmb x } , { \pmb y } )$ . We refer to such generative classifiers that use deep generative models as deep Bayes classifiers. + +In this paper, we consider a deep latent variable model (LVM) $p ( { \pmb x } , { \pmb z } , { \pmb y } )$ , which will be used for classification. In this case, the conditional distribution is p(x|y) = R p(x,z,y)dzR p(x,z,y)dzdx . Importantly, this leads to a conditional distribution $p ( { \pmb x } | { \pmb y } )$ that is not factorised (even when $p ( \pmb { x } | \pmb { z } , \pmb { y } )$ is), which is much more powerful than naive Bayes. Since maximum likelihood is intractable, we follow + +![](images/220140844094a3f088b10ef70c1787e4802498efb521d87e87e0c1adf30847b7.jpg) +Figure 1: A visualisation of the graphical models, including both Generative and Discriminative ones, as well as Fully connected and Bottleneck ones. The last character indicates the first node in the topological order of the graph. The colour encoding is the same as those in experiments. + +Kingma & Welling (2013) and Rezende et al. (2014) to introduce an amortised approximate posterior $q ( \boldsymbol { z } | \boldsymbol { x } , \boldsymbol { y } )$ , and train both $p$ and $q$ by maximising the variational lower-bound: + +$$ +\mathbb { E } _ { \mathcal { D } } [ \mathcal { L } _ { \mathrm { V I } } ( \pmb { x } , \pmb { y } ) ] = \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \mathbb { E } _ { q } \left[ \log \frac { p ( \pmb { x } _ { n } , \pmb { z } _ { n } , \pmb { y } _ { n } ) } { q ( \pmb { z } _ { n } | \pmb { x } _ { n } , \pmb { y } _ { n } ) } \right] . +$$ + +After training, the predicted class probability vector $\boldsymbol { y } ^ { * }$ for a future input $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } } \ast \mathbf { \ v { x } }$ is computed by an approximation to Bayes’ rule with importance sampling: + +$$ +p ( \pmb { y } ^ { * } | \pmb { x } ^ { * } ) \approx \mathrm { s o f t m a x } _ { c = 1 } ^ { C } \left[ \log \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \frac { p ( \pmb { x } ^ { * } , \pmb { z } _ { c } ^ { k } , \pmb { y } _ { c } ) } { q ( \pmb { z } _ { c } ^ { k } | \pmb { x } ^ { * } , \pmb { y } _ { c } ) } \right] , \quad \pmb { z } _ { c } ^ { k } \sim q ( \pmb { z } | \pmb { x } ^ { * } , \pmb { y } _ { c } ) . +$$ + +We evaluate the effect of the following factorisation structures on the robustness of the induced classifier from the generative model $p ( \pmb { x } , \pmb { z } , \pmb { y } )$ (see Figure 1). + +$$ +\begin{array} { l l l } { p ( { \pmb x } , { \pmb z } , { \pmb y } ) = p ( { \pmb z } ) p ( { \pmb y } | { \pmb z } ) p ( { \pmb x } | { \pmb z } , { \pmb y } ) \quad } & { ( { \mathrm { G F Z } } ) } \\ { p ( { \pmb x } , { \pmb z } , { \pmb y } ) = p _ { \mathcal { D } } ( { \pmb y } ) p ( { \pmb z } | { \pmb y } ) p ( { \pmb x } | { \pmb z } , { \pmb y } ) \quad } & { ( { \mathrm { G F Y } } ) } \\ { p ( { \pmb x } , { \pmb z } , { \pmb y } ) = p ( { \pmb z } ) p ( { \pmb y } | { \pmb z } ) p ( { \pmb x } | { \pmb z } ) \quad } & { ( { \mathrm { G B Z } } ) \quad } & { p ( { \pmb x } , { \pmb z } , { \pmb y } ) = p _ { \mathcal { D } } ( { \pmb x } ) p ( { \pmb z } | { \pmb x } ) p ( { \pmb y } | { \pmb z } ) } \\ { p ( { \pmb x } , { \pmb z } , { \pmb y } ) = p _ { \mathcal { D } } ( { \pmb y } ) p ( { \pmb z } | { \pmb y } ) p ( { \pmb x } | { \pmb z } ) \quad } & { ( { \mathrm { G B Y } } ) } \end{array} +$$ + +We use the initial character “G” to denote generative classifiers and “D” to denote discriminative classifiers. Models with the second character as “F” have a fully connected graph, while “B” models enforce the usage of the latent code $_ { z }$ as a bottleneck. The last character of the model name indicates the first node in topological order. Model DFZ is somewhat intermediate, as it builds a generative model for the inputs $_ { \textbf { \em x } }$ but also directly parameterises the conditional distribution $p ( \pmb { y } | \pmb { x } , z )$ . We do not test other architectures under this nomenclature, as either the graph contains directed cycles (e.g. $\pmb { x } \pmb { y } \pmb { z } \pmb { x } ,$ ), or $_ z$ is the last node in topological order (e.g. $\pmb { x } \pmb { y } , ( \pmb { x } , \pmb { y } ) \pmb { z } )$ so that the marginalisation of $_ z$ does not affect classification. + +# 3 DETECTING ADVERSARIAL ATTACKS WITH GENERATIVE CLASSIFIERS + +We propose detection methods for adversarial examples using generative classifier’s logit values. As an illustrating example, consider a labelled dataset of “cat” and “dog” images. If an adversarial image of a cat $\pmb { x } _ { \mathrm { a d v } }$ is incorrectly labelled as “dog”, then either this image is ambiguous, or, under a perfect generative model, the logit $\log p ( \pmb { x } _ { \mathrm { a d v } }$ , “dog”) will be significantly lower than normal. This means we can detect attacks using the logits $\log p ( { \pmb x } ^ { * } , { \pmb y } _ { c } ) , c = 1 , . . . , C$ computed on a test input $\mathbf { \boldsymbol { x } } ^ { * }$ , by comparing them with the logits computed on legitimate training inputs. + +Concretely, the proposed detection algorithms are as follows. We aim to reject both unlabelled input $_ { \textbf { \em x } }$ that have low probability under $p _ { \bar { D } } \bar { ( } \pmb { x } )$ , and labelled data $( { \pmb x } , { \pmb y } )$ that have low $p _ { \mathcal { D } } ( \pmb { x } , \pmb { y } )$ values. + +• Marginal detection: rejecting inputs that are far away from the manifold. One can select a threshold $\delta$ and reject an input $_ { \textbf { \em x } }$ if $- \log p ( { \pmb x } ) > \delta$ . To determine the threshold $\delta$ , we can compute $\bar { d } _ { \mathcal { D } } = \mathbb { E } _ { \pmb { x } \sim \mathcal { D } } [ - \log p ( \pmb { x } ) ]$ and $\sigma _ { \mathcal { D } } = \sqrt { \mathbb { V } _ { \pmb { x } \sim \mathcal { D } } [ \log p ( \pmb { x } ) ] }$ , then set $\delta = \bar { d } _ { D } + \alpha \sigma _ { D }$ . It is also possible to compute the statistics ${ \bar { d } } _ { p } , \sigma _ { p }$ on the images generated by the generative model accordingly. +Logit detection: rejecting inputs using joint density. Given a victim model $\pmb { y } = F ( \pmb { x } )$ , one can reject $_ { \textbf { \em x } }$ if $- \log p ( { \pmb x } , F ( { \pmb x } ) ) > \delta _ { \pmb y }$ . We can use the mean and variance statistics $\bar { d } _ { c } , \sigma _ { c }$ computed on $\log p ( { \pmb x } , { \pmb y } _ { c } )$ and select $\delta _ { y _ { c } } = \bar { d } _ { c } + \alpha \sigma _ { c }$ . + +![](images/63ac3a599fb3fd7f9b0ceeb289e5088441fbf7ba011a0d3f9a13300e966b86cc.jpg) +Figure 2: Visualising detection mechanisms. The scattered dots are training data points, with different classes shown in different colours (red for $c = 0$ and blue for $c = 1$ ). Same labels are manually assigned for inputs when the detection method requires $\textbf { { y } }$ . Decision regions are shown in the corresponding colours. Input points in the shaded area are rejected by detection. + +• Divergence detection: rejecting inputs with over- and/or under-confident predictions. + +Denote ${ \pmb p } ( { \pmb x } )$ as a $C$ -dimensional probability vector outputted by the classifier. For each class $c$ , we first collect the mean classification probability vector $\pmb { p } _ { c } = \mathbb { E } _ { ( \pmb { x } , \pmb { y } _ { c } ) \in \mathcal { D } } [ \pmb { p } ( \pmb { x } ) ]$ , then compute the mean $\bar { d } _ { c }$ and standard deviation $\sigma _ { c }$ on a selected divergence/distance measure $\bar { \mathrm { D } } [ p _ { c } | | p ( { \pmb x } ) ]$ for all $( \pmb { x } , \pmb { y } _ { c } ) \in \mathcal { D }$ . A test input $\pmb { x } ^ { * }$ with prediction label $c ^ { * } =$ arg max $\pmb { p } ( \pmb { x } ^ { * } )$ is rejected if $\mathrm { D } [ p _ { c ^ { * } } | | p ( { \pmb x } ^ { * } ) ] > \bar { d } _ { c ^ { * } } + \bar { \alpha \sigma } _ { c ^ { * } }$ . Therefore, an example $\pmb { x } ^ { * }$ will be rejected if the classifier is over-confident or under-confident (ambiguous inputs). + +When $\mathrm { D }$ is selected as the KL-divergence, we call this detection method $K L$ detection. +Other divergence/distance measures such as total variation (TV) can also be used. + +For better intuition, we visualise the detection mechanisms in Figure 2 with a synthetic “two rings” binary classification example. In this case we sample $2 \times 1 0 0 0$ training data points as: + +( $\begin{array} { r l } & { x , y ) \sim p _ { D } \Leftrightarrow y \sim \mathrm { B e r n } ( 0 . 5 ) , \theta \sim \mathrm { U n i f o r m } ( 0 , 2 \pi ) , x | y \sim \mathcal { N } ( x ; c _ { y } + r _ { y } [ c o s ( \theta ) , s i n ( \theta ) ] ^ { \mathrm { T } } , \sigma ^ { 2 } { \bf I } ) . } \end{array}$ We consider a generative classifier $p ( \pmb { x } , \pmb { y } ) = p ( \pmb { x } | \pmb { y } ) p _ { \mathscr { D } } ( \pmb { y } ) = \mathcal { N } ( \pmb { x } ; \pmb { \mu _ { y } } , \sigma ^ { 2 } \mathbf { I } ) \mathbf { B e r n } ( 0 . 5 )$ ) with $\mu _ { y } =$ $\operatorname { p r o j } ( x , \operatorname { r i n g } _ { y } ) = \arg \operatorname* { m i n } _ { \hat { x } \in \mathbb { R } ^ { 2 } }$ , $| | \hat { \mathbf { x } } - \pmb { c } _ { y } | | _ { 2 } = { r } _ { y } \ | | \pmb { x } - \hat { \pmb { x } } | | _ { 2 }$ . The $\delta$ thresholds are selected to achieve $1 0 \%$ false positive rates on training data. From the visualisations we see that inputs that are far away from the model manifold are rejected by marginal/logit detection. At the same time, logit detection rejects data points that are not on the manifold of the given class. KL/TV detection does not construct manifold-aware acceptance regions, which is as expected since the proposed divergence detection method does not require the classifier to be generative. However, both detection methods have some success in rejecting uncertain predictions, especially for TV, which also rejects ambiguous inputs (see the ring-cross regions in the last two plots). Combining all three methods, we see that the rejected inputs are either far away from the manifold, or are ambiguous. Again we emphasise that a suitable generative model is required to make the detection methods work in practice, since the data distribution $p _ { \mathcal { D } } ( \pmb { x } , \pmb { y } )$ is approximated by $p ( { \pmb x } , { \pmb y } )$ . + +Detection methods using logit values have also been proposed in Li & Gal (e.g. 2017); Feinman et al. (e.g. 2017). However it is unclear whether the logits values in discriminative classifiers have a clear semantic meaning, while the logit values in deep Bayes represent the log probability of generating the input $_ { \textbf { \em x } }$ given the class label ${ \mathbf { \nabla } } y = y _ { c }$ . Our approach is also distinct from previous “denoising” approaches (e.g. Song et al., 2018; Samangouei et al., 2018; Kurakin et al., 2018) that require training an additional generative model separately. The deep Bayes classifiers share the same generative model as the detection methods, meaning that detected adversarial examples are indeed far away from the classifier’s manifold (which is also an approximation to the data manifold). Therefore the claim “detection neural networks can be bypassed” (Carlini & Wagner, 2017a; Athalye et al., 2018) does not directly transfer to our approach, and we can use the marginal and logit detection methods to directly verify the “off-manifold” adversarial example conjecture. + +# 4 EXPERIMENTS + +We carry out a number of tests on the proposed deep Bayes classifiers (3), where $q ( z | \cdot )$ and $p ( z | \cdot )$ are factorised Gaussians, and the conditional probability $p ( { \pmb x } | \cdot )$ , if required, is parameterised by an $\ell _ { 2 }$ loss. Besides the LVM-based classifiers, we further train a Bayesian neural network (BNN) with Bernoulli dropout (dropout rate 0.3), as it has been shown in Li & Gal (2017) and Feinman et al. (2017) that BNNs are more robust than their deterministic counterparts. The constructed BNN has $2 \mathbf { x }$ more channels than LVM-based classifiers, making the comparison slightly “unfair”, as the BNN layers have more capacity. We use $K = 1 0$ Monte Carlo samples for all the classifiers. + +![](images/00ae51d305686cd331a0a614d4028a57298f23ffcfe3de3c5e0e296f5308977e.jpg) +Figure 3: Accuracies (column 1), detection rates (columns 2-4) and minimum $\ell _ { \mathrm { i n f } }$ perturbation (column 6) against white-box zero-knowledge $\ell _ { \infty }$ attacks on MNIST. The higher the better. The second from right most column visualises crafted adversarial examples on an image of digit “7”, with $\ell _ { \infty }$ distortion $\epsilon$ growing from 0.1 to 0.5. + +The adversarial attacks in test include both $\ell _ { \infty }$ and $\ell _ { 2 }$ untargeted attacks from CleverHans 2.0 library (Papernot et al., 2017a): fast gradient sign method (FGSM, Goodfellow et al., 2014), projected gradient descent (PGD, Madry et al., 2018), momentum iterative attack (MIM, Dong et al., 2017) and Carlini & Wagner $\ell _ { 2 }$ (CW, Carlini & Wagner, 2017a). Two metrics are reported: accuracy of the classifier on crafted adversarial examples, and detection rate on adversarial examples that have successfully caused the classifier to misclassify. This detection rate is defined as the true positive (TP) rate of finding an adversarial example, and the detection threshold is selected to achieve a $5 \%$ false positive rate on clean training data. + +The experiments are performed under various threat model settings. We further evaluate the transferability of crafted adversarial examples across different classifiers in the same way as done in Papernot et al. (2016a). We only provide visualisations in the main text; full table results can be found in the appendix. Readers are also referred to the appendix for further experiments, including a quantitative analysis of the bottleneck effect on robustness and detection. + +# 4.1 MNIST + +The first set of experiments evaluate the robustness of generative classifiers on MNIST. Here the image pixel values are normalised to [0, 1], and the LVM-based classifiers have $\dim ( z ) ~ = ~ 6 4$ . We first perform white-box zero-knowledge attacks, i.e. the attacker can differentiate through the classifier to craft adversarial examples, but he/she is not aware of the existence of the detector. Then we perform white-box perfect-knowledge attacks, where the attacker can differentiate through both the classifier and the detector, and he/she knows the usage of random $_ { z }$ samples by the VAE-based classifiers (Biggio et al., 2013; Carlini & Wagner, 2017b). Lastly we consider grey-box and blackbox attacks, and evaluate the robustness of generative classifiers against transferred attacks. + +White-box attacks (zero-knowledge, $\ell _ { \infty , }$ ) Results for $\ell _ { \infty }$ attacks are reported in Figure 3, and in general generative classifiers perform better in terms of victim accuracy and minimum perturbation of the attacks.1 By contrast, DFX & DFZ are not robust to the weakest attack (FGSM) even when $\epsilon =$ 0.2 (where the adversarial examples are still visually close to the original digit “7”). Interestingly, DBX is the most robust against FGSM & $\mathbf { M I M } ^ { 2 }$ , which agrees with the preliminary tests in Alemi et al. (2017). But DBX is less robust to PGD, and here GBZ is a clear winner. These results show that the bottleneck is sometimes beneficial for better robustness of MNIST classifiers. + +For detection, generative classifiers have successfully detected the adversarial examples with $\epsilon \geq$ 0.3, which is reasonable as the visual distortion is already significant. Importantly, generative classifier’s victim accuracy decreases as the $\ell _ { \infty }$ distortion $\epsilon$ increases, but at the same time the TP rates for marginal and logit detection also increase. Therefore these $\ell _ { \infty }$ attacks fail to find near-manifold adversarial examples that fool both the classifier and the detection methods. DFZ, as an intermediate between generative and discriminative classifiers, has worse robustness results, but has good detection performance for the marginal and logit metrics. This is because with softmax activation, the marginal distribution $p ( { \pmb x } )$ is dropped, but in marginal/logit detection $p ( { \pmb x } )$ is still in use. + +Table 1: White-box zero-knowledge CW- $\cdot \ell _ { 2 }$ attack results. Here, accuracy (adv) measures classifying the adversarial inputs to the original classes. + +
acc.(clean)acc. (adv)l2 dist.TP KL
BNN99.12%24.40%2.12995.31
GFZ98.55%28.58%2.66395.37
GBZ97.45%81.51%2.44691.01
GFY99.15%28.64%2.73296.03
GBY98.72%32.72%2.73594.46
DFX99.10%20.31%2.09599.96
DBX98.87%30.19%1.80696.76
DFZ99.10%13.60%2.18899.57
+ +![](images/7155749b68688828aa6066f627f9693ca5bbdeeaa7d46d33450c322e20420f9b.jpg) +Figure 4: Visualising the clean inputs and the CW adversarial examples crafted on GFZ, digits in red rectangles show significant ambiguity. + +White-box attack (zero-knowledge, $\ell _ { 2 }$ ) For the CW $\ell _ { 2 }$ attack, we performed a hyperparameter search for the loss-balancing parameter $c$ $\ell _ { 2 }$ distortion increases with larger $c$ ) in $\{ 0 . 1 , 1 , 1 0 , 1 0 0 , 1 0 0 0 \}$ on the first 100 test images, and we found $c = 1 0$ returns the highest success rate. Results are reported in Table 1. Although being successful on fooling many classifiers, CW failed on attacking GBZ. Also, the mean $\ell _ { 2 }$ distortions of the successful attacks on generative classifiers are significantly larger. Furthermore, we found the success of the attack on other generative classifiers is mainly due to the ambiguity of the crafted adversarial images. As visualised in Figure 4 (also see Figure D.1 in appendix), the induced distortion from CW leads to ambiguous digits which sit at the perceptual boundary between the original and the adversarial classes. With the KL detection method, all classifiers achieve $> 9 5 \%$ detection rates, which is as expected as the default CW attack configuration, by construction, generates adversarial examples that lead to minimal difference between the logit values of the most and the second most probable classes. + +White-box attack (perfect-knowledge, $\ell _ { \infty }$ ) The PGD-based perfect-knowledge $\ell _ { \infty }$ attack is designed following (Carlini & Wagner, 2017b): we construct an (approximate) Bayes classifier $p _ { k } ( \pmb { y } | \pmb { x } )$ using (3) for each set of samples $\{ z _ { c } ^ { k } \} _ { c = 1 } ^ { C }$ , and minimize the following with PGD: + +$$ +\mathcal { L } ( \eta ) = \sum _ { k = 1 } ^ { K } \log p _ { k } ( \boldsymbol { y } | \boldsymbol { x } + \eta ) + \lambda _ { \mathrm { d e t e c t } } \operatorname* { m a x } ( 0 , \Phi ( \boldsymbol { x } + \eta , \boldsymbol { y } ) - \delta ) . +$$ + +The detection statistic $\Phi ( { \pmb x } + { \pmb \eta } , { \pmb y } )$ is $- \log p ( { \pmb x } + { \pmb \eta } )$ for marginal detection, and $\delta$ is the corresponding threshold computed on training data. For logit/KL detection, the detection statistics and thresholds are constructed accordingly. We label the three attacks against the marginal, logit and KL based detection schemes ’-PKM’, ’-PKL’ and ’-PKK’ respectively. When we only have knowledge of the $K$ samples and not the detection method we call this attack ’-PK0’ (ie $\lambda _ { \mathrm { d e t e c t } } { = } 0$ in Eq. 4). + +Results are visualised in Figure 5. We see that although the attacker can reduce detection levels, this comes with the trade-off of increasing accuracy, suggesting that a perfect-knowledge adversary cannot break both the classifier and detector working in tandem. + +Sanity checks on gradient masking Athalye et al. (2018) claimed that if a successful defence against white-box attacks is due to gradient masking, then this defence is likely to be less effective against grey-box/black-box attacks, as they do not differentiate through the victim classifier and the defence mechanism (Papernot et al., 2017b). Therefore, we consider two transfer attacks based on distilling the victim classifier using a “student” CNN which has no gradient masking. The two attacks differ in their threat models: in the grey-box setting the attacker has access to the output probability vectors of the classifiers on the training data, while in the black-box setting the attacker has access to queried labels only. For the latter black-box setting, we follow Papernot et al. (2017b) to train a substitute CNN using Jacobian-based dataset augmentation, and we refer to appendix B.1 for the detailed algorithm. The grey-box substitutes achieve $> 9 9 \%$ agreement with the victims on test data, and the black-box substitutes obtained $\sim 9 6 \%$ accuracy on test data. + +![](images/92fbc68f077b0912c224d9aa79a22a044fe5d692681f9182450b3127b5751d73.jpg) +Figure 5: Accuracy and detection rates of DBX, GBY, and GBZ against PGD-based perfectknowledge attack $( \epsilon = 0 . 2 )$ ) on MNIST. The solid area denotes accuracy and the hatched area denotes detection rate with each considered detector. Zero knowledge attacks are labelled ’-ZK’, other attack labels are described in the main text. + +![](images/a9d0675f4478e57a95c5946f6c2e9e26184ee8ec7905a49a23ceeb8edb588c24.jpg) +Figure 6: Accuracy and detection rates against distillation-based attacks on MNIST. The higher the better. We only present generative classifiers’ results here, for full results see appendix E. + +Figure 6 shows the accuracy and detection metrics on transferred $\ell _ { \infty }$ attacks crafted on the substitute models, with a comparison to their white-box counterparts. Note that these crafted attacks achieve $\sim$ $1 0 0 \%$ success rates on fooling the substitute models when $\epsilon \geq 0 . 2$ (see appendix E). However, they do not transfer very well to the generative classifiers. Importantly, for a fixed $\epsilon$ setting, the whitebox attacks achieve significantly higher success rates (i.e. lower victim accuracies) than their grey/black-box counterparts, and the gap is at least $> 2 0 \%$ for $\epsilon \leq 0 . 3$ (see Table 2). We further present the mean minimum $\ell _ { \infty }$ perturbation in Table 2, and we see that the minimum perturbation obtained by grey-/black-box attacks are significantly higher than those obtained by white-box attacks. + +All these results suggest that the robustness of generative classifiers is unlikely to be caused by gradient masking. Again on the detection side, the detection rates increase as the perturbation size increase, and they are near $1 0 0 \%$ for $\epsilon = 0 . 3$ . This means that most of the successfully transferred adversarial images are off the generative classifier’s manifold (as a proxy to the data manifold). + +SPSA (evolutionary strategies) We consider another black-box setting that only assumes access to the logit values of the prediction given an input. We use the SPSA $\ell _ { \infty }$ attack (Uesato et al., 2018), which is similar to the white-box zero-knowledge attacks, except that gradients are numerically estimated using the logit values from the victim classifier. Results in Table 3 clearly show that SPSA performs much worse on generative classifiers when compared to white-box PGD. Again this means gradient masking is unlikely to be responsible for the improved robustness of generative classifiers, as utilising the exact gradient yielded improved results. + +Table 2: Mean minimum $\ell _ { \infty }$ perturbation (in red, computed on $\epsilon \in \{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 \} )$ and victim accuracy (in blue, for $\epsilon \leq 0 . 3 )$ for $\ell _ { \infty }$ attacks on MNIST. We manually assign the min. perturbation $\epsilon = 0 . 6$ to inputs that all attacks failed to find adversarial perturbations. + +
AttackGFZGBZGFYGBY
PGD(white-box)0.23 /7.71%0.30 /30.78%0.21 / 5.52%0.23 / 8.89%
MIM(white-box)0.24 / 9.02%0.21 /4.97%0.22 / 6.72%0.21 /1.54%
PGD(grey-box)0.37 /51.08%0.36 / 50.64%0.38 / 53.29%0.36 / 48.66%
MIM(grey-box)0.34 / 43.00%0.33 / 40.94%0.34 / 46.64%0.33 /40.06%
PGD(black-box)0.40 / 61.93%0.42 /66.75%0.38 / 56.35%0.43 / 68.50%
MIM(black-box)0.36 / 50.44%0.38 / 59.86%0.36 / 48.07%0.39 / 61.78%
+ +![](images/539526329232367bbefc7f3dde85c40b740d2252d0a5700694c8065b953efdca.jpg) +Figure 7: Results on cross-model transfer attacks on MNIST. The horizontal axis corresponds to the source victim that the adversarial examples are crafted on, and the vertical axis corresponds to the target victim that the attacks are transferred to. The higher (i.e. the lighter) the better. + +Cross-model attack transferability Finally, we report in Figure 7 the transferability results of the crafted adversarial examples between different models (Papernot et al., 2016a). Here we take adversarial examples crafted in the white-box setting with PGD and MIM $\acute { \epsilon } = 0 . 3$ ), and transfer successful attacks to other classifiers. The transfer is effective between generative classifiers but not from generative to discriminative (and vice versa). The attacks crafted on DBX do not transfer in general, while at the same time, DBX is the least robust model in this case. Furthermore, the generative classifiers obtain very high detection rates on all transferred attacks $( > 9 5 \% )$ . In summary, generative classifiers are more robust against the tested transfer attacks across different models. + +# 4.2 CIFAR PLANE-VS-FROG BINARY CLASSIFICATION + +We consider the same set of evaluations on CIFAR-10, in order to validate the robustness of generative classifiers on natural images (c.f. Carlini & Wagner, 2017b). Unfortunately, we failed to train fully generative classifiers with comparable test accuracies to discriminative CNNs (typically $> 8 0 \%$ ): the clean accuracies for GFZ & GFY are all $< 5 0 \%$ . Even when using the conditional PixelC $\mathrm { N N } { + } { + }$ (Salimans et al., 2017) (which uses much deeper networks), the clean accuracy on the test data is $7 2 . 4 \%$ . Instead, we consider a simpler binary classification problem and construct a dataset containing CIFAR images from the “airplane” and “frog” categories. The images in this dataset are scaled to [0, 1]. On this dataset, the generative classifiers use $\mathrm { d i m } ( z ) = 1 2 8$ and obtain $> 9 0 \%$ clean test accuracy (see appendix). The attacks are performed on the test images that all models initially correctly classify, leading to a test set of 1577 instances. Due to the page limit, we only present white-box zero-knowledge attacks here, and discuss further attacks in the appendix. + +White-box attacks (zero-knowledge, $\ell _ { \infty }$ ) We present the white-box $\ell _ { \infty }$ attack results in Figure 8, where the distortion strengths are selected as $\epsilon \in \lbrace 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \rbrace$ . Again, generative classifiers are more robust than the discriminative ones, and GBZ is the most robust, much better than the others when $\epsilon \geq 0 . 0 5$ . BNN is significantly better than other discriminative VAE-based classifiers, presumably due to higher randomness. Detection results are less satisfactory: marginal/logit detection fail to detect attacks with $\epsilon = 0 . 1$ (which attain both high success rate and induce visually perceptible distortion). KL detection performs better, and interestingly, discriminative classifiers dominate in this metric. These results suggest that the $\ell _ { 2 }$ likelihood might not be best suited for modelling natural images (c.f. Larsen et al., 2016; van den Oord et al., 2016). Still, the minimum distortion required to fool generative classifiers are much higher than that for discriminative ones, indeed the visual distortion of the adversarial examples on generative classifiers are more significant. + +![](images/efeb93cf87777d23548d2d32db60dccd59abcfdfbb5946443a93be80204c004d.jpg) +Figure 8: Accuracy, detection rates and minimum $\ell _ { \mathrm { i n f } }$ perturbations against white-box zeroknowledge $\ell _ { \infty }$ attacks on the CIFAR plane-vs-frog dataset. The higher the better. The second from right most column visualises crafted adversarial examples on an image of a plane, with $\ell _ { \infty }$ distortion $\epsilon \in \{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \}$ . + +![](images/0dd2d70f1cc4886126c26f92b994f6f2f74de3b1a925e6ac9f2d7859b1aa8cd2.jpg) +Figure 9: Accuracy and detection rates against white-box zero-knowledge CW attacks on the CIFAR plane-vs-frog dataset. The higher the better. + +White-box attack (zero-knowledge, $\ell _ { 2 }$ ) As the test set is relatively small, we directly perform CW attacks on the test data with $c \in \{ 0 . 1 , 1 , 1 0 , 1 0 0 , 1 0 0 0 \}$ . Results are visualised in Figure 9. The generative classifiers are significantly more robust than the others (with the best being GBZ), and the mean $\ell _ { 2 }$ distortions computed on successful attacks are also significantly higher. The TP rates are low for marginal and logit detection, which is reasonable as the crafted images are visually similar to the clean ones. Note that the distortion for the attacks on generative classifiers is perceptible. These results indicate that this CW attack is ineffective when attacking generative classifiers. + +4.3 FULL CIFAR-10: COMBINING DEEP BAYES AND DISCRIMINATIVE FEATURES + +The final experiment examines the robustness of CIFAR-10 multi-class classifiers, with the generative classifiers trained on discriminative visual features. To do this, we download a pretrained deep convolutional net3 with VGG16-like architecture (Simonyan & Zisserman, 2014), and use its feature representation $\phi ( { \pmb x } )$ as the input to the VAE-based classifiers: $p ( { \pmb y } | { \pmb x } ) = p ( { \pmb y } | \phi ( { \pmb x } ) )$ . The classifiers in test include GBZ, GBY and DBX. We use fully-connected neural networks for these classifiers, and select from VGG16 the $9 ^ { \mathrm { t h } }$ convolution layer (CONV9) and the first fully connected layer after convolution (FC1) as the feature layers to ensure $\sim 9 0 \%$ test accuracy (see appendix). + +Results on white-box zero-knowledge $\ell _ { \infty }$ attacks are visualised in Figure 10. Compared with the VGG16 baseline, we see clear improvements in robustness and detection for all VAE-based classifiers. In particular, the generative classifiers GBZ and GBY are overall better than DBX. More importantly, generative classifiers based on CONV9 features are significantly more robust than those based on FC1 features. In contrast, for DBX, which is discriminative, the robustness results are very similar, indicating that the level of feature representation has little effect. These results suggest that one can achieve both high clean accuracy and better robustness against adversaries by combining discriminatively learned visual features and generative classifiers. + +![](images/312124c69aaa2404a21fddcb8a35efb3d4d839cf4d73b5393739a6dacb03296e.jpg) +Figure 10: Accuracy and detection rates against white-box zero-knowledge $\ell _ { \infty }$ attacks on CIFAR10. The higher the better. Note that results for the DBX classifiers are almost identical. + +# 5 DISCUSSION + +We have proposed deep Bayes as a generative classifier that uses deep latent variable models to model the joint distribution of input-output pairs. We have given evidence, on multiple classification tasks, that generative classifiers are more robust to adversarial attacks than discriminative classifiers. Furthermore, the logit in generative classifiers has a well defined meaning and can be used to detect attacks, even when the classifier is fooled. + +Our results corroborate with the Bayesian neural network literature, in particular (Li & Gal, 2017; Feinman et al., 2017; Carlini & Wagner, 2017b), in showing that modelling unobserved variables are effective for defending against adversarial attacks4. Concurrent to us, Schott et al. (2018) also demonstrated the robustness of generative classifiers on MNIST, in which the logits are computed by a tempered version of the variational lower-bound. However, their approach requires thousands of random $_ z$ samples and tens of optimisation steps to approximate $\log p ( { \pmb x } | { \pmb y } )$ for every input-output pair $( { \pmb x } , { \pmb y } )$ , making it less scalable than our importance sampling technique to large datasets and big architectures. Indeed, we have scaled our approach to CIFAR-10, a natural image dataset, and the robustness results are consistent with those on MNIST. In addition, we have also shown that the structure of the graphical model has a significant impact on robustness: deep LVM-based generative classifiers generally outperform the (randomised) discriminative ones. + +While we have given strong evidence to suggest that generative classifiers are more robust to current adversarial attacks, we do not wish to claim that these models will be robust to all possible attacks. Aside from many recent attacks being designed specifically for discriminative neural networks, there is also evidence for the fragility of generative models; e.g. naive Bayes as a standard approach for spam filtering is well-known to be fragile (Dalvi et al., 2004; Huang et al., 2011), and very recently Tabacof et al. (2016); Kos et al. (2017); Creswell et al. (2017) also designed attacks for (unconditional) VAE-type models. However, generative classifiers can be made more robust too, to counter these weaknesses. Dalvi et al. (2004) have shown that generative classifiers can be made more secure if aware of the attack strategy, and Biggio et al. (2011; 2014) further improved naive Bayes’ robustness by modelling the conditional distribution of the adversarial inputs. These approaches are similar to the adversarial training of discriminative classifiers, and efficient ways for doing so with generative classifiers can be an interesting research direction. + +But even with this note of caution, we believe this work offers exciting avenues for future work. Using generative classifiers offers an interesting way to evaluate generative models and can drive improvements in their ability to tackle high-dimensional datasets, where traditionally generative classifiers have been less accurate than discriminative classifiers (Efron, 1975; Ng & Jordan, 2002). In addition, the combination of generative and discriminative models investigated in this paper is a compelling direction for future research. Overall, we believe that progress on generative classifiers can inspire better designs of attack, defence and detection techniques. + +# REFERENCES + +Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. Deep variational information bottleneck. In International Conference on Learning Representations, 2017. + +Moustafa Alzantot, Yash Sharma, Supriyo Chakraborty, and Mani Srivastava. Genattack: Practical black-box attacks with gradient-free optimization. arXiv preprint arXiv:1805.11090, 2018a. + +Moustafa Alzantot, Yash Sharma, Ahmed Elgohary, Bo-Jhang Ho, Mani Srivastava, and Kai-Wei Chang. Generating natural language adversarial examples. arXiv preprint arXiv:1804.07998, 2018b. + +Anish Athalye, Nicholas Carlini, and David Wagner. 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Adversarial images for variational autoencoders. arXiv preprint arXiv:1612.00155, 2016. + +Florian Tramer, Alexey Kurakin, Nicolas Papernot, Ian Goodfellow, Dan Boneh, and Patrick Mc- \` Daniel. Ensemble adversarial training: Attacks and defenses. arXiv preprint arXiv:1705.07204, 2017. + +Jonathan Uesato, Brendan O’Donoghue, Aaron van den Oord, and Pushmeet Kohli. Adversarial risk and the dangers of evaluating against weak attacks. arXiv preprint arXiv:1802.05666, 2018. + +Aaron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In International Conference on Machine Learning, pp. 1747–1756, 2016. + +# A MODEL ARCHITECTURES + +MNIST experiments The VAEs are constructed with convolutional encoders and deconvolutional generators. More specifically, the encoder network for $q ( \boldsymbol { z } | \boldsymbol { x } , \boldsymbol { y } )$ is the same across all VAE-based classifiers. It starts with a 3-layer convolutional neural network with $5 \times 5$ filters and 64 channels, with a max-pooling operation after each convolution. Then, the convolutional network is followed by a MLP with 2 hidden layers, each with 500 units, to produce the mean and variance parameters of $q$ . The label $\textbf { { y } }$ is injected into the MLP at the first hidden layer, as a one hot encoding (i.e. for MNIST, the first hidden layer has $5 0 0 { + } 1 0 $ units). The latent dimension is $\mathrm { d i m } ( z ) = 6 4$ . + +The $p$ models’ architectures are the following: + +GFZ: For $p ( \pmb { y } | \pmb { z } )$ we use a MLP with 1 hidden layer composed of 500 units. For $p ( { \pmb x } | { \pmb y } , z )$ we used an MLP with 2 hidden layers, each with 500 units, and $4 \times 4 \times 6 4$ dimension output, followed by a 3-layer deconvolutional network with $5 \times 5$ kernel size, stride 2 and [64, 64, 1] channels. +GFY: We use an MLP with 1 hidden layer composed of 500 units for $p ( z | \boldsymbol { y } )$ , and the same architecture as GFZ for $p ( { \pmb x } | { \pmb y } , z )$ . +DFZ: We use almost the same deconvolutional network architecture for $p ( { \pmb x } | z )$ as GFZ’s $p ( { \pmb x } | { \pmb y } , z )$ network, except that the input is $_ z$ only. For $p ( \pmb { y } | \pmb { x } , z )$ we use almost the same architecture as $q ( \boldsymbol { z } | \boldsymbol { x } , \boldsymbol { y } )$ except that the injected input to the MLP is $_ z$ and the MLP output is the set of logit values for $\textbf { { y } }$ . +DFX: We use the same architecture as G3 for $p ( \pmb { y } | \pmb { x } , z )$ . The network for $p ( \boldsymbol { z } | \boldsymbol { x } )$ is almost identical except that there is no injected input to the MLP, and the network returns the mean and variance parameters for $q ( \boldsymbol { z } | \boldsymbol { x } )$ . +DBX: We use GFZ’s architecture for $p ( \pmb { y } | \pmb { z } )$ and DFX’s architecture for $p ( \boldsymbol { z } | \boldsymbol { x } )$ . +GBY: We use GFY’s architecture for $p ( z | \mathbf { \boldsymbol { y } } )$ and DFZ’s architecture for $p ( { \pmb x } | { \pmb z } )$ . +GBZ: We use GFZ’s architecture for $p ( \pmb { y } | \pmb { z } )$ and DFZ’s architecture for $p ( { \pmb x } | z )$ . + +The BNN has almost the same architecture as the encoder network $q$ , except that it uses $2 \mathbf { x }$ the hidden units/channels, and the last layer is 10 dimensions. Note that here we used dropout as it is convenient to implement, and we expect better approximate inference methods (such as stochastic gradient MCMC) to return better results for robustness and detection. + +CIFAR plane-vs-frog experiments The model architectures are almost the same as used in MNIST experiments, except that the hidden layer dimensions for the MLP layers are increased to 1000. For the encoder $q$ , the channels are increased to [64, 128, 256]. For the $p$ models, the deconvolutional networks have different channel values, [128, 64, 3], and the MLP before the deconvolution outputs a $4 \times 4 \times 2 5 6$ vector (before reshaping). The BNN has $2 \mathbf { x }$ the channels but still uses 1000 hidden units. + +CIFAR-10 experiments The pre-trained VGG16 network is downloaded from https:// github.com/geifmany/cifar-vgg, where the CONV9 and FC1 layers correspond to: + +CONV9: https://github.com/geifmany/cifar-vgg/blob/master/ cifar10vgg.py#L82 FC1: https://github.com/geifmany/cifar-vgg/blob/master/ cifar10vgg.py#L109 + +The VAE-based classifiers build fully connected networks on top of the extracted features, and use $\mathrm { d i m } ( z ) = 1 2 8$ for bottleneck. The encoder $q ( \boldsymbol { z } | \phi ( \boldsymbol { x } ) , \boldsymbol { y } )$ has the network architectures $[ \mathrm { d i m } ( \phi ( { \pmb x } ) ) $ $+ \dim ( y )$ , 1000, 1000, $\mathrm { d i m } ( z ) \times 2 ]$ , and we use the same encoder architecture across all classifiers. The decoder architectures are as follows: + +DBX: We use an MLP of layers $[ \dim ( z ) , ~ 1 0 0 0 , ~ \dim ( y ) ]$ for $p ( \pmb { y } | \pmb { z } )$ and an MLP of layers $[ \mathrm { d i m } ( \phi ( { \pmb x } ) )$ , 1000, 1000, $\mathrm { d i m } ( z ) \times 2 ]$ for $p ( \pmb { z } | \phi ( \pmb { x } ) )$ . + +GBZ: We use an MLP of layers $[ \mathrm { d i m } ( z )$ , 1000, 1000, $\dim ( y ) ]$ for $p ( \pmb { y } | \pmb { z } )$ and an MLP of layers [dim(z), 1000, 1000, $\mathrm { d i m } ( \phi ( { \pmb x } ) ) ]$ for $p ( \phi ( { \pmb x } ) | z )$ . + +GBY: We use an MLP of layers $[ \mathrm { d i m } ( y )$ , 1000, $\mathrm { d i m } ( z ) \times 2 ]$ for $p ( z | \boldsymbol { y } )$ and GBZ’s architecture for $p ( \phi ( { \pmb x } ) | z )$ . + +# B ATTACK SETTINGS + +We use the Cleverhans package to perform attacks. We use the default hyper-parameters, if not specifically stated. + +PGD: We perform the attack for 40 iterations with step-size 0.01. + +MIM: We perform the attack for 40 iterations with step-size 0.01 and decay factor 1.0. + +CW- $\ell _ { 2 }$ : We use learning rate 0.01 and confidence 0, and we optimise the loss for 1000 iterations. + +SPSA: We use almost the same hyper-parameters as in Uesato et al. (2018) except for the number of samples for gradient estimates. In detail, we perform the attack for 100 iterations with perturbation size 0.01, Adam learning rate 0.01, stopping threshold -5.0 and 2000 samples for each gradient estimate. + +# B.1 JACOBIAN-BASED DATASET AUGMENTATION + +The black-box distillation attack is based on Papernot et al. (2017b), which trains a substitute CNN using Jacobian-based dataset augmentation. Assume $\boldsymbol { y } = \boldsymbol { F } ( \boldsymbol { x } )$ is the output one-hot vector of the victim, and ${ \pmb p } ( { \pmb x } )$ is the probability vector output of the substitute model, then at the $t ^ { \mathrm { { t h } } }$ outerloop, we train the substitute CNN on dataset $\mathcal { D } _ { t } \overset { \cdot } { = } \{ ( \boldsymbol { x } _ { n } , \boldsymbol { y } _ { n } ) \}$ with queried ${ \bf { { y } } } _ { n }$ for 10 epochs, and augment the dataset by + +$$ +\mathcal { D } _ { t + 1 } = \mathcal { D } _ { t } \cup \big \{ \big ( \hat { x } , F ( \hat { x } ) \big ) \mid \hat { x } = x + \lambda \nabla _ { x } p ( x ) ^ { \mathrm { T } } y , \big ( x , y \big ) \in \mathcal { D } _ { t } \big \} . +$$ + +We initialise $\mathcal { D } _ { 1 }$ with $2 0 0 \times 1 0$ datapoints from the MNIST test set, select $\lambda = 0 . 1$ , and run the algorithm for 6 outer-loops. On MNIST, this results in 64, 000 queried inputs, and $\sim 9 6 \%$ accuracy of the substitute model on test data. On CIFAR binary classification, we use $2 0 0 \times 2$ datapoints for the inital query set $\mathcal { D } _ { 1 }$ , resulting in 12, 800 queries in total. The substitutes achieved almost the same accuracy as their corresponding victim models on clean test datapoints. + +# C FURTHER EXPERIMENTS + +# C.1 FURTHER EXPERIMENTS ON CIFAR BINARY CLASSIFICATION + +White-box attack (perfect-knowledge, $\ell _ { \infty }$ ) Figure C.1 shows the perfect knowledge attack on the CIFAR binary classification task. Again we see that although the attack is effective for the detection schemes, it comes with the price of decreased mis-classification rates. Interestingly GBY seems to be robust to this attack, where the accuracies on the crafted adversarial examples increase. + +Sanity checks on gradient masking We conducted the same sets of transferred $\ell _ { \infty }$ attack experiments, and presents the results in Figure C.2 and Table C.1. Again that these crafted attacks achieve $\sim 1 0 0 \%$ success rates on fooling the substitute models when $\epsilon \geq 0 . 1$ . Similar to the MNIST experiments, these adversarial examples do not transfer very well to the generative classifiers, and for a fixed $\epsilon$ setting, the white-box attacks achieve significantly higher success rates than their grey-/black-box counterparts (with the gap at $\epsilon \leq 0 . 1$ around $3 0 \%$ ). Furthermore, the minimum perturbation obtained by grey-/black-box attacks are significantly higher than those obtained by white-box attacks. For detection, the detection rates are relatively low at $\epsilon \leq 0 . 1$ (where the classifiers achieved high accuracy). But the detection rates increase significantly for $\epsilon = 0 . 2$ , where the victim accuracies also drop. + +All these results suggest that the robustness of generative classifiers is unlikely to be caused by gradient masking. Also most of the successfully transferred adversarial images are off the generative classifier’s manifold (as a proxy to the data manifold). + +![](images/5152312abaec5bc13db7e41eea936e83e591f26cf23328722a6ed5def0474647.jpg) +Figure C.1: Accuracy and detection rates of DBX, GBY, and GBZ against PGD-based perfectknowledge attack $\epsilon = 0 . 1$ ) on CIFAR binary task. The solid area denotes accuracy and the hatched area denotes detection rate with each considered detector. Zero knowledge attacks are labelled ’- ZK’, other attack labels are the same as for the MNIST plot (Figure 5). + +![](images/efd469fdcaf929c8a6615157ea860f31d2e71ae5eb8fbeb99dfc0c6d149f3ee1.jpg) +Figure C.2: Accuracy and detection rates against distillation-based attacks on CIFAR plane-vs-frog binary classification. The higher the better. We only present generative classifiers’ results here, for full results see appendix E. + +Table C.1: Mean minimum $\ell _ { \infty }$ perturbation (in red, computed on $\epsilon \in \{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \} )$ ) and victim accuracy (in blue, for $\epsilon \leq 0 . 1 $ ) for $\ell _ { \infty }$ attacks on CIFAR plane-vs-frog binary classification. We manually assign the min. perturbation $\epsilon = 0 . 3$ to inputs that all attacks failed to find adversarial perturbations. + +
AttackGFZGBZGFYGBY
PGD(white-box)0.11 / 21.81%0.20 / 65.63%0.11 / 25.81%0.11 / 25.24%
MIM(white-box)0.09 / 15.22%0.13 /37.60%0.10 /16.4%90.09 / 14.39%
PGD(grey-box)0.15 / 50.48%0.23 / 77.30%0.16 / 54.66%0.17 / 57.96%
MIM(grey-box)0.15 /47.62%0.21 / 75.71%0.15 / 51.11%0.16 / 53.84%
PGD(black-box)0.19 / 68.36%0.23 / 79.45%0.20 / 70.13%0.19 / 67.98%
MIM(black-box)0.18 / 66.39%0.23 / 78.38%0.19 / 68.42%0.19 / 66.52%
+ +Table C.2: Accuracy and detection rates against black-box SPSA attack $\epsilon = 0 . 0 5 )$ on CIFAR plane-vs-frog, with a comparision to white-box PGD. The higher the better. + +
GFZGBZGBYGFYDFXDBXDFZ
PGD victim acc67.7%83.9%67.9%67.3%0.3%4.2%0.4%
SPSA victim acc96.4%95.2%96.4%96.3%0.4%87.5%5.1%
SPSA TP logit10.0%17.1%15.5%10.0%N/AN/A4.5%
+ +![](images/c2212cb826926ce9d453f97e43195c2f01d46651d793cf484348cc807383b02c.jpg) +Figure C.3: Results on cross-model transfer attacks on the CIFAR plane-vs-frog dataset. The horizontal axis corresponds to the source victim, and the vertical axis corresponds to the target victim. The higher (lighter) the better. + +SPSA (evolutionary strategies) Similarly we perform the SPSA attack (Uesato et al., 2018) on the CIFAR binary classification task. The results for $\epsilon = 0 . 0 5$ are presented in Table C.2, with a comparison to the white-box PGD attacks. Again we see that SPSA fails to attack generative classifiers, and the bottleneck discriminative classifier DBX is significantly more robust than the discriminative ones with fully-connected graphical models. + +Cross-model attack transferability Finally, we present the CIFAR cross-model attack transferability results in Figure C.3, and here we select $\epsilon = 0 . 1$ instead. Again, transferred attacks are less effective across the victim models. However, the TP rates for logit detection are significantly lower than in the MNIST case (also see Figure 8). Nevertheless, the detection rates for the “discriminative to generative” transfer are considerably higher. Combined with the accuracy results, we see that discriminative models as substitutes are ineffective in the transferred attack setting. + +# C.2 QUANTIFYING THE EFFECT OF THE BOTTLENECK LAYER + +We see from the main text that classifiers with bottleneck structure may be preferred for resisting adversarial examples. To quantify this bottleneck effect, we train on MNIST models DBX, GBZ and GBY with $_ { z }$ dimensions in $\{ \dot { 1 } 6 , 3 2 , 6 4 , 1 2 8 \}$ (the main text experiments use $\dim ( z ) = 6 4 $ ).The clean test accuracy is shown in Table C.3, showing that all models in test perform reasonably well. + +Table C.3: Clean test accuracy on MNIST classification (with varied bottleneck layer sizes). + +
dim(z) = 16dim(z) = 32dim(z)= 64dim(z)=128
DBX99.11%99.01%98.98%98.91%
GBZ97.11%97.08%97.45%96.62%
GBY98.82%98.95%98.72%98.75%
+ +We repeat the same white-box zero-knowledge $\ell _ { \infty }$ attack experiments as done in the main text, where results are presented in Figure C.4 and Tables E.25, E.26 and E.27. It is clear that for discriminative classifiers, DBX, the models become less robust as the bottleneck dimension $\mathrm { d i m } ( z )$ increases. Interestingly DBX classifiers seem to be very robust against FGSM attacks, which agrees with the results in Alemi et al. (2017). For the generative ones, we also observe similar trends (although much less significant) of decreased robustness for GBY classifiers, and for GBZ the trend is unclear, presumably due to local optimum issues in optimisation. In summary, GBZ classifiers are generally more robust compared to GBY classifiers. More importantly, when the accuracy of generative classifiers on adversarial images decreases to zero, the detection rates with marginal/logit detection increases to $1 0 0 \%$ . This clearly shows that the three attacks tested here cannot fool the generative classifiers without being detected. + +![](images/ac97b61ab1e7245d921f06ec5438eb1e3ec5e1ff1b11e353e9a44c52679e086b.jpg) +Figure C.4: Accuracy and detection rates against white-box zero-knowledge $\ell _ { \infty }$ attacks on MNIST, with varied bottleneck layer sizes. + +Table D.1: Clean test accuracy on CIFAR plane-vs-frog classification. + +
BNNGFZGFYDFZDFXDBXGBZGBY
97.00%91.60%91.20%94.85%95.65%96.00%89.35%90.65
+ +# D ADDITIONAL RESULTS + +We visualise in Figure D.1 the crafted adversarial images using white-box CW attack. +We present in Table D.1 the clean accuracy on CIFAR plane-vs-frog test images (2000 in total). +We present in Table D.2 the clean accuracy on CIFAR-10 test images. + +Table D.2: Clean test accuracy on CIFAR-10 classification. + +
VGG16GBZ-FC1GBY-FC1DBX-FC1GBZ-CONV9GBY-CONV9DBX-CONV9
93.59%92.55%93.21%93.49%91.76%88.33%93.21%
+ +![](images/dbcf0906a736b6a88bfdb52b23677d578009e609628af44f312d5c13c3ee6558.jpg) +Figure D.1: Visualising the clean inputs of MNIST and the CW adversarial examples crafted on all the classifiers. + +# E RESULTS IN TABLES + +We present in tables the full results of the experiments. +See Tables E.1 to E.9 for the white-box attacks. +See Tables E.10 to E.15 for the grey-box attacks. +See Tables E.16 to E.21 for the black-box attacks. +See Tables E.22 to E.24 for CIFAR-10 results with VGG-based classifiers. +See Tables E.25 to E.27 for bottleneck effect quantification results. + +Table E. 1 : FGSM white-box zero-knowledge attack results on MNIST. + +
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN92.467.840.526.220.4N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.094.295.495.596.5
GFZ94.274.538.912.95.743.679.8100.0100.0100.056.489.699.9100.0100.089.291.792.292.592.2
GBZ92.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GFY94.374.846.521.710.553.192.6100.0100.0100.066.297.9100.0100.0100.090.893.193.894.294.2
GBY93.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
DFX70.114.81.00.40.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.693.893.893.594.3
DBX91.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DFZ75.219.02.41.11.012.650.6100.0100.0100.026.965.7100.0100.0100.093.195.194.594.994.8
+ +Table E.2: PGD white-box zero-knowledge attack results on MNIST. + +
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN83.212.00.50.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.192.188.720.40.2
GFZ86.737.77.71.20.343.271.891.499.4100.055.877.194.899.6100.090.392.190.990.489.7
GBZ85.157.432.519.311.633.750.384.499.7100.052.266.391.599.8100.090.091.591.591.791.8
GFY79.727.45.61.20.358.187.998.0100.0100.068.792.299.3100.0100.092.692.590.790.785.4
GBY86.735.99.01.70.445.376.194.699.7100.056.979.395.699.9100.090.192.191.691.791.4
DFX47.60.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.889.813.331.442.8
DBX DFZ58.0 49.618.36.01.30.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
1.00.00.00.011.844.394.1100.0100.025.957.694.599.9100.093.989.812.122.928.0
+ +Table E.3: MIM white-box zero-knowledge attack results on MNIST. + +
acc.(adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN82.07.20.10.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.291.172.064.57.1
GFZ87.040.49.01.21.243.583.898.499.499.457.689.398.699.299.291.592.392.090.690.6
GBZ79.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GFY80.830.16.81.41.456.093.799.8100.0100.068.196.699.9100.0100.090.992.492.091.691.6
GBY84.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
DFX48.40.80.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.289.69.217.517.5
DBX66.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DFZ50.51.20.00.00.011.950.799.4100.0100.026.463.299.0100.0100.094.389.811.517.317.3
+ +Table E.4: Perfect knowledge attacks on MNIST. This is done using the PGD attack with $\epsilon = 0 . 2$ . ‘ZK’ stands for zero knowledge, ‘PK0’ where you have knowledge of the $K$ samples but not of the detection system. ‘PKM’ ‘PKL’ and ‘PKK’ are attacks where you have knowledge of the K samples and of the marginal, logit and KL detection mechanisms respectively. + +
ZKPKK
acc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KL
DBX18.3N/AN/A9417N/AN/A93.1N/AN/AN/AN/AN/AN/AN/AN/A35.9N/AN/A93.9
GBZ57.450.366.391.535.582.290.791.447.936.567.491.344.650.569.991.04573.084.891.5
GBY35.976.179.392.12290.891.992.143.847.061.991.152.056.466.990.632.889.291.893.3
+ +Table E.5: FGSM white-box zero-knowledge attack results on CIFAR plane-vs-frog binary classification. + +
acc. (adv)TP marginal
E0.010.020.050.100.200.010.020.050.100.20 0.010.020.050.100.200.010.020.050.100.20
BNN98.293.258.514.56.3N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A58.655.165.554.058.5
GFZ97.194.781.856.431.411.410.15.9 17.499.311.410.46.817.299.358.640.135.941.050.6
GBZ95.193.587.174.962.026.018.615.5 38.699.626.320.319.141.999.635.541.547.043.745.6
GFY96.594.280.756.732.09.69.26.9 18.899.017.613.18.120.899.146.643.736.139.549.3
GBY96.192.982.060.536.317.215.18.1 28.599.220.218.110.031.199.249.748.039.238.747.3
DFX83.842.20.70.00.0N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A63.366.648.497.7100.0
DBX90.978.650.731.718.3N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A50.758.756.859.757.2
DFZ83.952.12.90.00.06.63.82.23.9 60.66.94.22.43.360.457.562.354.866.799.8
+ +Table E.6: PGD white-box zero-knowledge atack results on CIFAR plane-vs-frog binary classification. + +
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN97.986.719.71.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A41.759.455.768.498.9
GFZ98.093.967.721.73.53.36.65.67.832.95.09.56.08.233.037.545.134.544.057.7
GBZ94.693.783.967.452.819.817.912.814.343.122.819.517.417.044.332.139.340.237.133.2
GFY98.495.067.925.84.14.26.76.47.835.73.18.27.67.833.831.243.632.939.653.3
GBY96.492.967.325.76.911.18.47.911.641.413.111.59.212.240.343.043.033.639.152.2
DFX82.735.70.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A64.164.769.6100.0100.0
DBX83.334.64.20.70.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A59.859.565.178.793.4
DFZ82.436.90.40.00.04.93.85.112.391.16.54.15.512.391.356.360.354.799.7100.0
+ +Table E.7 : MIM white-box zero-knowledge attack results on CIFAR plane-vs-frog binary classification. + +
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN96.984.618.70.90.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A35.954.756.362.478.8
GFZ96.991.558.115.21.96.56.26.410.686.110.08.36.611.384.531.039.032.147.277.1
GBZ92.589.471.540.417.915.013.211.024.095.916.414.812.825.196.042.537.531.037.455.0
GFY97.692.359.516.52.35.66.96.313.388.15.76.26.813.887.623.841.433.843.974.6
GBY95.188.856.514.61.315.19.58.619.297.315.310.79.918.597.051.143.231.746.787.6
DFX82.535.20.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A65.567.169.6100.0100.0
DBX82.538.86.11.20.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A54.660.061.578.888.5
DFZ81.536.10.40.00.05.43.95.013.496.96.24.15.513.896.861.158.555.399.9100.0
+ +Table E.8: CW white-box zero-knowledge atack results on CIFAR plane-vs-frog binary classification. + +
acc. (adv)TP marginalTP logitTP KL
E0.101.0010.00100.001000.000.101.0010.001100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
BNN93.766.237.332.347.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A60.980.382.272.080.0
GFZ99.595.676.577.978.30.04.52.63.03.40.04.63.03.64.229.230.425.627.129.0
GBZ96.093.688.988.789.016.711.57.87.77.918.913.79.39.19.441.151.440.434.732.0
GFY99.895.978.880.080.50.06.64.14.75.10.06.64.84.95.475.037.326.026.431.1
GBY97.593.176.777.378.015.814.66.07.17.318.913.96.07.07.942.634.327.231.629.0
DFX82.644.234.36.10.6N/AN/AN/AN/AN/AN/AN/A N/AN/AN/A100.0100.0100.082.595.0
DBX96.572.226.36.316.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A61.679.485.262.267.9
DFZ94.572.329.96.317.97.35.84.34.14.214.16.95.04.44.482.697.397.183.080.2
+ +Table E.9: Perfect knowledge attacks on CIFAR binary task. This is done using the PGD attack with $\epsilon = 0 . 1$ . ‘ZK’ stands for zero knowledge, ‘PK0’ where you have knowledge of the $K$ samples but not of the detection system. ‘PKM’ , ‘PKL’ , and ‘PKK’ are attacks where you have knowledge of the K samples and of the marginal, logit and KL detection mechanisms respectively. + +
PKK
acc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KL
DBX0.7N/AN/A78.70.7N/A82.5N/AN/AN/AN/AN/AN/AN/A0.6N/AN/A 27.682.6
GBZ67.414.317.037.15740.645.14958.49.618.546.958.51.87.847.859.925.542.4
GBY25.711.612.239.128.522.022.64929.55.67.645.929.20.81.845.427.915.615.945
+ +Table E. 1 0: Grey-box PGD attack results on MNIST. + +
substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ86.714.30.00.00.096.683.351.326.817.249.573.899.7100.0100.0
GBZ81.29.10.00.00.093.580.155.335.025.246.773.399.6100.0100.0
GFY84.86.40.00.00.096.782.355.233.524.058.788.8100.0100.0100.0
GBY86.715.00.00.00.095.881.651.027.518.150.178.599.9100.0100.0
DFX74.25.20.50.00.091.757.719.64.31.4N/AN/AN/AN/AN/A
DBX80.65.10.00.00.093.259.222.510.98.2N/AN/AN/AN/AN/A
DFZ69.63.40.30.00.091.957.221.46.32.633.655.191.7100.0100.0
+ +Table E. 1 1 : Grey-box MIM attack results on MNIST. + +
substitute accvictim accTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ87.216.10.00.00.096.582.442.814.23.750.682.5100.0100.0100.0
GBZ81.89.10.00.00.093.378.645.317.05.246.281.7100.0100.0100.0
GFY85.26.90.00.00.096.680.848.018.97.459.896.0100.0100.0100.0
GBY87.015.90.00.00.095.779.941.813.04.051.487.1100.0100.0100.0
DFX76.512.33.62.12.191.556.122.29.99.9N/AN/AN/AN/AN/A
DBX85.119.90.60.10.193.457.014.16.36.3N/AN/AN/AN/AN/A
DFZ70.93.20.00.00.091.854.616.42.70.334.563.898.9100.0100.0
+ +Table E.12: Grey-box CW attack results on MNIST. + +
substitute accvictim accTP logit
E0.101.0010.00100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
GFZ98.44.60.00.00.098.896.996.093.490.253.444.241.640.643.7
GBZ98.463.20.00.00.097.495.492.988.684.545.939.733.031.238.6
GFY98.10.90.00.00.099.097.697.095.794.061.055.249.249.552.7
GBY98.35.30.00.00.098.796.896.093.891.254.241.537.638.743.6
DFX85.10.00.00.00.097.693.391.290.489.8N/AN/AN/AN/AN/A
DBX97.346.00.40.00.098.793.988.485.576.3N/AN/AN/AN/AN/A
DFZ80.50.00.00.00.097.294.692.391.591.232.928.628.136.544.5
+ +Table E. 1 3 : Grey-box PGD attack results on CIFAR plane-vs-frog binary classification. + +
substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ96.788.141.74.50.197.094.984.048.38.38.411.26.17.549.8
GBZ92.983.137.85.50.095.394.891.883.165.615.512.810.919.979.8
GFY96.988.133.73.30.196.995.985.354.212.72.62.05.08.561.2
GBY95.285.632.42.80.196.895.386.459.818.521.814.310.512.575.2
DFX95.780.810.10.20.099.196.779.428.30.9N/AN/AN/AN/AN/A
DBX95.780.530.51.70.099.598.291.165.121.8N/AN/AN/AN/AN/A
DFZ96.581.910.50.30.099.196.976.926.21.34.516.56.18.559.6
+ +Table E.14: Grey-box MIM attck results on CIFAR plane-vs-frog binary classification. + +
substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ96.387.942.14.70.197.194.883.044.84.98.711.15.88.575.1
GBZ92.982.536.94.90.095.594.791.481.749.514.812.613.921.997.8
GFY96.887.934.13.50.196.995.684.649.16.92.62.06.710.381.1
GBY95.185.532.02.60.196.895.585.756.210.921.814.89.213.497.6
DFX95.580.510.50.20.099.196.577.324.00.2N/AN/AN/AN/AN/A
DBX95.681.134.52.20.099.598.389.358.99.9N/AN/AN/AN/AN/A
DFZ96.381.510.70.50.099.196.974.921.90.34.516.66.49.980.2
+ +Table E.15: Grey-box CW attack results on CIFAR plane-vs-frog binary classification. + +
substitute accvictim accTP logit
E0.101.0010.00100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
GFZ98.761.90.00.00.098.695.894.393.893.211.710.210.99.98.5
GBZ96.369.83.30.00.095.995.794.794.794.513.512.712.313.714.4
GFY98.454.40.00.00.098.196.195.395.194.30.02.23.74.67.6
GBY97.652.92.20.00.097.795.694.594.393.725.815.714.213.718.0
DFX64.40.00.00.00.098.597.297.196.795.5N/AN/AN/AN/AN/A
DBX91.356.20.30.00.099.799.099.098.998.1N/AN/AN/AN/AN/A
DFZ80.30.00.00.00.099.297.997.597.395.40.015.714.68.86.1
+ +Table E. 1 6 : Black-box PGD attack results on MNIST. + +
substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ44.20.70.00.00.097.388.462.733.821.848.761.092.399.8100.0
GBZ7.40.00.00.00.094.887.170.752.941.351.068.697.7100.0100.0
GFY49.40.80.00.00.097.486.558.131.921.052.570.897.499.8100.0
GBY21.70.00.00.00.096.989.370.049.737.851.270.898.2100.0100.0
DFX49.21.30.00.00.091.452.113.92.20.7N/AN/AN/AN/AN/A
DBX43.10.70.00.00.095.067.026.29.76.5N/AN/AN/AN/AN/A
DFZ53.81.70.00.00.092.956.315.23.41.534.752.894.099.9100.0
+ +Table E. 1 7 : Black-box MIM attack results on MNIST. + +
substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ45.52.80.00.00.097.286.650.317.017.051.168.097.599.599.5
GBZ8.40.00.00.00.094.785.963.535.217.151.274.499.0100.0100.0
GFY53.12.50.30.10.197.283.749.220.120.155.180.699.099.999.9
GBY24.50.00.00.00.096.888.362.831.713.349.377.699.9100.0100.0
DFX51.32.30.10.00.091.451.713.31.91.9N/AN/AN/AN/AN/A
DBX47.81.60.10.00.094.867.023.27.67.6N/AN/AN/AN/AN/A
DFZ56.12.90.10.00.092.756.013.12.82.835.259.997.6100.0100.0
+ +Table E.18: Black-box CW attack results on MNIST. + +
substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.0010.00100.001000.00
GFZ65.20.30.00.00.098.898.797.4 94.392.052.3 50.839.644.755.3
GBZ76.60.40.00.00.097.397.295.9 92.790.445.746.141.540.345.7
GFY88.43.80.00.00.099.098.998.0 94.892.060.059.251.947.657.2
GBY76.01.30.00.00.098.798.697.3 95.193.553.452.241.839.746.7
DFX82.40.00.00.00.098.896.8 92.486.284.9N/AN/AN/AN/AN/A
DBX82.50.60.00.00.098.998.495.5 85.283.1N/AN/AN/AN/AN/A
DFZ86.50.20.00.00.098.894.589.3 81.579.342.925.522.327.738.2
+ +Table E. 1 9 : Black-box PGD attack results on CIFAR plane-vs-frog binary classification. + +
substitute accvictim accTP logit
E0.010.020.050.100.200.010.02 0.050.100.200.010.020.050.100.20
GFZ95.589.945.56.80.197.695.688.768.829.76.26.76.010.871.8
GBZ92.084.133.83.50.095.694.992.785.065.916.714.612.817.976.8
GFY95.389.138.72.50.097.396.690.072.935.50.02.36.810.472.3
GBY91.880.529.76.10.297.295.990.270.830.321.816.49.613.677.3
DFX89.174.519.10.90.099.798.792.664.89.9N/AN/AN/AN/AN/A
DBX85.576.442.54.50.099.699.094.877.524.5N/AN/AN/AN/AN/A
DFZ85.573.521.11.50.099.599.293.870.115.10.05.08.915.985.1
+ +Table E.2O: Black-box MIM attack results on CIFAR plane-vs-frog binary classification. + +
substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ95.589.746.47.70.297.595.788.566.520.56.26.75.912.782.7
GBZ91.884.033.13.40.095.694.992.385.160.716.715.514.221.697.9
GFY95.388.940.53.10.097.396.789.570.631.90.02.47.612.588.0
GBY91.780.229.55.90.197.395.989.669.026.122.216.19.714.095.9
DFX89.074.519.40.90.099.698.792.163.010.9N/AN/AN/AN/AN/A
DBX85.576.342.74.50.099.699.095.176.527.2N/AN/AN/AN/AN/A
DFZ85.373.421.41.60.099.699.192.967.714.90.04.57.514.291.9
+ +Table E.21: Black-box CW attack results on CIFAR plane-vs-frog binary clasification. + +
substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.00TP logit 10.00100.001000.00
GFZ96.341.50.50.00.098.794.793.4 93.392.313.3 8.67.77.78.9
GBZ94.973.31.40.00.095.995.3 94.694.594.513.511.811.813.213.1
GFY96.462.01.30.00.098.196.7 94.894.794.10.02.86.56.16.8
GBY93.533.61.70.00.097.795.9 95.495.495.025.816.415.918.619.8
DFX90.310.90.00.00.099.998.8 98.598.597.9N/AN/AN/AN/AN/A
DBX87.231.80.00.00.099.997.1 94.593.994.7N/AN/AN/AN/AN/A
DFZ84.922.70.00.00.099.999.0 98.398.397.80.03.82.32.311.0
+ +Table E.22: FGSM white-box zero-knowledge attack results on CIFAR- 1 0. + +
acc. (adv) 0.020.050.100.20TP marginalTP logitTP KL
E0.010.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1644.525.816.611.910.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A95.195.797.097.597.1
GBZ-FC163.749.734.318.912.760.062.480.690.994.163.765.080.695.098.490.190.892.392.692.6
GBY-FC164.950.036.119.511.352.259.080.490.192.554.661.980.395.698.490.791.392.192.490.3
DBX-FC157.345.731.816.310.8N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.893.294.094.392.8
GBZ-CONV981.674.158.548.945.813.717.028.476.090.218.320.631.764.371.691.791.591.793.386.4
GBY-CONV975.866.039.722.517.514.018.229.474.488.917.720.825.855.268.989.689.990.089.189.2
DBX-CONV959.045.731.817.811.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.794.194.995.293.7
+ +Table E.23: PGD white-box zero-knowledge attack results on CIFAR-10 + +
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1618.80.60.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.991.867.469.966.4
GBZ-FC137.76.10.10.00.030.722.276.092.094.133.524.977.091.790.290.893.797.398.595.9
GBY-FC131.28.52.61.50.632.020.382.395.197.636.629.584.193.790.490.394.197.797.396.9
DBX-FC123.41.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.991.05.67.45.9
GBZ-CONV961.138.324.210.22.319.260.396.798.999.620.160.396.898.398.790.393.197.798.799.6
GBY-CONV966.526.63.20.40.017.055.695.799.299.617.852.795.398.697.689.691.898.799.496.9
DBX-CONV924.03.00.30.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.393.199.399.699.2
+ +Table E.24: MIM white-box zero-knowledge attack results on CIFAR-10. + +
acc. (adv)TP marginalTP logit
E0.010.02（0.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.10 0.20
VGG1613.10.40.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A92.292.472.377.179.0
GBZ-FC124.52.70.10.00.034.840.988.595.896.0 38.844.090.496.496.691.395.095.797.996.8
GBY-FC133.019.611.87.24.929.138.795.799.199.3 38.449.296.399.399.490.795.097.897.997.9
DBX-FC117.30.90.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.189.69.614.516.0
GBZ-CONV957.643.529.124.122.627.478.599.499.6 99.729.178.399.699.799.890.294.497.598.098.2
GBY-CONV939.118.14.31.91.525.670.498.699.699.8 28.271.198.699.799.889.694.599.499.899.7
DBX-CONV919.23.20.50.40.4N/AN/AN/AN/AN/A N/AN/AN/AN/AN/A92.196.799.9100.0100.0
+ +Table E.25 : FGSM white-box zero-knowledge attack results on MNIST (with varied bottleneck layer sizes) . + +
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1692.685.676.464.752.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A90.392.492.392.794.2
DBX-3292.684.471.157.246.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.292.392.894.995.0
DBX-6491.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DBX-12887.847.620.112.710.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A94.497.697.897.196.4
GBZ-1692.075.852.534.021.728.771.599.9100.0100.049.286.399.9100.0100.090.890.692.091.990.9
GBZ-3291.174.452.433.421.626.573.599.7100.0100.049.487.299.9100.0100.091.090.691.891.491.6
GBZ-6492.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GBZ-12890.876.857.538.924.826.663.099.6100.0100.044.978.799.8100.0100.087.990.190.791.190.6
GBY-1694.277.749.423.912.141.984.1100.0100.0100.052.991.9100.0100.0100.086.692.093.093.192.6
GBY-3294.576.945.420.09.641.483.3100.0100.0100.056.392.8100.0100.0100.089.091.693.393.293.0
GBY-6493.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
GBY-12893.072.942.218.59.037.4：71.6100.0100.0100.050.181.999.9100.0100.090.791.4 91.891.492.0
+ +Table E.26: PGD white-box zero-knowledge atack results on MNIST(with varied bottleneck layer sizes). + +
acc. (adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1669.836.616.05.91.8N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.291.992.893.393.7
DBX-3263.526.812.56.22.7N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A91.592.192.792.993.5
DBX-6458.018.36.01.30.2N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
DBX-12842.33.01.10.30.2N/AN/A N/AN/AN/AN/AN/AN/AN/AN/A97.197.198.498.498.6
GBZ-1688.362.837.423.115.527.135.871.9 98.9100.046.353.082.399.3100.089.691.792.391.491.8
GBZ-3287.161.940.228.020.424.039.176.798.9 100.041.556.485.499.7100.091.391.091.191.391.1
GBZ-6485.157.432.519.311.633.750.384.4 99.7100.052.266.391.599.8100.090.091.591.591.791.8
GBZ-12884.457.536.825.216.825.435.466.1 96.6100.041.550.576.497.7100.088.690.490.490.090.6
GBY-1690.449.715.04.21.642.664.789.9 99.3100.051.170.892.899.5100.091.993.191.990.791.0
GBY-3288.439.89.51.80.645.674.392.8 99.5100.056.878.294.899.8100.090.892.291.791.090.6
GBY-6486.735.99.01.70.445.376.194.6 99.7100.056.979.395.699.9100.090.192.191.691.791.4
GBY-12883.335.18.72.30.839.163.986.7 98.3100.051.769.189.298.9100.089.890.790.489.789.5
+ +Table E.27 : MIM white-box zero-knowledge attack results on MNIST (with varied bottleneck layer sizes) . + +
acc. (adv)TP marginalTP logitTP KL
E0.10（0.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1672.237.120.714.711.7N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.392.893.493.9
DBX-3266.727.616.812.510.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.592.493.293.5
DBX-6466.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DBX-12842.31.30.20.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A97.297.698.698.999.1
GBZ-1683.836.48.92.30.926.979.899.7100.0100.043.388.099.8100.0100.091.692.191.291.692.6
GBZ-3282.735.59.53.01.126.683.299.7100.0100.045.890.099.8100.0100.090.391.791.492.592.6
GBZ-6479.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GBZ-12879.232.18.72.61.226.168.599.2100.0100.041.279.199.4100.0100.089.791.591.491.492.5
GBY-1689.234.63.50.20.044.888.199.8100.0100.055.991.799.7100.0100.089.492.591.090.636.9
GBY-3286.725.81.60.10.045.191.2100.0100.0100.058.694.0100.0100.0100.090.991.891.1 68.631.1
GBY-6484.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
GBY-12882.123.71.90.00.040.483.099.7100.0100.051.787.599.6100.0100.091.291.289.882.844.4
\ No newline at end of file diff --git a/parse/train/HygUOoC5KX/HygUOoC5KX_content_list.json b/parse/train/HygUOoC5KX/HygUOoC5KX_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..29e3d3e0d70648d7d53c6c53dd9787f64909d87d --- /dev/null +++ b/parse/train/HygUOoC5KX/HygUOoC5KX_content_list.json @@ -0,0 +1,2532 @@ +[ + { + "type": "text", + "text": "ARE GENERATIVE CLASSIFIERS MORE ROBUST TO ADVERSARIAL ATTACKS? ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 184, + 171, + 398, + 196 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 233, + 544, + 250 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There is a rising interest in studying the robustness of deep neural network classifiers against adversaries, with both advanced attack and defence techniques being actively developed. However, most recent work focuses on discriminative classifiers, which only model the conditional distribution of the labels given the inputs. In this paper, we propose and investigate the deep Bayes classifier, which improves classical naive Bayes with conditional deep generative models. We further develop detection methods for adversarial examples, which reject inputs with low likelihood under the generative model. Experimental results suggest that deep Bayes classifiers are more robust than deep discriminative classifiers, and that the proposed detection methods are effective against many recently proposed attacks. ", + "bbox": [ + 233, + 266, + 764, + 405 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 433, + 336, + 449 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks have been shown to be vulnerable to adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2014). The latest attack techniques can easily fool a deep neural network with imperceptible perturbations (Goodfellow et al., 2014; Papernot et al., 2016b; Carlini & Wagner, 2017a; Kurakin et al., 2016; Madry et al., 2018; Chen et al., 2017a), even in the black-box case, where the attacker does not have access to the network’s weights (Papernot et al., 2017b; Chen et al., 2017b; Alzantot et al., 2018a). Adversarial attacks are serious security threats to machine learning systems, threatening applications beyond image classification (Carlini & Wagner, 2018; Alzantot et al., 2018b). ", + "bbox": [ + 174, + 465, + 825, + 577 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To address this outstanding security issue, researchers have proposed defence mechanisms against adversarial attacks. Adversarial training, which augments the training data with adversarially perturbed inputs, has shown moderate success at defending against recently proposed attack techniques (Szegedy et al., 2013; Goodfellow et al., 2014; Tramer et al., 2017; Madry et al., 2018). In addition, \\` recent advances in Bayesian neural networks have demonstrated that uncertainty estimates can be used to detect adversarial attacks (Li & Gal, 2017; Feinman et al., 2017; Louizos & Welling, 2017; Smith & Gal, 2018). Another notable category of defence techniques involves the usage of generative models. For example, Gu & Rigazio (2014) used an auto-encoder to denoise the inputs before feeding them to the classifier. This denoising approach has been extensively investigated, and the “denoisers” in usage include generative adversarial networks (Samangouei et al., 2018), PixelCNNs (Song et al., 2018) and denoising auto-encoders (Kurakin et al., 2018). These developments rely on the “off-manifold” conjecture, that is, that adversarial examples are far away from the data manifold, although Gilmer et al. (2018) has challenged this observation with a synthetic example. ", + "bbox": [ + 174, + 583, + 825, + 763 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Surprisingly, much less recent work has investigated the robustness of generative classifiers (Ng & Jordan, 2002) against adversarial attacks for multi-class classification, where such classifiers explicitly model the conditional distribution of the inputs given labels. Typically, a generative classifier produces predictions by comparing between the likelihood of the labels for a given input, which is closely related to the “distance” of the input to the data manifold associated with a class. Therefore, generative classifiers should be robust to many recently proposed adversarial attacks if the “off-manifold” conjecture holds for many real-world applications. Unfortunately, many generative classifiers in popular use, including naive Bayes and linear discriminant analysis (Fisher, 1936), perform poorly on natural image classification tasks, making it difficult to verify the “off-manifold” conjecture and the robustness of generative classifiers with these tools. In recent work, $\\mathbf { k }$ -nearest neighbors (Cover & Hart, 1967), a method which shares many similarities with generative classifiers, has been significantly improved in handling natural images by leveraging deep feature representations (Papernot & McDaniel, 2018). To the best of our knowledge, an approach which targets a similar contribution has not yet been proposed for generative classifiers. ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Are generative classifiers more robust to recently proposed adversarial attack techniques? To answer this, we improve the naive Bayes algorithm by using conditional deep generative models, and evaluate the conjecture on the proposed generative classifier. In summary, our contributions include: ", + "bbox": [ + 176, + 152, + 821, + 195 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose deep Bayes as an extension of naive Bayes, in which the conditional distribution of an input, given a label, is parameterised by a deep latent variable model (LVM). We learn the LVM with the variational auto-encoder algorithm (Kingma & Welling, 2013; Rezende et al., 2014), and approximate Bayes’ rule using importance sampling. We propose three detection methods for adversarial perturbations. The first two use the learned generative model as a proxy of the data manifold, and reject inputs that are far away from it. The third computes statistics for the classifier’s output probability vector on training data, and rejects inputs that lead to under-confident predictions. We evaluate the robustness of the proposed generative classifier on the MNIST multi-class and CIFAR binary classification tasks. We also improve the robustness of deep neural networks on CIFAR-10 multi-class classification, by fusing discriminatively learned visual feature representations with the proposed generative classifiers. We further compare the generative classifiers with a number of discriminative classifiers, including Bayesian neural networks and discriminative latent variable models. ", + "bbox": [ + 215, + 202, + 825, + 402 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 DEEP BAYES: CONDITIONAL DEEP LVM AS A GENERATIVE CLASSIFIER ", + "text_level": 1, + "bbox": [ + 173, + 421, + 795, + 438 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Denote $p _ { \\mathcal { D } } ( \\pmb { x } , \\pmb { y } )$ the data distribution for the input $\\pmb { x } \\in \\mathbb { R } ^ { D }$ and label $\\pmb { y } \\in \\{ \\pmb { y } _ { c } | c = 1 , . . . , C \\}$ , where $\\mathbf { \\nabla } \\mathbf { \\textbf { { y } } } _ { c }$ denotes the one-hot encoding vector for class $c$ . For a given $\\pmb { x } \\in \\mathbb { R } ^ { D }$ we can define the ground-truth label by ", + "bbox": [ + 176, + 452, + 823, + 494 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/3f25ff578bc04df392a7988c6a6dc97d88adc9ae33acaabaf75af79046a08371.jpg", + "text": "$$\n\\pmb { y } \\sim p _ { \\mathcal { D } } ( \\pmb { y } | \\pmb { x } ) \\quad \\mathrm { ~ i f ~ } \\pmb { x } \\in \\mathrm { s u p p } ( p _ { \\mathcal { D } } ( \\pmb { x } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 493, + 624, + 511 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We assume the data distribution $p _ { \\mathcal { D } } ( \\pmb { x } , \\pmb { y } )$ follows the manifold assumption: for every class $c$ , the conditional distribution $p _ { \\mathcal { D } } ( \\pmb { x } | \\pmb { y } _ { c } )$ has a low-dimensional manifold support $\\mathcal { M } _ { c } = \\operatorname { s u p p } ( p _ { \\mathcal { D } } ( \\pmb { x } | \\pmb { y } _ { c } ) )$ . Therefore the training dataset $\\mathcal { D } = \\{ ( \\pmb { x } ^ { ( n ) } , \\pmb { y } ^ { ( n ) } ) \\} _ { n = 1 } ^ { N }$ is generated as the following: ", + "bbox": [ + 174, + 513, + 825, + 559 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/2a77f67cf8384fba506bbd19f7556c0d4078815562b8a102848afc4a6562b663.jpg", + "text": "$$\n( { \\pmb x } ^ { ( n ) } , { \\pmb y } ^ { ( n ) } ) \\sim p _ { \\mathcal { D } } ( { \\pmb x } , { \\pmb y } ) \\Leftrightarrow { \\pmb y } ^ { ( n ) } \\sim p _ { \\mathcal { D } } ( { \\pmb y } ) , { \\pmb x } ^ { ( n ) } \\sim p _ { \\mathcal { D } } ( { \\pmb x } | { \\pmb y } ) .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 564, + 704, + 585 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A (Bayesian) generative classifier first builds a generative model $p ( { \\pmb x } , { \\pmb y } ) = p ( { \\pmb x } | { \\pmb y } ) p ( { \\pmb y } )$ , and then, in prediction time, predicts the label $\\boldsymbol { y } ^ { * }$ of a test input $\\pmb { x } ^ { * }$ using Bayes’ rule, ", + "bbox": [ + 171, + 598, + 825, + 627 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f29efb96ed0cb8682de51ef56c004bc8a253a0b3ecf8158ce68a20aa27dad6d4.jpg", + "text": "$$\np ( { \\pmb y } ^ { * } | { \\pmb x } ^ { * } ) = \\frac { p ( { \\pmb x } ^ { * } | { \\pmb y } ^ { * } ) p ( { \\pmb y } ^ { * } ) } { p ( { \\pmb x } ^ { * } ) } = \\mathrm { s o f t m a x } _ { c = 1 } ^ { C } \\left[ \\log p ( { \\pmb x } ^ { * } , { \\pmb y } _ { c } ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 302, + 632, + 694, + 666 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathrm { s o f t m a x } _ { c = 1 } ^ { C }$ denotes the softmax operator over the $c$ axis. Therefore, the output probability vector is computed analogously to many deep discriminative classifiers which use softmax activation in the output layer, so many existing attacks can be tested directly. However, unlike discriminative classifiers, the “logit” values prior to softmax activation have a clear meaning here, which is the (approximated) log joint distribution $\\log p ( { \\pmb x } ^ { * } , { \\pmb y } _ { c } )$ of input $\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } } \\ast \\mathbf { \\ v { x } }$ conditioned on a given label $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { c } }$ . Therefore, one can also analyse the logit values to determine whether the unseen pair $( { \\boldsymbol { \\it x } } ^ { * } , { \\boldsymbol { \\it y } } ^ { * } )$ is legitimate, a utility which will be discussed further in later sections. ", + "bbox": [ + 173, + 674, + 825, + 773 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Naive Bayes is perhaps the most well-known generative classifier; it assumes a factorised distribution for the conditional generator, i.e. $\\begin{array} { r } { p ( \\pmb { x } | \\pmb { y } ) = \\prod _ { d = 1 } ^ { D } p ( x _ { d } | \\pmb { y } ) } \\end{array}$ . However, this factorisation assumption is inappropriate for e.g. image and speech data. Fortunately, we can leverage the recent advances in generative modelling and apply a deep generative model for the joint distribution $p ( { \\pmb x } , { \\pmb y } )$ . We refer to such generative classifiers that use deep generative models as deep Bayes classifiers. ", + "bbox": [ + 174, + 780, + 825, + 853 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we consider a deep latent variable model (LVM) $p ( { \\pmb x } , { \\pmb z } , { \\pmb y } )$ , which will be used for classification. In this case, the conditional distribution is p(x|y) = R p(x,z,y)dzR p(x,z,y)dzdx . Importantly, this leads to a conditional distribution $p ( { \\pmb x } | { \\pmb y } )$ that is not factorised (even when $p ( \\pmb { x } | \\pmb { z } , \\pmb { y } )$ is), which is much more powerful than naive Bayes. Since maximum likelihood is intractable, we follow ", + "bbox": [ + 174, + 858, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/220140844094a3f088b10ef70c1787e4802498efb521d87e87e0c1adf30847b7.jpg", + "image_caption": [ + "Figure 1: A visualisation of the graphical models, including both Generative and Discriminative ones, as well as Fully connected and Bottleneck ones. The last character indicates the first node in the topological order of the graph. The colour encoding is the same as those in experiments. " + ], + "image_footnote": [], + "bbox": [ + 184, + 82, + 815, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Kingma & Welling (2013) and Rezende et al. (2014) to introduce an amortised approximate posterior $q ( \\boldsymbol { z } | \\boldsymbol { x } , \\boldsymbol { y } )$ , and train both $p$ and $q$ by maximising the variational lower-bound: ", + "bbox": [ + 174, + 239, + 820, + 267 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8595a26beeaccab141d9c8e5a93936164246668454ae05c1346ee9439c9131c9.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathcal { D } } [ \\mathcal { L } _ { \\mathrm { V I } } ( \\pmb { x } , \\pmb { y } ) ] = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathbb { E } _ { q } \\left[ \\log \\frac { p ( \\pmb { x } _ { n } , \\pmb { z } _ { n } , \\pmb { y } _ { n } ) } { q ( \\pmb { z } _ { n } | \\pmb { x } _ { n } , \\pmb { y } _ { n } ) } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 271, + 666, + 315 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "After training, the predicted class probability vector $\\boldsymbol { y } ^ { * }$ for a future input $\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } } \\ast \\mathbf { \\ v { x } }$ is computed by an approximation to Bayes’ rule with importance sampling: ", + "bbox": [ + 173, + 318, + 823, + 345 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/560c74e0c17165355389a17fb009ee0200ce947f55d59660afffb5b430a01bfd.jpg", + "text": "$$\np ( \\pmb { y } ^ { * } | \\pmb { x } ^ { * } ) \\approx \\mathrm { s o f t m a x } _ { c = 1 } ^ { C } \\left[ \\log \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\frac { p ( \\pmb { x } ^ { * } , \\pmb { z } _ { c } ^ { k } , \\pmb { y } _ { c } ) } { q ( \\pmb { z } _ { c } ^ { k } | \\pmb { x } ^ { * } , \\pmb { y } _ { c } ) } \\right] , \\quad \\pmb { z } _ { c } ^ { k } \\sim q ( \\pmb { z } | \\pmb { x } ^ { * } , \\pmb { y } _ { c } ) .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 348, + 745, + 392 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We evaluate the effect of the following factorisation structures on the robustness of the induced classifier from the generative model $p ( \\pmb { x } , \\pmb { z } , \\pmb { y } )$ (see Figure 1). ", + "bbox": [ + 173, + 402, + 823, + 431 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a40d72b853603654df5f89bdcbfc275e308d7d9144d685a9c5b815fb3f20ae58.jpg", + "text": "$$\n\\begin{array} { l l l } { p ( { \\pmb x } , { \\pmb z } , { \\pmb y } ) = p ( { \\pmb z } ) p ( { \\pmb y } | { \\pmb z } ) p ( { \\pmb x } | { \\pmb z } , { \\pmb y } ) \\quad } & { ( { \\mathrm { G F Z } } ) } \\\\ { p ( { \\pmb x } , { \\pmb z } , { \\pmb y } ) = p _ { \\mathcal { D } } ( { \\pmb y } ) p ( { \\pmb z } | { \\pmb y } ) p ( { \\pmb x } | { \\pmb z } , { \\pmb y } ) \\quad } & { ( { \\mathrm { G F Y } } ) } \\\\ { p ( { \\pmb x } , { \\pmb z } , { \\pmb y } ) = p ( { \\pmb z } ) p ( { \\pmb y } | { \\pmb z } ) p ( { \\pmb x } | { \\pmb z } ) \\quad } & { ( { \\mathrm { G B Z } } ) \\quad } & { p ( { \\pmb x } , { \\pmb z } , { \\pmb y } ) = p _ { \\mathcal { D } } ( { \\pmb x } ) p ( { \\pmb z } | { \\pmb x } ) p ( { \\pmb y } | { \\pmb z } ) } \\\\ { p ( { \\pmb x } , { \\pmb z } , { \\pmb y } ) = p _ { \\mathcal { D } } ( { \\pmb y } ) p ( { \\pmb z } | { \\pmb y } ) p ( { \\pmb x } | { \\pmb z } ) \\quad } & { ( { \\mathrm { G B Y } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 176, + 435, + 766, + 508 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We use the initial character “G” to denote generative classifiers and “D” to denote discriminative classifiers. Models with the second character as “F” have a fully connected graph, while “B” models enforce the usage of the latent code $_ { z }$ as a bottleneck. The last character of the model name indicates the first node in topological order. Model DFZ is somewhat intermediate, as it builds a generative model for the inputs $_ { \\textbf { \\em x } }$ but also directly parameterises the conditional distribution $p ( \\pmb { y } | \\pmb { x } , z )$ . We do not test other architectures under this nomenclature, as either the graph contains directed cycles (e.g. $\\pmb { x } \\pmb { y } \\pmb { z } \\pmb { x } ,$ ), or $_ z$ is the last node in topological order (e.g. $\\pmb { x } \\pmb { y } , ( \\pmb { x } , \\pmb { y } ) \\pmb { z } )$ so that the marginalisation of $_ z$ does not affect classification. ", + "bbox": [ + 173, + 512, + 825, + 625 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 DETECTING ADVERSARIAL ATTACKS WITH GENERATIVE CLASSIFIERS ", + "text_level": 1, + "bbox": [ + 173, + 645, + 782, + 659 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We propose detection methods for adversarial examples using generative classifier’s logit values. As an illustrating example, consider a labelled dataset of “cat” and “dog” images. If an adversarial image of a cat $\\pmb { x } _ { \\mathrm { a d v } }$ is incorrectly labelled as “dog”, then either this image is ambiguous, or, under a perfect generative model, the logit $\\log p ( \\pmb { x } _ { \\mathrm { a d v } }$ , “dog”) will be significantly lower than normal. This means we can detect attacks using the logits $\\log p ( { \\pmb x } ^ { * } , { \\pmb y } _ { c } ) , c = 1 , . . . , C$ computed on a test input $\\mathbf { \\boldsymbol { x } } ^ { * }$ , by comparing them with the logits computed on legitimate training inputs. ", + "bbox": [ + 173, + 674, + 825, + 758 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Concretely, the proposed detection algorithms are as follows. We aim to reject both unlabelled input $_ { \\textbf { \\em x } }$ that have low probability under $p _ { \\bar { D } } \\bar { ( } \\pmb { x } )$ , and labelled data $( { \\pmb x } , { \\pmb y } )$ that have low $p _ { \\mathcal { D } } ( \\pmb { x } , \\pmb { y } )$ values. ", + "bbox": [ + 173, + 765, + 823, + 794 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Marginal detection: rejecting inputs that are far away from the manifold. One can select a threshold $\\delta$ and reject an input $_ { \\textbf { \\em x } }$ if $- \\log p ( { \\pmb x } ) > \\delta$ . To determine the threshold $\\delta$ , we can compute $\\bar { d } _ { \\mathcal { D } } = \\mathbb { E } _ { \\pmb { x } \\sim \\mathcal { D } } [ - \\log p ( \\pmb { x } ) ]$ and $\\sigma _ { \\mathcal { D } } = \\sqrt { \\mathbb { V } _ { \\pmb { x } \\sim \\mathcal { D } } [ \\log p ( \\pmb { x } ) ] }$ , then set $\\delta = \\bar { d } _ { D } + \\alpha \\sigma _ { D }$ . It is also possible to compute the statistics ${ \\bar { d } } _ { p } , \\sigma _ { p }$ on the images generated by the generative model accordingly. \nLogit detection: rejecting inputs using joint density. Given a victim model $\\pmb { y } = F ( \\pmb { x } )$ , one can reject $_ { \\textbf { \\em x } }$ if $- \\log p ( { \\pmb x } , F ( { \\pmb x } ) ) > \\delta _ { \\pmb y }$ . We can use the mean and variance statistics $\\bar { d } _ { c } , \\sigma _ { c }$ computed on $\\log p ( { \\pmb x } , { \\pmb y } _ { c } )$ and select $\\delta _ { y _ { c } } = \\bar { d } _ { c } + \\alpha \\sigma _ { c }$ . ", + "bbox": [ + 215, + 804, + 826, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/63ac3a599fb3fd7f9b0ceeb289e5088441fbf7ba011a0d3f9a13300e966b86cc.jpg", + "image_caption": [ + "Figure 2: Visualising detection mechanisms. The scattered dots are training data points, with different classes shown in different colours (red for $c = 0$ and blue for $c = 1$ ). Same labels are manually assigned for inputs when the detection method requires $\\textbf { { y } }$ . Decision regions are shown in the corresponding colours. Input points in the shaded area are rejected by detection. " + ], + "image_footnote": [], + "bbox": [ + 176, + 83, + 820, + 161 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Divergence detection: rejecting inputs with over- and/or under-confident predictions. ", + "bbox": [ + 212, + 248, + 794, + 262 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Denote ${ \\pmb p } ( { \\pmb x } )$ as a $C$ -dimensional probability vector outputted by the classifier. For each class $c$ , we first collect the mean classification probability vector $\\pmb { p } _ { c } = \\mathbb { E } _ { ( \\pmb { x } , \\pmb { y } _ { c } ) \\in \\mathcal { D } } [ \\pmb { p } ( \\pmb { x } ) ]$ , then compute the mean $\\bar { d } _ { c }$ and standard deviation $\\sigma _ { c }$ on a selected divergence/distance measure $\\bar { \\mathrm { D } } [ p _ { c } | | p ( { \\pmb x } ) ]$ for all $( \\pmb { x } , \\pmb { y } _ { c } ) \\in \\mathcal { D }$ . A test input $\\pmb { x } ^ { * }$ with prediction label $c ^ { * } =$ arg max $\\pmb { p } ( \\pmb { x } ^ { * } )$ is rejected if $\\mathrm { D } [ p _ { c ^ { * } } | | p ( { \\pmb x } ^ { * } ) ] > \\bar { d } _ { c ^ { * } } + \\bar { \\alpha \\sigma } _ { c ^ { * } }$ . Therefore, an example $\\pmb { x } ^ { * }$ will be rejected if the classifier is over-confident or under-confident (ambiguous inputs). ", + "bbox": [ + 222, + 263, + 825, + 348 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "When $\\mathrm { D }$ is selected as the KL-divergence, we call this detection method $K L$ detection. \nOther divergence/distance measures such as total variation (TV) can also be used. ", + "bbox": [ + 233, + 349, + 821, + 378 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For better intuition, we visualise the detection mechanisms in Figure 2 with a synthetic “two rings” binary classification example. In this case we sample $2 \\times 1 0 0 0$ training data points as: ", + "bbox": [ + 176, + 390, + 821, + 419 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "( $\\begin{array} { r l } & { x , y ) \\sim p _ { D } \\Leftrightarrow y \\sim \\mathrm { B e r n } ( 0 . 5 ) , \\theta \\sim \\mathrm { U n i f o r m } ( 0 , 2 \\pi ) , x | y \\sim \\mathcal { N } ( x ; c _ { y } + r _ { y } [ c o s ( \\theta ) , s i n ( \\theta ) ] ^ { \\mathrm { T } } , \\sigma ^ { 2 } { \\bf I } ) . } \\end{array}$ We consider a generative classifier $p ( \\pmb { x } , \\pmb { y } ) = p ( \\pmb { x } | \\pmb { y } ) p _ { \\mathscr { D } } ( \\pmb { y } ) = \\mathcal { N } ( \\pmb { x } ; \\pmb { \\mu _ { y } } , \\sigma ^ { 2 } \\mathbf { I } ) \\mathbf { B e r n } ( 0 . 5 )$ ) with $\\mu _ { y } =$ $\\operatorname { p r o j } ( x , \\operatorname { r i n g } _ { y } ) = \\arg \\operatorname* { m i n } _ { \\hat { x } \\in \\mathbb { R } ^ { 2 } }$ , $| | \\hat { \\mathbf { x } } - \\pmb { c } _ { y } | | _ { 2 } = { r } _ { y } \\ | | \\pmb { x } - \\hat { \\pmb { x } } | | _ { 2 }$ . The $\\delta$ thresholds are selected to achieve $1 0 \\%$ false positive rates on training data. From the visualisations we see that inputs that are far away from the model manifold are rejected by marginal/logit detection. At the same time, logit detection rejects data points that are not on the manifold of the given class. KL/TV detection does not construct manifold-aware acceptance regions, which is as expected since the proposed divergence detection method does not require the classifier to be generative. However, both detection methods have some success in rejecting uncertain predictions, especially for TV, which also rejects ambiguous inputs (see the ring-cross regions in the last two plots). Combining all three methods, we see that the rejected inputs are either far away from the manifold, or are ambiguous. Again we emphasise that a suitable generative model is required to make the detection methods work in practice, since the data distribution $p _ { \\mathcal { D } } ( \\pmb { x } , \\pmb { y } )$ is approximated by $p ( { \\pmb x } , { \\pmb y } )$ . ", + "bbox": [ + 173, + 422, + 825, + 617 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Detection methods using logit values have also been proposed in Li & Gal (e.g. 2017); Feinman et al. (e.g. 2017). However it is unclear whether the logits values in discriminative classifiers have a clear semantic meaning, while the logit values in deep Bayes represent the log probability of generating the input $_ { \\textbf { \\em x } }$ given the class label ${ \\mathbf { \\nabla } } y = y _ { c }$ . Our approach is also distinct from previous “denoising” approaches (e.g. Song et al., 2018; Samangouei et al., 2018; Kurakin et al., 2018) that require training an additional generative model separately. The deep Bayes classifiers share the same generative model as the detection methods, meaning that detected adversarial examples are indeed far away from the classifier’s manifold (which is also an approximation to the data manifold). Therefore the claim “detection neural networks can be bypassed” (Carlini & Wagner, 2017a; Athalye et al., 2018) does not directly transfer to our approach, and we can use the marginal and logit detection methods to directly verify the “off-manifold” adversarial example conjecture. ", + "bbox": [ + 173, + 622, + 825, + 775 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 796, + 326, + 810 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We carry out a number of tests on the proposed deep Bayes classifiers (3), where $q ( z | \\cdot )$ and $p ( z | \\cdot )$ are factorised Gaussians, and the conditional probability $p ( { \\pmb x } | \\cdot )$ , if required, is parameterised by an $\\ell _ { 2 }$ loss. Besides the LVM-based classifiers, we further train a Bayesian neural network (BNN) with Bernoulli dropout (dropout rate 0.3), as it has been shown in Li & Gal (2017) and Feinman et al. (2017) that BNNs are more robust than their deterministic counterparts. The constructed BNN has $2 \\mathbf { x }$ more channels than LVM-based classifiers, making the comparison slightly “unfair”, as the BNN layers have more capacity. We use $K = 1 0$ Monte Carlo samples for all the classifiers. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/00ae51d305686cd331a0a614d4028a57298f23ffcfe3de3c5e0e296f5308977e.jpg", + "image_caption": [ + "Figure 3: Accuracies (column 1), detection rates (columns 2-4) and minimum $\\ell _ { \\mathrm { i n f } }$ perturbation (column 6) against white-box zero-knowledge $\\ell _ { \\infty }$ attacks on MNIST. The higher the better. The second from right most column visualises crafted adversarial examples on an image of digit “7”, with $\\ell _ { \\infty }$ distortion $\\epsilon$ growing from 0.1 to 0.5. " + ], + "image_footnote": [], + "bbox": [ + 183, + 80, + 813, + 295 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The adversarial attacks in test include both $\\ell _ { \\infty }$ and $\\ell _ { 2 }$ untargeted attacks from CleverHans 2.0 library (Papernot et al., 2017a): fast gradient sign method (FGSM, Goodfellow et al., 2014), projected gradient descent (PGD, Madry et al., 2018), momentum iterative attack (MIM, Dong et al., 2017) and Carlini & Wagner $\\ell _ { 2 }$ (CW, Carlini & Wagner, 2017a). Two metrics are reported: accuracy of the classifier on crafted adversarial examples, and detection rate on adversarial examples that have successfully caused the classifier to misclassify. This detection rate is defined as the true positive (TP) rate of finding an adversarial example, and the detection threshold is selected to achieve a $5 \\%$ false positive rate on clean training data. ", + "bbox": [ + 174, + 369, + 825, + 482 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The experiments are performed under various threat model settings. We further evaluate the transferability of crafted adversarial examples across different classifiers in the same way as done in Papernot et al. (2016a). We only provide visualisations in the main text; full table results can be found in the appendix. Readers are also referred to the appendix for further experiments, including a quantitative analysis of the bottleneck effect on robustness and detection. ", + "bbox": [ + 174, + 488, + 825, + 558 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 MNIST ", + "text_level": 1, + "bbox": [ + 174, + 575, + 269, + 589 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first set of experiments evaluate the robustness of generative classifiers on MNIST. Here the image pixel values are normalised to [0, 1], and the LVM-based classifiers have $\\dim ( z ) ~ = ~ 6 4$ . We first perform white-box zero-knowledge attacks, i.e. the attacker can differentiate through the classifier to craft adversarial examples, but he/she is not aware of the existence of the detector. Then we perform white-box perfect-knowledge attacks, where the attacker can differentiate through both the classifier and the detector, and he/she knows the usage of random $_ { z }$ samples by the VAE-based classifiers (Biggio et al., 2013; Carlini & Wagner, 2017b). Lastly we consider grey-box and blackbox attacks, and evaluate the robustness of generative classifiers against transferred attacks. ", + "bbox": [ + 174, + 601, + 825, + 713 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "White-box attacks (zero-knowledge, $\\ell _ { \\infty , }$ ) Results for $\\ell _ { \\infty }$ attacks are reported in Figure 3, and in general generative classifiers perform better in terms of victim accuracy and minimum perturbation of the attacks.1 By contrast, DFX & DFZ are not robust to the weakest attack (FGSM) even when $\\epsilon =$ 0.2 (where the adversarial examples are still visually close to the original digit “7”). Interestingly, DBX is the most robust against FGSM & $\\mathbf { M I M } ^ { 2 }$ , which agrees with the preliminary tests in Alemi et al. (2017). But DBX is less robust to PGD, and here GBZ is a clear winner. These results show that the bottleneck is sometimes beneficial for better robustness of MNIST classifiers. ", + "bbox": [ + 173, + 727, + 825, + 824 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For detection, generative classifiers have successfully detected the adversarial examples with $\\epsilon \\geq$ 0.3, which is reasonable as the visual distortion is already significant. Importantly, generative classifier’s victim accuracy decreases as the $\\ell _ { \\infty }$ distortion $\\epsilon$ increases, but at the same time the TP rates for marginal and logit detection also increase. Therefore these $\\ell _ { \\infty }$ attacks fail to find near-manifold adversarial examples that fool both the classifier and the detection methods. DFZ, as an intermediate between generative and discriminative classifiers, has worse robustness results, but has good detection performance for the marginal and logit metrics. This is because with softmax activation, the marginal distribution $p ( { \\pmb x } )$ is dropped, but in marginal/logit detection $p ( { \\pmb x } )$ is still in use. ", + "bbox": [ + 174, + 832, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/a6cd977b40ec50f4d8cef617d2ed066a4f0f2539c7a01da22877e077825eabee.jpg", + "table_caption": [ + "Table 1: White-box zero-knowledge CW- $\\cdot \\ell _ { 2 }$ attack results. Here, accuracy (adv) measures classifying the adversarial inputs to the original classes. " + ], + "table_footnote": [], + "table_body": "
acc.(clean)acc. (adv)l2 dist.TP KL
BNN99.12%24.40%2.12995.31
GFZ98.55%28.58%2.66395.37
GBZ97.45%81.51%2.44691.01
GFY99.15%28.64%2.73296.03
GBY98.72%32.72%2.73594.46
DFX99.10%20.31%2.09599.96
DBX98.87%30.19%1.80696.76
DFZ99.10%13.60%2.18899.57
", + "bbox": [ + 176, + 145, + 488, + 251 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/7155749b68688828aa6066f627f9693ca5bbdeeaa7d46d33450c322e20420f9b.jpg", + "image_caption": [ + "Figure 4: Visualising the clean inputs and the CW adversarial examples crafted on GFZ, digits in red rectangles show significant ambiguity. " + ], + "image_footnote": [], + "bbox": [ + 514, + 82, + 820, + 205 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 268, + 825, + 339 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "White-box attack (zero-knowledge, $\\ell _ { 2 }$ ) For the CW $\\ell _ { 2 }$ attack, we performed a hyperparameter search for the loss-balancing parameter $c$ $\\ell _ { 2 }$ distortion increases with larger $c$ ) in $\\{ 0 . 1 , 1 , 1 0 , 1 0 0 , 1 0 0 0 \\}$ on the first 100 test images, and we found $c = 1 0$ returns the highest success rate. Results are reported in Table 1. Although being successful on fooling many classifiers, CW failed on attacking GBZ. Also, the mean $\\ell _ { 2 }$ distortions of the successful attacks on generative classifiers are significantly larger. Furthermore, we found the success of the attack on other generative classifiers is mainly due to the ambiguity of the crafted adversarial images. As visualised in Figure 4 (also see Figure D.1 in appendix), the induced distortion from CW leads to ambiguous digits which sit at the perceptual boundary between the original and the adversarial classes. With the KL detection method, all classifiers achieve $> 9 5 \\%$ detection rates, which is as expected as the default CW attack configuration, by construction, generates adversarial examples that lead to minimal difference between the logit values of the most and the second most probable classes. ", + "bbox": [ + 173, + 354, + 825, + 521 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "White-box attack (perfect-knowledge, $\\ell _ { \\infty }$ ) The PGD-based perfect-knowledge $\\ell _ { \\infty }$ attack is designed following (Carlini & Wagner, 2017b): we construct an (approximate) Bayes classifier $p _ { k } ( \\pmb { y } | \\pmb { x } )$ using (3) for each set of samples $\\{ z _ { c } ^ { k } \\} _ { c = 1 } ^ { C }$ , and minimize the following with PGD: ", + "bbox": [ + 174, + 536, + 825, + 580 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/b88daad51d04d59a32e86b673f8a35394d357ad5eed2fba0b6ba30c89e3ba829.jpg", + "text": "$$\n\\mathcal { L } ( \\eta ) = \\sum _ { k = 1 } ^ { K } \\log p _ { k } ( \\boldsymbol { y } | \\boldsymbol { x } + \\eta ) + \\lambda _ { \\mathrm { d e t e c t } } \\operatorname* { m a x } ( 0 , \\Phi ( \\boldsymbol { x } + \\eta , \\boldsymbol { y } ) - \\delta ) .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 585, + 714, + 630 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The detection statistic $\\Phi ( { \\pmb x } + { \\pmb \\eta } , { \\pmb y } )$ is $- \\log p ( { \\pmb x } + { \\pmb \\eta } )$ for marginal detection, and $\\delta$ is the corresponding threshold computed on training data. For logit/KL detection, the detection statistics and thresholds are constructed accordingly. We label the three attacks against the marginal, logit and KL based detection schemes ’-PKM’, ’-PKL’ and ’-PKK’ respectively. When we only have knowledge of the $K$ samples and not the detection method we call this attack ’-PK0’ (ie $\\lambda _ { \\mathrm { d e t e c t } } { = } 0$ in Eq. 4). ", + "bbox": [ + 174, + 636, + 825, + 707 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Results are visualised in Figure 5. We see that although the attacker can reduce detection levels, this comes with the trade-off of increasing accuracy, suggesting that a perfect-knowledge adversary cannot break both the classifier and detector working in tandem. ", + "bbox": [ + 174, + 713, + 823, + 756 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Sanity checks on gradient masking Athalye et al. (2018) claimed that if a successful defence against white-box attacks is due to gradient masking, then this defence is likely to be less effective against grey-box/black-box attacks, as they do not differentiate through the victim classifier and the defence mechanism (Papernot et al., 2017b). Therefore, we consider two transfer attacks based on distilling the victim classifier using a “student” CNN which has no gradient masking. The two attacks differ in their threat models: in the grey-box setting the attacker has access to the output probability vectors of the classifiers on the training data, while in the black-box setting the attacker has access to queried labels only. For the latter black-box setting, we follow Papernot et al. (2017b) to train a substitute CNN using Jacobian-based dataset augmentation, and we refer to appendix B.1 for the detailed algorithm. The grey-box substitutes achieve $> 9 9 \\%$ agreement with the victims on test data, and the black-box substitutes obtained $\\sim 9 6 \\%$ accuracy on test data. ", + "bbox": [ + 173, + 770, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/92fbc68f077b0912c224d9aa79a22a044fe5d692681f9182450b3127b5751d73.jpg", + "image_caption": [ + "Figure 5: Accuracy and detection rates of DBX, GBY, and GBZ against PGD-based perfectknowledge attack $( \\epsilon = 0 . 2 )$ ) on MNIST. The solid area denotes accuracy and the hatched area denotes detection rate with each considered detector. Zero knowledge attacks are labelled ’-ZK’, other attack labels are described in the main text. " + ], + "image_footnote": [], + "bbox": [ + 191, + 84, + 803, + 222 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/a9d0675f4478e57a95c5946f6c2e9e26184ee8ec7905a49a23ceeb8edb588c24.jpg", + "image_caption": [ + "Figure 6: Accuracy and detection rates against distillation-based attacks on MNIST. The higher the better. We only present generative classifiers’ results here, for full results see appendix E. " + ], + "image_footnote": [], + "bbox": [ + 181, + 295, + 812, + 458 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 6 shows the accuracy and detection metrics on transferred $\\ell _ { \\infty }$ attacks crafted on the substitute models, with a comparison to their white-box counterparts. Note that these crafted attacks achieve $\\sim$ $1 0 0 \\%$ success rates on fooling the substitute models when $\\epsilon \\geq 0 . 2$ (see appendix E). However, they do not transfer very well to the generative classifiers. Importantly, for a fixed $\\epsilon$ setting, the whitebox attacks achieve significantly higher success rates (i.e. lower victim accuracies) than their grey/black-box counterparts, and the gap is at least $> 2 0 \\%$ for $\\epsilon \\leq 0 . 3$ (see Table 2). We further present the mean minimum $\\ell _ { \\infty }$ perturbation in Table 2, and we see that the minimum perturbation obtained by grey-/black-box attacks are significantly higher than those obtained by white-box attacks. ", + "bbox": [ + 173, + 507, + 825, + 619 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "All these results suggest that the robustness of generative classifiers is unlikely to be caused by gradient masking. Again on the detection side, the detection rates increase as the perturbation size increase, and they are near $1 0 0 \\%$ for $\\epsilon = 0 . 3$ . This means that most of the successfully transferred adversarial images are off the generative classifier’s manifold (as a proxy to the data manifold). ", + "bbox": [ + 174, + 626, + 825, + 683 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "SPSA (evolutionary strategies) We consider another black-box setting that only assumes access to the logit values of the prediction given an input. We use the SPSA $\\ell _ { \\infty }$ attack (Uesato et al., 2018), which is similar to the white-box zero-knowledge attacks, except that gradients are numerically estimated using the logit values from the victim classifier. Results in Table 3 clearly show that SPSA performs much worse on generative classifiers when compared to white-box PGD. Again this means gradient masking is unlikely to be responsible for the improved robustness of generative classifiers, as utilising the exact gradient yielded improved results. ", + "bbox": [ + 173, + 699, + 825, + 741 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e0fd024672c265a62c064673aa45aad74f7ff0c4aaa2e243918fbac3930f75f8.jpg", + "table_caption": [ + "Table 2: Mean minimum $\\ell _ { \\infty }$ perturbation (in red, computed on $\\epsilon \\in \\{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 \\} )$ and victim accuracy (in blue, for $\\epsilon \\leq 0 . 3 )$ for $\\ell _ { \\infty }$ attacks on MNIST. We manually assign the min. perturbation $\\epsilon = 0 . 6$ to inputs that all attacks failed to find adversarial perturbations. " + ], + "table_footnote": [], + "table_body": "
AttackGFZGBZGFYGBY
PGD(white-box)0.23 /7.71%0.30 /30.78%0.21 / 5.52%0.23 / 8.89%
MIM(white-box)0.24 / 9.02%0.21 /4.97%0.22 / 6.72%0.21 /1.54%
PGD(grey-box)0.37 /51.08%0.36 / 50.64%0.38 / 53.29%0.36 / 48.66%
MIM(grey-box)0.34 / 43.00%0.33 / 40.94%0.34 / 46.64%0.33 /40.06%
PGD(black-box)0.40 / 61.93%0.42 /66.75%0.38 / 56.35%0.43 / 68.50%
MIM(black-box)0.36 / 50.44%0.38 / 59.86%0.36 / 48.07%0.39 / 61.78%
", + "bbox": [ + 240, + 811, + 756, + 921 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/539526329232367bbefc7f3dde85c40b740d2252d0a5700694c8065b953efdca.jpg", + "image_caption": [ + "Figure 7: Results on cross-model transfer attacks on MNIST. The horizontal axis corresponds to the source victim that the adversarial examples are crafted on, and the vertical axis corresponds to the target victim that the attacks are transferred to. The higher (i.e. the lighter) the better. " + ], + "image_footnote": [], + "bbox": [ + 178, + 113, + 816, + 313 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 377, + 825, + 433 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Cross-model attack transferability Finally, we report in Figure 7 the transferability results of the crafted adversarial examples between different models (Papernot et al., 2016a). Here we take adversarial examples crafted in the white-box setting with PGD and MIM $\\acute { \\epsilon } = 0 . 3$ ), and transfer successful attacks to other classifiers. The transfer is effective between generative classifiers but not from generative to discriminative (and vice versa). The attacks crafted on DBX do not transfer in general, while at the same time, DBX is the least robust model in this case. Furthermore, the generative classifiers obtain very high detection rates on all transferred attacks $( > 9 5 \\% )$ . In summary, generative classifiers are more robust against the tested transfer attacks across different models. ", + "bbox": [ + 174, + 448, + 825, + 559 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 CIFAR PLANE-VS-FROG BINARY CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 577, + 560, + 590 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We consider the same set of evaluations on CIFAR-10, in order to validate the robustness of generative classifiers on natural images (c.f. Carlini & Wagner, 2017b). Unfortunately, we failed to train fully generative classifiers with comparable test accuracies to discriminative CNNs (typically $> 8 0 \\%$ ): the clean accuracies for GFZ & GFY are all $< 5 0 \\%$ . Even when using the conditional PixelC $\\mathrm { N N } { + } { + }$ (Salimans et al., 2017) (which uses much deeper networks), the clean accuracy on the test data is $7 2 . 4 \\%$ . Instead, we consider a simpler binary classification problem and construct a dataset containing CIFAR images from the “airplane” and “frog” categories. The images in this dataset are scaled to [0, 1]. On this dataset, the generative classifiers use $\\mathrm { d i m } ( z ) = 1 2 8$ and obtain $> 9 0 \\%$ clean test accuracy (see appendix). The attacks are performed on the test images that all models initially correctly classify, leading to a test set of 1577 instances. Due to the page limit, we only present white-box zero-knowledge attacks here, and discuss further attacks in the appendix. ", + "bbox": [ + 174, + 603, + 825, + 755 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "White-box attacks (zero-knowledge, $\\ell _ { \\infty }$ ) We present the white-box $\\ell _ { \\infty }$ attack results in Figure 8, where the distortion strengths are selected as $\\epsilon \\in \\lbrace 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \\rbrace$ . Again, generative classifiers are more robust than the discriminative ones, and GBZ is the most robust, much better than the others when $\\epsilon \\geq 0 . 0 5$ . BNN is significantly better than other discriminative VAE-based classifiers, presumably due to higher randomness. Detection results are less satisfactory: marginal/logit detection fail to detect attacks with $\\epsilon = 0 . 1$ (which attain both high success rate and induce visually perceptible distortion). KL detection performs better, and interestingly, discriminative classifiers dominate in this metric. These results suggest that the $\\ell _ { 2 }$ likelihood might not be best suited for modelling natural images (c.f. Larsen et al., 2016; van den Oord et al., 2016). Still, the minimum distortion required to fool generative classifiers are much higher than that for discriminative ones, indeed the visual distortion of the adversarial examples on generative classifiers are more significant. ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/efeb93cf87777d23548d2d32db60dccd59abcfdfbb5946443a93be80204c004d.jpg", + "image_caption": [ + "Figure 8: Accuracy, detection rates and minimum $\\ell _ { \\mathrm { i n f } }$ perturbations against white-box zeroknowledge $\\ell _ { \\infty }$ attacks on the CIFAR plane-vs-frog dataset. The higher the better. The second from right most column visualises crafted adversarial examples on an image of a plane, with $\\ell _ { \\infty }$ distortion $\\epsilon \\in \\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \\}$ . " + ], + "image_footnote": [], + "bbox": [ + 183, + 80, + 813, + 295 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/0dd2d70f1cc4886126c26f92b994f6f2f74de3b1a925e6ac9f2d7859b1aa8cd2.jpg", + "image_caption": [ + "Figure 9: Accuracy and detection rates against white-box zero-knowledge CW attacks on the CIFAR plane-vs-frog dataset. The higher the better. " + ], + "image_footnote": [], + "bbox": [ + 179, + 369, + 820, + 453 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "White-box attack (zero-knowledge, $\\ell _ { 2 }$ ) As the test set is relatively small, we directly perform CW attacks on the test data with $c \\in \\{ 0 . 1 , 1 , 1 0 , 1 0 0 , 1 0 0 0 \\}$ . Results are visualised in Figure 9. The generative classifiers are significantly more robust than the others (with the best being GBZ), and the mean $\\ell _ { 2 }$ distortions computed on successful attacks are also significantly higher. The TP rates are low for marginal and logit detection, which is reasonable as the crafted images are visually similar to the clean ones. Note that the distortion for the attacks on generative classifiers is perceptible. These results indicate that this CW attack is ineffective when attacking generative classifiers. ", + "bbox": [ + 174, + 520, + 825, + 618 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 FULL CIFAR-10: COMBINING DEEP BAYES AND DISCRIMINATIVE FEATURES ", + "bbox": [ + 174, + 642, + 745, + 656 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The final experiment examines the robustness of CIFAR-10 multi-class classifiers, with the generative classifiers trained on discriminative visual features. To do this, we download a pretrained deep convolutional net3 with VGG16-like architecture (Simonyan & Zisserman, 2014), and use its feature representation $\\phi ( { \\pmb x } )$ as the input to the VAE-based classifiers: $p ( { \\pmb y } | { \\pmb x } ) = p ( { \\pmb y } | \\phi ( { \\pmb x } ) )$ . The classifiers in test include GBZ, GBY and DBX. We use fully-connected neural networks for these classifiers, and select from VGG16 the $9 ^ { \\mathrm { t h } }$ convolution layer (CONV9) and the first fully connected layer after convolution (FC1) as the feature layers to ensure $\\sim 9 0 \\%$ test accuracy (see appendix). ", + "bbox": [ + 174, + 671, + 825, + 768 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Results on white-box zero-knowledge $\\ell _ { \\infty }$ attacks are visualised in Figure 10. Compared with the VGG16 baseline, we see clear improvements in robustness and detection for all VAE-based classifiers. In particular, the generative classifiers GBZ and GBY are overall better than DBX. More importantly, generative classifiers based on CONV9 features are significantly more robust than those based on FC1 features. In contrast, for DBX, which is discriminative, the robustness results are very similar, indicating that the level of feature representation has little effect. These results suggest that one can achieve both high clean accuracy and better robustness against adversaries by combining discriminatively learned visual features and generative classifiers. ", + "bbox": [ + 173, + 775, + 825, + 887 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/312124c69aaa2404a21fddcb8a35efb3d4d839cf4d73b5393739a6dacb03296e.jpg", + "image_caption": [ + "Figure 10: Accuracy and detection rates against white-box zero-knowledge $\\ell _ { \\infty }$ attacks on CIFAR10. The higher the better. Note that results for the DBX classifiers are almost identical. " + ], + "image_footnote": [], + "bbox": [ + 267, + 83, + 718, + 299 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 351, + 310, + 366 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have proposed deep Bayes as a generative classifier that uses deep latent variable models to model the joint distribution of input-output pairs. We have given evidence, on multiple classification tasks, that generative classifiers are more robust to adversarial attacks than discriminative classifiers. Furthermore, the logit in generative classifiers has a well defined meaning and can be used to detect attacks, even when the classifier is fooled. ", + "bbox": [ + 174, + 385, + 825, + 454 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our results corroborate with the Bayesian neural network literature, in particular (Li & Gal, 2017; Feinman et al., 2017; Carlini & Wagner, 2017b), in showing that modelling unobserved variables are effective for defending against adversarial attacks4. Concurrent to us, Schott et al. (2018) also demonstrated the robustness of generative classifiers on MNIST, in which the logits are computed by a tempered version of the variational lower-bound. However, their approach requires thousands of random $_ z$ samples and tens of optimisation steps to approximate $\\log p ( { \\pmb x } | { \\pmb y } )$ for every input-output pair $( { \\pmb x } , { \\pmb y } )$ , making it less scalable than our importance sampling technique to large datasets and big architectures. Indeed, we have scaled our approach to CIFAR-10, a natural image dataset, and the robustness results are consistent with those on MNIST. In addition, we have also shown that the structure of the graphical model has a significant impact on robustness: deep LVM-based generative classifiers generally outperform the (randomised) discriminative ones. ", + "bbox": [ + 174, + 462, + 825, + 614 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "While we have given strong evidence to suggest that generative classifiers are more robust to current adversarial attacks, we do not wish to claim that these models will be robust to all possible attacks. Aside from many recent attacks being designed specifically for discriminative neural networks, there is also evidence for the fragility of generative models; e.g. naive Bayes as a standard approach for spam filtering is well-known to be fragile (Dalvi et al., 2004; Huang et al., 2011), and very recently Tabacof et al. (2016); Kos et al. (2017); Creswell et al. (2017) also designed attacks for (unconditional) VAE-type models. However, generative classifiers can be made more robust too, to counter these weaknesses. Dalvi et al. (2004) have shown that generative classifiers can be made more secure if aware of the attack strategy, and Biggio et al. (2011; 2014) further improved naive Bayes’ robustness by modelling the conditional distribution of the adversarial inputs. These approaches are similar to the adversarial training of discriminative classifiers, and efficient ways for doing so with generative classifiers can be an interesting research direction. ", + "bbox": [ + 174, + 621, + 825, + 787 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "But even with this note of caution, we believe this work offers exciting avenues for future work. Using generative classifiers offers an interesting way to evaluate generative models and can drive improvements in their ability to tackle high-dimensional datasets, where traditionally generative classifiers have been less accurate than discriminative classifiers (Efron, 1975; Ng & Jordan, 2002). In addition, the combination of generative and discriminative models investigated in this paper is a compelling direction for future research. Overall, we believe that progress on generative classifiers can inspire better designs of attack, defence and detection techniques. ", + "bbox": [ + 174, + 795, + 825, + 892 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. Deep variational information bottleneck. In International Conference on Learning Representations, 2017. 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", + "bbox": [ + 174, + 652, + 821, + 683 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A MODEL ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 419, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "MNIST experiments The VAEs are constructed with convolutional encoders and deconvolutional generators. More specifically, the encoder network for $q ( \\boldsymbol { z } | \\boldsymbol { x } , \\boldsymbol { y } )$ is the same across all VAE-based classifiers. It starts with a 3-layer convolutional neural network with $5 \\times 5$ filters and 64 channels, with a max-pooling operation after each convolution. Then, the convolutional network is followed by a MLP with 2 hidden layers, each with 500 units, to produce the mean and variance parameters of $q$ . The label $\\textbf { { y } }$ is injected into the MLP at the first hidden layer, as a one hot encoding (i.e. for MNIST, the first hidden layer has $5 0 0 { + } 1 0 $ units). The latent dimension is $\\mathrm { d i m } ( z ) = 6 4$ . ", + "bbox": [ + 173, + 133, + 825, + 232 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The $p$ models’ architectures are the following: ", + "bbox": [ + 176, + 238, + 477, + 253 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "GFZ: For $p ( \\pmb { y } | \\pmb { z } )$ we use a MLP with 1 hidden layer composed of 500 units. For $p ( { \\pmb x } | { \\pmb y } , z )$ we used an MLP with 2 hidden layers, each with 500 units, and $4 \\times 4 \\times 6 4$ dimension output, followed by a 3-layer deconvolutional network with $5 \\times 5$ kernel size, stride 2 and [64, 64, 1] channels. \nGFY: We use an MLP with 1 hidden layer composed of 500 units for $p ( z | \\boldsymbol { y } )$ , and the same architecture as GFZ for $p ( { \\pmb x } | { \\pmb y } , z )$ . \nDFZ: We use almost the same deconvolutional network architecture for $p ( { \\pmb x } | z )$ as GFZ’s $p ( { \\pmb x } | { \\pmb y } , z )$ network, except that the input is $_ z$ only. For $p ( \\pmb { y } | \\pmb { x } , z )$ we use almost the same architecture as $q ( \\boldsymbol { z } | \\boldsymbol { x } , \\boldsymbol { y } )$ except that the injected input to the MLP is $_ z$ and the MLP output is the set of logit values for $\\textbf { { y } }$ . \nDFX: We use the same architecture as G3 for $p ( \\pmb { y } | \\pmb { x } , z )$ . The network for $p ( \\boldsymbol { z } | \\boldsymbol { x } )$ is almost identical except that there is no injected input to the MLP, and the network returns the mean and variance parameters for $q ( \\boldsymbol { z } | \\boldsymbol { x } )$ . \nDBX: We use GFZ’s architecture for $p ( \\pmb { y } | \\pmb { z } )$ and DFX’s architecture for $p ( \\boldsymbol { z } | \\boldsymbol { x } )$ . \nGBY: We use GFY’s architecture for $p ( z | \\mathbf { \\boldsymbol { y } } )$ and DFZ’s architecture for $p ( { \\pmb x } | { \\pmb z } )$ . \nGBZ: We use GFZ’s architecture for $p ( \\pmb { y } | \\pmb { z } )$ and DFZ’s architecture for $p ( { \\pmb x } | z )$ . ", + "bbox": [ + 183, + 263, + 826, + 529 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The BNN has almost the same architecture as the encoder network $q$ , except that it uses $2 \\mathbf { x }$ the hidden units/channels, and the last layer is 10 dimensions. Note that here we used dropout as it is convenient to implement, and we expect better approximate inference methods (such as stochastic gradient MCMC) to return better results for robustness and detection. ", + "bbox": [ + 174, + 537, + 821, + 593 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "CIFAR plane-vs-frog experiments The model architectures are almost the same as used in MNIST experiments, except that the hidden layer dimensions for the MLP layers are increased to 1000. For the encoder $q$ , the channels are increased to [64, 128, 256]. For the $p$ models, the deconvolutional networks have different channel values, [128, 64, 3], and the MLP before the deconvolution outputs a $4 \\times 4 \\times 2 5 6$ vector (before reshaping). The BNN has $2 \\mathbf { x }$ the channels but still uses 1000 hidden units. ", + "bbox": [ + 173, + 611, + 825, + 694 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "CIFAR-10 experiments The pre-trained VGG16 network is downloaded from https:// github.com/geifmany/cifar-vgg, where the CONV9 and FC1 layers correspond to: ", + "bbox": [ + 173, + 710, + 823, + 739 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "CONV9: https://github.com/geifmany/cifar-vgg/blob/master/ cifar10vgg.py#L82 FC1: https://github.com/geifmany/cifar-vgg/blob/master/ cifar10vgg.py#L109 ", + "bbox": [ + 171, + 752, + 725, + 815 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The VAE-based classifiers build fully connected networks on top of the extracted features, and use $\\mathrm { d i m } ( z ) = 1 2 8$ for bottleneck. The encoder $q ( \\boldsymbol { z } | \\phi ( \\boldsymbol { x } ) , \\boldsymbol { y } )$ has the network architectures $[ \\mathrm { d i m } ( \\phi ( { \\pmb x } ) ) $ $+ \\dim ( y )$ , 1000, 1000, $\\mathrm { d i m } ( z ) \\times 2 ]$ , and we use the same encoder architecture across all classifiers. The decoder architectures are as follows: ", + "bbox": [ + 174, + 827, + 825, + 882 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "DBX: We use an MLP of layers $[ \\dim ( z ) , ~ 1 0 0 0 , ~ \\dim ( y ) ]$ for $p ( \\pmb { y } | \\pmb { z } )$ and an MLP of layers $[ \\mathrm { d i m } ( \\phi ( { \\pmb x } ) )$ , 1000, 1000, $\\mathrm { d i m } ( z ) \\times 2 ]$ for $p ( \\pmb { z } | \\phi ( \\pmb { x } ) )$ . ", + "bbox": [ + 181, + 895, + 821, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "GBZ: We use an MLP of layers $[ \\mathrm { d i m } ( z )$ , 1000, 1000, $\\dim ( y ) ]$ for $p ( \\pmb { y } | \\pmb { z } )$ and an MLP of layers [dim(z), 1000, 1000, $\\mathrm { d i m } ( \\phi ( { \\pmb x } ) ) ]$ for $p ( \\phi ( { \\pmb x } ) | z )$ . ", + "bbox": [ + 181, + 103, + 825, + 133 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "GBY: We use an MLP of layers $[ \\mathrm { d i m } ( y )$ , 1000, $\\mathrm { d i m } ( z ) \\times 2 ]$ for $p ( z | \\boldsymbol { y } )$ and GBZ’s architecture for $p ( \\phi ( { \\pmb x } ) | z )$ . ", + "bbox": [ + 184, + 136, + 823, + 166 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B ATTACK SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 185, + 364, + 202 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We use the Cleverhans package to perform attacks. We use the default hyper-parameters, if not specifically stated. ", + "bbox": [ + 176, + 217, + 823, + 246 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "PGD: We perform the attack for 40 iterations with step-size 0.01. ", + "bbox": [ + 173, + 252, + 604, + 267 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "MIM: We perform the attack for 40 iterations with step-size 0.01 and decay factor 1.0. ", + "bbox": [ + 173, + 273, + 740, + 289 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "CW- $\\ell _ { 2 }$ : We use learning rate 0.01 and confidence 0, and we optimise the loss for 1000 iterations. ", + "bbox": [ + 173, + 295, + 807, + 310 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "SPSA: We use almost the same hyper-parameters as in Uesato et al. (2018) except for the number of samples for gradient estimates. In detail, we perform the attack for 100 iterations with perturbation size 0.01, Adam learning rate 0.01, stopping threshold -5.0 and 2000 samples for each gradient estimate. ", + "bbox": [ + 173, + 315, + 825, + 372 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.1 JACOBIAN-BASED DATASET AUGMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 390, + 522, + 404 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The black-box distillation attack is based on Papernot et al. (2017b), which trains a substitute CNN using Jacobian-based dataset augmentation. Assume $\\boldsymbol { y } = \\boldsymbol { F } ( \\boldsymbol { x } )$ is the output one-hot vector of the victim, and ${ \\pmb p } ( { \\pmb x } )$ is the probability vector output of the substitute model, then at the $t ^ { \\mathrm { { t h } } }$ outerloop, we train the substitute CNN on dataset $\\mathcal { D } _ { t } \\overset { \\cdot } { = } \\{ ( \\boldsymbol { x } _ { n } , \\boldsymbol { y } _ { n } ) \\}$ with queried ${ \\bf { { y } } } _ { n }$ for 10 epochs, and augment the dataset by ", + "bbox": [ + 174, + 415, + 825, + 486 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c44002978a53ef0602ba9656edb7b3cf391d6584628476a1383527a3292d7f5b.jpg", + "text": "$$\n\\mathcal { D } _ { t + 1 } = \\mathcal { D } _ { t } \\cup \\big \\{ \\big ( \\hat { x } , F ( \\hat { x } ) \\big ) \\mid \\hat { x } = x + \\lambda \\nabla _ { x } p ( x ) ^ { \\mathrm { T } } y , \\big ( x , y \\big ) \\in \\mathcal { D } _ { t } \\big \\} .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 491, + 715, + 510 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We initialise $\\mathcal { D } _ { 1 }$ with $2 0 0 \\times 1 0$ datapoints from the MNIST test set, select $\\lambda = 0 . 1$ , and run the algorithm for 6 outer-loops. On MNIST, this results in 64, 000 queried inputs, and $\\sim 9 6 \\%$ accuracy of the substitute model on test data. On CIFAR binary classification, we use $2 0 0 \\times 2$ datapoints for the inital query set $\\mathcal { D } _ { 1 }$ , resulting in 12, 800 queries in total. The substitutes achieved almost the same accuracy as their corresponding victim models on clean test datapoints. ", + "bbox": [ + 174, + 516, + 825, + 587 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C FURTHER EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 606, + 411, + 622 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C.1 FURTHER EXPERIMENTS ON CIFAR BINARY CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 637, + 642, + 652 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "White-box attack (perfect-knowledge, $\\ell _ { \\infty }$ ) Figure C.1 shows the perfect knowledge attack on the CIFAR binary classification task. Again we see that although the attack is effective for the detection schemes, it comes with the price of decreased mis-classification rates. Interestingly GBY seems to be robust to this attack, where the accuracies on the crafted adversarial examples increase. ", + "bbox": [ + 174, + 664, + 825, + 719 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Sanity checks on gradient masking We conducted the same sets of transferred $\\ell _ { \\infty }$ attack experiments, and presents the results in Figure C.2 and Table C.1. Again that these crafted attacks achieve $\\sim 1 0 0 \\%$ success rates on fooling the substitute models when $\\epsilon \\geq 0 . 1$ . Similar to the MNIST experiments, these adversarial examples do not transfer very well to the generative classifiers, and for a fixed $\\epsilon$ setting, the white-box attacks achieve significantly higher success rates than their grey-/black-box counterparts (with the gap at $\\epsilon \\leq 0 . 1$ around $3 0 \\%$ ). Furthermore, the minimum perturbation obtained by grey-/black-box attacks are significantly higher than those obtained by white-box attacks. For detection, the detection rates are relatively low at $\\epsilon \\leq 0 . 1$ (where the classifiers achieved high accuracy). But the detection rates increase significantly for $\\epsilon = 0 . 2$ , where the victim accuracies also drop. ", + "bbox": [ + 174, + 734, + 825, + 875 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "All these results suggest that the robustness of generative classifiers is unlikely to be caused by gradient masking. Also most of the successfully transferred adversarial images are off the generative classifier’s manifold (as a proxy to the data manifold). ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/5152312abaec5bc13db7e41eea936e83e591f26cf23328722a6ed5def0474647.jpg", + "image_caption": [ + "Figure C.1: Accuracy and detection rates of DBX, GBY, and GBZ against PGD-based perfectknowledge attack $\\epsilon = 0 . 1$ ) on CIFAR binary task. The solid area denotes accuracy and the hatched area denotes detection rate with each considered detector. Zero knowledge attacks are labelled ’- ZK’, other attack labels are the same as for the MNIST plot (Figure 5). " + ], + "image_footnote": [], + "bbox": [ + 192, + 140, + 805, + 279 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/efd469fdcaf929c8a6615157ea860f31d2e71ae5eb8fbeb99dfc0c6d149f3ee1.jpg", + "image_caption": [ + "Figure C.2: Accuracy and detection rates against distillation-based attacks on CIFAR plane-vs-frog binary classification. The higher the better. We only present generative classifiers’ results here, for full results see appendix E. " + ], + "image_footnote": [], + "bbox": [ + 181, + 424, + 812, + 588 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/ad700c3a274bdfabaf861bf517ad2d9df80e265fa79dd62b518b995d8092f2ff.jpg", + "table_caption": [ + "Table C.1: Mean minimum $\\ell _ { \\infty }$ perturbation (in red, computed on $\\epsilon \\in \\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \\} )$ ) and victim accuracy (in blue, for $\\epsilon \\leq 0 . 1 $ ) for $\\ell _ { \\infty }$ attacks on CIFAR plane-vs-frog binary classification. We manually assign the min. perturbation $\\epsilon = 0 . 3$ to inputs that all attacks failed to find adversarial perturbations. " + ], + "table_footnote": [], + "table_body": "
AttackGFZGBZGFYGBY
PGD(white-box)0.11 / 21.81%0.20 / 65.63%0.11 / 25.81%0.11 / 25.24%
MIM(white-box)0.09 / 15.22%0.13 /37.60%0.10 /16.4%90.09 / 14.39%
PGD(grey-box)0.15 / 50.48%0.23 / 77.30%0.16 / 54.66%0.17 / 57.96%
MIM(grey-box)0.15 /47.62%0.21 / 75.71%0.15 / 51.11%0.16 / 53.84%
PGD(black-box)0.19 / 68.36%0.23 / 79.45%0.20 / 70.13%0.19 / 67.98%
MIM(black-box)0.18 / 66.39%0.23 / 78.38%0.19 / 68.42%0.19 / 66.52%
", + "bbox": [ + 240, + 773, + 756, + 883 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/05941d1d37a465318a19b333ee210964ee3f328c3ec80af16d2151b21b4c212a.jpg", + "table_caption": [ + "Table C.2: Accuracy and detection rates against black-box SPSA attack $\\epsilon = 0 . 0 5 )$ on CIFAR plane-vs-frog, with a comparision to white-box PGD. The higher the better. " + ], + "table_footnote": [], + "table_body": "
GFZGBZGBYGFYDFXDBXDFZ
PGD victim acc67.7%83.9%67.9%67.3%0.3%4.2%0.4%
SPSA victim acc96.4%95.2%96.4%96.3%0.4%87.5%5.1%
SPSA TP logit10.0%17.1%15.5%10.0%N/AN/A4.5%
", + "bbox": [ + 223, + 141, + 772, + 213 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/c2212cb826926ce9d453f97e43195c2f01d46651d793cf484348cc807383b02c.jpg", + "image_caption": [ + "Figure C.3: Results on cross-model transfer attacks on the CIFAR plane-vs-frog dataset. The horizontal axis corresponds to the source victim, and the vertical axis corresponds to the target victim. The higher (lighter) the better. " + ], + "image_footnote": [], + "bbox": [ + 174, + 229, + 821, + 351 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "SPSA (evolutionary strategies) Similarly we perform the SPSA attack (Uesato et al., 2018) on the CIFAR binary classification task. The results for $\\epsilon = 0 . 0 5$ are presented in Table C.2, with a comparison to the white-box PGD attacks. Again we see that SPSA fails to attack generative classifiers, and the bottleneck discriminative classifier DBX is significantly more robust than the discriminative ones with fully-connected graphical models. ", + "bbox": [ + 173, + 428, + 825, + 498 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Cross-model attack transferability Finally, we present the CIFAR cross-model attack transferability results in Figure C.3, and here we select $\\epsilon = 0 . 1$ instead. Again, transferred attacks are less effective across the victim models. However, the TP rates for logit detection are significantly lower than in the MNIST case (also see Figure 8). Nevertheless, the detection rates for the “discriminative to generative” transfer are considerably higher. Combined with the accuracy results, we see that discriminative models as substitutes are ineffective in the transferred attack setting. ", + "bbox": [ + 173, + 513, + 825, + 598 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.2 QUANTIFYING THE EFFECT OF THE BOTTLENECK LAYER ", + "text_level": 1, + "bbox": [ + 181, + 617, + 604, + 631 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We see from the main text that classifiers with bottleneck structure may be preferred for resisting adversarial examples. To quantify this bottleneck effect, we train on MNIST models DBX, GBZ and GBY with $_ { z }$ dimensions in $\\{ \\dot { 1 } 6 , 3 2 , 6 4 , 1 2 8 \\}$ (the main text experiments use $\\dim ( z ) = 6 4 $ ).The clean test accuracy is shown in Table C.3, showing that all models in test perform reasonably well. ", + "bbox": [ + 174, + 642, + 825, + 700 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/fb730c8aea9e853f23bbeeb113988771173ea29b3716a47400ad668d4926df9c.jpg", + "table_caption": [ + "Table C.3: Clean test accuracy on MNIST classification (with varied bottleneck layer sizes). " + ], + "table_footnote": [], + "table_body": "
dim(z) = 16dim(z) = 32dim(z)= 64dim(z)=128
DBX99.11%99.01%98.98%98.91%
GBZ97.11%97.08%97.45%96.62%
GBY98.82%98.95%98.72%98.75%
", + "bbox": [ + 254, + 739, + 743, + 797 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We repeat the same white-box zero-knowledge $\\ell _ { \\infty }$ attack experiments as done in the main text, where results are presented in Figure C.4 and Tables E.25, E.26 and E.27. It is clear that for discriminative classifiers, DBX, the models become less robust as the bottleneck dimension $\\mathrm { d i m } ( z )$ increases. Interestingly DBX classifiers seem to be very robust against FGSM attacks, which agrees with the results in Alemi et al. (2017). For the generative ones, we also observe similar trends (although much less significant) of decreased robustness for GBY classifiers, and for GBZ the trend is unclear, presumably due to local optimum issues in optimisation. In summary, GBZ classifiers are generally more robust compared to GBY classifiers. More importantly, when the accuracy of generative classifiers on adversarial images decreases to zero, the detection rates with marginal/logit detection increases to $1 0 0 \\%$ . This clearly shows that the three attacks tested here cannot fool the generative classifiers without being detected. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/ac97b61ab1e7245d921f06ec5438eb1e3ec5e1ff1b11e353e9a44c52679e086b.jpg", + "image_caption": [ + "Figure C.4: Accuracy and detection rates against white-box zero-knowledge $\\ell _ { \\infty }$ attacks on MNIST, with varied bottleneck layer sizes. " + ], + "image_footnote": [], + "bbox": [ + 228, + 99, + 767, + 297 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/da2ab9e2617b7023b114b6661d3a22afd6ef69bb833ae82a948b11f22d58d298.jpg", + "table_caption": [ + "Table D.1: Clean test accuracy on CIFAR plane-vs-frog classification. " + ], + "table_footnote": [], + "table_body": "
BNNGFZGFYDFZDFXDBXGBZGBY
97.00%91.60%91.20%94.85%95.65%96.00%89.35%90.65
", + "bbox": [ + 220, + 377, + 774, + 409 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 434, + 825, + 477 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 497, + 398, + 513 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We visualise in Figure D.1 the crafted adversarial images using white-box CW attack. \nWe present in Table D.1 the clean accuracy on CIFAR plane-vs-frog test images (2000 in total). \nWe present in Table D.2 the clean accuracy on CIFAR-10 test images. ", + "bbox": [ + 173, + 527, + 799, + 587 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/f6e398af395dfda26afc7838c84d43f9574e6d7c5701d027bb141d304f9650e8.jpg", + "table_caption": [ + "Table D.2: Clean test accuracy on CIFAR-10 classification. " + ], + "table_footnote": [], + "table_body": "
VGG16GBZ-FC1GBY-FC1DBX-FC1GBZ-CONV9GBY-CONV9DBX-CONV9
93.59%92.55%93.21%93.49%91.76%88.33%93.21%
", + "bbox": [ + 169, + 890, + 843, + 921 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/dbcf0906a736b6a88bfdb52b23677d578009e609628af44f312d5c13c3ee6558.jpg", + "image_caption": [ + "Figure D.1: Visualising the clean inputs of MNIST and the CW adversarial examples crafted on all the classifiers. " + ], + "image_footnote": [], + "bbox": [ + 232, + 267, + 769, + 723 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E RESULTS IN TABLES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 377, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We present in tables the full results of the experiments. \nSee Tables E.1 to E.9 for the white-box attacks. \nSee Tables E.10 to E.15 for the grey-box attacks. \nSee Tables E.16 to E.21 for the black-box attacks. \nSee Tables E.22 to E.24 for CIFAR-10 results with VGG-based classifiers. \nSee Tables E.25 to E.27 for bottleneck effect quantification results. ", + "bbox": [ + 173, + 132, + 661, + 256 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/13a389dd514e00a5ac50d3a2c85cbfc7210a7ac1d65af7b331edf5e67cc0c8dd.jpg", + "table_caption": [ + "Table E. 1 : FGSM white-box zero-knowledge attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN92.467.840.526.220.4N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.094.295.495.596.5
GFZ94.274.538.912.95.743.679.8100.0100.0100.056.489.699.9100.0100.089.291.792.292.592.2
GBZ92.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GFY94.374.846.521.710.553.192.6100.0100.0100.066.297.9100.0100.0100.090.893.193.894.294.2
GBY93.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
DFX70.114.81.00.40.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.693.893.893.594.3
DBX91.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DFZ75.219.02.41.11.012.650.6100.0100.0100.026.965.7100.0100.0100.093.195.194.594.994.8
", + "bbox": [ + 170, + 215, + 804, + 375 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/0c121ec19495bb0fe8fe320e043b4a5146be46413952c7eccd8119ea1f017fc3.jpg", + "table_caption": [ + "Table E.2: PGD white-box zero-knowledge attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN83.212.00.50.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.192.188.720.40.2
GFZ86.737.77.71.20.343.271.891.499.4100.055.877.194.899.6100.090.392.190.990.489.7
GBZ85.157.432.519.311.633.750.384.499.7100.052.266.391.599.8100.090.091.591.591.791.8
GFY79.727.45.61.20.358.187.998.0100.0100.068.792.299.3100.0100.092.692.590.790.785.4
GBY86.735.99.01.70.445.376.194.699.7100.056.979.395.699.9100.090.192.191.691.791.4
DFX47.60.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.889.813.331.442.8
DBX DFZ58.0 49.618.36.01.30.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
1.00.00.00.011.844.394.1100.0100.025.957.694.599.9100.093.989.812.122.928.0
", + "bbox": [ + 174, + 434, + 797, + 596 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/6d3f886df0f38f545c7679ec27eb874829c595320fd002edce766693610d75ca.jpg", + "table_caption": [ + "Table E.3: MIM white-box zero-knowledge attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
acc.(adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN82.07.20.10.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.291.172.064.57.1
GFZ87.040.49.01.21.243.583.898.499.499.457.689.398.699.299.291.592.392.090.690.6
GBZ79.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GFY80.830.16.81.41.456.093.799.8100.0100.068.196.699.9100.0100.090.992.492.091.691.6
GBY84.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
DFX48.40.80.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.289.69.217.517.5
DBX66.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DFZ50.51.20.00.00.011.950.799.4100.0100.026.463.299.0100.0100.094.389.811.517.317.3
", + "bbox": [ + 172, + 655, + 800, + 815 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/bb08de53e89e454f80650546110f2c977f67863295c3fa8f6fed8e6221402821.jpg", + "table_caption": [ + "Table E.4: Perfect knowledge attacks on MNIST. This is done using the PGD attack with $\\epsilon = 0 . 2$ . ‘ZK’ stands for zero knowledge, ‘PK0’ where you have knowledge of the $K$ samples but not of the detection system. ‘PKM’ ‘PKL’ and ‘PKK’ are attacks where you have knowledge of the K samples and of the marginal, logit and KL detection mechanisms respectively. " + ], + "table_footnote": [], + "table_body": "
ZKPKK
acc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KL
DBX18.3N/AN/A9417N/AN/A93.1N/AN/AN/AN/AN/AN/AN/AN/A35.9N/AN/A93.9
GBZ57.450.366.391.535.582.290.791.447.936.567.491.344.650.569.991.04573.084.891.5
GBY35.976.179.392.12290.891.992.143.847.061.991.152.056.466.990.632.889.291.893.3
", + "bbox": [ + 71, + 258, + 912, + 341 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/e0c7744832753cf7a05cd70afc998f5fb15062962639fce97c6b30c6d96d6a6f.jpg", + "table_caption": [ + "Table E.5: FGSM white-box zero-knowledge attack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
acc. (adv)TP marginal
E0.010.020.050.100.200.010.020.050.100.20 0.010.020.050.100.200.010.020.050.100.20
BNN98.293.258.514.56.3N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A58.655.165.554.058.5
GFZ97.194.781.856.431.411.410.15.9 17.499.311.410.46.817.299.358.640.135.941.050.6
GBZ95.193.587.174.962.026.018.615.5 38.699.626.320.319.141.999.635.541.547.043.745.6
GFY96.594.280.756.732.09.69.26.9 18.899.017.613.18.120.899.146.643.736.139.549.3
GBY96.192.982.060.536.317.215.18.1 28.599.220.218.110.031.199.249.748.039.238.747.3
DFX83.842.20.70.00.0N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A63.366.648.497.7100.0
DBX90.978.650.731.718.3N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A50.758.756.859.757.2
DFZ83.952.12.90.00.06.63.82.23.9 60.66.94.22.43.360.457.562.354.866.799.8
", + "bbox": [ + 184, + 415, + 790, + 575 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/540c5962e323a6fa3c4cfc9aaf10816117104161b1c09e8ef9c4aee39d9e7f79.jpg", + "table_caption": [ + "Table E.6: PGD white-box zero-knowledge atack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN97.986.719.71.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A41.759.455.768.498.9
GFZ98.093.967.721.73.53.36.65.67.832.95.09.56.08.233.037.545.134.544.057.7
GBZ94.693.783.967.452.819.817.912.814.343.122.819.517.417.044.332.139.340.237.133.2
GFY98.495.067.925.84.14.26.76.47.835.73.18.27.67.833.831.243.632.939.653.3
GBY96.492.967.325.76.911.18.47.911.641.413.111.59.212.240.343.043.033.639.152.2
DFX82.735.70.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A64.164.769.6100.0100.0
DBX83.334.64.20.70.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A59.859.565.178.793.4
DFZ82.436.90.40.00.04.93.85.112.391.16.54.15.512.391.356.360.354.799.7100.0
", + "bbox": [ + 181, + 648, + 792, + 808 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/bde5b59973f2f77e1f53dd5ff17d0d35ff424ffce2b12078719971ab9242a073.jpg", + "table_caption": [ + "Table E.7 : MIM white-box zero-knowledge attack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN96.984.618.70.90.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A35.954.756.362.478.8
GFZ96.991.558.115.21.96.56.26.410.686.110.08.36.611.384.531.039.032.147.277.1
GBZ92.589.471.540.417.915.013.211.024.095.916.414.812.825.196.042.537.531.037.455.0
GFY97.692.359.516.52.35.66.96.313.388.15.76.26.813.887.623.841.433.843.974.6
GBY95.188.856.514.61.315.19.58.619.297.315.310.79.918.597.051.143.231.746.787.6
DFX82.535.20.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A65.567.169.6100.0100.0
DBX82.538.86.11.20.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A54.660.061.578.888.5
DFZ81.536.10.40.00.05.43.95.013.496.96.24.15.513.896.861.158.555.399.9100.0
", + "bbox": [ + 181, + 222, + 792, + 382 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/22a7d9eeabd52a6b307ca07eed81f72a515f402c0a6e0daf63e5de24e0b5a180.jpg", + "table_caption": [ + "Table E.8: CW white-box zero-knowledge atack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
acc. (adv)TP marginalTP logitTP KL
E0.101.0010.00100.001000.000.101.0010.001100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
BNN93.766.237.332.347.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A60.980.382.272.080.0
GFZ99.595.676.577.978.30.04.52.63.03.40.04.63.03.64.229.230.425.627.129.0
GBZ96.093.688.988.789.016.711.57.87.77.918.913.79.39.19.441.151.440.434.732.0
GFY99.895.978.880.080.50.06.64.14.75.10.06.64.84.95.475.037.326.026.431.1
GBY97.593.176.777.378.015.814.66.07.17.318.913.96.07.07.942.634.327.231.629.0
DFX82.644.234.36.10.6N/AN/AN/AN/AN/AN/AN/A N/AN/AN/A100.0100.0100.082.595.0
DBX96.572.226.36.316.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A61.679.485.262.267.9
DFZ94.572.329.96.317.97.35.84.34.14.214.16.95.04.44.482.697.397.183.080.2
", + "bbox": [ + 116, + 455, + 857, + 616 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/0aa86c3f2f459dad9cf4a535bc6c2d478f236425effb2f356ee14be0972f19a7.jpg", + "table_caption": [ + "Table E.9: Perfect knowledge attacks on CIFAR binary task. This is done using the PGD attack with $\\epsilon = 0 . 1$ . ‘ZK’ stands for zero knowledge, ‘PK0’ where you have knowledge of the $K$ samples but not of the detection system. ‘PKM’ , ‘PKL’ , and ‘PKK’ are attacks where you have knowledge of the K samples and of the marginal, logit and KL detection mechanisms respectively. " + ], + "table_footnote": [], + "table_body": "
PKK
acc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KL
DBX0.7N/AN/A78.70.7N/A82.5N/AN/AN/AN/AN/AN/AN/A0.6N/AN/A 27.682.6
GBZ67.414.317.037.15740.645.14958.49.618.546.958.51.87.847.859.925.542.4
GBY25.711.612.239.128.522.022.64929.55.67.645.929.20.81.845.427.915.615.945
", + "bbox": [ + 74, + 725, + 912, + 808 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/3e20d9ddf5da2c4016d841bf1cc46fa85fb7ca7af16cac078fd437d6f2cc4efc.jpg", + "table_caption": [ + "Table E. 1 0: Grey-box PGD attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ86.714.30.00.00.096.683.351.326.817.249.573.899.7100.0100.0
GBZ81.29.10.00.00.093.580.155.335.025.246.773.399.6100.0100.0
GFY84.86.40.00.00.096.782.355.233.524.058.788.8100.0100.0100.0
GBY86.715.00.00.00.095.881.651.027.518.150.178.599.9100.0100.0
DFX74.25.20.50.00.091.757.719.64.31.4N/AN/AN/AN/AN/A
DBX80.65.10.00.00.093.259.222.510.98.2N/AN/AN/AN/AN/A
DFZ69.63.40.30.00.091.957.221.46.32.633.655.191.7100.0100.0
", + "bbox": [ + 250, + 222, + 723, + 367 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/8c3d81d591c93f12d791ffb08ed4d7edc2304827e0d94876715b533350a246c8.jpg", + "table_caption": [ + "Table E. 1 1 : Grey-box MIM attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim accTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ87.216.10.00.00.096.582.442.814.23.750.682.5100.0100.0100.0
GBZ81.89.10.00.00.093.378.645.317.05.246.281.7100.0100.0100.0
GFY85.26.90.00.00.096.680.848.018.97.459.896.0100.0100.0100.0
GBY87.015.90.00.00.095.779.941.813.04.051.487.1100.0100.0100.0
DFX76.512.33.62.12.191.556.122.29.99.9N/AN/AN/AN/AN/A
DBX85.119.90.60.10.193.457.014.16.36.3N/AN/AN/AN/AN/A
DFZ70.93.20.00.00.091.854.616.42.70.334.563.898.9100.0100.0
", + "bbox": [ + 251, + 442, + 723, + 588 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/d43e3c3e08cc551a53d00674ff3f9ef3c2789cc54425bbad12ff2b3afa28ba78.jpg", + "table_caption": [ + "Table E.12: Grey-box CW attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim accTP logit
E0.101.0010.00100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
GFZ98.44.60.00.00.098.896.996.093.490.253.444.241.640.643.7
GBZ98.463.20.00.00.097.495.492.988.684.545.939.733.031.238.6
GFY98.10.90.00.00.099.097.697.095.794.061.055.249.249.552.7
GBY98.35.30.00.00.098.796.896.093.891.254.241.537.638.743.6
DFX85.10.00.00.00.097.693.391.290.489.8N/AN/AN/AN/AN/A
DBX97.346.00.40.00.098.793.988.485.576.3N/AN/AN/AN/AN/A
DFZ80.50.00.00.00.097.294.692.391.591.232.928.628.136.544.5
", + "bbox": [ + 212, + 663, + 762, + 808 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/3770e5de03d715a04ccc91867b52f3d31aedcf1ddf77758ce3b806141a9df3e9.jpg", + "table_caption": [ + "Table E. 1 3 : Grey-box PGD attack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ96.788.141.74.50.197.094.984.048.38.38.411.26.17.549.8
GBZ92.983.137.85.50.095.394.891.883.165.615.512.810.919.979.8
GFY96.988.133.73.30.196.995.985.354.212.72.62.05.08.561.2
GBY95.285.632.42.80.196.895.386.459.818.521.814.310.512.575.2
DFX95.780.810.10.20.099.196.779.428.30.9N/AN/AN/AN/AN/A
DBX95.780.530.51.70.099.598.291.165.121.8N/AN/AN/AN/AN/A
DFZ96.581.910.50.30.099.196.976.926.21.34.516.56.18.559.6
", + "bbox": [ + 260, + 223, + 714, + 367 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/c7a32a3ee4f8103a04a02af4d798574bdf60d4ca925310d7f639170a3fb2fbac.jpg", + "table_caption": [ + "Table E.14: Grey-box MIM attck results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ96.387.942.14.70.197.194.883.044.84.98.711.15.88.575.1
GBZ92.982.536.94.90.095.594.791.481.749.514.812.613.921.997.8
GFY96.887.934.13.50.196.995.684.649.16.92.62.06.710.381.1
GBY95.185.532.02.60.196.895.585.756.210.921.814.89.213.497.6
DFX95.580.510.50.20.099.196.577.324.00.2N/AN/AN/AN/AN/A
DBX95.681.134.52.20.099.598.389.358.99.9N/AN/AN/AN/AN/A
DFZ96.381.510.70.50.099.196.974.921.90.34.516.66.49.980.2
", + "bbox": [ + 260, + 442, + 714, + 588 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/6b1cc18428ca2414a0cbecc7c49f2b0b68044aa5ca5af3da61296a635bf14112.jpg", + "table_caption": [ + "Table E.15: Grey-box CW attack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim accTP logit
E0.101.0010.00100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
GFZ98.761.90.00.00.098.695.894.393.893.211.710.210.99.98.5
GBZ96.369.83.30.00.095.995.794.794.794.513.512.712.313.714.4
GFY98.454.40.00.00.098.196.195.395.194.30.02.23.74.67.6
GBY97.652.92.20.00.097.795.694.594.393.725.815.714.213.718.0
DFX64.40.00.00.00.098.597.297.196.795.5N/AN/AN/AN/AN/A
DBX91.356.20.30.00.099.799.099.098.998.1N/AN/AN/AN/AN/A
DFZ80.30.00.00.00.099.297.997.597.395.40.015.714.68.86.1
", + "bbox": [ + 212, + 663, + 762, + 808 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/2c43bedd2493e2c0116d115e799e5e1ab141747e0677d367f7d5965ab6feb06e.jpg", + "table_caption": [ + "Table E. 1 6 : Black-box PGD attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ44.20.70.00.00.097.388.462.733.821.848.761.092.399.8100.0
GBZ7.40.00.00.00.094.887.170.752.941.351.068.697.7100.0100.0
GFY49.40.80.00.00.097.486.558.131.921.052.570.897.499.8100.0
GBY21.70.00.00.00.096.989.370.049.737.851.270.898.2100.0100.0
DFX49.21.30.00.00.091.452.113.92.20.7N/AN/AN/AN/AN/A
DBX43.10.70.00.00.095.067.026.29.76.5N/AN/AN/AN/AN/A
DFZ53.81.70.00.00.092.956.315.23.41.534.752.894.099.9100.0
", + "bbox": [ + 253, + 222, + 719, + 367 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/cf7698a940572ba97f63d53f83cbc9ebb0b6194ea62178d6117355d3db4ae076.jpg", + "table_caption": [ + "Table E. 1 7 : Black-box MIM attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ45.52.80.00.00.097.286.650.317.017.051.168.097.599.599.5
GBZ8.40.00.00.00.094.785.963.535.217.151.274.499.0100.0100.0
GFY53.12.50.30.10.197.283.749.220.120.155.180.699.099.999.9
GBY24.50.00.00.00.096.888.362.831.713.349.377.699.9100.0100.0
DFX51.32.30.10.00.091.451.713.31.91.9N/AN/AN/AN/AN/A
DBX47.81.60.10.00.094.867.023.27.67.6N/AN/AN/AN/AN/A
DFZ56.12.90.10.00.092.756.013.12.82.835.259.997.6100.0100.0
", + "bbox": [ + 253, + 442, + 719, + 588 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/97c05ee54ca6147bed36597f991cc005648c07e9e18d7bbcf2682b13fc87bf83.jpg", + "table_caption": [ + "Table E.18: Black-box CW attack results on MNIST. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.0010.00100.001000.00
GFZ65.20.30.00.00.098.898.797.4 94.392.052.3 50.839.644.755.3
GBZ76.60.40.00.00.097.397.295.9 92.790.445.746.141.540.345.7
GFY88.43.80.00.00.099.098.998.0 94.892.060.059.251.947.657.2
GBY76.01.30.00.00.098.798.697.3 95.193.553.452.241.839.746.7
DFX82.40.00.00.00.098.896.8 92.486.284.9N/AN/AN/AN/AN/A
DBX82.50.60.00.00.098.998.495.5 85.283.1N/AN/AN/AN/AN/A
DFZ86.50.20.00.00.098.894.589.3 81.579.342.925.522.327.738.2
", + "bbox": [ + 212, + 663, + 761, + 808 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/e319a20972933961b8f960bc13c336ba99c90bdee74be629a25521dc6714222c.jpg", + "table_caption": [ + "Table E. 1 9 : Black-box PGD attack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim accTP logit
E0.010.020.050.100.200.010.02 0.050.100.200.010.020.050.100.20
GFZ95.589.945.56.80.197.695.688.768.829.76.26.76.010.871.8
GBZ92.084.133.83.50.095.694.992.785.065.916.714.612.817.976.8
GFY95.389.138.72.50.097.396.690.072.935.50.02.36.810.472.3
GBY91.880.529.76.10.297.295.990.270.830.321.816.49.613.677.3
DFX89.174.519.10.90.099.798.792.664.89.9N/AN/AN/AN/AN/A
DBX85.576.442.54.50.099.699.094.877.524.5N/AN/AN/AN/AN/A
DFZ85.573.521.11.50.099.599.293.870.115.10.05.08.915.985.1
", + "bbox": [ + 260, + 223, + 714, + 367 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/7b251889e625f5dd4596e8402204a6af2ca32b03eb2bcd38bf0968d0f1a42696.jpg", + "table_caption": [ + "Table E.2O: Black-box MIM attack results on CIFAR plane-vs-frog binary classification. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ95.589.746.47.70.297.595.788.566.520.56.26.75.912.782.7
GBZ91.884.033.13.40.095.694.992.385.160.716.715.514.221.697.9
GFY95.388.940.53.10.097.396.789.570.631.90.02.47.612.588.0
GBY91.780.229.55.90.197.395.989.669.026.122.216.19.714.095.9
DFX89.074.519.40.90.099.698.792.163.010.9N/AN/AN/AN/AN/A
DBX85.576.342.74.50.099.699.095.176.527.2N/AN/AN/AN/AN/A
DFZ85.373.421.41.60.099.699.192.967.714.90.04.57.514.291.9
", + "bbox": [ + 260, + 442, + 715, + 588 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/067bb95485b95f9d2dd81422dd4200a76ebfa59f57ad0c72011b41d58ea018d4.jpg", + "table_caption": [ + "Table E.21: Black-box CW attack results on CIFAR plane-vs-frog binary clasification. " + ], + "table_footnote": [], + "table_body": "
substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.00TP logit 10.00100.001000.00
GFZ96.341.50.50.00.098.794.793.4 93.392.313.3 8.67.77.78.9
GBZ94.973.31.40.00.095.995.3 94.694.594.513.511.811.813.213.1
GFY96.462.01.30.00.098.196.7 94.894.794.10.02.86.56.16.8
GBY93.533.61.70.00.097.795.9 95.495.495.025.816.415.918.619.8
DFX90.310.90.00.00.099.998.8 98.598.597.9N/AN/AN/AN/AN/A
DBX87.231.80.00.00.099.997.1 94.593.994.7N/AN/AN/AN/AN/A
DFZ84.922.70.00.00.099.999.0 98.398.397.80.03.82.32.311.0
", + "bbox": [ + 212, + 663, + 762, + 808 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/32736db02f59012838a2cbb0ddefca64422817539a41315ee85d363bdac44a2a.jpg", + "table_caption": [ + "Table E.22: FGSM white-box zero-knowledge attack results on CIFAR- 1 0. " + ], + "table_footnote": [], + "table_body": "
acc. (adv) 0.020.050.100.20TP marginalTP logitTP KL
E0.010.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1644.525.816.611.910.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A95.195.797.097.597.1
GBZ-FC163.749.734.318.912.760.062.480.690.994.163.765.080.695.098.490.190.892.392.692.6
GBY-FC164.950.036.119.511.352.259.080.490.192.554.661.980.395.698.490.791.392.192.490.3
DBX-FC157.345.731.816.310.8N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.893.294.094.392.8
GBZ-CONV981.674.158.548.945.813.717.028.476.090.218.320.631.764.371.691.791.591.793.386.4
GBY-CONV975.866.039.722.517.514.018.229.474.488.917.720.825.855.268.989.689.990.089.189.2
DBX-CONV959.045.731.817.811.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.794.194.995.293.7
", + "bbox": [ + 166, + 223, + 808, + 367 + ], + "page_idx": 27 + }, + { + "type": "table", + "img_path": "images/74cb85fa8228d6c4c1b69f9aa9c5b01b47277ed6b864db9980f356b0ddc29e61.jpg", + "table_caption": [ + "Table E.23: PGD white-box zero-knowledge attack results on CIFAR-10 " + ], + "table_footnote": [], + "table_body": "
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1618.80.60.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.991.867.469.966.4
GBZ-FC137.76.10.10.00.030.722.276.092.094.133.524.977.091.790.290.893.797.398.595.9
GBY-FC131.28.52.61.50.632.020.382.395.197.636.629.584.193.790.490.394.197.797.396.9
DBX-FC123.41.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.991.05.67.45.9
GBZ-CONV961.138.324.210.22.319.260.396.798.999.620.160.396.898.398.790.393.197.798.799.6
GBY-CONV966.526.63.20.40.017.055.695.799.299.617.852.795.398.697.689.691.898.799.496.9
DBX-CONV924.03.00.30.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.393.199.399.699.2
", + "bbox": [ + 166, + 442, + 808, + 588 + ], + "page_idx": 27 + }, + { + "type": "table", + "img_path": "images/25c2b7bc2ca7263bb2a4012b3fd682ba9ea99fbef41be0e8fae22940ac955f93.jpg", + "table_caption": [ + "Table E.24: MIM white-box zero-knowledge attack results on CIFAR-10. " + ], + "table_footnote": [], + "table_body": "
acc. (adv)TP marginalTP logit
E0.010.02（0.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.10 0.20
VGG1613.10.40.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A92.292.472.377.179.0
GBZ-FC124.52.70.10.00.034.840.988.595.896.0 38.844.090.496.496.691.395.095.797.996.8
GBY-FC133.019.611.87.24.929.138.795.799.199.3 38.449.296.399.399.490.795.097.897.997.9
DBX-FC117.30.90.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.189.69.614.516.0
GBZ-CONV957.643.529.124.122.627.478.599.499.6 99.729.178.399.699.799.890.294.497.598.098.2
GBY-CONV939.118.14.31.91.525.670.498.699.699.8 28.271.198.699.799.889.694.599.499.899.7
DBX-CONV919.23.20.50.40.4N/AN/AN/AN/AN/A N/AN/AN/AN/AN/A92.196.799.9100.0100.0
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acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1692.685.676.464.752.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A90.392.492.392.794.2
DBX-3292.684.471.157.246.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.292.392.894.995.0
DBX-6491.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DBX-12887.847.620.112.710.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A94.497.697.897.196.4
GBZ-1692.075.852.534.021.728.771.599.9100.0100.049.286.399.9100.0100.090.890.692.091.990.9
GBZ-3291.174.452.433.421.626.573.599.7100.0100.049.487.299.9100.0100.091.090.691.891.491.6
GBZ-6492.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GBZ-12890.876.857.538.924.826.663.099.6100.0100.044.978.799.8100.0100.087.990.190.791.190.6
GBY-1694.277.749.423.912.141.984.1100.0100.0100.052.991.9100.0100.0100.086.692.093.093.192.6
GBY-3294.576.945.420.09.641.483.3100.0100.0100.056.392.8100.0100.0100.089.091.693.393.293.0
GBY-6493.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
GBY-12893.072.942.218.59.037.4：71.6100.0100.0100.050.181.999.9100.0100.090.791.4 91.891.492.0
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acc. (adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1669.836.616.05.91.8N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.291.992.893.393.7
DBX-3263.526.812.56.22.7N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A91.592.192.792.993.5
DBX-6458.018.36.01.30.2N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
DBX-12842.33.01.10.30.2N/AN/A N/AN/AN/AN/AN/AN/AN/AN/A97.197.198.498.498.6
GBZ-1688.362.837.423.115.527.135.871.9 98.9100.046.353.082.399.3100.089.691.792.391.491.8
GBZ-3287.161.940.228.020.424.039.176.798.9 100.041.556.485.499.7100.091.391.091.191.391.1
GBZ-6485.157.432.519.311.633.750.384.4 99.7100.052.266.391.599.8100.090.091.591.591.791.8
GBZ-12884.457.536.825.216.825.435.466.1 96.6100.041.550.576.497.7100.088.690.490.490.090.6
GBY-1690.449.715.04.21.642.664.789.9 99.3100.051.170.892.899.5100.091.993.191.990.791.0
GBY-3288.439.89.51.80.645.674.392.8 99.5100.056.878.294.899.8100.090.892.291.791.090.6
GBY-6486.735.99.01.70.445.376.194.6 99.7100.056.979.395.699.9100.090.192.191.691.791.4
GBY-12883.335.18.72.30.839.163.986.7 98.3100.051.769.189.298.9100.089.890.790.489.789.5
", + "bbox": [ + 170, + 568, + 804, + 792 + ], + "page_idx": 28 + }, + { + "type": "table", + "img_path": "images/2b4c473b7b6f9c8747097e3df9f38d91c8aafa10a6917786cc753b7ae3ac1ca2.jpg", + "table_caption": [ + "Table E.27 : MIM white-box zero-knowledge attack results on MNIST (with varied bottleneck layer sizes) . " + ], + "table_footnote": [], + "table_body": "
acc. (adv)TP marginalTP logitTP KL
E0.10（0.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1672.237.120.714.711.7N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.392.893.493.9
DBX-3266.727.616.812.510.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.592.493.293.5
DBX-6466.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DBX-12842.31.30.20.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A97.297.698.698.999.1
GBZ-1683.836.48.92.30.926.979.899.7100.0100.043.388.099.8100.0100.091.692.191.291.692.6
GBZ-3282.735.59.53.01.126.683.299.7100.0100.045.890.099.8100.0100.090.391.791.492.592.6
GBZ-6479.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GBZ-12879.232.18.72.61.226.168.599.2100.0100.041.279.199.4100.0100.089.791.591.491.492.5
GBY-1689.234.63.50.20.044.888.199.8100.0100.055.991.799.7100.0100.089.492.591.090.636.9
GBY-3286.725.81.60.10.045.191.2100.0100.0100.058.694.0100.0100.0100.090.991.891.1 68.631.1
GBY-6484.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
GBY-12882.123.71.90.00.040.483.099.7100.0100.051.787.599.6100.0100.091.291.289.882.844.4
", + "bbox": [ + 159, + 405, + 815, + 627 + ], + "page_idx": 29 + } +] \ No newline at end of file diff --git a/parse/train/HygUOoC5KX/HygUOoC5KX_middle.json b/parse/train/HygUOoC5KX/HygUOoC5KX_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..5f653048b3c18e70224db39ec588e490e1dce062 --- /dev/null +++ b/parse/train/HygUOoC5KX/HygUOoC5KX_middle.json @@ -0,0 +1,59049 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 77, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 77, + 505, + 98 + ], + "score": 1.0, + "content": "ARE GENERATIVE CLASSIFIERS MORE ROBUST TO", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 299, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 299, + 116 + ], + "score": 1.0, + "content": "ADVERSARIAL ATTACKS?", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 136, + 244, + 156 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 145, + 245, + 157 + ], + "spans": [ + { + "bbox": [ + 111, + 145, + 245, + 157 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 185, + 333, + 198 + ], + "lines": [ + { + "bbox": [ + 277, + 185, + 335, + 199 + ], + "spans": [ + { + "bbox": [ + 277, + 185, + 335, + 199 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 211, + 468, + 321 + ], + "lines": [ + { + "bbox": [ + 142, + 212, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 142, + 212, + 469, + 223 + ], + "score": 1.0, + "content": "There is a rising interest in studying the robustness of deep neural network classi-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 470, + 235 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 470, + 235 + ], + "score": 1.0, + "content": "fiers against adversaries, with both advanced attack and defence techniques being", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 233, + 469, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 245 + ], + "score": 1.0, + "content": "actively developed. However, most recent work focuses on discriminative classi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 469, + 257 + ], + "score": 1.0, + "content": "fiers, which only model the conditional distribution of the labels given the inputs.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 140, + 254, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 140, + 254, + 469, + 268 + ], + "score": 1.0, + "content": "In this paper, we propose and investigate the deep Bayes classifier, which im-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 470, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 278 + ], + "score": 1.0, + "content": "proves classical naive Bayes with conditional deep generative models. We further", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "score": 1.0, + "content": "develop detection methods for adversarial examples, which reject inputs with low", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 301 + ], + "score": 1.0, + "content": "likelihood under the generative model. Experimental results suggest that deep", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 299, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 142, + 299, + 469, + 311 + ], + "score": 1.0, + "content": "Bayes classifiers are more robust than deep discriminative classifiers, and that the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "score": 1.0, + "content": "proposed detection methods are effective against many recently proposed attacks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 206, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 208, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 208, + 359 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "Deep neural networks have been shown to be vulnerable to adversarial examples (Szegedy et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "2013; Goodfellow et al., 2014). The latest attack techniques can easily fool a deep neural network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "score": 1.0, + "content": "with imperceptible perturbations (Goodfellow et al., 2014; Papernot et al., 2016b; Carlini & Wagner,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 400, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 415 + ], + "score": 1.0, + "content": "2017a; Kurakin et al., 2016; Madry et al., 2018; Chen et al., 2017a), even in the black-box case,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "where the attacker does not have access to the network’s weights (Papernot et al., 2017b; Chen", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "et al., 2017b; Alzantot et al., 2018a). Adversarial attacks are serious security threats to machine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "learning systems, threatening applications beyond image classification (Carlini & Wagner, 2018;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 201, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 201, + 458 + ], + "score": 1.0, + "content": "Alzantot et al., 2018b).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "To address this outstanding security issue, researchers have proposed defence mechanisms against", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 504, + 487 + ], + "score": 1.0, + "content": "adversarial attacks. Adversarial training, which augments the training data with adversarially per-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "turbed inputs, has shown moderate success at defending against recently proposed attack techniques", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "(Szegedy et al., 2013; Goodfellow et al., 2014; Tramer et al., 2017; Madry et al., 2018). In addition, `", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "recent advances in Bayesian neural networks have demonstrated that uncertainty estimates can be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "used to detect adversarial attacks (Li & Gal, 2017; Feinman et al., 2017; Louizos & Welling, 2017;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "Smith & Gal, 2018). Another notable category of defence techniques involves the usage of genera-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "tive models. For example, Gu & Rigazio (2014) used an auto-encoder to denoise the inputs before", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "feeding them to the classifier. 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(2018) has challenged this observation with a synthetic example.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Surprisingly, much less recent work has investigated the robustness of generative classifiers (Ng &", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Jordan, 2002) against adversarial attacks for multi-class classification, where such classifiers explic-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "itly model the conditional distribution of the inputs given labels. 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There-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "fore, generative classifiers should be robust to many recently proposed adversarial attacks if the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "“off-manifold” conjecture holds for many real-world applications. Unfortunately, many generative", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "classifiers in popular use, including naive Bayes and linear discriminant analysis (Fisher, 1936),", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "perform poorly on natural image classification tasks, making it difficult to verify the “off-manifold”", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 465, + 722 + ], + "score": 1.0, + "content": "conjecture and the robustness of generative classifiers with these tools. 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However, most recent work focuses on discriminative classi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 469, + 257 + ], + "score": 1.0, + "content": "fiers, which only model the conditional distribution of the labels given the inputs.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 140, + 254, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 140, + 254, + 469, + 268 + ], + "score": 1.0, + "content": "In this paper, we propose and investigate the deep Bayes classifier, which im-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 470, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 278 + ], + "score": 1.0, + "content": "proves classical naive Bayes with conditional deep generative models. We further", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "score": 1.0, + "content": "develop detection methods for adversarial examples, which reject inputs with low", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 301 + ], + "score": 1.0, + "content": "likelihood under the generative model. Experimental results suggest that deep", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 299, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 142, + 299, + 469, + 311 + ], + "score": 1.0, + "content": "Bayes classifiers are more robust than deep discriminative classifiers, and that the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "score": 1.0, + "content": "proposed detection methods are effective against many recently proposed attacks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 140, + 212, + 470, + 323 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 206, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 208, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 208, + 359 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "Deep neural networks have been shown to be vulnerable to adversarial examples (Szegedy et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "2013; Goodfellow et al., 2014). The latest attack techniques can easily fool a deep neural network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "score": 1.0, + "content": "with imperceptible perturbations (Goodfellow et al., 2014; Papernot et al., 2016b; Carlini & Wagner,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 400, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 415 + ], + "score": 1.0, + "content": "2017a; Kurakin et al., 2016; Madry et al., 2018; Chen et al., 2017a), even in the black-box case,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "where the attacker does not have access to the network’s weights (Papernot et al., 2017b; Chen", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "et al., 2017b; Alzantot et al., 2018a). Adversarial attacks are serious security threats to machine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "learning systems, threatening applications beyond image classification (Carlini & Wagner, 2018;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 201, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 201, + 458 + ], + "score": 1.0, + "content": "Alzantot et al., 2018b).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 369, + 506, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "To address this outstanding security issue, researchers have proposed defence mechanisms against", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 504, + 487 + ], + "score": 1.0, + "content": "adversarial attacks. Adversarial training, which augments the training data with adversarially per-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "turbed inputs, has shown moderate success at defending against recently proposed attack techniques", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "(Szegedy et al., 2013; Goodfellow et al., 2014; Tramer et al., 2017; Madry et al., 2018). In addition, `", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "recent advances in Bayesian neural networks have demonstrated that uncertainty estimates can be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "used to detect adversarial attacks (Li & Gal, 2017; Feinman et al., 2017; Louizos & Welling, 2017;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "Smith & Gal, 2018). Another notable category of defence techniques involves the usage of genera-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "tive models. For example, Gu & Rigazio (2014) used an auto-encoder to denoise the inputs before", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "feeding them to the classifier. This denoising approach has been extensively investigated, and the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 559, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 575 + ], + "score": 1.0, + "content": "“denoisers” in usage include generative adversarial networks (Samangouei et al., 2018), PixelCNNs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "(Song et al., 2018) and denoising auto-encoders (Kurakin et al., 2018). These developments rely on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "the “off-manifold” conjecture, that is, that adversarial examples are far away from the data manifold,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 455, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 455, + 607 + ], + "score": 1.0, + "content": "although Gilmer et al. (2018) has challenged this observation with a synthetic example.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 462, + 506, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Surprisingly, much less recent work has investigated the robustness of generative classifiers (Ng &", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Jordan, 2002) against adversarial attacks for multi-class classification, where such classifiers explic-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "itly model the conditional distribution of the inputs given labels. 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There-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "fore, generative classifiers should be robust to many recently proposed adversarial attacks if the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "“off-manifold” conjecture holds for many real-world applications. Unfortunately, many generative", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "classifiers in popular use, including naive Bayes and linear discriminant analysis (Fisher, 1936),", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "perform poorly on natural image classification tasks, making it difficult to verify the “off-manifold”", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 465, + 722 + ], + "score": 1.0, + "content": "conjecture and the robustness of generative classifiers with these tools. In recent work,", + "type": "text" + }, + { + "bbox": [ + 465, + 710, + 473, + 720 + ], + "score": 0.45, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "-nearest", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "neighbors (Cover & Hart, 1967), a method which shares many similarities with generative classi-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "fiers, has been significantly improved in handling natural images by leveraging deep feature repre-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "sentations (Papernot & McDaniel, 2018). To the best of our knowledge, an approach which targets", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 402, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 402, + 117 + ], + "score": 1.0, + "content": "a similar contribution has not yet been proposed for generative classifiers.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 611, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "fiers, has been significantly improved in handling natural images by leveraging deep feature repre-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "sentations (Papernot & McDaniel, 2018). To the best of our knowledge, an approach which targets", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 402, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 402, + 117 + ], + "score": 1.0, + "content": "a similar contribution has not yet been proposed for generative classifiers.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 108, + 121, + 503, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 504, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 504, + 134 + ], + "score": 1.0, + "content": "Are generative classifiers more robust to recently proposed adversarial attack techniques? To an-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "swer this, we improve the naive Bayes algorithm by using conditional deep generative models, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "evaluate the conjecture on the proposed generative classifier. In summary, our contributions include:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 132, + 160, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 132, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 132, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "• We propose deep Bayes as an extension of naive Bayes, in which the conditional distribu-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 142, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "tion of an input, given a label, is parameterised by a deep latent variable model (LVM).", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 142, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "We learn the LVM with the variational auto-encoder algorithm (Kingma & Welling, 2013;", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 193, + 461, + 207 + ], + "spans": [ + { + "bbox": [ + 141, + 193, + 461, + 207 + ], + "score": 1.0, + "content": "Rezende et al., 2014), and approximate Bayes’ rule using importance sampling.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 138, + 207, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 138, + 207, + 506, + 219 + ], + "score": 1.0, + "content": "We propose three detection methods for adversarial perturbations. The first two use the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 218, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 141, + 218, + 506, + 230 + ], + "score": 1.0, + "content": "learned generative model as a proxy of the data manifold, and reject inputs that are far", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 227, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 141, + 227, + 506, + 242 + ], + "score": 1.0, + "content": "away from it. The third computes statistics for the classifier’s output probability vector on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 240, + 430, + 251 + ], + "spans": [ + { + "bbox": [ + 142, + 240, + 430, + 251 + ], + "score": 1.0, + "content": "training data, and rejects inputs that lead to under-confident predictions.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 139, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 139, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "We evaluate the robustness of the proposed generative classifier on the MNIST multi-class", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 142, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "and CIFAR binary classification tasks. 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We refer", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 664, + 455, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 455, + 676 + ], + "score": 1.0, + "content": "to such generative classifiers that use deep generative models as deep Bayes classifiers.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 364, + 694 + ], + "score": 1.0, + "content": "In this paper, we consider a deep latent variable model (LVM)", + "type": "text" + }, + { + "bbox": [ + 365, + 681, + 406, + 693 + ], + "score": 0.9, + "content": "p ( { \\pmb x } , { \\pmb z } , { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 680, + 506, + 694 + ], + "score": 1.0, + "content": ", which will be used for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 102, + 691, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 102, + 691, + 454, + 712 + ], + "score": 1.0, + "content": "classification. 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To an-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "swer this, we improve the naive Bayes algorithm by using conditional deep generative models, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "evaluate the conjecture on the proposed generative classifier. 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In this case, the conditional distribution is p(x|y) = R p(x,z,y)dzR p(x,z,y)dzdx .", + "type": "text" + }, + { + "bbox": [ + 449, + 693, + 506, + 708 + ], + "score": 1.0, + "content": "Importantly,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 259, + 723 + ], + "score": 1.0, + "content": "this leads to a conditional distribution", + "type": "text" + }, + { + "bbox": [ + 259, + 709, + 288, + 722 + ], + "score": 0.94, + "content": "p ( { \\pmb x } | { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 708, + 422, + 723 + ], + "score": 1.0, + "content": "that is not factorised (even when", + "type": "text" + }, + { + "bbox": [ + 422, + 710, + 461, + 722 + ], + "score": 0.91, + "content": "p ( \\pmb { x } | \\pmb { z } , \\pmb { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "is), which", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "is much more powerful than naive Bayes. 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The scattered dots are training data points, with dif-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 312, + 163 + ], + "score": 1.0, + "content": "ferent classes shown in different colours (red for", + "type": "text" + }, + { + "bbox": [ + 312, + 151, + 341, + 161 + ], + "score": 0.9, + "content": "c = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 151, + 397, + 163 + ], + "score": 1.0, + "content": "and blue for", + "type": "text" + }, + { + "bbox": [ + 398, + 151, + 426, + 161 + ], + "score": 0.88, + "content": "c = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "). 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For each", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 140, + 218, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 140, + 218, + 164, + 233 + ], + "score": 1.0, + "content": "class", + "type": "text" + }, + { + "bbox": [ + 165, + 221, + 170, + 229 + ], + "score": 0.67, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 218, + 407, + 233 + ], + "score": 1.0, + "content": ", we first collect the mean classification probability vector", + "type": "text" + }, + { + "bbox": [ + 408, + 219, + 501, + 232 + ], + "score": 0.92, + "content": "\\pmb { p } _ { c } = \\mathbb { E } _ { ( \\pmb { x } , \\pmb { y } _ { c } ) \\in \\mathcal { D } } [ \\pmb { p } ( \\pmb { x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 218, + 506, + 233 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 243, + 244 + ], + "score": 1.0, + "content": "then compute the mean", + "type": "text" + }, + { + "bbox": [ + 243, + 231, + 253, + 243 + ], + "score": 0.88, + "content": "\\bar { d } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 231, + 352, + 244 + ], + "score": 1.0, + "content": "and standard deviation", + "type": "text" + }, + { + "bbox": [ + 352, + 234, + 363, + 243 + ], + "score": 0.84, + "content": "\\sigma _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "on a selected divergence/distance", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 243, + 504, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 179, + 255 + ], + "score": 1.0, + "content": "measure", + "type": "text" + }, + { + "bbox": [ + 180, + 243, + 229, + 255 + ], + "score": 0.91, + "content": "\\bar { \\mathrm { D } } [ p _ { c } | | p ( { \\pmb x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 243, + 262, + 255 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 262, + 243, + 316, + 254 + ], + "score": 0.89, + "content": "( \\pmb { x } , \\pmb { y } _ { c } ) \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 243, + 377, + 255 + ], + "score": 1.0, + "content": ". A test input", + "type": "text" + }, + { + "bbox": [ + 377, + 243, + 389, + 253 + ], + "score": 0.86, + "content": "\\pmb { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 243, + 480, + 255 + ], + "score": 1.0, + "content": "with prediction label", + "type": "text" + }, + { + "bbox": [ + 481, + 243, + 504, + 253 + ], + "score": 0.88, + "content": "c ^ { * } =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 178, + 265 + ], + "score": 1.0, + "content": "arg max", + "type": "text" + }, + { + "bbox": [ + 178, + 254, + 204, + 266 + ], + "score": 0.91, + "content": "\\pmb { p } ( \\pmb { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 254, + 257, + 265 + ], + "score": 1.0, + "content": "is rejected if", + "type": "text" + }, + { + "bbox": [ + 258, + 253, + 376, + 266 + ], + "score": 0.91, + "content": "\\mathrm { D } [ p _ { c ^ { * } } | | p ( { \\pmb x } ^ { * } ) ] > \\bar { d } _ { c ^ { * } } + \\bar { \\alpha \\sigma } _ { c ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 254, + 473, + 265 + ], + "score": 1.0, + "content": ". Therefore, an example", + "type": "text" + }, + { + "bbox": [ + 473, + 254, + 486, + 264 + ], + "score": 0.87, + "content": "\\pmb { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 254, + 505, + 265 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 264, + 477, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 264, + 477, + 277 + ], + "score": 1.0, + "content": "be rejected if the classifier is over-confident or under-confident (ambiguous inputs).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 143, + 277, + 503, + 300 + ], + "lines": [ + { + "bbox": [ + 142, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 142, + 276, + 169, + 290 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 169, + 277, + 179, + 288 + ], + "score": 0.25, + "content": "\\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 276, + 447, + 290 + ], + "score": 1.0, + "content": "is selected as the KL-divergence, we call this detection method", + "type": "text" + }, + { + "bbox": [ + 447, + 278, + 462, + 288 + ], + "score": 0.52, + "content": "K L", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "detection.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 288, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 288, + 470, + 301 + ], + "score": 1.0, + "content": "Other divergence/distance measures such as total variation (TV) can also be used.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 108, + 309, + 503, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 504, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 504, + 322 + ], + "score": 1.0, + "content": "For better intuition, we visualise the detection mechanisms in Figure 2 with a synthetic “two rings”", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 453, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 320, + 333 + ], + "score": 1.0, + "content": "binary classification example. In this case we sample", + "type": "text" + }, + { + "bbox": [ + 320, + 321, + 359, + 331 + ], + "score": 0.9, + "content": "2 \\times 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 320, + 453, + 333 + ], + "score": 1.0, + "content": "training data points as:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 111, + 350 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 112, + 336, + 505, + 349 + ], + "score": 0.76, + "content": "\\begin{array} { r l } & { x , y ) \\sim p _ { D } \\Leftrightarrow y \\sim \\mathrm { B e r n } ( 0 . 5 ) , \\theta \\sim \\mathrm { U n i f o r m } ( 0 , 2 \\pi ) , x | y \\sim \\mathcal { N } ( x ; c _ { y } + r _ { y } [ c o s ( \\theta ) , s i n ( \\theta ) ] ^ { \\mathrm { T } } , \\sigma ^ { 2 } { \\bf I } ) . } \\end{array}", + "type": "inline_equation", + "image_path": "413f78cca08b70e46a942fcda790ad76ea84a234caec23c9601ffa791449849c.jpg" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 245, + 368 + ], + "score": 1.0, + "content": "We consider a generative classifier", + "type": "text" + }, + { + "bbox": [ + 245, + 353, + 456, + 366 + ], + "score": 0.81, + "content": "p ( \\pmb { x } , \\pmb { y } ) = p ( \\pmb { x } | \\pmb { y } ) p _ { \\mathscr { D } } ( \\pmb { y } ) = \\mathcal { N } ( \\pmb { x } ; \\pmb { \\mu _ { y } } , \\sigma ^ { 2 } \\mathbf { I } ) \\mathbf { B e r n } ( 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 351, + 479, + 368 + ], + "score": 1.0, + "content": ") with", + "type": "text" + }, + { + "bbox": [ + 479, + 354, + 505, + 366 + ], + "score": 0.87, + "content": "\\mu _ { y } =", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 364, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 233, + 378 + ], + "score": 0.86, + "content": "\\operatorname { p r o j } ( x , \\operatorname { r i n g } _ { y } ) = \\arg \\operatorname* { m i n } _ { \\hat { x } \\in \\mathbb { R } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 364, + 236, + 381 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 237, + 366, + 330, + 379 + ], + "score": 0.89, + "content": "| | \\hat { \\mathbf { x } } - \\pmb { c } _ { y } | | _ { 2 } = { r } _ { y } \\ | | \\pmb { x } - \\hat { \\pmb { x } } | | _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 364, + 357, + 381 + ], + "score": 1.0, + "content": ". The", + "type": "text" + }, + { + "bbox": [ + 357, + 366, + 363, + 375 + ], + "score": 0.76, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 364, + 507, + 381 + ], + "score": 1.0, + "content": "thresholds are selected to achieve", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 126, + 389 + ], + "score": 0.87, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "false positive rates on training data. From the visualisations we see that inputs that are far away", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "from the model manifold are rejected by marginal/logit detection. At the same time, logit detection", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "rejects data points that are not on the manifold of the given class. KL/TV detection does not construct", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "manifold-aware acceptance regions, which is as expected since the proposed divergence detection", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "method does not require the classifier to be generative. However, both detection methods have some", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "success in rejecting uncertain predictions, especially for TV, which also rejects ambiguous inputs", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "(see the ring-cross regions in the last two plots). Combining all three methods, we see that the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "rejected inputs are either far away from the manifold, or are ambiguous. Again we emphasise that a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "suitable generative model is required to make the detection methods work in practice, since the data", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 308, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 154, + 489 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 477, + 192, + 488 + ], + "score": 0.94, + "content": "p _ { \\mathcal { D } } ( \\pmb { x } , \\pmb { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 475, + 273, + 489 + ], + "score": 1.0, + "content": "is approximated by", + "type": "text" + }, + { + "bbox": [ + 273, + 476, + 304, + 488 + ], + "score": 0.93, + "content": "p ( { \\pmb x } , { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 475, + 308, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 493, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "Detection methods using logit values have also been proposed in Li & Gal (e.g. 2017); Feinman et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "(e.g. 2017). However it is unclear whether the logits values in discriminative classifiers have a clear", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "semantic meaning, while the logit values in deep Bayes represent the log probability of generating", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 526, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 145, + 540 + ], + "score": 1.0, + "content": "the input", + "type": "text" + }, + { + "bbox": [ + 145, + 528, + 153, + 537 + ], + "score": 0.69, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 526, + 239, + 540 + ], + "score": 1.0, + "content": "given the class label", + "type": "text" + }, + { + "bbox": [ + 239, + 528, + 271, + 538 + ], + "score": 0.89, + "content": "{ \\mathbf { \\nabla } } y = y _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 526, + 504, + 540 + ], + "score": 1.0, + "content": ". Our approach is also distinct from previous “denoising”", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "approaches (e.g. Song et al., 2018; Samangouei et al., 2018; Kurakin et al., 2018) that require training", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "an additional generative model separately. The deep Bayes classifiers share the same generative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "model as the detection methods, meaning that detected adversarial examples are indeed far away", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "from the classifier’s manifold (which is also an approximation to the data manifold). Therefore the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "claim “detection neural networks can be bypassed” (Carlini & Wagner, 2017a; Athalye et al., 2018)", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "does not directly transfer to our approach, and we can use the marginal and logit detection methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 603, + 381, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 381, + 615 + ], + "score": 1.0, + "content": "to directly verify the “off-manifold” adversarial example conjecture.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 631, + 200, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 202, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 202, + 645 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 434, + 668 + ], + "score": 1.0, + "content": "We carry out a number of tests on the proposed deep Bayes classifiers (3), where", + "type": "text" + }, + { + "bbox": [ + 434, + 655, + 460, + 667 + ], + "score": 0.92, + "content": "q ( z | \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 654, + 478, + 668 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 654, + 504, + 667 + ], + "score": 0.91, + "content": "p ( z | \\cdot )", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 335, + 678 + ], + "score": 1.0, + "content": "are factorised Gaussians, and the conditional probability", + "type": "text" + }, + { + "bbox": [ + 335, + 666, + 361, + 678 + ], + "score": 0.92, + "content": "p ( { \\pmb x } | \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", if required, is parameterised by an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 116, + 688 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "loss. Besides the LVM-based classifiers, we further train a Bayesian neural network (BNN) with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Bernoulli dropout (dropout rate 0.3), as it has been shown in Li & Gal (2017) and Feinman et al.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "(2017) that BNNs are more robust than their deterministic counterparts. The constructed BNN has", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 118, + 720 + ], + "score": 0.73, + "content": "2 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "more channels than LVM-based classifiers, making the comparison slightly “unfair”, as the BNN", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 455, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 246, + 733 + ], + "score": 1.0, + "content": "layers have more capacity. We use", + "type": "text" + }, + { + "bbox": [ + 246, + 721, + 280, + 731 + ], + "score": 0.89, + "content": "K = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 721, + 455, + 733 + ], + "score": 1.0, + "content": "Monte Carlo samples for all the classifiers.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 66, + 502, + 128 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 66, + 502, + 128 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 66, + 502, + 128 + ], + "spans": [ + { + "bbox": [ + 108, + 66, + 502, + 128 + ], + "score": 0.957, + "type": "image", + "image_path": "63ac3a599fb3fd7f9b0ceeb289e5088441fbf7ba011a0d3f9a13300e966b86cc.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 66, + 502, + 86.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 86.66666666666667, + 502, + 107.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 107.33333333333334, + 502, + 128.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 140, + 505, + 184 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 140, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 504, + 151 + ], + "score": 1.0, + "content": "Figure 2: Visualising detection mechanisms. The scattered dots are training data points, with dif-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 312, + 163 + ], + "score": 1.0, + "content": "ferent classes shown in different colours (red for", + "type": "text" + }, + { + "bbox": [ + 312, + 151, + 341, + 161 + ], + "score": 0.9, + "content": "c = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 151, + 397, + 163 + ], + "score": 1.0, + "content": "and blue for", + "type": "text" + }, + { + "bbox": [ + 398, + 151, + 426, + 161 + ], + "score": 0.88, + "content": "c = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "). Same labels are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 369, + 174 + ], + "score": 1.0, + "content": "manually assigned for inputs when the detection method requires", + "type": "text" + }, + { + "bbox": [ + 370, + 163, + 377, + 173 + ], + "score": 0.69, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 162, + 505, + 174 + ], + "score": 1.0, + "content": ". Decision regions are shown in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 173, + 442, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 442, + 185 + ], + "score": 1.0, + "content": "the corresponding colours. Input points in the shaded area are rejected by detection.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 130, + 197, + 486, + 208 + ], + "lines": [ + { + "bbox": [ + 131, + 196, + 487, + 210 + ], + "spans": [ + { + "bbox": [ + 131, + 196, + 487, + 210 + ], + "score": 1.0, + "content": "• Divergence detection: rejecting inputs with over- and/or under-confident predictions.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 131, + 196, + 487, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 136, + 209, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 142, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 142, + 208, + 174, + 220 + ], + "score": 1.0, + "content": "Denote", + "type": "text" + }, + { + "bbox": [ + 174, + 208, + 196, + 220 + ], + "score": 0.92, + "content": "{ \\pmb p } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 208, + 217, + 220 + ], + "score": 1.0, + "content": "as a", + "type": "text" + }, + { + "bbox": [ + 217, + 208, + 226, + 218 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "-dimensional probability vector outputted by the classifier. For each", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 140, + 218, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 140, + 218, + 164, + 233 + ], + "score": 1.0, + "content": "class", + "type": "text" + }, + { + "bbox": [ + 165, + 221, + 170, + 229 + ], + "score": 0.67, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 218, + 407, + 233 + ], + "score": 1.0, + "content": ", we first collect the mean classification probability vector", + "type": "text" + }, + { + "bbox": [ + 408, + 219, + 501, + 232 + ], + "score": 0.92, + "content": "\\pmb { p } _ { c } = \\mathbb { E } _ { ( \\pmb { x } , \\pmb { y } _ { c } ) \\in \\mathcal { D } } [ \\pmb { p } ( \\pmb { x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 218, + 506, + 233 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 243, + 244 + ], + "score": 1.0, + "content": "then compute the mean", + "type": "text" + }, + { + "bbox": [ + 243, + 231, + 253, + 243 + ], + "score": 0.88, + "content": "\\bar { d } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 231, + 352, + 244 + ], + "score": 1.0, + "content": "and standard deviation", + "type": "text" + }, + { + "bbox": [ + 352, + 234, + 363, + 243 + ], + "score": 0.84, + "content": "\\sigma _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "on a selected divergence/distance", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 243, + 504, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 179, + 255 + ], + "score": 1.0, + "content": "measure", + "type": "text" + }, + { + "bbox": [ + 180, + 243, + 229, + 255 + ], + "score": 0.91, + "content": "\\bar { \\mathrm { D } } [ p _ { c } | | p ( { \\pmb x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 243, + 262, + 255 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 262, + 243, + 316, + 254 + ], + "score": 0.89, + "content": "( \\pmb { x } , \\pmb { y } _ { c } ) \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 243, + 377, + 255 + ], + "score": 1.0, + "content": ". A test input", + "type": "text" + }, + { + "bbox": [ + 377, + 243, + 389, + 253 + ], + "score": 0.86, + "content": "\\pmb { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 243, + 480, + 255 + ], + "score": 1.0, + "content": "with prediction label", + "type": "text" + }, + { + "bbox": [ + 481, + 243, + 504, + 253 + ], + "score": 0.88, + "content": "c ^ { * } =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 178, + 265 + ], + "score": 1.0, + "content": "arg max", + "type": "text" + }, + { + "bbox": [ + 178, + 254, + 204, + 266 + ], + "score": 0.91, + "content": "\\pmb { p } ( \\pmb { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 254, + 257, + 265 + ], + "score": 1.0, + "content": "is rejected if", + "type": "text" + }, + { + "bbox": [ + 258, + 253, + 376, + 266 + ], + "score": 0.91, + "content": "\\mathrm { D } [ p _ { c ^ { * } } | | p ( { \\pmb x } ^ { * } ) ] > \\bar { d } _ { c ^ { * } } + \\bar { \\alpha \\sigma } _ { c ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 254, + 473, + 265 + ], + "score": 1.0, + "content": ". Therefore, an example", + "type": "text" + }, + { + "bbox": [ + 473, + 254, + 486, + 264 + ], + "score": 0.87, + "content": "\\pmb { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 254, + 505, + 265 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 264, + 477, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 264, + 477, + 277 + ], + "score": 1.0, + "content": "be rejected if the classifier is over-confident or under-confident (ambiguous inputs).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 140, + 208, + 506, + 277 + ] + }, + { + "type": "list", + "bbox": [ + 143, + 277, + 503, + 300 + ], + "lines": [ + { + "bbox": [ + 142, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 142, + 276, + 169, + 290 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 169, + 277, + 179, + 288 + ], + "score": 0.25, + "content": "\\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 276, + 447, + 290 + ], + "score": 1.0, + "content": "is selected as the KL-divergence, we call this detection method", + "type": "text" + }, + { + "bbox": [ + 447, + 278, + 462, + 288 + ], + "score": 0.52, + "content": "K L", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "detection.", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 142, + 288, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 288, + 470, + 301 + ], + "score": 1.0, + "content": "Other divergence/distance measures such as total variation (TV) can also be used.", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 14.5, + "bbox_fs": [ + 142, + 276, + 505, + 301 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 309, + 503, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 504, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 504, + 322 + ], + "score": 1.0, + "content": "For better intuition, we visualise the detection mechanisms in Figure 2 with a synthetic “two rings”", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 453, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 320, + 333 + ], + "score": 1.0, + "content": "binary classification example. In this case we sample", + "type": "text" + }, + { + "bbox": [ + 320, + 321, + 359, + 331 + ], + "score": 0.9, + "content": "2 \\times 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 320, + 453, + 333 + ], + "score": 1.0, + "content": "training data points as:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 308, + 504, + 333 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 111, + 350 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 112, + 336, + 505, + 349 + ], + "score": 0.76, + "content": "\\begin{array} { r l } & { x , y ) \\sim p _ { D } \\Leftrightarrow y \\sim \\mathrm { B e r n } ( 0 . 5 ) , \\theta \\sim \\mathrm { U n i f o r m } ( 0 , 2 \\pi ) , x | y \\sim \\mathcal { N } ( x ; c _ { y } + r _ { y } [ c o s ( \\theta ) , s i n ( \\theta ) ] ^ { \\mathrm { T } } , \\sigma ^ { 2 } { \\bf I } ) . } \\end{array}", + "type": "inline_equation", + "image_path": "413f78cca08b70e46a942fcda790ad76ea84a234caec23c9601ffa791449849c.jpg" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 245, + 368 + ], + "score": 1.0, + "content": "We consider a generative classifier", + "type": "text" + }, + { + "bbox": [ + 245, + 353, + 456, + 366 + ], + "score": 0.81, + "content": "p ( \\pmb { x } , \\pmb { y } ) = p ( \\pmb { x } | \\pmb { y } ) p _ { \\mathscr { D } } ( \\pmb { y } ) = \\mathcal { N } ( \\pmb { x } ; \\pmb { \\mu _ { y } } , \\sigma ^ { 2 } \\mathbf { I } ) \\mathbf { B e r n } ( 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 351, + 479, + 368 + ], + "score": 1.0, + "content": ") with", + "type": "text" + }, + { + "bbox": [ + 479, + 354, + 505, + 366 + ], + "score": 0.87, + "content": "\\mu _ { y } =", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 364, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 233, + 378 + ], + "score": 0.86, + "content": "\\operatorname { p r o j } ( x , \\operatorname { r i n g } _ { y } ) = \\arg \\operatorname* { m i n } _ { \\hat { x } \\in \\mathbb { R } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 364, + 236, + 381 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 237, + 366, + 330, + 379 + ], + "score": 0.89, + "content": "| | \\hat { \\mathbf { x } } - \\pmb { c } _ { y } | | _ { 2 } = { r } _ { y } \\ | | \\pmb { x } - \\hat { \\pmb { x } } | | _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 364, + 357, + 381 + ], + "score": 1.0, + "content": ". The", + "type": "text" + }, + { + "bbox": [ + 357, + 366, + 363, + 375 + ], + "score": 0.76, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 364, + 507, + 381 + ], + "score": 1.0, + "content": "thresholds are selected to achieve", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 126, + 389 + ], + "score": 0.87, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "false positive rates on training data. From the visualisations we see that inputs that are far away", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "from the model manifold are rejected by marginal/logit detection. At the same time, logit detection", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "rejects data points that are not on the manifold of the given class. KL/TV detection does not construct", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "manifold-aware acceptance regions, which is as expected since the proposed divergence detection", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "method does not require the classifier to be generative. However, both detection methods have some", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "success in rejecting uncertain predictions, especially for TV, which also rejects ambiguous inputs", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "(see the ring-cross regions in the last two plots). Combining all three methods, we see that the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "rejected inputs are either far away from the manifold, or are ambiguous. Again we emphasise that a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "suitable generative model is required to make the detection methods work in practice, since the data", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 308, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 154, + 489 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 477, + 192, + 488 + ], + "score": 0.94, + "content": "p _ { \\mathcal { D } } ( \\pmb { x } , \\pmb { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 475, + 273, + 489 + ], + "score": 1.0, + "content": "is approximated by", + "type": "text" + }, + { + "bbox": [ + 273, + 476, + 304, + 488 + ], + "score": 0.93, + "content": "p ( { \\pmb x } , { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 475, + 308, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 335, + 507, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 493, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "Detection methods using logit values have also been proposed in Li & Gal (e.g. 2017); Feinman et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "(e.g. 2017). However it is unclear whether the logits values in discriminative classifiers have a clear", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "semantic meaning, while the logit values in deep Bayes represent the log probability of generating", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 526, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 145, + 540 + ], + "score": 1.0, + "content": "the input", + "type": "text" + }, + { + "bbox": [ + 145, + 528, + 153, + 537 + ], + "score": 0.69, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 526, + 239, + 540 + ], + "score": 1.0, + "content": "given the class label", + "type": "text" + }, + { + "bbox": [ + 239, + 528, + 271, + 538 + ], + "score": 0.89, + "content": "{ \\mathbf { \\nabla } } y = y _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 526, + 504, + 540 + ], + "score": 1.0, + "content": ". Our approach is also distinct from previous “denoising”", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "approaches (e.g. Song et al., 2018; Samangouei et al., 2018; Kurakin et al., 2018) that require training", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "an additional generative model separately. The deep Bayes classifiers share the same generative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "model as the detection methods, meaning that detected adversarial examples are indeed far away", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "from the classifier’s manifold (which is also an approximation to the data manifold). Therefore the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "claim “detection neural networks can be bypassed” (Carlini & Wagner, 2017a; Athalye et al., 2018)", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "does not directly transfer to our approach, and we can use the marginal and logit detection methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 603, + 381, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 381, + 615 + ], + "score": 1.0, + "content": "to directly verify the “off-manifold” adversarial example conjecture.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 493, + 506, + 615 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 631, + 200, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 202, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 202, + 645 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 434, + 668 + ], + "score": 1.0, + "content": "We carry out a number of tests on the proposed deep Bayes classifiers (3), where", + "type": "text" + }, + { + "bbox": [ + 434, + 655, + 460, + 667 + ], + "score": 0.92, + "content": "q ( z | \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 654, + 478, + 668 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 654, + 504, + 667 + ], + "score": 0.91, + "content": "p ( z | \\cdot )", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 335, + 678 + ], + "score": 1.0, + "content": "are factorised Gaussians, and the conditional probability", + "type": "text" + }, + { + "bbox": [ + 335, + 666, + 361, + 678 + ], + "score": 0.92, + "content": "p ( { \\pmb x } | \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", if required, is parameterised by an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 116, + 688 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "loss. Besides the LVM-based classifiers, we further train a Bayesian neural network (BNN) with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Bernoulli dropout (dropout rate 0.3), as it has been shown in Li & Gal (2017) and Feinman et al.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "(2017) that BNNs are more robust than their deterministic counterparts. The constructed BNN has", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 118, + 720 + ], + "score": 0.73, + "content": "2 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "more channels than LVM-based classifiers, making the comparison slightly “unfair”, as the BNN", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 455, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 246, + 733 + ], + "score": 1.0, + "content": "layers have more capacity. We use", + "type": "text" + }, + { + "bbox": [ + 246, + 721, + 280, + 731 + ], + "score": 0.89, + "content": "K = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 721, + 455, + 733 + ], + "score": 1.0, + "content": "Monte Carlo samples for all the classifiers.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 654, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 64, + 498, + 234 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 64, + 498, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 64, + 498, + 234 + ], + "spans": [ + { + "bbox": [ + 112, + 64, + 498, + 234 + ], + "score": 0.97, + "type": "image", + "image_path": "00ae51d305686cd331a0a614d4028a57298f23ffcfe3de3c5e0e296f5308977e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 64, + 498, + 120.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 120.66666666666666, + 498, + 177.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 177.33333333333331, + 498, + 233.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 240, + 505, + 284 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 435, + 252 + ], + "score": 1.0, + "content": "Figure 3: Accuracies (column 1), detection rates (columns 2-4) and minimum", + "type": "text" + }, + { + "bbox": [ + 435, + 240, + 451, + 251 + ], + "score": 0.86, + "content": "\\ell _ { \\mathrm { i n f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "perturbation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 251, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 299, + 262 + ], + "score": 1.0, + "content": "(column 6) against white-box zero-knowledge", + "type": "text" + }, + { + "bbox": [ + 300, + 251, + 314, + 262 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 251, + 505, + 262 + ], + "score": 1.0, + "content": "attacks on MNIST. The higher the better. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 262, + 504, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 504, + 275 + ], + "score": 1.0, + "content": "second from right most column visualises crafted adversarial examples on an image of digit “7”,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 273, + 289, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 126, + 285 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 273, + 140, + 284 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 273, + 182, + 285 + ], + "score": 1.0, + "content": "distortion", + "type": "text" + }, + { + "bbox": [ + 182, + 275, + 189, + 282 + ], + "score": 0.52, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 273, + 289, + 285 + ], + "score": 1.0, + "content": "growing from 0.1 to 0.5.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 276, + 307 + ], + "score": 1.0, + "content": "The adversarial attacks in test include both", + "type": "text" + }, + { + "bbox": [ + 276, + 294, + 290, + 305 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 293, + 307, + 307 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 294, + 317, + 305 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "untargeted attacks from CleverHans 2.0 library", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "(Papernot et al., 2017a): fast gradient sign method (FGSM, Goodfellow et al., 2014), projected", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "score": 1.0, + "content": "gradient descent (PGD, Madry et al., 2018), momentum iterative attack (MIM, Dong et al., 2017)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 327, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 198, + 339 + ], + "score": 1.0, + "content": "and Carlini & Wagner", + "type": "text" + }, + { + "bbox": [ + 199, + 327, + 209, + 338 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 327, + 506, + 339 + ], + "score": 1.0, + "content": "(CW, Carlini & Wagner, 2017a). Two metrics are reported: accuracy of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "the classifier on crafted adversarial examples, and detection rate on adversarial examples that have", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 349, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 504, + 360 + ], + "score": 1.0, + "content": "successfully caused the classifier to misclassify. This detection rate is defined as the true positive", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 360, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 489, + 371 + ], + "score": 1.0, + "content": "(TP) rate of finding an adversarial example, and the detection threshold is selected to achieve a", + "type": "text" + }, + { + "bbox": [ + 489, + 360, + 504, + 370 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 370, + 270, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 270, + 383 + ], + "score": 1.0, + "content": "false positive rate on clean training data.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "The experiments are performed under various threat model settings. We further evaluate the trans-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "ferability of crafted adversarial examples across different classifiers in the same way as done in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "Papernot et al. (2016a). We only provide visualisations in the main text; full table results can be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 419, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 435 + ], + "score": 1.0, + "content": "found in the appendix. Readers are also referred to the appendix for further experiments, including", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 407, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 407, + 444 + ], + "score": 1.0, + "content": "a quantitative analysis of the bottleneck effect on robustness and detection.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 456, + 165, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 167, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 167, + 469 + ], + "score": 1.0, + "content": "4.1 MNIST", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "The first set of experiments evaluate the robustness of generative classifiers on MNIST. Here the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 442, + 499 + ], + "score": 1.0, + "content": "image pixel values are normalised to [0, 1], and the LVM-based classifiers have", + "type": "text" + }, + { + "bbox": [ + 442, + 487, + 501, + 500 + ], + "score": 0.91, + "content": "\\dim ( z ) ~ = ~ 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 488, + 505, + 499 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "We first perform white-box zero-knowledge attacks, i.e. the attacker can differentiate through the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 510, + 504, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 504, + 521 + ], + "score": 1.0, + "content": "classifier to craft adversarial examples, but he/she is not aware of the existence of the detector. Then", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "we perform white-box perfect-knowledge attacks, where the attacker can differentiate through both", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 386, + 544 + ], + "score": 1.0, + "content": "the classifier and the detector, and he/she knows the usage of random", + "type": "text" + }, + { + "bbox": [ + 386, + 533, + 394, + 542 + ], + "score": 0.68, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "samples by the VAE-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "classifiers (Biggio et al., 2013; Carlini & Wagner, 2017b). Lastly we consider grey-box and black-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 553, + 473, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 473, + 565 + ], + "score": 1.0, + "content": "box attacks, and evaluate the robustness of generative classifiers against transferred attacks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 261, + 590 + ], + "score": 1.0, + "content": "White-box attacks (zero-knowledge,", + "type": "text" + }, + { + "bbox": [ + 262, + 577, + 276, + 588 + ], + "score": 0.85, + "content": "\\ell _ { \\infty , }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 576, + 333, + 590 + ], + "score": 1.0, + "content": ") Results for", + "type": "text" + }, + { + "bbox": [ + 334, + 577, + 347, + 588 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "attacks are reported in Figure 3, and in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "general generative classifiers perform better in terms of victim accuracy and minimum perturbation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 597, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 487, + 612 + ], + "score": 1.0, + "content": "of the attacks.1 By contrast, DFX & DFZ are not robust to the weakest attack (FGSM) even when", + "type": "text" + }, + { + "bbox": [ + 488, + 600, + 505, + 609 + ], + "score": 0.83, + "content": "\\epsilon =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 608, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 505, + 623 + ], + "score": 1.0, + "content": "0.2 (where the adversarial examples are still visually close to the original digit “7”). Interestingly,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 275, + 633 + ], + "score": 1.0, + "content": "DBX is the most robust against FGSM &", + "type": "text" + }, + { + "bbox": [ + 276, + 620, + 302, + 631 + ], + "score": 0.52, + "content": "\\mathbf { M I M } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 619, + 505, + 633 + ], + "score": 1.0, + "content": ", which agrees with the preliminary tests in Alemi", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "et al. (2017). But DBX is less robust to PGD, and here GBZ is a clear winner. These results show", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 642, + 451, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 451, + 654 + ], + "score": 1.0, + "content": "that the bottleneck is sometimes beneficial for better robustness of MNIST classifiers.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 486, + 673 + ], + "score": 1.0, + "content": "For detection, generative classifiers have successfully detected the adversarial examples with", + "type": "text" + }, + { + "bbox": [ + 487, + 659, + 505, + 671 + ], + "score": 0.84, + "content": "\\epsilon \\geq", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "0.3, which is reasonable as the visual distortion is already significant. 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The higher the better. 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Two metrics are reported: accuracy of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "the classifier on crafted adversarial examples, and detection rate on adversarial examples that have", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 349, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 504, + 360 + ], + "score": 1.0, + "content": "successfully caused the classifier to misclassify. This detection rate is defined as the true positive", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 360, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 489, + 371 + ], + "score": 1.0, + "content": "(TP) rate of finding an adversarial example, and the detection threshold is selected to achieve a", + "type": "text" + }, + { + "bbox": [ + 489, + 360, + 504, + 370 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 370, + 270, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 270, + 383 + ], + "score": 1.0, + "content": "false positive rate on clean training data.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 293, + 506, + 383 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "The experiments are performed under various threat model settings. We further evaluate the trans-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "ferability of crafted adversarial examples across different classifiers in the same way as done in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "Papernot et al. (2016a). We only provide visualisations in the main text; full table results can be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 419, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 435 + ], + "score": 1.0, + "content": "found in the appendix. Readers are also referred to the appendix for further experiments, including", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 407, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 407, + 444 + ], + "score": 1.0, + "content": "a quantitative analysis of the bottleneck effect on robustness and detection.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 386, + 505, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 456, + 165, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 167, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 167, + 469 + ], + "score": 1.0, + "content": "4.1 MNIST", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "The first set of experiments evaluate the robustness of generative classifiers on MNIST. Here the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 442, + 499 + ], + "score": 1.0, + "content": "image pixel values are normalised to [0, 1], and the LVM-based classifiers have", + "type": "text" + }, + { + "bbox": [ + 442, + 487, + 501, + 500 + ], + "score": 0.91, + "content": "\\dim ( z ) ~ = ~ 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 488, + 505, + 499 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "We first perform white-box zero-knowledge attacks, i.e. the attacker can differentiate through the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 510, + 504, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 504, + 521 + ], + "score": 1.0, + "content": "classifier to craft adversarial examples, but he/she is not aware of the existence of the detector. Then", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "we perform white-box perfect-knowledge attacks, where the attacker can differentiate through both", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 386, + 544 + ], + "score": 1.0, + "content": "the classifier and the detector, and he/she knows the usage of random", + "type": "text" + }, + { + "bbox": [ + 386, + 533, + 394, + 542 + ], + "score": 0.68, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "samples by the VAE-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "classifiers (Biggio et al., 2013; Carlini & Wagner, 2017b). Lastly we consider grey-box and black-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 553, + 473, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 473, + 565 + ], + "score": 1.0, + "content": "box attacks, and evaluate the robustness of generative classifiers against transferred attacks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 476, + 505, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 261, + 590 + ], + "score": 1.0, + "content": "White-box attacks (zero-knowledge,", + "type": "text" + }, + { + "bbox": [ + 262, + 577, + 276, + 588 + ], + "score": 0.85, + "content": "\\ell _ { \\infty , }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 576, + 333, + 590 + ], + "score": 1.0, + "content": ") Results for", + "type": "text" + }, + { + "bbox": [ + 334, + 577, + 347, + 588 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "attacks are reported in Figure 3, and in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "general generative classifiers perform better in terms of victim accuracy and minimum perturbation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 597, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 487, + 612 + ], + "score": 1.0, + "content": "of the attacks.1 By contrast, DFX & DFZ are not robust to the weakest attack (FGSM) even when", + "type": "text" + }, + { + "bbox": [ + 488, + 600, + 505, + 609 + ], + "score": 0.83, + "content": "\\epsilon =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 608, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 505, + 623 + ], + "score": 1.0, + "content": "0.2 (where the adversarial examples are still visually close to the original digit “7”). Interestingly,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 275, + 633 + ], + "score": 1.0, + "content": "DBX is the most robust against FGSM &", + "type": "text" + }, + { + "bbox": [ + 276, + 620, + 302, + 631 + ], + "score": 0.52, + "content": "\\mathbf { M I M } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 619, + 505, + 633 + ], + "score": 1.0, + "content": ", which agrees with the preliminary tests in Alemi", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "et al. (2017). But DBX is less robust to PGD, and here GBZ is a clear winner. These results show", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 642, + 451, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 451, + 654 + ], + "score": 1.0, + "content": "that the bottleneck is sometimes beneficial for better robustness of MNIST classifiers.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 576, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 486, + 673 + ], + "score": 1.0, + "content": "For detection, generative classifiers have successfully detected the adversarial examples with", + "type": "text" + }, + { + "bbox": [ + 487, + 659, + 505, + 671 + ], + "score": 0.84, + "content": "\\epsilon \\geq", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "0.3, which is reasonable as the visual distortion is already significant. Importantly, generative clas-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 267, + 694 + ], + "score": 1.0, + "content": "sifier’s victim accuracy decreases as the", + "type": "text" + }, + { + "bbox": [ + 268, + 682, + 281, + 693 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 681, + 323, + 694 + ], + "score": 1.0, + "content": "distortion", + "type": "text" + }, + { + "bbox": [ + 324, + 684, + 329, + 691 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "increases, but at the same time the TP rates", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 356, + 225 + ], + "score": 1.0, + "content": "for marginal and logit detection also increase. Therefore these", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 357, + 214, + 371, + 225 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 371, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "attacks fail to find near-manifold", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "adversarial examples that fool both the classifier and the detection methods. 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For logit/KL detection, the detection statistics and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "thresholds are constructed accordingly. We label the three attacks against the marginal, logit and KL", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "score": 1.0, + "content": "based detection schemes ’-PKM’, ’-PKL’ and ’-PKK’ respectively. When we only have knowledge", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 548, + 490, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 132, + 561 + ], + "score": 1.0, + "content": "of the", + "type": "text" + }, + { + "bbox": [ + 132, + 549, + 142, + 558 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 548, + 414, + 561 + ], + "score": 1.0, + "content": "samples and not the detection method we call this attack ’-PK0’ (ie", + "type": "text" + }, + { + "bbox": [ + 414, + 549, + 449, + 559 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { d e t e c t } } { = } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 548, + 490, + 561 + ], + "score": 1.0, + "content": "in Eq. 4).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "Results are visualised in Figure 5. We see that although the attacker can reduce detection levels,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "score": 1.0, + "content": "this comes with the trade-off of increasing accuracy, suggesting that a perfect-knowledge adversary", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 587, + 364, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 364, + 600 + ], + "score": 1.0, + "content": "cannot break both the classifier and detector working in tandem.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Sanity checks on gradient masking Athalye et al. (2018) claimed that if a successful defence", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "against white-box attacks is due to gradient masking, then this defence is likely to be less effective", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "against grey-box/black-box attacks, as they do not differentiate through the victim classifier and", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "the defence mechanism (Papernot et al., 2017b). Therefore, we consider two transfer attacks based", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "on distilling the victim classifier using a “student” CNN which has no gradient masking. The two", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "attacks differ in their threat models: in the grey-box setting the attacker has access to the output", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "probability vectors of the classifiers on the training data, while in the black-box setting the attacker", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "has access to queried labels only. For the latter black-box setting, we follow Papernot et al. (2017b)", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "to train a substitute CNN using Jacobian-based dataset augmentation, and we refer to appendix B.1", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 348, + 722 + ], + "score": 1.0, + "content": "for the detailed algorithm. 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acc.(clean)acc. (adv)l2 dist.TP KL
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With the KL detec-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 247, + 392 + ], + "score": 1.0, + "content": "tion method, all classifiers achieve", + "type": "text" + }, + { + "bbox": [ + 248, + 380, + 279, + 390 + ], + "score": 0.89, + "content": "> 9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "detection rates, which is as expected as the default CW", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "attack configuration, by construction, generates adversarial examples that lead to minimal difference", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 402, + 407, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 407, + 414 + ], + "score": 1.0, + "content": "between the logit values of the most and the second most probable classes.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 280, + 506, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 275, + 439 + ], + "score": 1.0, + "content": "White-box attack (perfect-knowledge,", + "type": "text" + }, + { + "bbox": [ + 276, + 426, + 289, + 437 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 424, + 450, + 439 + ], + "score": 1.0, + "content": ") The PGD-based perfect-knowledge", + "type": "text" + }, + { + "bbox": [ + 451, + 426, + 465, + 437 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 424, + 506, + 439 + ], + "score": 1.0, + "content": "attack is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "designed following (Carlini & Wagner, 2017b): we construct an (approximate) Bayes classifier", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 442, + 475, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 141, + 460 + ], + "score": 0.92, + "content": "p _ { k } ( \\pmb { y } | \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 442, + 274, + 465 + ], + "score": 1.0, + "content": "using (3) for each set of samples", + "type": "text" + }, + { + "bbox": [ + 275, + 447, + 310, + 460 + ], + "score": 0.92, + "content": "\\{ z _ { c } ^ { k } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 442, + 475, + 465 + ], + "score": 1.0, + "content": ", and minimize the following with PGD:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 424, + 506, + 465 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 464, + 437, + 499 + ], + "lines": [ + { + "bbox": [ + 173, + 464, + 437, + 499 + ], + "spans": [ + { + "bbox": [ + 173, + 464, + 437, + 499 + ], + "score": 0.94, + "content": "\\mathcal { L } ( \\eta ) = \\sum _ { k = 1 } ^ { K } \\log p _ { k } ( \\boldsymbol { y } | \\boldsymbol { x } + \\eta ) + \\lambda _ { \\mathrm { d e t e c t } } \\operatorname* { m a x } ( 0 , \\Phi ( \\boldsymbol { x } + \\eta , \\boldsymbol { y } ) - \\delta ) .", + "type": "interline_equation", + "image_path": "b88daad51d04d59a32e86b673f8a35394d357ad5eed2fba0b6ba30c89e3ba829.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 173, + 464, + 437, + 475.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 173, + 475.6666666666667, + 437, + 487.33333333333337 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 173, + 487.33333333333337, + 437, + 499.00000000000006 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 199, + 517 + ], + "score": 1.0, + "content": "The detection statistic", + "type": "text" + }, + { + "bbox": [ + 199, + 504, + 252, + 516 + ], + "score": 0.92, + "content": "\\Phi ( { \\pmb x } + { \\pmb \\eta } , { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 504, + 264, + 517 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 265, + 504, + 329, + 516 + ], + "score": 0.91, + "content": "- \\log p ( { \\pmb x } + { \\pmb \\eta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 504, + 445, + 517 + ], + "score": 1.0, + "content": "for marginal detection, and", + "type": "text" + }, + { + "bbox": [ + 445, + 505, + 451, + 514 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "is the corre-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "sponding threshold computed on training data. For logit/KL detection, the detection statistics and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "thresholds are constructed accordingly. We label the three attacks against the marginal, logit and KL", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "score": 1.0, + "content": "based detection schemes ’-PKM’, ’-PKL’ and ’-PKK’ respectively. When we only have knowledge", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 548, + 490, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 132, + 561 + ], + "score": 1.0, + "content": "of the", + "type": "text" + }, + { + "bbox": [ + 132, + 549, + 142, + 558 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 548, + 414, + 561 + ], + "score": 1.0, + "content": "samples and not the detection method we call this attack ’-PK0’ (ie", + "type": "text" + }, + { + "bbox": [ + 414, + 549, + 449, + 559 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { d e t e c t } } { = } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 548, + 490, + 561 + ], + "score": 1.0, + "content": "in Eq. 4).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 504, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "Results are visualised in Figure 5. We see that although the attacker can reduce detection levels,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "score": 1.0, + "content": "this comes with the trade-off of increasing accuracy, suggesting that a perfect-knowledge adversary", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 587, + 364, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 364, + 600 + ], + "score": 1.0, + "content": "cannot break both the classifier and detector working in tandem.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 565, + 505, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Sanity checks on gradient masking Athalye et al. (2018) claimed that if a successful defence", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "against white-box attacks is due to gradient masking, then this defence is likely to be less effective", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "against grey-box/black-box attacks, as they do not differentiate through the victim classifier and", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "the defence mechanism (Papernot et al., 2017b). Therefore, we consider two transfer attacks based", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "on distilling the victim classifier using a “student” CNN which has no gradient masking. The two", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "attacks differ in their threat models: in the grey-box setting the attacker has access to the output", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "probability vectors of the classifiers on the training data, while in the black-box setting the attacker", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "has access to queried labels only. For the latter black-box setting, we follow Papernot et al. (2017b)", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "to train a substitute CNN using Jacobian-based dataset augmentation, and we refer to appendix B.1", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 348, + 722 + ], + "score": 1.0, + "content": "for the detailed algorithm. The grey-box substitutes achieve", + "type": "text" + }, + { + "bbox": [ + 349, + 709, + 380, + 720 + ], + "score": 0.89, + "content": "> 9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "agreement with the victims on", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 720, + 419, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 299, + 733 + ], + "score": 1.0, + "content": "test data, and the black-box substitutes obtained", + "type": "text" + }, + { + "bbox": [ + 299, + 720, + 330, + 731 + ], + "score": 0.89, + "content": "\\sim 9 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 720, + 419, + 733 + ], + "score": 1.0, + "content": "accuracy on test data.", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 56, + "bbox_fs": [ + 105, + 611, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 67, + 492, + 176 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 67, + 492, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 67, + 492, + 176 + ], + "spans": [ + { + "bbox": [ + 117, + 67, + 492, + 176 + ], + "score": 0.968, + "type": "image", + "image_path": "92fbc68f077b0912c224d9aa79a22a044fe5d692681f9182450b3127b5751d73.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 67, + 492, + 103.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 103.33333333333334, + 492, + 139.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 139.66666666666669, + 492, + 176.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 181, + 505, + 226 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "Figure 5: Accuracy and detection rates of DBX, GBY, and GBZ against PGD-based perfect-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 185, + 205 + ], + "score": 1.0, + "content": "knowledge attack", + "type": "text" + }, + { + "bbox": [ + 186, + 193, + 225, + 204 + ], + "score": 0.86, + "content": "( \\epsilon = 0 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 192, + 505, + 205 + ], + "score": 1.0, + "content": ") on MNIST. The solid area denotes accuracy and the hatched area", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "denotes detection rate with each considered detector. Zero knowledge attacks are labelled ’-ZK’,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 215, + 303, + 226 + ], + "spans": [ + { + "bbox": [ + 107, + 215, + 303, + 226 + ], + "score": 1.0, + "content": "other attack labels are described in the main text.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 111, + 234, + 497, + 363 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 234, + 497, + 363 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 234, + 497, + 363 + ], + "spans": [ + { + "bbox": [ + 111, + 234, + 497, + 363 + ], + "score": 0.968, + "type": "image", + "image_path": "a9d0675f4478e57a95c5946f6c2e9e26184ee8ec7905a49a23ceeb8edb588c24.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 111, + 234, + 497, + 277.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 111, + 277.0, + 497, + 320.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 320.0, + 497, + 363.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 367, + 505, + 390 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "Figure 6: Accuracy and detection rates against distillation-based attacks on MNIST. The higher the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 378, + 465, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 465, + 391 + ], + "score": 1.0, + "content": "better. We only present generative classifiers’ results here, for full results see appendix E.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 9.25 + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 363, + 415 + ], + "score": 1.0, + "content": "Figure 6 shows the accuracy and detection metrics on transferred", + "type": "text" + }, + { + "bbox": [ + 363, + 402, + 378, + 413 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "attacks crafted on the substitute", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 494, + 425 + ], + "score": 1.0, + "content": "models, with a comparison to their white-box counterparts. Note that these crafted attacks achieve", + "type": "text" + }, + { + "bbox": [ + 495, + 415, + 505, + 423 + ], + "score": 0.65, + "content": "\\sim", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 107, + 424, + 131, + 435 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 423, + 340, + 437 + ], + "score": 1.0, + "content": "success rates on fooling the substitute models when", + "type": "text" + }, + { + "bbox": [ + 340, + 425, + 372, + 435 + ], + "score": 0.9, + "content": "\\epsilon \\geq 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "(see appendix E). However, they", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 422, + 447 + ], + "score": 1.0, + "content": "do not transfer very well to the generative classifiers. Importantly, for a fixed", + "type": "text" + }, + { + "bbox": [ + 422, + 437, + 429, + 445 + ], + "score": 0.64, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "setting, the white-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 445, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 460 + ], + "score": 1.0, + "content": "box attacks achieve significantly higher success rates (i.e. lower victim accuracies) than their grey-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 293, + 470 + ], + "score": 1.0, + "content": "/black-box counterparts, and the gap is at least", + "type": "text" + }, + { + "bbox": [ + 293, + 457, + 324, + 468 + ], + "score": 0.9, + "content": "> 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 456, + 339, + 470 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 339, + 457, + 371, + 468 + ], + "score": 0.9, + "content": "\\epsilon \\leq 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "(see Table 2). We further present", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 186, + 480 + ], + "score": 1.0, + "content": "the mean minimum", + "type": "text" + }, + { + "bbox": [ + 186, + 468, + 200, + 479 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "perturbation in Table 2, and we see that the minimum perturbation obtained", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 479, + 477, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 477, + 491 + ], + "score": 1.0, + "content": "by grey-/black-box attacks are significantly higher than those obtained by white-box attacks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "All these results suggest that the robustness of generative classifiers is unlikely to be caused by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "score": 1.0, + "content": "gradient masking. Again on the detection side, the detection rates increase as the perturbation size", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 216, + 529 + ], + "score": 1.0, + "content": "increase, and they are near", + "type": "text" + }, + { + "bbox": [ + 216, + 518, + 241, + 528 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 518, + 256, + 529 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 256, + 518, + 289, + 528 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 518, + 505, + 529 + ], + "score": 1.0, + "content": ". This means that most of the successfully transferred", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 529, + 488, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 488, + 542 + ], + "score": 1.0, + "content": "adversarial images are off the generative classifier’s manifold (as a proxy to the data manifold).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "SPSA (evolutionary strategies) We consider another black-box setting that only assumes access", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 378, + 578 + ], + "score": 1.0, + "content": "to the logit values of the prediction given an input. We use the SPSA", + "type": "text" + }, + { + "bbox": [ + 379, + 565, + 393, + 576 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "attack (Uesato et al., 2018),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "which is similar to the white-box zero-knowledge attacks, except that gradients are numerically", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "table", + "bbox": [ + 147, + 643, + 463, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 607, + 504, + 641 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 212, + 620 + ], + "score": 1.0, + "content": "Table 2: Mean minimum", + "type": "text" + }, + { + "bbox": [ + 213, + 608, + 226, + 618 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 606, + 370, + 620 + ], + "score": 1.0, + "content": "perturbation (in red, computed on", + "type": "text" + }, + { + "bbox": [ + 370, + 607, + 484, + 619 + ], + "score": 0.9, + "content": "\\epsilon \\in \\{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 223, + 631 + ], + "score": 1.0, + "content": "victim accuracy (in blue, for", + "type": "text" + }, + { + "bbox": [ + 223, + 619, + 257, + 629 + ], + "score": 0.87, + "content": "\\epsilon \\leq 0 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 617, + 274, + 631 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 274, + 619, + 288, + 630 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "attacks on MNIST. We manually assign the min. per-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 629, + 431, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 145, + 642 + ], + "score": 1.0, + "content": "turbation", + "type": "text" + }, + { + "bbox": [ + 145, + 630, + 176, + 640 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 629, + 431, + 642 + ], + "score": 1.0, + "content": "to inputs that all attacks failed to find adversarial perturbations.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 147, + 643, + 463, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 643, + 463, + 730 + ], + "spans": [ + { + "bbox": [ + 147, + 643, + 463, + 730 + ], + "score": 0.98, + "html": "
AttackGFZGBZGFYGBY
PGD(white-box)0.23 /7.71%0.30 /30.78%0.21 / 5.52%0.23 / 8.89%
MIM(white-box)0.24 / 9.02%0.21 /4.97%0.22 / 6.72%0.21 /1.54%
PGD(grey-box)0.37 /51.08%0.36 / 50.64%0.38 / 53.29%0.36 / 48.66%
MIM(grey-box)0.34 / 43.00%0.33 / 40.94%0.34 / 46.64%0.33 /40.06%
PGD(black-box)0.40 / 61.93%0.42 /66.75%0.38 / 56.35%0.43 / 68.50%
MIM(black-box)0.36 / 50.44%0.38 / 59.86%0.36 / 48.07%0.39 / 61.78%
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The solid area denotes accuracy and the hatched area", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "denotes detection rate with each considered detector. Zero knowledge attacks are labelled ’-ZK’,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 215, + 303, + 226 + ], + "spans": [ + { + "bbox": [ + 107, + 215, + 303, + 226 + ], + "score": 1.0, + "content": "other attack labels are described in the main text.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 111, + 234, + 497, + 363 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 234, + 497, + 363 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 234, + 497, + 363 + ], + "spans": [ + { + "bbox": [ + 111, + 234, + 497, + 363 + ], + "score": 0.968, + "type": "image", + "image_path": "a9d0675f4478e57a95c5946f6c2e9e26184ee8ec7905a49a23ceeb8edb588c24.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 111, + 234, + 497, + 277.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 111, + 277.0, + 497, + 320.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 320.0, + 497, + 363.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 367, + 505, + 390 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "Figure 6: Accuracy and detection rates against distillation-based attacks on MNIST. The higher the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 378, + 465, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 465, + 391 + ], + "score": 1.0, + "content": "better. We only present generative classifiers’ results here, for full results see appendix E.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 9.25 + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 363, + 415 + ], + "score": 1.0, + "content": "Figure 6 shows the accuracy and detection metrics on transferred", + "type": "text" + }, + { + "bbox": [ + 363, + 402, + 378, + 413 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "attacks crafted on the substitute", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 494, + 425 + ], + "score": 1.0, + "content": "models, with a comparison to their white-box counterparts. Note that these crafted attacks achieve", + "type": "text" + }, + { + "bbox": [ + 495, + 415, + 505, + 423 + ], + "score": 0.65, + "content": "\\sim", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 107, + 424, + 131, + 435 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 423, + 340, + 437 + ], + "score": 1.0, + "content": "success rates on fooling the substitute models when", + "type": "text" + }, + { + "bbox": [ + 340, + 425, + 372, + 435 + ], + "score": 0.9, + "content": "\\epsilon \\geq 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "(see appendix E). However, they", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 422, + 447 + ], + "score": 1.0, + "content": "do not transfer very well to the generative classifiers. Importantly, for a fixed", + "type": "text" + }, + { + "bbox": [ + 422, + 437, + 429, + 445 + ], + "score": 0.64, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "setting, the white-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 445, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 460 + ], + "score": 1.0, + "content": "box attacks achieve significantly higher success rates (i.e. lower victim accuracies) than their grey-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 293, + 470 + ], + "score": 1.0, + "content": "/black-box counterparts, and the gap is at least", + "type": "text" + }, + { + "bbox": [ + 293, + 457, + 324, + 468 + ], + "score": 0.9, + "content": "> 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 456, + 339, + 470 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 339, + 457, + 371, + 468 + ], + "score": 0.9, + "content": "\\epsilon \\leq 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "(see Table 2). We further present", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 186, + 480 + ], + "score": 1.0, + "content": "the mean minimum", + "type": "text" + }, + { + "bbox": [ + 186, + 468, + 200, + 479 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "perturbation in Table 2, and we see that the minimum perturbation obtained", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 479, + 477, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 477, + 491 + ], + "score": 1.0, + "content": "by grey-/black-box attacks are significantly higher than those obtained by white-box attacks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 402, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "All these results suggest that the robustness of generative classifiers is unlikely to be caused by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "score": 1.0, + "content": "gradient masking. Again on the detection side, the detection rates increase as the perturbation size", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 216, + 529 + ], + "score": 1.0, + "content": "increase, and they are near", + "type": "text" + }, + { + "bbox": [ + 216, + 518, + 241, + 528 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 518, + 256, + 529 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 256, + 518, + 289, + 528 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 518, + 505, + 529 + ], + "score": 1.0, + "content": ". This means that most of the successfully transferred", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 529, + 488, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 488, + 542 + ], + "score": 1.0, + "content": "adversarial images are off the generative classifier’s manifold (as a proxy to the data manifold).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 495, + 506, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "SPSA (evolutionary strategies) We consider another black-box setting that only assumes access", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 378, + 578 + ], + "score": 1.0, + "content": "to the logit values of the prediction given an input. We use the SPSA", + "type": "text" + }, + { + "bbox": [ + 379, + 565, + 393, + 576 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "attack (Uesato et al., 2018),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "which is similar to the white-box zero-knowledge attacks, except that gradients are numerically", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "estimated using the logit values from the victim classifier. Results in Table 3 clearly show that", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "SPSA performs much worse on generative classifiers when compared to white-box PGD. Again", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "this means gradient masking is unlikely to be responsible for the improved robustness of generative", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 333, + 373, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 373, + 344 + ], + "score": 1.0, + "content": "classifiers, as utilising the exact gradient yielded improved results.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 553, + 506, + 589 + ] + }, + { + "type": "table", + "bbox": [ + 147, + 643, + 463, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 607, + 504, + 641 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 212, + 620 + ], + "score": 1.0, + "content": "Table 2: Mean minimum", + "type": "text" + }, + { + "bbox": [ + 213, + 608, + 226, + 618 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 606, + 370, + 620 + ], + "score": 1.0, + "content": "perturbation (in red, computed on", + "type": "text" + }, + { + "bbox": [ + 370, + 607, + 484, + 619 + ], + "score": 0.9, + "content": "\\epsilon \\in \\{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 223, + 631 + ], + "score": 1.0, + "content": "victim accuracy (in blue, for", + "type": "text" + }, + { + "bbox": [ + 223, + 619, + 257, + 629 + ], + "score": 0.87, + "content": "\\epsilon \\leq 0 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 617, + 274, + 631 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 274, + 619, + 288, + 630 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "attacks on MNIST. We manually assign the min. per-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 629, + 431, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 145, + 642 + ], + "score": 1.0, + "content": "turbation", + "type": "text" + }, + { + "bbox": [ + 145, + 630, + 176, + 640 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 629, + 431, + 642 + ], + "score": 1.0, + "content": "to inputs that all attacks failed to find adversarial perturbations.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 147, + 643, + 463, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 643, + 463, + 730 + ], + "spans": [ + { + "bbox": [ + 147, + 643, + 463, + 730 + ], + "score": 0.98, + "html": "
AttackGFZGBZGFYGBY
PGD(white-box)0.23 /7.71%0.30 /30.78%0.21 / 5.52%0.23 / 8.89%
MIM(white-box)0.24 / 9.02%0.21 /4.97%0.22 / 6.72%0.21 /1.54%
PGD(grey-box)0.37 /51.08%0.36 / 50.64%0.38 / 53.29%0.36 / 48.66%
MIM(grey-box)0.34 / 43.00%0.33 / 40.94%0.34 / 46.64%0.33 /40.06%
PGD(black-box)0.40 / 61.93%0.42 /66.75%0.38 / 56.35%0.43 / 68.50%
MIM(black-box)0.36 / 50.44%0.38 / 59.86%0.36 / 48.07%0.39 / 61.78%
", + "type": "table", + "image_path": "e0fd024672c265a62c064673aa45aad74f7ff0c4aaa2e243918fbac3930f75f8.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 147, + 643, + 463, + 672.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 147, + 672.0, + 463, + 701.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 147, + 701.0, + 463, + 730.0 + ], + "spans": [], + "index": 32 + } + ] + } + ], + "index": 29.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 90, + 500, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 90, + 500, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 90, + 500, + 248 + ], + "spans": [ + { + "bbox": [ + 109, + 90, + 500, + 248 + ], + "score": 0.743, + "type": "image", + "image_path": "539526329232367bbefc7f3dde85c40b740d2252d0a5700694c8065b953efdca.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 90, + 500, + 142.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 142.66666666666666, + 500, + 195.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 195.33333333333331, + 500, + 247.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 255, + 505, + 289 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "Figure 7: Results on cross-model transfer attacks on MNIST. The horizontal axis corresponds to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 267, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 278 + ], + "score": 1.0, + "content": "the source victim that the adversarial examples are crafted on, and the vertical axis corresponds to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 277, + 462, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 462, + 290 + ], + "score": 1.0, + "content": "the target victim that the attacks are transferred to. The higher (i.e. the lighter) the better.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "estimated using the logit values from the victim classifier. Results in Table 3 clearly show that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "SPSA performs much worse on generative classifiers when compared to white-box PGD. Again", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "this means gradient masking is unlikely to be responsible for the improved robustness of generative", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 333, + 373, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 373, + 344 + ], + "score": 1.0, + "content": "classifiers, as utilising the exact gradient yielded improved results.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 506, + 367 + ], + "score": 1.0, + "content": "Cross-model attack transferability Finally, we report in Figure 7 the transferability results of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "the crafted adversarial examples between different models (Papernot et al., 2016a). Here we take", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 378, + 504, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 412, + 389 + ], + "score": 1.0, + "content": "adversarial examples crafted in the white-box setting with PGD and MIM", + "type": "text" + }, + { + "bbox": [ + 412, + 378, + 447, + 388 + ], + "score": 0.84, + "content": "\\acute { \\epsilon } = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 378, + 504, + 389 + ], + "score": 1.0, + "content": "), and transfer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "score": 1.0, + "content": "successful attacks to other classifiers. The transfer is effective between generative classifiers but not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "from generative to discriminative (and vice versa). The attacks crafted on DBX do not transfer in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "general, while at the same time, DBX is the least robust model in this case. Furthermore, the gen-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 420, + 410, + 435 + ], + "score": 1.0, + "content": "erative classifiers obtain very high detection rates on all transferred attacks", + "type": "text" + }, + { + "bbox": [ + 411, + 421, + 445, + 432 + ], + "score": 0.86, + "content": "( > 9 5 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 420, + 505, + 435 + ], + "score": 1.0, + "content": ". In summary,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 432, + 488, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 488, + 444 + ], + "score": 1.0, + "content": "generative classifiers are more robust against the tested transfer attacks across different models.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 457, + 343, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 345, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 345, + 470 + ], + "score": 1.0, + "content": "4.2 CIFAR PLANE-VS-FROG BINARY CLASSIFICATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "We consider the same set of evaluations on CIFAR-10, in order to validate the robustness of gen-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "score": 1.0, + "content": "erative classifiers on natural images (c.f. Carlini & Wagner, 2017b). Unfortunately, we failed to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "train fully generative classifiers with comparable test accuracies to discriminative CNNs (typically", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 139, + 522 + ], + "score": 0.86, + "content": "> 8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 510, + 331, + 523 + ], + "score": 1.0, + "content": "): the clean accuracies for GFZ & GFY are all", + "type": "text" + }, + { + "bbox": [ + 331, + 510, + 362, + 521 + ], + "score": 0.89, + "content": "< 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 510, + 505, + 523 + ], + "score": 1.0, + "content": ". Even when using the conditional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 135, + 534 + ], + "score": 1.0, + "content": "PixelC", + "type": "text" + }, + { + "bbox": [ + 135, + 522, + 161, + 532 + ], + "score": 0.35, + "content": "\\mathrm { N N } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "(Salimans et al., 2017) (which uses much deeper networks), the clean accuracy on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 168, + 546 + ], + "score": 1.0, + "content": "the test data is", + "type": "text" + }, + { + "bbox": [ + 168, + 532, + 195, + 543 + ], + "score": 0.88, + "content": "7 2 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 532, + 506, + 546 + ], + "score": 1.0, + "content": ". Instead, we consider a simpler binary classification problem and construct", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "a dataset containing CIFAR images from the “airplane” and “frog” categories. The images in this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 399, + 567 + ], + "score": 1.0, + "content": "dataset are scaled to [0, 1]. On this dataset, the generative classifiers use", + "type": "text" + }, + { + "bbox": [ + 400, + 554, + 459, + 567 + ], + "score": 0.92, + "content": "\\mathrm { d i m } ( z ) = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "and obtain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 138, + 576 + ], + "score": 0.88, + "content": "> 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "clean test accuracy (see appendix). The attacks are performed on the test images that all", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "models initially correctly classify, leading to a test set of 1577 instances. Due to the page limit, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 587, + 494, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 494, + 600 + ], + "score": 1.0, + "content": "only present white-box zero-knowledge attacks here, and discuss further attacks in the appendix.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 262, + 624 + ], + "score": 1.0, + "content": "White-box attacks (zero-knowledge,", + "type": "text" + }, + { + "bbox": [ + 263, + 611, + 277, + 622 + ], + "score": 0.85, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 610, + 394, + 624 + ], + "score": 1.0, + "content": ") We present the white-box", + "type": "text" + }, + { + "bbox": [ + 394, + 612, + 408, + 622 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "attack results in Figure", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 300, + 634 + ], + "score": 1.0, + "content": "8, where the distortion strengths are selected as", + "type": "text" + }, + { + "bbox": [ + 301, + 622, + 426, + 634 + ], + "score": 0.9, + "content": "\\epsilon \\in \\lbrace 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \\rbrace", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 622, + 505, + 634 + ], + "score": 1.0, + "content": ". Again, generative", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "score": 1.0, + "content": "classifiers are more robust than the discriminative ones, and GBZ is the most robust, much better than", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 173, + 656 + ], + "score": 1.0, + "content": "the others when", + "type": "text" + }, + { + "bbox": [ + 173, + 644, + 210, + 655 + ], + "score": 0.89, + "content": "\\epsilon \\geq 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 643, + 505, + 656 + ], + "score": 1.0, + "content": ". BNN is significantly better than other discriminative VAE-based classi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "fiers, presumably due to higher randomness. Detection results are less satisfactory: marginal/logit", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 248, + 678 + ], + "score": 1.0, + "content": "detection fail to detect attacks with", + "type": "text" + }, + { + "bbox": [ + 248, + 666, + 279, + 676 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "(which attain both high success rate and induce visually", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "perceptible distortion). KL detection performs better, and interestingly, discriminative classifiers", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 333, + 700 + ], + "score": 1.0, + "content": "dominate in this metric. These results suggest that the", + "type": "text" + }, + { + "bbox": [ + 333, + 688, + 343, + 699 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "likelihood might not be best suited for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "modelling natural images (c.f. Larsen et al., 2016; van den Oord et al., 2016). Still, the minimum", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "distortion required to fool generative classifiers are much higher than that for discriminative ones,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "indeed the visual distortion of the adversarial examples on generative classifiers are more significant.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 90, + 500, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 90, + 500, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 90, + 500, + 248 + ], + "spans": [ + { + "bbox": [ + 109, + 90, + 500, + 248 + ], + "score": 0.743, + "type": "image", + "image_path": "539526329232367bbefc7f3dde85c40b740d2252d0a5700694c8065b953efdca.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 90, + 500, + 142.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 142.66666666666666, + 500, + 195.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 195.33333333333331, + 500, + 247.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 255, + 505, + 289 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "Figure 7: Results on cross-model transfer attacks on MNIST. The horizontal axis corresponds to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 267, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 278 + ], + "score": 1.0, + "content": "the source victim that the adversarial examples are crafted on, and the vertical axis corresponds to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 277, + 462, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 462, + 290 + ], + "score": 1.0, + "content": "the target victim that the attacks are transferred to. The higher (i.e. the lighter) the better.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 343 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 299, + 505, + 344 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 506, + 367 + ], + "score": 1.0, + "content": "Cross-model attack transferability Finally, we report in Figure 7 the transferability results of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "the crafted adversarial examples between different models (Papernot et al., 2016a). Here we take", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 378, + 504, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 412, + 389 + ], + "score": 1.0, + "content": "adversarial examples crafted in the white-box setting with PGD and MIM", + "type": "text" + }, + { + "bbox": [ + 412, + 378, + 447, + 388 + ], + "score": 0.84, + "content": "\\acute { \\epsilon } = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 378, + 504, + 389 + ], + "score": 1.0, + "content": "), and transfer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "score": 1.0, + "content": "successful attacks to other classifiers. The transfer is effective between generative classifiers but not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "from generative to discriminative (and vice versa). The attacks crafted on DBX do not transfer in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "general, while at the same time, DBX is the least robust model in this case. Furthermore, the gen-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 420, + 410, + 435 + ], + "score": 1.0, + "content": "erative classifiers obtain very high detection rates on all transferred attacks", + "type": "text" + }, + { + "bbox": [ + 411, + 421, + 445, + 432 + ], + "score": 0.86, + "content": "( > 9 5 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 420, + 505, + 435 + ], + "score": 1.0, + "content": ". In summary,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 432, + 488, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 488, + 444 + ], + "score": 1.0, + "content": "generative classifiers are more robust against the tested transfer attacks across different models.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 356, + 506, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 457, + 343, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 345, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 345, + 470 + ], + "score": 1.0, + "content": "4.2 CIFAR PLANE-VS-FROG BINARY CLASSIFICATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "We consider the same set of evaluations on CIFAR-10, in order to validate the robustness of gen-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "score": 1.0, + "content": "erative classifiers on natural images (c.f. Carlini & Wagner, 2017b). Unfortunately, we failed to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "train fully generative classifiers with comparable test accuracies to discriminative CNNs (typically", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 139, + 522 + ], + "score": 0.86, + "content": "> 8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 510, + 331, + 523 + ], + "score": 1.0, + "content": "): the clean accuracies for GFZ & GFY are all", + "type": "text" + }, + { + "bbox": [ + 331, + 510, + 362, + 521 + ], + "score": 0.89, + "content": "< 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 510, + 505, + 523 + ], + "score": 1.0, + "content": ". Even when using the conditional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 135, + 534 + ], + "score": 1.0, + "content": "PixelC", + "type": "text" + }, + { + "bbox": [ + 135, + 522, + 161, + 532 + ], + "score": 0.35, + "content": "\\mathrm { N N } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "(Salimans et al., 2017) (which uses much deeper networks), the clean accuracy on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 168, + 546 + ], + "score": 1.0, + "content": "the test data is", + "type": "text" + }, + { + "bbox": [ + 168, + 532, + 195, + 543 + ], + "score": 0.88, + "content": "7 2 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 532, + 506, + 546 + ], + "score": 1.0, + "content": ". Instead, we consider a simpler binary classification problem and construct", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "a dataset containing CIFAR images from the “airplane” and “frog” categories. The images in this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 399, + 567 + ], + "score": 1.0, + "content": "dataset are scaled to [0, 1]. On this dataset, the generative classifiers use", + "type": "text" + }, + { + "bbox": [ + 400, + 554, + 459, + 567 + ], + "score": 0.92, + "content": "\\mathrm { d i m } ( z ) = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "and obtain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 138, + 576 + ], + "score": 0.88, + "content": "> 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "clean test accuracy (see appendix). The attacks are performed on the test images that all", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "models initially correctly classify, leading to a test set of 1577 instances. Due to the page limit, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 587, + 494, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 494, + 600 + ], + "score": 1.0, + "content": "only present white-box zero-knowledge attacks here, and discuss further attacks in the appendix.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 477, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 262, + 624 + ], + "score": 1.0, + "content": "White-box attacks (zero-knowledge,", + "type": "text" + }, + { + "bbox": [ + 263, + 611, + 277, + 622 + ], + "score": 0.85, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 610, + 394, + 624 + ], + "score": 1.0, + "content": ") We present the white-box", + "type": "text" + }, + { + "bbox": [ + 394, + 612, + 408, + 622 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "attack results in Figure", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 300, + 634 + ], + "score": 1.0, + "content": "8, where the distortion strengths are selected as", + "type": "text" + }, + { + "bbox": [ + 301, + 622, + 426, + 634 + ], + "score": 0.9, + "content": "\\epsilon \\in \\lbrace 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \\rbrace", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 622, + 505, + 634 + ], + "score": 1.0, + "content": ". Again, generative", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "score": 1.0, + "content": "classifiers are more robust than the discriminative ones, and GBZ is the most robust, much better than", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 173, + 656 + ], + "score": 1.0, + "content": "the others when", + "type": "text" + }, + { + "bbox": [ + 173, + 644, + 210, + 655 + ], + "score": 0.89, + "content": "\\epsilon \\geq 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 643, + 505, + 656 + ], + "score": 1.0, + "content": ". BNN is significantly better than other discriminative VAE-based classi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "fiers, presumably due to higher randomness. Detection results are less satisfactory: marginal/logit", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 248, + 678 + ], + "score": 1.0, + "content": "detection fail to detect attacks with", + "type": "text" + }, + { + "bbox": [ + 248, + 666, + 279, + 676 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "(which attain both high success rate and induce visually", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "perceptible distortion). KL detection performs better, and interestingly, discriminative classifiers", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 333, + 700 + ], + "score": 1.0, + "content": "dominate in this metric. These results suggest that the", + "type": "text" + }, + { + "bbox": [ + 333, + 688, + 343, + 699 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "likelihood might not be best suited for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "modelling natural images (c.f. Larsen et al., 2016; van den Oord et al., 2016). Still, the minimum", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "distortion required to fool generative classifiers are much higher than that for discriminative ones,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "indeed the visual distortion of the adversarial examples on generative classifiers are more significant.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 610, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 64, + 498, + 234 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 64, + 498, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 64, + 498, + 234 + ], + "spans": [ + { + "bbox": [ + 112, + 64, + 498, + 234 + ], + "score": 0.969, + "type": "image", + "image_path": "efeb93cf87777d23548d2d32db60dccd59abcfdfbb5946443a93be80204c004d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 64, + 498, + 120.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 120.66666666666666, + 498, + 177.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 177.33333333333331, + 498, + 233.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 239, + 505, + 284 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 239, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 325, + 253 + ], + "score": 1.0, + "content": "Figure 8: Accuracy, detection rates and minimum", + "type": "text" + }, + { + "bbox": [ + 325, + 240, + 341, + 251 + ], + "score": 0.86, + "content": "\\ell _ { \\mathrm { i n f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 239, + 505, + 253 + ], + "score": 1.0, + "content": "perturbations against white-box zero-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 155, + 263 + ], + "score": 1.0, + "content": "knowledge", + "type": "text" + }, + { + "bbox": [ + 155, + 251, + 169, + 262 + ], + "score": 0.92, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "attacks on the CIFAR plane-vs-frog dataset. The higher the better. 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Results are visualised in Figure 9. The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "generative classifiers are significantly more robust than the others (with the best being GBZ), and the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 131, + 458 + ], + "score": 1.0, + "content": "mean", + "type": "text" + }, + { + "bbox": [ + 131, + 446, + 141, + 457 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "distortions computed on successful attacks are also significantly higher. The TP rates are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "low for marginal and logit detection, which is reasonable as the crafted images are visually similar to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "the clean ones. Note that the distortion for the attacks on generative classifiers is perceptible. These", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 478, + 453, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 453, + 491 + ], + "score": 1.0, + "content": "results indicate that this CW attack is ineffective when attacking generative classifiers.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 456, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 457, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 457, + 522 + ], + "score": 1.0, + "content": "4.3 FULL CIFAR-10: COMBINING DEEP BAYES AND DISCRIMINATIVE FEATURES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 546 + ], + "score": 1.0, + "content": "The final experiment examines the robustness of CIFAR-10 multi-class classifiers, with the genera-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "tive classifiers trained on discriminative visual features. 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The classifiers", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "in test include GBZ, GBY and DBX. 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Compared with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "the VGG16 baseline, we see clear improvements in robustness and detection for all VAE-based", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "classifiers. In particular, the generative classifiers GBZ and GBY are overall better than DBX. 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The higher the better.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 9.25 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 260, + 425 + ], + "score": 1.0, + "content": "White-box attack (zero-knowledge,", + "type": "text" + }, + { + "bbox": [ + 261, + 414, + 270, + 424 + ], + "score": 0.78, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 412, + 505, + 425 + ], + "score": 1.0, + "content": ") As the test set is relatively small, we directly perform", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 237, + 436 + ], + "score": 1.0, + "content": "CW attacks on the test data with", + "type": "text" + }, + { + "bbox": [ + 237, + 424, + 345, + 436 + ], + "score": 0.89, + "content": "c \\in \\{ 0 . 1 , 1 , 1 0 , 1 0 0 , 1 0 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 424, + 505, + 436 + ], + "score": 1.0, + "content": ". Results are visualised in Figure 9. The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "generative classifiers are significantly more robust than the others (with the best being GBZ), and the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 131, + 458 + ], + "score": 1.0, + "content": "mean", + "type": "text" + }, + { + "bbox": [ + 131, + 446, + 141, + 457 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "distortions computed on successful attacks are also significantly higher. The TP rates are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "low for marginal and logit detection, which is reasonable as the crafted images are visually similar to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "the clean ones. Note that the distortion for the attacks on generative classifiers is perceptible. These", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 478, + 453, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 453, + 491 + ], + "score": 1.0, + "content": "results indicate that this CW attack is ineffective when attacking generative classifiers.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 412, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 456, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 457, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 457, + 522 + ], + "score": 1.0, + "content": "4.3 FULL CIFAR-10: COMBINING DEEP BAYES AND DISCRIMINATIVE FEATURES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 509, + 457, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 546 + ], + "score": 1.0, + "content": "The final experiment examines the robustness of CIFAR-10 multi-class classifiers, with the genera-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "tive classifiers trained on discriminative visual features. To do this, we download a pretrained deep", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 567 + ], + "score": 1.0, + "content": "convolutional net3 with VGG16-like architecture (Simonyan & Zisserman, 2014), and use its feature", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 165, + 578 + ], + "score": 1.0, + "content": "representation", + "type": "text" + }, + { + "bbox": [ + 165, + 565, + 188, + 577 + ], + "score": 0.92, + "content": "\\phi ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 565, + 355, + 578 + ], + "score": 1.0, + "content": "as the input to the VAE-based classifiers:", + "type": "text" + }, + { + "bbox": [ + 355, + 565, + 441, + 577 + ], + "score": 0.92, + "content": "p ( { \\pmb y } | { \\pmb x } ) = p ( { \\pmb y } | \\phi ( { \\pmb x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 565, + 505, + 578 + ], + "score": 1.0, + "content": ". The classifiers", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "in test include GBZ, GBY and DBX. We use fully-connected neural networks for these classifiers,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 220, + 600 + ], + "score": 1.0, + "content": "and select from VGG16 the", + "type": "text" + }, + { + "bbox": [ + 221, + 587, + 233, + 597 + ], + "score": 0.85, + "content": "9 ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "convolution layer (CONV9) and the first fully connected layer after", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 597, + 451, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 302, + 611 + ], + "score": 1.0, + "content": "convolution (FC1) as the feature layers to ensure", + "type": "text" + }, + { + "bbox": [ + 302, + 597, + 332, + 608 + ], + "score": 0.89, + "content": "\\sim 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 597, + 451, + 611 + ], + "score": 1.0, + "content": "test accuracy (see appendix).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 531, + 506, + 611 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 269, + 627 + ], + "score": 1.0, + "content": "Results on white-box zero-knowledge", + "type": "text" + }, + { + "bbox": [ + 269, + 615, + 283, + 626 + ], + "score": 0.92, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "attacks are visualised in Figure 10. Compared with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "the VGG16 baseline, we see clear improvements in robustness and detection for all VAE-based", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "classifiers. In particular, the generative classifiers GBZ and GBY are overall better than DBX. More", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "importantly, generative classifiers based on CONV9 features are significantly more robust than those", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "score": 1.0, + "content": "based on FC1 features. In contrast, for DBX, which is discriminative, the robustness results are very", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 670, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 682 + ], + "score": 1.0, + "content": "similar, indicating that the level of feature representation has little effect. These results suggest that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 104, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "one can achieve both high clean accuracy and better robustness against adversaries by combining", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 691, + 370, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 370, + 704 + ], + "score": 1.0, + "content": "discriminatively learned visual features and generative classifiers.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 614, + 506, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 164, + 66, + 440, + 237 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 164, + 66, + 440, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 164, + 66, + 440, + 237 + ], + "spans": [ + { + "bbox": [ + 164, + 66, + 440, + 237 + ], + "score": 0.971, + "type": "image", + "image_path": "312124c69aaa2404a21fddcb8a35efb3d4d839cf4d73b5393739a6dacb03296e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 164, + 66, + 440, + 123.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 164, + 123.0, + 440, + 180.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 164, + 180.0, + 440, + 237.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 243, + 503, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 412, + 255 + ], + "score": 1.0, + "content": "Figure 10: Accuracy and detection rates against white-box zero-knowledge", + "type": "text" + }, + { + "bbox": [ + 413, + 244, + 426, + 254 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "attacks on CIFAR-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 254, + 455, + 266 + ], + "spans": [ + { + "bbox": [ + 107, + 254, + 455, + 266 + ], + "score": 1.0, + "content": "10. The higher the better. Note that results for the DBX classifiers are almost identical.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 278, + 190, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 192, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 192, + 293 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "We have proposed deep Bayes as a generative classifier that uses deep latent variable models to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "model the joint distribution of input-output pairs. We have given evidence, on multiple classification", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "tasks, that generative classifiers are more robust to adversarial attacks than discriminative classifiers.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "Furthermore, the logit in generative classifiers has a well defined meaning and can be used to detect", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 349, + 275, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 275, + 360 + ], + "score": 1.0, + "content": "attacks, even when the classifier is fooled.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "Our results corroborate with the Bayesian neural network literature, in particular (Li & Gal, 2017;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "Feinman et al., 2017; Carlini & Wagner, 2017b), in showing that modelling unobserved variables", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "are effective for defending against adversarial attacks4. Concurrent to us, Schott et al. (2018) also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "demonstrated the robustness of generative classifiers on MNIST, in which the logits are computed by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "a tempered version of the variational lower-bound. However, their approach requires thousands of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 139, + 435 + ], + "score": 1.0, + "content": "random", + "type": "text" + }, + { + "bbox": [ + 140, + 423, + 147, + 431 + ], + "score": 0.67, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 421, + 368, + 435 + ], + "score": 1.0, + "content": "samples and tens of optimisation steps to approximate", + "type": "text" + }, + { + "bbox": [ + 369, + 421, + 413, + 433 + ], + "score": 0.93, + "content": "\\log p ( { \\pmb x } | { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "for every input-output", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 126, + 444 + ], + "score": 1.0, + "content": "pair", + "type": "text" + }, + { + "bbox": [ + 126, + 432, + 151, + 444 + ], + "score": 0.91, + "content": "( { \\pmb x } , { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 432, + 506, + 444 + ], + "score": 1.0, + "content": ", making it less scalable than our importance sampling technique to large datasets and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "big architectures. Indeed, we have scaled our approach to CIFAR-10, a natural image dataset, and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "the robustness results are consistent with those on MNIST. In addition, we have also shown that the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 465, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 504, + 476 + ], + "score": 1.0, + "content": "structure of the graphical model has a significant impact on robustness: deep LVM-based generative", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 477, + 388, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 388, + 488 + ], + "score": 1.0, + "content": "classifiers generally outperform the (randomised) discriminative ones.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "score": 1.0, + "content": "While we have given strong evidence to suggest that generative classifiers are more robust to current", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 504, + 504, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 504, + 515 + ], + "score": 1.0, + "content": "adversarial attacks, we do not wish to claim that these models will be robust to all possible attacks.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "Aside from many recent attacks being designed specifically for discriminative neural networks, there", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "is also evidence for the fragility of generative models; e.g. naive Bayes as a standard approach for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "spam filtering is well-known to be fragile (Dalvi et al., 2004; Huang et al., 2011), and very recently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "Tabacof et al. (2016); Kos et al. (2017); Creswell et al. (2017) also designed attacks for (uncondi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "tional) VAE-type models. However, generative classifiers can be made more robust too, to counter", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "these weaknesses. Dalvi et al. (2004) have shown that generative classifiers can be made more se-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 504, + 592 + ], + "score": 1.0, + "content": "cure if aware of the attack strategy, and Biggio et al. (2011; 2014) further improved naive Bayes’", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "robustness by modelling the conditional distribution of the adversarial inputs. These approaches are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "similar to the adversarial training of discriminative classifiers, and efficient ways for doing so with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 614, + 352, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 352, + 626 + ], + "score": 1.0, + "content": "generative classifiers can be an interesting research direction.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 504, + 642 + ], + "score": 1.0, + "content": "But even with this note of caution, we believe this work offers exciting avenues for future work.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "Using generative classifiers offers an interesting way to evaluate generative models and can drive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "improvements in their ability to tackle high-dimensional datasets, where traditionally generative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "classifiers have been less accurate than discriminative classifiers (Efron, 1975; Ng & Jordan, 2002).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "score": 1.0, + "content": "In addition, the combination of generative and discriminative models investigated in this paper is a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "compelling direction for future research. Overall, we believe that progress on generative classifiers", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 696, + 387, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 387, + 708 + ], + "score": 1.0, + "content": "can inspire better designs of attack, defence and detection techniques.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 721, + 455, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 457, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 457, + 733 + ], + "score": 1.0, + "content": "4In Bayesian neural networks, the network weights are treated as unobserved/latent variables.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 164, + 66, + 440, + 237 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 164, + 66, + 440, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 164, + 66, + 440, + 237 + ], + "spans": [ + { + "bbox": [ + 164, + 66, + 440, + 237 + ], + "score": 0.971, + "type": "image", + "image_path": "312124c69aaa2404a21fddcb8a35efb3d4d839cf4d73b5393739a6dacb03296e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 164, + 66, + 440, + 123.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 164, + 123.0, + 440, + 180.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 164, + 180.0, + 440, + 237.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 243, + 503, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 412, + 255 + ], + "score": 1.0, + "content": "Figure 10: Accuracy and detection rates against white-box zero-knowledge", + "type": "text" + }, + { + "bbox": [ + 413, + 244, + 426, + 254 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "attacks on CIFAR-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 254, + 455, + 266 + ], + "spans": [ + { + "bbox": [ + 107, + 254, + 455, + 266 + ], + "score": 1.0, + "content": "10. The higher the better. Note that results for the DBX classifiers are almost identical.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 278, + 190, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 192, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 192, + 293 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "We have proposed deep Bayes as a generative classifier that uses deep latent variable models to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "model the joint distribution of input-output pairs. We have given evidence, on multiple classification", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "tasks, that generative classifiers are more robust to adversarial attacks than discriminative classifiers.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "Furthermore, the logit in generative classifiers has a well defined meaning and can be used to detect", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 349, + 275, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 275, + 360 + ], + "score": 1.0, + "content": "attacks, even when the classifier is fooled.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 306, + 505, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "Our results corroborate with the Bayesian neural network literature, in particular (Li & Gal, 2017;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "Feinman et al., 2017; Carlini & Wagner, 2017b), in showing that modelling unobserved variables", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "are effective for defending against adversarial attacks4. Concurrent to us, Schott et al. (2018) also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "demonstrated the robustness of generative classifiers on MNIST, in which the logits are computed by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "a tempered version of the variational lower-bound. However, their approach requires thousands of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 139, + 435 + ], + "score": 1.0, + "content": "random", + "type": "text" + }, + { + "bbox": [ + 140, + 423, + 147, + 431 + ], + "score": 0.67, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 421, + 368, + 435 + ], + "score": 1.0, + "content": "samples and tens of optimisation steps to approximate", + "type": "text" + }, + { + "bbox": [ + 369, + 421, + 413, + 433 + ], + "score": 0.93, + "content": "\\log p ( { \\pmb x } | { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "for every input-output", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 126, + 444 + ], + "score": 1.0, + "content": "pair", + "type": "text" + }, + { + "bbox": [ + 126, + 432, + 151, + 444 + ], + "score": 0.91, + "content": "( { \\pmb x } , { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 432, + 506, + 444 + ], + "score": 1.0, + "content": ", making it less scalable than our importance sampling technique to large datasets and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "big architectures. Indeed, we have scaled our approach to CIFAR-10, a natural image dataset, and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "the robustness results are consistent with those on MNIST. In addition, we have also shown that the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 465, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 504, + 476 + ], + "score": 1.0, + "content": "structure of the graphical model has a significant impact on robustness: deep LVM-based generative", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 477, + 388, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 388, + 488 + ], + "score": 1.0, + "content": "classifiers generally outperform the (randomised) discriminative ones.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 366, + 506, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "score": 1.0, + "content": "While we have given strong evidence to suggest that generative classifiers are more robust to current", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 504, + 504, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 504, + 515 + ], + "score": 1.0, + "content": "adversarial attacks, we do not wish to claim that these models will be robust to all possible attacks.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "Aside from many recent attacks being designed specifically for discriminative neural networks, there", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "is also evidence for the fragility of generative models; e.g. naive Bayes as a standard approach for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "spam filtering is well-known to be fragile (Dalvi et al., 2004; Huang et al., 2011), and very recently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "Tabacof et al. 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(2011; 2014) further improved naive Bayes’", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "robustness by modelling the conditional distribution of the adversarial inputs. These approaches are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "similar to the adversarial training of discriminative classifiers, and efficient ways for doing so with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 614, + 352, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 352, + 626 + ], + "score": 1.0, + "content": "generative classifiers can be an interesting research direction.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 491, + 506, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 504, + 642 + ], + "score": 1.0, + "content": "But even with this note of caution, we believe this work offers exciting avenues for future work.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "Using generative classifiers offers an interesting way to evaluate generative models and can drive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "improvements in their ability to tackle high-dimensional datasets, where traditionally generative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "classifiers have been less accurate than discriminative classifiers (Efron, 1975; Ng & Jordan, 2002).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "score": 1.0, + "content": "In addition, the combination of generative and discriminative models investigated in this paper is a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "compelling direction for future research. Overall, we believe that progress on generative classifiers", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 696, + 387, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 387, + 708 + ], + "score": 1.0, + "content": "can inspire better designs of attack, defence and detection techniques.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 630, + 506, + 708 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 176, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 505, + 113 + ], + "score": 1.0, + "content": "Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. 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We use the default hyper-parameters, if not", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 183, + 182, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 182, + 195 + ], + "score": 1.0, + "content": "specifically stated.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 200, + 370, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 371, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 371, + 213 + ], + "score": 1.0, + "content": "PGD: We perform the attack for 40 iterations with step-size 0.01.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 453, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 455, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 455, + 230 + ], + "score": 1.0, + "content": "MIM: We perform the attack for 40 iterations with step-size 0.01 and decay factor 1.0.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 494, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 233, + 496, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 127, + 247 + ], + "score": 1.0, + "content": "CW-", + "type": "text" + }, + { + "bbox": [ + 127, + 234, + 136, + 245 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 233, + 496, + 247 + ], + "score": 1.0, + "content": ": We use learning rate 0.01 and confidence 0, and we optimise the loss for 1000 iterations.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "SPSA: We use almost the same hyper-parameters as in Uesato et al. (2018) except for the number of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "samples for gradient estimates. In detail, we perform the attack for 100 iterations with perturbation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "size 0.01, Adam learning rate 0.01, stopping threshold -5.0 and 2000 samples for each gradient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 283, + 146, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 146, + 296 + ], + "score": 1.0, + "content": "estimate.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 309, + 320, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 322, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 322, + 321 + ], + "score": 1.0, + "content": "B.1 JACOBIAN-BASED DATASET AUGMENTATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "The black-box distillation attack is based on Papernot et al. (2017b), which trains a substitute CNN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 326, + 353 + ], + "score": 1.0, + "content": "using Jacobian-based dataset augmentation. Assume", + "type": "text" + }, + { + "bbox": [ + 327, + 340, + 374, + 352 + ], + "score": 0.93, + "content": "\\boldsymbol { y } = \\boldsymbol { F } ( \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "is the output one-hot vector of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 170, + 364 + ], + "score": 1.0, + "content": "the victim, and", + "type": "text" + }, + { + "bbox": [ + 171, + 351, + 192, + 363 + ], + "score": 0.92, + "content": "{ \\pmb p } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 351, + 466, + 364 + ], + "score": 1.0, + "content": "is the probability vector output of the substitute model, then at the", + "type": "text" + }, + { + "bbox": [ + 466, + 351, + 477, + 362 + ], + "score": 0.87, + "content": "t ^ { \\mathrm { { t h } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "outer-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 287, + 375 + ], + "score": 1.0, + "content": "loop, we train the substitute CNN on dataset", + "type": "text" + }, + { + "bbox": [ + 288, + 362, + 360, + 375 + ], + "score": 0.94, + "content": "\\mathcal { D } _ { t } \\overset { \\cdot } { = } \\{ ( \\boldsymbol { x } _ { n } , \\boldsymbol { y } _ { n } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 361, + 414, + 375 + ], + "score": 1.0, + "content": "with queried", + "type": "text" + }, + { + "bbox": [ + 414, + 363, + 426, + 374 + ], + "score": 0.86, + "content": "{ \\bf { { y } } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "for 10 epochs, and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 374, + 200, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 200, + 387 + ], + "score": 1.0, + "content": "augment the dataset by", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 389, + 438, + 404 + ], + "lines": [ + { + "bbox": [ + 172, + 389, + 438, + 404 + ], + "spans": [ + { + "bbox": [ + 172, + 389, + 438, + 404 + ], + "score": 0.87, + "content": "\\mathcal { D } _ { t + 1 } = \\mathcal { D } _ { t } \\cup \\big \\{ \\big ( \\hat { x } , F ( \\hat { x } ) \\big ) \\mid \\hat { x } = x + \\lambda \\nabla _ { x } p ( x ) ^ { \\mathrm { T } } y , \\big ( x , y \\big ) \\in \\mathcal { D } _ { t } \\big \\} .", + "type": "interline_equation", + "image_path": "c44002978a53ef0602ba9656edb7b3cf391d6584628476a1383527a3292d7f5b.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 172, + 389, + 438, + 404 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 160, + 421 + ], + "score": 1.0, + "content": "We initialise", + "type": "text" + }, + { + "bbox": [ + 160, + 410, + 174, + 420 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 410, + 197, + 421 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 197, + 410, + 237, + 420 + ], + "score": 0.89, + "content": "2 0 0 \\times 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 410, + 415, + 421 + ], + "score": 1.0, + "content": "datapoints from the MNIST test set, select", + "type": "text" + }, + { + "bbox": [ + 415, + 410, + 451, + 420 + ], + "score": 0.9, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 410, + 505, + 421 + ], + "score": 1.0, + "content": ", and run the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 435, + 433 + ], + "score": 1.0, + "content": "algorithm for 6 outer-loops. On MNIST, this results in 64, 000 queried inputs, and", + "type": "text" + }, + { + "bbox": [ + 436, + 420, + 466, + 431 + ], + "score": 0.9, + "content": "\\sim 9 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 425, + 443 + ], + "score": 1.0, + "content": "of the substitute model on test data. On CIFAR binary classification, we use", + "type": "text" + }, + { + "bbox": [ + 425, + 432, + 460, + 442 + ], + "score": 0.89, + "content": "2 0 0 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 432, + 505, + 443 + ], + "score": 1.0, + "content": "datapoints", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 197, + 455 + ], + "score": 1.0, + "content": "for the inital query set", + "type": "text" + }, + { + "bbox": [ + 198, + 443, + 211, + 453 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 442, + 505, + 455 + ], + "score": 1.0, + "content": ", resulting in 12, 800 queries in total. The substitutes achieved almost the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 453, + 415, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 415, + 465 + ], + "score": 1.0, + "content": "same accuracy as their corresponding victim models on clean test datapoints.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 252, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 253, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 253, + 495 + ], + "score": 1.0, + "content": "C FURTHER EXPERIMENTS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 505, + 393, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 394, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 394, + 518 + ], + "score": 1.0, + "content": "C.1 FURTHER EXPERIMENTS ON CIFAR BINARY CLASSIFICATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 526, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 272, + 540 + ], + "score": 1.0, + "content": "White-box attack (perfect-knowledge,", + "type": "text" + }, + { + "bbox": [ + 272, + 527, + 286, + 537 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 525, + 506, + 540 + ], + "score": 1.0, + "content": ") Figure C.1 shows the perfect knowledge attack on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "the CIFAR binary classification task. Again we see that although the attack is effective for the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "detection schemes, it comes with the price of decreased mis-classification rates. Interestingly GBY", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "seems to be robust to this attack, where the accuracies on the crafted adversarial examples increase.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 446, + 595 + ], + "score": 1.0, + "content": "Sanity checks on gradient masking We conducted the same sets of transferred", + "type": "text" + }, + { + "bbox": [ + 446, + 583, + 460, + 594 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "attack ex-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "periments, and presents the results in Figure C.2 and Table C.1. Again that these crafted attacks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 140, + 617 + ], + "score": 1.0, + "content": "achieve", + "type": "text" + }, + { + "bbox": [ + 140, + 605, + 178, + 615 + ], + "score": 0.89, + "content": "\\sim 1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 604, + 399, + 617 + ], + "score": 1.0, + "content": "success rates on fooling the substitute models when", + "type": "text" + }, + { + "bbox": [ + 400, + 605, + 437, + 616 + ], + "score": 0.9, + "content": "\\epsilon \\geq 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 604, + 505, + 617 + ], + "score": 1.0, + "content": ". Similar to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "MNIST experiments, these adversarial examples do not transfer very well to the generative classi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 190, + 639 + ], + "score": 1.0, + "content": "fiers, and for a fixed", + "type": "text" + }, + { + "bbox": [ + 190, + 628, + 196, + 637 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "setting, the white-box attacks achieve significantly higher success rates than", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 314, + 650 + ], + "score": 1.0, + "content": "their grey-/black-box counterparts (with the gap at", + "type": "text" + }, + { + "bbox": [ + 314, + 638, + 348, + 649 + ], + "score": 0.89, + "content": "\\epsilon \\leq 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 637, + 380, + 650 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 381, + 638, + 401, + 649 + ], + "score": 0.86, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "). Furthermore, the mini-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "mum perturbation obtained by grey-/black-box attacks are significantly higher than those obtained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 422, + 673 + ], + "score": 1.0, + "content": "by white-box attacks. For detection, the detection rates are relatively low at", + "type": "text" + }, + { + "bbox": [ + 422, + 660, + 457, + 671 + ], + "score": 0.9, + "content": "\\epsilon \\leq 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "(where the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 443, + 684 + ], + "score": 1.0, + "content": "classifiers achieved high accuracy). But the detection rates increase significantly for", + "type": "text" + }, + { + "bbox": [ + 443, + 671, + 474, + 681 + ], + "score": 0.88, + "content": "\\epsilon = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 670, + 505, + 684 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 681, + 235, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 235, + 694 + ], + "score": 1.0, + "content": "the victim accuracies also drop.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "All these results suggest that the robustness of generative classifiers is unlikely to be caused by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "gradient masking. Also most of the successfully transferred adversarial images are off the generative", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 324, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 324, + 732 + ], + "score": 1.0, + "content": "classifier’s manifold (as a proxy to the data manifold).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + 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], + "score": 1.0, + "content": "We use the Cleverhans package to perform attacks. We use the default hyper-parameters, if not", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 183, + 182, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 182, + 195 + ], + "score": 1.0, + "content": "specifically stated.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 170, + 506, + 195 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 200, + 370, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 371, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 371, + 213 + ], + "score": 1.0, + "content": "PGD: We perform the attack for 40 iterations with step-size 0.01.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 199, + 371, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 453, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 455, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 455, + 230 + ], + "score": 1.0, + "content": "MIM: We perform the attack for 40 iterations with step-size 0.01 and decay factor 1.0.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 216, + 455, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 494, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 233, + 496, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 127, + 247 + ], + "score": 1.0, + "content": "CW-", + "type": "text" + }, + { + "bbox": [ + 127, + 234, + 136, + 245 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 233, + 496, + 247 + ], + "score": 1.0, + "content": ": We use learning rate 0.01 and confidence 0, and we optimise the loss for 1000 iterations.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 233, + 496, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "SPSA: We use almost the same hyper-parameters as in Uesato et al. (2018) except for the number of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "samples for gradient estimates. In detail, we perform the attack for 100 iterations with perturbation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "size 0.01, Adam learning rate 0.01, stopping threshold -5.0 and 2000 samples for each gradient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 283, + 146, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 146, + 296 + ], + "score": 1.0, + "content": "estimate.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 250, + 506, + 296 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 309, + 320, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 322, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 322, + 321 + ], + "score": 1.0, + "content": "B.1 JACOBIAN-BASED DATASET AUGMENTATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "The black-box distillation attack is based on Papernot et al. (2017b), which trains a substitute CNN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 326, + 353 + ], + "score": 1.0, + "content": "using Jacobian-based dataset augmentation. Assume", + "type": "text" + }, + { + "bbox": [ + 327, + 340, + 374, + 352 + ], + "score": 0.93, + "content": "\\boldsymbol { y } = \\boldsymbol { F } ( \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "is the output one-hot vector of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 170, + 364 + ], + "score": 1.0, + "content": "the victim, and", + "type": "text" + }, + { + "bbox": [ + 171, + 351, + 192, + 363 + ], + "score": 0.92, + "content": "{ \\pmb p } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 351, + 466, + 364 + ], + "score": 1.0, + "content": "is the probability vector output of the substitute model, then at the", + "type": "text" + }, + { + "bbox": [ + 466, + 351, + 477, + 362 + ], + "score": 0.87, + "content": "t ^ { \\mathrm { { t h } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "outer-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 287, + 375 + ], + "score": 1.0, + "content": "loop, we train the substitute CNN on dataset", + "type": "text" + }, + { + "bbox": [ + 288, + 362, + 360, + 375 + ], + "score": 0.94, + "content": "\\mathcal { D } _ { t } \\overset { \\cdot } { = } \\{ ( \\boldsymbol { x } _ { n } , \\boldsymbol { y } _ { n } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 361, + 414, + 375 + ], + "score": 1.0, + "content": "with queried", + "type": "text" + }, + { + "bbox": [ + 414, + 363, + 426, + 374 + ], + "score": 0.86, + "content": "{ \\bf { { y } } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "for 10 epochs, and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 374, + 200, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 200, + 387 + ], + "score": 1.0, + "content": "augment the dataset by", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 329, + 506, + 387 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 389, + 438, + 404 + ], + "lines": [ + { + "bbox": [ + 172, + 389, + 438, + 404 + ], + "spans": [ + { + "bbox": [ + 172, + 389, + 438, + 404 + ], + "score": 0.87, + "content": "\\mathcal { D } _ { t + 1 } = \\mathcal { D } _ { t } \\cup \\big \\{ \\big ( \\hat { x } , F ( \\hat { x } ) \\big ) \\mid \\hat { x } = x + \\lambda \\nabla _ { x } p ( x ) ^ { \\mathrm { T } } y , \\big ( x , y \\big ) \\in \\mathcal { D } _ { t } \\big \\} .", + "type": "interline_equation", + "image_path": "c44002978a53ef0602ba9656edb7b3cf391d6584628476a1383527a3292d7f5b.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 172, + 389, + 438, + 404 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 160, + 421 + ], + "score": 1.0, + "content": "We initialise", + "type": "text" + }, + { + "bbox": [ + 160, + 410, + 174, + 420 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 410, + 197, + 421 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 197, + 410, + 237, + 420 + ], + "score": 0.89, + "content": "2 0 0 \\times 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 410, + 415, + 421 + ], + "score": 1.0, + "content": "datapoints from the MNIST test set, select", + "type": "text" + }, + { + "bbox": [ + 415, + 410, + 451, + 420 + ], + "score": 0.9, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 410, + 505, + 421 + ], + "score": 1.0, + "content": ", and run the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 435, + 433 + ], + "score": 1.0, + "content": "algorithm for 6 outer-loops. On MNIST, this results in 64, 000 queried inputs, and", + "type": "text" + }, + { + "bbox": [ + 436, + 420, + 466, + 431 + ], + "score": 0.9, + "content": "\\sim 9 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 425, + 443 + ], + "score": 1.0, + "content": "of the substitute model on test data. On CIFAR binary classification, we use", + "type": "text" + }, + { + "bbox": [ + 425, + 432, + 460, + 442 + ], + "score": 0.89, + "content": "2 0 0 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 432, + 505, + 443 + ], + "score": 1.0, + "content": "datapoints", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 197, + 455 + ], + "score": 1.0, + "content": "for the inital query set", + "type": "text" + }, + { + "bbox": [ + 198, + 443, + 211, + 453 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 442, + 505, + 455 + ], + "score": 1.0, + "content": ", resulting in 12, 800 queries in total. The substitutes achieved almost the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 453, + 415, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 415, + 465 + ], + "score": 1.0, + "content": "same accuracy as their corresponding victim models on clean test datapoints.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 410, + 505, + 465 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 252, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 253, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 253, + 495 + ], + "score": 1.0, + "content": "C FURTHER EXPERIMENTS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 505, + 393, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 394, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 394, + 518 + ], + "score": 1.0, + "content": "C.1 FURTHER EXPERIMENTS ON CIFAR BINARY CLASSIFICATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 526, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 272, + 540 + ], + "score": 1.0, + "content": "White-box attack (perfect-knowledge,", + "type": "text" + }, + { + "bbox": [ + 272, + 527, + 286, + 537 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 525, + 506, + 540 + ], + "score": 1.0, + "content": ") Figure C.1 shows the perfect knowledge attack on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "the CIFAR binary classification task. Again we see that although the attack is effective for the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "detection schemes, it comes with the price of decreased mis-classification rates. Interestingly GBY", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "seems to be robust to this attack, where the accuracies on the crafted adversarial examples increase.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 525, + 506, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 446, + 595 + ], + "score": 1.0, + "content": "Sanity checks on gradient masking We conducted the same sets of transferred", + "type": "text" + }, + { + "bbox": [ + 446, + 583, + 460, + 594 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "attack ex-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "periments, and presents the results in Figure C.2 and Table C.1. Again that these crafted attacks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 140, + 617 + ], + "score": 1.0, + "content": "achieve", + "type": "text" + }, + { + "bbox": [ + 140, + 605, + 178, + 615 + ], + "score": 0.89, + "content": "\\sim 1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 604, + 399, + 617 + ], + "score": 1.0, + "content": "success rates on fooling the substitute models when", + "type": "text" + }, + { + "bbox": [ + 400, + 605, + 437, + 616 + ], + "score": 0.9, + "content": "\\epsilon \\geq 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 604, + 505, + 617 + ], + "score": 1.0, + "content": ". Similar to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "MNIST experiments, these adversarial examples do not transfer very well to the generative classi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 190, + 639 + ], + "score": 1.0, + "content": "fiers, and for a fixed", + "type": "text" + }, + { + "bbox": [ + 190, + 628, + 196, + 637 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "setting, the white-box attacks achieve significantly higher success rates than", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 314, + 650 + ], + "score": 1.0, + "content": "their grey-/black-box counterparts (with the gap at", + "type": "text" + }, + { + "bbox": [ + 314, + 638, + 348, + 649 + ], + "score": 0.89, + "content": "\\epsilon \\leq 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 637, + 380, + 650 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 381, + 638, + 401, + 649 + ], + "score": 0.86, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "). Furthermore, the mini-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "mum perturbation obtained by grey-/black-box attacks are significantly higher than those obtained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 422, + 673 + ], + "score": 1.0, + "content": "by white-box attacks. For detection, the detection rates are relatively low at", + "type": "text" + }, + { + "bbox": [ + 422, + 660, + 457, + 671 + ], + "score": 0.9, + "content": "\\epsilon \\leq 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "(where the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 443, + 684 + ], + "score": 1.0, + "content": "classifiers achieved high accuracy). But the detection rates increase significantly for", + "type": "text" + }, + { + "bbox": [ + 443, + 671, + 474, + 681 + ], + "score": 0.88, + "content": "\\epsilon = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 670, + 505, + 684 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 681, + 235, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 235, + 694 + ], + "score": 1.0, + "content": "the victim accuracies also drop.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 582, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "All these results suggest that the robustness of generative classifiers is unlikely to be caused by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "gradient masking. 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AttackGFZGBZGFYGBY
PGD(white-box)0.11 / 21.81%0.20 / 65.63%0.11 / 25.81%0.11 / 25.24%
MIM(white-box)0.09 / 15.22%0.13 /37.60%0.10 /16.4%90.09 / 14.39%
PGD(grey-box)0.15 / 50.48%0.23 / 77.30%0.16 / 54.66%0.17 / 57.96%
MIM(grey-box)0.15 /47.62%0.21 / 75.71%0.15 / 51.11%0.16 / 53.84%
PGD(black-box)0.19 / 68.36%0.23 / 79.45%0.20 / 70.13%0.19 / 67.98%
MIM(black-box)0.18 / 66.39%0.23 / 78.38%0.19 / 68.42%0.19 / 66.52%
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The higher the better. We only present generative classifiers’ results here, for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 492, + 217, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 217, + 504 + ], + "score": 1.0, + "content": "full results see appendix E.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "table", + "bbox": [ + 147, + 613, + 463, + 700 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 558, + 505, + 604 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 219, + 572 + ], + "score": 1.0, + "content": "Table C.1: Mean minimum", + "type": "text" + }, + { + "bbox": [ + 219, + 560, + 234, + 570 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 558, + 374, + 572 + ], + "score": 1.0, + "content": "perturbation (in red, computed on", + "type": "text" + }, + { + "bbox": [ + 375, + 559, + 502, + 571 + ], + "score": 0.88, + "content": "\\epsilon \\in \\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 558, + 506, + 572 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 242, + 582 + ], + "score": 1.0, + "content": "and victim accuracy (in blue, for", + "type": "text" + }, + { + "bbox": [ + 242, + 570, + 277, + 582 + ], + "score": 0.87, + "content": "\\epsilon \\leq 0 . 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 570, + 295, + 582 + ], + "score": 1.0, + "content": ") for", + "type": "text" + }, + { + "bbox": [ + 295, + 570, + 309, + 581 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "attacks on CIFAR plane-vs-frog binary classifi-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 581, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 312, + 593 + ], + "score": 1.0, + "content": "cation. We manually assign the min. perturbation", + "type": "text" + }, + { + "bbox": [ + 312, + 582, + 347, + 592 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 581, + 505, + 593 + ], + "score": 1.0, + "content": "to inputs that all attacks failed to find", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 593, + 210, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 210, + 605 + ], + "score": 1.0, + "content": "adversarial perturbations.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "table_body", + "bbox": [ + 147, + 613, + 463, + 700 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 613, + 463, + 700 + ], + "spans": [ + { + "bbox": [ + 147, + 613, + 463, + 700 + ], + "score": 0.982, + "html": "
AttackGFZGBZGFYGBY
PGD(white-box)0.11 / 21.81%0.20 / 65.63%0.11 / 25.81%0.11 / 25.24%
MIM(white-box)0.09 / 15.22%0.13 /37.60%0.10 /16.4%90.09 / 14.39%
PGD(grey-box)0.15 / 50.48%0.23 / 77.30%0.16 / 54.66%0.17 / 57.96%
MIM(grey-box)0.15 /47.62%0.21 / 75.71%0.15 / 51.11%0.16 / 53.84%
PGD(black-box)0.19 / 68.36%0.23 / 79.45%0.20 / 70.13%0.19 / 67.98%
MIM(black-box)0.18 / 66.39%0.23 / 78.38%0.19 / 68.42%0.19 / 66.52%
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GFZGBZGBYGFYDFXDBXDFZ
PGD victim acc67.7%83.9%67.9%67.3%0.3%4.2%0.4%
SPSA victim acc96.4%95.2%96.4%96.3%0.4%87.5%5.1%
SPSA TP logit10.0%17.1%15.5%10.0%N/AN/A4.5%
", + "type": "table", + "image_path": "05941d1d37a465318a19b333ee210964ee3f328c3ec80af16d2151b21b4c212a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 137, + 112, + 473, + 131.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 137, + 131.0, + 473, + 150.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 137, + 150.0, + 473, + 169.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "image", + "bbox": [ + 107, + 182, + 503, + 278 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 182, + 503, + 278 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 182, + 503, + 278 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 503, + 278 + ], + "score": 0.899, + "type": "image", + "image_path": "c2212cb826926ce9d453f97e43195c2f01d46651d793cf484348cc807383b02c.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 107, + 182, + 503, + 214.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 107, + 214.0, + 503, + 246.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 107, + 246.0, + 503, + 278.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 291, + 504, + 324 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "Figure C.3: Results on cross-model transfer attacks on the CIFAR plane-vs-frog dataset. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "horizontal axis corresponds to the source victim, and the vertical axis corresponds to the target", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 313, + 260, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 260, + 325 + ], + "score": 1.0, + "content": "victim. The higher (lighter) the better.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "SPSA (evolutionary strategies) Similarly we perform the SPSA attack (Uesato et al., 2018) on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 327, + 363 + ], + "score": 1.0, + "content": "the CIFAR binary classification task. The results for", + "type": "text" + }, + { + "bbox": [ + 327, + 351, + 368, + 361 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "are presented in Table C.2, with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "a comparison to the white-box PGD attacks. Again we see that SPSA fails to attack generative", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "classifiers, and the bottleneck discriminative classifier DBX is significantly more robust than the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 384, + 344, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 344, + 396 + ], + "score": 1.0, + "content": "discriminative ones with fully-connected graphical models.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 407, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "Cross-model attack transferability Finally, we present the CIFAR cross-model attack transfer-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 298, + 431 + ], + "score": 1.0, + "content": "ability results in Figure C.3, and here we select", + "type": "text" + }, + { + "bbox": [ + 299, + 420, + 331, + 429 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "instead. Again, transferred attacks are less", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "effective across the victim models. However, the TP rates for logit detection are significantly lower", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "than in the MNIST case (also see Figure 8). Nevertheless, the detection rates for the “discriminative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "to generative” transfer are considerably higher. Combined with the accuracy results, we see that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 461, + 439, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 439, + 476 + ], + "score": 1.0, + "content": "discriminative models as substitutes are ineffective in the transferred attack setting.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 111, + 489, + 370, + 500 + ], + "lines": [ + { + "bbox": [ + 108, + 488, + 373, + 503 + ], + "spans": [ + { + "bbox": [ + 108, + 488, + 373, + 503 + ], + "score": 1.0, + "content": "C.2 QUANTIFYING THE EFFECT OF THE BOTTLENECK LAYER", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "We see from the main text that classifiers with bottleneck structure may be preferred for resisting", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 521, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 532 + ], + "score": 1.0, + "content": "adversarial examples. To quantify this bottleneck effect, we train on MNIST models DBX, GBZ", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 167, + 545 + ], + "score": 1.0, + "content": "and GBY with", + "type": "text" + }, + { + "bbox": [ + 167, + 533, + 175, + 542 + ], + "score": 0.73, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 532, + 234, + 545 + ], + "score": 1.0, + "content": "dimensions in", + "type": "text" + }, + { + "bbox": [ + 234, + 532, + 303, + 544 + ], + "score": 0.88, + "content": "\\{ \\dot { 1 } 6 , 3 2 , 6 4 , 1 2 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 532, + 429, + 545 + ], + "score": 1.0, + "content": "(the main text experiments use", + "type": "text" + }, + { + "bbox": [ + 429, + 532, + 483, + 544 + ], + "score": 0.91, + "content": "\\dim ( z ) = 6 4 ", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 532, + 505, + 545 + ], + "score": 1.0, + "content": ").The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 542, + 502, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 502, + 556 + ], + "score": 1.0, + "content": "clean test accuracy is shown in Table C.3, showing that all models in test perform reasonably well.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "table", + "bbox": [ + 156, + 586, + 455, + 632 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 120, + 565, + 489, + 577 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 119, + 564, + 490, + 579 + ], + "spans": [ + { + "bbox": [ + 119, + 564, + 490, + 579 + ], + "score": 1.0, + "content": "Table C.3: Clean test accuracy on MNIST classification (with varied bottleneck layer sizes).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "table_body", + "bbox": [ + 156, + 586, + 455, + 632 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 156, + 586, + 455, + 632 + ], + "spans": [ + { + "bbox": [ + 156, + 586, + 455, + 632 + ], + "score": 0.974, + "html": "
dim(z) = 16dim(z) = 32dim(z)= 64dim(z)=128
DBX99.11%99.01%98.98%98.91%
GBZ97.11%97.08%97.45%96.62%
GBY98.82%98.95%98.72%98.75%
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It is clear that for dis-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 473, + 678 + ], + "score": 1.0, + "content": "criminative classifiers, DBX, the models become less robust as the bottleneck dimension", + "type": "text" + }, + { + "bbox": [ + 473, + 666, + 504, + 678 + ], + "score": 0.81, + "content": "\\mathrm { d i m } ( z )", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "increases. Interestingly DBX classifiers seem to be very robust against FGSM attacks, which agrees", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "with the results in Alemi et al. (2017). For the generative ones, we also observe similar trends (al-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "though much less significant) of decreased robustness for GBY classifiers, and for GBZ the trend", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "is unclear, presumably due to local optimum issues in optimisation. In summary, GBZ classifiers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "are generally more robust compared to GBY classifiers. More importantly, when the accuracy of", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 137, + 112, + 473, + 169 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 414, + 93 + ], + "score": 1.0, + "content": "Table C.2: Accuracy and detection rates against black-box SPSA attack", + "type": "text" + }, + { + "bbox": [ + 414, + 81, + 456, + 91 + ], + "score": 0.81, + "content": "\\epsilon = 0 . 0 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "on CIFAR", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 410, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 410, + 104 + ], + "score": 1.0, + "content": "plane-vs-frog, with a comparision to white-box PGD. The higher the better.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 137, + 112, + 473, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 112, + 473, + 169 + ], + "spans": [ + { + "bbox": [ + 137, + 112, + 473, + 169 + ], + "score": 0.923, + "html": "
GFZGBZGBYGFYDFXDBXDFZ
PGD victim acc67.7%83.9%67.9%67.3%0.3%4.2%0.4%
SPSA victim acc96.4%95.2%96.4%96.3%0.4%87.5%5.1%
SPSA TP logit10.0%17.1%15.5%10.0%N/AN/A4.5%
", + "type": "table", + "image_path": "05941d1d37a465318a19b333ee210964ee3f328c3ec80af16d2151b21b4c212a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 137, + 112, + 473, + 131.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 137, + 131.0, + 473, + 150.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 137, + 150.0, + 473, + 169.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "image", + "bbox": [ + 107, + 182, + 503, + 278 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 182, + 503, + 278 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 182, + 503, + 278 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 503, + 278 + ], + "score": 0.899, + "type": "image", + "image_path": "c2212cb826926ce9d453f97e43195c2f01d46651d793cf484348cc807383b02c.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 107, + 182, + 503, + 214.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 107, + 214.0, + 503, + 246.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 107, + 246.0, + 503, + 278.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 291, + 504, + 324 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "Figure C.3: Results on cross-model transfer attacks on the CIFAR plane-vs-frog dataset. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "horizontal axis corresponds to the source victim, and the vertical axis corresponds to the target", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 313, + 260, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 260, + 325 + ], + "score": 1.0, + "content": "victim. The higher (lighter) the better.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "SPSA (evolutionary strategies) Similarly we perform the SPSA attack (Uesato et al., 2018) on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 327, + 363 + ], + "score": 1.0, + "content": "the CIFAR binary classification task. The results for", + "type": "text" + }, + { + "bbox": [ + 327, + 351, + 368, + 361 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "are presented in Table C.2, with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "a comparison to the white-box PGD attacks. Again we see that SPSA fails to attack generative", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "classifiers, and the bottleneck discriminative classifier DBX is significantly more robust than the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 384, + 344, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 344, + 396 + ], + "score": 1.0, + "content": "discriminative ones with fully-connected graphical models.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 339, + 506, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 407, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "Cross-model attack transferability Finally, we present the CIFAR cross-model attack transfer-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 298, + 431 + ], + "score": 1.0, + "content": "ability results in Figure C.3, and here we select", + "type": "text" + }, + { + "bbox": [ + 299, + 420, + 331, + 429 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "instead. Again, transferred attacks are less", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "effective across the victim models. However, the TP rates for logit detection are significantly lower", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "than in the MNIST case (also see Figure 8). Nevertheless, the detection rates for the “discriminative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "to generative” transfer are considerably higher. Combined with the accuracy results, we see that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 461, + 439, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 439, + 476 + ], + "score": 1.0, + "content": "discriminative models as substitutes are ineffective in the transferred attack setting.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 407, + 506, + 476 + ] + }, + { + "type": "title", + "bbox": [ + 111, + 489, + 370, + 500 + ], + "lines": [ + { + "bbox": [ + 108, + 488, + 373, + 503 + ], + "spans": [ + { + "bbox": [ + 108, + 488, + 373, + 503 + ], + "score": 1.0, + "content": "C.2 QUANTIFYING THE EFFECT OF THE BOTTLENECK LAYER", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "We see from the main text that classifiers with bottleneck structure may be preferred for resisting", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 521, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 532 + ], + "score": 1.0, + "content": "adversarial examples. To quantify this bottleneck effect, we train on MNIST models DBX, GBZ", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 167, + 545 + ], + "score": 1.0, + "content": "and GBY with", + "type": "text" + }, + { + "bbox": [ + 167, + 533, + 175, + 542 + ], + "score": 0.73, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 532, + 234, + 545 + ], + "score": 1.0, + "content": "dimensions in", + "type": "text" + }, + { + "bbox": [ + 234, + 532, + 303, + 544 + ], + "score": 0.88, + "content": "\\{ \\dot { 1 } 6 , 3 2 , 6 4 , 1 2 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 532, + 429, + 545 + ], + "score": 1.0, + "content": "(the main text experiments use", + "type": "text" + }, + { + "bbox": [ + 429, + 532, + 483, + 544 + ], + "score": 0.91, + "content": "\\dim ( z ) = 6 4 ", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 532, + 505, + 545 + ], + "score": 1.0, + "content": ").The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 542, + 502, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 502, + 556 + ], + "score": 1.0, + "content": "clean test accuracy is shown in Table C.3, showing that all models in test perform reasonably well.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 509, + 506, + 556 + ] + }, + { + "type": "table", + "bbox": [ + 156, + 586, + 455, + 632 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 120, + 565, + 489, + 577 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 119, + 564, + 490, + 579 + ], + "spans": [ + { + "bbox": [ + 119, + 564, + 490, + 579 + ], + "score": 1.0, + "content": "Table C.3: Clean test accuracy on MNIST classification (with varied bottleneck layer sizes).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "table_body", + "bbox": [ + 156, + 586, + 455, + 632 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 156, + 586, + 455, + 632 + ], + "spans": [ + { + "bbox": [ + 156, + 586, + 455, + 632 + ], + "score": 0.974, + "html": "
dim(z) = 16dim(z) = 32dim(z)= 64dim(z)=128
DBX99.11%99.01%98.98%98.91%
GBZ97.11%97.08%97.45%96.62%
GBY98.82%98.95%98.72%98.75%
", + "type": "table", + "image_path": "fb730c8aea9e853f23bbeeb113988771173ea29b3716a47400ad668d4926df9c.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 156, + 586, + 455, + 601.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 156, + 601.3333333333334, + 455, + 616.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 156, + 616.6666666666667, + 455, + 632.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "index": 28.0 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 305, + 656 + ], + "score": 1.0, + "content": "We repeat the same white-box zero-knowledge", + "type": "text" + }, + { + "bbox": [ + 305, + 644, + 320, + 655 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "attack experiments as done in the main text,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "where results are presented in Figure C.4 and Tables E.25, E.26 and E.27. It is clear that for dis-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 473, + 678 + ], + "score": 1.0, + "content": "criminative classifiers, DBX, the models become less robust as the bottleneck dimension", + "type": "text" + }, + { + "bbox": [ + 473, + 666, + 504, + 678 + ], + "score": 0.81, + "content": "\\mathrm { d i m } ( z )", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "increases. Interestingly DBX classifiers seem to be very robust against FGSM attacks, which agrees", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "with the results in Alemi et al. (2017). For the generative ones, we also observe similar trends (al-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "though much less significant) of decreased robustness for GBY classifiers, and for GBZ the trend", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "is unclear, presumably due to local optimum issues in optimisation. In summary, GBZ classifiers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "are generally more robust compared to GBY classifiers. More importantly, when the accuracy of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "generative classifiers on adversarial images decreases to zero, the detection rates with marginal/logit", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 197, + 367 + ], + "score": 1.0, + "content": "detection increases to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 197, + 355, + 221, + 366 + ], + "score": 0.9, + "content": "1 0 0 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 222, + 355, + 505, + 367 + ], + "score": 1.0, + "content": ". This clearly shows that the three attacks tested here cannot fool the", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 367, + 287, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 287, + 378 + ], + "score": 1.0, + "content": "generative classifiers without being detected.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 140, + 79, + 470, + 236 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 140, + 79, + 470, + 236 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 79, + 470, + 236 + ], + "spans": [ + { + "bbox": [ + 140, + 79, + 470, + 236 + ], + "score": 0.972, + "type": "image", + "image_path": "ac97b61ab1e7245d921f06ec5438eb1e3ec5e1ff1b11e353e9a44c52679e086b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 140, + 79, + 470, + 131.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 140, + 131.33333333333334, + 470, + 183.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 140, + 183.66666666666669, + 470, + 236.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 245, + 504, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 413, + 258 + ], + "score": 1.0, + "content": "Figure C.4: Accuracy and detection rates against white-box zero-knowledge", + "type": "text" + }, + { + "bbox": [ + 413, + 246, + 427, + 256 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "attacks on MNIST,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 256, + 244, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 244, + 268 + ], + "score": 1.0, + "content": "with varied bottleneck layer sizes.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "table", + "bbox": [ + 135, + 299, + 474, + 324 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 164, + 279, + 445, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 164, + 277, + 446, + 293 + ], + "spans": [ + { + "bbox": [ + 164, + 277, + 446, + 293 + ], + "score": 1.0, + "content": "Table D.1: Clean test accuracy on CIFAR plane-vs-frog classification.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 135, + 299, + 474, + 324 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 299, + 474, + 324 + ], + "spans": [ + { + "bbox": [ + 135, + 299, + 474, + 324 + ], + "score": 0.957, + "html": "
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GFY94.374.846.521.710.553.192.6100.0100.0100.066.297.9100.0100.0100.090.893.193.894.294.2
GBY93.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
DFX70.114.81.00.40.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.693.893.893.594.3
DBX91.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DFZ75.219.02.41.11.012.650.6100.0100.0100.026.965.7100.0100.0100.093.195.194.594.994.8
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acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN83.212.00.50.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.192.188.720.40.2
GFZ86.737.77.71.20.343.271.891.499.4100.055.877.194.899.6100.090.392.190.990.489.7
GBZ85.157.432.519.311.633.750.384.499.7100.052.266.391.599.8100.090.091.591.591.791.8
GFY79.727.45.61.20.358.187.998.0100.0100.068.792.299.3100.0100.092.692.590.790.785.4
GBY86.735.99.01.70.445.376.194.699.7100.056.979.395.699.9100.090.192.191.691.791.4
DFX47.60.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.889.813.331.442.8
DBX DFZ58.0 49.618.36.01.30.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
1.00.00.00.011.844.394.1100.0100.025.957.694.599.9100.093.989.812.122.928.0
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acc.(adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN82.07.20.10.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.291.172.064.57.1
GFZ87.040.49.01.21.243.583.898.499.499.457.689.398.699.299.291.592.392.090.690.6
GBZ79.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GFY80.830.16.81.41.456.093.799.8100.0100.068.196.699.9100.0100.090.992.492.091.691.6
GBY84.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
DFX48.40.80.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.289.69.217.517.5
DBX66.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DFZ50.51.20.00.00.011.950.799.4100.0100.026.463.299.0100.0100.094.389.811.517.317.3
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acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN92.467.840.526.220.4N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.094.295.495.596.5
GFZ94.274.538.912.95.743.679.8100.0100.0100.056.489.699.9100.0100.089.291.792.292.592.2
GBZ92.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GFY94.374.846.521.710.553.192.6100.0100.0100.066.297.9100.0100.0100.090.893.193.894.294.2
GBY93.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
DFX70.114.81.00.40.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.693.893.893.594.3
DBX91.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DFZ75.219.02.41.11.012.650.6100.0100.0100.026.965.7100.0100.0100.093.195.194.594.994.8
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acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN83.212.00.50.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.192.188.720.40.2
GFZ86.737.77.71.20.343.271.891.499.4100.055.877.194.899.6100.090.392.190.990.489.7
GBZ85.157.432.519.311.633.750.384.499.7100.052.266.391.599.8100.090.091.591.591.791.8
GFY79.727.45.61.20.358.187.998.0100.0100.068.792.299.3100.0100.092.692.590.790.785.4
GBY86.735.99.01.70.445.376.194.699.7100.056.979.395.699.9100.090.192.191.691.791.4
DFX47.60.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.889.813.331.442.8
DBX DFZ58.0 49.618.36.01.30.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
1.00.00.00.011.844.394.1100.0100.025.957.694.599.9100.093.989.812.122.928.0
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acc.(adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN82.07.20.10.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.291.172.064.57.1
GFZ87.040.49.01.21.243.583.898.499.499.457.689.398.699.299.291.592.392.090.690.6
GBZ79.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GFY80.830.16.81.41.456.093.799.8100.0100.068.196.699.9100.0100.090.992.492.091.691.6
GBY84.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
DFX48.40.80.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.289.69.217.517.5
DBX66.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DFZ50.51.20.00.00.011.950.799.4100.0100.026.463.299.0100.0100.094.389.811.517.317.3
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ZKPKK
acc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KL
DBX18.3N/AN/A9417N/AN/A93.1N/AN/AN/AN/AN/AN/AN/AN/A35.9N/AN/A93.9
GBZ57.450.366.391.535.582.290.791.447.936.567.491.344.650.569.991.04573.084.891.5
GBY35.976.179.392.12290.891.992.143.847.061.991.152.056.466.990.632.889.291.893.3
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acc. (adv)TP marginal
E0.010.020.050.100.200.010.020.050.100.20 0.010.020.050.100.200.010.020.050.100.20
BNN98.293.258.514.56.3N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A58.655.165.554.058.5
GFZ97.194.781.856.431.411.410.15.9 17.499.311.410.46.817.299.358.640.135.941.050.6
GBZ95.193.587.174.962.026.018.615.5 38.699.626.320.319.141.999.635.541.547.043.745.6
GFY96.594.280.756.732.09.69.26.9 18.899.017.613.18.120.899.146.643.736.139.549.3
GBY96.192.982.060.536.317.215.18.1 28.599.220.218.110.031.199.249.748.039.238.747.3
DFX83.842.20.70.00.0N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A63.366.648.497.7100.0
DBX90.978.650.731.718.3N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A50.758.756.859.757.2
DFZ83.952.12.90.00.06.63.82.23.9 60.66.94.22.43.360.457.562.354.866.799.8
", + "type": "table", + "image_path": "e0c7744832753cf7a05cd70afc998f5fb15062962639fce97c6b30c6d96d6a6f.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 146, + 254, + 626, + 286.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 146, + 286.6666666666667, + 626, + 319.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 146, + 319.33333333333337, + 626, + 352.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + } + ], + "index": 7.0 + }, + { + "type": "table", + "bbox": [ + 144, + 397, + 628, + 495 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 177, + 376, + 593, + 388 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 177, + 375, + 593, + 390 + ], + "spans": [ + { + "bbox": [ + 177, + 375, + 593, + 390 + ], + "score": 0.99, + "content": " Table E.6: PGD white-box zero-knowledge atack results on CIFAR plane-vs-frog binary classification.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "table_body", + "bbox": [ + 144, + 397, + 628, + 495 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 144, + 397, + 628, + 495 + ], + "spans": [ + { + "bbox": [ + 144, + 397, + 628, + 495 + ], + "score": 0.983, + "html": "
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN97.986.719.71.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A41.759.455.768.498.9
GFZ98.093.967.721.73.53.36.65.67.832.95.09.56.08.233.037.545.134.544.057.7
GBZ94.693.783.967.452.819.817.912.814.343.122.819.517.417.044.332.139.340.237.133.2
GFY98.495.067.925.84.14.26.76.47.835.73.18.27.67.833.831.243.632.939.653.3
GBY96.492.967.325.76.911.18.47.911.641.413.111.59.212.240.343.043.033.639.152.2
DFX82.735.70.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A64.164.769.6100.0100.0
DBX83.334.64.20.70.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A59.859.565.178.793.4
DFZ82.436.90.40.00.04.93.85.112.391.16.54.15.512.391.356.360.354.799.7100.0
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ZKPKK
acc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KL
DBX18.3N/AN/A9417N/AN/A93.1N/AN/AN/AN/AN/AN/AN/AN/A35.9N/AN/A93.9
GBZ57.450.366.391.535.582.290.791.447.936.567.491.344.650.569.991.04573.084.891.5
GBY35.976.179.392.12290.891.992.143.847.061.991.152.056.466.990.632.889.291.893.3
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acc. (adv)TP marginal
E0.010.020.050.100.200.010.020.050.100.20 0.010.020.050.100.200.010.020.050.100.20
BNN98.293.258.514.56.3N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A58.655.165.554.058.5
GFZ97.194.781.856.431.411.410.15.9 17.499.311.410.46.817.299.358.640.135.941.050.6
GBZ95.193.587.174.962.026.018.615.5 38.699.626.320.319.141.999.635.541.547.043.745.6
GFY96.594.280.756.732.09.69.26.9 18.899.017.613.18.120.899.146.643.736.139.549.3
GBY96.192.982.060.536.317.215.18.1 28.599.220.218.110.031.199.249.748.039.238.747.3
DFX83.842.20.70.00.0N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A63.366.648.497.7100.0
DBX90.978.650.731.718.3N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A50.758.756.859.757.2
DFZ83.952.12.90.00.06.63.82.23.9 60.66.94.22.43.360.457.562.354.866.799.8
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acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN97.986.719.71.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A41.759.455.768.498.9
GFZ98.093.967.721.73.53.36.65.67.832.95.09.56.08.233.037.545.134.544.057.7
GBZ94.693.783.967.452.819.817.912.814.343.122.819.517.417.044.332.139.340.237.133.2
GFY98.495.067.925.84.14.26.76.47.835.73.18.27.67.833.831.243.632.939.653.3
GBY96.492.967.325.76.911.18.47.911.641.413.111.59.212.240.343.043.033.639.152.2
DFX82.735.70.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A64.164.769.6100.0100.0
DBX83.334.64.20.70.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A59.859.565.178.793.4
DFZ82.436.90.40.00.04.93.85.112.391.16.54.15.512.391.356.360.354.799.7100.0
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acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN96.984.618.70.90.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A35.954.756.362.478.8
GFZ96.991.558.115.21.96.56.26.410.686.110.08.36.611.384.531.039.032.147.277.1
GBZ92.589.471.540.417.915.013.211.024.095.916.414.812.825.196.042.537.531.037.455.0
GFY97.692.359.516.52.35.66.96.313.388.15.76.26.813.887.623.841.433.843.974.6
GBY95.188.856.514.61.315.19.58.619.297.315.310.79.918.597.051.143.231.746.787.6
DFX82.535.20.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A65.567.169.6100.0100.0
DBX82.538.86.11.20.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A54.660.061.578.888.5
DFZ81.536.10.40.00.05.43.95.013.496.96.24.15.513.896.861.158.555.399.9100.0
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acc. (adv)TP marginalTP logitTP KL
E0.101.0010.00100.001000.000.101.0010.001100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
BNN93.766.237.332.347.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A60.980.382.272.080.0
GFZ99.595.676.577.978.30.04.52.63.03.40.04.63.03.64.229.230.425.627.129.0
GBZ96.093.688.988.789.016.711.57.87.77.918.913.79.39.19.441.151.440.434.732.0
GFY99.895.978.880.080.50.06.64.14.75.10.06.64.84.95.475.037.326.026.431.1
GBY97.593.176.777.378.015.814.66.07.17.318.913.96.07.07.942.634.327.231.629.0
DFX82.644.234.36.10.6N/AN/AN/AN/AN/AN/AN/A N/AN/AN/A100.0100.0100.082.595.0
DBX96.572.226.36.316.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A61.679.485.262.267.9
DFZ94.572.329.96.317.97.35.84.34.14.214.16.95.04.44.482.697.397.183.080.2
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PKK
acc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KL
DBX0.7N/AN/A78.70.7N/A82.5N/AN/AN/AN/AN/AN/AN/A0.6N/AN/A 27.682.6
GBZ67.414.317.037.15740.645.14958.49.618.546.958.51.87.847.859.925.542.4
GBY25.711.612.239.128.522.022.64929.55.67.645.929.20.81.845.427.915.615.945
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acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN96.984.618.70.90.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A35.954.756.362.478.8
GFZ96.991.558.115.21.96.56.26.410.686.110.08.36.611.384.531.039.032.147.277.1
GBZ92.589.471.540.417.915.013.211.024.095.916.414.812.825.196.042.537.531.037.455.0
GFY97.692.359.516.52.35.66.96.313.388.15.76.26.813.887.623.841.433.843.974.6
GBY95.188.856.514.61.315.19.58.619.297.315.310.79.918.597.051.143.231.746.787.6
DFX82.535.20.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A65.567.169.6100.0100.0
DBX82.538.86.11.20.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A54.660.061.578.888.5
DFZ81.536.10.40.00.05.43.95.013.496.96.24.15.513.896.861.158.555.399.9100.0
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acc. (adv)TP marginalTP logitTP KL
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BNN93.766.237.332.347.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A60.980.382.272.080.0
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DFX82.644.234.36.10.6N/AN/AN/AN/AN/AN/AN/A N/AN/AN/A100.0100.0100.082.595.0
DBX96.572.226.36.316.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A61.679.485.262.267.9
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PKK
acc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KL
DBX0.7N/AN/A78.70.7N/A82.5N/AN/AN/AN/AN/AN/AN/A0.6N/AN/A 27.682.6
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substitute accvictimaccTP logit
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substitute accvictim accTP logit
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GFZ87.216.10.00.00.096.582.442.814.23.750.682.5100.0100.0100.0
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DFX76.512.33.62.12.191.556.122.29.99.9N/AN/AN/AN/AN/A
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substitute accvictim accTP logit
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GFZ98.44.60.00.00.098.896.996.093.490.253.444.241.640.643.7
GBZ98.463.20.00.00.097.495.492.988.684.545.939.733.031.238.6
GFY98.10.90.00.00.099.097.697.095.794.061.055.249.249.552.7
GBY98.35.30.00.00.098.796.896.093.891.254.241.537.638.743.6
DFX85.10.00.00.00.097.693.391.290.489.8N/AN/AN/AN/AN/A
DBX97.346.00.40.00.098.793.988.485.576.3N/AN/AN/AN/AN/A
DFZ80.50.00.00.00.097.294.692.391.591.232.928.628.136.544.5
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substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ86.714.30.00.00.096.683.351.326.817.249.573.899.7100.0100.0
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substitute accvictim accTP logit
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GFZ87.216.10.00.00.096.582.442.814.23.750.682.5100.0100.0100.0
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substitute accvictim accTP logit
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substitute accvictim accTP logit
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substitute accvictim accTP logit
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substitute accvictim accTP logit
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substitute accvictim accTP logit
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substitute accvictim accTP logit
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substitute accvictim accTP logit
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substitute accvictimaccTP logit
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substitute accvictimaccTP logit
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substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.0010.00100.001000.00
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substitute accvictimaccTP logit
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substitute accvictimaccTP logit
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substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.0010.00100.001000.00
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substitute accvictim accTP logit
E0.010.020.050.100.200.010.02 0.050.100.200.010.020.050.100.20
GFZ95.589.945.56.80.197.695.688.768.829.76.26.76.010.871.8
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GBY91.880.529.76.10.297.295.990.270.830.321.816.49.613.677.3
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substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
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DFX89.074.519.40.90.099.698.792.163.010.9N/AN/AN/AN/AN/A
DBX85.576.342.74.50.099.699.095.176.527.2N/AN/AN/AN/AN/A
DFZ85.373.421.41.60.099.699.192.967.714.90.04.57.514.291.9
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substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.00TP logit 10.00100.001000.00
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GBY93.533.61.70.00.097.795.9 95.495.495.025.816.415.918.619.8
DFX90.310.90.00.00.099.998.8 98.598.597.9N/AN/AN/AN/AN/A
DBX87.231.80.00.00.099.997.1 94.593.994.7N/AN/AN/AN/AN/A
DFZ84.922.70.00.00.099.999.0 98.398.397.80.03.82.32.311.0
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substitute accvictim accTP logit
E0.010.020.050.100.200.010.02 0.050.100.200.010.020.050.100.20
GFZ95.589.945.56.80.197.695.688.768.829.76.26.76.010.871.8
GBZ92.084.133.83.50.095.694.992.785.065.916.714.612.817.976.8
GFY95.389.138.72.50.097.396.690.072.935.50.02.36.810.472.3
GBY91.880.529.76.10.297.295.990.270.830.321.816.49.613.677.3
DFX89.174.519.10.90.099.798.792.664.89.9N/AN/AN/AN/AN/A
DBX85.576.442.54.50.099.699.094.877.524.5N/AN/AN/AN/AN/A
DFZ85.573.521.11.50.099.599.293.870.115.10.05.08.915.985.1
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substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ95.589.746.47.70.297.595.788.566.520.56.26.75.912.782.7
GBZ91.884.033.13.40.095.694.992.385.160.716.715.514.221.697.9
GFY95.388.940.53.10.097.396.789.570.631.90.02.47.612.588.0
GBY91.780.229.55.90.197.395.989.669.026.122.216.19.714.095.9
DFX89.074.519.40.90.099.698.792.163.010.9N/AN/AN/AN/AN/A
DBX85.576.342.74.50.099.699.095.176.527.2N/AN/AN/AN/AN/A
DFZ85.373.421.41.60.099.699.192.967.714.90.04.57.514.291.9
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substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.00TP logit 10.00100.001000.00
GFZ96.341.50.50.00.098.794.793.4 93.392.313.3 8.67.77.78.9
GBZ94.973.31.40.00.095.995.3 94.694.594.513.511.811.813.213.1
GFY96.462.01.30.00.098.196.7 94.894.794.10.02.86.56.16.8
GBY93.533.61.70.00.097.795.9 95.495.495.025.816.415.918.619.8
DFX90.310.90.00.00.099.998.8 98.598.597.9N/AN/AN/AN/AN/A
DBX87.231.80.00.00.099.997.1 94.593.994.7N/AN/AN/AN/AN/A
DFZ84.922.70.00.00.099.999.0 98.398.397.80.03.82.32.311.0
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acc. (adv) 0.020.050.100.20TP marginalTP logitTP KL
E0.010.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1644.525.816.611.910.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A95.195.797.097.597.1
GBZ-FC163.749.734.318.912.760.062.480.690.994.163.765.080.695.098.490.190.892.392.692.6
GBY-FC164.950.036.119.511.352.259.080.490.192.554.661.980.395.698.490.791.392.192.490.3
DBX-FC157.345.731.816.310.8N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.893.294.094.392.8
GBZ-CONV981.674.158.548.945.813.717.028.476.090.218.320.631.764.371.691.791.591.793.386.4
GBY-CONV975.866.039.722.517.514.018.229.474.488.917.720.825.855.268.989.689.990.089.189.2
DBX-CONV959.045.731.817.811.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.794.194.995.293.7
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acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1618.80.60.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.991.867.469.966.4
GBZ-FC137.76.10.10.00.030.722.276.092.094.133.524.977.091.790.290.893.797.398.595.9
GBY-FC131.28.52.61.50.632.020.382.395.197.636.629.584.193.790.490.394.197.797.396.9
DBX-FC123.41.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.991.05.67.45.9
GBZ-CONV961.138.324.210.22.319.260.396.798.999.620.160.396.898.398.790.393.197.798.799.6
GBY-CONV966.526.63.20.40.017.055.695.799.299.617.852.795.398.697.689.691.898.799.496.9
DBX-CONV924.03.00.30.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.393.199.399.699.2
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acc. (adv)TP marginalTP logit
E0.010.02（0.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.10 0.20
VGG1613.10.40.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A92.292.472.377.179.0
GBZ-FC124.52.70.10.00.034.840.988.595.896.0 38.844.090.496.496.691.395.095.797.996.8
GBY-FC133.019.611.87.24.929.138.795.799.199.3 38.449.296.399.399.490.795.097.897.997.9
DBX-FC117.30.90.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.189.69.614.516.0
GBZ-CONV957.643.529.124.122.627.478.599.499.6 99.729.178.399.699.799.890.294.497.598.098.2
GBY-CONV939.118.14.31.91.525.670.498.699.699.8 28.271.198.699.799.889.694.599.499.899.7
DBX-CONV919.23.20.50.40.4N/AN/AN/AN/AN/A N/AN/AN/AN/AN/A92.196.799.9100.0100.0
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acc. (adv) 0.020.050.100.20TP marginalTP logitTP KL
E0.010.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1644.525.816.611.910.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A95.195.797.097.597.1
GBZ-FC163.749.734.318.912.760.062.480.690.994.163.765.080.695.098.490.190.892.392.692.6
GBY-FC164.950.036.119.511.352.259.080.490.192.554.661.980.395.698.490.791.392.192.490.3
DBX-FC157.345.731.816.310.8N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.893.294.094.392.8
GBZ-CONV981.674.158.548.945.813.717.028.476.090.218.320.631.764.371.691.791.591.793.386.4
GBY-CONV975.866.039.722.517.514.018.229.474.488.917.720.825.855.268.989.689.990.089.189.2
DBX-CONV959.045.731.817.811.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.794.194.995.293.7
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acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1618.80.60.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.991.867.469.966.4
GBZ-FC137.76.10.10.00.030.722.276.092.094.133.524.977.091.790.290.893.797.398.595.9
GBY-FC131.28.52.61.50.632.020.382.395.197.636.629.584.193.790.490.394.197.797.396.9
DBX-FC123.41.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.991.05.67.45.9
GBZ-CONV961.138.324.210.22.319.260.396.798.999.620.160.396.898.398.790.393.197.798.799.6
GBY-CONV966.526.63.20.40.017.055.695.799.299.617.852.795.398.697.689.691.898.799.496.9
DBX-CONV924.03.00.30.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.393.199.399.699.2
", + "type": "table", + "image_path": "74cb85fa8228d6c4c1b69f9aa9c5b01b47277ed6b864db9980f356b0ddc29e61.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 132, + 271, + 640, + 300.6666666666667 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 132, + 300.6666666666667, + 640, + 330.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 132, + 330.33333333333337, + 640, + 360.00000000000006 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 5.0 + }, + { + "type": "table", + "bbox": [ + 127, + 406, + 644, + 495 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 238, + 385, + 533, + 397 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 237, + 385, + 533, + 398 + ], + "spans": [ + { + "bbox": [ + 237, + 385, + 533, + 398 + ], + "score": 0.99, + "content": "Table E.24: MIM white-box zero-knowledge attack results on CIFAR-10.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "table_body", + "bbox": [ + 127, + 406, + 644, + 495 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 127, + 406, + 644, + 495 + ], + "spans": [ + { + "bbox": [ + 127, + 406, + 644, + 495 + ], + "score": 0.98, + "html": "
acc. (adv)TP marginalTP logit
E0.010.02（0.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.10 0.20
VGG1613.10.40.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A92.292.472.377.179.0
GBZ-FC124.52.70.10.00.034.840.988.595.896.0 38.844.090.496.496.691.395.095.797.996.8
GBY-FC133.019.611.87.24.929.138.795.799.199.3 38.449.296.399.399.490.795.097.897.997.9
DBX-FC117.30.90.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.189.69.614.516.0
GBZ-CONV957.643.529.124.122.627.478.599.499.6 99.729.178.399.699.799.890.294.497.598.098.2
GBY-CONV939.118.14.31.91.525.670.498.699.699.8 28.271.198.699.799.889.694.599.499.899.7
DBX-CONV919.23.20.50.40.4N/AN/AN/AN/AN/A N/AN/AN/AN/AN/A92.196.799.9100.0100.0
", + "type": "table", + "image_path": "25c2b7bc2ca7263bb2a4012b3fd682ba9ea99fbef41be0e8fae22940ac955f93.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 127, + 406, + 644, + 435.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 127, + 435.6666666666667, + 644, + 465.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 127, + 465.33333333333337, + 644, + 495.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + } + ], + "index": 9.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 126, + 146, + 646, + 282 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 166, + 126, + 600, + 138 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 124, + 601, + 139 + ], + "spans": [ + { + "bbox": [ + 168, + 124, + 601, + 139 + ], + "score": 1.0, + "content": "Table E.25 : FGSM white-box zero-knowledge attack results on MNIST (with varied bottleneck layer sizes) .", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 126, + 146, + 646, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 146, + 646, + 282 + ], + "spans": [ + { + "bbox": [ + 126, + 146, + 646, + 282 + ], + "score": 0.985, + "html": "
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1692.685.676.464.752.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A90.392.492.392.794.2
DBX-3292.684.471.157.246.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.292.392.894.995.0
DBX-6491.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DBX-12887.847.620.112.710.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A94.497.697.897.196.4
GBZ-1692.075.852.534.021.728.771.599.9100.0100.049.286.399.9100.0100.090.890.692.091.990.9
GBZ-3291.174.452.433.421.626.573.599.7100.0100.049.487.299.9100.0100.091.090.691.891.491.6
GBZ-6492.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GBZ-12890.876.857.538.924.826.663.099.6100.0100.044.978.799.8100.0100.087.990.190.791.190.6
GBY-1694.277.749.423.912.141.984.1100.0100.0100.052.991.9100.0100.0100.086.692.093.093.192.6
GBY-3294.576.945.420.09.641.483.3100.0100.0100.056.392.8100.0100.0100.089.091.693.393.293.0
GBY-6493.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
GBY-12893.072.942.218.59.037.4：71.6100.0100.0100.050.181.999.9100.0100.090.791.4 91.891.492.0
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acc. (adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1669.836.616.05.91.8N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.291.992.893.393.7
DBX-3263.526.812.56.22.7N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A91.592.192.792.993.5
DBX-6458.018.36.01.30.2N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
DBX-12842.33.01.10.30.2N/AN/A N/AN/AN/AN/AN/AN/AN/AN/A97.197.198.498.498.6
GBZ-1688.362.837.423.115.527.135.871.9 98.9100.046.353.082.399.3100.089.691.792.391.491.8
GBZ-3287.161.940.228.020.424.039.176.798.9 100.041.556.485.499.7100.091.391.091.191.391.1
GBZ-6485.157.432.519.311.633.750.384.4 99.7100.052.266.391.599.8100.090.091.591.591.791.8
GBZ-12884.457.536.825.216.825.435.466.1 96.6100.041.550.576.497.7100.088.690.490.490.090.6
GBY-1690.449.715.04.21.642.664.789.9 99.3100.051.170.892.899.5100.091.993.191.990.791.0
GBY-3288.439.89.51.80.645.674.392.8 99.5100.056.878.294.899.8100.090.892.291.791.090.6
GBY-6486.735.99.01.70.445.376.194.6 99.7100.056.979.395.699.9100.090.192.191.691.791.4
GBY-12883.335.18.72.30.839.163.986.7 98.3100.051.769.189.298.9100.089.890.790.489.789.5
", + "type": "table", + "image_path": "576b993b0f77f5959a5d9abc2a897388689c88cd2e76b31db9ba90bfd6fb86a5.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 135, + 348, + 637, + 393.6666666666667 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 135, + 393.6666666666667, + 637, + 439.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 135, + 439.33333333333337, + 637, + 485.00000000000006 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 5.0 + } + ], + "page_idx": 28, + "page_size": [ + 792, + 612 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 30, + 299, + 40, + 311 + ], + "lines": [ + { + "bbox": [ + 29, + 299, + 41, + 313 + ], + "spans": [ + { + "bbox": [ + 29, + 299, + 41, + 313 + ], + "score": 1.0, + "content": "29", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 750, + 106, + 773, + 504 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 126, + 146, + 646, + 282 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 166, + 126, + 600, + 138 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 124, + 601, + 139 + ], + "spans": [ + { + "bbox": [ + 168, + 124, + 601, + 139 + ], + "score": 1.0, + "content": "Table E.25 : FGSM white-box zero-knowledge attack results on MNIST (with varied bottleneck layer sizes) .", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 126, + 146, + 646, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 146, + 646, + 282 + ], + "spans": [ + { + "bbox": [ + 126, + 146, + 646, + 282 + ], + "score": 0.985, + "html": "
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1692.685.676.464.752.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A90.392.492.392.794.2
DBX-3292.684.471.157.246.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.292.392.894.995.0
DBX-6491.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DBX-12887.847.620.112.710.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A94.497.697.897.196.4
GBZ-1692.075.852.534.021.728.771.599.9100.0100.049.286.399.9100.0100.090.890.692.091.990.9
GBZ-3291.174.452.433.421.626.573.599.7100.0100.049.487.299.9100.0100.091.090.691.891.491.6
GBZ-6492.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GBZ-12890.876.857.538.924.826.663.099.6100.0100.044.978.799.8100.0100.087.990.190.791.190.6
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GBY-6493.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
GBY-12893.072.942.218.59.037.4：71.6100.0100.0100.050.181.999.9100.0100.090.791.4 91.891.492.0
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acc. (adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1669.836.616.05.91.8N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.291.992.893.393.7
DBX-3263.526.812.56.22.7N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A91.592.192.792.993.5
DBX-6458.018.36.01.30.2N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
DBX-12842.33.01.10.30.2N/AN/A N/AN/AN/AN/AN/AN/AN/AN/A97.197.198.498.498.6
GBZ-1688.362.837.423.115.527.135.871.9 98.9100.046.353.082.399.3100.089.691.792.391.491.8
GBZ-3287.161.940.228.020.424.039.176.798.9 100.041.556.485.499.7100.091.391.091.191.391.1
GBZ-6485.157.432.519.311.633.750.384.4 99.7100.052.266.391.599.8100.090.091.591.591.791.8
GBZ-12884.457.536.825.216.825.435.466.1 96.6100.041.550.576.497.7100.088.690.490.490.090.6
GBY-1690.449.715.04.21.642.664.789.9 99.3100.051.170.892.899.5100.091.993.191.990.791.0
GBY-3288.439.89.51.80.645.674.392.8 99.5100.056.878.294.899.8100.090.892.291.791.090.6
GBY-6486.735.99.01.70.445.376.194.6 99.7100.056.979.395.699.9100.090.192.191.691.791.4
GBY-12883.335.18.72.30.839.163.986.7 98.3100.051.769.189.298.9100.089.890.790.489.789.5
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acc. (adv)TP marginalTP logitTP KL
E0.10（0.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1672.237.120.714.711.7N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.392.893.493.9
DBX-3266.727.616.812.510.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.592.493.293.5
DBX-6466.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DBX-12842.31.30.20.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A97.297.698.698.999.1
GBZ-1683.836.48.92.30.926.979.899.7100.0100.043.388.099.8100.0100.091.692.191.291.692.6
GBZ-3282.735.59.53.01.126.683.299.7100.0100.045.890.099.8100.0100.090.391.791.492.592.6
GBZ-6479.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GBZ-12879.232.18.72.61.226.168.599.2100.0100.041.279.199.4100.0100.089.791.591.491.492.5
GBY-1689.234.63.50.20.044.888.199.8100.0100.055.991.799.7100.0100.089.492.591.090.636.9
GBY-3286.725.81.60.10.045.191.2100.0100.0100.058.694.0100.0100.0100.090.991.891.1 68.631.1
GBY-6484.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
GBY-12882.123.71.90.00.040.483.099.7100.0100.051.787.599.6100.0100.091.291.289.882.844.4
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acc. (adv)TP marginalTP logitTP KL
E0.10（0.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
DBX-1672.237.120.714.711.7N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.392.893.493.9
DBX-3266.727.616.812.510.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.892.592.493.293.5
DBX-6466.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DBX-12842.31.30.20.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A97.297.698.698.999.1
GBZ-1683.836.48.92.30.926.979.899.7100.0100.043.388.099.8100.0100.091.692.191.291.692.6
GBZ-3282.735.59.53.01.126.683.299.7100.0100.045.890.099.8100.0100.090.391.791.492.592.6
GBZ-6479.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GBZ-12879.232.18.72.61.226.168.599.2100.0100.041.279.199.4100.0100.089.791.591.491.492.5
GBY-1689.234.63.50.20.044.888.199.8100.0100.055.991.799.7100.0100.089.492.591.090.636.9
GBY-3286.725.81.60.10.045.191.2100.0100.0100.058.694.0100.0100.0100.090.991.891.1 68.631.1
GBY-6484.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
GBY-12882.123.71.90.00.040.483.099.7100.0100.051.787.599.6100.0100.091.291.289.882.844.4
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MIM(white-box)0.24 / 9.02%0.21 /4.97%0.22 / 6.72%0.21 /1.54%
PGD(grey-box)0.37 /51.08%0.36 / 50.64%0.38 / 53.29%0.36 / 48.66%
MIM(grey-box)0.34 / 43.00%0.33 / 40.94%0.34 / 46.64%0.33 /40.06%
PGD(black-box)0.40 / 61.93%0.42 /66.75%0.38 / 56.35%0.43 / 68.50%
MIM(black-box)0.36 / 50.44%0.38 / 59.86%0.36 / 48.07%0.39 / 61.78%
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acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN92.467.840.526.220.4N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.094.295.495.596.5
GFZ94.274.538.912.95.743.679.8100.0100.0100.056.489.699.9100.0100.089.291.792.292.592.2
GBZ92.580.362.042.427.237.081.7100.0100.0100.057.693.5100.0100.0100.091.690.990.391.191.5
GFY94.374.846.521.710.553.192.6100.0100.0100.066.297.9100.0100.0100.090.893.193.894.294.2
GBY93.676.447.522.310.741.984.5100.0100.0100.057.792.5100.0100.0100.089.392.792.892.992.9
DFX70.114.81.00.40.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.693.893.893.594.3
DBX91.677.858.144.536.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.693.695.296.397.0
DFZ75.219.02.41.11.012.650.6100.0100.0100.026.965.7100.0100.0100.093.195.194.594.994.8
" + }, + { + "category_id": 5, + "poly": [ + 386, + 741, + 1758, + 741, + 1758, + 1014, + 386, + 1014 + ], + "score": 0.979, + "html": "
acc.(adv)TP marginalTP logitTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN83.212.00.50.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.192.188.720.40.2
GFZ86.737.77.71.20.343.271.891.499.4100.055.877.194.899.6100.090.392.190.990.489.7
GBZ85.157.432.519.311.633.750.384.499.7100.052.266.391.599.8100.090.091.591.591.791.8
GFY79.727.45.61.20.358.187.998.0100.0100.068.792.299.3100.0100.092.692.590.790.785.4
GBY86.735.99.01.70.445.376.194.699.7100.056.979.395.699.9100.090.192.191.691.791.4
DFX47.60.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.889.813.331.442.8
DBX DFZ58.0 49.618.36.01.30.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.794.094.193.784.8
1.00.00.00.011.844.394.1100.0100.025.957.694.599.9100.093.989.812.122.928.0
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acc.(adv)TP marginalTP KL
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
BNN82.07.20.10.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.291.172.064.57.1
GFZ87.040.49.01.21.243.583.898.499.499.457.689.398.699.299.291.592.392.090.690.6
GBZ79.627.45.61.50.536.186.3100.0100.0100.053.593.199.9100.0100.091.391.491.992.292.7
GFY80.830.16.81.41.456.093.799.8100.0100.068.196.699.9100.0100.090.992.492.091.691.6
GBY84.922.91.50.10.047.491.299.9100.0100.059.892.899.9100.0100.091.492.191.286.829.0
DFX48.40.80.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.289.69.217.517.5
DBX66.728.719.717.217.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.594.394.694.794.7
DFZ50.51.20.00.00.011.950.799.4100.0100.026.463.299.0100.0100.094.389.811.517.317.3
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acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN97.986.719.71.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A41.759.455.768.498.9
GFZ98.093.967.721.73.53.36.65.67.832.95.09.56.08.233.037.545.134.544.057.7
GBZ94.693.783.967.452.819.817.912.814.343.122.819.517.417.044.332.139.340.237.133.2
GFY98.495.067.925.84.14.26.76.47.835.73.18.27.67.833.831.243.632.939.653.3
GBY96.492.967.325.76.911.18.47.911.641.413.111.59.212.240.343.043.033.639.152.2
DFX82.735.70.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A64.164.769.6100.0100.0
DBX83.334.64.20.70.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A59.859.565.178.793.4
DFZ82.436.90.40.00.04.93.85.112.391.16.54.15.512.391.356.360.354.799.7100.0
" + }, + { + "category_id": 5, + "poly": [ + 407, + 707, + 1739, + 707, + 1739, + 979, + 407, + 979 + ], + "score": 0.982, + "html": "
acc. (adv)TP marginal
E0.010.020.050.100.200.010.020.050.100.20 0.010.020.050.100.200.010.020.050.100.20
BNN98.293.258.514.56.3N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A58.655.165.554.058.5
GFZ97.194.781.856.431.411.410.15.9 17.499.311.410.46.817.299.358.640.135.941.050.6
GBZ95.193.587.174.962.026.018.615.5 38.699.626.320.319.141.999.635.541.547.043.745.6
GFY96.594.280.756.732.09.69.26.9 18.899.017.613.18.120.899.146.643.736.139.549.3
GBY96.192.982.060.536.317.215.18.1 28.599.220.218.110.031.199.249.748.039.238.747.3
DFX83.842.20.70.00.0N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A63.366.648.497.7100.0
DBX90.978.650.731.718.3N/AN/AN/A N/AN/AN/AN/AN/AN/AN/A50.758.756.859.757.2
DFZ83.952.12.90.00.06.63.82.23.9 60.66.94.22.43.360.457.562.354.866.799.8
" + }, + { + "category_id": 5, + "poly": [ + 159, + 441, + 2010, + 441, + 2010, + 582, + 159, + 582 + ], + "score": 0.977, + "html": "
ZKPKK
acc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg.TP logitTP KL
DBX18.3N/AN/A9417N/AN/A93.1N/AN/AN/AN/AN/AN/AN/AN/A35.9N/AN/A93.9
GBZ57.450.366.391.535.582.290.791.447.936.567.491.344.650.569.991.04573.084.891.5
GBY35.976.179.392.12290.891.992.143.847.061.991.152.056.466.990.632.889.291.893.3
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acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
BNN96.984.618.70.90.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A35.954.756.362.478.8
GFZ96.991.558.115.21.96.56.26.410.686.110.08.36.611.384.531.039.032.147.277.1
GBZ92.589.471.540.417.915.013.211.024.095.916.414.812.825.196.042.537.531.037.455.0
GFY97.692.359.516.52.35.66.96.313.388.15.76.26.813.887.623.841.433.843.974.6
GBY95.188.856.514.61.315.19.58.619.297.315.310.79.918.597.051.143.231.746.787.6
DFX82.535.20.30.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A65.567.169.6100.0100.0
DBX82.538.86.11.20.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A54.660.061.578.888.5
DFZ81.536.10.40.00.05.43.95.013.496.96.24.15.513.896.861.158.555.399.9100.0
" + }, + { + "category_id": 5, + "poly": [ + 256, + 776, + 1888, + 776, + 1888, + 1049, + 256, + 1049 + ], + "score": 0.982, + "html": "
acc. (adv)TP marginalTP logitTP KL
E0.101.0010.00100.001000.000.101.0010.001100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
BNN93.766.237.332.347.6N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A60.980.382.272.080.0
GFZ99.595.676.577.978.30.04.52.63.03.40.04.63.03.64.229.230.425.627.129.0
GBZ96.093.688.988.789.016.711.57.87.77.918.913.79.39.19.441.151.440.434.732.0
GFY99.895.978.880.080.50.06.64.14.75.10.06.64.84.95.475.037.326.026.431.1
GBY97.593.176.777.378.015.814.66.07.17.318.913.96.07.07.942.634.327.231.629.0
DFX82.644.234.36.10.6N/AN/AN/AN/AN/AN/AN/A N/AN/AN/A100.0100.0100.082.595.0
DBX96.572.226.36.316.2N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A61.679.485.262.267.9
DFZ94.572.329.96.317.97.35.84.34.14.214.16.95.04.44.482.697.397.183.080.2
" + }, + { + "category_id": 5, + "poly": [ + 164, + 1235, + 2009, + 1235, + 2009, + 1376, + 164, + 1376 + ], + "score": 0.976, + "html": "
PKK
acc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KLacc.TP marg. N/ATP logitTP KLacc.TP marg.TP logitTP KL
DBX0.7N/AN/A78.70.7N/A82.5N/AN/AN/AN/AN/AN/AN/A0.6N/AN/A 27.682.6
GBZ67.414.317.037.15740.645.14958.49.618.546.958.51.87.847.859.925.542.4
GBY25.711.612.239.128.522.022.64929.55.67.645.929.20.81.845.427.915.615.945
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substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ86.714.30.00.00.096.683.351.326.817.249.573.899.7100.0100.0
GBZ81.29.10.00.00.093.580.155.335.025.246.773.399.6100.0100.0
GFY84.86.40.00.00.096.782.355.233.524.058.788.8100.0100.0100.0
GBY86.715.00.00.00.095.881.651.027.518.150.178.599.9100.0100.0
DFX74.25.20.50.00.091.757.719.64.31.4N/AN/AN/AN/AN/A
DBX80.65.10.00.00.093.259.222.510.98.2N/AN/AN/AN/AN/A
DFZ69.63.40.30.00.091.957.221.46.32.633.655.191.7100.0100.0
" + }, + { + "category_id": 5, + "poly": [ + 554, + 755, + 1592, + 755, + 1592, + 1001, + 554, + 1001 + ], + "score": 0.981, + "html": "
substitute accvictim accTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ87.216.10.00.00.096.582.442.814.23.750.682.5100.0100.0100.0
GBZ81.89.10.00.00.093.378.645.317.05.246.281.7100.0100.0100.0
GFY85.26.90.00.00.096.680.848.018.97.459.896.0100.0100.0100.0
GBY87.015.90.00.00.095.779.941.813.04.051.487.1100.0100.0100.0
DFX76.512.33.62.12.191.556.122.29.99.9N/AN/AN/AN/AN/A
DBX85.119.90.60.10.193.457.014.16.36.3N/AN/AN/AN/AN/A
DFZ70.93.20.00.00.091.854.616.42.70.334.563.898.9100.0100.0
" + }, + { + "category_id": 5, + "poly": [ + 467, + 1129, + 1680, + 1129, + 1680, + 1375, + 467, + 1375 + ], + "score": 0.981, + "html": "
substitute accvictim accTP logit
E0.101.0010.00100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
GFZ98.44.60.00.00.098.896.996.093.490.253.444.241.640.643.7
GBZ98.463.20.00.00.097.495.492.988.684.545.939.733.031.238.6
GFY98.10.90.00.00.099.097.697.095.794.061.055.249.249.552.7
GBY98.35.30.00.00.098.796.896.093.891.254.241.537.638.743.6
DFX85.10.00.00.00.097.693.391.290.489.8N/AN/AN/AN/AN/A
DBX97.346.00.40.00.098.793.988.485.576.3N/AN/AN/AN/AN/A
DFZ80.50.00.00.00.097.294.692.391.591.232.928.628.136.544.5
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substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ96.788.141.74.50.197.094.984.048.38.38.411.26.17.549.8
GBZ92.983.137.85.50.095.394.891.883.165.615.512.810.919.979.8
GFY96.988.133.73.30.196.995.985.354.212.72.62.05.08.561.2
GBY95.285.632.42.80.196.895.386.459.818.521.814.310.512.575.2
DFX95.780.810.10.20.099.196.779.428.30.9N/AN/AN/AN/AN/A
DBX95.780.530.51.70.099.598.291.165.121.8N/AN/AN/AN/AN/A
DFZ96.581.910.50.30.099.196.976.926.21.34.516.56.18.559.6
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substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ96.387.942.14.70.197.194.883.044.84.98.711.15.88.575.1
GBZ92.982.536.94.90.095.594.791.481.749.514.812.613.921.997.8
GFY96.887.934.13.50.196.995.684.649.16.92.62.06.710.381.1
GBY95.185.532.02.60.196.895.585.756.210.921.814.89.213.497.6
DFX95.580.510.50.20.099.196.577.324.00.2N/AN/AN/AN/AN/A
DBX95.681.134.52.20.099.598.389.358.99.9N/AN/AN/AN/AN/A
DFZ96.381.510.70.50.099.196.974.921.90.34.516.66.49.980.2
" + }, + { + "category_id": 5, + "poly": [ + 467, + 1129, + 1680, + 1129, + 1680, + 1375, + 467, + 1375 + ], + "score": 0.98, + "html": "
substitute accvictim accTP logit
E0.101.0010.00100.001000.000.101.0010.00100.001000.000.101.0010.00100.001000.00
GFZ98.761.90.00.00.098.695.894.393.893.211.710.210.99.98.5
GBZ96.369.83.30.00.095.995.794.794.794.513.512.712.313.714.4
GFY98.454.40.00.00.098.196.195.395.194.30.02.23.74.67.6
GBY97.652.92.20.00.097.795.694.594.393.725.815.714.213.718.0
DFX64.40.00.00.00.098.597.297.196.795.5N/AN/AN/AN/AN/A
DBX91.356.20.30.00.099.799.099.098.998.1N/AN/AN/AN/AN/A
DFZ80.30.00.00.00.099.297.997.597.395.40.015.714.68.86.1
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substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ44.20.70.00.00.097.388.462.733.821.848.761.092.399.8100.0
GBZ7.40.00.00.00.094.887.170.752.941.351.068.697.7100.0100.0
GFY49.40.80.00.00.097.486.558.131.921.052.570.897.499.8100.0
GBY21.70.00.00.00.096.989.370.049.737.851.270.898.2100.0100.0
DFX49.21.30.00.00.091.452.113.92.20.7N/AN/AN/AN/AN/A
DBX43.10.70.00.00.095.067.026.29.76.5N/AN/AN/AN/AN/A
DFZ53.81.70.00.00.092.956.315.23.41.534.752.894.099.9100.0
" + }, + { + "category_id": 5, + "poly": [ + 560, + 754, + 1585, + 754, + 1585, + 1001, + 560, + 1001 + ], + "score": 0.982, + "html": "
substitute accvictimaccTP logit
E0.100.200.300.400.500.100.200.300.400.500.100.200.300.400.50
GFZ45.52.80.00.00.097.286.650.317.017.051.168.097.599.599.5
GBZ8.40.00.00.00.094.785.963.535.217.151.274.499.0100.0100.0
GFY53.12.50.30.10.197.283.749.220.120.155.180.699.099.999.9
GBY24.50.00.00.00.096.888.362.831.713.349.377.699.9100.0100.0
DFX51.32.30.10.00.091.451.713.31.91.9N/AN/AN/AN/AN/A
DBX47.81.60.10.00.094.867.023.27.67.6N/AN/AN/AN/AN/A
DFZ56.12.90.10.00.092.756.013.12.82.835.259.997.6100.0100.0
" + }, + { + "category_id": 5, + "poly": [ + 468, + 1129, + 1677, + 1129, + 1677, + 1375, + 468, + 1375 + ], + "score": 0.981, + "html": "
substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.0010.00100.001000.00
GFZ65.20.30.00.00.098.898.797.4 94.392.052.3 50.839.644.755.3
GBZ76.60.40.00.00.097.397.295.9 92.790.445.746.141.540.345.7
GFY88.43.80.00.00.099.098.998.0 94.892.060.059.251.947.657.2
GBY76.01.30.00.00.098.798.697.3 95.193.553.452.241.839.746.7
DFX82.40.00.00.00.098.896.8 92.486.284.9N/AN/AN/AN/AN/A
DBX82.50.60.00.00.098.998.495.5 85.283.1N/AN/AN/AN/AN/A
DFZ86.50.20.00.00.098.894.589.3 81.579.342.925.522.327.738.2
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substitute accvictim accTP logit
E0.010.020.050.100.200.010.02 0.050.100.200.010.020.050.100.20
GFZ95.589.945.56.80.197.695.688.768.829.76.26.76.010.871.8
GBZ92.084.133.83.50.095.694.992.785.065.916.714.612.817.976.8
GFY95.389.138.72.50.097.396.690.072.935.50.02.36.810.472.3
GBY91.880.529.76.10.297.295.990.270.830.321.816.49.613.677.3
DFX89.174.519.10.90.099.798.792.664.89.9N/AN/AN/AN/AN/A
DBX85.576.442.54.50.099.699.094.877.524.5N/AN/AN/AN/AN/A
DFZ85.573.521.11.50.099.599.293.870.115.10.05.08.915.985.1
" + }, + { + "category_id": 5, + "poly": [ + 573, + 755, + 1575, + 755, + 1575, + 1001, + 573, + 1001 + ], + "score": 0.98, + "html": "
substitute accvictim accTP logit
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
GFZ95.589.746.47.70.297.595.788.566.520.56.26.75.912.782.7
GBZ91.884.033.13.40.095.694.992.385.160.716.715.514.221.697.9
GFY95.388.940.53.10.097.396.789.570.631.90.02.47.612.588.0
GBY91.780.229.55.90.197.395.989.669.026.122.216.19.714.095.9
DFX89.074.519.40.90.099.698.792.163.010.9N/AN/AN/AN/AN/A
DBX85.576.342.74.50.099.699.095.176.527.2N/AN/AN/AN/AN/A
DFZ85.373.421.41.60.099.699.192.967.714.90.04.57.514.291.9
" + }, + { + "category_id": 5, + "poly": [ + 467, + 1129, + 1679, + 1129, + 1679, + 1375, + 467, + 1375 + ], + "score": 0.98, + "html": "
substitute accvictim acc
E0.101.0010.00100.001000.000.101.0010.00 100.001000.000.101.00TP logit 10.00100.001000.00
GFZ96.341.50.50.00.098.794.793.4 93.392.313.3 8.67.77.78.9
GBZ94.973.31.40.00.095.995.3 94.694.594.513.511.811.813.213.1
GFY96.462.01.30.00.098.196.7 94.894.794.10.02.86.56.16.8
GBY93.533.61.70.00.097.795.9 95.495.495.025.816.415.918.619.8
DFX90.310.90.00.00.099.998.8 98.598.597.9N/AN/AN/AN/AN/A
DBX87.231.80.00.00.099.997.1 94.593.994.7N/AN/AN/AN/AN/A
DFZ84.922.70.00.00.099.999.0 98.398.397.80.03.82.32.311.0
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acc. (adv) 0.020.050.100.20TP marginalTP logitTP KL
E0.010.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1644.525.816.611.910.1N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A95.195.797.097.597.1
GBZ-FC163.749.734.318.912.760.062.480.690.994.163.765.080.695.098.490.190.892.392.692.6
GBY-FC164.950.036.119.511.352.259.080.490.192.554.661.980.395.698.490.791.392.192.490.3
DBX-FC157.345.731.816.310.8N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A92.893.294.094.392.8
GBZ-CONV981.674.158.548.945.813.717.028.476.090.218.320.631.764.371.691.791.591.793.386.4
GBY-CONV975.866.039.722.517.514.018.229.474.488.917.720.825.855.268.989.689.990.089.189.2
DBX-CONV959.045.731.817.811.3N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.794.194.995.293.7
" + }, + { + "category_id": 5, + "poly": [ + 369, + 754, + 1778, + 754, + 1778, + 1001, + 369, + 1001 + ], + "score": 0.98, + "html": "
acc. (adv)TP marginalTP logitTP KL
E0.010.020.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.100.20
VGG1618.80.60.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A93.991.867.469.966.4
GBZ-FC137.76.10.10.00.030.722.276.092.094.133.524.977.091.790.290.893.797.398.595.9
GBY-FC131.28.52.61.50.632.020.382.395.197.636.629.584.193.790.490.394.197.797.396.9
DBX-FC123.41.70.00.00.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.991.05.67.45.9
GBZ-CONV961.138.324.210.22.319.260.396.798.999.620.160.396.898.398.790.393.197.798.799.6
GBY-CONV966.526.63.20.40.017.055.695.799.299.617.852.795.398.697.689.691.898.799.496.9
DBX-CONV924.03.00.30.10.0N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A91.393.199.399.699.2
" + }, + { + "category_id": 5, + "poly": [ + 355, + 1129, + 1791, + 1129, + 1791, + 1375, + 355, + 1375 + ], + "score": 0.98, + "html": "
acc. (adv)TP marginalTP logit
E0.010.02（0.050.100.200.010.020.050.100.200.010.020.050.100.200.010.020.050.10 0.20
VGG1613.10.40.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A92.292.472.377.179.0
GBZ-FC124.52.70.10.00.034.840.988.595.896.0 38.844.090.496.496.691.395.095.797.996.8
GBY-FC133.019.611.87.24.929.138.795.799.199.3 38.449.296.399.399.490.795.097.897.997.9
DBX-FC117.30.90.00.00.0N/AN/AN/AN/A N/AN/AN/AN/AN/AN/A91.189.69.614.516.0
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However, solving the partial differential equations of incompressible fluids is a challenging task and traditional numerical approximation schemes come at high computational costs. Recent deep learning based approaches promise vast speed-ups but do not generalize to new fluid domains, require fluid simulation data for training, or rely on complex pipelines that outsource major parts of the fluid simulation to traditional methods. + +In this work, we propose a novel physics-constrained training approach that generalizes to new fluid domains, requires no fluid simulation data, and allows convolutional neural networks to map a fluid state from time-point $t$ to a subsequent state at time $t + d t$ in a single forward pass. This simplifies the pipeline to train and evaluate neural fluid models. After training, the framework yields models that are capable of fast fluid simulations and can handle various fluid phenomena including the Magnus effect and Kármán vortex streets. We present an interactive real-time demo to show the speed and generalization capabilities of our trained models. Moreover, the trained neural networks are efficient differentiable fluid solvers as they offer a differentiable update step to advance the fluid simulation in time. We exploit this fact in a proof-of-concept optimal control experiment. Our models significantly outperform a recent differentiable fluid solver in terms of computational speed and accuracy. + +# 1 INTRODUCTION + +Simulating the behavior of fluids by solving the incompressible Navier-Stokes equations is of great importance for a wide range of applications and accurate as well as fast fluid simulations are a long-standing research goal. On top of simulating the behavior of fluids, several applications such as sensitivity analysis of fluids or gradient-based control algorithms rely on differentiable fluid simulators that allow to propagate gradients throughout the simulation (Holl et al. (2020)). + +Recent advances in deep learning aim for fast and accurate fluid simulations but rely on vast datasets and / or do not generalize to new fluid domains. Kim et al. (2019) present a framework to learn parameterized fluid simulations and allow to interpolate efficiently in between such simulations. However, their work does not generalize to new domain geometries that lay outside the training data. Kim & Lee (2020) train a RNN-GAN that produces turbulent flow fields within a pipe domain, but do not show generalization results beyond pipe domains. Xie et al. (2018) introduce a tempoGAN to perform temporally consistent superresolution of smoke simulations. This allows to produce plausible high-resolution smoke-density fields for arbitrary low-resolution inputs, but our fluid model should output a complete fluid state description consisting of a velocity and a pressure field. Tompson et al. (2017) present how a Helmholtz projection step can be learned to accelerate Eulerian fluid simulations. This method generalizes to new domain geometries, but a particle tracer is needed to deal with the advection term of the Navier-Stokes equations. Furthermore, as Eulerian fluids do not model viscosity, effects like e.g. the Magnus effect or Kármán vortex streets cannot be simulated. Geneva & Zabaras (2020) propose a physics-informed framework to learn the entire update step for the Burgers equations in 1D and 2D, but no generalization results for new domain geometries are demonstrated. All of the aforementioned methods rely on the availability of vast amounts of data from fluid-solvers such as FEniCS, OpenFOAM or Mantaflow. Most of these methods do not generalize well or outsource a major part of the fluid simulation to traditional methods such as low-resolution fluid solvers or a particle tracer. + +In this work, we propose a novel unsupervised training framework to learn incompressible fluid dynamics from scratch. It does not require any simulated fluid-data (neither as ground truth data, nor to train an adversarial network, nor to initialize frames for a physics-constrained loss) and generalizes to fluid domains unseen during training. It allows CNNs to learn the entire update-step of mapping a fluid domain from time-point $t$ to $t + d t$ without having to rely on low resolution fluid-solvers or a particle-tracer. In fact, we will demonstrate that a physicsconstrained loss function combined with a simple strategy to recycle fluid-data generated by the neural network at training time suffices to teach CNNs fluid dynamics on increasingly realistic statistics of fluid states. This drastically simplifies the training pipeline. Fluid simulations get efficiently unrolled in time by recurrently applying the trained model on a fluid state. Furthermore, the fluid models include viscous friction and handle effects such as the Magnus effect and Kármán vortex streets. On top of that, we show by a gradient-based optimal control example how backpropagation through time can be used to differentiate the fluid simulation. Code and pretrained models are publicly available at https://github.com/aschethor/ Unsupervised_Deep_Learning_of_Incompressible_Fluid_Dynamics/. + +# 2 RELATED WORK + +In literature, several different approaches can be found that aim to approximate the dynamics of PDEs in general and fluids in particular with efficient, learning-based surrogate models. + +Lagrangian methods such as smoothed particle hydrodynamcs (SPH) Gingold & Monaghan (1977) handle fluids from the perspective of many individual particles that move with the velocity field. Following this approach, learning-based methods using regression forests by Ladický et al. (2015), graph neural networks by Mrowca et al. (2018); Li et al. (2019) and continuous convolutions by Ummenhofer et al. (2020) have been developed. In addition, Smooth Particle Networks (SP-Nets) by Schenck & Fox (2018) allow for differentiable fluid simulations within the Lagrangian frame of reference. These Lagrangian methods are particularly suitable when a fluid domain exhibits large, dynamic surfaces (e.g. waves or droplets). However, to simulate the dynamics within a fluid domain accurately, Eulerian methods, that treat the Navier-Stokes equations in a fixed frame of reference, are usually better suited. + +Continuous Eulerian methods allow for mesh-free solutions by mapping domain coordinates (e.g. $x , y , t )$ directly onto field values (e.g. velocity $\vec { v } \ : /$ pressure $p$ ) (Sirignano & Spiliopoulos (2018); Grohs et al. (2018); Khoo et al. (2019)). Recent applications focused on flow through porous media (Zhu & Zabaras (2018); Zhu et al. (2019); Tripathy & Bilionis (2018)), fluid modeling (Yang et al. (2016); Raissi et al. (2018)), turbulence modeling (Geneva & Zabaras (2019); Ling et al. (2016)) and modeling of molecular dynamics (Schöberl et al. (2019)). Training is usually based on physics-constrained loss functions that penalize residuals of the underlying PDEs. Similar to our approach, Raissi et al. (2019) uses vector potentials to obtain continuous divergence-free velocity fields to approximate the incompressible Navier-Stokes equations. Continuous methods return smooth, accurate results and can overcome the curse of dimensionality of discrete techniques in high-dimensional PDEs (Grohs et al. (2018)). However, these networks are trained on a specific domain and cannot generalize to new environments or be used in interactive scenarios. + +Discrete Eulerian methods, on the other hand, aim to solve the underlying PDEs on a grid and early work dates back to Harlow & Welch (1965) and Stam (1999). Accelerating such traditional works with deep learning techniques is a major field of research and all of the methods mentioned in the introduction fall into this category. Further methods include the approach by Thuerey et al. (2019) to learn solutions of the Reynolds-averaged Navier-Stokes equations for airfoil flows, but requires large amounts of training data and does not generalize beyond airfoil flows. In the work by Um et al. (2020), a correction step is learned that brings solutions of a low-resolution differentiable fluid solver closer to solutions of a high-resolution fluid simulation. However, generalization results for new domain geometries were not presented. The works of Mohan et al. (2020) and Kim et al. (2019) show that vector potentials are suitable to enforce the incompressibility constraint in fluids but do not generalize to new fluid domains beyond their training data. + +# 3 METHOD + +In this section, we briefly review the incompressible Navier-Stokes equations, which are to be solved by the neural network. Then, we explain how the Helmholtz decomposition can be exploited to ensure incompressibility within the fluid domain. Furthermore, we provide details of our discrete spatio-temporal fluid representation and introduce the fluid model. Afterwards, we formulate a physics-constrained loss function based on residuals of the Navier-Stokes equations and introduce a pressure regularization term for very high Reynolds numbers. Finally, we explain the unsupervised training strategy. + +# 3.1 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS + +Most fluids can be modeled with the incompressible Navier-Stokes equations - a set of non-linear equations that describe the interplay of a velocity field $\vec { v }$ and a pressure field $p$ within a fluid domain $\Omega$ : + +$$ +\begin{array} { r l r } { \boldsymbol { \nabla } \cdot \boldsymbol { \vec { v } } = 0 } & { \mathrm { i n c o m p r e s s i b i l i t y ~ o n ~ } \Omega } \\ { \rho \dot { \vec { v } } = \rho \left( \frac { \partial \vec { v } } { \partial t } + \left( \vec { v } \cdot \boldsymbol { \nabla } \right) \vec { v } \right) = - \boldsymbol { \nabla } p + \mu \Delta \vec { v } + \vec { f } } & { \mathrm { c o n s e r v a t i o n ~ o f ~ m o m e n t u m ~ o n ~ } \Omega } \end{array} +$$ + +Here, $\rho$ describes the fluid density and $\mu$ the viscosity. Equation 1 states that the fluid is incompressible and thus $\vec { v }$ is divergence-free. Equation 2 states that the change in momentum of fluid particles must correspond to the sum of forces that arise from the pressure gradient, viscous friction and external forces. Here, external forces on the fluid (such as e.g. gravity) can be neglected, so we set ${ \vec { f } } = 0$ . + +These incompressible Navier-Stokes equations shall be solved by a CNN given initial conditions $\vec { v } ^ { 0 }$ and $p ^ { 0 }$ at the beginning of the simulation and Dirichlet boundary conditions which constrain the velocity field at the domain boundary $\partial \Omega$ : + +$$ +\vec { v } = \vec { v } _ { d } \qquad \mathrm { D i r i c h l e t ~ b o u n d a r y ~ c o n d i t i o n ~ o n ~ } \partial \Omega +$$ + +# 3.2 HELMHOLTZ DECOMPOSITION + +A common method to ensure incompressibility of a fluid (see Equation 1) is to project the flow field onto the divergence-free part of its Helmholtz decomposition. The Helmholtz theorem states that every vector field $\vec { v }$ can be decomposed into a curl-free part $( \nabla q )$ and a divergence-free part $( \nabla \times \vec { a } )$ : + +$$ +\vec { v } = \nabla q + \nabla \times \vec { a } +$$ + +Note, that $\nabla \times ( \nabla q ) = \vec { 0 }$ and $\nabla \cdot ( \nabla \times { \vec { a } } ) = 0$ . The Helmholtz projection consists of solving the Poisson problem $\nabla \cdot \vec { v } = \Delta q$ for $q$ , followed by substracting $\nabla q$ from the original flow field. However, solving the Poisson equation on arbitrary domains comes at high computational costs for classical methods and one has to rely e.g. on conjugate gradient methods to approximate its solution. + +Here, we propose a different approach and directly try to learn a vector potential $\vec { a }$ with $\vec { v } = \nabla \times \vec { a }$ . This ensures that the network outputs a divergence-free velocity field within the domain $\Omega$ and automatically solves Equation 1. In this work, we consider 2D fluid simulations, so only the $\mathbf { Z }$ - component of $\vec { a }$ , $a _ { z }$ , is of interest since $v _ { z }$ and all derivatives with respect to the $z$ -axis are zero: + +$$ +\nabla \times { \vec { a } } = \left( { \partial _ { z } a _ { x } - \partial _ { z } a _ { y } } \right) = \left( { \partial _ { y } a _ { z } } \right) = \left( { \begin{array} { c } { v _ { x } } \\ { v _ { y } } \\ { 0 } \end{array} } \right) = { \vec { v } } +$$ + +# 3.3 DISCRETE SPATIO-TEMPORAL FLUID REPRESENTATION + +Marker-And-Cell (MAC) grid To solve the Navier-Stokes equations, we represent the relation between $a _ { z } , v _ { x } , v _ { y } , p$ on a 2D staggered marker-and-cell (MAC) grid (see Figure 1a). Therefore, we discretise time and space as follows: + +$$ +\vec { a } ( x , y , t ) = \left( \begin{array} { c } { 0 } \\ { 0 } \\ \left( a _ { z } \right) _ { i , j } ^ { t } \rule { 0 ex } { 5 ex } \right) ; \vec { v } ( x , y , t ) = \binom { \left( v _ { x } \right) _ { i , j } ^ { t } } { \left( v _ { y } \right) _ { i , j } ^ { t } } ; p ( x , y , t ) = p _ { i , j } ^ { t } \end{array} +$$ + +Obtaining gradient, divergence, Laplace and curl operations on this grid with finite differences is straight forward and can be efficiently implemented with convolutions (see appendix A). + +![](images/9b2df2899d456e4e435eca52239a4b16982712cebefb7099762ffba6f6c0e6f2.jpg) +Figure 1: MAC grid and diagram of the fluid model. + +Explicit, Implicit, Implicit-Explicit (IMEX) time integration methods The discretization of the time domain is needed to deal with the time-derivative of the velocity fiel d ∂\~v∂t in Equation 2, which becomes: + +$$ +\rho \left( \frac { \vec { v } ^ { t + d t } - \vec { v } ^ { t } } { d t } + \left( \vec { v } ^ { t ^ { \prime } } \cdot \nabla \right) \vec { v } ^ { t ^ { \prime } } \right) = - \nabla p ^ { t + d t } + \mu \Delta \vec { v } ^ { t ^ { \prime } } + \vec { f } +$$ + +The goal is to take as large as possible timesteps $d t$ while maintaining stable and accurate solutions. Stability and accuracy largely depend on the definition of $v ^ { t ^ { \prime } }$ . In literature, choosing $\boldsymbol { v } ^ { t ^ { \prime } } = \boldsymbol { v } ^ { t }$ is often referred to as explicit integration methods and frequently leads to unstable behavior. Choosing $v ^ { t ^ { \prime } } = v ^ { t + d t }$ is usually associated with implicit integration methods and gives stable solutions at the cost of numerical dissipation. Implicit-Explicit (IMEX) methods, which set $v ^ { t ^ { \prime } } = ( v ^ { t } + v ^ { t + d t } ) / 2$ are a compromise between both methods and considered to be more accurate but less stable than implicit methods. + +# 3.4 FLUID MODEL + +We represent the fluid dynamics by a recurrent model that maps the fluid state $p ^ { t } , \vec { a } ^ { t }$ for timestep $t$ and the domain description $\Omega ^ { t + d t } , \vec { v } _ { d } ^ { t + d t }$ to the fluid state $p ^ { t + d t } , \vec { a } ^ { t + d t }$ of the next timestep. Here, $p ^ { t }$ describes the pressure field and $\vec { a } ^ { t }$ describes the vector potential of . For $t = 0$ , we consider initial states $p ^ { 0 } = 0$ and $\vec { a } ^ { 0 } = \vec { 0 }$ , however, other initial conditions could be considered as well. $\Omega ^ { t + d t }$ is a binary mask that contains the domain geometry and is 1 for the fluid domain and 0 everywhere else. For the boundary of the domain, we simply take the inverse of $\Omega \colon \partial \Omega = 1 - \Omega$ . $\vec { v } _ { d } ^ { t + \dot { d } t }$ represents the Dirichlet boundary conditions and contains a velocity field that must be matched by ${ \vec { v } } ^ { t + d t }$ at the domain boundaries. Figure 1b shows a diagram of the fluid model. First, $\left( p ^ { t } , \vec { a } ^ { t } , \Omega ^ { t + d t } , \vec { v } _ { d } ^ { t + d t } \right)$ are taken to derive a slightly more meaningful feature representation that comprises $\left( p ^ { t } , a ^ { t } , \nabla \times a ^ { t } , \Omega ^ { t + d t } , \partial \Omega ^ { t + d t } , \Omega ^ { t + d t } \cdot \nabla \times a ^ { t } , \Omega ^ { t + d t } \cdot p ^ { t } , \partial \Omega ^ { t + d t } \cdot \vec { v } _ { d } ^ { t + d t } \right)$ . These features can be very efficiently computed with convolutions and are then fed into a U-Net (Ronneberger et al. (2015)) with a reduced number of channels (the exact network configuration can be found in appendix B). The mean of the U-Net output is set to 0 in order to keep $p$ and $\vec { a }$ well defined and prevent drifting offset values. Finally, the output is added to $p ^ { t }$ and $\vec { a } ^ { t }$ to obtain the updated fluid state $p ^ { t + d t }$ and $\vec { a } ^ { t + d t }$ . + +# 3.5 PHYSICS-CONSTRAINED LOSS FUNCTION + +Using the residuals of the Navier-Stokes equations (Equations 1 and 2), we can formulate the following loss terms on $\Omega$ and $\partial \Omega$ : + +$$ +\begin{array} { l l } { { L _ { d } = \displaystyle { \left\| \nabla \cdot \vec { v } \right\| ^ { 2 } } } } & { { \qquad \mathrm { ~ d i v e r g e n c e ~ l o s s ~ o n ~ } \Omega } } \\ { { L _ { p } = \displaystyle { \left\| \rho \left( \frac { \partial \vec { v } } { \partial t } + \left( \vec { v } \cdot \nabla \right) \vec { v } \right) + \nabla p - \mu \Delta \vec { v } - \vec { f } \right\| ^ { 2 } } } } & { { \qquad \mathrm { ~ m o m e n t u m ~ l o s s ~ o n ~ } \Omega } } \\ { { L _ { b } = \displaystyle { \left\| \vec { v } - \vec { v } _ { d } \right\| ^ { 2 } } } } & { { \qquad \mathrm { ~ b o u n d a r y ~ l o s s ~ o n ~ } \partial \Omega } } \end{array} +$$ + +Combining the described loss terms, we obtain the following loss function: + +$$ +L = \alpha L _ { d } + \beta L _ { p } + \gamma L _ { b } +$$ + +where $\alpha , \beta , \gamma$ are hyperparameters that weight the contributions of the different loss terms. Note that if we use a vector potential ${ \vec { v } } = \nabla \times { \vec { a } }$ , $L _ { d } = 0$ is automatically fulfilled and we can set $\alpha = 0$ . This loss function can be computed very efficiently with convolutions in $O ( N )$ (where $N =$ number of grid cells), whereas solving the Navier-Stokes equations explicitly would be computationally a lot more expensive. For detailed descriptions regarding the fully discretized loss-function, we refer to appendix A. + +# 3.6 PRESSURE REGULARIZATION + +For very high Reynolds numbers (see Equation 13) and inviscid flows, training becomes unstable as viscous friction cannot dissipate enough energy out of the system. This leads to unrealistic gradients in $\vec { v }$ and $p$ . For such cases, we introduce an additional regularization term for the loss function (11) that can be traded off with $L _ { p }$ to stabilize training: + +$$ +L _ { r } = \| \nabla p \| ^ { 2 } +$$ + +The intuition behind this regularization term is, that we want to penalize unrealistically high energies in the pressure field. + +# 3.7 TRAINING STRATEGY + +Training starts with initializing a pool $\{ \Omega _ { k } ^ { 0 } , ( v _ { d } ) _ { k } ^ { 0 } , ( a _ { z } ) _ { k } ^ { 0 } , p _ { k } ^ { 0 } \}$ of randomized domains $\Omega _ { k } ^ { 0 }$ and boundary conditions $( v _ { d } ) _ { k } ^ { 0 }$ as well as initial conditions for the vector potential and pressure fields that we both set to zero $( \check { ( a _ { z } ) } _ { k } ^ { 0 } = 0$ and $p _ { k } ^ { 0 } = 0$ ). The resolution of our training domains is $1 0 0 \mathrm { x } 3 0 0$ grid cells and example-domains of the training pool are shown in appendix C. Note that our training pool does not rely on any previously simulated fluid-data. + +At each training step, a random mini-batch $\{ \Omega _ { k } ^ { t } , ( v _ { d } ) _ { k } ^ { t } , ( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t } \} _ { \{ k \in \operatorname * { m i n i b a t c h } \} }$ is drawn from the pool and fed into the neural network which is designed to predict the velocity $( \vec { v } _ { k } ^ { t + d t } = \nabla \times \vec { a } _ { k } ^ { t + d t } )$ and pressure $( p _ { k } ^ { t + d t } )$ fields of the next time step. Based on a physics-constrained loss-function (Equation 11), we update the weights of the network using the Adam optimizer (Kingma & Ba (2015)). At the end of each training step, the pool is updated by replacing the old vector potential and pressure fields $( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t }$ by the newly predicted ones $( a _ { z } ) _ { k } ^ { t + \dot { d t } } , p _ { k } ^ { \dot { t } + d t }$ t, pt+dtk . This recycling strategy fills the training pool with more and more realistic fluid states as the model becomes better at simulating fluid dynamics. + +From time to time, old environments of the training pool are replaced by new randomized environments and the vector potential as well as the pressure fields are reset to 0. This increases the variance of the training pool and helps the neural network to learn "cold starts" from \~0-velocity and 0-pressure fields. + +Besides the fluid model described above, which we denote as $\vec { a }$ -Net in the following, we also trained an ablation model, $\vec { v }$ -Net, that directly learns to predict the velocity field without a vector potential. For the implementation of both models, we used the popular machine learning framework Pytorch and trained the models on a NVidia GeForce RTX 2080 Ti. Training converged after about 1 day. The hyperparameters in the loss-function for the $\vec { a }$ -Net were $\beta = 1$ and $\gamma = 2 0$ . The reason for choosing a higher weight for the loss term $L _ { b }$ than for $L _ { p }$ was the observation, that errors in $L _ { b }$ can lead to unrealistic flows leaking through boundaries. For the ablation study $\vec { v } \cdot \vec { }$ -Net), we used $\alpha = 1 0 0 , \beta = 1 , \gamma = 0 . 0 0 1$ . Here, we had to choose a very high weight for $L _ { d }$ to ensure incompressibility of the fluid, otherwise unrealistic source and sink effects start to appear. For $L _ { b }$ , on the other hand, we used a very low weight as the boundary conditions can be trivially learned by the $\vec { v }$ -Net. We used these parameter settings for all experiments. + +# 4 RESULTS + +To evaluate the potential of our method, we assess its ability to reproduce physical effects such as Kármán vortex streets and the Magnus effect. In addition, we demonstrate its generalization capability and real-time performance. Finally, we test the fluid models quantitatively. + +# 4.1 QUALITATIVE EVALUATION + +Qualitative analysis of wake dynamics Qualitative effects in fluid dynamics such as the wake dynamics behind an obstacle are closely related to the Reynolds number. It is a dimensionless quantity defined by: + +$$ +R e = \frac { \rho \left\| \vec { v } \right\| D } { \mu } +$$ + +Here, $\rho$ is the fluid density, $\lVert \vec { v } \rVert$ is the fluid speed, $D$ is the diameter of the obstacle, and $\mu$ is the viscosity. (We use the units of the grid). + +We retrained models for different values of $\mu$ and $\rho$ to compare the fluid behavior for a wide range of Reynolds numbers. Figure 2 shows, that the trained models are able to predict the wake dynamics behind an obstacle in good accordance with qualitative expectations from fluid dynamics. As a rule of thumb, for $R e \ll 1$ , the flow becomes time-reversible. This can be noticed in Figure 2a by the symmetry of the flow before and after the obstacle and the nearly constant pressure gradient within the pipe. Starting from $R e \approx 1 0$ , the flow is still laminar but a static wake is forming behind the obstacle (see Figure 2b). For Reynolds numbers $R e > \approx 9 0$ , Kármán vortex streets start to appear (see Figure 2c). A Kármán vortex street consists of clock and counterclockwise spinning vortices that are generated at the obstacle and then start moving in a regularly oscillating pattern with the flow. For very large Reynolds numbers or inviscid flows, the flow field becomes turbulent, which can be recognized by the irregular patterns behind the obstacle in Fig 2d. + +![](images/468c39a4bc3b15c1ad09df22f1ea2d017e585da0b07e13fdda3acc4af90342a2.jpg) +Figure 2: After training, our models are able to show correct wake flow dynamics for a wide range of different Reynolds numbers. $\dot { \boldsymbol { D } } = 3 0$ , $\| \vec { v } \| = 0 . 5 )$ . Streamlines indicate flow direction, linewidth indicates speed and colors represent the pressure field (blue: low pressure / yellow: high pressure). + +Magnus effect The Magnus effect appears when a flow interacts with a rotating body. It is widely known e.g. in sports such as soccer or tennis where spin is used to deflect the path of a ball. The reason for the deflection stems from a low pressure field where the surface of the object moves along flow direction and a high pressure field where the object surface moves against the flow. Figure 3a shows, that our models are able to reproduce the Magnus effect around a rotating cylinder. + +![](images/f016ae8a12c9f62a822e83f75f847c39d114a2759991e05eb174aa68cf62956a.jpg) +(a) Magnus effect on a clock-wise turning cylinder.(b) Generalization example: Note that the fluid model has never been confronted with wingprofiles during training. + +Figure 3: Our models feature the Magnus effect and generalize to new fluid domains. Further examples are presented in appendix D and the video. + +Analysis of generalization capability We tested the networks capability to generalize to objects not seen during training. Figure 3b shows the networks capability to meet boundary conditions of an airfoil and return a plausible pressure field that produces lift (see low pressure on top of wing). Note that in contrast to the approach by Thuerey et al. (2019), which learns simplified, time-averaged solutions of the Navier-Stokes equations, our method is able to simulate the full incompressible Navier-Stokes equations for an airfoil without relying on any ground truth data or having seen airfoilgeometries during training. In fact, the network was only trained on simple randomized domains as highlighted in appendix C and Figure 7. Possible reasons for the networks generalization capabilities are: + +• During training, the network gets confronted with an infinite number of different flowfields and randomized domain configurations because the training pool gets updated at every training step. This prevents the network from over-fitting. + +• The dynamics of a fluid-particle are mostly determined by its local neighborhood / surrounding particles. This means, the update step for a certain cell on the MAC grid is mostly determined by close / neighboring MAC-grid cells. Since more complicated shapes can be seen locally as a composition of basic shapes (e.g. the front of the wing can be locally regarded as a cylinder), it suffices to train on basic shapes that provide the network with enough examples to generalize to more complicated shapes. + +Further generalization examples are provided in appendix D. + +Real-time capability The fluid simulation can be easily parallelized and takes low computational costs as one time-integration step consists just of a single forward pass through a convolutional neural network. This enables for example interactive real-time simulations. We implemented a demo that allows to interact with a fluid by moving obstacles, rotating spheres and changing the flow speed within a pipe (see video in supplementary material and source code). Our method runs at 250 timesteps per second on a $1 0 0 \mathrm { x } 3 0 0$ grid. In the respective experiments, we used a NVidia GeForce RTX 2080 Ti consuming about $8 6 0 \mathrm { M B }$ of GPU memory. + +# 4.2 QUANTITATIVE EVALUATION + +We compare our method $\vec { a }$ -Net) quantitatively with PhiFlow by Holl et al. (2020). Phiflow is a recent, open source, differentiable fluid simulator based on a MAC grid data structure. Furthermore, we provide an ablation study ${ \vec { v } } .$ -Net) that does not make use of the Helmholtz decomposition but directly works on the velocity field $\vec { v }$ . + +Quantitative comparison of different fluid solvers is challenging, as their performance is highly dependent on factors like the geometry of the domain, fluid parameters such as viscosity or density, flow speed or the timestep of the integrator. As benchmarks for fluid simulations on MAC grids are not yet available, we built a simple toy domain on a $1 0 0 \mathrm { ~ x ~ } 1 0 0$ grid which simulates a flow around an obstacle within a pipe (more details are provided in appendix E). + +First, we compared the computational speed on a CPU and GPU by comparing the integration time-steps per second (see Table 1). The $\vec { v }$ -Net as well as the $\vec { a }$ -Net are significantly faster than PhiFlow (11x on CPU and $4 0 \mathrm { x }$ on GPU) as they do not rely on an iterative conjugate gradient solver but instead use a single forward pass through a convolutional neural network that can be easily parallelized on a GPU. To provide a fair comparison on $L _ { d }$ , we set the velocity field at the boundaries equal to $\vec { v _ { d } }$ . This enables us to compute $L _ { d }$ for the $\vec { a }$ -Net architecture on the domain boundaries which would otherwise have zero divergence everywhere. This way, $L _ { d }$ can be interpreted as a metric on how well the orthogonal components of the Dirichlet boundary conditions are met (i.e. no flow leaks through the boundaries). For $d t = 4$ , we outperformed Phiflow by several orders of magnitude. For both, $L _ { d }$ and $L _ { p }$ , the $\vec { a }$ -Net architecture significantly outperformed the more naive $\vec { v }$ -Net approach. + +Furthermore, we investigated stability by evaluating the evolution of $L _ { p }$ and $L _ { d }$ for the $\vec { a }$ -Net over time (see Figure 4). As the fluid state is initialized with $a _ { z } = 0$ and $p = 0$ , the $\vec { a }$ -Net has to perform a cold-start which is the reason for high $L _ { p }$ and $L _ { d }$ during the first circa 70 steps. Afterwards, the $\vec { a }$ -Net continues an accurate and stable fluid simulation. + +Table 1: Quantitative comparison of timesteps per second (TPS) on CPU / GPU as well as divergence loss and momentum loss for differentiable fluid solvers on a $1 0 0 \mathrm { x } 1 0 0$ grid for viscosity $\mu =$ 0.1, density $\rho = 4$ and timesteps of size $d t = 4$ . + +
MethodCPU[TPS]GPU[TPS]LdLp
PhiFlow7-6.2e-4-
U-Net (ours)823118.66e-74.87e-5
α-Net (ours)823115.44e-71.56e-5
+ +![](images/61d33028f9c357e03bf3a26def07202e3075dc25a8368f4cd3727ad576eaeaf4.jpg) +Figure 4: Long term stability of fluid simulations performed by the $\vec { a }$ -Net + +# 4.3 OPTIMAL CONTROL OF VORTEX SHEDDING FREQUENCY + +In this section, we present a proof-of-concept experiment that aims at controlling the shedding frequency of a Kármán vortex street behind an obstacle by changing the flow speed (see Figure 5a). To this end, we exploit our previously trained differentiable fluid models. + +![](images/12287e6b64712b1dba143620da30b36d8a493efa70292097fa075f8a7596b80d.jpg) + +(a) control setup (domain size: $2 0 0 \mathrm { x } 1 0 0$ grid cells) + +![](images/eb7c6bdf57710e1d85311fd4fa94bc4129112926d9bad99da48def5bc5fb1064.jpg) +Figure 5: The frequency of vortex streets can be controlled using our differentiable fluid models. + +First, we measure the y-component of the velocity field $v _ { y } ( t )$ behind an obstacle (see white box in Figure 5a) over 200 time steps. Then, we compute the frequency spectrum $V _ { y } ( f )$ of $v _ { y } ( t )$ using the fast Fourier transform (see Figure 5b). Now, we want to adjust the inflow $/$ outflow boundary conditions in $\vec { v _ { d } }$ such that $E [ | V _ { y } ( f ) | ^ { 2 } ] = \hat { f }$ . Here, $\hat { f }$ is the target frequency. To optimize $\Vec { v _ { d } }$ , we define a loss function $L = ( E [ | V _ { y } ( f ) | ^ { 2 } ] - \hat { f } ) ^ { 2 }$ and compute the gradients $\frac { \partial L } { \partial \vec { v } _ { d } }$ with backpropagation through time. This is possible since all parts of the loss function including the fluid simulation that is performed by our trained neural fluid model as well as the fast Fourier transform are differentiable. Computing the gradients with a standard automatic differentiation library (Pytorch) took 3.5 seconds for all 200 time steps on our $2 0 0 \mathrm { x } 1 0 0$ domain setup. This is considerably faster than the current state-of-the-art differentiable fluid solver by Takahashi et al. (2021) which takes 5.42 seconds for only 30 time steps on a smaller $1 2 8 \mathrm { x } 1 2 8$ grid. The update steps of $\Vec { v _ { d } }$ are done using the ADAMoptimizer and converge after approximately 70 iterations (see Figure 5c). We want to emphasize that differentiable fluid simulations are limited to scenarios with low Reynolds numbers as in the presence of turbulences, chaotic behavior will lead to exploding gradients. + +# 5 DISCUSSION AND OUTLOOK + +In this work, we present an unsupervised learning scheme for the incompressible Navier-Stokes equations and introduce a fluid model that uses a vector potential to output divergence-free velocity fields. Qualitative results of our trained fluid models are in good accordance with expectations from fluid dynamics for a wide range of Reynolds numbers and generalize to unknown fluid domains. 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Data-driven projection method in fluid simulation. Computer Animation and Virtual Worlds, 27(3-4):415–424, 2016. doi: 10.1002/cav.1695. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cav.1695. + +Yinhao Zhu and Nicholas Zabaras. Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification. Journal of Computational Physics, 366:415 – 447, 2018. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2018.04.018. URL http: //www.sciencedirect.com/science/article/pii/S0021999118302341. + +Yinhao Zhu, Nicholas Zabaras, Phaedon-Stelios Koutsourelakis, and Paris Perdikaris. Physicsconstrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. Journal of Computational Physics, 394:56 – 81, 2019. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2019.05.024. URL http://www.sciencedirect.com/ science/article/pii/S0021999119303559. + +# A PHYSICS-CONSTRAINED LOSS ON A MAC GRID + +As mentioned in Section 3.3 of the paper, our method relies on a staggered marker-and-cell grid representation for the vector potential as well as the velocity and pressure fields. In the following, we provide further details on how to apply this representation to learn incompressible fluid dynamics. + +To calculate the velocity field $\vec { v } = \nabla \times \vec { a }$ of a vector potential $\vec { a }$ on a MAC grid in 2D, we have to compute the curl as follows: + +$$ +\begin{array} { r } { ( v _ { x } ) _ { i , j } = ( a _ { z } ) _ { i + 1 , j } - ( a _ { z } ) _ { i , j } } \\ { ( v _ { y } ) _ { i , j } = ( a _ { z } ) _ { i , j } - ( a _ { z } ) _ { i , j + 1 } } \end{array} +$$ + +If this vector potential is inserted into the divergence operator on a MAC grid, we can show that $\nabla \cdot \vec { v } _ { i , j } = 0$ is indeed fulfilled: + +$$ +\begin{array} { r l } & { \nabla \cdot \vec { v } _ { i , j } = ( { v } _ { x } ) _ { i , j + 1 } - ( { v } _ { x } ) _ { i , j } + ( { v } _ { y } ) _ { i + 1 , j } - ( { v } _ { y } ) _ { i , j } } \\ & { \qquad = \begin{array} { r l } & { ( ( a _ { z } ) _ { i + 1 , j + 1 } - ( a _ { z } ) _ { i , j + 1 } ) - ( ( a _ { z } ) _ { i + 1 , j } - ( a _ { z } ) _ { i , j } ) } \\ & { \qquad + \left( ( a _ { z } ) _ { i + 1 , j } - ( a _ { z } ) _ { i + 1 , j + 1 } \right) - ( ( a _ { z } ) _ { i , j } - ( a _ { z } ) _ { i , j + 1 } ) } \\ & { \qquad = 0 } \end{array} } \end{array} +$$ + +Thus, for the $\vec { a }$ -Net, the incompressibility equation is automatically fulfilled and no further training on the divergence loss $L _ { d }$ is required. However, for the $\vec { v }$ -Net, the residuals of the divergence are still of importance: + +$$ +( R _ { d } ) _ { i , j } ^ { t + d t } = \nabla \cdot \vec { v } _ { i , j } ^ { t + d t } ( = 0 \mathrm { f o r } \vec { a } \ – \mathrm { N e t } ) +$$ + +The residuals of the momentum equation in $x$ -direction can be computed as follows: + +$$ +\begin{array} { r l } & { { R _ { p _ { x } } } ) _ { i , j } ^ { t + d t } = \rho ( \frac { ( v _ { x } ) _ { i , j } ^ { t + d t } - ( v _ { x } ) _ { i , j } ^ { t } } { d t } + ( v _ { x } ) _ { i , j } ^ { t ^ { \prime } } \cdot \frac { ( v _ { x } ) _ { i , j + 1 } ^ { t ^ { \prime } } - ( v _ { x } ) _ { i , j - 1 } ^ { t ^ { \prime } } } { 2 } } \\ & { \qquad + \frac { \frac { ( v _ { y } ) _ { i , j - 1 } ^ { t ^ { \prime } } + ( v _ { y } ) _ { i , j } ^ { t ^ { \prime } } } { 2 } \cdot ( ( v _ { x } ) _ { i , j } ^ { t ^ { \prime } } - ( v _ { x } ) _ { i - 1 , j } ^ { t ^ { \prime } } ) + \frac { ( v _ { y } ) _ { i + 1 , j - 1 } ^ { t ^ { \prime } } + ( v _ { y } ) _ { i + 1 , j } ^ { t ^ { \prime } } } { 2 } \cdot ( ( v _ { x } ) _ { i + 1 , j } ^ { t ^ { \prime } } - ( v _ { x } ) _ { i , j } ^ { t ^ { \prime } } } { ( \frac { 1 } { 2 } } } \\ & { \qquad + ( p _ { i , j } ^ { t + d t } - p _ { i , j - 1 } ^ { t + d t } ) - \mu \cdot \Delta ( v _ { x } ) _ { i , j } ^ { t ^ { \prime } } } \end{array} +$$ + +Here, we use the following isotropic Laplace operator: + +$$ +\begin{array} { c } { \Delta s _ { i , j } = \displaystyle \frac { 1 } { 4 } ( 1 * s _ { i - 1 , j - 1 } + 2 * s _ { i - 1 , j } + 1 * s _ { i - 1 , j + 1 } } \\ { + 2 * s _ { i , j - 1 } - 1 2 * s _ { i , j } + 2 * s _ { i , j + 1 } } \\ { + 1 * s _ { i + 1 , j - 1 } + 2 * s _ { i + 1 , j } + 1 * s _ { i + 1 , j + 1 } ) } \end{array} +$$ + +The derivation of the advection term for $R _ { p _ { x } }$ is a bit more complex since on a MAC grid, $v _ { x }$ and $v _ { y }$ are displaced by half a pixel in $x$ -direction and $y$ -direction. To obtain the residuals of the momentum equation in $y$ -direction, $( R _ { p _ { y } } ) _ { i , j }$ , one has to take $( R _ { p _ { x } } ) _ { i , j }$ and swap $x$ and $y$ and the indices respectively. + +Now, the discretized loss terms can be written as follows: + +$$ +\begin{array} { l } { { { \cal L } _ { d } ^ { t + d t } = \displaystyle \sum _ { i , j } \Omega _ { i , j } ^ { t + d t } ( ( R _ { d } ) _ { i , j } ^ { t + d t } ) ^ { 2 } } } \\ { { { \cal L } _ { p } ^ { t + d t } = \displaystyle \sum _ { i , j } \Omega _ { i , j } ^ { t + d t } \left( ( ( R _ { p _ { x } } ) _ { i , j } ^ { t + d t } ) ^ { 2 } + ( ( R _ { p _ { y } } ) _ { i , j } ^ { t + d t } ) ^ { 2 } \right) } } \\ { { { \cal L } _ { b } ^ { t + d t } = \displaystyle \sum _ { i , j } \partial \Omega _ { i , j } ^ { t + d t } \left. \vec { v } _ { d } ^ { t + d t } - \vec { v } ^ { t + d t } \right. ^ { 2 } } } \end{array} +$$ + +Note, that all mentioned operations can be efficiently implemented with convolutions. To obtain the final velocities on a square grid, we project the velocity fields of the MAC grid back onto the $\vec { a }$ -grid using linear interpolation: + +$$ +\vec { v } = \frac { 1 } { 2 } \left( { ( v _ { x } ) _ { i - 1 , j } } + { ( v _ { x } ) _ { i , j } } \right) +$$ + +# B NETWORK ARCHITECTURE + +Our fluid model is based on the U-Net architecture (Ronneberger et al. (2015)) with fewer channels (see Figure 6). As the pressure field and vector potential can have an arbitrary offset, we always normalize the mean of the pressure $( \Delta p )$ and vector potential $( \Delta a _ { z } )$ to 0 to keep these fields welldefined and prevent drifting offset values. + +![](images/a1472214f77f3cafefb7b390de489bc06eea4b6ab22438d1c71d7d3ffc0cd1ea.jpg) +Figure 6: U-Net architecture with fewer channels. + +# C EXAMPLES OF TRAINING DOMAINS + +The domains we used for training consist of $1 0 0 \times 3 0 0$ grids. We used 3 different randomized domains as exemplary depicted in Figure 7. First, we have boxes with randomized height and width that float on randomized paths inspired by Brownian motion in a pipe with randomized flow speed. Second, we have the same setup but replaced the boxes by cylinders with randomized radii and angular velocities in order to learn the Magnus effect. Finally, we have a folded pipe system with randomized flow speed, that is randomly flipped along the $x$ -axis. + +# D FURTHER EXAMPLES OF GENERALIZATION + +Note that the network was only trained on simple domain geometries as presented in appendix C. Still, as can be seen in Figure 8, the network is capable of generalizing to far more complicated domain geometries (e.g. shark, car). Figure 8c shows that it can generalize to multiple objects in the scene, although the training set contained at most one object per scene. And Figure 8d shows that we can alter the outer boundary conditions as well. For real-time simulations, please have a look at our source code and the supplementary video. + +# E QUANTITATIVE ANALYSIS: THE BENCHMARK PROBLEM + +Figure 9 shows the domain $\Omega$ and $v _ { d }$ on a $1 0 0 \times 1 0 0$ grid which was used as the benchmark problem for quantitative analysis. The flow speed for the inlet and outlet was set to 0.5. The timestep of the integrator was set to $d t = 4$ and the viscosity and fluid density were set to $\mu = 0 . 1$ and $\rho = 4$ respectively. + +![](images/0679b34b9e40dd6fbcd896e5dfe9d2350e866f3fa97de7241da416b4cf20acb7.jpg) +Figure 7: The left column shows $\Omega$ (in white) / $\partial \Omega$ (in black) and the right column shows $\vec { v _ { d } }$ for three examples of training domains. (Colors indicate the direction and magnitude of $\vec { v _ { d } }$ as depicted in Figure 9a) + +![](images/96b4681cd4da05c85f6888b7d4f74625dfc620d8aa2ce10458c046f72e55cfd8.jpg) +Figure 8: Our models generalize to various domain geometries, although being trained only on simple shapes (see Figure 7) + +# F QUALITATIVE COMPARISON OF $\vec { a }$ -NET AND $\vec { v }$ -NET + +We give a qualitative example to show the benefits of using a vector potential. Figure 10 demonstrates that the $\vec { a }$ -Net finds plausible solutions for a folded pipe domain while the $\vec { v }$ -Net looses most of the flow in the center of the domain. This is in good accordance with quantitative results shown in section 1. The folded pipe domain is particularly difficult to learn as the flow field contains long range dependencies to the inlet and outlet (as shown in the bottom row in Figure 7). + +# G TRAINING WITHOUT RESETTING ENVIRONMENTS + +We performed an ablation study to investigate what happens if we do not reset old environments from time to time and, thus, do not continuously present the fluid model with cold starts during training. Figure 11 shows that in this case, large error spikes appear in the validation curve. These error spikes appear since the model has troubles to perform a cold start as can be seen in Figure 11b: compared to a properly trained model (see Figure 4) the model takes longer to perform a cold start (ca 100 steps) and converges to a solution with high $L _ { p }$ - and $L _ { d } .$ - losses. By resetting the environments from time to time during training, we can prevent these error spikes as shown in Figure 11c. + +![](images/7ef32d47da39bc6f2709e48b8263109e11b93a309f24cdd7e9700f02552db9ad.jpg) +Figure 9: a) shows legend for $\scriptstyle { \vec { v _ { d } } }$ ; b) shows $\Omega$ (in white) $/ \partial \Omega$ (in black) for the benchmark problem; c) shows $\vec { v _ { d } }$ for the benchmark problem. (Colors indicate the direction of $\vec { v _ { d } }$ as depicted in a) + +![](images/c6c35a94812aee60c202ac85d9e3bb29901144c51d0e016529668a2acd428f16.jpg) +Figure 10: Qualitative comparison of $\vec { a }$ -Net and $\vec { v }$ -Net in a folded pipe domain + +![](images/e9613f5f266dea5d70e2be191186fc22a58c75552972d2b5b6f43b06f8bbc705.jpg) +Figure 11: a) ablation study without resetting environments: validation curve shows large error spikes during training; b) error spike: the fluid model takes longer to perform a cold start and converges to a solution with high losses; c) original training with resetting environments: validation curve is stable \ No newline at end of file diff --git a/parse/train/KUDUoRsEphu/KUDUoRsEphu_content_list.json b/parse/train/KUDUoRsEphu/KUDUoRsEphu_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..eb36124c5a6788802057ca78091b0e31bac17be7 --- /dev/null +++ b/parse/train/KUDUoRsEphu/KUDUoRsEphu_content_list.json @@ -0,0 +1,1861 @@ +[ + { + "type": "text", + "text": "LEARNING INCOMPRESSIBLE FLUID DYNAMICS FROM SCRATCH - TOWARDS FAST, DIFFERENTIABLE FLUID MODELS THAT GENERALIZE ", + "text_level": 1, + "bbox": [ + 174, + 98, + 823, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nils Wandel \nDepartment of Computer Science University of Bonn \nwandeln@cs.uni-bonn.de \nMichael Weinmann \nDepartment of Computer Science \nUniversity of Bonn \nmw@cs.uni-bonn.de \nReinhard Klein \nDepartment of Computer Science \nUniversity of Bonn \nrk@cs.uni-bonn.de ", + "bbox": [ + 184, + 194, + 405, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 593, + 194, + 813, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 184, + 272, + 405, + 328 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 364, + 544, + 380 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Fast and stable fluid simulations are an essential prerequisite for applications ranging from computer-generated imagery to computer-aided design in research and development. However, solving the partial differential equations of incompressible fluids is a challenging task and traditional numerical approximation schemes come at high computational costs. Recent deep learning based approaches promise vast speed-ups but do not generalize to new fluid domains, require fluid simulation data for training, or rely on complex pipelines that outsource major parts of the fluid simulation to traditional methods. ", + "bbox": [ + 233, + 396, + 764, + 507 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we propose a novel physics-constrained training approach that generalizes to new fluid domains, requires no fluid simulation data, and allows convolutional neural networks to map a fluid state from time-point $t$ to a subsequent state at time $t + d t$ in a single forward pass. This simplifies the pipeline to train and evaluate neural fluid models. After training, the framework yields models that are capable of fast fluid simulations and can handle various fluid phenomena including the Magnus effect and Kármán vortex streets. We present an interactive real-time demo to show the speed and generalization capabilities of our trained models. Moreover, the trained neural networks are efficient differentiable fluid solvers as they offer a differentiable update step to advance the fluid simulation in time. We exploit this fact in a proof-of-concept optimal control experiment. Our models significantly outperform a recent differentiable fluid solver in terms of computational speed and accuracy. ", + "bbox": [ + 233, + 511, + 764, + 690 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 718, + 336, + 733 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Simulating the behavior of fluids by solving the incompressible Navier-Stokes equations is of great importance for a wide range of applications and accurate as well as fast fluid simulations are a long-standing research goal. On top of simulating the behavior of fluids, several applications such as sensitivity analysis of fluids or gradient-based control algorithms rely on differentiable fluid simulators that allow to propagate gradients throughout the simulation (Holl et al. (2020)). ", + "bbox": [ + 174, + 750, + 823, + 819 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent advances in deep learning aim for fast and accurate fluid simulations but rely on vast datasets and / or do not generalize to new fluid domains. Kim et al. (2019) present a framework to learn parameterized fluid simulations and allow to interpolate efficiently in between such simulations. However, their work does not generalize to new domain geometries that lay outside the training data. Kim & Lee (2020) train a RNN-GAN that produces turbulent flow fields within a pipe domain, but do not show generalization results beyond pipe domains. Xie et al. (2018) introduce a tempoGAN to perform temporally consistent superresolution of smoke simulations. This allows to produce plausible high-resolution smoke-density fields for arbitrary low-resolution inputs, but our fluid model should output a complete fluid state description consisting of a velocity and a pressure field. Tompson et al. (2017) present how a Helmholtz projection step can be learned to accelerate Eulerian fluid simulations. This method generalizes to new domain geometries, but a particle tracer is needed to deal with the advection term of the Navier-Stokes equations. Furthermore, as Eulerian fluids do not model viscosity, effects like e.g. the Magnus effect or Kármán vortex streets cannot be simulated. Geneva & Zabaras (2020) propose a physics-informed framework to learn the entire update step for the Burgers equations in 1D and 2D, but no generalization results for new domain geometries are demonstrated. All of the aforementioned methods rely on the availability of vast amounts of data from fluid-solvers such as FEniCS, OpenFOAM or Mantaflow. Most of these methods do not generalize well or outsource a major part of the fluid simulation to traditional methods such as low-resolution fluid solvers or a particle tracer. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 270 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we propose a novel unsupervised training framework to learn incompressible fluid dynamics from scratch. It does not require any simulated fluid-data (neither as ground truth data, nor to train an adversarial network, nor to initialize frames for a physics-constrained loss) and generalizes to fluid domains unseen during training. It allows CNNs to learn the entire update-step of mapping a fluid domain from time-point $t$ to $t + d t$ without having to rely on low resolution fluid-solvers or a particle-tracer. In fact, we will demonstrate that a physicsconstrained loss function combined with a simple strategy to recycle fluid-data generated by the neural network at training time suffices to teach CNNs fluid dynamics on increasingly realistic statistics of fluid states. This drastically simplifies the training pipeline. Fluid simulations get efficiently unrolled in time by recurrently applying the trained model on a fluid state. Furthermore, the fluid models include viscous friction and handle effects such as the Magnus effect and Kármán vortex streets. On top of that, we show by a gradient-based optimal control example how backpropagation through time can be used to differentiate the fluid simulation. Code and pretrained models are publicly available at https://github.com/aschethor/ Unsupervised_Deep_Learning_of_Incompressible_Fluid_Dynamics/. ", + "bbox": [ + 174, + 277, + 825, + 486 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 505, + 344, + 521 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In literature, several different approaches can be found that aim to approximate the dynamics of PDEs in general and fluids in particular with efficient, learning-based surrogate models. ", + "bbox": [ + 174, + 537, + 823, + 565 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Lagrangian methods such as smoothed particle hydrodynamcs (SPH) Gingold & Monaghan (1977) handle fluids from the perspective of many individual particles that move with the velocity field. Following this approach, learning-based methods using regression forests by Ladický et al. (2015), graph neural networks by Mrowca et al. (2018); Li et al. (2019) and continuous convolutions by Ummenhofer et al. (2020) have been developed. In addition, Smooth Particle Networks (SP-Nets) by Schenck & Fox (2018) allow for differentiable fluid simulations within the Lagrangian frame of reference. These Lagrangian methods are particularly suitable when a fluid domain exhibits large, dynamic surfaces (e.g. waves or droplets). However, to simulate the dynamics within a fluid domain accurately, Eulerian methods, that treat the Navier-Stokes equations in a fixed frame of reference, are usually better suited. ", + "bbox": [ + 174, + 571, + 825, + 712 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Continuous Eulerian methods allow for mesh-free solutions by mapping domain coordinates (e.g. $x , y , t )$ directly onto field values (e.g. velocity $\\vec { v } \\ : /$ pressure $p$ ) (Sirignano & Spiliopoulos (2018); Grohs et al. (2018); Khoo et al. (2019)). Recent applications focused on flow through porous media (Zhu & Zabaras (2018); Zhu et al. (2019); Tripathy & Bilionis (2018)), fluid modeling (Yang et al. (2016); Raissi et al. (2018)), turbulence modeling (Geneva & Zabaras (2019); Ling et al. (2016)) and modeling of molecular dynamics (Schöberl et al. (2019)). Training is usually based on physics-constrained loss functions that penalize residuals of the underlying PDEs. Similar to our approach, Raissi et al. (2019) uses vector potentials to obtain continuous divergence-free velocity fields to approximate the incompressible Navier-Stokes equations. Continuous methods return smooth, accurate results and can overcome the curse of dimensionality of discrete techniques in high-dimensional PDEs (Grohs et al. (2018)). However, these networks are trained on a specific domain and cannot generalize to new environments or be used in interactive scenarios. ", + "bbox": [ + 174, + 718, + 825, + 885 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Discrete Eulerian methods, on the other hand, aim to solve the underlying PDEs on a grid and early work dates back to Harlow & Welch (1965) and Stam (1999). Accelerating such traditional works with deep learning techniques is a major field of research and all of the methods mentioned in the introduction fall into this category. Further methods include the approach by Thuerey et al. (2019) to learn solutions of the Reynolds-averaged Navier-Stokes equations for airfoil flows, but requires large amounts of training data and does not generalize beyond airfoil flows. In the work by Um et al. (2020), a correction step is learned that brings solutions of a low-resolution differentiable fluid solver closer to solutions of a high-resolution fluid simulation. However, generalization results for new domain geometries were not presented. The works of Mohan et al. (2020) and Kim et al. (2019) show that vector potentials are suitable to enforce the incompressibility constraint in fluids but do not generalize to new fluid domains beyond their training data. ", + "bbox": [ + 174, + 103, + 825, + 256 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 276, + 282, + 292 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we briefly review the incompressible Navier-Stokes equations, which are to be solved by the neural network. Then, we explain how the Helmholtz decomposition can be exploited to ensure incompressibility within the fluid domain. Furthermore, we provide details of our discrete spatio-temporal fluid representation and introduce the fluid model. Afterwards, we formulate a physics-constrained loss function based on residuals of the Navier-Stokes equations and introduce a pressure regularization term for very high Reynolds numbers. Finally, we explain the unsupervised training strategy. ", + "bbox": [ + 174, + 308, + 825, + 406 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ", + "text_level": 1, + "bbox": [ + 174, + 421, + 545, + 436 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Most fluids can be modeled with the incompressible Navier-Stokes equations - a set of non-linear equations that describe the interplay of a velocity field $\\vec { v }$ and a pressure field $p$ within a fluid domain $\\Omega$ : ", + "bbox": [ + 174, + 446, + 825, + 489 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c2739be49bcb70d4e71989d437fa9dc175763d8b140a742b3353373fd1da5aa6.jpg", + "text": "$$\n\\begin{array} { r l r } { \\boldsymbol { \\nabla } \\cdot \\boldsymbol { \\vec { v } } = 0 } & { \\mathrm { i n c o m p r e s s i b i l i t y ~ o n ~ } \\Omega } \\\\ { \\rho \\dot { \\vec { v } } = \\rho \\left( \\frac { \\partial \\vec { v } } { \\partial t } + \\left( \\vec { v } \\cdot \\boldsymbol { \\nabla } \\right) \\vec { v } \\right) = - \\boldsymbol { \\nabla } p + \\mu \\Delta \\vec { v } + \\vec { f } } & { \\mathrm { c o n s e r v a t i o n ~ o f ~ m o m e n t u m ~ o n ~ } \\Omega } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 513, + 774, + 571 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here, $\\rho$ describes the fluid density and $\\mu$ the viscosity. Equation 1 states that the fluid is incompressible and thus $\\vec { v }$ is divergence-free. Equation 2 states that the change in momentum of fluid particles must correspond to the sum of forces that arise from the pressure gradient, viscous friction and external forces. Here, external forces on the fluid (such as e.g. gravity) can be neglected, so we set ${ \\vec { f } } = 0$ . ", + "bbox": [ + 174, + 574, + 825, + 646 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "These incompressible Navier-Stokes equations shall be solved by a CNN given initial conditions $\\vec { v } ^ { 0 }$ and $p ^ { 0 }$ at the beginning of the simulation and Dirichlet boundary conditions which constrain the velocity field at the domain boundary $\\partial \\Omega$ : ", + "bbox": [ + 176, + 654, + 823, + 695 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/382b2c7eff65cdde5b09533719fcbe295bfe23f03b98979e602e16522a1b6084.jpg", + "text": "$$\n\\vec { v } = \\vec { v } _ { d } \\qquad \\mathrm { D i r i c h l e t ~ b o u n d a r y ~ c o n d i t i o n ~ o n ~ } \\partial \\Omega\n$$", + "text_format": "latex", + "bbox": [ + 294, + 700, + 699, + 718 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 HELMHOLTZ DECOMPOSITION ", + "text_level": 1, + "bbox": [ + 176, + 733, + 428, + 748 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A common method to ensure incompressibility of a fluid (see Equation 1) is to project the flow field onto the divergence-free part of its Helmholtz decomposition. The Helmholtz theorem states that every vector field $\\vec { v }$ can be decomposed into a curl-free part $( \\nabla q )$ and a divergence-free part $( \\nabla \\times \\vec { a } )$ : ", + "bbox": [ + 174, + 758, + 823, + 803 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/5d37c1b17de1cc04d458fc63441646666d5fed6f98220fdcdde403f6afb38e4d.jpg", + "text": "$$\n\\vec { v } = \\nabla q + \\nabla \\times \\vec { a }\n$$", + "text_format": "latex", + "bbox": [ + 441, + 808, + 558, + 824 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Note, that $\\nabla \\times ( \\nabla q ) = \\vec { 0 }$ and $\\nabla \\cdot ( \\nabla \\times { \\vec { a } } ) = 0$ . The Helmholtz projection consists of solving the Poisson problem $\\nabla \\cdot \\vec { v } = \\Delta q$ for $q$ , followed by substracting $\\nabla q$ from the original flow field. However, solving the Poisson equation on arbitrary domains comes at high computational costs for classical methods and one has to rely e.g. on conjugate gradient methods to approximate its solution. ", + "bbox": [ + 174, + 830, + 823, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here, we propose a different approach and directly try to learn a vector potential $\\vec { a }$ with $\\vec { v } = \\nabla \\times \\vec { a }$ . This ensures that the network outputs a divergence-free velocity field within the domain $\\Omega$ and automatically solves Equation 1. In this work, we consider 2D fluid simulations, so only the $\\mathbf { Z }$ - component of $\\vec { a }$ , $a _ { z }$ , is of interest since $v _ { z }$ and all derivatives with respect to the $z$ -axis are zero: ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 102, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e9bd2999fb9a6c2122ff0c0f5828b5f9ab50ac874726e203ce7b01db9ed3ba1a.jpg", + "text": "$$\n\\nabla \\times { \\vec { a } } = \\left( { \\partial _ { z } a _ { x } - \\partial _ { z } a _ { y } } \\right) = \\left( { \\partial _ { y } a _ { z } } \\right) = \\left( { \\begin{array} { c } { v _ { x } } \\\\ { v _ { y } } \\\\ { 0 } \\end{array} } \\right) = { \\vec { v } }\n$$", + "text_format": "latex", + "bbox": [ + 313, + 138, + 684, + 184 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 DISCRETE SPATIO-TEMPORAL FLUID REPRESENTATION ", + "text_level": 1, + "bbox": [ + 173, + 198, + 599, + 214 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Marker-And-Cell (MAC) grid To solve the Navier-Stokes equations, we represent the relation between $a _ { z } , v _ { x } , v _ { y } , p$ on a 2D staggered marker-and-cell (MAC) grid (see Figure 1a). Therefore, we discretise time and space as follows: ", + "bbox": [ + 173, + 224, + 826, + 267 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/aab624e339ff56bb8238810aeeac252a575df0efa30f328028aa0967a54307b7.jpg", + "text": "$$\n\\vec { a } ( x , y , t ) = \\left( \\begin{array} { c } { 0 } \\\\ { 0 } \\\\ \\left( a _ { z } \\right) _ { i , j } ^ { t } \\rule { 0 ex } { 5 ex } \\right) ; \\vec { v } ( x , y , t ) = \\binom { \\left( v _ { x } \\right) _ { i , j } ^ { t } } { \\left( v _ { y } \\right) _ { i , j } ^ { t } } ; p ( x , y , t ) = p _ { i , j } ^ { t } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 277, + 284, + 720, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Obtaining gradient, divergence, Laplace and curl operations on this grid with finite differences is straight forward and can be efficiently implemented with convolutions (see appendix A). ", + "bbox": [ + 173, + 335, + 826, + 364 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/9b2df2899d456e4e435eca52239a4b16982712cebefb7099762ffba6f6c0e6f2.jpg", + "image_caption": [ + "Figure 1: MAC grid and diagram of the fluid model. " + ], + "image_footnote": [], + "bbox": [ + 197, + 387, + 800, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Explicit, Implicit, Implicit-Explicit (IMEX) time integration methods The discretization of the time domain is needed to deal with the time-derivative of the velocity fiel d ∂\\~v∂t in Equation 2, which becomes: ", + "bbox": [ + 173, + 647, + 825, + 688 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a68a0861a6a57a77e0f3cab5216cc15f4e068bb6ec9724bb4787efe7195131a5.jpg", + "text": "$$\n\\rho \\left( \\frac { \\vec { v } ^ { t + d t } - \\vec { v } ^ { t } } { d t } + \\left( \\vec { v } ^ { t ^ { \\prime } } \\cdot \\nabla \\right) \\vec { v } ^ { t ^ { \\prime } } \\right) = - \\nabla p ^ { t + d t } + \\mu \\Delta \\vec { v } ^ { t ^ { \\prime } } + \\vec { f }\n$$", + "text_format": "latex", + "bbox": [ + 302, + 685, + 697, + 722 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The goal is to take as large as possible timesteps $d t$ while maintaining stable and accurate solutions. Stability and accuracy largely depend on the definition of $v ^ { t ^ { \\prime } }$ . In literature, choosing $\\boldsymbol { v } ^ { t ^ { \\prime } } = \\boldsymbol { v } ^ { t }$ is often referred to as explicit integration methods and frequently leads to unstable behavior. Choosing $v ^ { t ^ { \\prime } } = v ^ { t + d t }$ is usually associated with implicit integration methods and gives stable solutions at the cost of numerical dissipation. Implicit-Explicit (IMEX) methods, which set $v ^ { t ^ { \\prime } } = ( v ^ { t } + v ^ { t + d t } ) / 2$ are a compromise between both methods and considered to be more accurate but less stable than implicit methods. ", + "bbox": [ + 173, + 729, + 826, + 835 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.4 FLUID MODEL ", + "text_level": 1, + "bbox": [ + 174, + 103, + 316, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We represent the fluid dynamics by a recurrent model that maps the fluid state $p ^ { t } , \\vec { a } ^ { t }$ for timestep $t$ and the domain description $\\Omega ^ { t + d t } , \\vec { v } _ { d } ^ { t + d t }$ to the fluid state $p ^ { t + d t } , \\vec { a } ^ { t + d t }$ of the next timestep. Here, $p ^ { t }$ describes the pressure field and $\\vec { a } ^ { t }$ describes the vector potential of . For $t = 0$ , we consider initial states $p ^ { 0 } = 0$ and $\\vec { a } ^ { 0 } = \\vec { 0 }$ , however, other initial conditions could be considered as well. $\\Omega ^ { t + d t }$ is a binary mask that contains the domain geometry and is 1 for the fluid domain and 0 everywhere else. For the boundary of the domain, we simply take the inverse of $\\Omega \\colon \\partial \\Omega = 1 - \\Omega$ . $\\vec { v } _ { d } ^ { t + \\dot { d } t }$ represents the Dirichlet boundary conditions and contains a velocity field that must be matched by ${ \\vec { v } } ^ { t + d t }$ at the domain boundaries. Figure 1b shows a diagram of the fluid model. First, $\\left( p ^ { t } , \\vec { a } ^ { t } , \\Omega ^ { t + d t } , \\vec { v } _ { d } ^ { t + d t } \\right)$ are taken to derive a slightly more meaningful feature representation that comprises $\\left( p ^ { t } , a ^ { t } , \\nabla \\times a ^ { t } , \\Omega ^ { t + d t } , \\partial \\Omega ^ { t + d t } , \\Omega ^ { t + d t } \\cdot \\nabla \\times a ^ { t } , \\Omega ^ { t + d t } \\cdot p ^ { t } , \\partial \\Omega ^ { t + d t } \\cdot \\vec { v } _ { d } ^ { t + d t } \\right)$ . These features can be very efficiently computed with convolutions and are then fed into a U-Net (Ronneberger et al. (2015)) with a reduced number of channels (the exact network configuration can be found in appendix B). The mean of the U-Net output is set to 0 in order to keep $p$ and $\\vec { a }$ well defined and prevent drifting offset values. Finally, the output is added to $p ^ { t }$ and $\\vec { a } ^ { t }$ to obtain the updated fluid state $p ^ { t + d t }$ and $\\vec { a } ^ { t + d t }$ . ", + "bbox": [ + 173, + 130, + 826, + 349 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.5 PHYSICS-CONSTRAINED LOSS FUNCTION", + "text_level": 1, + "bbox": [ + 176, + 366, + 500, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Using the residuals of the Navier-Stokes equations (Equations 1 and 2), we can formulate the following loss terms on $\\Omega$ and $\\partial \\Omega$ : ", + "bbox": [ + 176, + 392, + 823, + 421 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/560a1fae369f8b42f48f2ab1f3a5be4107aa929f6c1c94893410c3404457bb26.jpg", + "text": "$$\n\\begin{array} { l l } { { L _ { d } = \\displaystyle { \\left\\| \\nabla \\cdot \\vec { v } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ d i v e r g e n c e ~ l o s s ~ o n ~ } \\Omega } } \\\\ { { L _ { p } = \\displaystyle { \\left\\| \\rho \\left( \\frac { \\partial \\vec { v } } { \\partial t } + \\left( \\vec { v } \\cdot \\nabla \\right) \\vec { v } \\right) + \\nabla p - \\mu \\Delta \\vec { v } - \\vec { f } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ m o m e n t u m ~ l o s s ~ o n ~ } \\Omega } } \\\\ { { L _ { b } = \\displaystyle { \\left\\| \\vec { v } - \\vec { v } _ { d } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ b o u n d a r y ~ l o s s ~ o n ~ } \\partial \\Omega } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 233, + 444, + 767, + 527 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Combining the described loss terms, we obtain the following loss function: ", + "bbox": [ + 173, + 539, + 665, + 554 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fcc301942512405e232beefdcdc19a6dfc51ec3a8249802eb2c2216684b8ac5c.jpg", + "text": "$$\nL = \\alpha L _ { d } + \\beta L _ { p } + \\gamma L _ { b }\n$$", + "text_format": "latex", + "bbox": [ + 418, + 561, + 580, + 579 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\alpha , \\beta , \\gamma$ are hyperparameters that weight the contributions of the different loss terms. Note that if we use a vector potential ${ \\vec { v } } = \\nabla \\times { \\vec { a } }$ , $L _ { d } = 0$ is automatically fulfilled and we can set $\\alpha = 0$ . This loss function can be computed very efficiently with convolutions in $O ( N )$ (where $N =$ number of grid cells), whereas solving the Navier-Stokes equations explicitly would be computationally a lot more expensive. For detailed descriptions regarding the fully discretized loss-function, we refer to appendix A. ", + "bbox": [ + 174, + 584, + 825, + 669 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.6 PRESSURE REGULARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 685, + 418, + 700 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For very high Reynolds numbers (see Equation 13) and inviscid flows, training becomes unstable as viscous friction cannot dissipate enough energy out of the system. This leads to unrealistic gradients in $\\vec { v }$ and $p$ . For such cases, we introduce an additional regularization term for the loss function (11) that can be traded off with $L _ { p }$ to stabilize training: ", + "bbox": [ + 174, + 712, + 823, + 767 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f76537da61c027d6456687ae5684be73967eba203194cb361484777f2199aea9.jpg", + "text": "$$\nL _ { r } = \\| \\nabla p \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 454, + 775, + 544, + 795 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The intuition behind this regularization term is, that we want to penalize unrealistically high energies in the pressure field. ", + "bbox": [ + 171, + 801, + 825, + 830 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.7 TRAINING STRATEGY ", + "text_level": 1, + "bbox": [ + 176, + 103, + 364, + 117 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training starts with initializing a pool $\\{ \\Omega _ { k } ^ { 0 } , ( v _ { d } ) _ { k } ^ { 0 } , ( a _ { z } ) _ { k } ^ { 0 } , p _ { k } ^ { 0 } \\}$ of randomized domains $\\Omega _ { k } ^ { 0 }$ and boundary conditions $( v _ { d } ) _ { k } ^ { 0 }$ as well as initial conditions for the vector potential and pressure fields that we both set to zero $( \\check { ( a _ { z } ) } _ { k } ^ { 0 } = 0$ and $p _ { k } ^ { 0 } = 0$ ). The resolution of our training domains is $1 0 0 \\mathrm { x } 3 0 0$ grid cells and example-domains of the training pool are shown in appendix C. Note that our training pool does not rely on any previously simulated fluid-data. ", + "bbox": [ + 174, + 128, + 825, + 200 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "At each training step, a random mini-batch $\\{ \\Omega _ { k } ^ { t } , ( v _ { d } ) _ { k } ^ { t } , ( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t } \\} _ { \\{ k \\in \\operatorname * { m i n i b a t c h } \\} }$ is drawn from the pool and fed into the neural network which is designed to predict the velocity $( \\vec { v } _ { k } ^ { t + d t } = \\nabla \\times \\vec { a } _ { k } ^ { t + d t } )$ and pressure $( p _ { k } ^ { t + d t } )$ fields of the next time step. Based on a physics-constrained loss-function (Equation 11), we update the weights of the network using the Adam optimizer (Kingma & Ba (2015)). At the end of each training step, the pool is updated by replacing the old vector potential and pressure fields $( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t }$ by the newly predicted ones $( a _ { z } ) _ { k } ^ { t + \\dot { d t } } , p _ { k } ^ { \\dot { t } + d t }$ t, pt+dtk . This recycling strategy fills the training pool with more and more realistic fluid states as the model becomes better at simulating fluid dynamics. ", + "bbox": [ + 174, + 205, + 825, + 313 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "From time to time, old environments of the training pool are replaced by new randomized environments and the vector potential as well as the pressure fields are reset to 0. This increases the variance of the training pool and helps the neural network to learn \"cold starts\" from \\~0-velocity and 0-pressure fields. ", + "bbox": [ + 174, + 319, + 825, + 377 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Besides the fluid model described above, which we denote as $\\vec { a }$ -Net in the following, we also trained an ablation model, $\\vec { v }$ -Net, that directly learns to predict the velocity field without a vector potential. For the implementation of both models, we used the popular machine learning framework Pytorch and trained the models on a NVidia GeForce RTX 2080 Ti. Training converged after about 1 day. The hyperparameters in the loss-function for the $\\vec { a }$ -Net were $\\beta = 1$ and $\\gamma = 2 0$ . The reason for choosing a higher weight for the loss term $L _ { b }$ than for $L _ { p }$ was the observation, that errors in $L _ { b }$ can lead to unrealistic flows leaking through boundaries. For the ablation study $\\vec { v } \\cdot \\vec { }$ -Net), we used $\\alpha = 1 0 0 , \\beta = 1 , \\gamma = 0 . 0 0 1$ . Here, we had to choose a very high weight for $L _ { d }$ to ensure incompressibility of the fluid, otherwise unrealistic source and sink effects start to appear. For $L _ { b }$ , on the other hand, we used a very low weight as the boundary conditions can be trivially learned by the $\\vec { v }$ -Net. We used these parameter settings for all experiments. ", + "bbox": [ + 174, + 383, + 825, + 536 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 559, + 281, + 575 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To evaluate the potential of our method, we assess its ability to reproduce physical effects such as Kármán vortex streets and the Magnus effect. In addition, we demonstrate its generalization capability and real-time performance. Finally, we test the fluid models quantitatively. ", + "bbox": [ + 174, + 592, + 825, + 635 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 QUALITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 654, + 405, + 667 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Qualitative analysis of wake dynamics Qualitative effects in fluid dynamics such as the wake dynamics behind an obstacle are closely related to the Reynolds number. It is a dimensionless quantity defined by: ", + "bbox": [ + 174, + 680, + 825, + 723 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/4dfd04cc28aff9cd79aa9924737756fe96be62393b9e0a5e81f5954f0c177749.jpg", + "text": "$$\nR e = \\frac { \\rho \\left\\| \\vec { v } \\right\\| D } { \\mu }\n$$", + "text_format": "latex", + "bbox": [ + 447, + 723, + 550, + 758 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here, $\\rho$ is the fluid density, $\\lVert \\vec { v } \\rVert$ is the fluid speed, $D$ is the diameter of the obstacle, and $\\mu$ is the viscosity. (We use the units of the grid). ", + "bbox": [ + 173, + 762, + 823, + 792 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We retrained models for different values of $\\mu$ and $\\rho$ to compare the fluid behavior for a wide range of Reynolds numbers. Figure 2 shows, that the trained models are able to predict the wake dynamics behind an obstacle in good accordance with qualitative expectations from fluid dynamics. As a rule of thumb, for $R e \\ll 1$ , the flow becomes time-reversible. This can be noticed in Figure 2a by the symmetry of the flow before and after the obstacle and the nearly constant pressure gradient within the pipe. Starting from $R e \\approx 1 0$ , the flow is still laminar but a static wake is forming behind the obstacle (see Figure 2b). For Reynolds numbers $R e > \\approx 9 0$ , Kármán vortex streets start to appear (see Figure 2c). A Kármán vortex street consists of clock and counterclockwise spinning vortices that are generated at the obstacle and then start moving in a regularly oscillating pattern with the flow. For very large Reynolds numbers or inviscid flows, the flow field becomes turbulent, which can be recognized by the irregular patterns behind the obstacle in Fig 2d. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/468c39a4bc3b15c1ad09df22f1ea2d017e585da0b07e13fdda3acc4af90342a2.jpg", + "image_caption": [ + "Figure 2: After training, our models are able to show correct wake flow dynamics for a wide range of different Reynolds numbers. $\\dot { \\boldsymbol { D } } = 3 0$ , $\\| \\vec { v } \\| = 0 . 5 )$ . Streamlines indicate flow direction, linewidth indicates speed and colors represent the pressure field (blue: low pressure / yellow: high pressure). " + ], + "image_footnote": [], + "bbox": [ + 209, + 162, + 789, + 381 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Magnus effect The Magnus effect appears when a flow interacts with a rotating body. It is widely known e.g. in sports such as soccer or tennis where spin is used to deflect the path of a ball. The reason for the deflection stems from a low pressure field where the surface of the object moves along flow direction and a high pressure field where the object surface moves against the flow. Figure 3a shows, that our models are able to reproduce the Magnus effect around a rotating cylinder. ", + "bbox": [ + 174, + 458, + 825, + 529 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f016ae8a12c9f62a822e83f75f847c39d114a2759991e05eb174aa68cf62956a.jpg", + "image_caption": [ + "(a) Magnus effect on a clock-wise turning cylinder.(b) Generalization example: Note that the fluid model has never been confronted with wingprofiles during training. " + ], + "image_footnote": [], + "bbox": [ + 204, + 559, + 790, + 633 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 3: Our models feature the Magnus effect and generalize to new fluid domains. Further examples are presented in appendix D and the video. ", + "bbox": [ + 176, + 690, + 823, + 717 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Analysis of generalization capability We tested the networks capability to generalize to objects not seen during training. Figure 3b shows the networks capability to meet boundary conditions of an airfoil and return a plausible pressure field that produces lift (see low pressure on top of wing). Note that in contrast to the approach by Thuerey et al. (2019), which learns simplified, time-averaged solutions of the Navier-Stokes equations, our method is able to simulate the full incompressible Navier-Stokes equations for an airfoil without relying on any ground truth data or having seen airfoilgeometries during training. In fact, the network was only trained on simple randomized domains as highlighted in appendix C and Figure 7. Possible reasons for the networks generalization capabilities are: ", + "bbox": [ + 174, + 743, + 825, + 868 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• During training, the network gets confronted with an infinite number of different flowfields and randomized domain configurations because the training pool gets updated at every training step. This prevents the network from over-fitting. ", + "bbox": [ + 217, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• The dynamics of a fluid-particle are mostly determined by its local neighborhood / surrounding particles. This means, the update step for a certain cell on the MAC grid is mostly determined by close / neighboring MAC-grid cells. Since more complicated shapes can be seen locally as a composition of basic shapes (e.g. the front of the wing can be locally regarded as a cylinder), it suffices to train on basic shapes that provide the network with enough examples to generalize to more complicated shapes. ", + "bbox": [ + 217, + 103, + 825, + 188 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Further generalization examples are provided in appendix D. ", + "bbox": [ + 174, + 209, + 570, + 223 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Real-time capability The fluid simulation can be easily parallelized and takes low computational costs as one time-integration step consists just of a single forward pass through a convolutional neural network. This enables for example interactive real-time simulations. We implemented a demo that allows to interact with a fluid by moving obstacles, rotating spheres and changing the flow speed within a pipe (see video in supplementary material and source code). Our method runs at 250 timesteps per second on a $1 0 0 \\mathrm { x } 3 0 0$ grid. In the respective experiments, we used a NVidia GeForce RTX 2080 Ti consuming about $8 6 0 \\mathrm { M B }$ of GPU memory. ", + "bbox": [ + 174, + 261, + 825, + 358 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 QUANTITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 397, + 415, + 411 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare our method $\\vec { a }$ -Net) quantitatively with PhiFlow by Holl et al. (2020). Phiflow is a recent, open source, differentiable fluid simulator based on a MAC grid data structure. Furthermore, we provide an ablation study ${ \\vec { v } } .$ -Net) that does not make use of the Helmholtz decomposition but directly works on the velocity field $\\vec { v }$ . ", + "bbox": [ + 174, + 431, + 825, + 487 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Quantitative comparison of different fluid solvers is challenging, as their performance is highly dependent on factors like the geometry of the domain, fluid parameters such as viscosity or density, flow speed or the timestep of the integrator. As benchmarks for fluid simulations on MAC grids are not yet available, we built a simple toy domain on a $1 0 0 \\mathrm { ~ x ~ } 1 0 0$ grid which simulates a flow around an obstacle within a pipe (more details are provided in appendix E). ", + "bbox": [ + 174, + 494, + 825, + 564 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "First, we compared the computational speed on a CPU and GPU by comparing the integration time-steps per second (see Table 1). The $\\vec { v }$ -Net as well as the $\\vec { a }$ -Net are significantly faster than PhiFlow (11x on CPU and $4 0 \\mathrm { x }$ on GPU) as they do not rely on an iterative conjugate gradient solver but instead use a single forward pass through a convolutional neural network that can be easily parallelized on a GPU. To provide a fair comparison on $L _ { d }$ , we set the velocity field at the boundaries equal to $\\vec { v _ { d } }$ . This enables us to compute $L _ { d }$ for the $\\vec { a }$ -Net architecture on the domain boundaries which would otherwise have zero divergence everywhere. This way, $L _ { d }$ can be interpreted as a metric on how well the orthogonal components of the Dirichlet boundary conditions are met (i.e. no flow leaks through the boundaries). For $d t = 4$ , we outperformed Phiflow by several orders of magnitude. For both, $L _ { d }$ and $L _ { p }$ , the $\\vec { a }$ -Net architecture significantly outperformed the more naive $\\vec { v }$ -Net approach. ", + "bbox": [ + 174, + 570, + 825, + 724 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Furthermore, we investigated stability by evaluating the evolution of $L _ { p }$ and $L _ { d }$ for the $\\vec { a }$ -Net over time (see Figure 4). As the fluid state is initialized with $a _ { z } = 0$ and $p = 0$ , the $\\vec { a }$ -Net has to perform a cold-start which is the reason for high $L _ { p }$ and $L _ { d }$ during the first circa 70 steps. Afterwards, the $\\vec { a }$ -Net continues an accurate and stable fluid simulation. ", + "bbox": [ + 174, + 731, + 825, + 786 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/57a7fb536fbdf6d2f343d51671835704f414615c5a03a13645e7d8050e19d8bb.jpg", + "table_caption": [ + "Table 1: Quantitative comparison of timesteps per second (TPS) on CPU / GPU as well as divergence loss and momentum loss for differentiable fluid solvers on a $1 0 0 \\mathrm { x } 1 0 0$ grid for viscosity $\\mu =$ 0.1, density $\\rho = 4$ and timesteps of size $d t = 4$ . " + ], + "table_footnote": [], + "table_body": "
MethodCPU[TPS]GPU[TPS]LdLp
PhiFlow7-6.2e-4-
U-Net (ours)823118.66e-74.87e-5
α-Net (ours)823115.44e-71.56e-5
", + "bbox": [ + 200, + 818, + 537, + 864 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/61d33028f9c357e03bf3a26def07202e3075dc25a8368f4cd3727ad576eaeaf4.jpg", + "image_caption": [ + "Figure 4: Long term stability of fluid simulations performed by the $\\vec { a }$ -Net " + ], + "image_footnote": [], + "bbox": [ + 589, + 794, + 813, + 890 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 OPTIMAL CONTROL OF VORTEX SHEDDING FREQUENCY ", + "text_level": 1, + "bbox": [ + 173, + 103, + 609, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we present a proof-of-concept experiment that aims at controlling the shedding frequency of a Kármán vortex street behind an obstacle by changing the flow speed (see Figure 5a). To this end, we exploit our previously trained differentiable fluid models. ", + "bbox": [ + 173, + 130, + 825, + 171 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/12287e6b64712b1dba143620da30b36d8a493efa70292097fa075f8a7596b80d.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 174, + 212, + 352, + 282 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) control setup (domain size: $2 0 0 \\mathrm { x } 1 0 0$ grid cells) ", + "bbox": [ + 174, + 309, + 356, + 334 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/eb7c6bdf57710e1d85311fd4fa94bc4129112926d9bad99da48def5bc5fb1064.jpg", + "image_caption": [ + "Figure 5: The frequency of vortex streets can be controlled using our differentiable fluid models. " + ], + "image_footnote": [], + "bbox": [ + 372, + 198, + 818, + 334 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "First, we measure the y-component of the velocity field $v _ { y } ( t )$ behind an obstacle (see white box in Figure 5a) over 200 time steps. Then, we compute the frequency spectrum $V _ { y } ( f )$ of $v _ { y } ( t )$ using the fast Fourier transform (see Figure 5b). Now, we want to adjust the inflow $/$ outflow boundary conditions in $\\vec { v _ { d } }$ such that $E [ | V _ { y } ( f ) | ^ { 2 } ] = \\hat { f }$ . Here, $\\hat { f }$ is the target frequency. To optimize $\\Vec { v _ { d } }$ , we define a loss function $L = ( E [ | V _ { y } ( f ) | ^ { 2 } ] - \\hat { f } ) ^ { 2 }$ and compute the gradients $\\frac { \\partial L } { \\partial \\vec { v } _ { d } }$ with backpropagation through time. This is possible since all parts of the loss function including the fluid simulation that is performed by our trained neural fluid model as well as the fast Fourier transform are differentiable. Computing the gradients with a standard automatic differentiation library (Pytorch) took 3.5 seconds for all 200 time steps on our $2 0 0 \\mathrm { x } 1 0 0$ domain setup. This is considerably faster than the current state-of-the-art differentiable fluid solver by Takahashi et al. (2021) which takes 5.42 seconds for only 30 time steps on a smaller $1 2 8 \\mathrm { x } 1 2 8$ grid. The update steps of $\\Vec { v _ { d } }$ are done using the ADAMoptimizer and converge after approximately 70 iterations (see Figure 5c). We want to emphasize that differentiable fluid simulations are limited to scenarios with low Reynolds numbers as in the presence of turbulences, chaotic behavior will lead to exploding gradients. ", + "bbox": [ + 173, + 375, + 825, + 577 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 DISCUSSION AND OUTLOOK ", + "text_level": 1, + "bbox": [ + 176, + 597, + 441, + 612 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we present an unsupervised learning scheme for the incompressible Navier-Stokes equations and introduce a fluid model that uses a vector potential to output divergence-free velocity fields. Qualitative results of our trained fluid models are in good accordance with expectations from fluid dynamics for a wide range of Reynolds numbers and generalize to unknown fluid domains. Quantitative assessment showed superior performance in terms of accuracy and speed compared to Phiflow and an ablation study that directly predicts the velocity field. We present a real-time demo and demonstrate how differentiability can be used in a proof-of-concept fluid control scenario. We believe that our fluid models can significantly speed up more sophisticated fluid control pipelines such as described by Holl et al. (2020). ", + "bbox": [ + 173, + 627, + 825, + 753 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "First experiments of extending this approach to 3D deliver encouraging results and are topic of future research. Furthermore, on top of Dirichlet boundary conditions, Neumann boundary conditions and multi-phase domains could be incorporated in future fluid models as well. 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", + "bbox": [ + 176, + 854, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A PHYSICS-CONSTRAINED LOSS ON A MAC GRID", + "text_level": 1, + "bbox": [ + 173, + 101, + 612, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "As mentioned in Section 3.3 of the paper, our method relies on a staggered marker-and-cell grid representation for the vector potential as well as the velocity and pressure fields. In the following, we provide further details on how to apply this representation to learn incompressible fluid dynamics. ", + "bbox": [ + 174, + 133, + 826, + 176 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "To calculate the velocity field $\\vec { v } = \\nabla \\times \\vec { a }$ of a vector potential $\\vec { a }$ on a MAC grid in 2D, we have to compute the curl as follows: ", + "bbox": [ + 174, + 181, + 823, + 210 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/7b765b97ec27f39cb3ef2684b813a64e2db36d5b197645326fd190e5a3cb7cb1.jpg", + "text": "$$\n\\begin{array} { r } { ( v _ { x } ) _ { i , j } = ( a _ { z } ) _ { i + 1 , j } - ( a _ { z } ) _ { i , j } } \\\\ { ( v _ { y } ) _ { i , j } = ( a _ { z } ) _ { i , j } - ( a _ { z } ) _ { i , j + 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 400, + 217, + 596, + 256 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "If this vector potential is inserted into the divergence operator on a MAC grid, we can show that $\\nabla \\cdot \\vec { v } _ { i , j } = 0$ is indeed fulfilled: ", + "bbox": [ + 173, + 265, + 823, + 295 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/27299d46665b7ad835bedbb79eebe13b79a69c1071708847f8a81582ba0fedd0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla \\cdot \\vec { v } _ { i , j } = ( { v } _ { x } ) _ { i , j + 1 } - ( { v } _ { x } ) _ { i , j } + ( { v } _ { y } ) _ { i + 1 , j } - ( { v } _ { y } ) _ { i , j } } \\\\ & { \\qquad = \\begin{array} { r l } & { ( ( a _ { z } ) _ { i + 1 , j + 1 } - ( a _ { z } ) _ { i , j + 1 } ) - ( ( a _ { z } ) _ { i + 1 , j } - ( a _ { z } ) _ { i , j } ) } \\\\ & { \\qquad + \\left( ( a _ { z } ) _ { i + 1 , j } - ( a _ { z } ) _ { i + 1 , j + 1 } \\right) - ( ( a _ { z } ) _ { i , j } - ( a _ { z } ) _ { i , j + 1 } ) } \\\\ & { \\qquad = 0 } \\end{array} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 299, + 705, + 376 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Thus, for the $\\vec { a }$ -Net, the incompressibility equation is automatically fulfilled and no further training on the divergence loss $L _ { d }$ is required. However, for the $\\vec { v }$ -Net, the residuals of the divergence are still of importance: ", + "bbox": [ + 174, + 388, + 821, + 430 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/85aebe665a6f8b22e1f6ab2835d641bc59f4f38e34b69dba8879592f1bae0fbc.jpg", + "text": "$$\n( R _ { d } ) _ { i , j } ^ { t + d t } = \\nabla \\cdot \\vec { v } _ { i , j } ^ { t + d t } ( = 0 \\mathrm { f o r } \\vec { a } \\ – \\mathrm { N e t } )\n$$", + "text_format": "latex", + "bbox": [ + 374, + 429, + 624, + 450 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The residuals of the momentum equation in $x$ -direction can be computed as follows: ", + "bbox": [ + 173, + 458, + 727, + 473 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/977088132e08193c468d875d3a44cd8ec98d7341e179d84247d2de5b4fc0cd44.jpg", + "text": "$$\n\\begin{array} { r l } & { { R _ { p _ { x } } } ) _ { i , j } ^ { t + d t } = \\rho ( \\frac { ( v _ { x } ) _ { i , j } ^ { t + d t } - ( v _ { x } ) _ { i , j } ^ { t } } { d t } + ( v _ { x } ) _ { i , j } ^ { t ^ { \\prime } } \\cdot \\frac { ( v _ { x } ) _ { i , j + 1 } ^ { t ^ { \\prime } } - ( v _ { x } ) _ { i , j - 1 } ^ { t ^ { \\prime } } } { 2 } } \\\\ & { \\qquad + \\frac { \\frac { ( v _ { y } ) _ { i , j - 1 } ^ { t ^ { \\prime } } + ( v _ { y } ) _ { i , j } ^ { t ^ { \\prime } } } { 2 } \\cdot ( ( v _ { x } ) _ { i , j } ^ { t ^ { \\prime } } - ( v _ { x } ) _ { i - 1 , j } ^ { t ^ { \\prime } } ) + \\frac { ( v _ { y } ) _ { i + 1 , j - 1 } ^ { t ^ { \\prime } } + ( v _ { y } ) _ { i + 1 , j } ^ { t ^ { \\prime } } } { 2 } \\cdot ( ( v _ { x } ) _ { i + 1 , j } ^ { t ^ { \\prime } } - ( v _ { x } ) _ { i , j } ^ { t ^ { \\prime } } } { ( \\frac { 1 } { 2 } } } \\\\ & { \\qquad + ( p _ { i , j } ^ { t + d t } - p _ { i , j - 1 } ^ { t + d t } ) - \\mu \\cdot \\Delta ( v _ { x } ) _ { i , j } ^ { t ^ { \\prime } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 478, + 839, + 607 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Here, we use the following isotropic Laplace operator: ", + "bbox": [ + 174, + 631, + 531, + 646 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c608899301ad0536233808bab885c3579407cbae54d8bc518e7a673af9e5a84c.jpg", + "text": "$$\n\\begin{array} { c } { \\Delta s _ { i , j } = \\displaystyle \\frac { 1 } { 4 } ( 1 * s _ { i - 1 , j - 1 } + 2 * s _ { i - 1 , j } + 1 * s _ { i - 1 , j + 1 } } \\\\ { + 2 * s _ { i , j - 1 } - 1 2 * s _ { i , j } + 2 * s _ { i , j + 1 } } \\\\ { + 1 * s _ { i + 1 , j - 1 } + 2 * s _ { i + 1 , j } + 1 * s _ { i + 1 , j + 1 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 651, + 673, + 719 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The derivation of the advection term for $R _ { p _ { x } }$ is a bit more complex since on a MAC grid, $v _ { x }$ and $v _ { y }$ are displaced by half a pixel in $x$ -direction and $y$ -direction. To obtain the residuals of the momentum equation in $y$ -direction, $( R _ { p _ { y } } ) _ { i , j }$ , one has to take $( R _ { p _ { x } } ) _ { i , j }$ and swap $x$ and $y$ and the indices respectively. ", + "bbox": [ + 173, + 729, + 825, + 787 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Now, the discretized loss terms can be written as follows: ", + "bbox": [ + 174, + 792, + 550, + 809 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c8b5a55ade2b30eea738910d0f4694d8eddab4a5166c7cb6fcb34248edd17ffd.jpg", + "text": "$$\n\\begin{array} { l } { { { \\cal L } _ { d } ^ { t + d t } = \\displaystyle \\sum _ { i , j } \\Omega _ { i , j } ^ { t + d t } ( ( R _ { d } ) _ { i , j } ^ { t + d t } ) ^ { 2 } } } \\\\ { { { \\cal L } _ { p } ^ { t + d t } = \\displaystyle \\sum _ { i , j } \\Omega _ { i , j } ^ { t + d t } \\left( ( ( R _ { p _ { x } } ) _ { i , j } ^ { t + d t } ) ^ { 2 } + ( ( R _ { p _ { y } } ) _ { i , j } ^ { t + d t } ) ^ { 2 } \\right) } } \\\\ { { { \\cal L } _ { b } ^ { t + d t } = \\displaystyle \\sum _ { i , j } \\partial \\Omega _ { i , j } ^ { t + d t } \\left. \\vec { v } _ { d } ^ { t + d t } - \\vec { v } ^ { t + d t } \\right. ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 814, + 669, + 922 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Note, that all mentioned operations can be efficiently implemented with convolutions. To obtain the final velocities on a square grid, we project the velocity fields of the MAC grid back onto the $\\vec { a }$ -grid using linear interpolation: ", + "bbox": [ + 176, + 103, + 823, + 145 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1eae517d940c36f0f181a8d541f42793760b25bacaefcf7eab28e5622a99d324.jpg", + "text": "$$\n\\vec { v } = \\frac { 1 } { 2 } \\left( { ( v _ { x } ) _ { i - 1 , j } } + { ( v _ { x } ) _ { i , j } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 400, + 146, + 598, + 181 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B NETWORK ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 199, + 431, + 215 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Our fluid model is based on the U-Net architecture (Ronneberger et al. (2015)) with fewer channels (see Figure 6). As the pressure field and vector potential can have an arbitrary offset, we always normalize the mean of the pressure $( \\Delta p )$ and vector potential $( \\Delta a _ { z } )$ to 0 to keep these fields welldefined and prevent drifting offset values. ", + "bbox": [ + 174, + 232, + 825, + 287 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/a1472214f77f3cafefb7b390de489bc06eea4b6ab22438d1c71d7d3ffc0cd1ea.jpg", + "image_caption": [ + "Figure 6: U-Net architecture with fewer channels. " + ], + "image_footnote": [], + "bbox": [ + 258, + 301, + 738, + 501 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C EXAMPLES OF TRAINING DOMAINS ", + "text_level": 1, + "bbox": [ + 174, + 556, + 508, + 573 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The domains we used for training consist of $1 0 0 \\times 3 0 0$ grids. We used 3 different randomized domains as exemplary depicted in Figure 7. First, we have boxes with randomized height and width that float on randomized paths inspired by Brownian motion in a pipe with randomized flow speed. Second, we have the same setup but replaced the boxes by cylinders with randomized radii and angular velocities in order to learn the Magnus effect. Finally, we have a folded pipe system with randomized flow speed, that is randomly flipped along the $x$ -axis. ", + "bbox": [ + 173, + 589, + 825, + 674 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D FURTHER EXAMPLES OF GENERALIZATION ", + "text_level": 1, + "bbox": [ + 173, + 695, + 570, + 712 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Note that the network was only trained on simple domain geometries as presented in appendix C. Still, as can be seen in Figure 8, the network is capable of generalizing to far more complicated domain geometries (e.g. shark, car). Figure 8c shows that it can generalize to multiple objects in the scene, although the training set contained at most one object per scene. And Figure 8d shows that we can alter the outer boundary conditions as well. For real-time simulations, please have a look at our source code and the supplementary video. ", + "bbox": [ + 174, + 728, + 825, + 813 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "E QUANTITATIVE ANALYSIS: THE BENCHMARK PROBLEM ", + "text_level": 1, + "bbox": [ + 174, + 834, + 673, + 852 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Figure 9 shows the domain $\\Omega$ and $v _ { d }$ on a $1 0 0 \\times 1 0 0$ grid which was used as the benchmark problem for quantitative analysis. The flow speed for the inlet and outlet was set to 0.5. The timestep of the integrator was set to $d t = 4$ and the viscosity and fluid density were set to $\\mu = 0 . 1$ and $\\rho = 4$ respectively. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/0679b34b9e40dd6fbcd896e5dfe9d2350e866f3fa97de7241da416b4cf20acb7.jpg", + "image_caption": [ + "Figure 7: The left column shows $\\Omega$ (in white) / $\\partial \\Omega$ (in black) and the right column shows $\\vec { v _ { d } }$ for three examples of training domains. (Colors indicate the direction and magnitude of $\\vec { v _ { d } }$ as depicted in Figure 9a) " + ], + "image_footnote": [], + "bbox": [ + 254, + 112, + 736, + 376 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/96b4681cd4da05c85f6888b7d4f74625dfc620d8aa2ce10458c046f72e55cfd8.jpg", + "image_caption": [ + "Figure 8: Our models generalize to various domain geometries, although being trained only on simple shapes (see Figure 7) " + ], + "image_footnote": [], + "bbox": [ + 209, + 444, + 790, + 650 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F QUALITATIVE COMPARISON OF $\\vec { a }$ -NET AND $\\vec { v }$ -NET ", + "text_level": 1, + "bbox": [ + 176, + 713, + 625, + 729 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We give a qualitative example to show the benefits of using a vector potential. Figure 10 demonstrates that the $\\vec { a }$ -Net finds plausible solutions for a folded pipe domain while the $\\vec { v }$ -Net looses most of the flow in the center of the domain. This is in good accordance with quantitative results shown in section 1. The folded pipe domain is particularly difficult to learn as the flow field contains long range dependencies to the inlet and outlet (as shown in the bottom row in Figure 7). ", + "bbox": [ + 174, + 746, + 825, + 815 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "G TRAINING WITHOUT RESETTING ENVIRONMENTS ", + "text_level": 1, + "bbox": [ + 174, + 837, + 619, + 852 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We performed an ablation study to investigate what happens if we do not reset old environments from time to time and, thus, do not continuously present the fluid model with cold starts during training. Figure 11 shows that in this case, large error spikes appear in the validation curve. These error spikes appear since the model has troubles to perform a cold start as can be seen in Figure 11b: compared to a properly trained model (see Figure 4) the model takes longer to perform a cold start (ca 100 steps) and converges to a solution with high $L _ { p }$ - and $L _ { d } .$ - losses. By resetting the environments from time to time during training, we can prevent these error spikes as shown in Figure 11c. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/7ef32d47da39bc6f2709e48b8263109e11b93a309f24cdd7e9700f02552db9ad.jpg", + "image_caption": [ + "Figure 9: a) shows legend for $\\scriptstyle { \\vec { v _ { d } } }$ ; b) shows $\\Omega$ (in white) $/ \\partial \\Omega$ (in black) for the benchmark problem; c) shows $\\vec { v _ { d } }$ for the benchmark problem. (Colors indicate the direction of $\\vec { v _ { d } }$ as depicted in a) " + ], + "image_footnote": [], + "bbox": [ + 302, + 112, + 691, + 242 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/c6c35a94812aee60c202ac85d9e3bb29901144c51d0e016529668a2acd428f16.jpg", + "image_caption": [ + "Figure 10: Qualitative comparison of $\\vec { a }$ -Net and $\\vec { v }$ -Net in a folded pipe domain " + ], + "image_footnote": [], + "bbox": [ + 181, + 309, + 813, + 411 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 463, + 825, + 506 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/e9613f5f266dea5d70e2be191186fc22a58c75552972d2b5b6f43b06f8bbc705.jpg", + "image_caption": [ + "Figure 11: a) ablation study without resetting environments: validation curve shows large error spikes during training; b) error spike: the fluid model takes longer to perform a cold start and converges to a solution with high losses; c) original training with resetting environments: validation curve is stable " + ], + "image_footnote": [], + "bbox": [ + 176, + 532, + 818, + 669 + ], + "page_idx": 14 + } +] \ No newline at end of file diff --git a/parse/train/KUDUoRsEphu/KUDUoRsEphu_middle.json b/parse/train/KUDUoRsEphu/KUDUoRsEphu_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..afd10cf073bf36f4433e6dc6324f28aef57294ab --- /dev/null +++ b/parse/train/KUDUoRsEphu/KUDUoRsEphu_middle.json @@ -0,0 +1,40684 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 506, + 98 + ], + "score": 1.0, + "content": "LEARNING INCOMPRESSIBLE FLUID DYNAMICS FROM", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 504, + 117 + ], + "score": 1.0, + "content": "SCRATCH - 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(2015),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "graph neural networks by Mrowca et al. (2018); Li et al. (2019) and continuous convolutions by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "Ummenhofer et al. (2020) have been developed. In addition, Smooth Particle Networks (SP-Nets)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "by Schenck & Fox (2018) allow for differentiable fluid simulations within the Lagrangian frame of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 518, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 534 + ], + "score": 1.0, + "content": "reference. These Lagrangian methods are particularly suitable when a fluid domain exhibits large,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "dynamic surfaces (e.g. waves or droplets). However, to simulate the dynamics within a fluid domain", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "accurately, Eulerian methods, that treat the Navier-Stokes equations in a fixed frame of reference,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 553, + 206, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 206, + 564 + ], + "score": 1.0, + "content": "are usually better suited.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 454, + 506, + 564 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 584 + ], + "score": 1.0, + "content": "Continuous Eulerian methods allow for mesh-free solutions by mapping domain coordinates (e.g.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 131, + 592 + ], + "score": 0.28, + "content": "x , y , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 580, + 298, + 594 + ], + "score": 1.0, + "content": "directly onto field values (e.g. velocity", + "type": "text" + }, + { + "bbox": [ + 298, + 580, + 312, + 591 + ], + "score": 0.77, + "content": "\\vec { v } \\ : /", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 580, + 349, + 594 + ], + "score": 1.0, + "content": "pressure", + "type": "text" + }, + { + "bbox": [ + 350, + 582, + 357, + 592 + ], + "score": 0.62, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 580, + 505, + 594 + ], + "score": 1.0, + "content": ") (Sirignano & Spiliopoulos (2018);", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "Grohs et al. (2018); Khoo et al. (2019)). Recent applications focused on flow through porous", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 601, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 506, + 616 + ], + "score": 1.0, + "content": "media (Zhu & Zabaras (2018); Zhu et al. (2019); Tripathy & Bilionis (2018)), fluid modeling", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "(Yang et al. (2016); Raissi et al. (2018)), turbulence modeling (Geneva & Zabaras (2019); Ling", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "et al. (2016)) and modeling of molecular dynamics (Schöberl et al. (2019)). Training is usually", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "based on physics-constrained loss functions that penalize residuals of the underlying PDEs. Similar", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "to our approach, Raissi et al. (2019) uses vector potentials to obtain continuous divergence-free", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "velocity fields to approximate the incompressible Navier-Stokes equations. Continuous methods", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "return smooth, accurate results and can overcome the curse of dimensionality of discrete techniques", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "in high-dimensional PDEs (Grohs et al. (2018)). However, these networks are trained on a specific", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 690, + 454, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 454, + 702 + ], + "score": 1.0, + "content": "domain and cannot generalize to new environments or be used in interactive scenarios.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 567, + 506, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Discrete Eulerian methods, on the other hand, aim to solve the underlying PDEs on a grid and early", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "work dates back to Harlow & Welch (1965) and Stam (1999). Accelerating such traditional works", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "with deep learning techniques is a major field of research and all of the methods mentioned in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "introduction fall into this category. Further methods include the approach by Thuerey et al. (2019)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "to learn solutions of the Reynolds-averaged Navier-Stokes equations for airfoil flows, but requires", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "large amounts of training data and does not generalize beyond airfoil flows. In the work by Um", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "et al. (2020), a correction step is learned that brings solutions of a low-resolution differentiable fluid", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "solver closer to solutions of a high-resolution fluid simulation. However, generalization results for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "new domain geometries were not presented. The works of Mohan et al. (2020) and Kim et al. (2019)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "show that vector potentials are suitable to enforce the incompressibility constraint in fluids but do", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 358, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 358, + 205 + ], + "score": 1.0, + "content": "not generalize to new fluid domains beyond their training data.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 219, + 173, + 232 + ], + "lines": [ + { + "bbox": [ + 104, + 217, + 175, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 175, + 235 + ], + "score": 1.0, + "content": "3 METHOD", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "In this section, we briefly review the incompressible Navier-Stokes equations, which are to be solved", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "by the neural network. Then, we explain how the Helmholtz decomposition can be exploited to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "ensure incompressibility within the fluid domain. Furthermore, we provide details of our discrete", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "spatio-temporal fluid representation and introduce the fluid model. Afterwards, we formulate a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "physics-constrained loss function based on residuals of the Navier-Stokes equations and introduce a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "pressure regularization term for very high Reynolds numbers. Finally, we explain the unsupervised", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 176, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 176, + 324 + ], + "score": 1.0, + "content": "training strategy.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 334, + 334, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 334, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 334, + 348 + ], + "score": 1.0, + "content": "3.1 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "Most fluids can be modeled with the incompressible Navier-Stokes equations - a set of non-linear", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 324, + 378 + ], + "score": 1.0, + "content": "equations that describe the interplay of a velocity field", + "type": "text" + }, + { + "bbox": [ + 324, + 367, + 331, + 376 + ], + "score": 0.82, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 366, + 410, + 378 + ], + "score": 1.0, + "content": "and a pressure field", + "type": "text" + }, + { + "bbox": [ + 410, + 368, + 417, + 378 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 366, + 504, + 378 + ], + "score": 1.0, + "content": "within a fluid domain", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 377, + 120, + 389 + ], + "spans": [ + { + "bbox": [ + 107, + 378, + 115, + 387 + ], + "score": 0.79, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 377, + 120, + 389 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 407, + 474, + 453 + ], + "lines": [ + { + "bbox": [ + 123, + 407, + 474, + 453 + ], + "spans": [ + { + "bbox": [ + 123, + 407, + 474, + 453 + ], + "score": 0.75, + "content": "\\begin{array} { r l r } { \\boldsymbol { \\nabla } \\cdot \\boldsymbol { \\vec { v } } = 0 } & { \\mathrm { i n c o m p r e s s i b i l i t y ~ o n ~ } \\Omega } \\\\ { \\rho \\dot { \\vec { v } } = \\rho \\left( \\frac { \\partial \\vec { v } } { \\partial t } + \\left( \\vec { v } \\cdot \\boldsymbol { \\nabla } \\right) \\vec { v } \\right) = - \\boldsymbol { \\nabla } p + \\mu \\Delta \\vec { v } + \\vec { f } } & { \\mathrm { c o n s e r v a t i o n ~ o f ~ m o m e n t u m ~ o n ~ } \\Omega } \\end{array}", + "type": "interline_equation", + "image_path": "c2739be49bcb70d4e71989d437fa9dc175763d8b140a742b3353373fd1da5aa6.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 123, + 407, + 474, + 422.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 123, + 422.3333333333333, + 474, + 437.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 123, + 437.66666666666663, + 474, + 452.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 133, + 468 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 134, + 457, + 141, + 467 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 454, + 280, + 468 + ], + "score": 1.0, + "content": "describes the fluid density and", + "type": "text" + }, + { + "bbox": [ + 280, + 457, + 288, + 467 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "the viscosity. Equation 1 states that the fluid is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 207, + 479 + ], + "score": 1.0, + "content": "incompressible and thus", + "type": "text" + }, + { + "bbox": [ + 208, + 466, + 215, + 476 + ], + "score": 0.82, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "is divergence-free. Equation 2 states that the change in momentum of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 476, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 490 + ], + "score": 1.0, + "content": "fluid particles must correspond to the sum of forces that arise from the pressure gradient, viscous", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "friction and external forces. Here, external forces on the fluid (such as e.g. gravity) can be neglected,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 499, + 175, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 145, + 513 + ], + "score": 1.0, + "content": "so we set", + "type": "text" + }, + { + "bbox": [ + 146, + 499, + 171, + 513 + ], + "score": 0.92, + "content": "{ \\vec { f } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 500, + 175, + 513 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 108, + 518, + 504, + 551 + ], + "lines": [ + { + "bbox": [ + 107, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 107, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "These incompressible Navier-Stokes equations shall be solved by a CNN given initial conditions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 107, + 529, + 117, + 539 + ], + "score": 0.86, + "content": "\\vec { v } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 527, + 135, + 542 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 136, + 528, + 146, + 540 + ], + "score": 0.89, + "content": "p ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "at the beginning of the simulation and Dirichlet boundary conditions which constrain the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 540, + 276, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 258, + 552 + ], + "score": 1.0, + "content": "velocity field at the domain boundary", + "type": "text" + }, + { + "bbox": [ + 258, + 540, + 272, + 550 + ], + "score": 0.82, + "content": "\\partial \\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 540, + 276, + 552 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 555, + 428, + 569 + ], + "lines": [ + { + "bbox": [ + 180, + 555, + 428, + 569 + ], + "spans": [ + { + "bbox": [ + 180, + 555, + 428, + 569 + ], + "score": 0.5, + "content": "\\vec { v } = \\vec { v } _ { d } \\qquad \\mathrm { D i r i c h l e t ~ b o u n d a r y ~ c o n d i t i o n ~ o n ~ } \\partial \\Omega", + "type": "interline_equation", + "image_path": "382b2c7eff65cdde5b09533719fcbe295bfe23f03b98979e602e16522a1b6084.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 180, + 555, + 428, + 569 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 581, + 262, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 262, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 262, + 594 + ], + "score": 1.0, + "content": "3.2 HELMHOLTZ DECOMPOSITION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 504, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "A common method to ensure incompressibility of a fluid (see Equation 1) is to project the flow field", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "onto the divergence-free part of its Helmholtz decomposition. The Helmholtz theorem states that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 177, + 637 + ], + "score": 1.0, + "content": "every vector field", + "type": "text" + }, + { + "bbox": [ + 177, + 624, + 184, + 634 + ], + "score": 0.8, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 624, + 343, + 637 + ], + "score": 1.0, + "content": "can be decomposed into a curl-free part", + "type": "text" + }, + { + "bbox": [ + 343, + 624, + 363, + 636 + ], + "score": 0.88, + "content": "( \\nabla q )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 624, + 470, + 637 + ], + "score": 1.0, + "content": "and a divergence-free part", + "type": "text" + }, + { + "bbox": [ + 470, + 624, + 501, + 635 + ], + "score": 0.92, + "content": "( \\nabla \\times \\vec { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 624, + 505, + 637 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 270, + 640, + 342, + 653 + ], + "lines": [ + { + "bbox": [ + 270, + 640, + 342, + 653 + ], + "spans": [ + { + "bbox": [ + 270, + 640, + 342, + 653 + ], + "score": 0.91, + "content": "\\vec { v } = \\nabla q + \\nabla \\times \\vec { a }", + "type": "interline_equation", + "image_path": "5d37c1b17de1cc04d458fc63441646666d5fed6f98220fdcdde403f6afb38e4d.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 270, + 640, + 342, + 653 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 150, + 674 + ], + "score": 1.0, + "content": "Note, that", + "type": "text" + }, + { + "bbox": [ + 151, + 658, + 217, + 672 + ], + "score": 0.92, + "content": "\\nabla \\times ( \\nabla q ) = \\vec { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 658, + 236, + 674 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 236, + 659, + 311, + 672 + ], + "score": 0.92, + "content": "\\nabla \\cdot ( \\nabla \\times { \\vec { a } } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 658, + 506, + 674 + ], + "score": 1.0, + "content": ". The Helmholtz projection consists of solving", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 192, + 684 + ], + "score": 1.0, + "content": "the Poisson problem", + "type": "text" + }, + { + "bbox": [ + 193, + 671, + 244, + 682 + ], + "score": 0.9, + "content": "\\nabla \\cdot \\vec { v } = \\Delta q", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 670, + 261, + 684 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 261, + 673, + 266, + 682 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 670, + 373, + 684 + ], + "score": 1.0, + "content": ", followed by substracting", + "type": "text" + }, + { + "bbox": [ + 374, + 671, + 388, + 682 + ], + "score": 0.88, + "content": "\\nabla q", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "from the original flow field.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "However, solving the Poisson equation on arbitrary domains comes at high computational costs for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 693, + 505, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 505, + 706 + ], + "score": 1.0, + "content": "classical methods and one has to rely e.g. on conjugate gradient methods to approximate its solution.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 428, + 722 + ], + "score": 1.0, + "content": "Here, we propose a different approach and directly try to learn a vector potential", + "type": "text" + }, + { + "bbox": [ + 429, + 710, + 435, + 720 + ], + "score": 0.81, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 709, + 456, + 722 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 456, + 709, + 501, + 720 + ], + "score": 0.91, + "content": "\\vec { v } = \\nabla \\times \\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 477, + 733 + ], + "score": 1.0, + "content": "This ensures that the network outputs a divergence-free velocity field within the domain", + "type": "text" + }, + { + "bbox": [ + 477, + 721, + 486, + 730 + ], + "score": 0.82, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Discrete Eulerian methods, on the other hand, aim to solve the underlying PDEs on a grid and early", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "work dates back to Harlow & Welch (1965) and Stam (1999). Accelerating such traditional works", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "with deep learning techniques is a major field of research and all of the methods mentioned in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "introduction fall into this category. Further methods include the approach by Thuerey et al. (2019)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "to learn solutions of the Reynolds-averaged Navier-Stokes equations for airfoil flows, but requires", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "large amounts of training data and does not generalize beyond airfoil flows. In the work by Um", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "et al. (2020), a correction step is learned that brings solutions of a low-resolution differentiable fluid", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "solver closer to solutions of a high-resolution fluid simulation. However, generalization results for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "new domain geometries were not presented. The works of Mohan et al. (2020) and Kim et al. (2019)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "show that vector potentials are suitable to enforce the incompressibility constraint in fluids but do", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 358, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 358, + 205 + ], + "score": 1.0, + "content": "not generalize to new fluid domains beyond their training data.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 82, + 506, + 205 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 219, + 173, + 232 + ], + "lines": [ + { + "bbox": [ + 104, + 217, + 175, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 175, + 235 + ], + "score": 1.0, + "content": "3 METHOD", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "In this section, we briefly review the incompressible Navier-Stokes equations, which are to be solved", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "by the neural network. Then, we explain how the Helmholtz decomposition can be exploited to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "ensure incompressibility within the fluid domain. Furthermore, we provide details of our discrete", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "spatio-temporal fluid representation and introduce the fluid model. Afterwards, we formulate a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "physics-constrained loss function based on residuals of the Navier-Stokes equations and introduce a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "pressure regularization term for very high Reynolds numbers. 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Equation 1 states that the fluid is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 207, + 479 + ], + "score": 1.0, + "content": "incompressible and thus", + "type": "text" + }, + { + "bbox": [ + 208, + 466, + 215, + 476 + ], + "score": 0.82, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "is divergence-free. 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\\mu \\Delta \\vec { v } - \\vec { f } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ m o m e n t u m ~ l o s s ~ o n ~ } \\Omega } } \\\\ { { L _ { b } = \\displaystyle { \\left\\| \\vec { v } - \\vec { v } _ { d } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ b o u n d a r y ~ l o s s ~ o n ~ } \\partial \\Omega } } \\end{array}", + "type": "interline_equation", + "image_path": "560a1fae369f8b42f48f2ab1f3a5be4107aa929f6c1c94893410c3404457bb26.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 143, + 352, + 470, + 374.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 143, + 374.0, + 470, + 396.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 143, + 396.0, + 470, + 418.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 407, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 408, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 408, + 442 + ], + "score": 1.0, + "content": "Combining the described loss terms, we obtain the following loss function:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 445, + 355, + 459 + ], + "lines": [ + { + "bbox": [ + 256, + 445, + 355, + 459 + ], + "spans": [ + { + "bbox": [ + 256, + 445, + 355, + 459 + ], + "score": 0.93, + "content": "L = \\alpha L _ { d } + \\beta L _ { p } + \\gamma L _ { b }", + "type": "interline_equation", + "image_path": "fcc301942512405e232beefdcdc19a6dfc51ec3a8249802eb2c2216684b8ac5c.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 256, + 445, + 355, + 459 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 134, + 477 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 464, + 163, + 475 + ], + "score": 0.92, + "content": "\\alpha , \\beta , \\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "are hyperparameters that weight the contributions of the different loss terms. Note", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 234, + 487 + ], + "score": 1.0, + "content": "that if we use a vector potential", + "type": "text" + }, + { + "bbox": [ + 234, + 475, + 279, + 486 + ], + "score": 0.86, + "content": "{ \\vec { v } } = \\nabla \\times { \\vec { a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 474, + 282, + 487 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 475, + 314, + 486 + ], + "score": 0.9, + "content": "L _ { d } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 474, + 475, + 487 + ], + "score": 1.0, + "content": "is automatically fulfilled and we can set", + "type": "text" + }, + { + "bbox": [ + 475, + 475, + 501, + 485 + ], + "score": 0.89, + "content": "\\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 474, + 505, + 487 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 396, + 498 + ], + "score": 1.0, + "content": "This loss function can be computed very efficiently with convolutions in", + "type": "text" + }, + { + "bbox": [ + 396, + 486, + 421, + 497 + ], + "score": 0.93, + "content": "O ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 484, + 452, + 498 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 453, + 486, + 472, + 496 + ], + "score": 0.78, + "content": "N =", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "number", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "of grid cells), whereas solving the Navier-Stokes equations explicitly would be computationally a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "lot more expensive. For detailed descriptions regarding the fully discretized loss-function, we refer", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 518, + 168, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 168, + 531 + ], + "score": 1.0, + "content": "to appendix A.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 256, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 544, + 257, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 257, + 556 + ], + "score": 1.0, + "content": "3.6 PRESSURE REGULARIZATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 504, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "For very high Reynolds numbers (see Equation 13) and inviscid flows, training becomes unstable as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "viscous friction cannot dissipate enough energy out of the system. This leads to unrealistic gradients", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 117, + 599 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 586, + 124, + 596 + ], + "score": 0.8, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 586, + 141, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 587, + 148, + 597 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 586, + 505, + 599 + ], + "score": 1.0, + "content": ". For such cases, we introduce an additional regularization term for the loss function (11)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 596, + 309, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 213, + 610 + ], + "score": 1.0, + "content": "that can be traded off with", + "type": "text" + }, + { + "bbox": [ + 214, + 597, + 226, + 609 + ], + "score": 0.89, + "content": "L _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 596, + 309, + 610 + ], + "score": 1.0, + "content": "to stabilize training:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 278, + 614, + 333, + 630 + ], + "lines": [ + { + "bbox": [ + 278, + 614, + 333, + 630 + ], + "spans": [ + { + "bbox": [ + 278, + 614, + 333, + 630 + ], + "score": 0.93, + "content": "L _ { r } = \\| \\nabla p \\| ^ { 2 }", + "type": "interline_equation", + "image_path": "f76537da61c027d6456687ae5684be73967eba203194cb361484777f2199aea9.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 278, + 614, + 333, + 630 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 635, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "The intuition behind this regularization term is, that we want to penalize unrealistically high energies", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 646, + 189, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 189, + 658 + ], + "score": 1.0, + "content": "in the pressure field.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 194, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 195, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 195, + 95 + ], + "score": 1.0, + "content": "3.4 FLUID MODEL", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 506, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 504, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 423, + 115 + ], + "score": 1.0, + "content": "We represent the fluid dynamics by a recurrent model that maps the fluid state", + "type": "text" + }, + { + "bbox": [ + 423, + 102, + 446, + 115 + ], + "score": 0.92, + "content": "p ^ { t } , \\vec { a } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 102, + 498, + 115 + ], + "score": 1.0, + "content": "for timestep", + "type": "text" + }, + { + "bbox": [ + 499, + 104, + 504, + 113 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 102, + 109, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 102, + 109, + 219, + 133 + ], + "score": 1.0, + "content": "and the domain description", + "type": "text" + }, + { + "bbox": [ + 220, + 114, + 271, + 127 + ], + "score": 0.27, + "content": "\\Omega ^ { t + d t } , \\vec { v } _ { d } ^ { t + d t }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 109, + 342, + 133 + ], + "score": 1.0, + "content": "to the fluid state", + "type": "text" + }, + { + "bbox": [ + 342, + 114, + 391, + 127 + ], + "score": 0.81, + "content": "p ^ { t + d t } , \\vec { a } ^ { t + d t }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 109, + 506, + 133 + ], + "score": 1.0, + "content": "of the next timestep. Here,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 116, + 138 + ], + "score": 0.88, + "content": "p ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 126, + 248, + 138 + ], + "score": 1.0, + "content": "describes the pressure field and", + "type": "text" + }, + { + "bbox": [ + 248, + 127, + 258, + 136 + ], + "score": 0.84, + "content": "\\vec { a } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 126, + 392, + 138 + ], + "score": 1.0, + "content": "describes the vector potential of", + "type": "text" + }, + { + "bbox": [ + 402, + 126, + 424, + 138 + ], + "score": 1.0, + "content": ". 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For the boundary of the domain, we simply take the inverse of", + "type": "text" + }, + { + "bbox": [ + 434, + 160, + 501, + 171 + ], + "score": 0.87, + "content": "\\Omega \\colon \\partial \\Omega = 1 - \\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 160, + 506, + 173 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 167, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 130, + 185 + ], + "score": 0.92, + "content": "\\vec { v } _ { d } ^ { t + \\dot { d } t }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 167, + 506, + 190 + ], + "score": 1.0, + "content": "represents the Dirichlet boundary conditions and contains a velocity field that must be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 183, + 159, + 198 + ], + "score": 1.0, + "content": "matched by", + "type": "text" + }, + { + "bbox": [ + 159, + 184, + 182, + 195 + ], + "score": 0.9, + "content": "{ \\vec { v } } ^ { t + d t }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 183, + 506, + 198 + ], + "score": 1.0, + "content": "at the domain boundaries. Figure 1b shows a diagram of the fluid model.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 102, + 191, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 102, + 191, + 132, + 217 + ], + "score": 1.0, + "content": "First,", + "type": "text" + }, + { + "bbox": [ + 132, + 196, + 217, + 210 + ], + "score": 0.94, + "content": "\\left( p ^ { t } , \\vec { a } ^ { t } , \\Omega ^ { t + d t } , \\vec { v } _ { d } ^ { t + d t } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 191, + 505, + 217 + ], + "score": 1.0, + "content": "are taken to derive a slightly more meaningful feature representation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 206, + 507, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 206, + 170, + 226 + ], + "score": 1.0, + "content": "that comprises", + "type": "text" + }, + { + "bbox": [ + 170, + 209, + 470, + 223 + ], + "score": 0.89, + "content": "\\left( p ^ { t } , a ^ { t } , \\nabla \\times a ^ { t } , \\Omega ^ { t + d t } , \\partial \\Omega ^ { t + d t } , \\Omega ^ { t + d t } \\cdot \\nabla \\times a ^ { t } , \\Omega ^ { t + d t } \\cdot p ^ { t } , \\partial \\Omega ^ { t + d t } \\cdot \\vec { v } _ { d } ^ { t + d t } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 206, + 507, + 226 + ], + "score": 1.0, + "content": ". These", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 222, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 506, + 234 + ], + "score": 1.0, + "content": "features can be very efficiently computed with convolutions and are then fed into a U-Net", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "(Ronneberger et al. (2015)) with a reduced number of channels (the exact network configuration", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 453, + 256 + ], + "score": 1.0, + "content": "can be found in appendix B). The mean of the U-Net output is set to 0 in order to keep", + "type": "text" + }, + { + "bbox": [ + 453, + 245, + 460, + 255 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 244, + 477, + 256 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 244, + 484, + 254 + ], + "score": 0.78, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "well", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 408, + 267 + ], + "score": 1.0, + "content": "defined and prevent drifting offset values. Finally, the output is added to", + "type": "text" + }, + { + "bbox": [ + 409, + 255, + 419, + 266 + ], + "score": 0.88, + "content": "p ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 254, + 438, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 255, + 449, + 265 + ], + "score": 0.86, + "content": "\\vec { a } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "to obtain the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 250, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 182, + 278 + ], + "score": 1.0, + "content": "updated fluid state", + "type": "text" + }, + { + "bbox": [ + 182, + 266, + 205, + 277 + ], + "score": 0.92, + "content": "p ^ { t + d t }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 265, + 223, + 278 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 266, + 246, + 276 + ], + "score": 0.9, + "content": "\\vec { a } ^ { t + d t }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 265, + 250, + 278 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8, + "bbox_fs": [ + 102, + 102, + 507, + 278 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 290, + 306, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 308, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 308, + 303 + ], + "score": 1.0, + "content": "3.5 PHYSICS-CONSTRAINED LOSS FUNCTION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 311, + 504, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 506, + 325 + ], + "score": 1.0, + "content": "Using the residuals of the Navier-Stokes equations (Equations 1 and 2), we can formulate the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 321, + 248, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 203, + 334 + ], + "score": 1.0, + "content": "following loss terms on", + "type": "text" + }, + { + "bbox": [ + 203, + 322, + 211, + 332 + ], + "score": 0.84, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 321, + 229, + 334 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 229, + 322, + 243, + 332 + ], + "score": 0.85, + "content": "\\partial \\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 321, + 248, + 334 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 310, + 506, + 334 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 352, + 470, + 418 + ], + "lines": [ + { + "bbox": [ + 143, + 352, + 470, + 418 + ], + "spans": [ + { + "bbox": [ + 143, + 352, + 470, + 418 + ], + "score": 0.84, + "content": "\\begin{array} { l l } { { L _ { d } = \\displaystyle { \\left\\| \\nabla \\cdot \\vec { v } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ d i v e r g e n c e ~ l o s s ~ o n ~ } \\Omega } } \\\\ { { L _ { p } = \\displaystyle { \\left\\| \\rho \\left( \\frac { \\partial \\vec { v } } { \\partial t } + \\left( \\vec { v } \\cdot \\nabla \\right) \\vec { v } \\right) + \\nabla p - \\mu \\Delta \\vec { v } - \\vec { f } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ m o m e n t u m ~ l o s s ~ o n ~ } \\Omega } } \\\\ { { L _ { b } = \\displaystyle { \\left\\| \\vec { v } - \\vec { v } _ { d } \\right\\| ^ { 2 } } } } & { { \\qquad \\mathrm { ~ b o u n d a r y ~ l o s s ~ o n ~ } \\partial \\Omega } } \\end{array}", + "type": "interline_equation", + "image_path": "560a1fae369f8b42f48f2ab1f3a5be4107aa929f6c1c94893410c3404457bb26.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 143, + 352, + 470, + 374.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 143, + 374.0, + 470, + 396.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 143, + 396.0, + 470, + 418.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 407, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 408, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 408, + 442 + ], + "score": 1.0, + "content": "Combining the described loss terms, we obtain the following loss function:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 426, + 408, + 442 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 445, + 355, + 459 + ], + "lines": [ + { + "bbox": [ + 256, + 445, + 355, + 459 + ], + "spans": [ + { + "bbox": [ + 256, + 445, + 355, + 459 + ], + "score": 0.93, + "content": "L = \\alpha L _ { d } + \\beta L _ { p } + \\gamma L _ { b }", + "type": "interline_equation", + "image_path": "fcc301942512405e232beefdcdc19a6dfc51ec3a8249802eb2c2216684b8ac5c.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 256, + 445, + 355, + 459 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 134, + 477 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 464, + 163, + 475 + ], + "score": 0.92, + "content": "\\alpha , \\beta , \\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "are hyperparameters that weight the contributions of the different loss terms. Note", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 234, + 487 + ], + "score": 1.0, + "content": "that if we use a vector potential", + "type": "text" + }, + { + "bbox": [ + 234, + 475, + 279, + 486 + ], + "score": 0.86, + "content": "{ \\vec { v } } = \\nabla \\times { \\vec { a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 474, + 282, + 487 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 475, + 314, + 486 + ], + "score": 0.9, + "content": "L _ { d } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 474, + 475, + 487 + ], + "score": 1.0, + "content": "is automatically fulfilled and we can set", + "type": "text" + }, + { + "bbox": [ + 475, + 475, + 501, + 485 + ], + "score": 0.89, + "content": "\\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 474, + 505, + 487 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 396, + 498 + ], + "score": 1.0, + "content": "This loss function can be computed very efficiently with convolutions in", + "type": "text" + }, + { + "bbox": [ + 396, + 486, + 421, + 497 + ], + "score": 0.93, + "content": "O ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 484, + 452, + 498 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 453, + 486, + 472, + 496 + ], + "score": 0.78, + "content": "N =", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "number", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "of grid cells), whereas solving the Navier-Stokes equations explicitly would be computationally a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "lot more expensive. For detailed descriptions regarding the fully discretized loss-function, we refer", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 518, + 168, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 168, + 531 + ], + "score": 1.0, + "content": "to appendix A.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 464, + 506, + 531 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 256, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 544, + 257, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 257, + 556 + ], + "score": 1.0, + "content": "3.6 PRESSURE REGULARIZATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 504, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "For very high Reynolds numbers (see Equation 13) and inviscid flows, training becomes unstable as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "viscous friction cannot dissipate enough energy out of the system. This leads to unrealistic gradients", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 117, + 599 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 586, + 124, + 596 + ], + "score": 0.8, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 586, + 141, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 587, + 148, + 597 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 586, + 505, + 599 + ], + "score": 1.0, + "content": ". For such cases, we introduce an additional regularization term for the loss function (11)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 596, + 309, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 213, + 610 + ], + "score": 1.0, + "content": "that can be traded off with", + "type": "text" + }, + { + "bbox": [ + 214, + 597, + 226, + 609 + ], + "score": 0.89, + "content": "L _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 596, + 309, + 610 + ], + "score": 1.0, + "content": "to stabilize training:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 564, + 506, + 610 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 278, + 614, + 333, + 630 + ], + "lines": [ + { + "bbox": [ + 278, + 614, + 333, + 630 + ], + "spans": [ + { + "bbox": [ + 278, + 614, + 333, + 630 + ], + "score": 0.93, + "content": "L _ { r } = \\| \\nabla p \\| ^ { 2 }", + "type": "interline_equation", + "image_path": "f76537da61c027d6456687ae5684be73967eba203194cb361484777f2199aea9.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 278, + 614, + 333, + 630 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 635, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "The intuition behind this regularization term is, that we want to penalize unrealistically high energies", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 646, + 189, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 189, + 658 + ], + "score": 1.0, + "content": "in the pressure field.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 633, + 505, + 658 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 223, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 225, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 225, + 95 + ], + "score": 1.0, + "content": "3.7 TRAINING STRATEGY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 273, + 117 + ], + "score": 1.0, + "content": "Training starts with initializing a pool", + "type": "text" + }, + { + "bbox": [ + 273, + 103, + 365, + 116 + ], + "score": 0.89, + "content": "\\{ \\Omega _ { k } ^ { 0 } , ( v _ { d } ) _ { k } ^ { 0 } , ( a _ { z } ) _ { k } ^ { 0 } , p _ { k } ^ { 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 102, + 471, + 117 + ], + "score": 1.0, + "content": "of randomized domains", + "type": "text" + }, + { + "bbox": [ + 471, + 103, + 485, + 116 + ], + "score": 0.9, + "content": "\\Omega _ { k } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 102, + 506, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 192, + 129 + ], + "score": 1.0, + "content": "boundary conditions", + "type": "text" + }, + { + "bbox": [ + 192, + 114, + 216, + 127 + ], + "score": 0.93, + "content": "( v _ { d } ) _ { k } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 113, + 506, + 129 + ], + "score": 1.0, + "content": "as well as initial conditions for the vector potential and pressure fields", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 123, + 504, + 141 + ], + "spans": [ + { + "bbox": [ + 104, + 123, + 203, + 141 + ], + "score": 1.0, + "content": "that we both set to zero", + "type": "text" + }, + { + "bbox": [ + 204, + 126, + 247, + 138 + ], + "score": 0.91, + "content": "( \\check { ( a _ { z } ) } _ { k } ^ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 123, + 264, + 141 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 125, + 294, + 138 + ], + "score": 0.91, + "content": "p _ { k } ^ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 123, + 468, + 141 + ], + "score": 1.0, + "content": "). The resolution of our training domains is", + "type": "text" + }, + { + "bbox": [ + 468, + 126, + 504, + 136 + ], + "score": 0.69, + "content": "1 0 0 \\mathrm { x } 3 0 0", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 150 + ], + "score": 1.0, + "content": "grid cells and example-domains of the training pool are shown in appendix C. Note that our training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 339, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 339, + 160 + ], + "score": 1.0, + "content": "pool does not rely on any previously simulated fluid-data.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 507, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 276, + 180 + ], + "score": 1.0, + "content": "At each training step, a random mini-batch", + "type": "text" + }, + { + "bbox": [ + 277, + 164, + 413, + 178 + ], + "score": 0.93, + "content": "\\{ \\Omega _ { k } ^ { t } , ( v _ { d } ) _ { k } ^ { t } , ( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t } \\} _ { \\{ k \\in \\operatorname * { m i n i b a t c h } \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 162, + 507, + 180 + ], + "score": 1.0, + "content": "is drawn from the pool", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 101, + 171, + 510, + 198 + ], + "spans": [ + { + "bbox": [ + 101, + 171, + 401, + 198 + ], + "score": 1.0, + "content": "and fed into the neural network which is designed to predict the velocity", + "type": "text" + }, + { + "bbox": [ + 401, + 177, + 486, + 191 + ], + "score": 0.92, + "content": "( \\vec { v } _ { k } ^ { t + d t } = \\nabla \\times \\vec { a } _ { k } ^ { t + d t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 171, + 510, + 198 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 101, + 185, + 509, + 207 + ], + "spans": [ + { + "bbox": [ + 101, + 185, + 142, + 207 + ], + "score": 1.0, + "content": "pressure", + "type": "text" + }, + { + "bbox": [ + 142, + 190, + 171, + 204 + ], + "score": 0.91, + "content": "( p _ { k } ^ { t + d t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 185, + 509, + 207 + ], + "score": 1.0, + "content": "fields of the next time step. Based on a physics-constrained loss-function (Equation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 506, + 215 + ], + "score": 1.0, + "content": "11), we update the weights of the network using the Adam optimizer (Kingma & Ba (2015)). At the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "end of each training step, the pool is updated by replacing the old vector potential and pressure fields", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 221, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 107, + 225, + 145, + 238 + ], + "score": 0.93, + "content": "( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 221, + 260, + 242 + ], + "score": 1.0, + "content": "by the newly predicted ones", + "type": "text" + }, + { + "bbox": [ + 260, + 224, + 321, + 238 + ], + "score": 0.92, + "content": "( a _ { z } ) _ { k } ^ { t + \\dot { d t } } , p _ { k } ^ { \\dot { t } + d t }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 221, + 506, + 241 + ], + "score": 1.0, + "content": "t, pt+dtk . This recycling strategy fills the training pool", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 502, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 502, + 249 + ], + "score": 1.0, + "content": "with more and more realistic fluid states as the model becomes better at simulating fluid dynamics.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "From time to time, old environments of the training pool are replaced by new randomized", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "environments and the vector potential as well as the pressure fields are reset to 0. This increases", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "the variance of the training pool and helps the neural network to learn \"cold starts\" from ~0-velocity", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 288, + 194, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 194, + 300 + ], + "score": 1.0, + "content": "and 0-pressure fields.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 350, + 317 + ], + "score": 1.0, + "content": "Besides the fluid model described above, which we denote as", + "type": "text" + }, + { + "bbox": [ + 350, + 305, + 356, + 315 + ], + "score": 0.77, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "-Net in the following, we also trained", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 183, + 327 + ], + "score": 1.0, + "content": "an ablation model,", + "type": "text" + }, + { + "bbox": [ + 183, + 316, + 190, + 326 + ], + "score": 0.78, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 316, + 504, + 327 + ], + "score": 1.0, + "content": "-Net, that directly learns to predict the velocity field without a vector potential.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 504, + 339 + ], + "score": 1.0, + "content": "For the implementation of both models, we used the popular machine learning framework Pytorch", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "and trained the models on a NVidia GeForce RTX 2080 Ti. Training converged after about 1 day.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 311, + 361 + ], + "score": 1.0, + "content": "The hyperparameters in the loss-function for the", + "type": "text" + }, + { + "bbox": [ + 311, + 349, + 318, + 358 + ], + "score": 0.77, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 348, + 361, + 361 + ], + "score": 1.0, + "content": "-Net were", + "type": "text" + }, + { + "bbox": [ + 361, + 349, + 393, + 360 + ], + "score": 0.92, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 348, + 412, + 361 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 349, + 448, + 360 + ], + "score": 0.91, + "content": "\\gamma = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 348, + 505, + 361 + ], + "score": 1.0, + "content": ". The reason", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 299, + 372 + ], + "score": 1.0, + "content": "for choosing a higher weight for the loss term", + "type": "text" + }, + { + "bbox": [ + 299, + 360, + 311, + 370 + ], + "score": 0.88, + "content": "L _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 358, + 348, + 372 + ], + "score": 1.0, + "content": "than for", + "type": "text" + }, + { + "bbox": [ + 348, + 360, + 361, + 371 + ], + "score": 0.89, + "content": "L _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "was the observation, that errors in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 107, + 371, + 119, + 381 + ], + "score": 0.86, + "content": "L _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 369, + 458, + 384 + ], + "score": 1.0, + "content": "can lead to unrealistic flows leaking through boundaries. For the ablation study", + "type": "text" + }, + { + "bbox": [ + 459, + 371, + 465, + 380 + ], + "score": 0.61, + "content": "\\vec { v } \\cdot \\vec { }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 369, + 506, + 384 + ], + "score": 1.0, + "content": "-Net), we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 128, + 394 + ], + "score": 1.0, + "content": "used", + "type": "text" + }, + { + "bbox": [ + 128, + 381, + 248, + 393 + ], + "score": 0.93, + "content": "\\alpha = 1 0 0 , \\beta = 1 , \\gamma = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 381, + 450, + 394 + ], + "score": 1.0, + "content": ". Here, we had to choose a very high weight for", + "type": "text" + }, + { + "bbox": [ + 451, + 382, + 464, + 392 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "to ensure", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 489, + 406 + ], + "score": 1.0, + "content": "incompressibility of the fluid, otherwise unrealistic source and sink effects start to appear. For", + "type": "text" + }, + { + "bbox": [ + 489, + 393, + 501, + 403 + ], + "score": 0.87, + "content": "L _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 392, + 505, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "score": 1.0, + "content": "on the other hand, we used a very low weight as the boundary conditions can be trivially learned by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 365, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 121, + 427 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 414, + 128, + 424 + ], + "score": 0.8, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 414, + 365, + 427 + ], + "score": 1.0, + "content": "-Net. We used these parameter settings for all experiments.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 443, + 172, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 174, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 174, + 458 + ], + "score": 1.0, + "content": "4 RESULTS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "To evaluate the potential of our method, we assess its ability to reproduce physical effects such", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "as Kármán vortex streets and the Magnus effect. In addition, we demonstrate its generalization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 448, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 448, + 505 + ], + "score": 1.0, + "content": "capability and real-time performance. Finally, we test the fluid models quantitatively.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 248, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 249, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 249, + 531 + ], + "score": 1.0, + "content": "4.1 QUALITATIVE EVALUATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "Qualitative analysis of wake dynamics Qualitative effects in fluid dynamics such as the wake", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "dynamics behind an obstacle are closely related to the Reynolds number. It is a dimensionless", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 561, + 190, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 190, + 576 + ], + "score": 1.0, + "content": "quantity defined by:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 573, + 337, + 601 + ], + "lines": [ + { + "bbox": [ + 274, + 573, + 337, + 601 + ], + "spans": [ + { + "bbox": [ + 274, + 573, + 337, + 601 + ], + "score": 0.95, + "content": "R e = \\frac { \\rho \\left\\| \\vec { v } \\right\\| D } { \\mu }", + "type": "interline_equation", + "image_path": "4dfd04cc28aff9cd79aa9924737756fe96be62393b9e0a5e81f5954f0c177749.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 274, + 573, + 337, + 601 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 131, + 618 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 607, + 138, + 617 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 604, + 221, + 618 + ], + "score": 1.0, + "content": "is the fluid density,", + "type": "text" + }, + { + "bbox": [ + 221, + 605, + 236, + 617 + ], + "score": 0.91, + "content": "\\lVert \\vec { v } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 604, + 313, + 618 + ], + "score": 1.0, + "content": "is the fluid speed,", + "type": "text" + }, + { + "bbox": [ + 314, + 605, + 324, + 615 + ], + "score": 0.86, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 604, + 471, + 618 + ], + "score": 1.0, + "content": "is the diameter of the obstacle, and", + "type": "text" + }, + { + "bbox": [ + 471, + 607, + 479, + 617 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 615, + 267, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 267, + 628 + ], + "score": 1.0, + "content": "viscosity. (We use the units of the grid).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 282, + 646 + ], + "score": 1.0, + "content": "We retrained models for different values of", + "type": "text" + }, + { + "bbox": [ + 282, + 635, + 290, + 644 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 632, + 308, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 635, + 315, + 645 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "to compare the fluid behavior for a wide range", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "of Reynolds numbers. Figure 2 shows, that the trained models are able to predict the wake dynamics", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "behind an obstacle in good accordance with qualitative expectations from fluid dynamics. As a rule", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 163, + 678 + ], + "score": 1.0, + "content": "of thumb, for", + "type": "text" + }, + { + "bbox": [ + 164, + 666, + 199, + 676 + ], + "score": 0.89, + "content": "R e \\ll 1", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 665, + 505, + 678 + ], + "score": 1.0, + "content": ", the flow becomes time-reversible. This can be noticed in Figure 2a by the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "symmetry of the flow before and after the obstacle and the nearly constant pressure gradient within", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 203, + 701 + ], + "score": 1.0, + "content": "the pipe. Starting from", + "type": "text" + }, + { + "bbox": [ + 204, + 688, + 242, + 698 + ], + "score": 0.89, + "content": "R e \\approx 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 687, + 506, + 701 + ], + "score": 1.0, + "content": ", the flow is still laminar but a static wake is forming behind the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 304, + 712 + ], + "score": 1.0, + "content": "obstacle (see Figure 2b). For Reynolds numbers", + "type": "text" + }, + { + "bbox": [ + 304, + 699, + 350, + 709 + ], + "score": 0.88, + "content": "R e > \\approx 9 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ", Kármán vortex streets start to appear", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(see Figure 2c). A Kármán vortex street consists of clock and counterclockwise spinning vortices", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "that are generated at the obstacle and then start moving in a regularly oscillating pattern with the", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 223, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 225, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 225, + 95 + ], + "score": 1.0, + "content": "3.7 TRAINING STRATEGY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 273, + 117 + ], + "score": 1.0, + "content": "Training starts with initializing a pool", + "type": "text" + }, + { + "bbox": [ + 273, + 103, + 365, + 116 + ], + "score": 0.89, + "content": "\\{ \\Omega _ { k } ^ { 0 } , ( v _ { d } ) _ { k } ^ { 0 } , ( a _ { z } ) _ { k } ^ { 0 } , p _ { k } ^ { 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 102, + 471, + 117 + ], + "score": 1.0, + "content": "of randomized domains", + "type": "text" + }, + { + "bbox": [ + 471, + 103, + 485, + 116 + ], + "score": 0.9, + "content": "\\Omega _ { k } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 102, + 506, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 192, + 129 + ], + "score": 1.0, + "content": "boundary conditions", + "type": "text" + }, + { + "bbox": [ + 192, + 114, + 216, + 127 + ], + "score": 0.93, + "content": "( v _ { d } ) _ { k } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 113, + 506, + 129 + ], + "score": 1.0, + "content": "as well as initial conditions for the vector potential and pressure fields", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 123, + 504, + 141 + ], + "spans": [ + { + "bbox": [ + 104, + 123, + 203, + 141 + ], + "score": 1.0, + "content": "that we both set to zero", + "type": "text" + }, + { + "bbox": [ + 204, + 126, + 247, + 138 + ], + "score": 0.91, + "content": "( \\check { ( a _ { z } ) } _ { k } ^ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 123, + 264, + 141 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 125, + 294, + 138 + ], + "score": 0.91, + "content": "p _ { k } ^ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 123, + 468, + 141 + ], + "score": 1.0, + "content": "). The resolution of our training domains is", + "type": "text" + }, + { + "bbox": [ + 468, + 126, + 504, + 136 + ], + "score": 0.69, + "content": "1 0 0 \\mathrm { x } 3 0 0", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 150 + ], + "score": 1.0, + "content": "grid cells and example-domains of the training pool are shown in appendix C. Note that our training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 339, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 339, + 160 + ], + "score": 1.0, + "content": "pool does not rely on any previously simulated fluid-data.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 102, + 506, + 160 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 507, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 276, + 180 + ], + "score": 1.0, + "content": "At each training step, a random mini-batch", + "type": "text" + }, + { + "bbox": [ + 277, + 164, + 413, + 178 + ], + "score": 0.93, + "content": "\\{ \\Omega _ { k } ^ { t } , ( v _ { d } ) _ { k } ^ { t } , ( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t } \\} _ { \\{ k \\in \\operatorname * { m i n i b a t c h } \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 162, + 507, + 180 + ], + "score": 1.0, + "content": "is drawn from the pool", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 101, + 171, + 510, + 198 + ], + "spans": [ + { + "bbox": [ + 101, + 171, + 401, + 198 + ], + "score": 1.0, + "content": "and fed into the neural network which is designed to predict the velocity", + "type": "text" + }, + { + "bbox": [ + 401, + 177, + 486, + 191 + ], + "score": 0.92, + "content": "( \\vec { v } _ { k } ^ { t + d t } = \\nabla \\times \\vec { a } _ { k } ^ { t + d t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 171, + 510, + 198 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 101, + 185, + 509, + 207 + ], + "spans": [ + { + "bbox": [ + 101, + 185, + 142, + 207 + ], + "score": 1.0, + "content": "pressure", + "type": "text" + }, + { + "bbox": [ + 142, + 190, + 171, + 204 + ], + "score": 0.91, + "content": "( p _ { k } ^ { t + d t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 185, + 509, + 207 + ], + "score": 1.0, + "content": "fields of the next time step. Based on a physics-constrained loss-function (Equation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 506, + 215 + ], + "score": 1.0, + "content": "11), we update the weights of the network using the Adam optimizer (Kingma & Ba (2015)). At the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "end of each training step, the pool is updated by replacing the old vector potential and pressure fields", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 221, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 107, + 225, + 145, + 238 + ], + "score": 0.93, + "content": "( a _ { z } ) _ { k } ^ { t } , p _ { k } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 221, + 260, + 242 + ], + "score": 1.0, + "content": "by the newly predicted ones", + "type": "text" + }, + { + "bbox": [ + 260, + 224, + 321, + 238 + ], + "score": 0.92, + "content": "( a _ { z } ) _ { k } ^ { t + \\dot { d t } } , p _ { k } ^ { \\dot { t } + d t }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 221, + 506, + 241 + ], + "score": 1.0, + "content": "t, pt+dtk . This recycling strategy fills the training pool", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 502, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 502, + 249 + ], + "score": 1.0, + "content": "with more and more realistic fluid states as the model becomes better at simulating fluid dynamics.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 101, + 162, + 510, + 249 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "From time to time, old environments of the training pool are replaced by new randomized", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "environments and the vector potential as well as the pressure fields are reset to 0. This increases", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "the variance of the training pool and helps the neural network to learn \"cold starts\" from ~0-velocity", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 288, + 194, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 194, + 300 + ], + "score": 1.0, + "content": "and 0-pressure fields.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 253, + 506, + 300 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 350, + 317 + ], + "score": 1.0, + "content": "Besides the fluid model described above, which we denote as", + "type": "text" + }, + { + "bbox": [ + 350, + 305, + 356, + 315 + ], + "score": 0.77, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "-Net in the following, we also trained", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 183, + 327 + ], + "score": 1.0, + "content": "an ablation model,", + "type": "text" + }, + { + "bbox": [ + 183, + 316, + 190, + 326 + ], + "score": 0.78, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 316, + 504, + 327 + ], + "score": 1.0, + "content": "-Net, that directly learns to predict the velocity field without a vector potential.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 504, + 339 + ], + "score": 1.0, + "content": "For the implementation of both models, we used the popular machine learning framework Pytorch", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "and trained the models on a NVidia GeForce RTX 2080 Ti. Training converged after about 1 day.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 311, + 361 + ], + "score": 1.0, + "content": "The hyperparameters in the loss-function for the", + "type": "text" + }, + { + "bbox": [ + 311, + 349, + 318, + 358 + ], + "score": 0.77, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 348, + 361, + 361 + ], + "score": 1.0, + "content": "-Net were", + "type": "text" + }, + { + "bbox": [ + 361, + 349, + 393, + 360 + ], + "score": 0.92, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 348, + 412, + 361 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 349, + 448, + 360 + ], + "score": 0.91, + "content": "\\gamma = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 348, + 505, + 361 + ], + "score": 1.0, + "content": ". The reason", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 299, + 372 + ], + "score": 1.0, + "content": "for choosing a higher weight for the loss term", + "type": "text" + }, + { + "bbox": [ + 299, + 360, + 311, + 370 + ], + "score": 0.88, + "content": "L _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 358, + 348, + 372 + ], + "score": 1.0, + "content": "than for", + "type": "text" + }, + { + "bbox": [ + 348, + 360, + 361, + 371 + ], + "score": 0.89, + "content": "L _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "was the observation, that errors in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 107, + 371, + 119, + 381 + ], + "score": 0.86, + "content": "L _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 369, + 458, + 384 + ], + "score": 1.0, + "content": "can lead to unrealistic flows leaking through boundaries. For the ablation study", + "type": "text" + }, + { + "bbox": [ + 459, + 371, + 465, + 380 + ], + "score": 0.61, + "content": "\\vec { v } \\cdot \\vec { }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 369, + 506, + 384 + ], + "score": 1.0, + "content": "-Net), we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 128, + 394 + ], + "score": 1.0, + "content": "used", + "type": "text" + }, + { + "bbox": [ + 128, + 381, + 248, + 393 + ], + "score": 0.93, + "content": "\\alpha = 1 0 0 , \\beta = 1 , \\gamma = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 381, + 450, + 394 + ], + "score": 1.0, + "content": ". Here, we had to choose a very high weight for", + "type": "text" + }, + { + "bbox": [ + 451, + 382, + 464, + 392 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "to ensure", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 489, + 406 + ], + "score": 1.0, + "content": "incompressibility of the fluid, otherwise unrealistic source and sink effects start to appear. For", + "type": "text" + }, + { + "bbox": [ + 489, + 393, + 501, + 403 + ], + "score": 0.87, + "content": "L _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 392, + 505, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 417 + ], + "score": 1.0, + "content": "on the other hand, we used a very low weight as the boundary conditions can be trivially learned by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 365, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 121, + 427 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 414, + 128, + 424 + ], + "score": 0.8, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 414, + 365, + 427 + ], + "score": 1.0, + "content": "-Net. We used these parameter settings for all experiments.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 304, + 506, + 427 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 443, + 172, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 174, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 174, + 458 + ], + "score": 1.0, + "content": "4 RESULTS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "To evaluate the potential of our method, we assess its ability to reproduce physical effects such", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "as Kármán vortex streets and the Magnus effect. In addition, we demonstrate its generalization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 448, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 448, + 505 + ], + "score": 1.0, + "content": "capability and real-time performance. Finally, we test the fluid models quantitatively.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 469, + 505, + 505 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 248, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 249, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 249, + 531 + ], + "score": 1.0, + "content": "4.1 QUALITATIVE EVALUATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "Qualitative analysis of wake dynamics Qualitative effects in fluid dynamics such as the wake", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "dynamics behind an obstacle are closely related to the Reynolds number. It is a dimensionless", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 561, + 190, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 190, + 576 + ], + "score": 1.0, + "content": "quantity defined by:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 540, + 506, + 576 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 573, + 337, + 601 + ], + "lines": [ + { + "bbox": [ + 274, + 573, + 337, + 601 + ], + "spans": [ + { + "bbox": [ + 274, + 573, + 337, + 601 + ], + "score": 0.95, + "content": "R e = \\frac { \\rho \\left\\| \\vec { v } \\right\\| D } { \\mu }", + "type": "interline_equation", + "image_path": "4dfd04cc28aff9cd79aa9924737756fe96be62393b9e0a5e81f5954f0c177749.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 274, + 573, + 337, + 601 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 131, + 618 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 607, + 138, + 617 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 604, + 221, + 618 + ], + "score": 1.0, + "content": "is the fluid density,", + "type": "text" + }, + { + "bbox": [ + 221, + 605, + 236, + 617 + ], + "score": 0.91, + "content": "\\lVert \\vec { v } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 604, + 313, + 618 + ], + "score": 1.0, + "content": "is the fluid speed,", + "type": "text" + }, + { + "bbox": [ + 314, + 605, + 324, + 615 + ], + "score": 0.86, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 604, + 471, + 618 + ], + "score": 1.0, + "content": "is the diameter of the obstacle, and", + "type": "text" + }, + { + "bbox": [ + 471, + 607, + 479, + 617 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 615, + 267, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 267, + 628 + ], + "score": 1.0, + "content": "viscosity. (We use the units of the grid).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 604, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 282, + 646 + ], + "score": 1.0, + "content": "We retrained models for different values of", + "type": "text" + }, + { + "bbox": [ + 282, + 635, + 290, + 644 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 632, + 308, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 635, + 315, + 645 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "to compare the fluid behavior for a wide range", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "of Reynolds numbers. Figure 2 shows, that the trained models are able to predict the wake dynamics", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "behind an obstacle in good accordance with qualitative expectations from fluid dynamics. As a rule", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 163, + 678 + ], + "score": 1.0, + "content": "of thumb, for", + "type": "text" + }, + { + "bbox": [ + 164, + 666, + 199, + 676 + ], + "score": 0.89, + "content": "R e \\ll 1", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 665, + 505, + 678 + ], + "score": 1.0, + "content": ", the flow becomes time-reversible. This can be noticed in Figure 2a by the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "symmetry of the flow before and after the obstacle and the nearly constant pressure gradient within", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 203, + 701 + ], + "score": 1.0, + "content": "the pipe. Starting from", + "type": "text" + }, + { + "bbox": [ + 204, + 688, + 242, + 698 + ], + "score": 0.89, + "content": "R e \\approx 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 687, + 506, + 701 + ], + "score": 1.0, + "content": ", the flow is still laminar but a static wake is forming behind the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 304, + 712 + ], + "score": 1.0, + "content": "obstacle (see Figure 2b). For Reynolds numbers", + "type": "text" + }, + { + "bbox": [ + 304, + 699, + 350, + 709 + ], + "score": 0.88, + "content": "R e > \\approx 9 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ", Kármán vortex streets start to appear", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(see Figure 2c). A Kármán vortex street consists of clock and counterclockwise spinning vortices", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "that are generated at the obstacle and then start moving in a regularly oscillating pattern with the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "flow. For very large Reynolds numbers or inviscid flows, the flow field becomes turbulent, which", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 399, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 399, + 106 + ], + "score": 1.0, + "content": "can be recognized by the irregular patterns behind the obstacle in Fig 2d.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 632, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "flow. For very large Reynolds numbers or inviscid flows, the flow field becomes turbulent, which", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 399, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 399, + 106 + ], + "score": 1.0, + "content": "can be recognized by the irregular patterns behind the obstacle in Fig 2d.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "image", + "bbox": [ + 128, + 129, + 483, + 302 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 129, + 483, + 302 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 129, + 483, + 302 + ], + "spans": [ + { + "bbox": [ + 128, + 129, + 483, + 302 + ], + "score": 0.977, + "type": "image", + "image_path": "468c39a4bc3b15c1ad09df22f1ea2d017e585da0b07e13fdda3acc4af90342a2.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 128, + 129, + 483, + 186.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 128, + 186.66666666666666, + 483, + 244.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 128, + 244.33333333333331, + 483, + 302.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 312, + 504, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 312, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 322 + ], + "score": 1.0, + "content": "Figure 2: After training, our models are able to show correct wake flow dynamics for a wide range of different", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 182, + 333 + ], + "score": 1.0, + "content": "Reynolds numbers.", + "type": "text" + }, + { + "bbox": [ + 182, + 322, + 214, + 332 + ], + "score": 0.74, + "content": "\\dot { \\boldsymbol { D } } = 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 321, + 217, + 333 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 218, + 321, + 259, + 333 + ], + "score": 0.64, + "content": "\\| \\vec { v } \\| = 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 321, + 505, + 333 + ], + "score": 1.0, + "content": ". Streamlines indicate flow direction, linewidth indicates speed and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 332, + 390, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 390, + 343 + ], + "score": 1.0, + "content": "colors represent the pressure field (blue: low pressure / yellow: high pressure).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 364, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 504, + 376 + ], + "score": 1.0, + "content": "Magnus effect The Magnus effect appears when a flow interacts with a rotating body. It is widely", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "known e.g. in sports such as soccer or tennis where spin is used to deflect the path of a ball. The", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 384, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 506, + 400 + ], + "score": 1.0, + "content": "reason for the deflection stems from a low pressure field where the surface of the object moves along", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "flow direction and a high pressure field where the object surface moves against the flow. Figure 3a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 408, + 468, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 468, + 420 + ], + "score": 1.0, + "content": "shows, that our models are able to reproduce the Magnus effect around a rotating cylinder.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "image", + "bbox": [ + 125, + 443, + 484, + 502 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 443, + 484, + 502 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 125, + 443, + 484, + 502 + ], + "spans": [ + { + "bbox": [ + 125, + 443, + 484, + 502 + ], + "score": 0.93, + "type": "image", + "image_path": "f016ae8a12c9f62a822e83f75f847c39d114a2759991e05eb174aa68cf62956a.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 125, + 443, + 484, + 462.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 125, + 462.6666666666667, + 484, + 482.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 125, + 482.33333333333337, + 484, + 502.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 122, + 507, + 490, + 538 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 122, + 507, + 490, + 519 + ], + "spans": [ + { + "bbox": [ + 122, + 507, + 490, + 519 + ], + "score": 1.0, + "content": "(a) Magnus effect on a clock-wise turning cylinder.(b) Generalization example: Note that the fluid", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 303, + 517, + 489, + 530 + ], + "spans": [ + { + "bbox": [ + 303, + 517, + 489, + 530 + ], + "score": 1.0, + "content": "model has never been confronted with wing-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 304, + 527, + 392, + 540 + ], + "spans": [ + { + "bbox": [ + 304, + 527, + 392, + 540 + ], + "score": 1.0, + "content": "profiles during training.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 108, + 547, + 504, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "Figure 3: Our models feature the Magnus effect and generalize to new fluid domains. Further examples are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 557, + 249, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 249, + 569 + ], + "score": 1.0, + "content": "presented in appendix D and the video.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 688 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "Analysis of generalization capability We tested the networks capability to generalize to objects", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "not seen during training. Figure 3b shows the networks capability to meet boundary conditions of an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "airfoil and return a plausible pressure field that produces lift (see low pressure on top of wing). Note", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "that in contrast to the approach by Thuerey et al. (2019), which learns simplified, time-averaged", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "solutions of the Navier-Stokes equations, our method is able to simulate the full incompressible", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 643, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 658 + ], + "score": 1.0, + "content": "Navier-Stokes equations for an airfoil without relying on any ground truth data or having seen airfoil-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "geometries during training. In fact, the network was only trained on simple randomized domains as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "highlighted in appendix C and Figure 7. 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Streamlines indicate flow direction, linewidth indicates speed and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 332, + 390, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 390, + 343 + ], + "score": 1.0, + "content": "colors represent the pressure field (blue: low pressure / yellow: high pressure).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 364, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 504, + 376 + ], + "score": 1.0, + "content": "Magnus effect The Magnus effect appears when a flow interacts with a rotating body. 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Figure 3b shows the networks capability to meet boundary conditions of an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "airfoil and return a plausible pressure field that produces lift (see low pressure on top of wing). Note", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "that in contrast to the approach by Thuerey et al. 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In the respective experiments, we used a NVidia", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 375, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 269, + 286 + ], + "score": 1.0, + "content": "GeForce RTX 2080 Ti consuming about", + "type": "text" + }, + { + "bbox": [ + 270, + 273, + 303, + 283 + ], + "score": 0.47, + "content": "8 6 0 \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 272, + 375, + 286 + ], + "score": 1.0, + "content": "of GPU memory.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 315, + 254, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 256, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 256, + 328 + ], + "score": 1.0, + "content": "4.2 QUANTITATIVE EVALUATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 214, + 355 + ], + "score": 1.0, + "content": "We compare our method", + "type": "text" + }, + { + "bbox": [ + 214, + 343, + 221, + 352 + ], + "score": 0.67, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "-Net) quantitatively with PhiFlow by Holl et al. (2020). Phiflow is a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "recent, open source, differentiable fluid simulator based on a MAC grid data structure. 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As benchmarks for fluid simulations on MAC grids are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 318, + 437 + ], + "score": 1.0, + "content": "not yet available, we built a simple toy domain on a", + "type": "text" + }, + { + "bbox": [ + 318, + 425, + 360, + 436 + ], + "score": 0.84, + "content": "1 0 0 \\mathrm { ~ x ~ } 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "grid which simulates a flow around", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 436, + 379, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 379, + 449 + ], + "score": 1.0, + "content": "an obstacle within a pipe (more details are provided in appendix E).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "First, we compared the computational speed on a CPU and GPU by comparing the integration", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 279, + 477 + ], + "score": 1.0, + "content": "time-steps per second (see Table 1). The", + "type": "text" + }, + { + "bbox": [ + 279, + 464, + 285, + 474 + ], + "score": 0.81, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 464, + 365, + 477 + ], + "score": 1.0, + "content": "-Net as well as the", + "type": "text" + }, + { + "bbox": [ + 365, + 464, + 371, + 474 + ], + "score": 0.77, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "-Net are significantly faster than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 213, + 488 + ], + "score": 1.0, + "content": "PhiFlow (11x on CPU and", + "type": "text" + }, + { + "bbox": [ + 214, + 475, + 230, + 486 + ], + "score": 0.62, + "content": "4 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "on GPU) as they do not rely on an iterative conjugate gradient solver", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "but instead use a single forward pass through a convolutional neural network that can be easily", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 325, + 510 + ], + "score": 1.0, + "content": "parallelized on a GPU. To provide a fair comparison on", + "type": "text" + }, + { + "bbox": [ + 325, + 497, + 337, + 508 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 497, + 506, + 510 + ], + "score": 1.0, + "content": ", we set the velocity field at the boundaries", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 142, + 520 + ], + "score": 1.0, + "content": "equal to", + "type": "text" + }, + { + "bbox": [ + 142, + 508, + 153, + 519 + ], + "score": 0.87, + "content": "\\vec { v _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 509, + 275, + 520 + ], + "score": 1.0, + "content": ". This enables us to compute", + "type": "text" + }, + { + "bbox": [ + 275, + 509, + 288, + 519 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 509, + 320, + 520 + ], + "score": 1.0, + "content": "for the", + "type": "text" + }, + { + "bbox": [ + 320, + 509, + 327, + 518 + ], + "score": 0.79, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "-Net architecture on the domain boundaries", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 393, + 532 + ], + "score": 1.0, + "content": "which would otherwise have zero divergence everywhere. This way,", + "type": "text" + }, + { + "bbox": [ + 394, + 519, + 407, + 530 + ], + "score": 0.88, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "can be interpreted as a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "metric on how well the orthogonal components of the Dirichlet boundary conditions are met (i.e.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 541, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 283, + 553 + ], + "score": 1.0, + "content": "no flow leaks through the boundaries). For", + "type": "text" + }, + { + "bbox": [ + 283, + 541, + 313, + 551 + ], + "score": 0.89, + "content": "d t = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 541, + 506, + 553 + ], + "score": 1.0, + "content": ", we outperformed Phiflow by several orders of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 195, + 565 + ], + "score": 1.0, + "content": "magnitude. 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MethodCPU[TPS]GPU[TPS]LdLp
PhiFlow7-6.2e-4-
U-Net (ours)823118.66e-74.87e-5
α-Net (ours)823115.44e-71.56e-5
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Since more complicated shapes", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "can be seen locally as a composition of basic shapes (e.g. the front of the wing can be", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "locally regarded as a cylinder), it suffices to train on basic shapes that provide the network", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 142, + 138, + 404, + 150 + ], + "spans": [ + { + "bbox": [ + 142, + 138, + 404, + 150 + ], + "score": 1.0, + "content": "with enough examples to generalize to more complicated shapes.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 132, + 82, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 166, + 349, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 351, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 351, + 180 + ], + "score": 1.0, + "content": "Further generalization examples are provided in appendix D.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 164, + 351, + 180 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Real-time capability The fluid simulation can be easily parallelized and takes low computational", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "costs as one time-integration step consists just of a single forward pass through a convolutional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "score": 1.0, + "content": "neural network. This enables for example interactive real-time simulations. We implemented a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "demo that allows to interact with a fluid by moving obstacles, rotating spheres and changing the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 250, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 264 + ], + "score": 1.0, + "content": "flow speed within a pipe (see video in supplementary material and source code). Our method runs", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 244, + 275 + ], + "score": 1.0, + "content": "at 250 timesteps per second on a", + "type": "text" + }, + { + "bbox": [ + 244, + 262, + 280, + 272 + ], + "score": 0.45, + "content": "1 0 0 \\mathrm { x } 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "grid. In the respective experiments, we used a NVidia", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 375, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 269, + 286 + ], + "score": 1.0, + "content": "GeForce RTX 2080 Ti consuming about", + "type": "text" + }, + { + "bbox": [ + 270, + 273, + 303, + 283 + ], + "score": 0.47, + "content": "8 6 0 \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 272, + 375, + 286 + ], + "score": 1.0, + "content": "of GPU memory.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 207, + 506, + 286 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 315, + 254, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 256, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 256, + 328 + ], + "score": 1.0, + "content": "4.2 QUANTITATIVE EVALUATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 214, + 355 + ], + "score": 1.0, + "content": "We compare our method", + "type": "text" + }, + { + "bbox": [ + 214, + 343, + 221, + 352 + ], + "score": 0.67, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "-Net) quantitatively with PhiFlow by Holl et al. (2020). Phiflow is a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "recent, open source, differentiable fluid simulator based on a MAC grid data structure. Furthermore,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 230, + 377 + ], + "score": 1.0, + "content": "we provide an ablation study", + "type": "text" + }, + { + "bbox": [ + 231, + 365, + 237, + 374 + ], + "score": 0.67, + "content": "{ \\vec { v } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "-Net) that does not make use of the Helmholtz decomposition but", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 258, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 248, + 387 + ], + "score": 1.0, + "content": "directly works on the velocity field", + "type": "text" + }, + { + "bbox": [ + 248, + 376, + 254, + 385 + ], + "score": 0.77, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 375, + 258, + 387 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 342, + 506, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "Quantitative comparison of different fluid solvers is challenging, as their performance is highly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "dependent on factors like the geometry of the domain, fluid parameters such as viscosity or density,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "flow speed or the timestep of the integrator. As benchmarks for fluid simulations on MAC grids are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 318, + 437 + ], + "score": 1.0, + "content": "not yet available, we built a simple toy domain on a", + "type": "text" + }, + { + "bbox": [ + 318, + 425, + 360, + 436 + ], + "score": 0.84, + "content": "1 0 0 \\mathrm { ~ x ~ } 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "grid which simulates a flow around", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 436, + 379, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 379, + 449 + ], + "score": 1.0, + "content": "an obstacle within a pipe (more details are provided in appendix E).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 393, + 505, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "First, we compared the computational speed on a CPU and GPU by comparing the integration", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 279, + 477 + ], + "score": 1.0, + "content": "time-steps per second (see Table 1). The", + "type": "text" + }, + { + "bbox": [ + 279, + 464, + 285, + 474 + ], + "score": 0.81, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 464, + 365, + 477 + ], + "score": 1.0, + "content": "-Net as well as the", + "type": "text" + }, + { + "bbox": [ + 365, + 464, + 371, + 474 + ], + "score": 0.77, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "-Net are significantly faster than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 213, + 488 + ], + "score": 1.0, + "content": "PhiFlow (11x on CPU and", + "type": "text" + }, + { + "bbox": [ + 214, + 475, + 230, + 486 + ], + "score": 0.62, + "content": "4 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "on GPU) as they do not rely on an iterative conjugate gradient solver", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "but instead use a single forward pass through a convolutional neural network that can be easily", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 325, + 510 + ], + "score": 1.0, + "content": "parallelized on a GPU. To provide a fair comparison on", + "type": "text" + }, + { + "bbox": [ + 325, + 497, + 337, + 508 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 497, + 506, + 510 + ], + "score": 1.0, + "content": ", we set the velocity field at the boundaries", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 142, + 520 + ], + "score": 1.0, + "content": "equal to", + "type": "text" + }, + { + "bbox": [ + 142, + 508, + 153, + 519 + ], + "score": 0.87, + "content": "\\vec { v _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 509, + 275, + 520 + ], + "score": 1.0, + "content": ". This enables us to compute", + "type": "text" + }, + { + "bbox": [ + 275, + 509, + 288, + 519 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 509, + 320, + 520 + ], + "score": 1.0, + "content": "for the", + "type": "text" + }, + { + "bbox": [ + 320, + 509, + 327, + 518 + ], + "score": 0.79, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "-Net architecture on the domain boundaries", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 393, + 532 + ], + "score": 1.0, + "content": "which would otherwise have zero divergence everywhere. This way,", + "type": "text" + }, + { + "bbox": [ + 394, + 519, + 407, + 530 + ], + "score": 0.88, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "can be interpreted as a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "metric on how well the orthogonal components of the Dirichlet boundary conditions are met (i.e.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 541, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 283, + 553 + ], + "score": 1.0, + "content": "no flow leaks through the boundaries). For", + "type": "text" + }, + { + "bbox": [ + 283, + 541, + 313, + 551 + ], + "score": 0.89, + "content": "d t = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 541, + 506, + 553 + ], + "score": 1.0, + "content": ", we outperformed Phiflow by several orders of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 195, + 565 + ], + "score": 1.0, + "content": "magnitude. For both,", + "type": "text" + }, + { + "bbox": [ + 195, + 552, + 208, + 563 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 551, + 226, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 552, + 239, + 564 + ], + "score": 0.89, + "content": "L _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 551, + 258, + 565 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 258, + 552, + 265, + 561 + ], + "score": 0.8, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "-Net architecture significantly outperformed the more naive", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 562, + 173, + 576 + ], + "spans": [ + { + "bbox": [ + 107, + 563, + 113, + 573 + ], + "score": 0.79, + "content": "\\vec { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 562, + 173, + 576 + ], + "score": 1.0, + "content": "-Net approach.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 453, + 506, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 384, + 592 + ], + "score": 1.0, + "content": "Furthermore, we investigated stability by evaluating the evolution of", + "type": "text" + }, + { + "bbox": [ + 384, + 580, + 397, + 592 + ], + "score": 0.88, + "content": "L _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 579, + 415, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 415, + 580, + 428, + 591 + ], + "score": 0.88, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 579, + 459, + 592 + ], + "score": 1.0, + "content": "for the", + "type": "text" + }, + { + "bbox": [ + 459, + 580, + 466, + 590 + ], + "score": 0.77, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "-Net over", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 591, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 328, + 602 + ], + "score": 1.0, + "content": "time (see Figure 4). As the fluid state is initialized with", + "type": "text" + }, + { + "bbox": [ + 329, + 591, + 358, + 601 + ], + "score": 0.91, + "content": "a _ { z } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 591, + 376, + 602 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 592, + 401, + 602 + ], + "score": 0.9, + "content": "p = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 591, + 419, + 602 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 419, + 591, + 425, + 600 + ], + "score": 0.78, + "content": "\\vec { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 591, + 505, + 602 + ], + "score": 1.0, + "content": "-Net has to perform", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 270, + 614 + ], + "score": 1.0, + "content": "a cold-start which is the reason for high", + "type": "text" + }, + { + "bbox": [ + 270, + 602, + 283, + 614 + ], + "score": 0.88, + "content": "L _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 601, + 301, + 614 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 602, + 315, + 613 + ], + "score": 0.89, + "content": "L _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "during the first circa 70 steps. 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MethodCPU[TPS]GPU[TPS]LdLp
PhiFlow7-6.2e-4-
U-Net (ours)823118.66e-74.87e-5
α-Net (ours)823115.44e-71.56e-5
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Then, we compute the frequency spectrum", + "type": "text" + }, + { + "bbox": [ + 417, + 308, + 443, + 321 + ], + "score": 0.92, + "content": "V _ { y } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 307, + 456, + 322 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 457, + 308, + 479, + 321 + ], + "score": 0.92, + "content": "v _ { y } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 307, + 506, + 322 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 425, + 333 + ], + "score": 1.0, + "content": "the fast Fourier transform (see Figure 5b). 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To optimize", + "type": "text" + }, + { + "bbox": [ + 475, + 332, + 486, + 343 + ], + "score": 0.87, + "content": "\\Vec { v _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 330, + 506, + 347 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 193, + 359 + ], + "score": 1.0, + "content": "define a loss function", + "type": "text" + }, + { + "bbox": [ + 193, + 344, + 291, + 358 + ], + "score": 0.92, + "content": "L = ( E [ | V _ { y } ( f ) | ^ { 2 } ] - \\hat { f } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 344, + 399, + 359 + ], + "score": 1.0, + "content": "and compute the gradients", + "type": "text" + }, + { + "bbox": [ + 399, + 344, + 415, + 360 + ], + "score": 0.94, + "content": "\\frac { \\partial L } { \\partial \\vec { v } _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 344, + 505, + 359 + ], + "score": 1.0, + "content": "with backpropagation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "through time. This is possible since all parts of the loss function including the fluid simulation that is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "performed by our trained neural fluid model as well as the fast Fourier transform are differentiable.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "Computing the gradients with a standard automatic differentiation library (Pytorch) took 3.5 seconds", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 227, + 403 + ], + "score": 1.0, + "content": "for all 200 time steps on our", + "type": "text" + }, + { + "bbox": [ + 227, + 390, + 264, + 401 + ], + "score": 0.52, + "content": "2 0 0 \\mathrm { x } 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "domain setup. This is considerably faster than the current", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "state-of-the-art differentiable fluid solver by Takahashi et al. (2021) which takes 5.42 seconds for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 236, + 425 + ], + "score": 1.0, + "content": "only 30 time steps on a smaller", + "type": "text" + }, + { + "bbox": [ + 237, + 413, + 273, + 424 + ], + "score": 0.41, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 412, + 379, + 425 + ], + "score": 1.0, + "content": "grid. 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We want to emphasize", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "that differentiable fluid simulations are limited to scenarios with low Reynolds numbers as in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 445, + 405, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 405, + 458 + ], + "score": 1.0, + "content": "presence of turbulences, chaotic behavior will lead to exploding gradients.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 270, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 272, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 272, + 488 + ], + "score": 1.0, + "content": "5 DISCUSSION AND OUTLOOK", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "In this work, we present an unsupervised learning scheme for the incompressible Navier-Stokes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "equations and introduce a fluid model that uses a vector potential to output divergence-free velocity", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 519, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 504, + 533 + ], + "score": 1.0, + "content": "fields. Qualitative results of our trained fluid models are in good accordance with expectations from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "score": 1.0, + "content": "fluid dynamics for a wide range of Reynolds numbers and generalize to unknown fluid domains.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "Quantitative assessment showed superior performance in terms of accuracy and speed compared to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "Phiflow and an ablation study that directly predicts the velocity field. We present a real-time demo", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "and demonstrate how differentiability can be used in a proof-of-concept fluid control scenario. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "believe that our fluid models can significantly speed up more sophisticated fluid control pipelines", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 585, + 264, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 264, + 598 + ], + "score": 1.0, + "content": "such as described by Holl et al. 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Furthermore, on top of Dirichlet boundary conditions, Neumann boundary conditions and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 403, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 403, + 636 + ], + "score": 1.0, + "content": "multi-phase domains could be incorporated in future fluid models as well.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 373, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 374, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 374, + 96 + ], + "score": 1.0, + "content": "4.3 OPTIMAL CONTROL OF VORTEX SHEDDING FREQUENCY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 136 + ], + "lines": [ + { + "bbox": [ + 104, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 104, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "In this section, we present a proof-of-concept experiment that aims at controlling the shedding", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "frequency of a Kármán vortex street behind an obstacle by changing the flow speed (see Figure", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 419, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 419, + 137 + ], + "score": 1.0, + "content": "5a). 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Then, we compute the frequency spectrum", + "type": "text" + }, + { + "bbox": [ + 417, + 308, + 443, + 321 + ], + "score": 0.92, + "content": "V _ { y } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 307, + 456, + 322 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 457, + 308, + 479, + 321 + ], + "score": 0.92, + "content": "v _ { y } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 307, + 506, + 322 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 425, + 333 + ], + "score": 1.0, + "content": "the fast Fourier transform (see Figure 5b). 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Here,", + "type": "text" + }, + { + "bbox": [ + 317, + 330, + 325, + 344 + ], + "score": 0.87, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 330, + 475, + 347 + ], + "score": 1.0, + "content": "is the target frequency. To optimize", + "type": "text" + }, + { + "bbox": [ + 475, + 332, + 486, + 343 + ], + "score": 0.87, + "content": "\\Vec { v _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 330, + 506, + 347 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 193, + 359 + ], + "score": 1.0, + "content": "define a loss function", + "type": "text" + }, + { + "bbox": [ + 193, + 344, + 291, + 358 + ], + "score": 0.92, + "content": "L = ( E [ | V _ { y } ( f ) | ^ { 2 } ] - \\hat { f } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 344, + 399, + 359 + ], + "score": 1.0, + "content": "and compute the gradients", + "type": "text" + }, + { + "bbox": [ + 399, + 344, + 415, + 360 + ], + "score": 0.94, + "content": "\\frac { \\partial L } { \\partial \\vec { v } _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 344, + 505, + 359 + ], + "score": 1.0, + "content": "with backpropagation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "through time. This is possible since all parts of the loss function including the fluid simulation that is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "performed by our trained neural fluid model as well as the fast Fourier transform are differentiable.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "Computing the gradients with a standard automatic differentiation library (Pytorch) took 3.5 seconds", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 227, + 403 + ], + "score": 1.0, + "content": "for all 200 time steps on our", + "type": "text" + }, + { + "bbox": [ + 227, + 390, + 264, + 401 + ], + "score": 0.52, + "content": "2 0 0 \\mathrm { x } 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "domain setup. This is considerably faster than the current", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "state-of-the-art differentiable fluid solver by Takahashi et al. (2021) which takes 5.42 seconds for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 236, + 425 + ], + "score": 1.0, + "content": "only 30 time steps on a smaller", + "type": "text" + }, + { + "bbox": [ + 237, + 413, + 273, + 424 + ], + "score": 0.41, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 412, + 379, + 425 + ], + "score": 1.0, + "content": "grid. The update steps of", + "type": "text" + }, + { + "bbox": [ + 379, + 413, + 390, + 424 + ], + "score": 0.89, + "content": "\\Vec { v _ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "are done using the ADAM-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "optimizer and converge after approximately 70 iterations (see Figure 5c). We want to emphasize", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "that differentiable fluid simulations are limited to scenarios with low Reynolds numbers as in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 445, + 405, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 405, + 458 + ], + "score": 1.0, + "content": "presence of turbulences, chaotic behavior will lead to exploding gradients.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 297, + 506, + 458 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 270, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 272, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 272, + 488 + ], + "score": 1.0, + "content": "5 DISCUSSION AND OUTLOOK", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "In this work, we present an unsupervised learning scheme for the incompressible Navier-Stokes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "equations and introduce a fluid model that uses a vector potential to output divergence-free velocity", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 519, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 504, + 533 + ], + "score": 1.0, + "content": "fields. Qualitative results of our trained fluid models are in good accordance with expectations from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "score": 1.0, + "content": "fluid dynamics for a wide range of Reynolds numbers and generalize to unknown fluid domains.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "Quantitative assessment showed superior performance in terms of accuracy and speed compared to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "Phiflow and an ablation study that directly predicts the velocity field. We present a real-time demo", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "and demonstrate how differentiability can be used in a proof-of-concept fluid control scenario. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "believe that our fluid models can significantly speed up more sophisticated fluid control pipelines", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 585, + 264, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 264, + 598 + ], + "score": 1.0, + "content": "such as described by Holl et al. (2020).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 498, + 506, + 598 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 602, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "First experiments of extending this approach to 3D deliver encouraging results and are topic of future", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "research. Furthermore, on top of Dirichlet boundary conditions, Neumann boundary conditions and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 403, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 403, + 636 + ], + "score": 1.0, + "content": "multi-phase domains could be incorporated in future fluid models as well.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 602, + 505, + 636 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 175, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "Nicholas Geneva and Nicholas Zabaras. 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First, we have boxes with randomized height and width", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "that float on randomized paths inspired by Brownian motion in a pipe with randomized flow speed.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Second, we have the same setup but replaced the boxes by cylinders with randomized radii and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "angular velocities in order to learn the Magnus effect. 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Figure 8c shows that it can generalize to multiple objects in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "scene, although the training set contained at most one object per scene. And Figure 8d shows that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "we can alter the outer boundary conditions as well. 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We used 3 different randomized", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "domains as exemplary depicted in Figure 7. First, we have boxes with randomized height and width", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "that float on randomized paths inspired by Brownian motion in a pipe with randomized flow speed.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Second, we have the same setup but replaced the boxes by cylinders with randomized radii and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "angular velocities in order to learn the Magnus effect. 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Figure 8c shows that it can generalize to multiple objects in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "scene, although the training set contained at most one object per scene. And Figure 8d shows that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "we can alter the outer boundary conditions as well. 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The flow speed for the inlet and outlet was set to 0.5. 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This is in good accordance with quantitative results", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "shown in section 1. 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State-of-the-art methods for off-policy learning, however, are based on inverse propensity score (IPS) weighting, which requires that the logging policy chooses all actions with non-zero probability for any context (i.e., full support). In real-world systems, this condition is often violated, and we show that existing off-policy learning methods based on IPS weighting can fail catastrophically. We therefore develop new off-policy contextual-bandit methods that can controllably and robustly learn even when the logging policy has deficient support. To this effect, we explore three approaches that provide various guarantees for safe learning despite the inherent limitations of support deficient data: restricting the action space, reward extrapolation, and restricting the policy space. We analyze the statistical and computational properties of these three approaches, and empirically evaluate their effectiveness in a series of experiments. We find that controlling the policy space is both computationally efficient and that it robustly leads to accurate policies. + +# 1 INTRODUCTION + +Many interactive systems (e.g., voice assistants, recommender systems, ad placement) can be modeled as contextual bandit problems (Langford & Zhang, 2008). In particular, each user request provides a context (e.g., user profile, query) for which the system selects an action (e.g., recommended product, presented ad) and receives a reward (e.g., purchase, click). Such contextual-bandit data is logged in large quantities as a by-product of normal system operation (Li et al., 2011; 2015; Joachims et al., 2017), making it an attractive and low-cost source of training data. With terabytes of such log data readily available in many online systems, a range of algorithms have been proposed for batch learning from such logged contextual-bandit feedback (Strehl et al., 2011; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019; London & Sandler, 2019). However, as we will argue below, these algorithms require an assumption about the log data that makes them unsuitable for many real-world applications. + +This assumption is typically referred to as the positivity or support assumption, and it is required by the Empirical Risk Minimization (ERM) objective that these algorithms optimize. Specifically, unlike in online learning for contextual bandits (Williams, 1992; Agarwal et al., 2014), batch learning from bandit feedback (BLBF) operates in the off-policy setting. During off-policy learning, the algorithm has to address the counterfactual question of how much reward each policy in the policy space would have received, if it had been used instead of the logging policy. To this effect, virtually all state-of-the-art off-policy learning methods for contextual-bandit problems rely on counterfactual estimators (Bottou et al., 2013; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019) that employ inverse propensity score (IPS) weighting to get an unbiased ERM objective. Unlike regression-based direct-modeling (DM) approaches that are often hampered by bias from model-misspecification, IPS allows a controllable bias-variance trade-off through clipping and other variance-regularization techniques (Strehl et al., 2011; Swaminathan & Joachims, 2015a; London & Sandler, 2019). + +Unfortunately, IPS and its variance-control mechanisms break down when the logging policy does not have full support – meaning that some actions have zero probability of being selected under the logging policy. In this case IPS can be highly biased. Full support is an unreasonable assumption in many real-world systems, especially when the action space is large and many actions have poor rewards. For example, in a recommender system with a large catalog (e.g. movies, music), it may be that less than $10 \%$ of the actions have support under the logging policy. We will show that existing learning algorithms can fail catastrophically on such support deficient data. + +In this paper, we develop new off-policy contextual-bandit algorithms that are specifically designed to deal with support deficient log data. Since support deficiency translates into blind spots where we do not have any knowledge about the rewards, accounting for these blind spots as part of learning is crucial for robust learning. We approach this problem from three perspectives. First, we explore restricting the action space to those actions that have support under the logging policy. Second, we explore imputation methods that extrapolate estimated rewards to those blind spots. And, third, we restrict the policy space to only those policies that have limited exposure to the blind spots. To make the latter approach computationally tractable, we define a new measure of Support Divergence between policies, show how it can be estimated efficiently without closed-form knowledge of the logging policy, and how it can be used as a constraint on the policy space. We analyze the statistical and computational properties of all three approaches and perform an extensive empirical evaluation. We find that restricting the policy space is particularly effective, since it is computationally efficient, empirically effective at learning good policies, and convenient to use in practice. + +# 2 RELATED WORK + +Most prior works on BLBF can be classified into two different approaches. The first – called Direct Model (DM) – is based on a reduction to supervised learning, where a regression estimate is trained to predict rewards (Beygelzimer & Langford, 2009). To derive a policy, the action with the highest predicted reward is chosen. A drawback of this simple approach is the bias that results from misspecification of the regression model. Since regression models are often substantially misspecified for real-world data, the DM approach often does not work well empirically. + +The second approach is based on policy learning via ERM with a counterfactual risk estimator. Inverse propensity score (IPS) weighting is one of the most popular estimators to be used as empirical risk. However, policy learning algorithms based on IPS and related estimators (Strehl et al., 2011; Swaminathan & Joachims, 2015a;b; Thomas & Brunskill, 2016; London & Sandler, 2019) require the assumption that the logging policy has full support for every policy in the policy space. One exception is the work of Liu et al. (2019). They relax the assumption to the existence of an optimal policy such that the logging policy covers the support of this optimal policy. However, this is an untestable assumption that does not provide guarantees for real-world applications. + +Our work proposes three approaches to addressing off-policy learning with support deficiency. First, our conservative extrapolation method is related to the method proposed by Liu et al. (2019). They focus on the correction of the state distribution by defining an augmented MDP, and pessimistic imputation is used to get an estimate for policy-gradient learning. Second, our method of restricting the policy space uses a surrogate for the support divergence of two policies that was previously used as control variate in the SNIPS estimator (Swaminathan & Joachims, 2015b). It also appeared in the Lagrangian formulation of the BanditNet objective (Joachims et al., 2018) and in the gradient update in REINFORCE algorithm (Williams, 1992). This connection gives interesting new insight that the baselines used in policy-gradient algorithms not only help to reduce variance in gradients (Greensmith et al., 2004), but that they also connect to the problem of support deficiency in the off-policy setting. + +# 3 OFF-POLICY LEARNING WITH DEFICIENT SUPPORT + +We start by formally defining the problem of learning a contextual-bandit policy in the BLBF setting. Input to the policy are contexts $x \in \mathcal { X }$ drawn i.i.d from a fixed but unknown distribution $P ( \mathcal X )$ . Given context $x$ , the system executes a possibly stochastic policy $\pi ( \mathcal { V } | x )$ that selects an action $y \in \mathcal { V }$ . For this context and action pair, the system observes a reward $r \in [ r _ { m i n } , r _ { m a x } ]$ from $P ( r | x , y )$ . Given a space of policies $\Pi$ , the reward of any policy $\pi \in \Pi$ is defined as + +$$ +R ( \pi ) = \underset { x } { \mathbb { E } } \underset { y \sim \pi ( y \mid x ) } { \mathbb { E } } \underset { r \sim P ( r \mid x , y ) } { \mathbb { E } } [ r ] . +$$ + +In the BLBF setting, the learning algorithm is given a dataset + +$$ +\mathcal { D } : = \{ x _ { i } , y _ { i } , r _ { i } , \pi _ { 0 } ( y _ { i } | x _ { i } ) \} _ { i = 1 } ^ { n } +$$ + +of past system interactions which consists of context-action-reward-propensity tuples. The propensity $\pi _ { 0 } ( y _ { i } | x _ { i } )$ is the probability of selecting action $y _ { i }$ for context $x _ { i }$ under the policy $\pi _ { 0 }$ that was used to log the data. We call $\pi _ { 0 }$ the logging policy, and we will discuss desired conditions on the stochasticity of $\pi _ { 0 }$ in the following. The goal of off-policy learning is to exploit the information in the logged data $\mathcal { D }$ to find a policy ${ \hat { \pi } } \in \Pi$ that has high reward $R ( { \hat { \pi } } )$ . + +Analogous to the ERM principle in supervised learning, off-policy learning algorithms typically optimize a counterfactual estimate $\hat { R } ( \pi )$ of $R ( \pi )$ as the training objective (Li et al., 2011; 2015; Bottou et al., 2013; Swaminathan & Joachims, 2015a). + +$$ +\hat { \pi } = \underset { \pi \in \Pi } { \arg \operatorname* { m a x } } [ \hat { R } ( \pi ) ] +$$ + +For conciseness, we ignore additional regularization terms in the objective (Swaminathan & Joachims, 2015a), since they are irrelevant to the main point of this paper. As counterfactual estimator $\hat { R } ( \pi )$ , most algorithms rely on some form of IPS weighting (Strehl et al., 2011; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a;b; Wang et al., 2017; Su et al., 2019) to correct the distribution mismatch between the logging policy $\pi _ { 0 }$ and each target policy $\pi \in \Pi$ . + +$$ +\hat { R } _ { I P S } ( \pi ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } . +$$ + +A crucial condition for the effectiveness of the IPS estimator (and similar estimators) is that the logging policy $\pi _ { 0 }$ assigns non-zero probability to all actions that have non-zero probability under the target policy $\pi$ we aim to evaluate. This condition is known as positivity or full support, and it is defined as follows. + +Definition 1 (Full support). The logging policy $\pi _ { 0 }$ is said to have full support for $\pi$ when $\pi _ { 0 } ( y | x ) >$ 0 for all actions $y \in \mathcal { V }$ and contexts $x \in \mathcal { X }$ for which $\pi ( y | x ) > 0$ . + +It is known that the IPS estimator is unbiased, $\mathbb { E } _ { \mathcal { D } } [ \hat { R } _ { I P S } ( \pi ) ] = R ( \pi )$ , if the logging policy $\pi _ { 0 }$ has full support for $\pi$ (Li et al., 2011). To ensure unbiased ERM, algorithms that use the IPS estimator require that the logging policy $\pi _ { 0 }$ has full support for all policies $\pi \in \Pi$ in the policy space. For sufficiently rich policy spaces, like deep-networks $f _ { w } ( x , y )$ with softmax outputs of the form + +$$ +\pi _ { w } ( y | x ) = \frac { e x p ( f _ { w } ( x , y ) ) } { \sum _ { y ^ { \prime } \in \mathcal { V } } e x p ( f _ { w } ( x , y ^ { \prime } ) ) } , +$$ + +this means that the logging policy $\pi _ { 0 }$ needs to assign non-zero probability to every action $y$ in every context $x$ . This is a strong condition that is not feasible in many real-world systems, especially if the action space is large and many actions have poor reward. + +If the support requirement is violated, ERM learning can fail catastrophically. We will show below that the underlying reason is bias, not excessive variance that could be remedied through clipping or variance regularization (Strehl et al., 2011; Swaminathan $\&$ Joachims, 2015a). To quantify how support deficient a logging policy is, we denote the set of unsupported actions for context $x$ under $\pi _ { 0 }$ as + +$$ +\mathcal { U } ( x , \pi _ { 0 } ) : = \{ y \in y | \pi _ { 0 } ( y | x ) = 0 \} . +$$ + +The bias of the IPS estimator is then characterized by the expected reward on the unsupported actions. + +Proposition 1. Given contexts $x \sim P ( \mathcal { X } )$ and logging policy $\pi _ { 0 } ( \mathcal { V } | x )$ , the bias of $\hat { R } _ { I P S }$ for target policy $\pi ( \mathcal { V } | x )$ is equal to the expected reward on the unsupported action sets, i.e., $b i a s ( \pi | \pi _ { 0 } ) =$ $\begin{array} { r } { \mathbb { E } _ { x } [ - \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \delta ( x , y ) ] } \end{array}$ . + +The proof is in Appendix A.1. From Proposition 1, it is clear that support deficient log data can drastically mislead ERM learning. To quantify the effect of support deficiency on ERM, we define the support divergence between a logging policy $\pi _ { 0 }$ and a target policy $\pi$ as follows. + +Definition 2 (Support Divergence). For contexts $x \sim P ( \mathcal { X } )$ and any corresponding pair of target policy $\pi$ and logging policy $\pi _ { 0 }$ , the Support Divergence is defined as + +$$ +\mathcal { D } _ { \mathcal { X } } \big ( \pi \big | \pi _ { 0 } \big ) : = \underset { x \sim P ( \mathcal { X } ) } { \mathbb { E } } \left[ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \right] . +$$ + +With this definition in hand, we can quantify the effect of support deficiency on ERM learning for a policy space $\Pi$ under logging policy $\pi _ { 0 }$ . + +Theorem 1. For any given hypothesis space $\Pi$ with logging policy $\pi _ { 0 } ~ \in ~ \Pi$ , there exists a reward distribution $\mathcal { P } _ { r }$ with support in $[ r _ { m i n } , r _ { m a x } ]$ such that in the limit of infinite training data, ERM using IPS over the logged data $\mathcal D \sim \dot { P } ( \mathcal X ) \times \pi _ { 0 } ( \cdot | \mathcal X ) \times \mathcal P _ { r }$ can select a policy $\begin{array} { r } { \hat { \pi } \in \arg \operatorname* { m a x } _ { \pi \in \Pi } \mathbb { E } _ { \mathcal { D } } [ \hat { R } _ { I P S } ( \pi ) ] } \end{array}$ that is at least $\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \operatorname* { m a x } _ { \pi \in \Pi } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) } \quad } & { { } } \end{array}$ suboptimal. + +The proof is in Appendix A.2. To illustrate the theorem, consider a problem with rewards $r \in$ $[ - 1 , 0 ]$ . Furthermore, consider a policy space $\Pi$ that contains a good policy $\pi _ { g }$ with $R ( \pi _ { g } ) = - 0 . 1$ and a bad policy $\pi _ { b }$ with $R ( \pi _ { b } ) = - 0 . 7 .$ . If policy $\pi _ { b }$ has support divergence $\mathcal { D } _ { \mathcal { X } } ( \pi _ { b } \vert \pi _ { 0 } ) = 0 . 6$ or larger, then ERM may return the bad $\pi _ { b }$ instead of $\pi _ { g }$ even with infinite amounts of training data. + +Note that it is sufficient to merely have one policy in $\Pi$ that has large support deficiency to achieve this suboptimality. It is therefore crucial to control the support divergence $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } )$ uniformly over all $\pi \in \Pi$ , or to account for the suboptimality it can induce. To this effect, we explore three approaches in the following. + +# 3.1 SAFE LEARNING BY RESTRICTING THE ACTION SPACE + +The first and arguably most direct approach to reducing $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } )$ is to disallow any action that has zero support under the logging policy. For the remaining action set, the logging policy has full support by definition. This restriction of the action set can be achieved by transforming each policy $\pi \in \Pi$ into a new policy that sets the probability of the unsupported actions to zero. + +$$ +\pi ( y | x ) \longrightarrow \bar { \pi } ( y | x ) : = \frac { \pi ( y | x ) \mathbb { 1 } _ { \{ y \notin \mathcal { U } ( x , \pi _ { 0 } ) \} } } { 1 - \sum _ { y ^ { \prime } \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y ^ { \prime } | x ) } +$$ + +This results in a new policy space $\bar { \Pi }$ . All $\bar { \pi } \in \bar { \Pi }$ have support divergence of zero $\mathcal { D } _ { \mathcal { X } } ( \bar { \pi } | \pi _ { 0 } ) = 0$ and ERM via IPS is guaranteed to be unbiased. + +While this transformation of the policy space from $\Pi$ to $\bar { \Pi }$ is conceptually straightforward, it has two potential drawbacks. First, restricting the action space without any exceptions may overly constrain the policies in $\bar { \Pi }$ . In particular, if the optimal action $y ^ { * }$ for a specific context $x$ does not have support under the logging policy, no $\bar { \pi } \in \dot { \bar { \Pi } }$ can ever choose $y ^ { * }$ even if there are many observations of similar $y$ ’s on similar context $x ^ { \prime }$ . The second drawback is computational. For every context $x$ during training and testing, the system needs to evaluate the logging policy $\pi _ { 0 } ( y | x )$ to compute the transformation from $\pi$ to $\bar { \pi }$ . This can be prohibitively expensive especially at test time, where – after multiple rounds of off-policy learning with data from previously learned policies – we would need to evaluate the whole sequence of previous logging policies to execute the learned policy. + +# 3.2 SAFE LEARNING THROUGH REWARD EXTRAPOLATION + +As illustrated above, support deficiency is a problem of blind spots where we lack information about the rewards of some actions in some contexts. Instead of disallowing the unsupported actions like in the previous section, an alternative is to extrapolate the observed rewards to fill in the blind spots. To this effect, we propose the following augmented IPS estimator that imputes an extrapolated reward $\hat { \delta } ( x , y )$ for each unsupported action $y \in \mathcal { U } ( x , \pi _ { 0 } )$ . + +$$ +\hat { R } _ { I P S } ^ { \delta } ( \pi ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left[ \frac { \pi ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \sum _ { y \in \mathcal { U } ( x _ { i } , \pi _ { 0 } ) } \pi ( y | x _ { i } ) \hat { \delta } ( x _ { i } , y ) \right] +$$ + +In the following proposition, we characterize the bias of the augmented IPS estimator for any given reward extrapolation $\hat { \delta } ( x , y )$ . We denote the mean of the reward $r$ for context $x$ and action $y$ with $\delta ( x , y ) = \mathbb { E } _ { r \sim P ( r | x , y ) } [ r ]$ . Furthermore, $\Delta ( x , y ) : = \hat { \delta } ( x , y ) - \delta ( x , y )$ denotes the error of the reward extrapolation for each $x$ and $y$ . + +Proposition 2. Given contexts $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ drawn i.i.d from the unknown distribution $P ( \mathcal X )$ , for action $y _ { i }$ drawn independently from logging policy $\pi _ { 0 }$ with probability $\pi _ { 0 } ( \mathcal { V } | x _ { i } )$ , the bias of the empirical risk defined in Equation (7) is $\begin{array} { r } { \mathbb E _ { x } [ \sum _ { y \in \mathcal U _ { x } ^ { \pi _ { 0 } } } \pi ( y | x ) \Delta ( x , y ) ] } \end{array}$ . + +In this way we can learn in the original action and policy space, but mitigate the effect of the support deficiency by explicitly incorporating the extrapolated reward $\hat { \delta } ( x , y )$ . We explore two choices for $\hat { \delta } ( x , y )$ in the following, which provide different types of guarantees. + +Conservative Extrapolation. To minimize the user impact of randomization in the logging policy, it is generally desirable to put zero probability on actions the are very likely to have low (or even catastrophic reward). This means that precisely those bad actions are likely to not be supported in the logging policy. A key danger of blind spots regarding those actions is that naive IPS training will inadvertently learn + +# Algorithm 1: Data Augmentation + +input: original logged dataset $\mathcal { D }$ , replaycount $k$ , +reward estimate $\hat { \delta } ( x , y )$ ; output: $\mathcal { D } ^ { \prime }$ ; +initialization: $\mathcal { D } ^ { \prime } = \varnothing$ ; +for $j = 1 , \dots , k$ do for $i = 1 , \ldots , n$ do Define $U _ { x _ { i } }$ to be the uniform distribution over $\mathcal { U } ( x _ { i } , \pi _ { 0 } )$ ; Draw $y \sim U _ { x _ { i } }$ ; $\begin{array} { r } { \mathcal { D } ^ { \prime } = \mathcal { D } ^ { \prime } \bigcup \{ x _ { i } , y , \hat { \delta } ( x _ { i } , y ) , \frac { 1 } { | \mathcal { U } ( x _ { i } , \pi _ { 0 } ) | } \} ; } \end{array}$ end +end + +a policy that selects those actions. This can be avoided by being maximally conservative about unsupported actions and imputing the lowest possible reward $\forall x , y \in \mathcal { U } ( x , \pi _ { 0 } ) : \hat { \delta } ( x , y ) = r _ { m i n }$ . Intuitively, by imposing the worst possible reward for the unsupported actions, the learning algorithm will aim to avoid these low-reward areas. However, unlike for the $\bar { \pi }$ policies resulting from the restricted action space, the learned policy is not strictly prohibited from choosing unsupported actions – it is merely made aware of the maximum loss that the action may incur. Note that for problems where $r _ { m i n } = 0$ , the naive IPS estimator is identical to conservative extrapolation since the second term in Equation (7) is zero. + +Regression Extrapolation. Instead of extrapolating with the worst-case reward, we may have additional prior knowledge in the form of a model-based estimate that reduces the bias. In particular, we explore using a regression estimate $\begin{array} { r } { \hat { \delta } = \arg \operatorname* { m i n } _ { \hat { \delta } ^ { \theta } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \hat { \delta } ^ { \theta } ( x _ { i } , y _ { i } ) - r _ { i } ) ^ { 2 } } \end{array}$ that extrapolates from the observed data $\mathcal { D }$ . Typically, comes from a parameterized class of regression functions. Other regression objectives could also be used, such as weighted linear regression that itself uses importance sampling as weights (Farajtabar et al., 2018). But, fundamentally, all regression approaches assume that the regression model is not misspecified and that it can thus extrapolate well. Note that the IPS part of Equation (7) can be changed to any estimators (with action set restricted on $\boldsymbol { \mathcal { U } } ( \boldsymbol { x } , \pi _ { 0 } ) ^ { c }$ for all $x$ ), and it turns out that doubly robust (Dud´ık et al., 2011) and CAB (Su et al., 2019) are special extensions of regression extrapolation that substitute the IPS part with their corresponding estimator. + +Efficient Approximation. Evaluating the augmented IPS estimator from Equation (7) can be computationally expensive if the number of unsupported actions $\boldsymbol { \mathcal { U } } ( \boldsymbol { x } , \pi _ { 0 } )$ is large. To overcome this problem, we propose to use sampling to estimate the expected reward on the unsupported action, which can be thought of as augmenting the dataset $\mathcal { D }$ with additional observations where the logging policy has zero support. In particular, we propose the data-augmentation procedure detailed in Algorithm 1. With the additional bandit data $\mathcal { D } ^ { \prime } = \{ x _ { j } ^ { \prime } , y _ { j } ^ { \prime } , \hat { \delta } ( x _ { j } ^ { \prime } , y _ { j } ^ { \prime } ) , p _ { j } ^ { \prime } \} _ { j = 1 } ^ { m }$ from Algorithm 1, the new objective is + +$$ +\underset { \pi \in \Pi } { \arg \operatorname* { m i n } } \left\{ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \frac { \pi ( y _ { j } ^ { \prime } | x _ { j } ^ { \prime } ) } { p _ { j } ^ { \prime } } \hat { \delta } ( x _ { j } ^ { \prime } , y _ { j } ^ { \prime } ) \right\} . +$$ + +In Appendix A.5, we show that the empirical risk in Equation (8) has the same expectation (over randomness in $\mathcal { D }$ and $\mathcal { D } ^ { \prime }$ ) as $\hat { R } _ { I P S } ^ { \delta } ( \mathcal { D } )$ and can thus serve as an approximation for Equation (7). + +# 3.3 SAFE LEARNING BY RESTRICTING THE POLICY SPACE + +As motivated by Theorem 1, the risk of learning from support deficient data scales with the maximum support divergence $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } )$ among the policies in the policy space $\Pi$ . Therefore, our third approach restricts the policy space to the subset $\Pi ^ { \kappa } \subset \Pi$ that contains the policies $\pi \in \Pi$ with an acceptably low support divergence $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } ) \le \kappa$ . + +$$ +\Pi ^ { \kappa } = \{ \pi | \pi \in \Pi \wedge { \mathscr { D } } \chi ( \pi | \pi _ { 0 } ) \leq \kappa \} +$$ + +The parameter $\kappa$ has an intuitive meaning. It specifies the maximum probability mass that a learned policy can place on unsupported actions. By limiting this to $\kappa$ , we limit the maximum bias of the ERM procedure according to Proposition 2 while not explicitly torquing the rewards like in conservative reward imputation. + +A key challenge, however, is implementing this restriction of the hypothesis space, such that the ERM learner $\hat { \pi } = \arg \operatorname* { m a x } _ { \pi \in \Pi ^ { \kappa } } [ \hat { R } _ { I P S } ( \pi ) ]$ only considers the subset $\Pi ^ { \kappa } \subset \Pi$ . In particular, we do not have access to the context distribution $P ( \mathcal X )$ for calculating $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } )$ , nor would it be possible to enumerate all $\pi \in \Pi$ to check the condition $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } ) ~ \le ~ \kappa$ , which itself requires a possibly infeasible iteration over all actions. The following theorem (with proof in Appendix A.3) gives us an efficient way of estimating and controlling $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } )$ without explicit knowledge of $P ( \mathcal X )$ or access to the logging policy $\pi _ { 0 }$ beyond the logged propensities. + +Theorem 2. For contexts $x _ { i }$ drawn i.i.d from $P ( \mathcal X )$ , action $y _ { i }$ drawn from logging policy $\pi _ { 0 }$ , we define SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) . For any policy $\pi$ it holds that + +$$ +\begin{array} { r } { \underset { x \sim P ( \mathcal { X } ) } { \mathbb { E } } \underset { y \sim \pi _ { 0 } ( \cdot | x ) } { \mathbb { E } } [ S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) ] + \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) = 1 } \end{array} +$$ + +Using this theorem, the following proposition (proof in Appendix A.4, empirically verified in Appendix B) gives us an efficient way of implementing the constraint $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } ) \le \kappa$ via $1 - S _ { D } ( \pi | \pi _ { 0 } )$ . + +Proposition 3. For any given $\kappa \in ( 0 , 1 )$ , $0 < \epsilon < \kappa / 2$ , let $p _ { m i n }$ denote the minimum propensity under supported set $p _ { m i n } ~ = ~ m a x _ { x , y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi _ { 0 } ( y | x )$ , then with probability larger than $1 - 2 \exp ( - 2 n \epsilon ^ { 2 } p _ { m i n } ^ { 2 } )$ , the constraint $1 - \kappa { + } \epsilon \le S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) \le 1 { - } \epsilon$ will ensure $0 \leq \mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } ) \leq \kappa$ . + +We can thus use $1 - S _ { D } ( \pi | \pi _ { 0 } )$ as a surrogate for $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } )$ in the training objective: + +$$ +\underset { \pi _ { w } \in \Pi } { \arg \operatorname* { m i n } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi _ { w } ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } . \mathrm { ~ s u b j e c t ~ t o ~ } 1 - \kappa + \epsilon \leq \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi _ { w } ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } \leq 1 - \epsilon +$$ + +Using Lagrange multipliers, an equivalent dual form of Equation (11) is: + +$$ +\operatorname* { m a x } _ { u _ { 1 } , u _ { 2 } \geq 0 } \operatorname* { m i n } _ { \pi _ { w } \in \Pi } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi _ { w } ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } ( r _ { i } + u _ { 1 } - u _ { 2 } ) - u _ { 1 } ( 1 - \epsilon ) + u _ { 2 } ( 1 - \kappa + \epsilon ) +$$ + +For each fixed $( u _ { 1 } , u _ { 2 } )$ pair, the inner minimization objective is ERM with IPS under a shift of the reward. Instead of maximizing over $( u _ { 1 } , u _ { 2 } )$ in the outer objective, we treat $( u _ { 1 } - u _ { 2 } )$ as a hyperparameter that we select on a validation set. We explore various estimators for this modelselection problem in Section 4. + +Note that, among the methods we proposed for dealing with support deficiency, this approach is the most efficient to implement, and it does not require access to the logging policy during training or testing. Furthermore, the form of the inner objective coincides with that of BanditNet (Joachims et al., 2018), which is known to work well for deep network training by controlling propensity overfitting (Swaminathan & Joachims, 2015a). + +# 4 EMPIRICAL EVALUATION + +We empirically evaluate the effectiveness and robustness of the three proposed approaches: restricting the action space, conservative and regression extrapolation, as well as restricting the policy space. The semi-synthetic experiments are based on two real-world datasets: one is the popular image classification dataset CIFAR10 (Krizhevsky et al.) and the other is the credit-card fraud dataset of Dal Pozzolo et al. (2015). We use the naive IPS estimator and the regression-based Direct Method (DM) as baselines. + +The experiments are set up as follows. We first create a train-validation-test split for both datasets. The training set is used to generate bandit datasets for learning, the validation set is used to generate bandit datasets for model selection, and the full-information test set serves as ground truth for evaluating the learned policies. To simulate bandit feedback for the CIFAR10 dataset, our experiment setup follows traditional supervised bandit conversion for multi-class classification datasets (Beygelzimer & Langford, 2009). To not only have bandit data with binary multi-class rewards, we choose a different methodology for the credit-card dataset by designating some features as corresponding to actions and rewards. More details are given in Appendix B. + +To get logging policies for generating bandit feedback, we start by training a softmax-policy as in Equation (4) on a subset of the full-information data. We then introduce a temperature parameter $\tau$ into the learned policy via $\tau f _ { w } ( x , y )$ to be able to control its stochasticity and support deficiency. In particular, we enforce zero support for some actions by clipping the propensities to 0 if they are below a threshold of $\epsilon = 0 . 0 1$ . The larger $\tau$ , the higher the support deficiency. Note that making the threshold at $\epsilon = 0 . 0 1$ allows us to control support while the variance of IPS stays bounded. This allows us to study support deficiency without having to worry about variance control. + +For both logging and target policies, we train softmax policies where $f _ { w } ( x , y )$ is a neural network. We use the ResNet20 architecture (He et al., 2016) for CIFAR10, and a fully-connected 2-layer network for the credit-card dataset. + +![](images/28a0537b9d42908064e6c5fb86cdcb47b4f7d8597ca2d9309b1cb944a9c7dace.jpg) +Figure 1: Learning results with varying support deficiency in the logging policy. + +How do the methods perform at different level of support deficiency? Results are shown in Figure 1. First, as expected, learning using naive IPS degrades on both datasets as we make the logging policy more peaked and the number of unsupported actions increases. Note that naive IPS coincides with Conservative Extrapolation, since both datasets are scaled to have a minimum reward of zero. In the rightmost column, however, we translated the rewards to $[ - 1 , 0 ]$ . This has a strong detrimental effect on naive IPS, as it is now overly optimistic about unsupported actions. Second, the approach of dealing with support deficiency by restricting the action space also performs poorly. The second row of plot sheds some light on this, as it shows the support divergence $\mathcal { D } _ { \mathcal { X } } ( \pi \vert \pi _ { 0 } )$ of the learned policy. It is zero for Action Restriction as expected, which means that bias is not the problem. Instead, as the number of unsupported actions increases, the best actions are more likely to be pruned and unavailable in the restricted policy space $\bar { \Pi }$ . Third, Regression Extrapolation performs better than Conservative Extrapolation on both datasets. In both cases, the DM model is quite good which also benefits Regression Extrapolation. However, on the credit-card dataset the regression seems better at ranking than at predicting the true reward, which explains why DM performs better than Regression Extrapolation. Fourth, the method that performs well most consistently is Policy Restriction. Unlike all the other IPS-based methods, it performs well even under the translated rewards in the third column of Figure 1. This is because the objective of Policy Restriction coincides with that of BanditNet (Joachims et al., 2018), which is known to remedy propensity overfitting due to the lack of equivariance of the IPS estimator (Swaminathan & Joachims, 2015b). + +How does the learning performance change with more training data? Results are shown in Figure 2. As the number of bandit examples increases, Policy Restriction, Regression Extrapolation and DM dominate over most of the range especially when the percentage of unsupported actions is large. Among the other methods, Action Restriction can take the least advantage of more data. This is plausible, since its maximum performance is limited by the available actions. For similar reasons, Conservative Extrapolation (and equivalently IPS) also flattens out, since it also tightly restricts the action space by imputing the minimum reward. + +![](images/aa8a5d1278cce3164dc98804179215126bb0ed224e646e59e7747093d8691b15.jpg) +Figure 2: Learning results with varying amounts of bandit data on CIFAR10 and credit-card dataset. + +
%Unsupp.OracleRegr.Extrap.DMCons.Extrap.SNIPS
450.8780.8780.8780.8780.876
600.8710.8710.8710.8710.871
700.8580.8580.8560.8580.858
770.8560.8540.8540.8560.856
800.8550.8550.8550.8380.849
+ +![](images/129795f238e4f9e260030195c39b917e0942e1c8abe6287722e9eee2e57b7f77.jpg) +Figure 3: Model selection performance on CIFAR10. + +How effective are the estimators for model selection? Most learning algorithms have hyperparameters, and we now evaluate how the estimators perform for this secondary learning problem. We specifically focus on the parameter $k : = u _ { 1 } - u _ { 2 }$ in Policy Restriction, since it controls how much the learned policies can step outside the region of support. The table on the left of Figure 3 shows the reward of the learned policy when performing model selection with the respective estimator. Oracle is the estimator that has access to the full-information validation set, and can thus be considered as a skyline. We also included the SNIPS estimator (Swaminathan & Joachims, 2015b), which imputes the average reward on the supported action for the unsupported actions (Gilotte et al., 2018). All estimators perform quite well for model selection on CIFAR, and the results are analogous for the credit-card data (see Appendix B.2). However, the plot to the right of Figure 3 reveals that SNIPS does not accurately reflect the shape of the Oracle curve. Both Regression Extrapolation and DM, however, are found to be sufficiently accurate for reliable model selection. + +# 5 DISCUSSION AND CONCLUSIONS + +We identified and analyzed how off-policy learning based on IPS weighting can suffer severely degraded learning performance when the logging policy is support deficient. To remedy this problem, we explored approaches that limit the impact of missing support through three different means: restricting the action space, reward extrapolation and restricting the policy space. We find that the most natural approach of restricting the action space is neither computationally efficient, nor does it learn accurate policies. Reward extrapolation through regression and restricting the policy space, however, both perform well and robustly even at high levels of support deficiency. Among those two methods, reward extrapolation has the potential drawback that we need to compute (and/or sample from) the complement of the logging policy, which can be computationally challenging. Furthermore, having to store all old logging policies is inconvenient in practice. This makes the approach of restricting the policy space particularly attractive, since it is computationally efficient and it does not require access to the logging policy beyond the logged propensity values. + +# REFERENCES + +Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In International Conference on Machine Learning (ICML), 2014. + +Alina Beygelzimer and John Langford. The offset tree for learning with partial labels. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 129–138. ACM, 2009. + +Leon Bottou, Jonas Peters, Joaquin Qui ´ nonero-Candela, Denis X Charles, D Max Chickering, Elon ˜ Portugaly, Dipankar Ray, Patrice Simard, and Ed Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. The Journal of Machine Learning Research, 14(1):3207–3260, 2013. + +Andrea Dal Pozzolo, Olivier Caelen, Reid A Johnson, and Gianluca Bontempi. Calibrating probability with undersampling for unbalanced classification. In 2015 IEEE Symposium Series on Computational Intelligence, pp. 159–166. IEEE, 2015. + +Miroslav Dud´ık, John Langford, and Lihong Li. Doubly robust policy evaluation and learning. In International Conference on Machine Learning (ICML), 2011. + +Mehrdad Farajtabar, Yinlam Chow, and Mohammad Ghavamzadeh. More robust doubly robust off-policy evaluation. In International Conference on Machine Learning, pp. 1446–1455, 2018. + +Alexandre Gilotte, Clement Calauz ´ enes, Thomas Nedelec, Alexandre Abraham, and Simon Doll \` e.´ Offline a/b testing for recommender systems. In Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining, pp. 198–206. ACM, 2018. + +Evan Greensmith, Peter L Bartlett, and Jonathan Baxter. Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5(Nov):1471–1530, 2004. + +Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. + +T. Joachims, A. Swaminathan, and T. Schnabel. Unbiased learning-to-rank with biased feedback. In ACM Conference on Web Search and Data Mining (WSDM), 2017. + +T. Joachims, A. Swaminathan, and M. de Rijke. Deep learning with logged bandit feedback. In International Conference on Learning Representations (ICLR), 2018. + +Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. Cifar-10 (canadian institute for advanced research). URL http://www.cs.toronto.edu/˜kriz/cifar.html. + +John Langford and Tong Zhang. The epoch-greedy algorithm for multi-armed bandits with side information. In Advances in neural information processing systems, pp. 817–824, 2008. + +Lihong Li, Wei Chu, John Langford, and Xuanhui Wang. Unbiased offline evaluation of contextualbandit-based news article recommendation algorithms. In Proceedings of the fourth ACM international conference on Web search and data mining, pp. 297–306. ACM, 2011. + +Lihong Li, Shunbao Chen, Jim Kleban, and Ankur Gupta. Counterfactual estimation and optimization of click metrics in search engines: A case study. In Proceedings of the 24th International Conference on World Wide Web, pp. 929–934. ACM, 2015. + +Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Off-policy policy gradient with state distribution correction. arXiv preprint arXiv:1904.08473, 2019. + +Ben London and Ted Sandler. Bayesian counterfactual risk minimization. In International Conference on Machine Learning, pp. 4125–4133, 2019. + +Alex Strehl, John Langford, Lihong Li, and Sham M Kakade. Learning from logged implicit exploration data. In Advances in Neural Information Processing Systems (NIPS), 2011. + +Yi Su, Lequn Wang, Michele Santacatterina, and Thorsten Joachims. Cab: Continuous adaptive blending for policy evaluation and learning. In International Conference on Machine Learning, pp. 6005–6014, 2019. + +A. Swaminathan and T. Joachims. Batch learning from logged bandit feedback through counterfactual risk minimization. Journal of Machine Learning Research (JMLR), 16:1731–1755, Sep 2015a. Special Issue in Memory of Alexey Chervonenkis. + +A. Swaminathan and T. Joachims. The self-normalized estimator for counterfactual learning. In Neural Information Processing Systems (NIPS), 2015b. + +Philip Thomas and Emma Brunskill. Data-efficient off-policy policy evaluation for reinforcement learning. In International Conference on Machine Learning (ICML), 2016. + +Yu-Xiang Wang, Alekh Agarwal, and Miroslav Dudik. Optimal and adaptive off-policy evaluation in contextual bandits. In International Conference on Machine Learning (ICML), 2017. + +Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992. + +# A APPENDIX: PROOFS + +In this appendix, we provide proofs of the main theorems and propositions. + +# A.1 PROOF OF PROPOSITION 1 + +Proposition 1. Given contexts $x \sim P ( \mathcal { X } )$ and logging policy $\pi _ { 0 } ( \mathcal { V } | x )$ , the bias of $\hat { R } _ { I P S }$ for target policy $\pi ( \mathcal { V } | x )$ is equal to the expected reward on the unsupported action sets, i.e., $b i a s ( \pi | \pi _ { 0 } ) =$ $\begin{array} { r } { \mathbb { E } _ { x } [ - \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \delta ( x , y ) ] } \end{array}$ . + +Proof. Recall $\delta ( x , y ) = \mathbb { E } _ { r } [ r ( x , y ) | x , y ]$ , and logged data $\mathcal { D } \sim \mathcal { P } _ { \mathcal { X } } \times \pi _ { 0 } ( \cdot | \mathcal { X } ) \times \mathcal { P } _ { r }$ . + +$$ +\begin{array} { l } { \displaystyle b i a s ( \pi | \pi _ { 0 } ) = \frac { \mathbb { E } [ \hat { R } _ { I P S } ( \pi ) ] - R ( \pi ) } { \mathcal { D } } } \\ { \displaystyle \quad = \frac { \mathbb { E } [ \sum _ { \scriptstyle x } } { y \in ( \mathcal { U } ( x , \pi _ { 0 } ) ) ^ { c } } \pi _ { 0 } ( y | x ) \frac { \pi ( y | x ) } { \pi _ { 0 } ( y | x ) } \delta ( x , y ) - \sum _ { y \in \mathcal { Y } } \pi ( y | x ) \delta ( x , y ) ] } \\ { \displaystyle \quad = \frac { \mathbb { E } [ - \sum _ { \scriptstyle x } } { x } - \pi ( y | x ) \delta ( x , y ) ] } \end{array} +$$ + +# A.2 PROOF OF THEOREM 1 + +Theorem 1. For any given hypothesis space $\Pi$ with logging policy $\pi _ { 0 } ~ \in ~ \Pi$ , there exists a reward distribution $\mathcal { P } _ { r }$ with support in $[ r _ { m i n } , r _ { m a x } ]$ such that in the limit of infinite training data, ERM using $I P S$ over the logged data $\mathcal D \sim \dot { P } ( \mathcal X ) \times \pi _ { 0 } ( \cdot | \mathcal X ) \times \mathcal P _ { r }$ can select a policy $\begin{array} { r } { \hat { \pi } \in \arg \operatorname* { m a x } _ { \pi \in \Pi } \mathbb { E } _ { \mathcal { D } } [ \hat { R } _ { I P S } ( \pi ) ] } \end{array}$ that is at least $\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \operatorname* { m a x } _ { \pi \in \Pi } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) } \quad } & { { } } \end{array}$ suboptimal. + +Proof. For any given hypothesis space $\Pi$ and logging policy $\pi _ { 0 }$ , define a deterministic reward distribution $\mathcal { P } _ { r }$ as the following: for all context $x$ , $r ( x , y ) = \delta ( x , y ) = r _ { m i n }$ for $y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c }$ and $r ( x , y ) ~ = ~ \delta ( x , y ) ~ = ~ r _ { m a x }$ for $y \in \mathcal { U } ( x , \pi _ { 0 } )$ . Let $\tilde { \pi } \in \arg \operatorname* { m a x } _ { \pi \in \Pi } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } )$ and $\pi ^ { * } \in \arg \operatorname* { m a x } _ { \pi \in \Pi } R ( \pi )$ , then we have the following lower bound for $R ( \pi ^ { * } )$ : + +$$ +\begin{array} { r l } & { R ( \pi ^ { * } ) \geq R ( \tilde { \pi } ) } \\ & { \qquad = \underset { x } { \mathbb { E } } [ \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } r _ { m a x } + \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } r _ { m i n } ] } \\ & { \qquad = r _ { m a x } \operatorname* { m a x } _ { \pi \in \Pi } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) + r _ { m i n } \big ( 1 - \underset { \pi \in \Pi } { \operatorname* { m a x } } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) \big ) } \end{array} +$$ + +where the first inequality follows from the definition of $\pi ^ { * }$ , the first and second equality is based on the specific reward distribution $\mathcal { P } _ { r }$ and the definition of $\tilde { \pi }$ . + +In the following we will show that for any $\hat { \pi }$ learned by the expectation of ERM (or in the limit of infinite amount data), i.e., πˆ ∈ arg max $\mathbb { E } _ { \mathcal { D } } [ \hat { R } _ { I P S } ^ { \mathcal { D } } ( \pi ) ]$ , $\hat { \pi }$ have the same support as $\pi _ { 0 }$ . + +$$ +\mathbb { E } [ \hat { R } _ { I P S } ( \pi ) ] = \mathbb { E } [ \sum _ { x } \pi ( y | x ) r _ { m i n } ] = r _ { m i n } \mathbb { E } [ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi ( y | x ) ] \le r _ { m i n } +$$ + +for all $\pi \in \Pi$ , then it is easy to see $\pi _ { 0 } \in \Pi$ is one of the solution of ERM. Actually for any $\hat { \pi } \in$ arg max $\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } [ \hat { R } _ { I P S } ^ { \mathcal { D } } ( \pi ) ] , \mathbb { E } _ { x } [ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi ( y | x ) ] = 1 } \end{array}$ and it gives us that any solution of the ERM has exactly the same support as $\pi _ { 0 }$ , then we have $R ( \hat { \pi } ) = r _ { m i n }$ for $\hat { \pi } \in \arg \operatorname* { m a x } \mathbb { E } _ { \mathcal { D } } [ \hat { R } _ { I P S } ^ { \mathcal { D } } ( \pi ) ]$ . + +Combining the lower bound for $R ( \pi ^ { * } )$ and $R ( \hat { \pi } ) = r _ { m i n }$ , we have + +$$ +\begin{array} { r l } & { R ( \pi ^ { * } ) - R ( \hat { \pi } ) \geq r _ { m a x } \displaystyle \operatorname* { m a x } _ { \pi \in \Pi } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) + r _ { m i n } ( 1 - \displaystyle \operatorname* { m a x } _ { \pi \in \Pi } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) ) - r _ { m i n } } \\ & { \quad \quad \quad \quad \quad = \left( r _ { m a x } - r _ { m i n } \right) \displaystyle \operatorname* { m a x } _ { \pi \in \Pi } \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) } \end{array} +$$ + +# A.3 PROOF OF THEOREM 2 + +Theorem 2. For contexts $x _ { i }$ drawn i.i.d from $P ( \mathcal X )$ , action $y _ { i }$ drawn from logging policy $\pi _ { 0 }$ , we define SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) . For any policy $\pi$ it holds that + +$$ +\begin{array} { r } { \underset { x \sim P ( \mathcal { X } ) } { \mathbb { E } } \underset { y \sim \pi _ { 0 } ( \cdot | x ) } { \mathbb { E } } [ S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) ] + \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) = 1 } \end{array} +$$ + +Proof. + +$$ +\begin{array} { r l } { \underset { x , y \sim \pi _ { 0 } } { \mathbb { E } } [ S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) ] + \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) = \underset { x } { \mathbb { E } } [ \ \underset { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } { \sum } \pi _ { 0 } ( y | x ) \frac { \pi ( y | x ) } { \pi _ { 0 } ( y | x ) } ] + \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) } & { } \\ { = \underset { x } { \mathbb { E } } [ \ \underset { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } { \sum } \pi ( y | x ) ] + \underset { x } { \mathbb { E } } [ \ \underset { y \in \mathcal { U } ( x , \pi _ { 0 } ) } { \sum } \pi ( y | x ) ] } & { } \\ { = \underset { x } { \mathbb { E } } [ \underset { y \in \mathcal { Y } } { \sum } \pi ( y | x ) ] = 1 } & { } \end{array} +$$ + +The first equality is based on definition of $S _ { D } ( \pi | \pi _ { 0 } )$ and the second equality is based on definition of support divergence. □ + +# A.4 PROOF OF PROPOSITION 3 + +Proposition 3. For any given $\kappa \in ( 0 , 1 )$ , $0 < \epsilon < \kappa / 2$ , let $p _ { m i n }$ denote the minimum propensity under supported set $p _ { m i n } ~ = ~ m a x _ { x , y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi _ { 0 } ( y | x )$ , then with probability larger than $1 - 2 \exp ( - 2 n \epsilon ^ { 2 } p _ { m i n } ^ { 2 } )$ , the constraint $1 - \kappa { + } \epsilon \le S _ { \mathscr { D } } ( \pi | \pi _ { 0 } ) \le 1 { - } \epsilon$ will ensure $0 \leq \mathcal { D } \boldsymbol { x } ( \pi | \pi _ { 0 } ) \leq \kappa$ . + +Proof. Recall SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) with $( x _ { i } , y _ { i } )$ draw i.i.d from $P ( \mathcal { X } ) \times \pi _ { 0 } ( \mathcal { Y } | \boldsymbol { x } )$ . From Appendix A.3, it is easy to see Ex,y∼π0(·|x)[ π(y|x)π0(y|x) ] $\begin{array} { r } { \mathbb { E } _ { x , y \sim \pi _ { 0 } ( \cdot | x ) } \big [ \frac { \pi ( y | x ) } { \pi _ { 0 } ( y | x ) } \big ] = 1 - \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) } \end{array}$ . Let $p _ { m i n }$ denote the smallest propensity under supported action set, $p _ { m i n } : = \mathrm { m i n } _ { x , y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi _ { 0 } ( y | x ) > 0$ , then the random variable $\frac { \pi ( y | x ) } { \pi _ { 0 } ( y | x ) }$ is strictly bounded between $[ 0 , \frac { 1 } { p _ { m i n } } ]$ . Applying Hoeffding’s bound gives: + +$$ +^ { \mathfrak { p } } ( { \mathcal { D } } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) < 1 - S _ { { \mathcal { D } } } ( \pi | \pi _ { 0 } ) - \epsilon ) = \mathbb { P } ( S _ { { \mathcal { D } } } ( \pi | \pi _ { 0 } ) - ( 1 - { \mathcal { D } } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) ) < - \epsilon ) \leq e x p ( - 2 n \epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) +$$ + +Since $S _ { \mathcal { D } ( \pi | \pi _ { 0 } ) } \le 1 - \epsilon$ gives $1 - S _ { \mathcal { D } } ( \pi \vert \pi _ { 0 } ) - \epsilon \geq 0$ , then we have + +$$ +\mathbb { P } ( \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) < 0 ) \le e x p ( - 2 n \epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) +$$ + +Similar for the other direction, Hoeffding’s bound gives: + +$$ +\mathbb { P } ( \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) > 1 - S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) + \epsilon ) = \mathbb { P } ( S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) - ( 1 - \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) ) > \epsilon ) \le e x p ( - 2 n \epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) +$$ + +Since $S _ { \mathcal { D } ( \pi | \pi _ { 0 } ) } \geq 1 + \epsilon - \kappa$ gives $\begin{array} { r } { 1 - S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) + \epsilon \le \kappa } \end{array}$ , then we have + +$$ +\begin{array} { r } { \mathbb { P } ( \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) \ge \kappa ) \le e x p ( - 2 n \epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \end{array} +$$ + +Combining the above, we have + +$$ +\begin{array} { r l } & { \mathbb { P } ( 0 \leq \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) \leq \kappa ) = 1 - \mathbb { P } ( \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) < 0 ) - \mathbb { P } ( \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } ) > \kappa ) } \\ & { \qquad \geq 1 - 2 e x p ( - 2 n \epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \end{array} +$$ + +# A.5 PROOF FOR EFFICIENT APPROXIMATION + +Claim 1. The empirical risk defined by in Equation (8) has the same expectation (over randomness in $\mathcal { D }$ and sampling) as $\hat { R } _ { I P S } ^ { \delta } ( \mathcal { D } )$ . + +Proof. Taking the expectation of empirical risk defined in Equation (8): + +$$ +\begin{array} { l l } { { \displaystyle { \mathbb E } \big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi \big ( y _ { i } | x _ { i } \big ) } { \pi _ { 0 } \big ( y _ { i } | x _ { i } \big ) } r _ { i } + \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \frac { \pi \big ( y _ { j } | x _ { j } \big ) } { p _ { j } } \hat { \delta } ( x _ { j } , y _ { j } ) \big ] } } \\ { { \displaystyle { = \mathbb E } \big [ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi _ { 0 } ( y | x ) \frac { \pi \big ( y | x \big ) } { \pi _ { 0 } ( y | x ) } \delta ( x , y ) \big ] + \frac { { \mathbb E } } { x } \big [ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \frac { 1 } { | \mathcal { U } ( x , \pi _ { 0 } ) | } \frac { \pi \big ( y | x \big ) } { \textstyle | \mathcal { U } ( x , \pi _ { 0 } ) | } \hat { \delta } ( x , y ) \big ] } } \\ { { \displaystyle { = \mathbb E } \big [ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi ( y | x ) \delta ( x , y ) \big ] + \frac { { \mathbb E } } { x } \big [ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \hat { \delta } ( x , y ) \big ] } } \end{array} +$$ + +Now we will show it has the same expectation with $\hat { R } _ { I P S } ^ { \delta } ( \pi )$ + +$$ +\begin{array} { r l } & { \frac { \mathbb { E } } { D } \big [ \frac { \pi \big ( y _ { 1 } | x _ { i } \big ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \displaystyle \sum _ { y \in \mathcal { U } ( x _ { \star } , \pi _ { 0 } ) } \pi ( y | x _ { i } ) \hat { \delta } ( x _ { i } , y ) \big ] } \\ & { = \frac { \mathbb { E } } { x } \big [ \underbrace { \mathbb { E } } _ { y \in \pi _ { 0 } } [ \frac { \pi } { \pi _ { 0 } ( y | x ) } \delta ( x , y ) ] + \displaystyle \sum _ { y ^ { \prime } \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y ^ { \prime } | x ) \hat { \delta } ( x , y ^ { \prime } ) \big ] } \\ & { = \frac { \mathbb { E } } { x } \big [ \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi _ { 0 } ( y | x ) \frac { \pi ( y | x ) } { \pi _ { 0 } ( y | x ) } \delta ( x , y ) + \displaystyle \sum _ { y ^ { \prime } \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y ^ { \prime } | x ) \hat { \delta } ( x , y ^ { \prime } ) \big ] } \\ & { = \frac { \mathbb { E } } { x } \big [ \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi ( y | x ) \delta ( x , y ) \big ] + \frac { \mathbb { E } } { x } \big [ \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \hat { \delta } ( x , y ) \big ] } \end{array} +$$ + +The proof is done by comparing Equation (23) and Equation (24). + +# A.6 PROOF OF PROPOSITION 2 + +Proposition 2. Given contexts $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ drawn i.i.d from the unknown distribution $P ( \mathcal X )$ , for action $y _ { i }$ drawn independently from logging policy $\pi _ { 0 }$ with probability $\pi _ { 0 } ( \mathcal { V } | x _ { i } )$ , the bias of the empirical risk defined in Equation (7) is $\begin{array} { r } { \mathbb E _ { x } [ \sum _ { y \in \mathcal U _ { x } ^ { \pi _ { 0 } } } \pi ( y | x ) \Delta ( x , y ) ] } \end{array}$ . + +Proof. From Appendix A.5, we are given the expectation of $\hat { R } _ { I P S } ^ { \delta } ( \pi )$ , and the bias is: + +$$ +\begin{array} { l } { { b i a s ( \hat { R } _ { I P S } ^ { \delta } ( \pi ) ) = { \displaystyle \mathbb { E } } [ \hat { R } _ { I P S } ^ { \delta } ( \pi ) ] - R ( \pi ) } } \\ { ~ = { \displaystyle \frac { { \mathbb { E } } } { x } } \big [ \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi ( y | x ) \delta ( x , y ) + \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \hat { \delta } ( x , y ) \big ] - R ( \pi ) } \\ { ~ = { \displaystyle \mathbb { E } } \big [ \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) ( \hat { \delta } ( x , y ) - \delta ( x , y ) ) \big ] } \\ { ~ = { \displaystyle \mathbb { E } } \big [ \displaystyle \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \Delta ( x , y ) \big ] } \end{array} +$$ + +The second equality is from Appendix A.5, the second equality is based on $\begin{array} { r l } { R ( \pi ) } & { { } = } \end{array}$ $\begin{array} { r } { \mathbb { E } _ { x } \left[ \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) ^ { c } } \pi ( y | x ) \delta ( x , y ) + \sum _ { y \in \mathcal { U } ( x , \pi _ { 0 } ) } \pi ( y | x ) \delta ( x , y ) \right] } \end{array}$ , and the last one is based on the definition of $\Delta ( x , y ) : = \hat { \delta } ( x , y ) - \delta ( x , y )$ for all $x \in \mathcal { X } , y \in \mathcal { Y }$ . □ + +# B APPENDIX: EXPERIMENTS + +In this section, we provide the experiment details and additional results to help promote reproducibility of this work. + +# B.1 EXPERIMENT SETUP DETAILS + +Datasets and baseline. We follow a 75:10:15 train-validation-test split for credit card fraud detection dataset, while for CIFAR10 already coming with a train-test split, we keep $10 \%$ of the training set as validation set. Baseline estimators are IPS and DM, the hyperparameters (learning rate, L2 regularization) are optimized for all the methods based on the validation set. + +Bandit data generation. For CIFAR10, given supervised data $\{ x _ { i } , y _ { i } ^ { * } \} _ { i = 1 } ^ { n }$ where $x _ { i }$ denotes the 3072 features and $y _ { i } ^ { * }$ denotes the correct label of data (ranging from 0 to 9), under logging policy $\pi _ { 0 }$ , the logged bandit data is generated by drawing $y _ { i } \sim \pi _ { 0 } ( \mathcal { V } | x _ { i } )$ , then a deterministic reward is defined as $\mathbb { I } _ { \{ y _ { i } = y _ { i } ^ { * } \} }$ . For the credit card fraud detection dataset, we throw away the class label and only use the features for each sample to generate bandit data. To be specific, for each sample with a 28-dimensional feature vector, we define the first 20 features as the contextual information, and use the remaining 8 features as the underlying true reward for 8 different actions (with normalization). + +Logging policy. For CIFAR, we learn the softmax logging policy on 35K full-information data points as a multi-class classification problem with cross-entropy loss. Similar as the experiments on BanditNet (Joachims et al., 2018), we adopt the conventional ResNet20 architecture but restrict training after a mere two epochs to derive a relative stochastic policy, since it will be easier to add temperature later to control its stochasticity and support deficiency. Similarly, for the credit card fraud detection dataset, the softmax logging policy is learned on 8K full-information data points by treating it as a multi-class classification problem using cross-entropy loss and the label being the action with the highest reward on this specific context. For CIFAR, the logging policy we trained has a $5 7 . 4 3 \%$ accuracy on the test-set; whereas for the credit card fraud detection dataset, the logging policy has an expected true reward of 0.71. + +Reward estimator. For each experiment, we train a different regression function using the full bandit dataset. We use the same architecture as the one used for off-policy learning - where the final layer is the size of the actions, specifying the reward for each action given a particular context. The regression function is trained using the MSE objective. + +![](images/51d656734af5ca7eebec76bcca0d315e378aa7a4b913d38454b7164ec98ffb1b.jpg) +Figure 4: Behaviour of $S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) + \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } )$ + +# B.2 SUPPLEMENTARY RESULTS + +Reliability of approximating support divergence using control variate. In this experiment, we empirically verify the reliability of estimating support divergence using control variate. The target policy is the uniform policy while the logging policies are varying in their support deficiency. Results are averaged over 10 runs and is shown in Figure 4. We investigate the behaviour of $S _ { \mathcal { D } } ( \pi | \pi _ { 0 } ) + \mathcal { D } _ { \mathcal { X } } ( \pi | \pi _ { 0 } )$ under different number of training data and different support deficiency of the corresponding logging policy. As we can see, the sum converges to 1 as the training data increases. Meanwhile, the variance decreases as shown in the right figure. For different support deficiency curves, the curve converges in a similar fashion and we conjecture it is due to the effect of clipping the propensity at the same threshold $\epsilon = 0 . 0 1$ , which makes $p _ { m i n } = 0 . 0 1$ in the bound shown in Proposition 3. + +![](images/2678c7d2bb519d2032262fa6ef15c3ce8481c96bf236def1f54a54d5ca71d2c6.jpg) +Figure 5: Model selection result for credit card fraud detection + +![](images/217e996b67f6df7a1b4bf8f2cf57d77113d0632e263176148aa1a303192a33e2.jpg) + +Model selection comparison for the credit card dataset. The model selection comparison over the credit card fraud detection dataset is demonstrated in Figure 5. Similar as the trend in Figure 3, SNIPS and Conservative Extrapolation exhibit a large bias, also SNIPS even can not reflect the shape of the Oracle curve. DM and Regression Extrapolation closely track the Oracle line, and they have the best performance when used in model selection, as seen in the left table of Figure 5. \ No newline at end of file diff --git a/parse/train/SklcyJBtvB/SklcyJBtvB_content_list.json b/parse/train/SklcyJBtvB/SklcyJBtvB_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d0c125d31667c4ff7ba024273d174a78d538c157 --- /dev/null +++ b/parse/train/SklcyJBtvB/SklcyJBtvB_content_list.json @@ -0,0 +1,2030 @@ +[ + { + "type": "text", + "text": "OFF-POLICY BANDITS WITH DEFICIENT SUPPORT ", + "text_level": 1, + "bbox": [ + 176, + 98, + 766, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 145, + 398, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 210, + 544, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Off-policy training of contextual-bandit policies is attractive in online systems (e.g. search, recommendation, ad placement), since it enables the reuse of large amounts of log data. State-of-the-art methods for off-policy learning, however, are based on inverse propensity score (IPS) weighting, which requires that the logging policy chooses all actions with non-zero probability for any context (i.e., full support). In real-world systems, this condition is often violated, and we show that existing off-policy learning methods based on IPS weighting can fail catastrophically. We therefore develop new off-policy contextual-bandit methods that can controllably and robustly learn even when the logging policy has deficient support. To this effect, we explore three approaches that provide various guarantees for safe learning despite the inherent limitations of support deficient data: restricting the action space, reward extrapolation, and restricting the policy space. We analyze the statistical and computational properties of these three approaches, and empirically evaluate their effectiveness in a series of experiments. We find that controlling the policy space is both computationally efficient and that it robustly leads to accurate policies. ", + "bbox": [ + 233, + 241, + 764, + 463 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 489, + 336, + 505 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many interactive systems (e.g., voice assistants, recommender systems, ad placement) can be modeled as contextual bandit problems (Langford & Zhang, 2008). In particular, each user request provides a context (e.g., user profile, query) for which the system selects an action (e.g., recommended product, presented ad) and receives a reward (e.g., purchase, click). Such contextual-bandit data is logged in large quantities as a by-product of normal system operation (Li et al., 2011; 2015; Joachims et al., 2017), making it an attractive and low-cost source of training data. With terabytes of such log data readily available in many online systems, a range of algorithms have been proposed for batch learning from such logged contextual-bandit feedback (Strehl et al., 2011; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019; London & Sandler, 2019). However, as we will argue below, these algorithms require an assumption about the log data that makes them unsuitable for many real-world applications. ", + "bbox": [ + 174, + 520, + 825, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This assumption is typically referred to as the positivity or support assumption, and it is required by the Empirical Risk Minimization (ERM) objective that these algorithms optimize. Specifically, unlike in online learning for contextual bandits (Williams, 1992; Agarwal et al., 2014), batch learning from bandit feedback (BLBF) operates in the off-policy setting. During off-policy learning, the algorithm has to address the counterfactual question of how much reward each policy in the policy space would have received, if it had been used instead of the logging policy. To this effect, virtually all state-of-the-art off-policy learning methods for contextual-bandit problems rely on counterfactual estimators (Bottou et al., 2013; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019) that employ inverse propensity score (IPS) weighting to get an unbiased ERM objective. Unlike regression-based direct-modeling (DM) approaches that are often hampered by bias from model-misspecification, IPS allows a controllable bias-variance trade-off through clipping and other variance-regularization techniques (Strehl et al., 2011; Swaminathan & Joachims, 2015a; London & Sandler, 2019). ", + "bbox": [ + 174, + 681, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Unfortunately, IPS and its variance-control mechanisms break down when the logging policy does not have full support – meaning that some actions have zero probability of being selected under the logging policy. In this case IPS can be highly biased. Full support is an unreasonable assumption in many real-world systems, especially when the action space is large and many actions have poor rewards. For example, in a recommender system with a large catalog (e.g. movies, music), it may be that less than $10 \\%$ of the actions have support under the logging policy. We will show that existing learning algorithms can fail catastrophically on such support deficient data. ", + "bbox": [ + 174, + 868, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we develop new off-policy contextual-bandit algorithms that are specifically designed to deal with support deficient log data. Since support deficiency translates into blind spots where we do not have any knowledge about the rewards, accounting for these blind spots as part of learning is crucial for robust learning. We approach this problem from three perspectives. First, we explore restricting the action space to those actions that have support under the logging policy. Second, we explore imputation methods that extrapolate estimated rewards to those blind spots. And, third, we restrict the policy space to only those policies that have limited exposure to the blind spots. To make the latter approach computationally tractable, we define a new measure of Support Divergence between policies, show how it can be estimated efficiently without closed-form knowledge of the logging policy, and how it can be used as a constraint on the policy space. We analyze the statistical and computational properties of all three approaches and perform an extensive empirical evaluation. We find that restricting the policy space is particularly effective, since it is computationally efficient, empirically effective at learning good policies, and convenient to use in practice. ", + "bbox": [ + 174, + 152, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 352, + 344, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Most prior works on BLBF can be classified into two different approaches. The first – called Direct Model (DM) – is based on a reduction to supervised learning, where a regression estimate is trained to predict rewards (Beygelzimer & Langford, 2009). To derive a policy, the action with the highest predicted reward is chosen. A drawback of this simple approach is the bias that results from misspecification of the regression model. Since regression models are often substantially misspecified for real-world data, the DM approach often does not work well empirically. ", + "bbox": [ + 174, + 382, + 823, + 467 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The second approach is based on policy learning via ERM with a counterfactual risk estimator. Inverse propensity score (IPS) weighting is one of the most popular estimators to be used as empirical risk. However, policy learning algorithms based on IPS and related estimators (Strehl et al., 2011; Swaminathan & Joachims, 2015a;b; Thomas & Brunskill, 2016; London & Sandler, 2019) require the assumption that the logging policy has full support for every policy in the policy space. One exception is the work of Liu et al. (2019). They relax the assumption to the existence of an optimal policy such that the logging policy covers the support of this optimal policy. However, this is an untestable assumption that does not provide guarantees for real-world applications. ", + "bbox": [ + 174, + 473, + 825, + 585 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work proposes three approaches to addressing off-policy learning with support deficiency. First, our conservative extrapolation method is related to the method proposed by Liu et al. (2019). They focus on the correction of the state distribution by defining an augmented MDP, and pessimistic imputation is used to get an estimate for policy-gradient learning. Second, our method of restricting the policy space uses a surrogate for the support divergence of two policies that was previously used as control variate in the SNIPS estimator (Swaminathan & Joachims, 2015b). It also appeared in the Lagrangian formulation of the BanditNet objective (Joachims et al., 2018) and in the gradient update in REINFORCE algorithm (Williams, 1992). This connection gives interesting new insight that the baselines used in policy-gradient algorithms not only help to reduce variance in gradients (Greensmith et al., 2004), but that they also connect to the problem of support deficiency in the off-policy setting. ", + "bbox": [ + 174, + 592, + 825, + 744 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 OFF-POLICY LEARNING WITH DEFICIENT SUPPORT ", + "text_level": 1, + "bbox": [ + 174, + 763, + 632, + 781 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We start by formally defining the problem of learning a contextual-bandit policy in the BLBF setting. Input to the policy are contexts $x \\in \\mathcal { X }$ drawn i.i.d from a fixed but unknown distribution $P ( \\mathcal X )$ . Given context $x$ , the system executes a possibly stochastic policy $\\pi ( \\mathcal { V } | x )$ that selects an action $y \\in \\mathcal { V }$ . For this context and action pair, the system observes a reward $r \\in [ r _ { m i n } , r _ { m a x } ]$ from $P ( r | x , y )$ . Given a space of policies $\\Pi$ , the reward of any policy $\\pi \\in \\Pi$ is defined as ", + "bbox": [ + 174, + 794, + 823, + 864 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/cc844e68665c2e4ffa8ae299fddc115cb87be6cf868f100649cc10522fe27df8.jpg", + "text": "$$\nR ( \\pi ) = \\underset { x } { \\mathbb { E } } \\underset { y \\sim \\pi ( y \\mid x ) } { \\mathbb { E } } \\underset { r \\sim P ( r \\mid x , y ) } { \\mathbb { E } } [ r ] .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 866, + 606, + 891 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the BLBF setting, the learning algorithm is given a dataset ", + "bbox": [ + 173, + 892, + 575, + 907 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/cc917c182d58be11a6073714fe38a53260f193e260a27f7d1212c8d3a05b988e.jpg", + "text": "$$\n\\mathcal { D } : = \\{ x _ { i } , y _ { i } , r _ { i } , \\pi _ { 0 } ( y _ { i } | x _ { i } ) \\} _ { i = 1 } ^ { n }\n$$", + "text_format": "latex", + "bbox": [ + 395, + 909, + 601, + 925 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "of past system interactions which consists of context-action-reward-propensity tuples. The propensity $\\pi _ { 0 } ( y _ { i } | x _ { i } )$ is the probability of selecting action $y _ { i }$ for context $x _ { i }$ under the policy $\\pi _ { 0 }$ that was used to log the data. We call $\\pi _ { 0 }$ the logging policy, and we will discuss desired conditions on the stochasticity of $\\pi _ { 0 }$ in the following. The goal of off-policy learning is to exploit the information in the logged data $\\mathcal { D }$ to find a policy ${ \\hat { \\pi } } \\in \\Pi$ that has high reward $R ( { \\hat { \\pi } } )$ . ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Analogous to the ERM principle in supervised learning, off-policy learning algorithms typically optimize a counterfactual estimate $\\hat { R } ( \\pi )$ of $R ( \\pi )$ as the training objective (Li et al., 2011; 2015; Bottou et al., 2013; Swaminathan & Joachims, 2015a). ", + "bbox": [ + 174, + 179, + 825, + 224 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e3b58cd496cef3671c86be4c1e71054edc436e42aee1fe5d47e0ebeade5554d8.jpg", + "text": "$$\n\\hat { \\pi } = \\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m a x } } [ \\hat { R } ( \\pi ) ]\n$$", + "text_format": "latex", + "bbox": [ + 429, + 228, + 566, + 256 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For conciseness, we ignore additional regularization terms in the objective (Swaminathan & Joachims, 2015a), since they are irrelevant to the main point of this paper. As counterfactual estimator $\\hat { R } ( \\pi )$ , most algorithms rely on some form of IPS weighting (Strehl et al., 2011; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a;b; Wang et al., 2017; Su et al., 2019) to correct the distribution mismatch between the logging policy $\\pi _ { 0 }$ and each target policy $\\pi \\in \\Pi$ . ", + "bbox": [ + 173, + 258, + 825, + 332 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2d8cffe0b06bb469b79cf452f32e68c3c96c63c9caf1a4e0c57f7536520747d8.jpg", + "text": "$$\n\\hat { R } _ { I P S } ( \\pi ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 335, + 604, + 376 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A crucial condition for the effectiveness of the IPS estimator (and similar estimators) is that the logging policy $\\pi _ { 0 }$ assigns non-zero probability to all actions that have non-zero probability under the target policy $\\pi$ we aim to evaluate. This condition is known as positivity or full support, and it is defined as follows. ", + "bbox": [ + 173, + 378, + 825, + 435 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 1 (Full support). The logging policy $\\pi _ { 0 }$ is said to have full support for $\\pi$ when $\\pi _ { 0 } ( y | x ) >$ 0 for all actions $y \\in \\mathcal { V }$ and contexts $x \\in \\mathcal { X }$ for which $\\pi ( y | x ) > 0$ . ", + "bbox": [ + 173, + 438, + 821, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "It is known that the IPS estimator is unbiased, $\\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] = R ( \\pi )$ , if the logging policy $\\pi _ { 0 }$ has full support for $\\pi$ (Li et al., 2011). To ensure unbiased ERM, algorithms that use the IPS estimator require that the logging policy $\\pi _ { 0 }$ has full support for all policies $\\pi \\in \\Pi$ in the policy space. For sufficiently rich policy spaces, like deep-networks $f _ { w } ( x , y )$ with softmax outputs of the form ", + "bbox": [ + 173, + 478, + 825, + 536 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/69fbd4c638d436ea0fd8a8b517589701ae75c671b82e68853958c5316b83b8b8.jpg", + "text": "$$\n\\pi _ { w } ( y | x ) = \\frac { e x p ( f _ { w } ( x , y ) ) } { \\sum _ { y ^ { \\prime } \\in \\mathcal { V } } e x p ( f _ { w } ( x , y ^ { \\prime } ) ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 379, + 540, + 617, + 577 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "this means that the logging policy $\\pi _ { 0 }$ needs to assign non-zero probability to every action $y$ in every context $x$ . This is a strong condition that is not feasible in many real-world systems, especially if the action space is large and many actions have poor reward. ", + "bbox": [ + 174, + 579, + 825, + 622 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "If the support requirement is violated, ERM learning can fail catastrophically. We will show below that the underlying reason is bias, not excessive variance that could be remedied through clipping or variance regularization (Strehl et al., 2011; Swaminathan $\\&$ Joachims, 2015a). To quantify how support deficient a logging policy is, we denote the set of unsupported actions for context $x$ under $\\pi _ { 0 }$ as ", + "bbox": [ + 173, + 627, + 825, + 698 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/fb7e1f3e0c702a131e358a03a19362fe2932c52e9a64ecb04540a8d59792f036.jpg", + "text": "$$\n\\mathcal { U } ( x , \\pi _ { 0 } ) : = \\{ y \\in y | \\pi _ { 0 } ( y | x ) = 0 \\} .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 696, + 614, + 713 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The bias of the IPS estimator is then characterized by the expected reward on the unsupported actions. ", + "bbox": [ + 173, + 715, + 823, + 743 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proposition 1. Given contexts $x \\sim P ( \\mathcal { X } )$ and logging policy $\\pi _ { 0 } ( \\mathcal { V } | x )$ , the bias of $\\hat { R } _ { I P S }$ for target policy $\\pi ( \\mathcal { V } | x )$ is equal to the expected reward on the unsupported action sets, i.e., $b i a s ( \\pi | \\pi _ { 0 } ) =$ $\\begin{array} { r } { \\mathbb { E } _ { x } [ - \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}$ . ", + "bbox": [ + 174, + 746, + 825, + 791 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The proof is in Appendix A.1. From Proposition 1, it is clear that support deficient log data can drastically mislead ERM learning. To quantify the effect of support deficiency on ERM, we define the support divergence between a logging policy $\\pi _ { 0 }$ and a target policy $\\pi$ as follows. ", + "bbox": [ + 173, + 801, + 825, + 844 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2 (Support Divergence). For contexts $x \\sim P ( \\mathcal { X } )$ and any corresponding pair of target policy $\\pi$ and logging policy $\\pi _ { 0 }$ , the Support Divergence is defined as ", + "bbox": [ + 173, + 847, + 825, + 876 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e6411aa184ef6ad7a01d465cc6e94f57c67796db9adc502772ec13ed13ba43a9.jpg", + "text": "$$\n\\mathcal { D } _ { \\mathcal { X } } \\big ( \\pi \\big | \\pi _ { 0 } \\big ) : = \\underset { x \\sim P ( \\mathcal { X } ) } { \\mathbb { E } } \\left[ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 878, + 643, + 929 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "With this definition in hand, we can quantify the effect of support deficiency on ERM learning for a policy space $\\Pi$ under logging policy $\\pi _ { 0 }$ . ", + "bbox": [ + 174, + 103, + 823, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1. For any given hypothesis space $\\Pi$ with logging policy $\\pi _ { 0 } ~ \\in ~ \\Pi$ , there exists a reward distribution $\\mathcal { P } _ { r }$ with support in $[ r _ { m i n } , r _ { m a x } ]$ such that in the limit of infinite training data, ERM using IPS over the logged data $\\mathcal D \\sim \\dot { P } ( \\mathcal X ) \\times \\pi _ { 0 } ( \\cdot | \\mathcal X ) \\times \\mathcal P _ { r }$ can select a policy $\\begin{array} { r } { \\hat { \\pi } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] } \\end{array}$ that is at least $\\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\quad } & { { } } \\end{array}$ suboptimal. ", + "bbox": [ + 174, + 136, + 825, + 195 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The proof is in Appendix A.2. To illustrate the theorem, consider a problem with rewards $r \\in$ $[ - 1 , 0 ]$ . Furthermore, consider a policy space $\\Pi$ that contains a good policy $\\pi _ { g }$ with $R ( \\pi _ { g } ) = - 0 . 1$ and a bad policy $\\pi _ { b }$ with $R ( \\pi _ { b } ) = - 0 . 7 .$ . If policy $\\pi _ { b }$ has support divergence $\\mathcal { D } _ { \\mathcal { X } } ( \\pi _ { b } \\vert \\pi _ { 0 } ) = 0 . 6$ or larger, then ERM may return the bad $\\pi _ { b }$ instead of $\\pi _ { g }$ even with infinite amounts of training data. ", + "bbox": [ + 174, + 204, + 825, + 262 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that it is sufficient to merely have one policy in $\\Pi$ that has large support deficiency to achieve this suboptimality. It is therefore crucial to control the support divergence $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )$ uniformly over all $\\pi \\in \\Pi$ , or to account for the suboptimality it can induce. To this effect, we explore three approaches in the following. ", + "bbox": [ + 173, + 267, + 825, + 325 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 SAFE LEARNING BY RESTRICTING THE ACTION SPACE ", + "text_level": 1, + "bbox": [ + 174, + 342, + 584, + 356 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The first and arguably most direct approach to reducing $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )$ is to disallow any action that has zero support under the logging policy. For the remaining action set, the logging policy has full support by definition. This restriction of the action set can be achieved by transforming each policy $\\pi \\in \\Pi$ into a new policy that sets the probability of the unsupported actions to zero. ", + "bbox": [ + 173, + 367, + 825, + 422 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9ee516459c258d49f7603e0d84225a234f07343a79f1f2a4c77f9ab3ab81133c.jpg", + "text": "$$\n\\pi ( y | x ) \\longrightarrow \\bar { \\pi } ( y | x ) : = \\frac { \\pi ( y | x ) \\mathbb { 1 } _ { \\{ y \\notin \\mathcal { U } ( x , \\pi _ { 0 } ) \\} } } { 1 - \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) }\n$$", + "text_format": "latex", + "bbox": [ + 338, + 428, + 661, + 467 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This results in a new policy space $\\bar { \\Pi }$ . All $\\bar { \\pi } \\in \\bar { \\Pi }$ have support divergence of zero $\\mathcal { D } _ { \\mathcal { X } } ( \\bar { \\pi } | \\pi _ { 0 } ) = 0$ and ERM via IPS is guaranteed to be unbiased. ", + "bbox": [ + 176, + 470, + 823, + 501 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "While this transformation of the policy space from $\\Pi$ to $\\bar { \\Pi }$ is conceptually straightforward, it has two potential drawbacks. First, restricting the action space without any exceptions may overly constrain the policies in $\\bar { \\Pi }$ . In particular, if the optimal action $y ^ { * }$ for a specific context $x$ does not have support under the logging policy, no $\\bar { \\pi } \\in \\dot { \\bar { \\Pi } }$ can ever choose $y ^ { * }$ even if there are many observations of similar $y$ ’s on similar context $x ^ { \\prime }$ . The second drawback is computational. For every context $x$ during training and testing, the system needs to evaluate the logging policy $\\pi _ { 0 } ( y | x )$ to compute the transformation from $\\pi$ to $\\bar { \\pi }$ . This can be prohibitively expensive especially at test time, where – after multiple rounds of off-policy learning with data from previously learned policies – we would need to evaluate the whole sequence of previous logging policies to execute the learned policy. ", + "bbox": [ + 173, + 506, + 825, + 633 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 SAFE LEARNING THROUGH REWARD EXTRAPOLATION", + "text_level": 1, + "bbox": [ + 174, + 650, + 588, + 664 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As illustrated above, support deficiency is a problem of blind spots where we lack information about the rewards of some actions in some contexts. Instead of disallowing the unsupported actions like in the previous section, an alternative is to extrapolate the observed rewards to fill in the blind spots. To this effect, we propose the following augmented IPS estimator that imputes an extrapolated reward $\\hat { \\delta } ( x , y )$ for each unsupported action $y \\in \\mathcal { U } ( x , \\pi _ { 0 } )$ . ", + "bbox": [ + 173, + 674, + 825, + 748 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/59c8e06f86b474751f538130c117f5249d38be5fc781dcfbb05d3dd759cb1464.jpg", + "text": "$$\n\\hat { R } _ { I P S } ^ { \\delta } ( \\pi ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left[ \\frac { \\pi ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \\sum _ { y \\in \\mathcal { U } ( x _ { i } , \\pi _ { 0 } ) } \\pi ( y | x _ { i } ) \\hat { \\delta } ( x _ { i } , y ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 289, + 753, + 707, + 803 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the following proposition, we characterize the bias of the augmented IPS estimator for any given reward extrapolation $\\hat { \\delta } ( x , y )$ . We denote the mean of the reward $r$ for context $x$ and action $y$ with $\\delta ( x , y ) = \\mathbb { E } _ { r \\sim P ( r | x , y ) } [ r ]$ . Furthermore, $\\Delta ( x , y ) : = \\hat { \\delta } ( x , y ) - \\delta ( x , y )$ denotes the error of the reward extrapolation for each $x$ and $y$ . ", + "bbox": [ + 173, + 806, + 825, + 868 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proposition 2. Given contexts $x _ { 1 } , x _ { 2 } , \\ldots , x _ { n }$ drawn i.i.d from the unknown distribution $P ( \\mathcal X )$ , for action $y _ { i }$ drawn independently from logging policy $\\pi _ { 0 }$ with probability $\\pi _ { 0 } ( \\mathcal { V } | x _ { i } )$ , the bias of the empirical risk defined in Equation (7) is $\\begin{array} { r } { \\mathbb E _ { x } [ \\sum _ { y \\in \\mathcal U _ { x } ^ { \\pi _ { 0 } } } \\pi ( y | x ) \\Delta ( x , y ) ] } \\end{array}$ . ", + "bbox": [ + 173, + 872, + 825, + 917 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this way we can learn in the original action and policy space, but mitigate the effect of the support deficiency by explicitly incorporating the extrapolated reward $\\hat { \\delta } ( x , y )$ . We explore two choices for $\\hat { \\delta } ( x , y )$ in the following, which provide different types of guarantees. ", + "bbox": [ + 174, + 103, + 483, + 193 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Conservative Extrapolation. To minimize the user impact of randomization in the logging policy, it is generally desirable to put zero probability on actions the are very likely to have low (or even catastrophic reward). This means that precisely those bad actions are likely to not be supported in the logging policy. A key danger of blind spots regarding those actions is that naive IPS training will inadvertently learn ", + "bbox": [ + 174, + 199, + 483, + 324 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1: Data Augmentation ", + "text_level": 1, + "bbox": [ + 500, + 108, + 723, + 123 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "input: original logged dataset $\\mathcal { D }$ , replaycount $k$ , \nreward estimate $\\hat { \\delta } ( x , y )$ ; output: $\\mathcal { D } ^ { \\prime }$ ; \ninitialization: $\\mathcal { D } ^ { \\prime } = \\varnothing$ ; \nfor $j = 1 , \\dots , k$ do for $i = 1 , \\ldots , n$ do Define $U _ { x _ { i } }$ to be the uniform distribution over $\\mathcal { U } ( x _ { i } , \\pi _ { 0 } )$ ; Draw $y \\sim U _ { x _ { i } }$ ; $\\begin{array} { r } { \\mathcal { D } ^ { \\prime } = \\mathcal { D } ^ { \\prime } \\bigcup \\{ x _ { i } , y , \\hat { \\delta } ( x _ { i } , y ) , \\frac { 1 } { | \\mathcal { U } ( x _ { i } , \\pi _ { 0 } ) | } \\} ; } \\end{array}$ end \nend ", + "bbox": [ + 498, + 126, + 818, + 289 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "a policy that selects those actions. This can be avoided by being maximally conservative about unsupported actions and imputing the lowest possible reward $\\forall x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) : \\hat { \\delta } ( x , y ) = r _ { m i n }$ . Intuitively, by imposing the worst possible reward for the unsupported actions, the learning algorithm will aim to avoid these low-reward areas. However, unlike for the $\\bar { \\pi }$ policies resulting from the restricted action space, the learned policy is not strictly prohibited from choosing unsupported actions – it is merely made aware of the maximum loss that the action may incur. Note that for problems where $r _ { m i n } = 0$ , the naive IPS estimator is identical to conservative extrapolation since the second term in Equation (7) is zero. ", + "bbox": [ + 173, + 324, + 825, + 436 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Regression Extrapolation. Instead of extrapolating with the worst-case reward, we may have additional prior knowledge in the form of a model-based estimate that reduces the bias. In particular, we explore using a regression estimate $\\begin{array} { r } { \\hat { \\delta } = \\arg \\operatorname* { m i n } _ { \\hat { \\delta } ^ { \\theta } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( \\hat { \\delta } ^ { \\theta } ( x _ { i } , y _ { i } ) - r _ { i } ) ^ { 2 } } \\end{array}$ that extrapolates from the observed data $\\mathcal { D }$ . Typically, comes from a parameterized class of regression functions. Other regression objectives could also be used, such as weighted linear regression that itself uses importance sampling as weights (Farajtabar et al., 2018). But, fundamentally, all regression approaches assume that the regression model is not misspecified and that it can thus extrapolate well. Note that the IPS part of Equation (7) can be changed to any estimators (with action set restricted on $\\boldsymbol { \\mathcal { U } } ( \\boldsymbol { x } , \\pi _ { 0 } ) ^ { c }$ for all $x$ ), and it turns out that doubly robust (Dud´ık et al., 2011) and CAB (Su et al., 2019) are special extensions of regression extrapolation that substitute the IPS part with their corresponding estimator. ", + "bbox": [ + 173, + 443, + 825, + 588 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Efficient Approximation. Evaluating the augmented IPS estimator from Equation (7) can be computationally expensive if the number of unsupported actions $\\boldsymbol { \\mathcal { U } } ( \\boldsymbol { x } , \\pi _ { 0 } )$ is large. To overcome this problem, we propose to use sampling to estimate the expected reward on the unsupported action, which can be thought of as augmenting the dataset $\\mathcal { D }$ with additional observations where the logging policy has zero support. In particular, we propose the data-augmentation procedure detailed in Algorithm 1. With the additional bandit data $\\mathcal { D } ^ { \\prime } = \\{ x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } , \\hat { \\delta } ( x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } ) , p _ { j } ^ { \\prime } \\} _ { j = 1 } ^ { m }$ from Algorithm 1, the new objective is ", + "bbox": [ + 173, + 594, + 825, + 695 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7cda3245e8cd39208c55d189a1f9e57ac5baf4ab6bced57093d53c8e79fd389e.jpg", + "text": "$$\n\\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m i n } } \\left\\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\frac { \\pi ( y _ { j } ^ { \\prime } | x _ { j } ^ { \\prime } ) } { p _ { j } ^ { \\prime } } \\hat { \\delta } ( x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } ) \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 696, + 702, + 746 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Appendix A.5, we show that the empirical risk in Equation (8) has the same expectation (over randomness in $\\mathcal { D }$ and $\\mathcal { D } ^ { \\prime }$ ) as $\\hat { R } _ { I P S } ^ { \\delta } ( \\mathcal { D } )$ and can thus serve as an approximation for Equation (7). ", + "bbox": [ + 171, + 747, + 828, + 779 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 SAFE LEARNING BY RESTRICTING THE POLICY SPACE ", + "text_level": 1, + "bbox": [ + 173, + 792, + 593, + 808 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As motivated by Theorem 1, the risk of learning from support deficient data scales with the maximum support divergence $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )$ among the policies in the policy space $\\Pi$ . Therefore, our third approach restricts the policy space to the subset $\\Pi ^ { \\kappa } \\subset \\Pi$ that contains the policies $\\pi \\in \\Pi$ with an acceptably low support divergence $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } ) \\le \\kappa$ . ", + "bbox": [ + 173, + 819, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2af08dcad4092a6ee5706a0b9b73bd6a07a13b61cf4f5db6fc366cc95d341f5d.jpg", + "text": "$$\n\\Pi ^ { \\kappa } = \\{ \\pi | \\pi \\in \\Pi \\wedge { \\mathscr { D } } \\chi ( \\pi | \\pi _ { 0 } ) \\leq \\kappa \\}\n$$", + "text_format": "latex", + "bbox": [ + 380, + 877, + 617, + 893 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The parameter $\\kappa$ has an intuitive meaning. It specifies the maximum probability mass that a learned policy can place on unsupported actions. By limiting this to $\\kappa$ , we limit the maximum bias of the ERM procedure according to Proposition 2 while not explicitly torquing the rewards like in conservative reward imputation. ", + "bbox": [ + 171, + 896, + 831, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A key challenge, however, is implementing this restriction of the hypothesis space, such that the ERM learner $\\hat { \\pi } = \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi ^ { \\kappa } } [ \\hat { R } _ { I P S } ( \\pi ) ]$ only considers the subset $\\Pi ^ { \\kappa } \\subset \\Pi$ . In particular, we do not have access to the context distribution $P ( \\mathcal X )$ for calculating $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )$ , nor would it be possible to enumerate all $\\pi \\in \\Pi$ to check the condition $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } ) ~ \\le ~ \\kappa$ , which itself requires a possibly infeasible iteration over all actions. The following theorem (with proof in Appendix A.3) gives us an efficient way of estimating and controlling $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )$ without explicit knowledge of $P ( \\mathcal X )$ or access to the logging policy $\\pi _ { 0 }$ beyond the logged propensities. ", + "bbox": [ + 173, + 138, + 825, + 239 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 2. For contexts $x _ { i }$ drawn i.i.d from $P ( \\mathcal X )$ , action $y _ { i }$ drawn from logging policy $\\pi _ { 0 }$ , we define SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) . For any policy $\\pi$ it holds that ", + "bbox": [ + 174, + 242, + 823, + 279 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c5da844e59a6ddd6d7938fe2c532c4e163ff2944041cf4f3ac48292fa0ff6663.jpg", + "text": "$$\n\\begin{array} { r } { \\underset { x \\sim P ( \\mathcal { X } ) } { \\mathbb { E } } \\underset { y \\sim \\pi _ { 0 } ( \\cdot | x ) } { \\mathbb { E } } [ S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) = 1 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 284, + 651, + 310 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Using this theorem, the following proposition (proof in Appendix A.4, empirically verified in Appendix B) gives us an efficient way of implementing the constraint $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } ) \\le \\kappa$ via $1 - S _ { D } ( \\pi | \\pi _ { 0 } )$ . ", + "bbox": [ + 173, + 324, + 825, + 353 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 3. For any given $\\kappa \\in ( 0 , 1 )$ , $0 < \\epsilon < \\kappa / 2$ , let $p _ { m i n }$ denote the minimum propensity under supported set $p _ { m i n } ~ = ~ m a x _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x )$ , then with probability larger than $1 - 2 \\exp ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )$ , the constraint $1 - \\kappa { + } \\epsilon \\le S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) \\le 1 { - } \\epsilon$ will ensure $0 \\leq \\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } ) \\leq \\kappa$ . ", + "bbox": [ + 174, + 369, + 825, + 416 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We can thus use $1 - S _ { D } ( \\pi | \\pi _ { 0 } )$ as a surrogate for $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )$ in the training objective: ", + "bbox": [ + 173, + 426, + 730, + 443 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/97452798e28f5e98c3a9e416c4f2fe8dee5b3da378c902f407fc26b2f2e5dbec.jpg", + "text": "$$\n\\underset { \\pi _ { w } \\in \\Pi } { \\arg \\operatorname* { m i n } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi _ { w } ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } . \\mathrm { ~ s u b j e c t ~ t o ~ } 1 - \\kappa + \\epsilon \\leq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi _ { w } ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } \\leq 1 - \\epsilon\n$$", + "text_format": "latex", + "bbox": [ + 241, + 449, + 758, + 491 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Using Lagrange multipliers, an equivalent dual form of Equation (11) is: ", + "bbox": [ + 174, + 496, + 650, + 511 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/59abd23ac5af8f5a1775769087f8319f75020aa1e291616c69f2f015ee7eb632.jpg", + "text": "$$\n\\operatorname* { m a x } _ { u _ { 1 } , u _ { 2 } \\geq 0 } \\operatorname* { m i n } _ { \\pi _ { w } \\in \\Pi } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi _ { w } ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } ( r _ { i } + u _ { 1 } - u _ { 2 } ) - u _ { 1 } ( 1 - \\epsilon ) + u _ { 2 } ( 1 - \\kappa + \\epsilon )\n$$", + "text_format": "latex", + "bbox": [ + 250, + 517, + 751, + 558 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For each fixed $( u _ { 1 } , u _ { 2 } )$ pair, the inner minimization objective is ERM with IPS under a shift of the reward. Instead of maximizing over $( u _ { 1 } , u _ { 2 } )$ in the outer objective, we treat $( u _ { 1 } - u _ { 2 } )$ as a hyperparameter that we select on a validation set. We explore various estimators for this modelselection problem in Section 4. ", + "bbox": [ + 173, + 564, + 823, + 621 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that, among the methods we proposed for dealing with support deficiency, this approach is the most efficient to implement, and it does not require access to the logging policy during training or testing. Furthermore, the form of the inner objective coincides with that of BanditNet (Joachims et al., 2018), which is known to work well for deep network training by controlling propensity overfitting (Swaminathan & Joachims, 2015a). ", + "bbox": [ + 174, + 627, + 825, + 698 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EMPIRICAL EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 718, + 415, + 734 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We empirically evaluate the effectiveness and robustness of the three proposed approaches: restricting the action space, conservative and regression extrapolation, as well as restricting the policy space. The semi-synthetic experiments are based on two real-world datasets: one is the popular image classification dataset CIFAR10 (Krizhevsky et al.) and the other is the credit-card fraud dataset of Dal Pozzolo et al. (2015). We use the naive IPS estimator and the regression-based Direct Method (DM) as baselines. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The experiments are set up as follows. We first create a train-validation-test split for both datasets. The training set is used to generate bandit datasets for learning, the validation set is used to generate bandit datasets for model selection, and the full-information test set serves as ground truth for evaluating the learned policies. To simulate bandit feedback for the CIFAR10 dataset, our experiment setup follows traditional supervised bandit conversion for multi-class classification datasets (Beygelzimer & Langford, 2009). To not only have bandit data with binary multi-class rewards, we choose a different methodology for the credit-card dataset by designating some features as corresponding to actions and rewards. More details are given in Appendix B. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 821, + 132 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To get logging policies for generating bandit feedback, we start by training a softmax-policy as in Equation (4) on a subset of the full-information data. We then introduce a temperature parameter $\\tau$ into the learned policy via $\\tau f _ { w } ( x , y )$ to be able to control its stochasticity and support deficiency. In particular, we enforce zero support for some actions by clipping the propensities to 0 if they are below a threshold of $\\epsilon = 0 . 0 1$ . The larger $\\tau$ , the higher the support deficiency. Note that making the threshold at $\\epsilon = 0 . 0 1$ allows us to control support while the variance of IPS stays bounded. This allows us to study support deficiency without having to worry about variance control. ", + "bbox": [ + 174, + 138, + 825, + 236 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For both logging and target policies, we train softmax policies where $f _ { w } ( x , y )$ is a neural network. We use the ResNet20 architecture (He et al., 2016) for CIFAR10, and a fully-connected 2-layer network for the credit-card dataset. ", + "bbox": [ + 174, + 243, + 825, + 285 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/28a0537b9d42908064e6c5fb86cdcb47b4f7d8597ca2d9309b1cb944a9c7dace.jpg", + "image_caption": [ + "Figure 1: Learning results with varying support deficiency in the logging policy. " + ], + "image_footnote": [], + "bbox": [ + 200, + 295, + 799, + 517 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "How do the methods perform at different level of support deficiency? Results are shown in Figure 1. First, as expected, learning using naive IPS degrades on both datasets as we make the logging policy more peaked and the number of unsupported actions increases. Note that naive IPS coincides with Conservative Extrapolation, since both datasets are scaled to have a minimum reward of zero. In the rightmost column, however, we translated the rewards to $[ - 1 , 0 ]$ . This has a strong detrimental effect on naive IPS, as it is now overly optimistic about unsupported actions. Second, the approach of dealing with support deficiency by restricting the action space also performs poorly. The second row of plot sheds some light on this, as it shows the support divergence $\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )$ of the learned policy. It is zero for Action Restriction as expected, which means that bias is not the problem. Instead, as the number of unsupported actions increases, the best actions are more likely to be pruned and unavailable in the restricted policy space $\\bar { \\Pi }$ . Third, Regression Extrapolation performs better than Conservative Extrapolation on both datasets. In both cases, the DM model is quite good which also benefits Regression Extrapolation. However, on the credit-card dataset the regression seems better at ranking than at predicting the true reward, which explains why DM performs better than Regression Extrapolation. Fourth, the method that performs well most consistently is Policy Restriction. Unlike all the other IPS-based methods, it performs well even under the translated rewards in the third column of Figure 1. This is because the objective of Policy Restriction coincides with that of BanditNet (Joachims et al., 2018), which is known to remedy propensity overfitting due to the lack of equivariance of the IPS estimator (Swaminathan & Joachims, 2015b). ", + "bbox": [ + 173, + 554, + 825, + 818 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "How does the learning performance change with more training data? Results are shown in Figure 2. As the number of bandit examples increases, Policy Restriction, Regression Extrapolation and DM dominate over most of the range especially when the percentage of unsupported actions is large. Among the other methods, Action Restriction can take the least advantage of more data. This is plausible, since its maximum performance is limited by the available actions. For similar reasons, Conservative Extrapolation (and equivalently IPS) also flattens out, since it also tightly restricts the action space by imputing the minimum reward. ", + "bbox": [ + 173, + 827, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/aa8a5d1278cce3164dc98804179215126bb0ed224e646e59e7747093d8691b15.jpg", + "image_caption": [ + "Figure 2: Learning results with varying amounts of bandit data on CIFAR10 and credit-card dataset. " + ], + "image_footnote": [], + "bbox": [ + 173, + 99, + 823, + 258 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e6ea670f0bc94f091c1f5ceebfda4a0e6a6f8a4223f3aa7556a3edc25a362410.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
%Unsupp.OracleRegr.Extrap.DMCons.Extrap.SNIPS
450.8780.8780.8780.8780.876
600.8710.8710.8710.8710.871
700.8580.8580.8560.8580.858
770.8560.8540.8540.8560.856
800.8550.8550.8550.8380.849
", + "bbox": [ + 176, + 310, + 545, + 406 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/129795f238e4f9e260030195c39b917e0942e1c8abe6287722e9eee2e57b7f77.jpg", + "image_caption": [ + "Figure 3: Model selection performance on CIFAR10. " + ], + "image_footnote": [], + "bbox": [ + 560, + 305, + 782, + 412 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "How effective are the estimators for model selection? Most learning algorithms have hyperparameters, and we now evaluate how the estimators perform for this secondary learning problem. We specifically focus on the parameter $k : = u _ { 1 } - u _ { 2 }$ in Policy Restriction, since it controls how much the learned policies can step outside the region of support. The table on the left of Figure 3 shows the reward of the learned policy when performing model selection with the respective estimator. Oracle is the estimator that has access to the full-information validation set, and can thus be considered as a skyline. We also included the SNIPS estimator (Swaminathan & Joachims, 2015b), which imputes the average reward on the supported action for the unsupported actions (Gilotte et al., 2018). All estimators perform quite well for model selection on CIFAR, and the results are analogous for the credit-card data (see Appendix B.2). However, the plot to the right of Figure 3 reveals that SNIPS does not accurately reflect the shape of the Oracle curve. Both Regression Extrapolation and DM, however, are found to be sufficiently accurate for reliable model selection. ", + "bbox": [ + 174, + 464, + 825, + 631 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 DISCUSSION AND CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 652, + 478, + 667 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We identified and analyzed how off-policy learning based on IPS weighting can suffer severely degraded learning performance when the logging policy is support deficient. To remedy this problem, we explored approaches that limit the impact of missing support through three different means: restricting the action space, reward extrapolation and restricting the policy space. We find that the most natural approach of restricting the action space is neither computationally efficient, nor does it learn accurate policies. Reward extrapolation through regression and restricting the policy space, however, both perform well and robustly even at high levels of support deficiency. Among those two methods, reward extrapolation has the potential drawback that we need to compute (and/or sample from) the complement of the logging policy, which can be computationally challenging. Furthermore, having to store all old logging policies is inconvenient in practice. This makes the approach of restricting the policy space particularly attractive, since it is computationally efficient and it does not require access to the logging policy beyond the logged propensity values. ", + "bbox": [ + 174, + 683, + 825, + 851 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 285, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In International Conference on Machine Learning (ICML), 2014. 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Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992. ", + "bbox": [ + 174, + 319, + 823, + 348 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A APPENDIX: PROOFS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 379, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In this appendix, we provide proofs of the main theorems and propositions. ", + "bbox": [ + 173, + 132, + 666, + 148 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.1 PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 176, + 164, + 400, + 179 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Proposition 1. Given contexts $x \\sim P ( \\mathcal { X } )$ and logging policy $\\pi _ { 0 } ( \\mathcal { V } | x )$ , the bias of $\\hat { R } _ { I P S }$ for target policy $\\pi ( \\mathcal { V } | x )$ is equal to the expected reward on the unsupported action sets, i.e., $b i a s ( \\pi | \\pi _ { 0 } ) =$ $\\begin{array} { r } { \\mathbb { E } _ { x } [ - \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}$ . ", + "bbox": [ + 173, + 189, + 825, + 236 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Proof. Recall $\\delta ( x , y ) = \\mathbb { E } _ { r } [ r ( x , y ) | x , y ]$ , and logged data $\\mathcal { D } \\sim \\mathcal { P } _ { \\mathcal { X } } \\times \\pi _ { 0 } ( \\cdot | \\mathcal { X } ) \\times \\mathcal { P } _ { r }$ . ", + "bbox": [ + 171, + 250, + 728, + 267 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/6d4b968aa6433203448a2cf1bd9a7ed25742e2c20557f6925eae882f01c2e21e.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle b i a s ( \\pi | \\pi _ { 0 } ) = \\frac { \\mathbb { E } [ \\hat { R } _ { I P S } ( \\pi ) ] - R ( \\pi ) } { \\mathcal { D } } } \\\\ { \\displaystyle \\quad = \\frac { \\mathbb { E } [ \\sum _ { \\scriptstyle x } } { y \\in ( \\mathcal { U } ( x , \\pi _ { 0 } ) ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) - \\sum _ { y \\in \\mathcal { Y } } \\pi ( y | x ) \\delta ( x , y ) ] } \\\\ { \\displaystyle \\quad = \\frac { \\mathbb { E } [ - \\sum _ { \\scriptstyle x } } { x } - \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 272, + 743, + 377 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.2 PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 176, + 411, + 375, + 426 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Theorem 1. For any given hypothesis space $\\Pi$ with logging policy $\\pi _ { 0 } ~ \\in ~ \\Pi$ , there exists a reward distribution $\\mathcal { P } _ { r }$ with support in $[ r _ { m i n } , r _ { m a x } ]$ such that in the limit of infinite training data, ERM using $I P S$ over the logged data $\\mathcal D \\sim \\dot { P } ( \\mathcal X ) \\times \\pi _ { 0 } ( \\cdot | \\mathcal X ) \\times \\mathcal P _ { r }$ can select a policy $\\begin{array} { r } { \\hat { \\pi } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] } \\end{array}$ that is at least $\\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\quad } & { { } } \\end{array}$ suboptimal. ", + "bbox": [ + 173, + 436, + 825, + 498 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Proof. For any given hypothesis space $\\Pi$ and logging policy $\\pi _ { 0 }$ , define a deterministic reward distribution $\\mathcal { P } _ { r }$ as the following: for all context $x$ , $r ( x , y ) = \\delta ( x , y ) = r _ { m i n }$ for $y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c }$ and $r ( x , y ) ~ = ~ \\delta ( x , y ) ~ = ~ r _ { m a x }$ for $y \\in \\mathcal { U } ( x , \\pi _ { 0 } )$ . Let $\\tilde { \\pi } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } )$ and $\\pi ^ { * } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } R ( \\pi )$ , then we have the following lower bound for $R ( \\pi ^ { * } )$ : ", + "bbox": [ + 174, + 511, + 825, + 569 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/44cc8f65fabbd7c0e4c93a0dd7aaac55936a365f9368c48eb8bcd0e6d222a090.jpg", + "text": "$$\n\\begin{array} { r l } & { R ( \\pi ^ { * } ) \\geq R ( \\tilde { \\pi } ) } \\\\ & { \\qquad = \\underset { x } { \\mathbb { E } } [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } r _ { m a x } + \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } r _ { m i n } ] } \\\\ & { \\qquad = r _ { m a x } \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) + r _ { m i n } \\big ( 1 - \\underset { \\pi \\in \\Pi } { \\operatorname* { m a x } } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) \\big ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 573, + 699, + 655 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where the first inequality follows from the definition of $\\pi ^ { * }$ , the first and second equality is based on the specific reward distribution $\\mathcal { P } _ { r }$ and the definition of $\\tilde { \\pi }$ . ", + "bbox": [ + 173, + 659, + 821, + 688 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In the following we will show that for any $\\hat { \\pi }$ learned by the expectation of ERM (or in the limit of infinite amount data), i.e., πˆ ∈ arg max $\\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ^ { \\mathcal { D } } ( \\pi ) ]$ , $\\hat { \\pi }$ have the same support as $\\pi _ { 0 }$ . ", + "bbox": [ + 173, + 694, + 825, + 726 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/2bd06cefbd4c5982bbb7016c09d27917dc0faf7c821c619a93ba7881b2699fa1.jpg", + "text": "$$\n\\mathbb { E } [ \\hat { R } _ { I P S } ( \\pi ) ] = \\mathbb { E } [ \\sum _ { x } \\pi ( y | x ) r _ { m i n } ] = r _ { m i n } \\mathbb { E } [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) ] \\le r _ { m i n }\n$$", + "text_format": "latex", + "bbox": [ + 240, + 731, + 759, + 767 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "for all $\\pi \\in \\Pi$ , then it is easy to see $\\pi _ { 0 } \\in \\Pi$ is one of the solution of ERM. Actually for any $\\hat { \\pi } \\in$ arg max $\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ^ { \\mathcal { D } } ( \\pi ) ] , \\mathbb { E } _ { x } [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) ] = 1 } \\end{array}$ and it gives us that any solution of the ERM has exactly the same support as $\\pi _ { 0 }$ , then we have $R ( \\hat { \\pi } ) = r _ { m i n }$ for $\\hat { \\pi } \\in \\arg \\operatorname* { m a x } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ^ { \\mathcal { D } } ( \\pi ) ]$ . ", + "bbox": [ + 173, + 771, + 825, + 823 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Combining the lower bound for $R ( \\pi ^ { * } )$ and $R ( \\hat { \\pi } ) = r _ { m i n }$ , we have ", + "bbox": [ + 173, + 827, + 611, + 843 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/c1469879419db866ae4edc51ff17ad1c69e75fcf4ff8a7ce21c424ca97af33ec.jpg", + "text": "$$\n\\begin{array} { r l } & { R ( \\pi ^ { * } ) - R ( \\hat { \\pi } ) \\geq r _ { m a x } \\displaystyle \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) + r _ { m i n } ( 1 - \\displaystyle \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) ) - r _ { m i n } } \\\\ & { \\quad \\quad \\quad \\quad \\quad = \\left( r _ { m a x } - r _ { m i n } \\right) \\displaystyle \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 848, + 753, + 898 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.3 PROOF OF THEOREM 2 ", + "text_level": 1, + "bbox": [ + 176, + 103, + 377, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Theorem 2. For contexts $x _ { i }$ drawn i.i.d from $P ( \\mathcal X )$ , action $y _ { i }$ drawn from logging policy $\\pi _ { 0 }$ , we define SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) . For any policy $\\pi$ it holds that ", + "bbox": [ + 173, + 128, + 823, + 166 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/5b6229bd4f51b77ebbac1398d1b9236d0893caf774847983ec8be9b3d6c62f31.jpg", + "text": "$$\n\\begin{array} { r } { \\underset { x \\sim P ( \\mathcal { X } ) } { \\mathbb { E } } \\underset { y \\sim \\pi _ { 0 } ( \\cdot | x ) } { \\mathbb { E } } [ S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) = 1 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 172, + 651, + 196 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 173, + 213, + 217, + 228 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c2a347450b0b0c96b5bb67e26bc59009d419610b7909cb3a4ba0439df5a79323.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { x , y \\sim \\pi _ { 0 } } { \\mathbb { E } } [ S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) = \\underset { x } { \\mathbb { E } } [ \\ \\underset { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } { \\sum } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } & { } \\\\ { = \\underset { x } { \\mathbb { E } } [ \\ \\underset { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } { \\sum } \\pi ( y | x ) ] + \\underset { x } { \\mathbb { E } } [ \\ \\underset { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } { \\sum } \\pi ( y | x ) ] } & { } \\\\ { = \\underset { x } { \\mathbb { E } } [ \\underset { y \\in \\mathcal { Y } } { \\sum } \\pi ( y | x ) ] = 1 } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 236, + 758, + 351 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The first equality is based on definition of $S _ { D } ( \\pi | \\pi _ { 0 } )$ and the second equality is based on definition of support divergence. □ ", + "bbox": [ + 173, + 357, + 823, + 386 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.4 PROOF OF PROPOSITION 3 ", + "text_level": 1, + "bbox": [ + 176, + 404, + 400, + 417 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proposition 3. For any given $\\kappa \\in ( 0 , 1 )$ , $0 < \\epsilon < \\kappa / 2$ , let $p _ { m i n }$ denote the minimum propensity under supported set $p _ { m i n } ~ = ~ m a x _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x )$ , then with probability larger than $1 - 2 \\exp ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )$ , the constraint $1 - \\kappa { + } \\epsilon \\le S _ { \\mathscr { D } } ( \\pi | \\pi _ { 0 } ) \\le 1 { - } \\epsilon$ will ensure $0 \\leq \\mathcal { D } \\boldsymbol { x } ( \\pi | \\pi _ { 0 } ) \\leq \\kappa$ . ", + "bbox": [ + 173, + 429, + 825, + 476 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof. Recall SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) with $( x _ { i } , y _ { i } )$ draw i.i.d from $P ( \\mathcal { X } ) \\times \\pi _ { 0 } ( \\mathcal { Y } | \\boldsymbol { x } )$ . From Appendix A.3, it is easy to see Ex,y∼π0(·|x)[ π(y|x)π0(y|x) ] $\\begin{array} { r } { \\mathbb { E } _ { x , y \\sim \\pi _ { 0 } ( \\cdot | x ) } \\big [ \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\big ] = 1 - \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\end{array}$ . Let $p _ { m i n }$ denote the smallest propensity under supported action set, $p _ { m i n } : = \\mathrm { m i n } _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) > 0$ , then the random variable $\\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) }$ is strictly bounded between $[ 0 , \\frac { 1 } { p _ { m i n } } ]$ . Applying Hoeffding’s bound gives: ", + "bbox": [ + 173, + 491, + 825, + 570 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/a349963948c3a1b2e05cf3d5e10778fe4e845ee4f4446e25baadcbc949e0d8e9.jpg", + "text": "$$\n^ { \\mathfrak { p } } ( { \\mathcal { D } } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 1 - S _ { { \\mathcal { D } } } ( \\pi | \\pi _ { 0 } ) - \\epsilon ) = \\mathbb { P } ( S _ { { \\mathcal { D } } } ( \\pi | \\pi _ { 0 } ) - ( 1 - { \\mathcal { D } } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) ) < - \\epsilon ) \\leq e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 577, + 826, + 595 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Since $S _ { \\mathcal { D } ( \\pi | \\pi _ { 0 } ) } \\le 1 - \\epsilon$ gives $1 - S _ { \\mathcal { D } } ( \\pi \\vert \\pi _ { 0 } ) - \\epsilon \\geq 0$ , then we have ", + "bbox": [ + 173, + 606, + 617, + 622 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/d8a1768dee920d611c15b86151b72f883ecc3b51214f8686cd821d3e6487e683.jpg", + "text": "$$\n\\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 0 ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 367, + 630, + 630, + 648 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Similar for the other direction, Hoeffding’s bound gives: ", + "bbox": [ + 173, + 655, + 545, + 671 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/0916b98880a7628259ac4f25c46a3966111975118a30c91717e97ffbc296ab7e.jpg", + "text": "$$\n\\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) > 1 - S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\epsilon ) = \\mathbb { P } ( S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) - ( 1 - \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) ) > \\epsilon ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 178, + 679, + 816, + 696 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Since $S _ { \\mathcal { D } ( \\pi | \\pi _ { 0 } ) } \\geq 1 + \\epsilon - \\kappa$ gives $\\begin{array} { r } { 1 - S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\epsilon \\le \\kappa } \\end{array}$ , then we have ", + "bbox": [ + 174, + 718, + 647, + 736 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/616cd1e7374fd963c892d6ffd47a17dcee013502f54988a3981b254f374abedf.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) \\ge \\kappa ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 742, + 630, + 762 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Combining the above, we have ", + "bbox": [ + 173, + 768, + 379, + 784 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/1c8784ddfc2083adf24fc1f7ba99f6a4ff60b9a6b83f25ae21d3e249a18f34d5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } ( 0 \\leq \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) \\leq \\kappa ) = 1 - \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 0 ) - \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) > \\kappa ) } \\\\ & { \\qquad \\geq 1 - 2 e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 790, + 733, + 830 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.5 PROOF FOR EFFICIENT APPROXIMATION ", + "text_level": 1, + "bbox": [ + 174, + 867, + 493, + 882 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Claim 1. The empirical risk defined by in Equation (8) has the same expectation (over randomness in $\\mathcal { D }$ and sampling) as $\\hat { R } _ { I P S } ^ { \\delta } ( \\mathcal { D } )$ . ", + "bbox": [ + 174, + 892, + 823, + 925 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof. Taking the expectation of empirical risk defined in Equation (8): ", + "bbox": [ + 173, + 102, + 643, + 119 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/e485bea8deb18f822ec7c8e3e8f8397a13e0917e30960c340d932b0f8d700269.jpg", + "text": "$$\n\\begin{array} { l l } { { \\displaystyle { \\mathbb E } \\big [ \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi \\big ( y _ { i } | x _ { i } \\big ) } { \\pi _ { 0 } \\big ( y _ { i } | x _ { i } \\big ) } r _ { i } + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\frac { \\pi \\big ( y _ { j } | x _ { j } \\big ) } { p _ { j } } \\hat { \\delta } ( x _ { j } , y _ { j } ) \\big ] } } \\\\ { { \\displaystyle { = \\mathbb E } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi \\big ( y | x \\big ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) \\big ] + \\frac { { \\mathbb E } } { x } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\frac { 1 } { | \\mathcal { U } ( x , \\pi _ { 0 } ) | } \\frac { \\pi \\big ( y | x \\big ) } { \\textstyle | \\mathcal { U } ( x , \\pi _ { 0 } ) | } \\hat { \\delta } ( x , y ) \\big ] } } \\\\ { { \\displaystyle { = \\mathbb E } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) \\big ] + \\frac { { \\mathbb E } } { x } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 125, + 767, + 248 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Now we will show it has the same expectation with $\\hat { R } _ { I P S } ^ { \\delta } ( \\pi )$ ", + "bbox": [ + 173, + 255, + 576, + 272 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/352dff5875f2a0f8b5de2aa38d0bba4030835c62b98de05c4201943b4bfa45fa.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { \\mathbb { E } } { D } \\big [ \\frac { \\pi \\big ( y _ { 1 } | x _ { i } \\big ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x _ { \\star } , \\pi _ { 0 } ) } \\pi ( y | x _ { i } ) \\hat { \\delta } ( x _ { i } , y ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\underbrace { \\mathbb { E } } _ { y \\in \\pi _ { 0 } } [ \\frac { \\pi } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) ] + \\displaystyle \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) \\hat { \\delta } ( x , y ^ { \\prime } ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) + \\displaystyle \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) \\hat { \\delta } ( x , y ^ { \\prime } ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) \\big ] + \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 279, + 728, + 441 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The proof is done by comparing Equation (23) and Equation (24). ", + "bbox": [ + 173, + 446, + 606, + 462 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.6 PROOF OF PROPOSITION 2 ", + "text_level": 1, + "bbox": [ + 176, + 478, + 401, + 492 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proposition 2. Given contexts $x _ { 1 } , x _ { 2 } , \\ldots , x _ { n }$ drawn i.i.d from the unknown distribution $P ( \\mathcal X )$ , for action $y _ { i }$ drawn independently from logging policy $\\pi _ { 0 }$ with probability $\\pi _ { 0 } ( \\mathcal { V } | x _ { i } )$ , the bias of the empirical risk defined in Equation (7) is $\\begin{array} { r } { \\mathbb E _ { x } [ \\sum _ { y \\in \\mathcal U _ { x } ^ { \\pi _ { 0 } } } \\pi ( y | x ) \\Delta ( x , y ) ] } \\end{array}$ . ", + "bbox": [ + 173, + 503, + 825, + 550 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. From Appendix A.5, we are given the expectation of $\\hat { R } _ { I P S } ^ { \\delta } ( \\pi )$ , and the bias is: ", + "bbox": [ + 173, + 565, + 738, + 583 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/2cef212a8c41541db37673064d4fa4e6251f9e604d511a5aecebfc41f828627f.jpg", + "text": "$$\n\\begin{array} { l } { { b i a s ( \\hat { R } _ { I P S } ^ { \\delta } ( \\pi ) ) = { \\displaystyle \\mathbb { E } } [ \\hat { R } _ { I P S } ^ { \\delta } ( \\pi ) ] - R ( \\pi ) } } \\\\ { ~ = { \\displaystyle \\frac { { \\mathbb { E } } } { x } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) + \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] - R ( \\pi ) } \\\\ { ~ = { \\displaystyle \\mathbb { E } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) ( \\hat { \\delta } ( x , y ) - \\delta ( x , y ) ) \\big ] } \\\\ { ~ = { \\displaystyle \\mathbb { E } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\Delta ( x , y ) \\big ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 589, + 758, + 727 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The second equality is from Appendix A.5, the second equality is based on $\\begin{array} { r l } { R ( \\pi ) } & { { } = } \\end{array}$ $\\begin{array} { r } { \\mathbb { E } _ { x } \\left[ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) + \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) \\right] } \\end{array}$ , and the last one is based on the definition of $\\Delta ( x , y ) : = \\hat { \\delta } ( x , y ) - \\delta ( x , y )$ for all $x \\in \\mathcal { X } , y \\in \\mathcal { Y }$ . □ ", + "bbox": [ + 173, + 733, + 823, + 784 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B APPENDIX: EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 431, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we provide the experiment details and additional results to help promote reproducibility of this work. ", + "bbox": [ + 174, + 133, + 823, + 162 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.1 EXPERIMENT SETUP DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 179, + 421, + 194 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Datasets and baseline. We follow a 75:10:15 train-validation-test split for credit card fraud detection dataset, while for CIFAR10 already coming with a train-test split, we keep $10 \\%$ of the training set as validation set. Baseline estimators are IPS and DM, the hyperparameters (learning rate, L2 regularization) are optimized for all the methods based on the validation set. ", + "bbox": [ + 174, + 205, + 825, + 262 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Bandit data generation. For CIFAR10, given supervised data $\\{ x _ { i } , y _ { i } ^ { * } \\} _ { i = 1 } ^ { n }$ where $x _ { i }$ denotes the 3072 features and $y _ { i } ^ { * }$ denotes the correct label of data (ranging from 0 to 9), under logging policy $\\pi _ { 0 }$ , the logged bandit data is generated by drawing $y _ { i } \\sim \\pi _ { 0 } ( \\mathcal { V } | x _ { i } )$ , then a deterministic reward is defined as $\\mathbb { I } _ { \\{ y _ { i } = y _ { i } ^ { * } \\} }$ . For the credit card fraud detection dataset, we throw away the class label and only use the features for each sample to generate bandit data. To be specific, for each sample with a 28-dimensional feature vector, we define the first 20 features as the contextual information, and use the remaining 8 features as the underlying true reward for 8 different actions (with normalization). ", + "bbox": [ + 173, + 277, + 825, + 376 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Logging policy. For CIFAR, we learn the softmax logging policy on 35K full-information data points as a multi-class classification problem with cross-entropy loss. Similar as the experiments on BanditNet (Joachims et al., 2018), we adopt the conventional ResNet20 architecture but restrict training after a mere two epochs to derive a relative stochastic policy, since it will be easier to add temperature later to control its stochasticity and support deficiency. Similarly, for the credit card fraud detection dataset, the softmax logging policy is learned on 8K full-information data points by treating it as a multi-class classification problem using cross-entropy loss and the label being the action with the highest reward on this specific context. For CIFAR, the logging policy we trained has a $5 7 . 4 3 \\%$ accuracy on the test-set; whereas for the credit card fraud detection dataset, the logging policy has an expected true reward of 0.71. ", + "bbox": [ + 173, + 391, + 825, + 530 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Reward estimator. For each experiment, we train a different regression function using the full bandit dataset. We use the same architecture as the one used for off-policy learning - where the final layer is the size of the actions, specifying the reward for each action given a particular context. The regression function is trained using the MSE objective. ", + "bbox": [ + 173, + 546, + 825, + 603 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/51d656734af5ca7eebec76bcca0d315e378aa7a4b913d38454b7164ec98ffb1b.jpg", + "image_caption": [ + "Figure 4: Behaviour of $S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } )$ " + ], + "image_footnote": [], + "bbox": [ + 173, + 616, + 823, + 760 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.2 SUPPLEMENTARY RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 814, + 403, + 828 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Reliability of approximating support divergence using control variate. In this experiment, we empirically verify the reliability of estimating support divergence using control variate. The target policy is the uniform policy while the logging policies are varying in their support deficiency. Results are averaged over 10 runs and is shown in Figure 4. We investigate the behaviour of $S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } )$ under different number of training data and different support deficiency of the corresponding logging policy. As we can see, the sum converges to 1 as the training data increases. Meanwhile, the variance decreases as shown in the right figure. For different support deficiency curves, the curve converges in a similar fashion and we conjecture it is due to the effect of clipping the propensity at the same threshold $\\epsilon = 0 . 0 1$ , which makes $p _ { m i n } = 0 . 0 1$ in the bound shown in Proposition 3. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/2678c7d2bb519d2032262fa6ef15c3ce8481c96bf236def1f54a54d5ca71d2c6.jpg", + "image_caption": [ + "Figure 5: Model selection result for credit card fraud detection " + ], + "image_footnote": [], + "bbox": [ + 187, + 103, + 560, + 227 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/217e996b67f6df7a1b4bf8f2cf57d77113d0632e263176148aa1a303192a33e2.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 578, + 103, + 808, + 227 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 280, + 825, + 335 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Model selection comparison for the credit card dataset. The model selection comparison over the credit card fraud detection dataset is demonstrated in Figure 5. Similar as the trend in Figure 3, SNIPS and Conservative Extrapolation exhibit a large bias, also SNIPS even can not reflect the shape of the Oracle curve. DM and Regression Extrapolation closely track the Oracle line, and they have the best performance when used in model selection, as seen in the left table of Figure 5. ", + "bbox": [ + 173, + 352, + 825, + 421 + ], + "page_idx": 14 + } +] \ No newline at end of file diff --git a/parse/train/SklcyJBtvB/SklcyJBtvB_middle.json b/parse/train/SklcyJBtvB/SklcyJBtvB_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..179f9fc5c3cba40df70ba71943317eebc5d36901 --- /dev/null +++ b/parse/train/SklcyJBtvB/SklcyJBtvB_middle.json @@ -0,0 +1,43979 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 469, + 96 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 473, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 473, + 99 + ], + "score": 1.0, + "content": "OFF-POLICY BANDITS WITH DEFICIENT SUPPORT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 115, + 244, + 137 + ], + "lines": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "spans": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "spans": [ + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 278, + 167, + 333, + 179 + ], + "lines": [ + { + "bbox": [ + 277, + 166, + 335, + 180 + ], + "spans": [ + { + "bbox": [ + 277, + 166, + 335, + 180 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 143, + 191, + 468, + 367 + ], + "lines": [ + { + "bbox": [ + 142, + 191, + 469, + 204 + ], + "spans": [ + { + "bbox": [ + 142, + 191, + 469, + 204 + ], + "score": 1.0, + "content": "Off-policy training of contextual-bandit policies is attractive in online systems", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 203, + 469, + 215 + ], + "spans": [ + { + "bbox": [ + 141, + 203, + 469, + 215 + ], + "score": 1.0, + "content": "(e.g. search, recommendation, ad placement), since it enables the reuse of large", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 213, + 470, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 470, + 227 + ], + "score": 1.0, + "content": "amounts of log data. State-of-the-art methods for off-policy learning, however, are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 223, + 469, + 238 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 469, + 238 + ], + "score": 1.0, + "content": "based on inverse propensity score (IPS) weighting, which requires that the logging", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "score": 1.0, + "content": "policy chooses all actions with non-zero probability for any context (i.e., full sup-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 246, + 470, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 470, + 258 + ], + "score": 1.0, + "content": "port). In real-world systems, this condition is often violated, and we show that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 257, + 470, + 269 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 470, + 269 + ], + "score": 1.0, + "content": "existing off-policy learning methods based on IPS weighting can fail catastroph-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "score": 1.0, + "content": "ically. We therefore develop new off-policy contextual-bandit methods that can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 279, + 469, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 469, + 292 + ], + "score": 1.0, + "content": "controllably and robustly learn even when the logging policy has deficient sup-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "score": 1.0, + "content": "port. To this effect, we explore three approaches that provide various guarantees", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "score": 1.0, + "content": "for safe learning despite the inherent limitations of support deficient data: restrict-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "score": 1.0, + "content": "ing the action space, reward extrapolation, and restricting the policy space. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "score": 1.0, + "content": "analyze the statistical and computational properties of these three approaches, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "score": 1.0, + "content": "empirically evaluate their effectiveness in a series of experiments. We find that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "score": 1.0, + "content": "controlling the policy space is both computationally efficient and that it robustly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 356, + 247, + 368 + ], + "spans": [ + { + "bbox": [ + 142, + 356, + 247, + 368 + ], + "score": 1.0, + "content": "leads to accurate policies.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 206, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 208, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 208, + 403 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 504, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 504, + 425 + ], + "score": 1.0, + "content": "Many interactive systems (e.g., voice assistants, recommender systems, ad placement) can be mod-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "eled as contextual bandit problems (Langford & Zhang, 2008). In particular, each user request", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "provides a context (e.g., user profile, query) for which the system selects an action (e.g., recom-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "mended product, presented ad) and receives a reward (e.g., purchase, click). Such contextual-bandit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "data is logged in large quantities as a by-product of normal system operation (Li et al., 2011; 2015;", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "Joachims et al., 2017), making it an attractive and low-cost source of training data. With terabytes of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "such log data readily available in many online systems, a range of algorithms have been proposed for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "batch learning from such logged contextual-bandit feedback (Strehl et al., 2011; Dud´ık et al., 2011;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "score": 1.0, + "content": "Swaminathan & Joachims, 2015a; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "London & Sandler, 2019). However, as we will argue below, these algorithms require an assumption", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 426, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 426, + 535 + ], + "score": 1.0, + "content": "about the log data that makes them unsuitable for many real-world applications.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "This assumption is typically referred to as the positivity or support assumption, and it is required", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "by the Empirical Risk Minimization (ERM) objective that these algorithms optimize. Specifically,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "unlike in online learning for contextual bandits (Williams, 1992; Agarwal et al., 2014), batch learn-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "ing from bandit feedback (BLBF) operates in the off-policy setting. During off-policy learning, the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "algorithm has to address the counterfactual question of how much reward each policy in the policy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "space would have received, if it had been used instead of the logging policy. To this effect, virtually", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "all state-of-the-art off-policy learning methods for contextual-bandit problems rely on counterfac-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "score": 1.0, + "content": "tual estimators (Bottou et al., 2013; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a; Thomas", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "& Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019) that employ inverse propensity score", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "(IPS) weighting to get an unbiased ERM objective. Unlike regression-based direct-modeling (DM)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "approaches that are often hampered by bias from model-misspecification, IPS allows a controllable", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "bias-variance trade-off through clipping and other variance-regularization techniques (Strehl et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 377, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 377, + 682 + ], + "score": 1.0, + "content": "2011; Swaminathan & Joachims, 2015a; London & Sandler, 2019).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Unfortunately, IPS and its variance-control mechanisms break down when the logging policy does", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "not have full support – meaning that some actions have zero probability of being selected under the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "logging policy. In this case IPS can be highly biased. Full support is an unreasonable assumption", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "in many real-world systems, especially when the action space is large and many actions have poor", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 469, + 96 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 473, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 473, + 99 + ], + "score": 1.0, + "content": "OFF-POLICY BANDITS WITH DEFICIENT SUPPORT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 115, + 244, + 137 + ], + "lines": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "spans": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "spans": [ + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 111, + 115, + 245, + 138 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 167, + 333, + 179 + ], + "lines": [ + { + "bbox": [ + 277, + 166, + 335, + 180 + ], + "spans": [ + { + "bbox": [ + 277, + 166, + 335, + 180 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 143, + 191, + 468, + 367 + ], + "lines": [ + { + "bbox": [ + 142, + 191, + 469, + 204 + ], + "spans": [ + { + "bbox": [ + 142, + 191, + 469, + 204 + ], + "score": 1.0, + "content": "Off-policy training of contextual-bandit policies is attractive in online systems", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 203, + 469, + 215 + ], + "spans": [ + { + "bbox": [ + 141, + 203, + 469, + 215 + ], + "score": 1.0, + "content": "(e.g. search, recommendation, ad placement), since it enables the reuse of large", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 213, + 470, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 470, + 227 + ], + "score": 1.0, + "content": "amounts of log data. State-of-the-art methods for off-policy learning, however, are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 223, + 469, + 238 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 469, + 238 + ], + "score": 1.0, + "content": "based on inverse propensity score (IPS) weighting, which requires that the logging", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "score": 1.0, + "content": "policy chooses all actions with non-zero probability for any context (i.e., full sup-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 246, + 470, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 470, + 258 + ], + "score": 1.0, + "content": "port). In real-world systems, this condition is often violated, and we show that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 257, + 470, + 269 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 470, + 269 + ], + "score": 1.0, + "content": "existing off-policy learning methods based on IPS weighting can fail catastroph-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "score": 1.0, + "content": "ically. We therefore develop new off-policy contextual-bandit methods that can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 279, + 469, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 469, + 292 + ], + "score": 1.0, + "content": "controllably and robustly learn even when the logging policy has deficient sup-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "score": 1.0, + "content": "port. To this effect, we explore three approaches that provide various guarantees", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "score": 1.0, + "content": "for safe learning despite the inherent limitations of support deficient data: restrict-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "score": 1.0, + "content": "ing the action space, reward extrapolation, and restricting the policy space. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "score": 1.0, + "content": "analyze the statistical and computational properties of these three approaches, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "score": 1.0, + "content": "empirically evaluate their effectiveness in a series of experiments. We find that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "score": 1.0, + "content": "controlling the policy space is both computationally efficient and that it robustly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 356, + 247, + 368 + ], + "spans": [ + { + "bbox": [ + 142, + 356, + 247, + 368 + ], + "score": 1.0, + "content": "leads to accurate policies.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 11.5, + "bbox_fs": [ + 141, + 191, + 470, + 368 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 206, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 208, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 208, + 403 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 504, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 504, + 425 + ], + "score": 1.0, + "content": "Many interactive systems (e.g., voice assistants, recommender systems, ad placement) can be mod-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "eled as contextual bandit problems (Langford & Zhang, 2008). In particular, each user request", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "provides a context (e.g., user profile, query) for which the system selects an action (e.g., recom-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "mended product, presented ad) and receives a reward (e.g., purchase, click). Such contextual-bandit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "data is logged in large quantities as a by-product of normal system operation (Li et al., 2011; 2015;", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "Joachims et al., 2017), making it an attractive and low-cost source of training data. With terabytes of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "such log data readily available in many online systems, a range of algorithms have been proposed for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "batch learning from such logged contextual-bandit feedback (Strehl et al., 2011; Dud´ık et al., 2011;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "score": 1.0, + "content": "Swaminathan & Joachims, 2015a; Thomas & Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "London & Sandler, 2019). However, as we will argue below, these algorithms require an assumption", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 426, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 426, + 535 + ], + "score": 1.0, + "content": "about the log data that makes them unsuitable for many real-world applications.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 413, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "This assumption is typically referred to as the positivity or support assumption, and it is required", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "by the Empirical Risk Minimization (ERM) objective that these algorithms optimize. Specifically,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "unlike in online learning for contextual bandits (Williams, 1992; Agarwal et al., 2014), batch learn-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "ing from bandit feedback (BLBF) operates in the off-policy setting. During off-policy learning, the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "algorithm has to address the counterfactual question of how much reward each policy in the policy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "space would have received, if it had been used instead of the logging policy. To this effect, virtually", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "all state-of-the-art off-policy learning methods for contextual-bandit problems rely on counterfac-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "score": 1.0, + "content": "tual estimators (Bottou et al., 2013; Dud´ık et al., 2011; Swaminathan & Joachims, 2015a; Thomas", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "& Brunskill, 2016; Farajtabar et al., 2018; Su et al., 2019) that employ inverse propensity score", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "(IPS) weighting to get an unbiased ERM objective. Unlike regression-based direct-modeling (DM)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "approaches that are often hampered by bias from model-misspecification, IPS allows a controllable", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "bias-variance trade-off through clipping and other variance-regularization techniques (Strehl et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 377, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 377, + 682 + ], + "score": 1.0, + "content": "2011; Swaminathan & Joachims, 2015a; London & Sandler, 2019).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 539, + 506, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Unfortunately, IPS and its variance-control mechanisms break down when the logging policy does", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "not have full support – meaning that some actions have zero probability of being selected under the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "logging policy. In this case IPS can be highly biased. Full support is an unreasonable assumption", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "in many real-world systems, especially when the action space is large and many actions have poor", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "rewards. For example, in a recommender system with a large catalog (e.g. movies, music), it may be", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 162, + 107 + ], + "score": 1.0, + "content": "that less than", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 162, + 94, + 181, + 104 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 181, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "of the actions have support under the logging policy. We will show that existing", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 408, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 408, + 117 + ], + "score": 1.0, + "content": "learning algorithms can fail catastrophically on such support deficient data.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "rewards. For example, in a recommender system with a large catalog (e.g. movies, music), it may be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 162, + 107 + ], + "score": 1.0, + "content": "that less than", + "type": "text" + }, + { + "bbox": [ + 162, + 94, + 181, + 104 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "of the actions have support under the logging policy. We will show that existing", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 408, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 408, + 117 + ], + "score": 1.0, + "content": "learning algorithms can fail catastrophically on such support deficient data.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "In this paper, we develop new off-policy contextual-bandit algorithms that are specifically designed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "to deal with support deficient log data. Since support deficiency translates into blind spots where we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 506, + 158 + ], + "score": 1.0, + "content": "do not have any knowledge about the rewards, accounting for these blind spots as part of learning", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 504, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 504, + 167 + ], + "score": 1.0, + "content": "is crucial for robust learning. We approach this problem from three perspectives. First, we explore", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "restricting the action space to those actions that have support under the logging policy. Second,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "we explore imputation methods that extrapolate estimated rewards to those blind spots. And, third,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "we restrict the policy space to only those policies that have limited exposure to the blind spots. To", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "make the latter approach computationally tractable, we define a new measure of Support Divergence", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "between policies, show how it can be estimated efficiently without closed-form knowledge of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "logging policy, and how it can be used as a constraint on the policy space. We analyze the statistical", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 504, + 244 + ], + "score": 1.0, + "content": "and computational properties of all three approaches and perform an extensive empirical evaluation.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "We find that restricting the policy space is particularly effective, since it is computationally efficient,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 429, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 429, + 266 + ], + "score": 1.0, + "content": "empirically effective at learning good policies, and convenient to use in practice.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 279, + 211, + 292 + ], + "lines": [ + { + "bbox": [ + 104, + 278, + 213, + 295 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 213, + 295 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 504, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "Most prior works on BLBF can be classified into two different approaches. The first – called Direct", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "Model (DM) – is based on a reduction to supervised learning, where a regression estimate is trained", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "to predict rewards (Beygelzimer & Langford, 2009). To derive a policy, the action with the highest", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "score": 1.0, + "content": "predicted reward is chosen. A drawback of this simple approach is the bias that results from mis-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "specification of the regression model. Since regression models are often substantially misspecified", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 408, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 408, + 371 + ], + "score": 1.0, + "content": "for real-world data, the DM approach often does not work well empirically.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "The second approach is based on policy learning via ERM with a counterfactual risk estimator. In-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "verse propensity score (IPS) weighting is one of the most popular estimators to be used as empirical", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "risk. However, policy learning algorithms based on IPS and related estimators (Strehl et al., 2011;", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "Swaminathan & Joachims, 2015a;b; Thomas & Brunskill, 2016; London & Sandler, 2019) require", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "the assumption that the logging policy has full support for every policy in the policy space. One", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "exception is the work of Liu et al. (2019). They relax the assumption to the existence of an optimal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "policy such that the logging policy covers the support of this optimal policy. However, this is an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 439, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 439, + 465 + ], + "score": 1.0, + "content": "untestable assumption that does not provide guarantees for real-world applications.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "Our work proposes three approaches to addressing off-policy learning with support deficiency. First,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 479, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 505, + 494 + ], + "score": 1.0, + "content": "our conservative extrapolation method is related to the method proposed by Liu et al. (2019). They", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "focus on the correction of the state distribution by defining an augmented MDP, and pessimistic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "score": 1.0, + "content": "imputation is used to get an estimate for policy-gradient learning. Second, our method of restricting", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "the policy space uses a surrogate for the support divergence of two policies that was previously used", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "as control variate in the SNIPS estimator (Swaminathan & Joachims, 2015b). It also appeared in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "the Lagrangian formulation of the BanditNet objective (Joachims et al., 2018) and in the gradient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "update in REINFORCE algorithm (Williams, 1992). This connection gives interesting new insight", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "that the baselines used in policy-gradient algorithms not only help to reduce variance in gradients", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "(Greensmith et al., 2004), but that they also connect to the problem of support deficiency in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 578, + 180, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 180, + 593 + ], + "score": 1.0, + "content": "off-policy setting.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 605, + 387, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 388, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 388, + 620 + ], + "score": 1.0, + "content": "3 OFF-POLICY LEARNING WITH DEFICIENT SUPPORT", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 504, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 644 + ], + "score": 1.0, + "content": "We start by formally defining the problem of learning a contextual-bandit policy in the BLBF setting.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 237, + 653 + ], + "score": 1.0, + "content": "Input to the policy are contexts", + "type": "text" + }, + { + "bbox": [ + 237, + 641, + 268, + 651 + ], + "score": 0.91, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 640, + 476, + 653 + ], + "score": 1.0, + "content": "drawn i.i.d from a fixed but unknown distribution", + "type": "text" + }, + { + "bbox": [ + 476, + 641, + 501, + 653 + ], + "score": 0.91, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 640, + 505, + 653 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 651, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 167, + 665 + ], + "score": 1.0, + "content": "Given context", + "type": "text" + }, + { + "bbox": [ + 167, + 654, + 174, + 662 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 651, + 382, + 665 + ], + "score": 1.0, + "content": ", the system executes a possibly stochastic policy", + "type": "text" + }, + { + "bbox": [ + 382, + 652, + 413, + 663 + ], + "score": 0.94, + "content": "\\pi ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 651, + 506, + 665 + ], + "score": 1.0, + "content": "that selects an action", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 661, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 137, + 674 + ], + "score": 0.89, + "content": "y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 661, + 407, + 677 + ], + "score": 1.0, + "content": ". For this context and action pair, the system observes a reward", + "type": "text" + }, + { + "bbox": [ + 407, + 663, + 481, + 675 + ], + "score": 0.91, + "content": "r \\in [ r _ { m i n } , r _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 661, + 505, + 677 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 673, + 444, + 686 + ], + "spans": [ + { + "bbox": [ + 107, + 674, + 146, + 686 + ], + "score": 0.92, + "content": "P ( r | x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 673, + 253, + 686 + ], + "score": 1.0, + "content": ". Given a space of policies", + "type": "text" + }, + { + "bbox": [ + 253, + 674, + 261, + 684 + ], + "score": 0.7, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 673, + 364, + 686 + ], + "score": 1.0, + "content": ", the reward of any policy", + "type": "text" + }, + { + "bbox": [ + 365, + 674, + 392, + 684 + ], + "score": 0.91, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 673, + 444, + 686 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 686, + 371, + 706 + ], + "lines": [ + { + "bbox": [ + 239, + 686, + 371, + 706 + ], + "spans": [ + { + "bbox": [ + 239, + 686, + 371, + 706 + ], + "score": 0.93, + "content": "R ( \\pi ) = \\underset { x } { \\mathbb { E } } \\underset { y \\sim \\pi ( y \\mid x ) } { \\mathbb { E } } \\underset { r \\sim P ( r \\mid x , y ) } { \\mathbb { E } } [ r ] .", + "type": "interline_equation", + "image_path": "cc844e68665c2e4ffa8ae299fddc115cb87be6cf868f100649cc10522fe27df8.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 239, + 686, + 371, + 706 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 707, + 352, + 719 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 353, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 353, + 720 + ], + "score": 1.0, + "content": "In the BLBF setting, the learning algorithm is given a dataset", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 720, + 368, + 733 + ], + "lines": [ + { + "bbox": [ + 242, + 720, + 368, + 733 + ], + "spans": [ + { + "bbox": [ + 242, + 720, + 368, + 733 + ], + "score": 0.91, + "content": "\\mathcal { D } : = \\{ x _ { i } , y _ { i } , r _ { i } , \\pi _ { 0 } ( y _ { i } | x _ { i } ) \\} _ { i = 1 } ^ { n }", + "type": "interline_equation", + "image_path": "cc917c182d58be11a6073714fe38a53260f193e260a27f7d1212c8d3a05b988e.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 242, + 720, + 368, + 733 + ], + "spans": [], + "index": 50 + } + ] + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 83, + 506, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "In this paper, we develop new off-policy contextual-bandit algorithms that are specifically designed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "to deal with support deficient log data. Since support deficiency translates into blind spots where we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 506, + 158 + ], + "score": 1.0, + "content": "do not have any knowledge about the rewards, accounting for these blind spots as part of learning", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 504, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 504, + 167 + ], + "score": 1.0, + "content": "is crucial for robust learning. We approach this problem from three perspectives. First, we explore", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "restricting the action space to those actions that have support under the logging policy. Second,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "we explore imputation methods that extrapolate estimated rewards to those blind spots. And, third,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "we restrict the policy space to only those policies that have limited exposure to the blind spots. To", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "make the latter approach computationally tractable, we define a new measure of Support Divergence", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "between policies, show how it can be estimated efficiently without closed-form knowledge of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "logging policy, and how it can be used as a constraint on the policy space. We analyze the statistical", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 504, + 244 + ], + "score": 1.0, + "content": "and computational properties of all three approaches and perform an extensive empirical evaluation.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "We find that restricting the policy space is particularly effective, since it is computationally efficient,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 429, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 429, + 266 + ], + "score": 1.0, + "content": "empirically effective at learning good policies, and convenient to use in practice.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 122, + 506, + 266 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 279, + 211, + 292 + ], + "lines": [ + { + "bbox": [ + 104, + 278, + 213, + 295 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 213, + 295 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 504, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "Most prior works on BLBF can be classified into two different approaches. The first – called Direct", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "Model (DM) – is based on a reduction to supervised learning, where a regression estimate is trained", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "to predict rewards (Beygelzimer & Langford, 2009). To derive a policy, the action with the highest", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "score": 1.0, + "content": "predicted reward is chosen. A drawback of this simple approach is the bias that results from mis-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "specification of the regression model. Since regression models are often substantially misspecified", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 408, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 408, + 371 + ], + "score": 1.0, + "content": "for real-world data, the DM approach often does not work well empirically.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 304, + 506, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "The second approach is based on policy learning via ERM with a counterfactual risk estimator. In-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "verse propensity score (IPS) weighting is one of the most popular estimators to be used as empirical", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "risk. However, policy learning algorithms based on IPS and related estimators (Strehl et al., 2011;", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "Swaminathan & Joachims, 2015a;b; Thomas & Brunskill, 2016; London & Sandler, 2019) require", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "the assumption that the logging policy has full support for every policy in the policy space. One", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "exception is the work of Liu et al. (2019). They relax the assumption to the existence of an optimal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "policy such that the logging policy covers the support of this optimal policy. However, this is an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 439, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 439, + 465 + ], + "score": 1.0, + "content": "untestable assumption that does not provide guarantees for real-world applications.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 376, + 506, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "Our work proposes three approaches to addressing off-policy learning with support deficiency. First,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 479, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 505, + 494 + ], + "score": 1.0, + "content": "our conservative extrapolation method is related to the method proposed by Liu et al. (2019). They", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "focus on the correction of the state distribution by defining an augmented MDP, and pessimistic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "score": 1.0, + "content": "imputation is used to get an estimate for policy-gradient learning. Second, our method of restricting", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "the policy space uses a surrogate for the support divergence of two policies that was previously used", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "as control variate in the SNIPS estimator (Swaminathan & Joachims, 2015b). It also appeared in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "the Lagrangian formulation of the BanditNet objective (Joachims et al., 2018) and in the gradient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "update in REINFORCE algorithm (Williams, 1992). This connection gives interesting new insight", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "that the baselines used in policy-gradient algorithms not only help to reduce variance in gradients", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "(Greensmith et al., 2004), but that they also connect to the problem of support deficiency in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 578, + 180, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 180, + 593 + ], + "score": 1.0, + "content": "off-policy setting.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 469, + 506, + 593 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 605, + 387, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 388, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 388, + 620 + ], + "score": 1.0, + "content": "3 OFF-POLICY LEARNING WITH DEFICIENT SUPPORT", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 504, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 644 + ], + "score": 1.0, + "content": "We start by formally defining the problem of learning a contextual-bandit policy in the BLBF setting.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 237, + 653 + ], + "score": 1.0, + "content": "Input to the policy are contexts", + "type": "text" + }, + { + "bbox": [ + 237, + 641, + 268, + 651 + ], + "score": 0.91, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 640, + 476, + 653 + ], + "score": 1.0, + "content": "drawn i.i.d from a fixed but unknown distribution", + "type": "text" + }, + { + "bbox": [ + 476, + 641, + 501, + 653 + ], + "score": 0.91, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 640, + 505, + 653 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 651, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 167, + 665 + ], + "score": 1.0, + "content": "Given context", + "type": "text" + }, + { + "bbox": [ + 167, + 654, + 174, + 662 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 651, + 382, + 665 + ], + "score": 1.0, + "content": ", the system executes a possibly stochastic policy", + "type": "text" + }, + { + "bbox": [ + 382, + 652, + 413, + 663 + ], + "score": 0.94, + "content": "\\pi ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 651, + 506, + 665 + ], + "score": 1.0, + "content": "that selects an action", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 661, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 137, + 674 + ], + "score": 0.89, + "content": "y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 661, + 407, + 677 + ], + "score": 1.0, + "content": ". For this context and action pair, the system observes a reward", + "type": "text" + }, + { + "bbox": [ + 407, + 663, + 481, + 675 + ], + "score": 0.91, + "content": "r \\in [ r _ { m i n } , r _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 661, + 505, + 677 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 673, + 444, + 686 + ], + "spans": [ + { + "bbox": [ + 107, + 674, + 146, + 686 + ], + "score": 0.92, + "content": "P ( r | x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 673, + 253, + 686 + ], + "score": 1.0, + "content": ". 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We call", + "type": "text" + }, + { + "bbox": [ + 227, + 106, + 238, + 115 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "the logging policy, and we will discuss desired conditions on the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 170, + 127 + ], + "score": 1.0, + "content": "stochasticity of", + "type": "text" + }, + { + "bbox": [ + 170, + 117, + 182, + 127 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "in the following. The goal of off-policy learning is to exploit the information in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 380, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 170, + 138 + ], + "score": 1.0, + "content": "the logged data", + "type": "text" + }, + { + "bbox": [ + 170, + 127, + 179, + 136 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 127, + 243, + 138 + ], + "score": 1.0, + "content": "to find a policy", + "type": "text" + }, + { + "bbox": [ + 243, + 127, + 270, + 137 + ], + "score": 0.9, + "content": "{ \\hat { \\pi } } \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 127, + 354, + 138 + ], + "score": 1.0, + "content": "that has high reward", + "type": "text" + }, + { + "bbox": [ + 354, + 126, + 376, + 138 + ], + "score": 0.92, + "content": "R ( { \\hat { \\pi } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 127, + 380, + 138 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 142, + 505, + 178 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "Analogous to the ERM principle in supervised learning, off-policy learning algorithms typically", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 249, + 168 + ], + "score": 1.0, + "content": "optimize a counterfactual estimate", + "type": "text" + }, + { + "bbox": [ + 249, + 154, + 272, + 168 + ], + "score": 0.94, + "content": "\\hat { R } ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 156, + 286, + 168 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 286, + 155, + 308, + 168 + ], + "score": 0.92, + "content": "R ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "as the training objective (Li et al., 2011; 2015;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 327, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 327, + 179 + ], + "score": 1.0, + "content": "Bottou et al., 2013; Swaminathan & Joachims, 2015a).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 181, + 347, + 203 + ], + "lines": [ + { + "bbox": [ + 263, + 181, + 347, + 203 + ], + "spans": [ + { + "bbox": [ + 263, + 181, + 347, + 203 + ], + "score": 0.95, + "content": "\\hat { \\pi } = \\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m a x } } [ \\hat { R } ( \\pi ) ]", + "type": "interline_equation", + "image_path": "e3b58cd496cef3671c86be4c1e71054edc436e42aee1fe5d47e0ebeade5554d8.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 263, + 181, + 347, + 203 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "For conciseness, we ignore additional regularization terms in the objective (Swaminathan &", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "Joachims, 2015a), since they are irrelevant to the main point of this paper. As counterfactual es-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 137, + 242 + ], + "score": 1.0, + "content": "timator", + "type": "text" + }, + { + "bbox": [ + 138, + 228, + 160, + 241 + ], + "score": 0.93, + "content": "\\hat { R } ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 228, + 505, + 242 + ], + "score": 1.0, + "content": ", most algorithms rely on some form of IPS weighting (Strehl et al., 2011; Dud´ık et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "score": 1.0, + "content": "2011; Swaminathan & Joachims, 2015a;b; Wang et al., 2017; Su et al., 2019) to correct the distribu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 410, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 276, + 264 + ], + "score": 1.0, + "content": "tion mismatch between the logging policy", + "type": "text" + }, + { + "bbox": [ + 276, + 253, + 288, + 263 + ], + "score": 0.85, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 250, + 379, + 264 + ], + "score": 1.0, + "content": "and each target policy", + "type": "text" + }, + { + "bbox": [ + 379, + 252, + 406, + 262 + ], + "score": 0.9, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 250, + 410, + 264 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 266, + 370, + 298 + ], + "lines": [ + { + "bbox": [ + 240, + 266, + 370, + 298 + ], + "spans": [ + { + "bbox": [ + 240, + 266, + 370, + 298 + ], + "score": 0.95, + "content": "\\hat { R } _ { I P S } ( \\pi ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } .", + "type": "interline_equation", + "image_path": "2d8cffe0b06bb469b79cf452f32e68c3c96c63c9caf1a4e0c57f7536520747d8.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 266, + 370, + 282.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 240, + 282.0, + 370, + 298.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "A crucial condition for the effectiveness of the IPS estimator (and similar estimators) is that the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 168, + 324 + ], + "score": 1.0, + "content": "logging policy", + "type": "text" + }, + { + "bbox": [ + 168, + 314, + 180, + 323 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "assigns non-zero probability to all actions that have non-zero probability under", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 173, + 335 + ], + "score": 1.0, + "content": "the target policy", + "type": "text" + }, + { + "bbox": [ + 173, + 325, + 180, + 333 + ], + "score": 0.7, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "we aim to evaluate. This condition is known as positivity or full support, and it is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 184, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 184, + 346 + ], + "score": 1.0, + "content": "defined as follows.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 503, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 295, + 361 + ], + "score": 1.0, + "content": "Definition 1 (Full support). The logging policy", + "type": "text" + }, + { + "bbox": [ + 296, + 349, + 307, + 359 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 346, + 429, + 361 + ], + "score": 1.0, + "content": "is said to have full support for", + "type": "text" + }, + { + "bbox": [ + 429, + 350, + 436, + 358 + ], + "score": 0.69, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 346, + 460, + 361 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 461, + 348, + 505, + 360 + ], + "score": 0.91, + "content": "\\pi _ { 0 } ( y | x ) >", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 358, + 371, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 172, + 371 + ], + "score": 1.0, + "content": "0 for all actions", + "type": "text" + }, + { + "bbox": [ + 172, + 359, + 199, + 370 + ], + "score": 0.91, + "content": "y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 358, + 252, + 371 + ], + "score": 1.0, + "content": "and contexts", + "type": "text" + }, + { + "bbox": [ + 252, + 360, + 280, + 370 + ], + "score": 0.89, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 358, + 321, + 371 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 322, + 358, + 368, + 371 + ], + "score": 0.93, + "content": "\\pi ( y | x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 358, + 371, + 371 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 294, + 393 + ], + "score": 1.0, + "content": "It is known that the IPS estimator is unbiased,", + "type": "text" + }, + { + "bbox": [ + 294, + 379, + 388, + 393 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] = R ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 380, + 476, + 393 + ], + "score": 1.0, + "content": ", if the logging policy", + "type": "text" + }, + { + "bbox": [ + 476, + 382, + 488, + 392 + ], + "score": 0.85, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 169, + 404 + ], + "score": 1.0, + "content": "full support for", + "type": "text" + }, + { + "bbox": [ + 170, + 393, + 177, + 402 + ], + "score": 0.72, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "(Li et al., 2011). To ensure unbiased ERM, algorithms that use the IPS estimator", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 232, + 415 + ], + "score": 1.0, + "content": "require that the logging policy", + "type": "text" + }, + { + "bbox": [ + 232, + 404, + 244, + 414 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 403, + 374, + 415 + ], + "score": 1.0, + "content": "has full support for all policies", + "type": "text" + }, + { + "bbox": [ + 374, + 403, + 403, + 413 + ], + "score": 0.91, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "in the policy space. For", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 479, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 308, + 426 + ], + "score": 1.0, + "content": "sufficiently rich policy spaces, like deep-networks", + "type": "text" + }, + { + "bbox": [ + 308, + 414, + 344, + 426 + ], + "score": 0.93, + "content": "f _ { w } ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 414, + 479, + 426 + ], + "score": 1.0, + "content": "with softmax outputs of the form", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 428, + 378, + 457 + ], + "lines": [ + { + "bbox": [ + 232, + 428, + 378, + 457 + ], + "spans": [ + { + "bbox": [ + 232, + 428, + 378, + 457 + ], + "score": 0.95, + "content": "\\pi _ { w } ( y | x ) = \\frac { e x p ( f _ { w } ( x , y ) ) } { \\sum _ { y ^ { \\prime } \\in \\mathcal { V } } e x p ( f _ { w } ( x , y ^ { \\prime } ) ) } ,", + "type": "interline_equation", + "image_path": "69fbd4c638d436ea0fd8a8b517589701ae75c671b82e68853958c5316b83b8b8.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 428, + 378, + 442.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 232, + 442.5, + 378, + 457.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 243, + 473 + ], + "score": 1.0, + "content": "this means that the logging policy", + "type": "text" + }, + { + "bbox": [ + 243, + 461, + 254, + 471 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 458, + 463, + 473 + ], + "score": 1.0, + "content": "needs to assign non-zero probability to every action", + "type": "text" + }, + { + "bbox": [ + 463, + 462, + 470, + 471 + ], + "score": 0.82, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 458, + 505, + 473 + ], + "score": 1.0, + "content": "in every", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 138, + 483 + ], + "score": 1.0, + "content": "context", + "type": "text" + }, + { + "bbox": [ + 139, + 473, + 145, + 480 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 471, + 506, + 483 + ], + "score": 1.0, + "content": ". This is a strong condition that is not feasible in many real-world systems, especially if", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 482, + 349, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 349, + 494 + ], + "score": 1.0, + "content": "the action space is large and many actions have poor reward.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "If the support requirement is violated, ERM learning can fail catastrophically. We will show below", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 524 + ], + "score": 1.0, + "content": "that the underlying reason is bias, not excessive variance that could be remedied through clipping", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 349, + 533 + ], + "score": 1.0, + "content": "or variance regularization (Strehl et al., 2011; Swaminathan", + "type": "text" + }, + { + "bbox": [ + 349, + 521, + 358, + 531 + ], + "score": 0.36, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "Joachims, 2015a). To quantify how", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 471, + 544 + ], + "score": 1.0, + "content": "support deficient a logging policy is, we denote the set of unsupported actions for context", + "type": "text" + }, + { + "bbox": [ + 471, + 533, + 478, + 541 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 544, + 131, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 118, + 553 + ], + "score": 0.82, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 544, + 131, + 554 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 552, + 376, + 565 + ], + "lines": [ + { + "bbox": [ + 233, + 552, + 376, + 565 + ], + "spans": [ + { + "bbox": [ + 233, + 552, + 376, + 565 + ], + "score": 0.92, + "content": "\\mathcal { U } ( x , \\pi _ { 0 } ) : = \\{ y \\in y | \\pi _ { 0 } ( y | x ) = 0 \\} .", + "type": "interline_equation", + "image_path": "fb7e1f3e0c702a131e358a03a19362fe2932c52e9a64ecb04540a8d59792f036.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 233, + 552, + 376, + 565 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "The bias of the IPS estimator is then characterized by the expected reward on the unsupported", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 576, + 141, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 141, + 590 + ], + "score": 1.0, + "content": "actions.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 232, + 605 + ], + "score": 1.0, + "content": "Proposition 1. Given contexts", + "type": "text" + }, + { + "bbox": [ + 232, + 591, + 276, + 604 + ], + "score": 0.93, + "content": "x \\sim P ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 590, + 355, + 605 + ], + "score": 1.0, + "content": "and logging policy", + "type": "text" + }, + { + "bbox": [ + 356, + 591, + 390, + 604 + ], + "score": 0.94, + "content": "\\pi _ { 0 } ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 590, + 439, + 605 + ], + "score": 1.0, + "content": ", the bias of", + "type": "text" + }, + { + "bbox": [ + 439, + 590, + 463, + 603 + ], + "score": 0.9, + "content": "\\hat { R } _ { I P S }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 590, + 505, + 605 + ], + "score": 1.0, + "content": "for target", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 133, + 616 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 134, + 603, + 165, + 614 + ], + "score": 0.9, + "content": "\\pi ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 602, + 447, + 616 + ], + "score": 1.0, + "content": "is equal to the expected reward on the unsupported action sets, i.e.,", + "type": "text" + }, + { + "bbox": [ + 447, + 603, + 505, + 615 + ], + "score": 0.88, + "content": "b i a s ( \\pi | \\pi _ { 0 } ) =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 612, + 244, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 239, + 628 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x } [ - \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 612, + 244, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "The proof is in Appendix A.1. From Proposition 1, it is clear that support deficient log data can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "score": 1.0, + "content": "drastically mislead ERM learning. 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For contexts", + "type": "text" + }, + { + "bbox": [ + 306, + 671, + 351, + 683 + ], + "score": 0.91, + "content": "x \\sim P ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 669, + 505, + 685 + ], + "score": 1.0, + "content": "and any corresponding pair of target", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 682, + 383, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 133, + 695 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 133, + 684, + 141, + 692 + ], + "score": 0.6, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 682, + 219, + 695 + ], + "score": 1.0, + "content": "and logging policy", + "type": "text" + }, + { + "bbox": [ + 219, + 683, + 230, + 693 + ], + "score": 0.83, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 682, + 383, + 695 + ], + "score": 1.0, + "content": ", the Support Divergence is defined as", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 696, + 394, + 736 + ], + "lines": [ + { + "bbox": [ + 217, + 696, + 394, + 736 + ], + "spans": [ + { + "bbox": [ + 217, + 696, + 394, + 736 + ], + "score": 0.94, + "content": "\\mathcal { D } _ { \\mathcal { X } } \\big ( \\pi \\big | \\pi _ { 0 } \\big ) : = \\underset { x \\sim P ( \\mathcal { X } ) } { \\mathbb { E } } \\left[ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\right] .", + "type": "interline_equation", + "image_path": "e6411aa184ef6ad7a01d465cc6e94f57c67796db9adc502772ec13ed13ba43a9.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 217, + 696, + 394, + 709.3333333333334 + ], + "spans": [], + "index": 47 + }, + { + "bbox": [ + 217, + 709.3333333333334, + 394, + 722.6666666666667 + ], + "spans": [], + "index": 48 + }, + { + "bbox": [ + 217, + 722.6666666666667, + 394, + 736.0000000000001 + ], + "spans": [], + "index": 49 + } + ] + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 94 + ], + "score": 1.0, + "content": "of past system interactions which consists of context-action-reward-propensity tuples. 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We call", + "type": "text" + }, + { + "bbox": [ + 227, + 106, + 238, + 115 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "the logging policy, and we will discuss desired conditions on the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 170, + 127 + ], + "score": 1.0, + "content": "stochasticity of", + "type": "text" + }, + { + "bbox": [ + 170, + 117, + 182, + 127 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "in the following. The goal of off-policy learning is to exploit the information in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 380, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 170, + 138 + ], + "score": 1.0, + "content": "the logged data", + "type": "text" + }, + { + "bbox": [ + 170, + 127, + 179, + 136 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 127, + 243, + 138 + ], + "score": 1.0, + "content": "to find a policy", + "type": "text" + }, + { + "bbox": [ + 243, + 127, + 270, + 137 + ], + "score": 0.9, + "content": "{ \\hat { \\pi } } \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 127, + 354, + 138 + ], + "score": 1.0, + "content": "that has high reward", + "type": "text" + }, + { + "bbox": [ + 354, + 126, + 376, + 138 + ], + "score": 0.92, + "content": "R ( { \\hat { \\pi } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 127, + 380, + 138 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 82, + 505, + 138 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 142, + 505, + 178 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "Analogous to the ERM principle in supervised learning, off-policy learning algorithms typically", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 249, + 168 + ], + "score": 1.0, + "content": "optimize a counterfactual estimate", + "type": "text" + }, + { + "bbox": [ + 249, + 154, + 272, + 168 + ], + "score": 0.94, + "content": "\\hat { R } ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 156, + 286, + 168 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 286, + 155, + 308, + 168 + ], + "score": 0.92, + "content": "R ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "as the training objective (Li et al., 2011; 2015;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 327, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 327, + 179 + ], + "score": 1.0, + "content": "Bottou et al., 2013; Swaminathan & Joachims, 2015a).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 143, + 505, + 179 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 181, + 347, + 203 + ], + "lines": [ + { + "bbox": [ + 263, + 181, + 347, + 203 + ], + "spans": [ + { + "bbox": [ + 263, + 181, + 347, + 203 + ], + "score": 0.95, + "content": "\\hat { \\pi } = \\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m a x } } [ \\hat { R } ( \\pi ) ]", + "type": "interline_equation", + "image_path": "e3b58cd496cef3671c86be4c1e71054edc436e42aee1fe5d47e0ebeade5554d8.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 263, + 181, + 347, + 203 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "For conciseness, we ignore additional regularization terms in the objective (Swaminathan &", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "Joachims, 2015a), since they are irrelevant to the main point of this paper. As counterfactual es-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 137, + 242 + ], + "score": 1.0, + "content": "timator", + "type": "text" + }, + { + "bbox": [ + 138, + 228, + 160, + 241 + ], + "score": 0.93, + "content": "\\hat { R } ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 228, + 505, + 242 + ], + "score": 1.0, + "content": ", most algorithms rely on some form of IPS weighting (Strehl et al., 2011; Dud´ık et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "score": 1.0, + "content": "2011; Swaminathan & Joachims, 2015a;b; Wang et al., 2017; Su et al., 2019) to correct the distribu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 410, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 276, + 264 + ], + "score": 1.0, + "content": "tion mismatch between the logging policy", + "type": "text" + }, + { + "bbox": [ + 276, + 253, + 288, + 263 + ], + "score": 0.85, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 250, + 379, + 264 + ], + "score": 1.0, + "content": "and each target policy", + "type": "text" + }, + { + "bbox": [ + 379, + 252, + 406, + 262 + ], + "score": 0.9, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 250, + 410, + 264 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 206, + 505, + 264 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 266, + 370, + 298 + ], + "lines": [ + { + "bbox": [ + 240, + 266, + 370, + 298 + ], + "spans": [ + { + "bbox": [ + 240, + 266, + 370, + 298 + ], + "score": 0.95, + "content": "\\hat { R } _ { I P S } ( \\pi ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } .", + "type": "interline_equation", + "image_path": "2d8cffe0b06bb469b79cf452f32e68c3c96c63c9caf1a4e0c57f7536520747d8.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 266, + 370, + 282.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 240, + 282.0, + 370, + 298.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "A crucial condition for the effectiveness of the IPS estimator (and similar estimators) is that the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 168, + 324 + ], + "score": 1.0, + "content": "logging policy", + "type": "text" + }, + { + "bbox": [ + 168, + 314, + 180, + 323 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "assigns non-zero probability to all actions that have non-zero probability under", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 173, + 335 + ], + "score": 1.0, + "content": "the target policy", + "type": "text" + }, + { + "bbox": [ + 173, + 325, + 180, + 333 + ], + "score": 0.7, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "we aim to evaluate. This condition is known as positivity or full support, and it is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 184, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 184, + 346 + ], + "score": 1.0, + "content": "defined as follows.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 301, + 505, + 346 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 503, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 295, + 361 + ], + "score": 1.0, + "content": "Definition 1 (Full support). The logging policy", + "type": "text" + }, + { + "bbox": [ + 296, + 349, + 307, + 359 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 346, + 429, + 361 + ], + "score": 1.0, + "content": "is said to have full support for", + "type": "text" + }, + { + "bbox": [ + 429, + 350, + 436, + 358 + ], + "score": 0.69, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 346, + 460, + 361 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 461, + 348, + 505, + 360 + ], + "score": 0.91, + "content": "\\pi _ { 0 } ( y | x ) >", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 358, + 371, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 172, + 371 + ], + "score": 1.0, + "content": "0 for all actions", + "type": "text" + }, + { + "bbox": [ + 172, + 359, + 199, + 370 + ], + "score": 0.91, + "content": "y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 358, + 252, + 371 + ], + "score": 1.0, + "content": "and contexts", + "type": "text" + }, + { + "bbox": [ + 252, + 360, + 280, + 370 + ], + "score": 0.89, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 358, + 321, + 371 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 322, + 358, + 368, + 371 + ], + "score": 0.93, + "content": "\\pi ( y | x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 358, + 371, + 371 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 346, + 505, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 294, + 393 + ], + "score": 1.0, + "content": "It is known that the IPS estimator is unbiased,", + "type": "text" + }, + { + "bbox": [ + 294, + 379, + 388, + 393 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] = R ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 380, + 476, + 393 + ], + "score": 1.0, + "content": ", if the logging policy", + "type": "text" + }, + { + "bbox": [ + 476, + 382, + 488, + 392 + ], + "score": 0.85, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 169, + 404 + ], + "score": 1.0, + "content": "full support for", + "type": "text" + }, + { + "bbox": [ + 170, + 393, + 177, + 402 + ], + "score": 0.72, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "(Li et al., 2011). To ensure unbiased ERM, algorithms that use the IPS estimator", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 232, + 415 + ], + "score": 1.0, + "content": "require that the logging policy", + "type": "text" + }, + { + "bbox": [ + 232, + 404, + 244, + 414 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 403, + 374, + 415 + ], + "score": 1.0, + "content": "has full support for all policies", + "type": "text" + }, + { + "bbox": [ + 374, + 403, + 403, + 413 + ], + "score": 0.91, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "in the policy space. For", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 479, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 308, + 426 + ], + "score": 1.0, + "content": "sufficiently rich policy spaces, like deep-networks", + "type": "text" + }, + { + "bbox": [ + 308, + 414, + 344, + 426 + ], + "score": 0.93, + "content": "f _ { w } ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 414, + 479, + 426 + ], + "score": 1.0, + "content": "with softmax outputs of the form", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 379, + 505, + 426 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 428, + 378, + 457 + ], + "lines": [ + { + "bbox": [ + 232, + 428, + 378, + 457 + ], + "spans": [ + { + "bbox": [ + 232, + 428, + 378, + 457 + ], + "score": 0.95, + "content": "\\pi _ { w } ( y | x ) = \\frac { e x p ( f _ { w } ( x , y ) ) } { \\sum _ { y ^ { \\prime } \\in \\mathcal { V } } e x p ( f _ { w } ( x , y ^ { \\prime } ) ) } ,", + "type": "interline_equation", + "image_path": "69fbd4c638d436ea0fd8a8b517589701ae75c671b82e68853958c5316b83b8b8.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 428, + 378, + 442.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 232, + 442.5, + 378, + 457.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 243, + 473 + ], + "score": 1.0, + "content": "this means that the logging policy", + "type": "text" + }, + { + "bbox": [ + 243, + 461, + 254, + 471 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 458, + 463, + 473 + ], + "score": 1.0, + "content": "needs to assign non-zero probability to every action", + "type": "text" + }, + { + "bbox": [ + 463, + 462, + 470, + 471 + ], + "score": 0.82, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 458, + 505, + 473 + ], + "score": 1.0, + "content": "in every", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 138, + 483 + ], + "score": 1.0, + "content": "context", + "type": "text" + }, + { + "bbox": [ + 139, + 473, + 145, + 480 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 471, + 506, + 483 + ], + "score": 1.0, + "content": ". This is a strong condition that is not feasible in many real-world systems, especially if", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 482, + 349, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 349, + 494 + ], + "score": 1.0, + "content": "the action space is large and many actions have poor reward.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 458, + 506, + 494 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "If the support requirement is violated, ERM learning can fail catastrophically. We will show below", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 524 + ], + "score": 1.0, + "content": "that the underlying reason is bias, not excessive variance that could be remedied through clipping", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 349, + 533 + ], + "score": 1.0, + "content": "or variance regularization (Strehl et al., 2011; Swaminathan", + "type": "text" + }, + { + "bbox": [ + 349, + 521, + 358, + 531 + ], + "score": 0.36, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "Joachims, 2015a). To quantify how", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 471, + 544 + ], + "score": 1.0, + "content": "support deficient a logging policy is, we denote the set of unsupported actions for context", + "type": "text" + }, + { + "bbox": [ + 471, + 533, + 478, + 541 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 544, + 131, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 118, + 553 + ], + "score": 0.82, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 544, + 131, + 554 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 498, + 506, + 554 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 552, + 376, + 565 + ], + "lines": [ + { + "bbox": [ + 233, + 552, + 376, + 565 + ], + "spans": [ + { + "bbox": [ + 233, + 552, + 376, + 565 + ], + "score": 0.92, + "content": "\\mathcal { U } ( x , \\pi _ { 0 } ) : = \\{ y \\in y | \\pi _ { 0 } ( y | x ) = 0 \\} .", + "type": "interline_equation", + "image_path": "fb7e1f3e0c702a131e358a03a19362fe2932c52e9a64ecb04540a8d59792f036.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 233, + 552, + 376, + 565 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "The bias of the IPS estimator is then characterized by the expected reward on the unsupported", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 576, + 141, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 141, + 590 + ], + "score": 1.0, + "content": "actions.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 565, + 505, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 232, + 605 + ], + "score": 1.0, + "content": "Proposition 1. Given contexts", + "type": "text" + }, + { + "bbox": [ + 232, + 591, + 276, + 604 + ], + "score": 0.93, + "content": "x \\sim P ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 590, + 355, + 605 + ], + "score": 1.0, + "content": "and logging policy", + "type": "text" + }, + { + "bbox": [ + 356, + 591, + 390, + 604 + ], + "score": 0.94, + "content": "\\pi _ { 0 } ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 590, + 439, + 605 + ], + "score": 1.0, + "content": ", the bias of", + "type": "text" + }, + { + "bbox": [ + 439, + 590, + 463, + 603 + ], + "score": 0.9, + "content": "\\hat { R } _ { I P S }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 590, + 505, + 605 + ], + "score": 1.0, + "content": "for target", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 133, + 616 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 134, + 603, + 165, + 614 + ], + "score": 0.9, + "content": "\\pi ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 602, + 447, + 616 + ], + "score": 1.0, + "content": "is equal to the expected reward on the unsupported action sets, i.e.,", + "type": "text" + }, + { + "bbox": [ + 447, + 603, + 505, + 615 + ], + "score": 0.88, + "content": "b i a s ( \\pi | \\pi _ { 0 } ) =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 612, + 244, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 239, + 628 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x } [ - \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 612, + 244, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 590, + 505, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "The proof is in Appendix A.1. From Proposition 1, it is clear that support deficient log data can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "score": 1.0, + "content": "drastically mislead ERM learning. To quantify the effect of support deficiency on ERM, we define", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 657, + 446, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 302, + 671 + ], + "score": 1.0, + "content": "the support divergence between a logging policy", + "type": "text" + }, + { + "bbox": [ + 302, + 659, + 313, + 668 + ], + "score": 0.85, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 657, + 390, + 671 + ], + "score": 1.0, + "content": "and a target policy", + "type": "text" + }, + { + "bbox": [ + 390, + 659, + 398, + 667 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 657, + 446, + 671 + ], + "score": 1.0, + "content": "as follows.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 635, + 505, + 671 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 669, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 306, + 685 + ], + "score": 1.0, + "content": "Definition 2 (Support Divergence). For contexts", + "type": "text" + }, + { + "bbox": [ + 306, + 671, + 351, + 683 + ], + "score": 0.91, + "content": "x \\sim P ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 669, + 505, + 685 + ], + "score": 1.0, + "content": "and any corresponding pair of target", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 682, + 383, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 133, + 695 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 133, + 684, + 141, + 692 + ], + "score": 0.6, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 682, + 219, + 695 + ], + "score": 1.0, + "content": "and logging policy", + "type": "text" + }, + { + "bbox": [ + 219, + 683, + 230, + 693 + ], + "score": 0.83, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 682, + 383, + 695 + ], + "score": 1.0, + "content": ", the Support Divergence is defined as", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 669, + 505, + 695 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 696, + 394, + 736 + ], + "lines": [ + { + "bbox": [ + 217, + 696, + 394, + 736 + ], + "spans": [ + { + "bbox": [ + 217, + 696, + 394, + 736 + ], + "score": 0.94, + "content": "\\mathcal { D } _ { \\mathcal { X } } \\big ( \\pi \\big | \\pi _ { 0 } \\big ) : = \\underset { x \\sim P ( \\mathcal { X } ) } { \\mathbb { E } } \\left[ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\right] .", + "type": "interline_equation", + "image_path": "e6411aa184ef6ad7a01d465cc6e94f57c67796db9adc502772ec13ed13ba43a9.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 217, + 696, + 394, + 709.3333333333334 + ], + "spans": [], + "index": 47 + }, + { + "bbox": [ + 217, + 709.3333333333334, + 394, + 722.6666666666667 + ], + "spans": [], + "index": 48 + }, + { + "bbox": [ + 217, + 722.6666666666667, + 394, + 736.0000000000001 + ], + "spans": [], + "index": 49 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "With this definition in hand, we can quantify the effect of support deficiency on ERM learning for a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 269, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 158, + 107 + ], + "score": 1.0, + "content": "policy space", + "type": "text" + }, + { + "bbox": [ + 158, + 94, + 167, + 104 + ], + "score": 0.38, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 93, + 253, + 107 + ], + "score": 1.0, + "content": "under logging policy", + "type": "text" + }, + { + "bbox": [ + 254, + 96, + 265, + 105 + ], + "score": 0.82, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 93, + 269, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 108, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 506, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 302, + 121 + ], + "score": 1.0, + "content": "Theorem 1. For any given hypothesis space", + "type": "text" + }, + { + "bbox": [ + 302, + 108, + 311, + 118 + ], + "score": 0.56, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 108, + 399, + 121 + ], + "score": 1.0, + "content": "with logging policy", + "type": "text" + }, + { + "bbox": [ + 400, + 108, + 439, + 119 + ], + "score": 0.9, + "content": "\\pi _ { 0 } ~ \\in ~ \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 108, + 506, + 121 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 189, + 132 + ], + "score": 1.0, + "content": "reward distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 119, + 203, + 130 + ], + "score": 0.87, + "content": "\\mathcal { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 118, + 275, + 132 + ], + "score": 1.0, + "content": "with support in", + "type": "text" + }, + { + "bbox": [ + 276, + 119, + 327, + 131 + ], + "score": 0.9, + "content": "[ r _ { m i n } , r _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 118, + 505, + 132 + ], + "score": 1.0, + "content": "such that in the limit of infinite training", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 293, + 143 + ], + "score": 1.0, + "content": "data, ERM using IPS over the logged data", + "type": "text" + }, + { + "bbox": [ + 293, + 130, + 420, + 142 + ], + "score": 0.91, + "content": "\\mathcal D \\sim \\dot { P } ( \\mathcal X ) \\times \\pi _ { 0 } ( \\cdot | \\mathcal X ) \\times \\mathcal P _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 129, + 505, + 143 + ], + "score": 1.0, + "content": "can select a policy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 141, + 487, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 235, + 155 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\pi } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 142, + 295, + 156 + ], + "score": 1.0, + "content": "that is at least", + "type": "text" + }, + { + "bbox": [ + 295, + 143, + 436, + 155 + ], + "score": 0.87, + "content": "\\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\quad } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 142, + 487, + 156 + ], + "score": 1.0, + "content": "suboptimal.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 162, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 486, + 175 + ], + "score": 1.0, + "content": "The proof is in Appendix A.2. To illustrate the theorem, consider a problem with rewards", + "type": "text" + }, + { + "bbox": [ + 486, + 164, + 505, + 174 + ], + "score": 0.84, + "content": "r \\in", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 173, + 504, + 187 + ], + "spans": [ + { + "bbox": [ + 107, + 174, + 135, + 186 + ], + "score": 0.9, + "content": "[ - 1 , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 173, + 289, + 187 + ], + "score": 1.0, + "content": ". Furthermore, consider a policy space", + "type": "text" + }, + { + "bbox": [ + 290, + 174, + 299, + 185 + ], + "score": 0.46, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 173, + 409, + 187 + ], + "score": 1.0, + "content": "that contains a good policy", + "type": "text" + }, + { + "bbox": [ + 410, + 176, + 421, + 186 + ], + "score": 0.85, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 173, + 443, + 187 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 443, + 174, + 504, + 186 + ], + "score": 0.88, + "content": "R ( \\pi _ { g } ) = - 0 . 1", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 175, + 198 + ], + "score": 1.0, + "content": "and a bad policy", + "type": "text" + }, + { + "bbox": [ + 176, + 186, + 187, + 196 + ], + "score": 0.85, + "content": "\\pi _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 186, + 208, + 198 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 209, + 185, + 271, + 197 + ], + "score": 0.92, + "content": "R ( \\pi _ { b } ) = - 0 . 7 .", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 186, + 312, + 198 + ], + "score": 1.0, + "content": ". If policy", + "type": "text" + }, + { + "bbox": [ + 312, + 187, + 323, + 196 + ], + "score": 0.85, + "content": "\\pi _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 186, + 419, + 198 + ], + "score": 1.0, + "content": "has support divergence", + "type": "text" + }, + { + "bbox": [ + 420, + 186, + 493, + 197 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi _ { b } \\vert \\pi _ { 0 } ) = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 496, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 255, + 209 + ], + "score": 1.0, + "content": "larger, then ERM may return the bad", + "type": "text" + }, + { + "bbox": [ + 255, + 198, + 266, + 207 + ], + "score": 0.83, + "content": "\\pi _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 196, + 309, + 209 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 309, + 198, + 320, + 209 + ], + "score": 0.88, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 196, + 496, + 209 + ], + "score": 1.0, + "content": "even with infinite amounts of training data.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 320, + 225 + ], + "score": 1.0, + "content": "Note that it is sufficient to merely have one policy in", + "type": "text" + }, + { + "bbox": [ + 320, + 213, + 329, + 223 + ], + "score": 0.47, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "that has large support deficiency to achieve", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 224, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 417, + 236 + ], + "score": 1.0, + "content": "this suboptimality. It is therefore crucial to control the support divergence", + "type": "text" + }, + { + "bbox": [ + 417, + 224, + 460, + 236 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 225, + 504, + 236 + ], + "score": 1.0, + "content": "uniformly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 140, + 247 + ], + "score": 1.0, + "content": "over all", + "type": "text" + }, + { + "bbox": [ + 140, + 235, + 169, + 245 + ], + "score": 0.89, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 235, + 505, + 247 + ], + "score": 1.0, + "content": ", or to account for the suboptimality it can induce. To this effect, we explore three", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 222, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 222, + 259 + ], + "score": 1.0, + "content": "approaches in the following.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 271, + 358, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 361, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 361, + 283 + ], + "score": 1.0, + "content": "3.1 SAFE LEARNING BY RESTRICTING THE ACTION SPACE", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 339, + 304 + ], + "score": 1.0, + "content": "The first and arguably most direct approach to reducing", + "type": "text" + }, + { + "bbox": [ + 339, + 291, + 381, + 303 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "is to disallow any action that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "has zero support under the logging policy. For the remaining action set, the logging policy has full", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "support by definition. This restriction of the action set can be achieved by transforming each policy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 324, + 442, + 336 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 133, + 334 + ], + "score": 0.89, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 324, + 442, + 336 + ], + "score": 1.0, + "content": "into a new policy that sets the probability of the unsupported actions to zero.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 339, + 405, + 370 + ], + "lines": [ + { + "bbox": [ + 207, + 339, + 405, + 370 + ], + "spans": [ + { + "bbox": [ + 207, + 339, + 405, + 370 + ], + "score": 0.95, + "content": "\\pi ( y | x ) \\longrightarrow \\bar { \\pi } ( y | x ) : = \\frac { \\pi ( y | x ) \\mathbb { 1 } _ { \\{ y \\notin \\mathcal { U } ( x , \\pi _ { 0 } ) \\} } } { 1 - \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) }", + "type": "interline_equation", + "image_path": "9ee516459c258d49f7603e0d84225a234f07343a79f1f2a4c77f9ab3ab81133c.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 339, + 405, + 354.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 207, + 354.5, + 405, + 370.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 373, + 504, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 241, + 388 + ], + "score": 1.0, + "content": "This results in a new policy space", + "type": "text" + }, + { + "bbox": [ + 242, + 374, + 250, + 384 + ], + "score": 0.7, + "content": "\\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 373, + 269, + 388 + ], + "score": 1.0, + "content": ". All", + "type": "text" + }, + { + "bbox": [ + 270, + 373, + 297, + 385 + ], + "score": 0.89, + "content": "\\bar { \\pi } \\in \\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 373, + 426, + 388 + ], + "score": 1.0, + "content": "have support divergence of zero", + "type": "text" + }, + { + "bbox": [ + 426, + 374, + 487, + 387 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\bar { \\pi } | \\pi _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 373, + 505, + 388 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 385, + 279, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 279, + 397 + ], + "score": 1.0, + "content": "ERM via IPS is guaranteed to be unbiased.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 307, + 416 + ], + "score": 1.0, + "content": "While this transformation of the policy space from", + "type": "text" + }, + { + "bbox": [ + 308, + 402, + 317, + 412 + ], + "score": 0.53, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 401, + 327, + 416 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 327, + 402, + 336, + 412 + ], + "score": 0.81, + "content": "\\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 401, + 506, + 416 + ], + "score": 1.0, + "content": "is conceptually straightforward, it has two", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "potential drawbacks. First, restricting the action space without any exceptions may overly constrain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 169, + 437 + ], + "score": 1.0, + "content": "the policies in", + "type": "text" + }, + { + "bbox": [ + 169, + 424, + 178, + 434 + ], + "score": 0.79, + "content": "\\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 425, + 331, + 437 + ], + "score": 1.0, + "content": ". In particular, if the optimal action", + "type": "text" + }, + { + "bbox": [ + 331, + 425, + 342, + 436 + ], + "score": 0.88, + "content": "y ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 425, + 435, + 437 + ], + "score": 1.0, + "content": "for a specific context", + "type": "text" + }, + { + "bbox": [ + 436, + 426, + 443, + 434 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "does not have", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 253, + 448 + ], + "score": 1.0, + "content": "support under the logging policy, no", + "type": "text" + }, + { + "bbox": [ + 254, + 434, + 281, + 446 + ], + "score": 0.92, + "content": "\\bar { \\pi } \\in \\dot { \\bar { \\Pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 435, + 348, + 448 + ], + "score": 1.0, + "content": "can ever choose", + "type": "text" + }, + { + "bbox": [ + 348, + 436, + 360, + 447 + ], + "score": 0.87, + "content": "y ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "even if there are many observations", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 149, + 459 + ], + "score": 1.0, + "content": "of similar", + "type": "text" + }, + { + "bbox": [ + 149, + 447, + 156, + 458 + ], + "score": 0.73, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 446, + 240, + 459 + ], + "score": 1.0, + "content": "’s on similar context", + "type": "text" + }, + { + "bbox": [ + 240, + 446, + 250, + 456 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 446, + 497, + 459 + ], + "score": 1.0, + "content": ". 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Given contexts", + "type": "text" + }, + { + "bbox": [ + 232, + 693, + 291, + 703 + ], + "score": 0.88, + "content": "x _ { 1 } , x _ { 2 } , \\ldots , x _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 690, + 462, + 704 + ], + "score": 1.0, + "content": "drawn i.i.d from the unknown distribution", + "type": "text" + }, + { + "bbox": [ + 462, + 691, + 487, + 703 + ], + "score": 0.91, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 690, + 506, + 704 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 135, + 715 + ], + "score": 1.0, + "content": "action", + "type": "text" + }, + { + "bbox": [ + 135, + 704, + 144, + 713 + ], + "score": 0.81, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 701, + 318, + 715 + ], + "score": 1.0, + "content": "drawn independently from logging policy", + "type": "text" + }, + { + "bbox": [ + 318, + 703, + 330, + 713 + ], + "score": 0.77, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 701, + 400, + 715 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 400, + 702, + 438, + 714 + ], + "score": 0.93, + "content": "\\pi _ { 0 } ( \\mathcal { V } | x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 701, + 505, + 715 + ], + "score": 1.0, + "content": ", the bias of the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 712, + 387, + 728 + ], + "spans": [ + { + "bbox": [ + 104, + 712, + 268, + 728 + ], + "score": 1.0, + "content": "empirical risk defined in Equation (7) is", + "type": "text" + }, + { + "bbox": [ + 269, + 713, + 382, + 726 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb E _ { x } [ \\sum _ { y \\in \\mathcal U _ { x } ^ { \\pi _ { 0 } } } \\pi ( y | x ) \\Delta ( x , y ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 712, + 387, + 728 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "With this definition in hand, we can quantify the effect of support deficiency on ERM learning for a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 269, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 158, + 107 + ], + "score": 1.0, + "content": "policy space", + "type": "text" + }, + { + "bbox": [ + 158, + 94, + 167, + 104 + ], + "score": 0.38, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 93, + 253, + 107 + ], + "score": 1.0, + "content": "under logging policy", + "type": "text" + }, + { + "bbox": [ + 254, + 96, + 265, + 105 + ], + "score": 0.82, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 93, + 269, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 506, + 107 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 108, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 506, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 302, + 121 + ], + "score": 1.0, + "content": "Theorem 1. For any given hypothesis space", + "type": "text" + }, + { + "bbox": [ + 302, + 108, + 311, + 118 + ], + "score": 0.56, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 108, + 399, + 121 + ], + "score": 1.0, + "content": "with logging policy", + "type": "text" + }, + { + "bbox": [ + 400, + 108, + 439, + 119 + ], + "score": 0.9, + "content": "\\pi _ { 0 } ~ \\in ~ \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 108, + 506, + 121 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 189, + 132 + ], + "score": 1.0, + "content": "reward distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 119, + 203, + 130 + ], + "score": 0.87, + "content": "\\mathcal { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 118, + 275, + 132 + ], + "score": 1.0, + "content": "with support in", + "type": "text" + }, + { + "bbox": [ + 276, + 119, + 327, + 131 + ], + "score": 0.9, + "content": "[ r _ { m i n } , r _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 118, + 505, + 132 + ], + "score": 1.0, + "content": "such that in the limit of infinite training", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 293, + 143 + ], + "score": 1.0, + "content": "data, ERM using IPS over the logged data", + "type": "text" + }, + { + "bbox": [ + 293, + 130, + 420, + 142 + ], + "score": 0.91, + "content": "\\mathcal D \\sim \\dot { P } ( \\mathcal X ) \\times \\pi _ { 0 } ( \\cdot | \\mathcal X ) \\times \\mathcal P _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 129, + 505, + 143 + ], + "score": 1.0, + "content": "can select a policy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 141, + 487, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 235, + 155 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\pi } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 142, + 295, + 156 + ], + "score": 1.0, + "content": "that is at least", + "type": "text" + }, + { + "bbox": [ + 295, + 143, + 436, + 155 + ], + "score": 0.87, + "content": "\\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\quad } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 142, + 487, + 156 + ], + "score": 1.0, + "content": "suboptimal.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 108, + 506, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 162, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 486, + 175 + ], + "score": 1.0, + "content": "The proof is in Appendix A.2. To illustrate the theorem, consider a problem with rewards", + "type": "text" + }, + { + "bbox": [ + 486, + 164, + 505, + 174 + ], + "score": 0.84, + "content": "r \\in", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 173, + 504, + 187 + ], + "spans": [ + { + "bbox": [ + 107, + 174, + 135, + 186 + ], + "score": 0.9, + "content": "[ - 1 , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 173, + 289, + 187 + ], + "score": 1.0, + "content": ". 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If policy", + "type": "text" + }, + { + "bbox": [ + 312, + 187, + 323, + 196 + ], + "score": 0.85, + "content": "\\pi _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 186, + 419, + 198 + ], + "score": 1.0, + "content": "has support divergence", + "type": "text" + }, + { + "bbox": [ + 420, + 186, + 493, + 197 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi _ { b } \\vert \\pi _ { 0 } ) = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 496, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 255, + 209 + ], + "score": 1.0, + "content": "larger, then ERM may return the bad", + "type": "text" + }, + { + "bbox": [ + 255, + 198, + 266, + 207 + ], + "score": 0.83, + "content": "\\pi _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 196, + 309, + 209 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 309, + 198, + 320, + 209 + ], + "score": 0.88, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 196, + 496, + 209 + ], + "score": 1.0, + "content": "even with infinite amounts of training data.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 163, + 505, + 209 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 320, + 225 + ], + "score": 1.0, + "content": "Note that it is sufficient to merely have one policy in", + "type": "text" + }, + { + "bbox": [ + 320, + 213, + 329, + 223 + ], + "score": 0.47, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "that has large support deficiency to achieve", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 224, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 417, + 236 + ], + "score": 1.0, + "content": "this suboptimality. It is therefore crucial to control the support divergence", + "type": "text" + }, + { + "bbox": [ + 417, + 224, + 460, + 236 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 225, + 504, + 236 + ], + "score": 1.0, + "content": "uniformly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 140, + 247 + ], + "score": 1.0, + "content": "over all", + "type": "text" + }, + { + "bbox": [ + 140, + 235, + 169, + 245 + ], + "score": 0.89, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 235, + 505, + 247 + ], + "score": 1.0, + "content": ", or to account for the suboptimality it can induce. To this effect, we explore three", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 222, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 222, + 259 + ], + "score": 1.0, + "content": "approaches in the following.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 213, + 505, + 259 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 271, + 358, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 361, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 361, + 283 + ], + "score": 1.0, + "content": "3.1 SAFE LEARNING BY RESTRICTING THE ACTION SPACE", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 339, + 304 + ], + "score": 1.0, + "content": "The first and arguably most direct approach to reducing", + "type": "text" + }, + { + "bbox": [ + 339, + 291, + 381, + 303 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "is to disallow any action that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "has zero support under the logging policy. For the remaining action set, the logging policy has full", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "support by definition. This restriction of the action set can be achieved by transforming each policy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 324, + 442, + 336 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 133, + 334 + ], + "score": 0.89, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 324, + 442, + 336 + ], + "score": 1.0, + "content": "into a new policy that sets the probability of the unsupported actions to zero.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 290, + 506, + 336 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 339, + 405, + 370 + ], + "lines": [ + { + "bbox": [ + 207, + 339, + 405, + 370 + ], + "spans": [ + { + "bbox": [ + 207, + 339, + 405, + 370 + ], + "score": 0.95, + "content": "\\pi ( y | x ) \\longrightarrow \\bar { \\pi } ( y | x ) : = \\frac { \\pi ( y | x ) \\mathbb { 1 } _ { \\{ y \\notin \\mathcal { U } ( x , \\pi _ { 0 } ) \\} } } { 1 - \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) }", + "type": "interline_equation", + "image_path": "9ee516459c258d49f7603e0d84225a234f07343a79f1f2a4c77f9ab3ab81133c.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 339, + 405, + 354.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 207, + 354.5, + 405, + 370.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 373, + 504, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 241, + 388 + ], + "score": 1.0, + "content": "This results in a new policy space", + "type": "text" + }, + { + "bbox": [ + 242, + 374, + 250, + 384 + ], + "score": 0.7, + "content": "\\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 373, + 269, + 388 + ], + "score": 1.0, + "content": ". All", + "type": "text" + }, + { + "bbox": [ + 270, + 373, + 297, + 385 + ], + "score": 0.89, + "content": "\\bar { \\pi } \\in \\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 373, + 426, + 388 + ], + "score": 1.0, + "content": "have support divergence of zero", + "type": "text" + }, + { + "bbox": [ + 426, + 374, + 487, + 387 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\bar { \\pi } | \\pi _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 373, + 505, + 388 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 385, + 279, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 279, + 397 + ], + "score": 1.0, + "content": "ERM via IPS is guaranteed to be unbiased.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 106, + 373, + 505, + 397 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 307, + 416 + ], + "score": 1.0, + "content": "While this transformation of the policy space from", + "type": "text" + }, + { + "bbox": [ + 308, + 402, + 317, + 412 + ], + "score": 0.53, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 401, + 327, + 416 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 327, + 402, + 336, + 412 + ], + "score": 0.81, + "content": "\\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 401, + 506, + 416 + ], + "score": 1.0, + "content": "is conceptually straightforward, it has two", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "potential drawbacks. First, restricting the action space without any exceptions may overly constrain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 169, + 437 + ], + "score": 1.0, + "content": "the policies in", + "type": "text" + }, + { + "bbox": [ + 169, + 424, + 178, + 434 + ], + "score": 0.79, + "content": "\\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 425, + 331, + 437 + ], + "score": 1.0, + "content": ". 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The second drawback is computational. For every context", + "type": "text" + }, + { + "bbox": [ + 497, + 448, + 504, + 456 + ], + "score": 0.69, + "content": "x", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 408, + 470 + ], + "score": 1.0, + "content": "during training and testing, the system needs to evaluate the logging policy", + "type": "text" + }, + { + "bbox": [ + 409, + 457, + 441, + 469 + ], + "score": 0.93, + "content": "\\pi _ { 0 } ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "to compute the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 189, + 480 + ], + "score": 1.0, + "content": "transformation from", + "type": "text" + }, + { + "bbox": [ + 189, + 470, + 196, + 478 + ], + "score": 0.71, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 468, + 207, + 480 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 207, + 469, + 214, + 478 + ], + "score": 0.78, + "content": "\\bar { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 468, + 506, + 480 + ], + "score": 1.0, + "content": ". This can be prohibitively expensive especially at test time, where – after", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "multiple rounds of off-policy learning with data from previously learned policies – we would need", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 490, + 464, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 464, + 504 + ], + "score": 1.0, + "content": "to evaluate the whole sequence of previous logging policies to execute the learned policy.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 401, + 506, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 515, + 360, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 361, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 361, + 527 + ], + "score": 1.0, + "content": "3.2 SAFE LEARNING THROUGH REWARD EXTRAPOLATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 549 + ], + "score": 1.0, + "content": "As illustrated above, support deficiency is a problem of blind spots where we lack information about", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "the rewards of some actions in some contexts. 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To minimize the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 298, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 298, + 182 + ], + "score": 1.0, + "content": "user impact of randomization in the logging", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 180, + 297, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 297, + 192 + ], + "score": 1.0, + "content": "policy, it is generally desirable to put zero prob-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 297, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 297, + 203 + ], + "score": 1.0, + "content": "ability on actions the are very likely to have", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 201, + 298, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 298, + 213 + ], + "score": 1.0, + "content": "low (or even catastrophic reward). This means", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 297, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 297, + 225 + ], + "score": 1.0, + "content": "that precisely those bad actions are likely to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 297, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 297, + 236 + ], + "score": 1.0, + "content": "not be supported in the logging policy. A key", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 298, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 298, + 247 + ], + "score": 1.0, + "content": "danger of blind spots regarding those actions is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 297, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 297, + 258 + ], + "score": 1.0, + "content": "that naive IPS training will inadvertently learn", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 306, + 86, + 443, + 98 + ], + "lines": [ + { + "bbox": [ + 304, + 86, + 443, + 99 + ], + "spans": [ + { + "bbox": [ + 304, + 86, + 443, + 99 + ], + "score": 1.0, + "content": "Algorithm 1: Data Augmentation", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 305, + 100, + 501, + 229 + ], + "lines": [ + { + "bbox": [ + 304, + 99, + 497, + 112 + ], + "spans": [ + { + "bbox": [ + 304, + 99, + 424, + 112 + ], + "score": 1.0, + "content": "input: original logged dataset", + "type": "text" + }, + { + "bbox": [ + 424, + 100, + 433, + 110 + ], + "score": 0.73, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 99, + 486, + 112 + ], + "score": 1.0, + "content": ", replaycount", + "type": "text" + }, + { + "bbox": [ + 487, + 101, + 493, + 110 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 99, + 497, + 112 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 308, + 111, + 456, + 125 + ], + "spans": [ + { + "bbox": [ + 308, + 112, + 375, + 125 + ], + "score": 1.0, + "content": "reward estimate", + "type": "text" + }, + { + "bbox": [ + 375, + 111, + 404, + 124 + ], + "score": 0.92, + "content": "\\hat { \\delta } ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 112, + 439, + 125 + ], + "score": 1.0, + "content": "; 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} \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 319, + 205, + 338, + 217 + ], + "spans": [ + { + "bbox": [ + 319, + 205, + 338, + 217 + ], + "score": 1.0, + "content": "end", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 303, + 217, + 323, + 228 + ], + "spans": [ + { + "bbox": [ + 303, + 217, + 323, + 228 + ], + "score": 1.0, + "content": "end", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "score": 1.0, + "content": "a policy that selects those actions. This can be avoided by being maximally conservative about", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 267, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 356, + 283 + ], + "score": 1.0, + "content": "unsupported actions and imputing the lowest possible reward", + "type": "text" + }, + { + "bbox": [ + 356, + 267, + 501, + 281 + ], + "score": 0.92, + "content": "\\forall x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) : \\hat { \\delta } ( x , y ) = r _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 267, + 506, + 283 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "Intuitively, by imposing the worst possible reward for the unsupported actions, the learning algo-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 401, + 303 + ], + "score": 1.0, + "content": "rithm will aim to avoid these low-reward areas. However, unlike for the", + "type": "text" + }, + { + "bbox": [ + 402, + 292, + 410, + 301 + ], + "score": 0.74, + "content": "\\bar { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "policies resulting from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "the restricted action space, the learned policy is not strictly prohibited from choosing unsupported", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "actions – it is merely made aware of the maximum loss that the action may incur. Note that for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 173, + 337 + ], + "score": 1.0, + "content": "problems where", + "type": "text" + }, + { + "bbox": [ + 174, + 325, + 215, + 335 + ], + "score": 0.91, + "content": "r _ { m i n } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 325, + 505, + 337 + ], + "score": 1.0, + "content": ", the naive IPS estimator is identical to conservative extrapolation since", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 335, + 266, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 266, + 347 + ], + "score": 1.0, + "content": "the second term in Equation (7) is zero.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 504, + 364 + ], + "score": 1.0, + "content": "Regression Extrapolation. Instead of extrapolating with the worst-case reward, we may have addi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "tional prior knowledge in the form of a model-based estimate that reduces the bias. In particular, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 368, + 509, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 368, + 249, + 395 + ], + "score": 1.0, + "content": "explore using a regression estimate", + "type": "text" + }, + { + "bbox": [ + 249, + 374, + 414, + 388 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\hat { \\delta } = \\arg \\operatorname* { m i n } _ { \\hat { \\delta } ^ { \\theta } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( \\hat { \\delta } ^ { \\theta } ( x _ { i } , y _ { i } ) - r _ { i } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 368, + 509, + 395 + ], + "score": 1.0, + "content": "that extrapolates from", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 387, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 178, + 403 + ], + "score": 1.0, + "content": "the observed data", + "type": "text" + }, + { + "bbox": [ + 178, + 389, + 188, + 399 + ], + "score": 0.66, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 387, + 233, + 403 + ], + "score": 1.0, + "content": ". Typically,", + "type": "text" + }, + { + "bbox": [ + 245, + 387, + 506, + 403 + ], + "score": 1.0, + "content": "comes from a parameterized class of regression functions. Other", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "regression objectives could also be used, such as weighted linear regression that itself uses impor-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "tance sampling as weights (Farajtabar et al., 2018). But, fundamentally, all regression approaches", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "assume that the regression model is not misspecified and that it can thus extrapolate well. Note that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 433, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 463, + 446 + ], + "score": 1.0, + "content": "the IPS part of Equation (7) can be changed to any estimators (with action set restricted on", + "type": "text" + }, + { + "bbox": [ + 463, + 433, + 504, + 445 + ], + "score": 0.92, + "content": "\\boldsymbol { \\mathcal { U } } ( \\boldsymbol { x } , \\pi _ { 0 } ) ^ { c }", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 132, + 457 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 132, + 446, + 139, + 454 + ], + "score": 0.69, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "), and it turns out that doubly robust (Dud´ık et al., 2011) and CAB (Su et al., 2019) are special", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 454, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 504, + 468 + ], + "score": 1.0, + "content": "extensions of regression extrapolation that substitute the IPS part with their corresponding estimator.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "Efficient Approximation. Evaluating the augmented IPS estimator from Equation (7) can be com-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 354, + 496 + ], + "score": 1.0, + "content": "putationally expensive if the number of unsupported actions", + "type": "text" + }, + { + "bbox": [ + 355, + 483, + 391, + 495 + ], + "score": 0.93, + "content": "\\boldsymbol { \\mathcal { U } } ( \\boldsymbol { x } , \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "is large. To overcome this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "problem, we propose to use sampling to estimate the expected reward on the unsupported action,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 503, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 316, + 519 + ], + "score": 1.0, + "content": "which can be thought of as augmenting the dataset", + "type": "text" + }, + { + "bbox": [ + 316, + 505, + 325, + 515 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 503, + 505, + 519 + ], + "score": 1.0, + "content": "with additional observations where the log-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "ging policy has zero support. In particular, we propose the data-augmentation procedure detailed in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 526, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 286, + 543 + ], + "score": 1.0, + "content": "Algorithm 1. With the additional bandit data", + "type": "text" + }, + { + "bbox": [ + 287, + 527, + 413, + 542 + ], + "score": 0.93, + "content": "\\mathcal { D } ^ { \\prime } = \\{ x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } , \\hat { \\delta } ( x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } ) , p _ { j } ^ { \\prime } \\} _ { j = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 526, + 506, + 543 + ], + "score": 1.0, + "content": "from Algorithm 1, the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 540, + 173, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 173, + 552 + ], + "score": 1.0, + "content": "new objective is", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 552, + 430, + 591 + ], + "lines": [ + { + "bbox": [ + 182, + 552, + 430, + 591 + ], + "spans": [ + { + "bbox": [ + 182, + 552, + 430, + 591 + ], + "score": 0.94, + "content": "\\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m i n } } \\left\\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\frac { \\pi ( y _ { j } ^ { \\prime } | x _ { j } ^ { \\prime } ) } { p _ { j } ^ { \\prime } } \\hat { \\delta } ( x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } ) \\right\\} .", + "type": "interline_equation", + "image_path": "7cda3245e8cd39208c55d189a1f9e57ac5baf4ab6bced57093d53c8e79fd389e.jpg" + } + ] + } + ], + "index": 53, + "virtual_lines": [ + { + "bbox": [ + 182, + 552, + 430, + 565.0 + ], + "spans": [], + "index": 52 + }, + { + "bbox": [ + 182, + 565.0, + 430, + 578.0 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 182, + 578.0, + 430, + 591.0 + ], + "spans": [], + "index": 54 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 592, + 507, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "In Appendix A.5, we show that the empirical risk in Equation (8) has the same expectation (over", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 603, + 492, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 167, + 618 + ], + "score": 1.0, + "content": "randomness in", + "type": "text" + }, + { + "bbox": [ + 167, + 605, + 176, + 614 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 603, + 194, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 194, + 604, + 206, + 614 + ], + "score": 0.82, + "content": "\\mathcal { D } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 603, + 221, + 618 + ], + "score": 1.0, + "content": ") as", + "type": "text" + }, + { + "bbox": [ + 222, + 603, + 262, + 617 + ], + "score": 0.94, + "content": "\\hat { R } _ { I P S } ^ { \\delta } ( \\mathcal { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 603, + 492, + 618 + ], + "score": 1.0, + "content": "and can thus serve as an approximation for Equation (7).", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 55.5 + }, + { + "type": "title", + "bbox": [ + 106, + 628, + 363, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 364, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 364, + 641 + ], + "score": 1.0, + "content": "3.3 SAFE LEARNING BY RESTRICTING THE POLICY SPACE", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 57 + }, + { + "type": "text", + "bbox": [ + 106, + 649, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "As motivated by Theorem 1, the risk of learning from support deficient data scales with the maxi-", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 208, + 673 + ], + "score": 1.0, + "content": "mum support divergence", + "type": "text" + }, + { + "bbox": [ + 208, + 660, + 251, + 672 + ], + "score": 0.94, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 659, + 409, + 673 + ], + "score": 1.0, + "content": "among the policies in the policy space", + "type": "text" + }, + { + "bbox": [ + 409, + 660, + 418, + 670 + ], + "score": 0.45, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". 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This can be avoided by being maximally conservative about", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 267, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 356, + 283 + ], + "score": 1.0, + "content": "unsupported actions and imputing the lowest possible reward", + "type": "text" + }, + { + "bbox": [ + 356, + 267, + 501, + 281 + ], + "score": 0.92, + "content": "\\forall x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) : \\hat { \\delta } ( x , y ) = r _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 267, + 506, + 283 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "Intuitively, by imposing the worst possible reward for the unsupported actions, the learning algo-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 401, + 303 + ], + "score": 1.0, + "content": "rithm will aim to avoid these low-reward areas. However, unlike for the", + "type": "text" + }, + { + "bbox": [ + 402, + 292, + 410, + 301 + ], + "score": 0.74, + "content": "\\bar { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "policies resulting from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "the restricted action space, the learned policy is not strictly prohibited from choosing unsupported", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "actions – it is merely made aware of the maximum loss that the action may incur. Note that for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 173, + 337 + ], + "score": 1.0, + "content": "problems where", + "type": "text" + }, + { + "bbox": [ + 174, + 325, + 215, + 335 + ], + "score": 0.91, + "content": "r _ { m i n } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 325, + 505, + 337 + ], + "score": 1.0, + "content": ", the naive IPS estimator is identical to conservative extrapolation since", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 335, + 266, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 266, + 347 + ], + "score": 1.0, + "content": "the second term in Equation (7) is zero.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 256, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 504, + 364 + ], + "score": 1.0, + "content": "Regression Extrapolation. Instead of extrapolating with the worst-case reward, we may have addi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "tional prior knowledge in the form of a model-based estimate that reduces the bias. In particular, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 368, + 509, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 368, + 249, + 395 + ], + "score": 1.0, + "content": "explore using a regression estimate", + "type": "text" + }, + { + "bbox": [ + 249, + 374, + 414, + 388 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\hat { \\delta } = \\arg \\operatorname* { m i n } _ { \\hat { \\delta } ^ { \\theta } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( \\hat { \\delta } ^ { \\theta } ( x _ { i } , y _ { i } ) - r _ { i } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 368, + 509, + 395 + ], + "score": 1.0, + "content": "that extrapolates from", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 387, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 178, + 403 + ], + "score": 1.0, + "content": "the observed data", + "type": "text" + }, + { + "bbox": [ + 178, + 389, + 188, + 399 + ], + "score": 0.66, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 387, + 233, + 403 + ], + "score": 1.0, + "content": ". Typically,", + "type": "text" + }, + { + "bbox": [ + 245, + 387, + 506, + 403 + ], + "score": 1.0, + "content": "comes from a parameterized class of regression functions. Other", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "regression objectives could also be used, such as weighted linear regression that itself uses impor-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "tance sampling as weights (Farajtabar et al., 2018). But, fundamentally, all regression approaches", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "assume that the regression model is not misspecified and that it can thus extrapolate well. Note that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 433, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 463, + 446 + ], + "score": 1.0, + "content": "the IPS part of Equation (7) can be changed to any estimators (with action set restricted on", + "type": "text" + }, + { + "bbox": [ + 463, + 433, + 504, + 445 + ], + "score": 0.92, + "content": "\\boldsymbol { \\mathcal { U } } ( \\boldsymbol { x } , \\pi _ { 0 } ) ^ { c }", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 132, + 457 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 132, + 446, + 139, + 454 + ], + "score": 0.69, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "), and it turns out that doubly robust (Dud´ık et al., 2011) and CAB (Su et al., 2019) are special", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 454, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 504, + 468 + ], + "score": 1.0, + "content": "extensions of regression extrapolation that substitute the IPS part with their corresponding estimator.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 351, + 509, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "Efficient Approximation. Evaluating the augmented IPS estimator from Equation (7) can be com-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 354, + 496 + ], + "score": 1.0, + "content": "putationally expensive if the number of unsupported actions", + "type": "text" + }, + { + "bbox": [ + 355, + 483, + 391, + 495 + ], + "score": 0.93, + "content": "\\boldsymbol { \\mathcal { U } } ( \\boldsymbol { x } , \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "is large. To overcome this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "problem, we propose to use sampling to estimate the expected reward on the unsupported action,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 503, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 316, + 519 + ], + "score": 1.0, + "content": "which can be thought of as augmenting the dataset", + "type": "text" + }, + { + "bbox": [ + 316, + 505, + 325, + 515 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 503, + 505, + 519 + ], + "score": 1.0, + "content": "with additional observations where the log-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "ging policy has zero support. In particular, we propose the data-augmentation procedure detailed in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 526, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 286, + 543 + ], + "score": 1.0, + "content": "Algorithm 1. With the additional bandit data", + "type": "text" + }, + { + "bbox": [ + 287, + 527, + 413, + 542 + ], + "score": 0.93, + "content": "\\mathcal { D } ^ { \\prime } = \\{ x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } , \\hat { \\delta } ( x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } ) , p _ { j } ^ { \\prime } \\} _ { j = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 526, + 506, + 543 + ], + "score": 1.0, + "content": "from Algorithm 1, the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 540, + 173, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 173, + 552 + ], + "score": 1.0, + "content": "new objective is", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48, + "bbox_fs": [ + 104, + 471, + 506, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 552, + 430, + 591 + ], + "lines": [ + { + "bbox": [ + 182, + 552, + 430, + 591 + ], + "spans": [ + { + "bbox": [ + 182, + 552, + 430, + 591 + ], + "score": 0.94, + "content": "\\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m i n } } \\left\\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\frac { \\pi ( y _ { j } ^ { \\prime } | x _ { j } ^ { \\prime } ) } { p _ { j } ^ { \\prime } } \\hat { \\delta } ( x _ { j } ^ { \\prime } , y _ { j } ^ { \\prime } ) \\right\\} .", + "type": "interline_equation", + "image_path": "7cda3245e8cd39208c55d189a1f9e57ac5baf4ab6bced57093d53c8e79fd389e.jpg" + } + ] + } + ], + "index": 53, + "virtual_lines": [ + { + "bbox": [ + 182, + 552, + 430, + 565.0 + ], + "spans": [], + "index": 52 + }, + { + "bbox": [ + 182, + 565.0, + 430, + 578.0 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 182, + 578.0, + 430, + 591.0 + ], + "spans": [], + "index": 54 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 592, + 507, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "In Appendix A.5, we show that the empirical risk in Equation (8) has the same expectation (over", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 603, + 492, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 167, + 618 + ], + "score": 1.0, + "content": "randomness in", + "type": "text" + }, + { + "bbox": [ + 167, + 605, + 176, + 614 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 603, + 194, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 194, + 604, + 206, + 614 + ], + "score": 0.82, + "content": "\\mathcal { D } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 603, + 221, + 618 + ], + "score": 1.0, + "content": ") as", + "type": "text" + }, + { + "bbox": [ + 222, + 603, + 262, + 617 + ], + "score": 0.94, + "content": "\\hat { R } _ { I P S } ^ { \\delta } ( \\mathcal { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 603, + 492, + 618 + ], + "score": 1.0, + "content": "and can thus serve as an approximation for Equation (7).", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 55.5, + "bbox_fs": [ + 105, + 591, + 506, + 618 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 628, + 363, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 364, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 364, + 641 + ], + "score": 1.0, + "content": "3.3 SAFE LEARNING BY RESTRICTING THE POLICY SPACE", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 57 + }, + { + "type": "text", + "bbox": [ + 106, + 649, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "As motivated by Theorem 1, the risk of learning from support deficient data scales with the maxi-", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 208, + 673 + ], + "score": 1.0, + "content": "mum support divergence", + "type": "text" + }, + { + "bbox": [ + 208, + 660, + 251, + 672 + ], + "score": 0.94, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 659, + 409, + 673 + ], + "score": 1.0, + "content": "among the policies in the policy space", + "type": "text" + }, + { + "bbox": [ + 409, + 660, + 418, + 670 + ], + "score": 0.45, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". Therefore, our third", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 302, + 684 + ], + "score": 1.0, + "content": "approach restricts the policy space to the subset", + "type": "text" + }, + { + "bbox": [ + 302, + 671, + 338, + 681 + ], + "score": 0.9, + "content": "\\Pi ^ { \\kappa } \\subset \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 671, + 442, + 684 + ], + "score": 1.0, + "content": "that contains the policies", + "type": "text" + }, + { + "bbox": [ + 442, + 671, + 471, + 681 + ], + "score": 0.9, + "content": "\\pi \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "with an", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 681, + 313, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 247, + 695 + ], + "score": 1.0, + "content": "acceptably low support divergence", + "type": "text" + }, + { + "bbox": [ + 247, + 682, + 309, + 694 + ], + "score": 0.9, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } ) \\le \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 681, + 313, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 59.5, + "bbox_fs": [ + 105, + 649, + 506, + 695 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 695, + 378, + 708 + ], + "lines": [ + { + "bbox": [ + 233, + 695, + 378, + 708 + ], + "spans": [ + { + "bbox": [ + 233, + 695, + 378, + 708 + ], + "score": 0.91, + "content": "\\Pi ^ { \\kappa } = \\{ \\pi | \\pi \\in \\Pi \\wedge { \\mathscr { D } } \\chi ( \\pi | \\pi _ { 0 } ) \\leq \\kappa \\}", + "type": "interline_equation", + "image_path": "2af08dcad4092a6ee5706a0b9b73bd6a07a13b61cf4f5db6fc366cc95d341f5d.jpg" + } + ] + } + ], + "index": 62, + "virtual_lines": [ + { + "bbox": [ + 233, + 695, + 378, + 708 + ], + "spans": [], + "index": 62 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 710, + 509, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 166, + 722 + ], + "score": 1.0, + "content": "The parameter", + "type": "text" + }, + { + "bbox": [ + 167, + 712, + 174, + 720 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "has an intuitive meaning. It specifies the maximum probability mass that a learned", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 363, + 732 + ], + "score": 1.0, + "content": "policy can place on unsupported actions. 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The following theorem (with proof in Appendix A.3) gives us", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 504, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 296, + 179 + ], + "score": 1.0, + "content": "an efficient way of estimating and controlling", + "type": "text" + }, + { + "bbox": [ + 297, + 167, + 339, + 179 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 166, + 466, + 179 + ], + "score": 1.0, + "content": "without explicit knowledge of", + "type": "text" + }, + { + "bbox": [ + 467, + 167, + 492, + 179 + ], + "score": 0.92, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 166, + 504, + 179 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 178, + 361, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 219, + 191 + ], + "score": 1.0, + "content": "access to the logging policy", + "type": "text" + }, + { + "bbox": [ + 219, + 179, + 231, + 189 + ], + "score": 0.83, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 178, + 361, + 191 + ], + "score": 1.0, + "content": "beyond the logged propensities.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 504, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 190, + 504, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 214, + 206 + ], + "score": 1.0, + "content": "Theorem 2. For contexts", + "type": "text" + }, + { + "bbox": [ + 214, + 194, + 224, + 204 + ], + "score": 0.82, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 190, + 295, + 206 + ], + "score": 1.0, + "content": "drawn i.i.d from", + "type": "text" + }, + { + "bbox": [ + 295, + 192, + 320, + 205 + ], + "score": 0.92, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 190, + 353, + 206 + ], + "score": 1.0, + "content": ", action", + "type": "text" + }, + { + "bbox": [ + 353, + 194, + 362, + 204 + ], + "score": 0.81, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 190, + 475, + 206 + ], + "score": 1.0, + "content": "drawn from logging policy", + "type": "text" + }, + { + "bbox": [ + 475, + 195, + 487, + 204 + ], + "score": 0.81, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 190, + 504, + 206 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 202, + 379, + 223 + ], + "spans": [ + { + "bbox": [ + 103, + 202, + 263, + 223 + ], + "score": 1.0, + "content": "define SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) .", + "type": "text" + }, + { + "bbox": [ + 257, + 205, + 320, + 219 + ], + "score": 1.0, + "content": "For any policy", + "type": "text" + }, + { + "bbox": [ + 320, + 208, + 328, + 216 + ], + "score": 0.71, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 205, + 379, + 219 + ], + "score": 1.0, + "content": "it holds that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 225, + 399, + 246 + ], + "lines": [ + { + "bbox": [ + 213, + 225, + 399, + 246 + ], + "spans": [ + { + "bbox": [ + 213, + 225, + 399, + 246 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\underset { x \\sim P ( \\mathcal { X } ) } { \\mathbb { E } } \\underset { y \\sim \\pi _ { 0 } ( \\cdot | x ) } { \\mathbb { E } } [ S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) = 1 } \\end{array}", + "type": "interline_equation", + "image_path": "c5da844e59a6ddd6d7938fe2c532c4e163ff2944041cf4f3ac48292fa0ff6663.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 213, + 225, + 399, + 246 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 271 + ], + "score": 1.0, + "content": "Using this theorem, the following proposition (proof in Appendix A.4, empirically verified in Ap-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 369, + 281 + ], + "score": 1.0, + "content": "pendix B) gives us an efficient way of implementing the constraint", + "type": "text" + }, + { + "bbox": [ + 370, + 268, + 431, + 280 + ], + "score": 0.92, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } ) \\le \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 268, + 446, + 281 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 447, + 268, + 501, + 280 + ], + "score": 0.92, + "content": "1 - S _ { D } ( \\pi | \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 268, + 504, + 281 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 232, + 308 + ], + "score": 1.0, + "content": "Proposition 3. For any given", + "type": "text" + }, + { + "bbox": [ + 232, + 294, + 277, + 306 + ], + "score": 0.48, + "content": "\\kappa \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 292, + 282, + 308 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 282, + 294, + 343, + 306 + ], + "score": 0.75, + "content": "0 < \\epsilon < \\kappa / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 292, + 361, + 308 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 361, + 295, + 383, + 306 + ], + "score": 0.85, + "content": "p _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 292, + 505, + 308 + ], + "score": 1.0, + "content": "denote the minimum propen-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 304, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 213, + 320 + ], + "score": 1.0, + "content": "sity under supported set", + "type": "text" + }, + { + "bbox": [ + 213, + 306, + 356, + 318 + ], + "score": 0.87, + "content": "p _ { m i n } ~ = ~ m a x _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 304, + 506, + 320 + ], + "score": 1.0, + "content": ", then with probability larger than", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 316, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 107, + 317, + 198, + 330 + ], + "score": 0.92, + "content": "1 - 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Instead of maximizing over", + "type": "text" + }, + { + "bbox": [ + 275, + 459, + 308, + 471 + ], + "score": 0.93, + "content": "( u _ { 1 } , u _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 458, + 441, + 471 + ], + "score": 1.0, + "content": "in the outer objective, we treat", + "type": "text" + }, + { + "bbox": [ + 441, + 459, + 484, + 471 + ], + "score": 0.92, + "content": "( u _ { 1 } - u _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "as a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 470, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 481 + ], + "score": 1.0, + "content": "hyperparameter that we select on a validation set. 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Furthermore, the form of the inner objective coincides with that of BanditNet (Joachims", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 529, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 504, + 545 + ], + "score": 1.0, + "content": "et al., 2018), which is known to work well for deep network training by controlling propensity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 541, + 295, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 295, + 554 + ], + "score": 1.0, + "content": "overfitting (Swaminathan & Joachims, 2015a).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 108, + 569, + 254, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 256, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 256, + 584 + ], + "score": 1.0, + "content": "4 EMPIRICAL EVALUATION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "We empirically evaluate the effectiveness and robustness of the three proposed approaches: restrict-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "score": 1.0, + "content": "ing the action space, conservative and regression extrapolation, as well as restricting the policy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "space. The semi-synthetic experiments are based on two real-world datasets: one is the popular im-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "age classification dataset CIFAR10 (Krizhevsky et al.) and the other is the credit-card fraud dataset", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "of Dal Pozzolo et al. (2015). We use the naive IPS estimator and the regression-based Direct Method", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 649, + 184, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 184, + 661 + ], + "score": 1.0, + "content": "(DM) as baselines.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "The experiments are set up as follows. 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To simulate bandit feedback for the CIFAR10 dataset, our experi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 271, + 722 + ], + "score": 1.0, + "content": "ment setup follows traditional supervised", + "type": "text" + }, + { + "bbox": [ + 271, + 711, + 284, + 720 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "bandit conversion for multi-class classification datasets", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "(Beygelzimer & Langford, 2009). 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For contexts", + "type": "text" + }, + { + "bbox": [ + 214, + 194, + 224, + 204 + ], + "score": 0.82, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 190, + 295, + 206 + ], + "score": 1.0, + "content": "drawn i.i.d from", + "type": "text" + }, + { + "bbox": [ + 295, + 192, + 320, + 205 + ], + "score": 0.92, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 190, + 353, + 206 + ], + "score": 1.0, + "content": ", action", + "type": "text" + }, + { + "bbox": [ + 353, + 194, + 362, + 204 + ], + "score": 0.81, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 190, + 475, + 206 + ], + "score": 1.0, + "content": "drawn from logging policy", + "type": "text" + }, + { + "bbox": [ + 475, + 195, + 487, + 204 + ], + "score": 0.81, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 190, + 504, + 206 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 202, + 379, + 223 + ], + "spans": [ + { + "bbox": [ + 103, + 202, + 263, + 223 + ], + "score": 1.0, + "content": "define SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi) .", + "type": "text" + }, + { + "bbox": [ + 257, + 205, + 320, + 219 + ], + "score": 1.0, + "content": "For any policy", + "type": "text" + }, + { + "bbox": [ + 320, + 208, + 328, + 216 + ], + "score": 0.71, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 205, + 379, + 219 + ], + "score": 1.0, + "content": "it holds that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 190, + 504, + 223 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 225, + 399, + 246 + ], + "lines": [ + { + "bbox": [ + 213, + 225, + 399, + 246 + ], + "spans": [ + { + "bbox": [ + 213, + 225, + 399, + 246 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\underset { x \\sim P ( \\mathcal { X } ) } { \\mathbb { E } } \\underset { y \\sim \\pi _ { 0 } ( \\cdot | x ) } { \\mathbb { E } } [ S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) = 1 } \\end{array}", + "type": "interline_equation", + "image_path": "c5da844e59a6ddd6d7938fe2c532c4e163ff2944041cf4f3ac48292fa0ff6663.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 213, + 225, + 399, + 246 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 271 + ], + "score": 1.0, + "content": "Using this theorem, the following proposition (proof in Appendix A.4, empirically verified in Ap-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 369, + 281 + ], + "score": 1.0, + "content": "pendix B) gives us an efficient way of implementing the constraint", + "type": "text" + }, + { + "bbox": [ + 370, + 268, + 431, + 280 + ], + "score": 0.92, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } ) \\le \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 268, + 446, + 281 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 447, + 268, + 501, + 280 + ], + "score": 0.92, + "content": "1 - S _ { D } ( \\pi | \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 268, + 504, + 281 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 255, + 505, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 232, + 308 + ], + "score": 1.0, + "content": "Proposition 3. 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Instead of maximizing over", + "type": "text" + }, + { + "bbox": [ + 275, + 459, + 308, + 471 + ], + "score": 0.93, + "content": "( u _ { 1 } , u _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 458, + 441, + 471 + ], + "score": 1.0, + "content": "in the outer objective, we treat", + "type": "text" + }, + { + "bbox": [ + 441, + 459, + 484, + 471 + ], + "score": 0.92, + "content": "( u _ { 1 } - u _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "as a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 470, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 481 + ], + "score": 1.0, + "content": "hyperparameter that we select on a validation set. We explore various estimators for this model-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 233, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 233, + 492 + ], + "score": 1.0, + "content": "selection problem in Section 4.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 447, + 506, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 497, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "Note that, among the methods we proposed for dealing with support deficiency, this approach is the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "most efficient to implement, and it does not require access to the logging policy during training or", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "testing. Furthermore, the form of the inner objective coincides with that of BanditNet (Joachims", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 529, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 504, + 545 + ], + "score": 1.0, + "content": "et al., 2018), which is known to work well for deep network training by controlling propensity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 541, + 295, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 295, + 554 + ], + "score": 1.0, + "content": "overfitting (Swaminathan & Joachims, 2015a).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 498, + 505, + 554 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 569, + 254, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 256, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 256, + 584 + ], + "score": 1.0, + "content": "4 EMPIRICAL EVALUATION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "We empirically evaluate the effectiveness and robustness of the three proposed approaches: restrict-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "score": 1.0, + "content": "ing the action space, conservative and regression extrapolation, as well as restricting the policy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "space. The semi-synthetic experiments are based on two real-world datasets: one is the popular im-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "age classification dataset CIFAR10 (Krizhevsky et al.) and the other is the credit-card fraud dataset", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "of Dal Pozzolo et al. (2015). We use the naive IPS estimator and the regression-based Direct Method", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 649, + 184, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 184, + 661 + ], + "score": 1.0, + "content": "(DM) as baselines.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 593, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "The experiments are set up as follows. We first create a train-validation-test split for both datasets.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "The training set is used to generate bandit datasets for learning, the validation set is used to gener-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "ate bandit datasets for model selection, and the full-information test set serves as ground truth for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "evaluating the learned policies. To simulate bandit feedback for the CIFAR10 dataset, our experi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 271, + 722 + ], + "score": 1.0, + "content": "ment setup follows traditional supervised", + "type": "text" + }, + { + "bbox": [ + 271, + 711, + 284, + 720 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "bandit conversion for multi-class classification datasets", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "(Beygelzimer & Langford, 2009). To not only have bandit data with binary multi-class rewards, we", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 504, + 95 + ], + "score": 1.0, + "content": "choose a different methodology for the credit-card dataset by designating some features as corre-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 396, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 396, + 106 + ], + "score": 1.0, + "content": "sponding to actions and rewards. More details are given in Appendix B.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 666, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 504, + 95 + ], + "score": 1.0, + "content": "choose a different methodology for the credit-card dataset by designating some features as corre-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 396, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 396, + 106 + ], + "score": 1.0, + "content": "sponding to actions and rewards. More details are given in Appendix B.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "To get logging policies for generating bandit feedback, we start by training a softmax-policy as in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 504, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 497, + 135 + ], + "score": 1.0, + "content": "Equation (4) on a subset of the full-information data. We then introduce a temperature parameter", + "type": "text" + }, + { + "bbox": [ + 497, + 123, + 504, + 131 + ], + "score": 0.68, + "content": "\\tau", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 216, + 146 + ], + "score": 1.0, + "content": "into the learned policy via", + "type": "text" + }, + { + "bbox": [ + 216, + 132, + 257, + 144 + ], + "score": 0.94, + "content": "\\tau f _ { w } ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 131, + 505, + 146 + ], + "score": 1.0, + "content": "to be able to control its stochasticity and support deficiency.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "In particular, we enforce zero support for some actions by clipping the propensities to 0 if they are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 192, + 168 + ], + "score": 1.0, + "content": "below a threshold of", + "type": "text" + }, + { + "bbox": [ + 193, + 154, + 231, + 165 + ], + "score": 0.9, + "content": "\\epsilon = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 153, + 281, + 168 + ], + "score": 1.0, + "content": ". The larger", + "type": "text" + }, + { + "bbox": [ + 281, + 156, + 288, + 164 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 153, + 506, + 168 + ], + "score": 1.0, + "content": ", the higher the support deficiency. Note that making", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 170, + 178 + ], + "score": 1.0, + "content": "the threshold at", + "type": "text" + }, + { + "bbox": [ + 170, + 166, + 206, + 176 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "allows us to control support while the variance of IPS stays bounded. This", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 448, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 448, + 189 + ], + "score": 1.0, + "content": "allows us to study support deficiency without having to worry about variance control.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 104, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 386, + 206 + ], + "score": 1.0, + "content": "For both logging and target policies, we train softmax policies where", + "type": "text" + }, + { + "bbox": [ + 386, + 193, + 421, + 205 + ], + "score": 0.93, + "content": "f _ { w } ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "is a neural network.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "We use the ResNet20 architecture (He et al., 2016) for CIFAR10, and a fully-connected 2-layer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 249, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 249, + 227 + ], + "score": 1.0, + "content": "network for the credit-card dataset.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "image", + "bbox": [ + 123, + 234, + 489, + 410 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 123, + 234, + 489, + 410 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 234, + 489, + 410 + ], + "spans": [ + { + "bbox": [ + 123, + 234, + 489, + 410 + ], + "score": 0.974, + "type": "image", + "image_path": "28a0537b9d42908064e6c5fb86cdcb47b4f7d8597ca2d9309b1cb944a9c7dace.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 123, + 234, + 489, + 292.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 123, + 292.6666666666667, + 489, + 351.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 123, + 351.33333333333337, + 489, + 410.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 144, + 420, + 466, + 432 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 416, + 467, + 436 + ], + "spans": [ + { + "bbox": [ + 143, + 416, + 467, + 436 + ], + "score": 1.0, + "content": "Figure 1: Learning results with varying support deficiency in the logging policy.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + } + ], + "index": 14.0 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "How do the methods perform at different level of support deficiency? Results are shown in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "Figure 1. First, as expected, learning using naive IPS degrades on both datasets as we make the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "logging policy more peaked and the number of unsupported actions increases. Note that naive IPS", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "coincides with Conservative Extrapolation, since both datasets are scaled to have a minimum reward", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 399, + 497 + ], + "score": 1.0, + "content": "of zero. In the rightmost column, however, we translated the rewards to", + "type": "text" + }, + { + "bbox": [ + 399, + 484, + 428, + 496 + ], + "score": 0.91, + "content": "[ - 1 , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 482, + 506, + 497 + ], + "score": 1.0, + "content": ". This has a strong", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "detrimental effect on naive IPS, as it is now overly optimistic about unsupported actions. Second,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "the approach of dealing with support deficiency by restricting the action space also performs poorly.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 449, + 529 + ], + "score": 1.0, + "content": "The second row of plot sheds some light on this, as it shows the support divergence", + "type": "text" + }, + { + "bbox": [ + 450, + 516, + 492, + 529 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "the learned policy. It is zero for Action Restriction as expected, which means that bias is not the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "problem. Instead, as the number of unsupported actions increases, the best actions are more likely to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 325, + 562 + ], + "score": 1.0, + "content": "be pruned and unavailable in the restricted policy space", + "type": "text" + }, + { + "bbox": [ + 325, + 549, + 334, + 560 + ], + "score": 0.49, + "content": "\\bar { \\Pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 549, + 505, + 562 + ], + "score": 1.0, + "content": ". Third, Regression Extrapolation performs", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "better than Conservative Extrapolation on both datasets. In both cases, the DM model is quite good", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "which also benefits Regression Extrapolation. However, on the credit-card dataset the regression", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "seems better at ranking than at predicting the true reward, which explains why DM performs better", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "than Regression Extrapolation. Fourth, the method that performs well most consistently is Policy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "score": 1.0, + "content": "Restriction. Unlike all the other IPS-based methods, it performs well even under the translated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "rewards in the third column of Figure 1. This is because the objective of Policy Restriction coincides", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "with that of BanditNet (Joachims et al., 2018), which is known to remedy propensity overfitting due", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 638, + 441, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 441, + 649 + ], + "score": 1.0, + "content": "to the lack of equivariance of the IPS estimator (Swaminathan & Joachims, 2015b).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "How does the learning performance change with more training data? Results are shown in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Figure 2. As the number of bandit examples increases, Policy Restriction, Regression Extrapolation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "and DM dominate over most of the range especially when the percentage of unsupported actions is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "large. Among the other methods, Action Restriction can take the least advantage of more data. This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "is plausible, since its maximum performance is limited by the available actions. For similar reasons,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Conservative Extrapolation (and equivalently IPS) also flattens out, since it also tightly restricts the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 297, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 297, + 732 + ], + "score": 1.0, + "content": "action space by imputing the minimum reward.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 106, + 81, + 504, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "To get logging policies for generating bandit feedback, we start by training a softmax-policy as in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 504, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 497, + 135 + ], + "score": 1.0, + "content": "Equation (4) on a subset of the full-information data. We then introduce a temperature parameter", + "type": "text" + }, + { + "bbox": [ + 497, + 123, + 504, + 131 + ], + "score": 0.68, + "content": "\\tau", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 216, + 146 + ], + "score": 1.0, + "content": "into the learned policy via", + "type": "text" + }, + { + "bbox": [ + 216, + 132, + 257, + 144 + ], + "score": 0.94, + "content": "\\tau f _ { w } ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 131, + 505, + 146 + ], + "score": 1.0, + "content": "to be able to control its stochasticity and support deficiency.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "In particular, we enforce zero support for some actions by clipping the propensities to 0 if they are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 192, + 168 + ], + "score": 1.0, + "content": "below a threshold of", + "type": "text" + }, + { + "bbox": [ + 193, + 154, + 231, + 165 + ], + "score": 0.9, + "content": "\\epsilon = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 153, + 281, + 168 + ], + "score": 1.0, + "content": ". The larger", + "type": "text" + }, + { + "bbox": [ + 281, + 156, + 288, + 164 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 153, + 506, + 168 + ], + "score": 1.0, + "content": ", the higher the support deficiency. Note that making", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 170, + 178 + ], + "score": 1.0, + "content": "the threshold at", + "type": "text" + }, + { + "bbox": [ + 170, + 166, + 206, + 176 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "allows us to control support while the variance of IPS stays bounded. This", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 448, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 448, + 189 + ], + "score": 1.0, + "content": "allows us to study support deficiency without having to worry about variance control.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 110, + 506, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 104, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 386, + 206 + ], + "score": 1.0, + "content": "For both logging and target policies, we train softmax policies where", + "type": "text" + }, + { + "bbox": [ + 386, + 193, + 421, + 205 + ], + "score": 0.93, + "content": "f _ { w } ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "is a neural network.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "We use the ResNet20 architecture (He et al., 2016) for CIFAR10, and a fully-connected 2-layer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 249, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 249, + 227 + ], + "score": 1.0, + "content": "network for the credit-card dataset.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 192, + 505, + 227 + ] + }, + { + "type": "image", + "bbox": [ + 123, + 234, + 489, + 410 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 123, + 234, + 489, + 410 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 234, + 489, + 410 + ], + "spans": [ + { + "bbox": [ + 123, + 234, + 489, + 410 + ], + "score": 0.974, + "type": "image", + "image_path": "28a0537b9d42908064e6c5fb86cdcb47b4f7d8597ca2d9309b1cb944a9c7dace.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 123, + 234, + 489, + 292.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 123, + 292.6666666666667, + 489, + 351.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 123, + 351.33333333333337, + 489, + 410.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 144, + 420, + 466, + 432 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 416, + 467, + 436 + ], + "spans": [ + { + "bbox": [ + 143, + 416, + 467, + 436 + ], + "score": 1.0, + "content": "Figure 1: Learning results with varying support deficiency in the logging policy.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + } + ], + "index": 14.0 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "How do the methods perform at different level of support deficiency? Results are shown in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "Figure 1. First, as expected, learning using naive IPS degrades on both datasets as we make the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "logging policy more peaked and the number of unsupported actions increases. Note that naive IPS", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "coincides with Conservative Extrapolation, since both datasets are scaled to have a minimum reward", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 399, + 497 + ], + "score": 1.0, + "content": "of zero. In the rightmost column, however, we translated the rewards to", + "type": "text" + }, + { + "bbox": [ + 399, + 484, + 428, + 496 + ], + "score": 0.91, + "content": "[ - 1 , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 482, + 506, + 497 + ], + "score": 1.0, + "content": ". This has a strong", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "detrimental effect on naive IPS, as it is now overly optimistic about unsupported actions. Second,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "the approach of dealing with support deficiency by restricting the action space also performs poorly.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 449, + 529 + ], + "score": 1.0, + "content": "The second row of plot sheds some light on this, as it shows the support divergence", + "type": "text" + }, + { + "bbox": [ + 450, + 516, + 492, + 529 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { \\mathcal { X } } ( \\pi \\vert \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "the learned policy. 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Third, Regression Extrapolation performs", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "better than Conservative Extrapolation on both datasets. In both cases, the DM model is quite good", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "which also benefits Regression Extrapolation. However, on the credit-card dataset the regression", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "seems better at ranking than at predicting the true reward, which explains why DM performs better", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "than Regression Extrapolation. Fourth, the method that performs well most consistently is Policy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "score": 1.0, + "content": "Restriction. Unlike all the other IPS-based methods, it performs well even under the translated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "rewards in the third column of Figure 1. This is because the objective of Policy Restriction coincides", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "with that of BanditNet (Joachims et al., 2018), which is known to remedy propensity overfitting due", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 638, + 441, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 441, + 649 + ], + "score": 1.0, + "content": "to the lack of equivariance of the IPS estimator (Swaminathan & Joachims, 2015b).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 440, + 506, + 649 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "How does the learning performance change with more training data? Results are shown in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Figure 2. As the number of bandit examples increases, Policy Restriction, Regression Extrapolation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "and DM dominate over most of the range especially when the percentage of unsupported actions is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "large. Among the other methods, Action Restriction can take the least advantage of more data. This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "is plausible, since its maximum performance is limited by the available actions. 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Machine learning, 8(3-4):229–256, 1992.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 253, + 505, + 277 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 232, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 232, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 232, + 96 + ], + "score": 1.0, + "content": "A APPENDIX: PROOFS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 408, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 409, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 409, + 119 + ], + "score": 1.0, + "content": "In this appendix, we provide proofs of the main theorems and propositions.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 130, + 245, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 246, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 246, + 144 + ], + "score": 1.0, + "content": "A.1 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 104, + 149, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 104, + 149, + 232, + 165 + ], + "score": 1.0, + "content": "Proposition 1. Given contexts", + "type": "text" + }, + { + "bbox": [ + 232, + 151, + 276, + 163 + ], + "score": 0.92, + "content": "x \\sim P ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 149, + 355, + 165 + ], + "score": 1.0, + "content": "and logging policy", + "type": "text" + }, + { + "bbox": [ + 356, + 151, + 390, + 163 + ], + "score": 0.92, + "content": "\\pi _ { 0 } ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 149, + 439, + 165 + ], + "score": 1.0, + "content": ", the bias of", + "type": "text" + }, + { + "bbox": [ + 439, + 150, + 463, + 162 + ], + "score": 0.87, + "content": "\\hat { R } _ { I P S }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 149, + 506, + 165 + ], + "score": 1.0, + "content": "for target", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 134, + 176 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 134, + 162, + 165, + 174 + ], + "score": 0.89, + "content": "\\pi ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 162, + 447, + 176 + ], + "score": 1.0, + "content": "is equal to the expected reward on the unsupported action sets, i.e.,", + "type": "text" + }, + { + "bbox": [ + 447, + 163, + 505, + 174 + ], + "score": 0.88, + "content": "b i a s ( \\pi | \\pi _ { 0 } ) =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 170, + 245, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 239, + 187 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x } [ - \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 170, + 245, + 190 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 198, + 446, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 447, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 164, + 213 + ], + "score": 1.0, + "content": "Proof. Recall", + "type": "text" + }, + { + "bbox": [ + 165, + 199, + 269, + 211 + ], + "score": 0.93, + "content": "\\delta ( x , y ) = \\mathbb { E } _ { r } [ r ( x , y ) | x , y ]", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 197, + 339, + 213 + ], + "score": 1.0, + "content": ", and logged data", + "type": "text" + }, + { + "bbox": [ + 339, + 199, + 442, + 211 + ], + "score": 0.92, + "content": "\\mathcal { D } \\sim \\mathcal { P } _ { \\mathcal { X } } \\times \\pi _ { 0 } ( \\cdot | \\mathcal { X } ) \\times \\mathcal { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 197, + 447, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 216, + 455, + 299 + ], + "lines": [ + { + "bbox": [ + 155, + 216, + 455, + 299 + ], + "spans": [ + { + "bbox": [ + 155, + 216, + 455, + 299 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle b i a s ( \\pi | \\pi _ { 0 } ) = \\frac { \\mathbb { E } [ \\hat { R } _ { I P S } ( \\pi ) ] - R ( \\pi ) } { \\mathcal { D } } } \\\\ { \\displaystyle \\quad = \\frac { \\mathbb { E } [ \\sum _ { \\scriptstyle x } } { y \\in ( \\mathcal { U } ( x , \\pi _ { 0 } ) ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) - \\sum _ { y \\in \\mathcal { Y } } \\pi ( y | x ) \\delta ( x , y ) ] } \\\\ { \\displaystyle \\quad = \\frac { \\mathbb { E } [ - \\sum _ { \\scriptstyle x } } { x } - \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "6d4b968aa6433203448a2cf1bd9a7ed25742e2c20557f6925eae882f01c2e21e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 155, + 216, + 455, + 243.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 155, + 243.66666666666666, + 455, + 271.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 155, + 271.3333333333333, + 455, + 299.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 326, + 230, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 231, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 231, + 339 + ], + "score": 1.0, + "content": "A.2 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 507, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 302, + 360 + ], + "score": 1.0, + "content": "Theorem 1. For any given hypothesis space", + "type": "text" + }, + { + "bbox": [ + 302, + 348, + 311, + 357 + ], + "score": 0.71, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 345, + 399, + 360 + ], + "score": 1.0, + "content": "with logging policy", + "type": "text" + }, + { + "bbox": [ + 400, + 347, + 439, + 358 + ], + "score": 0.9, + "content": "\\pi _ { 0 } ~ \\in ~ \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 345, + 507, + 360 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 357, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 189, + 373 + ], + "score": 1.0, + "content": "reward distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 359, + 203, + 369 + ], + "score": 0.88, + "content": "\\mathcal { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 357, + 275, + 373 + ], + "score": 1.0, + "content": "with support in", + "type": "text" + }, + { + "bbox": [ + 276, + 358, + 327, + 370 + ], + "score": 0.91, + "content": "[ r _ { m i n } , r _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 357, + 506, + 373 + ], + "score": 1.0, + "content": "such that in the limit of infinite training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 182, + 383 + ], + "score": 1.0, + "content": "data, ERM using", + "type": "text" + }, + { + "bbox": [ + 182, + 370, + 199, + 379 + ], + "score": 0.25, + "content": "I P S", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 368, + 293, + 383 + ], + "score": 1.0, + "content": "over the logged data", + "type": "text" + }, + { + "bbox": [ + 293, + 369, + 420, + 381 + ], + "score": 0.89, + "content": "\\mathcal D \\sim \\dot { P } ( \\mathcal X ) \\times \\pi _ { 0 } ( \\cdot | \\mathcal X ) \\times \\mathcal P _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "can select a policy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 380, + 488, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 235, + 395 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\hat { \\pi } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 381, + 294, + 396 + ], + "score": 1.0, + "content": "that is at least", + "type": "text" + }, + { + "bbox": [ + 295, + 382, + 436, + 394 + ], + "score": 0.84, + "content": "\\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\quad } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 381, + 488, + 396 + ], + "score": 1.0, + "content": "suboptimal.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 104, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 404, + 264, + 419 + ], + "score": 1.0, + "content": "Proof. 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Actually for any", + "type": "text" + }, + { + "bbox": [ + 486, + 612, + 505, + 623 + ], + "score": 0.85, + "content": "\\hat { \\pi } \\in", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 619, + 508, + 643 + ], + "spans": [ + { + "bbox": [ + 102, + 619, + 144, + 643 + ], + "score": 1.0, + "content": "arg max", + "type": "text" + }, + { + "bbox": [ + 144, + 623, + 323, + 639 + ], + "score": 0.79, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ^ { \\mathcal { D } } ( \\pi ) ] , \\mathbb { E } _ { x } [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) ] = 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 619, + 508, + 643 + ], + "score": 1.0, + "content": "and it gives us that any solution of the ERM", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 636, + 491, + 654 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 234, + 654 + ], + "score": 1.0, + "content": "has exactly the same support as", + "type": "text" + }, + { + "bbox": [ + 234, + 641, + 245, + 651 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 636, + 304, + 654 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + }, + { + "bbox": [ + 304, + 639, + 360, + 651 + ], + "score": 0.92, + "content": "R ( \\hat { \\pi } ) = r _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 636, + 375, + 654 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 375, + 637, + 487, + 652 + ], + "score": 0.64, + "content": "\\hat { \\pi } \\in \\arg \\operatorname* { m a x } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ^ { \\mathcal { D } } ( \\pi ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 636, + 491, + 654 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 374, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 375, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 235, + 671 + ], + "score": 1.0, + "content": "Combining the lower bound for", + "type": "text" + }, + { + "bbox": [ + 235, + 656, + 262, + 668 + ], + "score": 0.93, + "content": "R ( \\pi ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 654, + 280, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 656, + 336, + 669 + ], + "score": 0.92, + "content": "R ( \\hat { \\pi } ) = r _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 654, + 375, + 671 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 672, + 461, + 712 + ], + "lines": [ + { + "bbox": [ + 149, + 672, + 461, + 712 + ], + "spans": [ + { + "bbox": [ + 149, + 672, + 461, + 712 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { R ( \\pi ^ { * } ) - R ( \\hat { \\pi } ) \\geq r _ { m a x } \\displaystyle \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) + r _ { m i n } ( 1 - \\displaystyle \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) ) - r _ { m i n } } \\\\ & { \\quad \\quad \\quad \\quad \\quad = \\left( r _ { m a x } - r _ { m i n } \\right) \\displaystyle \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "c1469879419db866ae4edc51ff17ad1c69e75fcf4ff8a7ce21c424ca97af33ec.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 149, + 672, + 461, + 685.3333333333334 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 149, + 685.3333333333334, + 461, + 698.6666666666667 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 149, + 698.6666666666667, + 461, + 712.0000000000001 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 309, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 721, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 722, + 505, + 731 + ], + "spans": [ + { + "bbox": [ + 496, + 722, + 505, + 731 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 302, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 496, + 303, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 496, + 303, + 505, + 313 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 232, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 232, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 232, + 96 + ], + "score": 1.0, + "content": "A APPENDIX: PROOFS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 408, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 409, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 409, + 119 + ], + "score": 1.0, + "content": "In this appendix, we provide proofs of the main theorems and propositions.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 105, + 409, + 119 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 130, + 245, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 246, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 246, + 144 + ], + "score": 1.0, + "content": "A.1 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 104, + 149, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 104, + 149, + 232, + 165 + ], + "score": 1.0, + "content": "Proposition 1. Given contexts", + "type": "text" + }, + { + "bbox": [ + 232, + 151, + 276, + 163 + ], + "score": 0.92, + "content": "x \\sim P ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 149, + 355, + 165 + ], + "score": 1.0, + "content": "and logging policy", + "type": "text" + }, + { + "bbox": [ + 356, + 151, + 390, + 163 + ], + "score": 0.92, + "content": "\\pi _ { 0 } ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 149, + 439, + 165 + ], + "score": 1.0, + "content": ", the bias of", + "type": "text" + }, + { + "bbox": [ + 439, + 150, + 463, + 162 + ], + "score": 0.87, + "content": "\\hat { R } _ { I P S }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 149, + 506, + 165 + ], + "score": 1.0, + "content": "for target", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 134, + 176 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 134, + 162, + 165, + 174 + ], + "score": 0.89, + "content": "\\pi ( \\mathcal { V } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 162, + 447, + 176 + ], + "score": 1.0, + "content": "is equal to the expected reward on the unsupported action sets, i.e.,", + "type": "text" + }, + { + "bbox": [ + 447, + 163, + 505, + 174 + ], + "score": 0.88, + "content": "b i a s ( \\pi | \\pi _ { 0 } ) =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 170, + 245, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 239, + 187 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x } [ - \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 170, + 245, + 190 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 149, + 506, + 190 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 198, + 446, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 447, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 164, + 213 + ], + "score": 1.0, + "content": "Proof. Recall", + "type": "text" + }, + { + "bbox": [ + 165, + 199, + 269, + 211 + ], + "score": 0.93, + "content": "\\delta ( x , y ) = \\mathbb { E } _ { r } [ r ( x , y ) | x , y ]", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 197, + 339, + 213 + ], + "score": 1.0, + "content": ", and logged data", + "type": "text" + }, + { + "bbox": [ + 339, + 199, + 442, + 211 + ], + "score": 0.92, + "content": "\\mathcal { D } \\sim \\mathcal { P } _ { \\mathcal { X } } \\times \\pi _ { 0 } ( \\cdot | \\mathcal { X } ) \\times \\mathcal { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 197, + 447, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 197, + 447, + 213 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 216, + 455, + 299 + ], + "lines": [ + { + "bbox": [ + 155, + 216, + 455, + 299 + ], + "spans": [ + { + "bbox": [ + 155, + 216, + 455, + 299 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle b i a s ( \\pi | \\pi _ { 0 } ) = \\frac { \\mathbb { E } [ \\hat { R } _ { I P S } ( \\pi ) ] - R ( \\pi ) } { \\mathcal { D } } } \\\\ { \\displaystyle \\quad = \\frac { \\mathbb { E } [ \\sum _ { \\scriptstyle x } } { y \\in ( \\mathcal { U } ( x , \\pi _ { 0 } ) ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) - \\sum _ { y \\in \\mathcal { Y } } \\pi ( y | x ) \\delta ( x , y ) ] } \\\\ { \\displaystyle \\quad = \\frac { \\mathbb { E } [ - \\sum _ { \\scriptstyle x } } { x } - \\pi ( y | x ) \\delta ( x , y ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "6d4b968aa6433203448a2cf1bd9a7ed25742e2c20557f6925eae882f01c2e21e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 155, + 216, + 455, + 243.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 155, + 243.66666666666666, + 455, + 271.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 155, + 271.3333333333333, + 455, + 299.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 326, + 230, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 231, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 231, + 339 + ], + "score": 1.0, + "content": "A.2 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 507, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 302, + 360 + ], + "score": 1.0, + "content": "Theorem 1. For any given hypothesis space", + "type": "text" + }, + { + "bbox": [ + 302, + 348, + 311, + 357 + ], + "score": 0.71, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 345, + 399, + 360 + ], + "score": 1.0, + "content": "with logging policy", + "type": "text" + }, + { + "bbox": [ + 400, + 347, + 439, + 358 + ], + "score": 0.9, + "content": "\\pi _ { 0 } ~ \\in ~ \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 345, + 507, + 360 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 357, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 189, + 373 + ], + "score": 1.0, + "content": "reward distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 359, + 203, + 369 + ], + "score": 0.88, + "content": "\\mathcal { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 357, + 275, + 373 + ], + "score": 1.0, + "content": "with support in", + "type": "text" + }, + { + "bbox": [ + 276, + 358, + 327, + 370 + ], + "score": 0.91, + "content": "[ r _ { m i n } , r _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 357, + 506, + 373 + ], + "score": 1.0, + "content": "such that in the limit of infinite training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 182, + 383 + ], + "score": 1.0, + "content": "data, ERM using", + "type": "text" + }, + { + "bbox": [ + 182, + 370, + 199, + 379 + ], + "score": 0.25, + "content": "I P S", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 368, + 293, + 383 + ], + "score": 1.0, + "content": "over the logged data", + "type": "text" + }, + { + "bbox": [ + 293, + 369, + 420, + 381 + ], + "score": 0.89, + "content": "\\mathcal D \\sim \\dot { P } ( \\mathcal X ) \\times \\pi _ { 0 } ( \\cdot | \\mathcal X ) \\times \\mathcal P _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "can select a policy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 380, + 488, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 235, + 395 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\hat { \\pi } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\mathcal { D } } [ \\hat { R } _ { I P S } ( \\pi ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 381, + 294, + 396 + ], + "score": 1.0, + "content": "that is at least", + "type": "text" + }, + { + "bbox": [ + 295, + 382, + 436, + 394 + ], + "score": 0.84, + "content": "\\begin{array} { r l } { { ( r _ { m a x } - r _ { m i n } ) \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\quad } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 381, + 488, + 396 + ], + "score": 1.0, + "content": "suboptimal.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 345, + 507, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 104, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 404, + 264, + 419 + ], + "score": 1.0, + "content": "Proof. 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}, + { + "bbox": [ + 327, + 563, + 334, + 573 + ], + "score": 0.69, + "content": "\\hat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 559, + 437, + 577 + ], + "score": 1.0, + "content": "have the same support as", + "type": "text" + }, + { + "bbox": [ + 438, + 565, + 448, + 574 + ], + "score": 0.86, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 559, + 453, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 549, + 506, + 577 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 579, + 465, + 608 + ], + "lines": [ + { + "bbox": [ + 147, + 579, + 465, + 608 + ], + "spans": [ + { + "bbox": [ + 147, + 579, + 465, + 608 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\hat { R } _ { I P S } ( \\pi ) ] = \\mathbb { E } [ \\sum _ { x } \\pi ( y | x ) r _ { m i n } ] = r _ { m i n } \\mathbb { E } [ \\sum _ { y \\in \\mathcal { U } 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For any given", + "type": "text" + }, + { + "bbox": [ + 232, + 341, + 277, + 353 + ], + "score": 0.45, + "content": "\\kappa \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 339, + 282, + 355 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 282, + 341, + 342, + 352 + ], + "score": 0.7, + "content": "0 < \\epsilon < \\kappa / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 339, + 361, + 355 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 361, + 342, + 383, + 352 + ], + "score": 0.86, + "content": "p _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 339, + 505, + 355 + ], + "score": 1.0, + "content": "denote the minimum propen-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 350, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 350, + 213, + 367 + ], + "score": 1.0, + "content": "sity under supported set", + "type": "text" + }, + { + "bbox": [ + 213, + 353, + 356, + 365 + ], + "score": 0.87, + "content": "p _ { m i n } ~ = ~ m a x _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 350, + 506, + 367 + ], + "score": 1.0, + "content": ", then with probability larger than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 363, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 198, + 377 + ], + "score": 0.9, + "content": "1 - 2 \\exp ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 363, + 259, + 379 + ], + "score": 1.0, + "content": ", the constraint", + "type": "text" + }, + { + "bbox": [ + 259, + 365, + 375, + 376 + ], + "score": 0.89, + "content": "1 - \\kappa { + } \\epsilon \\le S _ { \\mathscr { D } } ( \\pi | \\pi _ { 0 } ) \\le 1 { - } \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 363, + 421, + 379 + ], + "score": 1.0, + "content": "will ensure", + "type": "text" + }, + { + "bbox": [ + 422, + 364, + 501, + 376 + ], + "score": 0.95, + "content": "0 \\leq \\mathcal { D } \\boldsymbol { x } ( \\pi | \\pi _ { 0 } ) \\leq \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 363, + 506, + 379 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 101, + 382, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 101, + 382, + 293, + 412 + ], + "score": 1.0, + "content": "Proof. Recall SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi)", + "type": "text" + }, + { + "bbox": [ + 288, + 389, + 310, + 405 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 311, + 392, + 340, + 405 + ], + "score": 0.9, + "content": "( x _ { i } , y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 389, + 405, + 405 + ], + "score": 1.0, + "content": "draw i.i.d from", + "type": "text" + }, + { + "bbox": [ + 405, + 392, + 476, + 405 + ], + "score": 0.93, + "content": "P ( \\mathcal { X } ) \\times \\pi _ { 0 } ( \\mathcal { Y } | \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 389, + 505, + 405 + ], + "score": 1.0, + "content": ". From", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 315, + 425 + ], + "score": 1.0, + "content": "Appendix A.3, it is easy to see Ex,y∼π0(·|x)[ π(y|x)π0(y|x) ]", + "type": "text" + }, + { + "bbox": [ + 230, + 407, + 384, + 423 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x , y \\sim \\pi _ { 0 } ( \\cdot | x ) } \\big [ \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\big ] = 1 - \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 405, + 403, + 424 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 404, + 410, + 425, + 420 + ], + "score": 0.88, + "content": "p _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 405, + 506, + 424 + ], + "score": 1.0, + "content": "denote the smallest", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 421, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 267, + 436 + ], + "score": 1.0, + "content": "propensity under supported action set,", + "type": "text" + }, + { + "bbox": [ + 267, + 423, + 430, + 435 + ], + "score": 0.84, + "content": "p _ { m i n } : = \\mathrm { m i n } _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 421, + 506, + 436 + ], + "score": 1.0, + "content": ", then the random", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 434, + 468, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 141, + 453 + ], + "score": 1.0, + "content": "variable", + "type": "text" + }, + { + "bbox": [ + 141, + 434, + 170, + 452 + ], + "score": 0.94, + "content": "\\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 434, + 284, + 453 + ], + "score": 1.0, + "content": "is strictly bounded between", + "type": "text" + }, + { + "bbox": [ + 284, + 436, + 320, + 451 + ], + "score": 0.92, + "content": "[ 0 , \\frac { 1 } { p _ { m i n } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 434, + 468, + 453 + ], + "score": 1.0, + "content": ". 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\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 479, + 226, + 495 + ], + "score": 1.0, + "content": "gives", + "type": "text" + }, + { + "bbox": [ + 226, + 481, + 319, + 493 + ], + "score": 0.91, + "content": "1 - S _ { \\mathcal { D } } ( \\pi \\vert \\pi _ { 0 } ) - \\epsilon \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 479, + 378, + 495 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 499, + 386, + 514 + ], + "lines": [ + { + "bbox": [ + 225, + 499, + 386, + 514 + ], + "spans": [ + { + "bbox": [ + 225, + 499, + 386, + 514 + ], + "score": 0.91, + "content": "\\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 0 ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )", + "type": "interline_equation", + "image_path": "d8a1768dee920d611c15b86151b72f883ecc3b51214f8686cd821d3e6487e683.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 225, + 499, + 386, + 514 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 334, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 333, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 333, + 533 + ], + "score": 1.0, + "content": "Similar for the other direction, Hoeffding’s bound gives:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 538, + 500, + 552 + ], + "lines": [ + { + "bbox": [ + 109, + 538, + 500, + 552 + ], + "spans": [ + { + "bbox": [ + 109, + 538, + 500, + 552 + ], + "score": 0.88, + "content": "\\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) > 1 - S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\epsilon ) = \\mathbb { P } ( S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) - ( 1 - \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) ) > \\epsilon ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )", + "type": "interline_equation", + "image_path": "0916b98880a7628259ac4f25c46a3966111975118a30c91717e97ffbc296ab7e.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 109, + 538, + 500, + 552 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 396, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 397, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 131, + 585 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 570, + 219, + 583 + ], + "score": 0.92, + "content": "S _ { \\mathcal { D } ( \\pi | \\pi _ { 0 } ) } \\geq 1 + \\epsilon - \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 569, + 244, + 585 + ], + "score": 1.0, + "content": "gives", + "type": "text" + }, + { + "bbox": [ + 244, + 570, + 338, + 582 + ], + "score": 0.91, + "content": "\\begin{array} { r } { 1 - S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\epsilon \\le \\kappa } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 569, + 397, + 585 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 588, + 386, + 604 + ], + "lines": [ + { + "bbox": [ + 224, + 588, + 386, + 604 + ], + "spans": [ + { + "bbox": [ + 224, + 588, + 386, + 604 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) \\ge \\kappa ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "616cd1e7374fd963c892d6ffd47a17dcee013502f54988a3981b254f374abedf.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 224, + 588, + 386, + 604 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 232, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 233, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 233, + 622 + ], + "score": 1.0, + "content": "Combining the above, we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 626, + 449, + 658 + ], + "lines": [ + { + "bbox": [ + 162, + 626, + 449, + 658 + ], + "spans": [ + { + "bbox": [ + 162, + 626, + 449, + 658 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbb { P } ( 0 \\leq \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) \\leq \\kappa ) = 1 - \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 0 ) - \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) > \\kappa ) } \\\\ & { \\qquad \\geq 1 - 2 e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "1c8784ddfc2083adf24fc1f7ba99f6a4ff60b9a6b83f25ae21d3e249a18f34d5.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 162, + 626, + 449, + 658 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 687, + 302, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 303, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 303, + 700 + ], + "score": 1.0, + "content": "A.5 PROOF FOR EFFICIENT APPROXIMATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 707, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "score": 1.0, + "content": "Claim 1. The empirical risk defined by in Equation (8) has the same expectation (over randomness", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 719, + 242, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 117, + 734 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 721, + 126, + 730 + ], + "score": 0.71, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 719, + 199, + 734 + ], + "score": 1.0, + "content": "and sampling) as", + "type": "text" + }, + { + "bbox": [ + 199, + 720, + 238, + 733 + ], + "score": 0.93, + "content": "\\hat { R } _ { I P S } ^ { \\delta } ( \\mathcal { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 719, + 242, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 309, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 662, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 496, + 663, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 496, + 663, + 505, + 673 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 231, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 231, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 231, + 95 + ], + "score": 1.0, + "content": "A.3 PROOF OF THEOREM 2", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 504, + 132 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 214, + 117 + ], + "score": 1.0, + "content": "Theorem 2. For contexts", + "type": "text" + }, + { + "bbox": [ + 214, + 105, + 224, + 114 + ], + "score": 0.81, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 101, + 295, + 117 + ], + "score": 1.0, + "content": "drawn i.i.d from", + "type": "text" + }, + { + "bbox": [ + 295, + 103, + 320, + 115 + ], + "score": 0.91, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 101, + 353, + 117 + ], + "score": 1.0, + "content": ", action", + "type": "text" + }, + { + "bbox": [ + 353, + 105, + 362, + 115 + ], + "score": 0.79, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 101, + 476, + 117 + ], + "score": 1.0, + "content": "drawn from logging policy", + "type": "text" + }, + { + "bbox": [ + 476, + 108, + 486, + 114 + ], + "score": 0.88, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 101, + 505, + 117 + ], + "score": 1.0, + "content": ", we", + "type": 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464, + 278 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\underset { x , y \\sim \\pi _ { 0 } } { \\mathbb { E } } [ S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) = \\underset { x } { \\mathbb { E } } [ \\ \\underset { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } { \\sum } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } ] + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } & { } \\\\ { = \\underset { x } { \\mathbb { E } } [ \\ \\underset { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } { \\sum } \\pi ( y | x ) ] + \\underset { x } { \\mathbb { E } } [ \\ \\underset { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } { \\sum } \\pi ( y | x ) ] } & { } \\\\ { = \\underset { x } { \\mathbb { E } } [ \\underset { y \\in \\mathcal { Y } } { \\sum } \\pi ( y | x ) ] = 1 } & { } \\end{array}", + "type": "interline_equation", + "image_path": 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For any given", + "type": "text" + }, + { + "bbox": [ + 232, + 341, + 277, + 353 + ], + "score": 0.45, + "content": "\\kappa \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 339, + 282, + 355 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 282, + 341, + 342, + 352 + ], + "score": 0.7, + "content": "0 < \\epsilon < \\kappa / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 339, + 361, + 355 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 361, + 342, + 383, + 352 + ], + "score": 0.86, + "content": "p _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 339, + 505, + 355 + ], + "score": 1.0, + "content": "denote the minimum propen-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 350, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 350, + 213, + 367 + ], + "score": 1.0, + "content": "sity under supported set", + "type": "text" + }, + { + "bbox": [ + 213, + 353, + 356, + 365 + ], + "score": 0.87, + "content": "p _ { m i n } ~ = ~ m a x _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 350, + 506, + 367 + ], + "score": 1.0, + "content": ", then with probability larger than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 363, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 198, + 377 + ], + "score": 0.9, + "content": "1 - 2 \\exp ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 363, + 259, + 379 + ], + "score": 1.0, + "content": ", the constraint", + "type": "text" + }, + { + "bbox": [ + 259, + 365, + 375, + 376 + ], + "score": 0.89, + "content": "1 - \\kappa { + } \\epsilon \\le S _ { \\mathscr { D } } ( \\pi | \\pi _ { 0 } ) \\le 1 { - } \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 363, + 421, + 379 + ], + "score": 1.0, + "content": "will ensure", + "type": "text" + }, + { + "bbox": [ + 422, + 364, + 501, + 376 + ], + "score": 0.95, + "content": "0 \\leq \\mathcal { D } \\boldsymbol { x } ( \\pi | \\pi _ { 0 } ) \\leq \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 363, + 506, + 379 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 339, + 506, + 379 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 101, + 382, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 101, + 382, + 293, + 412 + ], + "score": 1.0, + "content": "Proof. Recall SD(π|π0) = 1n Pni=1 π(yi|xi)π0(yi|xi)", + "type": "text" + }, + { + "bbox": [ + 288, + 389, + 310, + 405 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 311, + 392, + 340, + 405 + ], + "score": 0.9, + "content": "( x _ { i } , y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 389, + 405, + 405 + ], + "score": 1.0, + "content": "draw i.i.d from", + "type": "text" + }, + { + "bbox": [ + 405, + 392, + 476, + 405 + ], + "score": 0.93, + "content": "P ( \\mathcal { X } ) \\times \\pi _ { 0 } ( \\mathcal { Y } | \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 389, + 505, + 405 + ], + "score": 1.0, + "content": ". From", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 315, + 425 + ], + "score": 1.0, + "content": "Appendix A.3, it is easy to see Ex,y∼π0(·|x)[ π(y|x)π0(y|x) ]", + "type": "text" + }, + { + "bbox": [ + 230, + 407, + 384, + 423 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x , y \\sim \\pi _ { 0 } ( \\cdot | x ) } \\big [ \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\big ] = 1 - \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 405, + 403, + 424 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 404, + 410, + 425, + 420 + ], + "score": 0.88, + "content": "p _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 405, + 506, + 424 + ], + "score": 1.0, + "content": "denote the smallest", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 421, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 267, + 436 + ], + "score": 1.0, + "content": "propensity under supported action set,", + "type": "text" + }, + { + "bbox": [ + 267, + 423, + 430, + 435 + ], + "score": 0.84, + "content": "p _ { m i n } : = \\mathrm { m i n } _ { x , y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 421, + 506, + 436 + ], + "score": 1.0, + "content": ", then the random", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 434, + 468, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 141, + 453 + ], + "score": 1.0, + "content": "variable", + "type": "text" + }, + { + "bbox": [ + 141, + 434, + 170, + 452 + ], + "score": 0.94, + "content": "\\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 434, + 284, + 453 + ], + "score": 1.0, + "content": "is strictly bounded between", + "type": "text" + }, + { + "bbox": [ + 284, + 436, + 320, + 451 + ], + "score": 0.92, + "content": "[ 0 , \\frac { 1 } { p _ { m i n } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 434, + 468, + 453 + ], + "score": 1.0, + "content": ". Applying Hoeffding’s bound gives:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 101, + 382, + 506, + 453 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 457, + 506, + 472 + ], + "lines": [ + { + "bbox": [ + 111, + 457, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 111, + 457, + 506, + 472 + ], + "score": 0.88, + "content": "^ { \\mathfrak { p } } ( { \\mathcal { D } } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 1 - S _ { { \\mathcal { D } } } ( \\pi | \\pi _ { 0 } ) - \\epsilon ) = \\mathbb { P } ( S _ { { \\mathcal { D } } } ( \\pi | \\pi _ { 0 } ) - ( 1 - { \\mathcal { D } } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) ) < - \\epsilon ) \\leq e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )", + "type": "interline_equation", + "image_path": "a349963948c3a1b2e05cf3d5e10778fe4e845ee4f4446e25baadcbc949e0d8e9.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 111, + 457, + 506, + 472 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 378, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 378, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 131, + 495 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 481, + 202, + 494 + ], + "score": 0.91, + "content": "S _ { \\mathcal { D } ( \\pi | \\pi _ { 0 } ) } \\le 1 - \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 479, + 226, + 495 + ], + "score": 1.0, + "content": "gives", + "type": "text" + }, + { + "bbox": [ + 226, + 481, + 319, + 493 + ], + "score": 0.91, + "content": "1 - S _ { \\mathcal { D } } ( \\pi \\vert \\pi _ { 0 } ) - \\epsilon \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 479, + 378, + 495 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 479, + 378, + 495 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 499, + 386, + 514 + ], + "lines": [ + { + "bbox": [ + 225, + 499, + 386, + 514 + ], + "spans": [ + { + "bbox": [ + 225, + 499, + 386, + 514 + ], + "score": 0.91, + "content": "\\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 0 ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )", + "type": "interline_equation", + "image_path": "d8a1768dee920d611c15b86151b72f883ecc3b51214f8686cd821d3e6487e683.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 225, + 499, + 386, + 514 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 334, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 333, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 333, + 533 + ], + "score": 1.0, + "content": "Similar for the other direction, Hoeffding’s bound gives:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 518, + 333, + 533 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 538, + 500, + 552 + ], + "lines": [ + { + "bbox": [ + 109, + 538, + 500, + 552 + ], + "spans": [ + { + "bbox": [ + 109, + 538, + 500, + 552 + ], + "score": 0.88, + "content": "\\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) > 1 - S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\epsilon ) = \\mathbb { P } ( S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) - ( 1 - \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) ) > \\epsilon ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } )", + "type": "interline_equation", + "image_path": "0916b98880a7628259ac4f25c46a3966111975118a30c91717e97ffbc296ab7e.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 109, + 538, + 500, + 552 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 396, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 397, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 131, + 585 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 570, + 219, + 583 + ], + "score": 0.92, + "content": "S _ { \\mathcal { D } ( \\pi | \\pi _ { 0 } ) } \\geq 1 + \\epsilon - \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 569, + 244, + 585 + ], + "score": 1.0, + "content": "gives", + "type": "text" + }, + { + "bbox": [ + 244, + 570, + 338, + 582 + ], + "score": 0.91, + "content": "\\begin{array} { r } { 1 - S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\epsilon \\le \\kappa } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 569, + 397, + 585 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 569, + 397, + 585 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 588, + 386, + 604 + ], + "lines": [ + { + "bbox": [ + 224, + 588, + 386, + 604 + ], + "spans": [ + { + "bbox": [ + 224, + 588, + 386, + 604 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) \\ge \\kappa ) \\le e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "616cd1e7374fd963c892d6ffd47a17dcee013502f54988a3981b254f374abedf.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 224, + 588, + 386, + 604 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 232, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 233, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 233, + 622 + ], + "score": 1.0, + "content": "Combining the above, we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 607, + 233, + 622 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 626, + 449, + 658 + ], + "lines": [ + { + "bbox": [ + 162, + 626, + 449, + 658 + ], + "spans": [ + { + "bbox": [ + 162, + 626, + 449, + 658 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbb { P } ( 0 \\leq \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) \\leq \\kappa ) = 1 - \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) < 0 ) - \\mathbb { P } ( \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } ) > \\kappa ) } \\\\ & { \\qquad \\geq 1 - 2 e x p ( - 2 n \\epsilon ^ { 2 } p _ { m i n } ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "1c8784ddfc2083adf24fc1f7ba99f6a4ff60b9a6b83f25ae21d3e249a18f34d5.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 162, + 626, + 449, + 658 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 687, + 302, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 303, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 303, + 700 + ], + "score": 1.0, + "content": "A.5 PROOF FOR EFFICIENT APPROXIMATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 707, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "score": 1.0, + "content": "Claim 1. The empirical risk defined by in Equation (8) has the same expectation (over randomness", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 719, + 242, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 117, + 734 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 721, + 126, + 730 + ], + "score": 0.71, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 719, + 199, + 734 + ], + "score": 1.0, + "content": "and sampling) as", + "type": "text" + }, + { + "bbox": [ + 199, + 720, + 238, + 733 + ], + "score": 0.93, + "content": "\\hat { R } _ { I P S } ^ { \\delta } ( \\mathcal { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 719, + 242, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 707, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 394, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 395, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 395, + 97 + ], + "score": 1.0, + "content": "Proof. Taking the expectation of empirical risk defined in Equation (8):", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 99, + 470, + 197 + ], + "lines": [ + { + "bbox": [ + 125, + 99, + 470, + 197 + ], + "spans": [ + { + "bbox": [ + 125, + 99, + 470, + 197 + ], + "score": 0.95, + "content": "\\begin{array} { l l } { { \\displaystyle { \\mathbb E } \\big [ \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi \\big ( y _ { i } | x _ { i } \\big ) } { \\pi _ { 0 } \\big ( y _ { i } | x _ { i } \\big ) } r _ { i } + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\frac { \\pi \\big ( y _ { j } | x _ { j } \\big ) } { p _ { j } } \\hat { \\delta } ( x _ { j } , y _ { j } ) \\big ] } } \\\\ { { \\displaystyle { = \\mathbb E } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi \\big ( y | x \\big ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) \\big ] + \\frac { { \\mathbb E } } { x } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\frac { 1 } { | \\mathcal { U } ( x , \\pi _ { 0 } ) | } \\frac { \\pi \\big ( y | x \\big ) } { \\textstyle | \\mathcal { U } ( x , \\pi _ { 0 } ) | } \\hat { \\delta } ( x , y ) \\big ] } } \\\\ { { \\displaystyle { = \\mathbb E } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) \\big ] + \\frac { { \\mathbb E } } { x } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] } } \\end{array}", + "type": "interline_equation", + "image_path": "e485bea8deb18f822ec7c8e3e8f8397a13e0917e30960c340d932b0f8d700269.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 125, + 99, + 470, + 131.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 125, + 131.66666666666666, + 470, + 164.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 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i } + \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x _ { \\star } , \\pi _ { 0 } ) } \\pi ( y | x _ { i } ) \\hat { \\delta } ( x _ { i } , y ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\underbrace { \\mathbb { E } } _ { y \\in \\pi _ { 0 } } [ \\frac { \\pi } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) ] + \\displaystyle \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) \\hat { \\delta } ( x , y ^ { \\prime } ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) + \\displaystyle \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) \\hat { \\delta } ( x , y ^ { \\prime } ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) \\big ] + \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] } \\end{array}", + "type": "interline_equation", + "image_path": "352dff5875f2a0f8b5de2aa38d0bba4030835c62b98de05c4201943b4bfa45fa.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 165, + 221, + 446, + 264.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 165, + 264.0, + 446, + 307.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 165, + 307.0, + 446, + 350.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 371, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 372, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 372, + 369 + ], + "score": 1.0, + "content": "The proof is done by comparing Equation (23) and Equation (24).", + "type": "text" + } + ], + "index": 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Given contexts", + "type": "text" + }, + { + "bbox": [ + 232, + 402, + 291, + 411 + ], + "score": 0.88, + "content": "x _ { 1 } , x _ { 2 } , \\ldots , x _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 399, + 461, + 414 + ], + "score": 1.0, + "content": "drawn i.i.d from the unknown distribution", + "type": "text" + }, + { + "bbox": [ + 462, + 400, + 487, + 412 + ], + "score": 0.91, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 399, + 506, + 414 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 135, + 424 + ], + "score": 1.0, + "content": "action", + "type": "text" + }, + { + "bbox": [ + 135, + 412, + 144, + 423 + ], + "score": 0.77, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 410, + 318, + 424 + ], + "score": 1.0, + "content": "drawn independently from logging policy", + "type": "text" + }, + { + "bbox": [ + 318, + 412, + 330, + 422 + ], + "score": 0.82, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 410, + 399, + 424 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 400, + 411, + 438, + 423 + ], + "score": 0.93, + "content": "\\pi _ { 0 } ( \\mathcal { V } | x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 410, + 506, + 424 + ], + "score": 1.0, + "content": ", the bias of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 421, + 386, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 268, + 437 + ], + "score": 1.0, + "content": "empirical risk defined in Equation (7) is", + "type": "text" + }, + { + "bbox": [ + 269, + 422, + 382, + 435 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathbb E _ { x } [ \\sum _ { y \\in \\mathcal U _ { x } ^ { \\pi _ { 0 } } } \\pi ( y | x ) \\Delta ( x , y ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 421, + 386, + 437 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 452, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 453, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 348, + 464 + ], + "score": 1.0, + "content": "Proof. From Appendix A.5, we are given the expectation of", + "type": "text" + }, + { + "bbox": [ + 349, + 448, + 387, + 462 + ], + "score": 0.93, + "content": "\\hat { R } _ { I P S } ^ { \\delta } ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 448, + 453, + 464 + ], + "score": 1.0, + "content": ", and the bias is:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 467, + 464, + 576 + ], + "lines": [ + { + "bbox": [ + 129, + 467, + 464, + 576 + ], + "spans": [ + { + "bbox": [ + 129, + 467, + 464, + 576 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { b i a s ( \\hat { R } _ { I P S } ^ { \\delta } ( \\pi ) ) = { \\displaystyle \\mathbb { E } } [ \\hat { R } _ { I P S } ^ { \\delta } ( \\pi ) ] - R ( \\pi ) } } \\\\ { ~ = { \\displaystyle \\frac { { \\mathbb { E } } } { x } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) + \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] - R ( \\pi ) } \\\\ { ~ = { \\displaystyle \\mathbb { E } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) ( \\hat { \\delta } ( x , y ) - \\delta ( x , y ) ) \\big ] } \\\\ { ~ = { \\displaystyle \\mathbb { E } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\Delta ( x , y ) \\big ] } \\end{array}", + "type": "interline_equation", + "image_path": "2cef212a8c41541db37673064d4fa4e6251f9e604d511a5aecebfc41f828627f.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 129, + 467, + 464, + 503.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 129, + 503.3333333333333, + 464, + 539.6666666666666 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 129, + 539.6666666666666, + 464, + 576.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 504, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 462, + 594 + ], + "score": 1.0, + "content": "The second equality is from Appendix A.5, the second equality is based on", + "type": "text" + }, + { + "bbox": [ + 463, + 581, + 505, + 594 + ], + "score": 0.87, + "content": "\\begin{array} { r l } { R ( \\pi ) } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 590, + 507, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 357, + 607 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x } \\left[ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) + \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 590, + 507, + 610 + ], + "score": 1.0, + "content": ", and the last one is based on the def-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 146, + 621 + ], + "score": 1.0, + "content": "inition of", + "type": "text" + }, + { + "bbox": [ + 146, + 607, + 262, + 621 + ], + "score": 0.92, + "content": "\\Delta ( x , y ) : = \\hat { \\delta } ( x , y ) - \\delta ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 606, + 290, + 621 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 291, + 609, + 347, + 620 + ], + "score": 0.9, + "content": "x \\in \\mathcal { X } , y \\in \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 606, + 351, + 621 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 607, + 506, + 621 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 354, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 496, + 356, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 496, + 356, + 505, + 366 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 394, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 395, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 395, + 97 + ], + "score": 1.0, + "content": "Proof. Taking the expectation of empirical risk defined in Equation (8):", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 395, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 99, + 470, + 197 + ], + "lines": [ + { + "bbox": [ + 125, + 99, + 470, + 197 + ], + "spans": [ + { + "bbox": [ + 125, + 99, + 470, + 197 + ], + "score": 0.95, + "content": "\\begin{array} { l l } { { \\displaystyle { \\mathbb E } \\big [ \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi \\big ( y _ { i } | x _ { i } \\big ) } { \\pi _ { 0 } \\big ( y _ { i } | x _ { i } \\big ) } r _ { i } + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\frac { \\pi \\big ( y _ { j } | x _ { j } \\big ) } { p _ { j } } \\hat { \\delta } ( x _ { j } , y _ { j } ) \\big ] } } \\\\ { { \\displaystyle { = \\mathbb E } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi \\big ( y | x \\big ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) \\big ] + \\frac { { \\mathbb E } } { x } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\frac { 1 } { | \\mathcal { U } ( x , \\pi _ { 0 } ) | } \\frac { \\pi \\big ( y | x \\big ) } { \\textstyle | \\mathcal { U } ( x , \\pi _ { 0 } ) | } \\hat { \\delta } ( x , y ) \\big ] } } \\\\ { { \\displaystyle { = \\mathbb E } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) \\big ] + \\frac { { \\mathbb E } } { x } \\big [ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] } } \\end{array}", + "type": "interline_equation", + "image_path": "e485bea8deb18f822ec7c8e3e8f8397a13e0917e30960c340d932b0f8d700269.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 125, + 99, + 470, + 131.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 125, + 131.66666666666666, + 470, + 164.33333333333331 + ], + 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{ \\pi \\big ( y _ { 1 } | x _ { i } \\big ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } r _ { i } + \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x _ { \\star } , \\pi _ { 0 } ) } \\pi ( y | x _ { i } ) \\hat { \\delta } ( x _ { i } , y ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\underbrace { \\mathbb { E } } _ { y \\in \\pi _ { 0 } } [ \\frac { \\pi } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) ] + \\displaystyle \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) \\hat { \\delta } ( x , y ^ { \\prime } ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi _ { 0 } ( y | x ) \\frac { \\pi ( y | x ) } { \\pi _ { 0 } ( y | x ) } \\delta ( x , y ) + \\displaystyle \\sum _ { y ^ { \\prime } \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y ^ { \\prime } | x ) \\hat { \\delta } ( x , y ^ { \\prime } ) \\big ] } \\\\ & { = \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) \\big ] + \\frac { \\mathbb { E } } { x } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] } \\end{array}", + "type": "interline_equation", + "image_path": "352dff5875f2a0f8b5de2aa38d0bba4030835c62b98de05c4201943b4bfa45fa.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 165, + 221, + 446, + 264.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 165, + 264.0, + 446, + 307.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 165, + 307.0, + 446, + 350.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 371, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 372, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 372, + 369 + ], + "score": 1.0, + "content": "The proof is done by comparing Equation (23) and Equation (24).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 352, + 372, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 379, + 246, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 247, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 247, + 392 + ], + "score": 1.0, + "content": "A.6 PROOF OF PROPOSITION 2", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 232, + 414 + ], + "score": 1.0, + "content": "Proposition 2. Given contexts", + "type": "text" + }, + { + "bbox": [ + 232, + 402, + 291, + 411 + ], + "score": 0.88, + "content": "x _ { 1 } , x _ { 2 } , \\ldots , x _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 399, + 461, + 414 + ], + "score": 1.0, + "content": "drawn i.i.d from the unknown distribution", + "type": "text" + }, + { + "bbox": [ + 462, + 400, + 487, + 412 + ], + "score": 0.91, + "content": "P ( \\mathcal X )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 399, + 506, + 414 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 135, + 424 + ], + "score": 1.0, + "content": "action", + "type": "text" + }, + { + "bbox": [ + 135, + 412, + 144, + 423 + ], + "score": 0.77, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 410, + 318, + 424 + ], + "score": 1.0, + "content": "drawn independently from logging policy", + "type": "text" + }, + { + "bbox": [ + 318, + 412, + 330, + 422 + ], + "score": 0.82, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 410, + 399, + 424 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 400, + 411, + 438, + 423 + ], + "score": 0.93, + "content": "\\pi _ { 0 } ( \\mathcal { V } | x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 410, + 506, + 424 + ], + "score": 1.0, + "content": ", the bias of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 421, + 386, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 268, + 437 + ], + "score": 1.0, + "content": "empirical risk defined in Equation (7) is", + "type": "text" + }, + { + "bbox": [ + 269, + 422, + 382, + 435 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathbb E _ { x } [ \\sum _ { y \\in \\mathcal U _ { x } ^ { \\pi _ { 0 } } } \\pi ( y | x ) \\Delta ( x , y ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 421, + 386, + 437 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 399, + 506, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 452, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 453, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 348, + 464 + ], + "score": 1.0, + "content": "Proof. From Appendix A.5, we are given the expectation of", + "type": "text" + }, + { + "bbox": [ + 349, + 448, + 387, + 462 + ], + "score": 0.93, + "content": "\\hat { R } _ { I P S } ^ { \\delta } ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 448, + 453, + 464 + ], + "score": 1.0, + "content": ", and the bias is:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 448, + 453, + 464 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 467, + 464, + 576 + ], + "lines": [ + { + "bbox": [ + 129, + 467, + 464, + 576 + ], + "spans": [ + { + "bbox": [ + 129, + 467, + 464, + 576 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { b i a s ( \\hat { R } _ { I P S } ^ { \\delta } ( \\pi ) ) = { \\displaystyle \\mathbb { E } } [ \\hat { R } _ { I P S } ^ { \\delta } ( \\pi ) ] - R ( \\pi ) } } \\\\ { ~ = { \\displaystyle \\frac { { \\mathbb { E } } } { x } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) + \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\hat { \\delta } ( x , y ) \\big ] - R ( \\pi ) } \\\\ { ~ = { \\displaystyle \\mathbb { E } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) ( \\hat { \\delta } ( x , y ) - \\delta ( x , y ) ) \\big ] } \\\\ { ~ = { \\displaystyle \\mathbb { E } } \\big [ \\displaystyle \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\Delta ( x , y ) \\big ] } \\end{array}", + "type": "interline_equation", + "image_path": "2cef212a8c41541db37673064d4fa4e6251f9e604d511a5aecebfc41f828627f.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 129, + 467, + 464, + 503.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 129, + 503.3333333333333, + 464, + 539.6666666666666 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 129, + 539.6666666666666, + 464, + 576.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 504, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 462, + 594 + ], + "score": 1.0, + "content": "The second equality is from Appendix A.5, the second equality is based on", + "type": "text" + }, + { + "bbox": [ + 463, + 581, + 505, + 594 + ], + "score": 0.87, + "content": "\\begin{array} { r l } { R ( \\pi ) } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 590, + 507, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 357, + 607 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { x } \\left[ \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) ^ { c } } \\pi ( y | x ) \\delta ( x , y ) + \\sum _ { y \\in \\mathcal { U } ( x , \\pi _ { 0 } ) } \\pi ( y | x ) \\delta ( x , y ) \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 590, + 507, + 610 + ], + "score": 1.0, + "content": ", and the last one is based on the def-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 146, + 621 + ], + "score": 1.0, + "content": "inition of", + "type": "text" + }, + { + "bbox": [ + 146, + 607, + 262, + 621 + ], + "score": 0.92, + "content": "\\Delta ( x , y ) : = \\hat { \\delta } ( x , y ) - \\delta ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 606, + 290, + 621 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 291, + 609, + 347, + 620 + ], + "score": 0.9, + "content": "x \\in \\mathcal { X } , y \\in \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 606, + 351, + 621 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 607, + 506, + 621 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 579, + 507, + 621 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 264, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 264, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 264, + 95 + ], + "score": 1.0, + "content": "B APPENDIX: EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 504, + 119 + ], + "score": 1.0, + "content": "In this section, we provide the experiment details and additional results to help promote reproducibil-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 173, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 173, + 130 + ], + "score": 1.0, + "content": "ity of this work.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 142, + 258, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 259, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 259, + 155 + ], + "score": 1.0, + "content": "B.1 EXPERIMENT SETUP DETAILS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "Datasets and baseline. We follow a 75:10:15 train-validation-test split for credit card fraud detec-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 173, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 425, + 188 + ], + "score": 1.0, + "content": "tion dataset, while for CIFAR10 already coming with a train-test split, we keep", + "type": "text" + }, + { + "bbox": [ + 425, + 175, + 445, + 185 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 173, + 505, + 188 + ], + "score": 1.0, + "content": "of the training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "set as validation set. Baseline estimators are IPS and DM, the hyperparameters (learning rate, L2", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 196, + 413, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 413, + 209 + ], + "score": 1.0, + "content": "regularization) are optimized for all the methods based on the validation set.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 220, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 103, + 217, + 508, + 237 + ], + "spans": [ + { + "bbox": [ + 103, + 217, + 369, + 237 + ], + "score": 1.0, + "content": "Bandit data generation. For CIFAR10, given supervised data", + "type": "text" + }, + { + "bbox": [ + 369, + 221, + 416, + 233 + ], + "score": 0.93, + "content": "\\{ x _ { i } , y _ { i } ^ { * } \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 217, + 444, + 237 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 445, + 223, + 455, + 232 + ], + "score": 0.85, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 217, + 508, + 237 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 230, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 181, + 245 + ], + "score": 1.0, + "content": "3072 features and", + "type": "text" + }, + { + "bbox": [ + 181, + 232, + 192, + 244 + ], + "score": 0.88, + "content": "y _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 230, + 505, + 245 + ], + "score": 1.0, + "content": "denotes the correct label of data (ranging from 0 to 9), under logging policy", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 242, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 118, + 254 + ], + "score": 0.83, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 242, + 316, + 256 + ], + "score": 1.0, + "content": ", the logged bandit data is generated by drawing", + "type": "text" + }, + { + "bbox": [ + 316, + 243, + 378, + 255 + ], + "score": 0.93, + "content": "y _ { i } \\sim \\pi _ { 0 } ( \\mathcal { V } | x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 242, + 506, + 256 + ], + "score": 1.0, + "content": ", then a deterministic reward is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 252, + 507, + 269 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 150, + 269 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + }, + { + "bbox": [ + 150, + 254, + 187, + 267 + ], + "score": 0.92, + "content": "\\mathbb { I } _ { \\{ y _ { i } = y _ { i } ^ { * } \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 252, + 507, + 269 + ], + "score": 1.0, + "content": ". 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In this experiment, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "empirically verify the reliability of estimating support divergence using control variate. The tar-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "get policy is the uniform policy while the logging policies are varying in their support deficiency.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Results are averaged over 10 runs and is shown in Figure 4. We investigate the behaviour of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 203, + 722 + ], + "score": 0.92, + "content": "S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "under different number of training data and different support deficiency", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "of the corresponding logging policy. As we can see, the sum converges to 1 as the training data", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 264, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 264, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 264, + 95 + ], + "score": 1.0, + "content": "B APPENDIX: EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 504, + 119 + ], + "score": 1.0, + "content": "In this section, we provide the experiment details and additional results to help promote reproducibil-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 173, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 173, + 130 + ], + "score": 1.0, + "content": "ity of this work.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 106, + 504, + 130 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 142, + 258, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 259, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 259, + 155 + ], + "score": 1.0, + "content": "B.1 EXPERIMENT SETUP DETAILS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "Datasets and baseline. We follow a 75:10:15 train-validation-test split for credit card fraud detec-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 173, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 425, + 188 + ], + "score": 1.0, + "content": "tion dataset, while for CIFAR10 already coming with a train-test split, we keep", + "type": "text" + }, + { + "bbox": [ + 425, + 175, + 445, + 185 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 173, + 505, + 188 + ], + "score": 1.0, + "content": "of the training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "set as validation set. Baseline estimators are IPS and DM, the hyperparameters (learning rate, L2", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 196, + 413, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 413, + 209 + ], + "score": 1.0, + "content": "regularization) are optimized for all the methods based on the validation set.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 163, + 505, + 209 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 220, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 103, + 217, + 508, + 237 + ], + "spans": [ + { + "bbox": [ + 103, + 217, + 369, + 237 + ], + "score": 1.0, + "content": "Bandit data generation. For CIFAR10, given supervised data", + "type": "text" + }, + { + "bbox": [ + 369, + 221, + 416, + 233 + ], + "score": 0.93, + "content": "\\{ x _ { i } , y _ { i } ^ { * } \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 217, + 444, + 237 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 445, + 223, + 455, + 232 + ], + "score": 0.85, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 217, + 508, + 237 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 230, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 181, + 245 + ], + "score": 1.0, + "content": "3072 features and", + "type": "text" + }, + { + "bbox": [ + 181, + 232, + 192, + 244 + ], + "score": 0.88, + "content": "y _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 230, + 505, + 245 + ], + "score": 1.0, + "content": "denotes the correct label of data (ranging from 0 to 9), under logging policy", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 242, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 118, + 254 + ], + "score": 0.83, + "content": "\\pi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 242, + 316, + 256 + ], + "score": 1.0, + "content": ", the logged bandit data is generated by drawing", + "type": "text" + }, + { + "bbox": [ + 316, + 243, + 378, + 255 + ], + "score": 0.93, + "content": "y _ { i } \\sim \\pi _ { 0 } ( \\mathcal { V } | x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 242, + 506, + 256 + ], + "score": 1.0, + "content": ", then a deterministic reward is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 252, + 507, + 269 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 150, + 269 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + }, + { + "bbox": [ + 150, + 254, + 187, + 267 + ], + "score": 0.92, + "content": "\\mathbb { I } _ { \\{ y _ { i } = y _ { i } ^ { * } \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 252, + 507, + 269 + ], + "score": 1.0, + "content": ". For the credit card fraud detection dataset, we throw away the class label and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "only use the features for each sample to generate bandit data. To be specific, for each sample with a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "28-dimensional feature vector, we define the first 20 features as the contextual information, and use", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 287, + 501, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 501, + 299 + ], + "score": 1.0, + "content": "the remaining 8 features as the underlying true reward for 8 different actions (with normalization).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 103, + 217, + 508, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 104, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "Logging policy. For CIFAR, we learn the softmax logging policy on 35K full-information data", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 506, + 335 + ], + "score": 1.0, + "content": "points as a multi-class classification problem with cross-entropy loss. Similar as the experiments", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "on BanditNet (Joachims et al., 2018), we adopt the conventional ResNet20 architecture but restrict", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "training after a mere two epochs to derive a relative stochastic policy, since it will be easier to add", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "temperature later to control its stochasticity and support deficiency. 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For each experiment, we train a different regression function using the full", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "bandit dataset. We use the same architecture as the one used for off-policy learning - where the final", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "layer is the size of the actions, specifying the reward for each action given a particular context. 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In this experiment, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "empirically verify the reliability of estimating support divergence using control variate. The tar-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "get policy is the uniform policy while the logging policies are varying in their support deficiency.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Results are averaged over 10 runs and is shown in Figure 4. We investigate the behaviour of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 203, + 722 + ], + "score": 0.92, + "content": "S _ { \\mathcal { D } } ( \\pi | \\pi _ { 0 } ) + \\mathcal { D } _ { \\mathcal { X } } ( \\pi | \\pi _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "under different number of training data and different support deficiency", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "of the corresponding logging policy. As we can see, the sum converges to 1 as the training data", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "increases. Meanwhile, the variance decreases as shown in the right figure. 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0000000000000000000000000000000000000000..031d19195292b81e9fd812130a3dd5e1eb75f8b0 --- /dev/null +++ b/parse/train/ajOrOhQOsYx/ajOrOhQOsYx_content_list.json @@ -0,0 +1,16363 @@ +[ + { + "type": "text", + "text": "A WIGNER-ECKART THEOREM FOR GROUP EQUIVARIANT CONVOLUTION KERNELS ", + "text_level": 1, + "bbox": [ + 176, + 99, + 750, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Leon Lang∗ \nAMLab, CSL \nUniversity of Amsterdam \nl.lang@uva.nl ", + "bbox": [ + 183, + 172, + 352, + 228 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Maurice Weiler AMLab, QUVA Lab University of Amsterdam m.weiler.ml@gmail.com ", + "bbox": [ + 486, + 172, + 696, + 228 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 266, + 544, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with $G$ -steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the $G$ -steerability constraint has been derived, it has to date only been solved for specific use cases – a general characterization of $G$ - steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of $G$ being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerable kernel spaces are fully understood and parameterized in terms of 1) generalized reduced matrix elements, 2) ClebschGordan coefficients, and 3) harmonic basis functions on homogeneous spaces. ", + "bbox": [ + 233, + 297, + 764, + 505 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 534, + 334, + 549 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Undoubtedly, symmetries play a central role in the formulation of physical theories. Any imposed symmetry greatly reduces the set of admissible physical laws and dynamics. Specifically in quantum mechanics, the Hilbert space of a system is equipped with a group representation which specifies the transformation law of system states. Quantum mechanical operators, which map between different states, are required to respect these transformation laws. That is, any symmetry transformation of a state on which they act should lead to a corresponding transformation of the resulting state after their action. This requirement imposes a symmetry constraint on the operators themselves – only specific operators can map between a given pair of states. ", + "bbox": [ + 174, + 565, + 825, + 678 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The situation in equivariant deep learning is remarkably similar to that in physics. Instead of a physical system, one considers in this case some learning task subject to symmetries. For instance, image segmentation is usually assumed to be translationally symmetric: a shift of the input image should lead to a corresponding shift of the predicted segmentation mask. Convolutional networks guarantee this property via their inherent translation equivariance. The role of the quantum states is in equivariant deep learning taken by the features in each layer, which are due to the enforced equivariance endowed with some transformation law. The analog of quantum mechanical operators, mapping between states, is the neural connectivity, mapping between features of consecutive layers. As in the case of operators, there is a symmetry (equivariance) constraint on the neural connectivity – only specific connectivity patterns guarantee a correct transformation law of the resulting features. ", + "bbox": [ + 174, + 684, + 825, + 823 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work we are considering group equivariant convolutional networks (GCNNs), which are convolutional networks that are equivariant w.r.t. symmetries of the space on which the convolution is performed. Typical examples are isometry equivariant CNNs on Euclidean spaces (Weiler & Cesa, 2019) or spherical CNNs (Cohen et al., 2018). Many different formulations of GCNNs have been proposed, however, it has recently been shown that $H$ -equivariant GCNNs on homogeneous spaces ", + "bbox": [ + 174, + 830, + 823, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "$H / G$ can in a fairly general setting be understood as performing convolutions with $G$ -steerable kernels (Cohen et al., 2019b). Convolutional weight sharing hereby guarantees the equivariance under “translations” of the space while $G$ -steerability is a constraint on the convolution kernel that ensures its equivariance under the action of the stabilizer subgroup $G \\ : < \\ : H$ . Although the space of $G$ - steerable kernels has been characterized for specific choices of groups $G$ and feature transformation laws, i.e., group representations $\\rho$ , see Section 5, no general solution was known so far. This work characterizes the solution space for arbitrary compact groups $G$ . ", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our solution is motivated by the close resemblance of the $G$ -steerability kernel constraint to the defining constraint of spherical tensor operators (or more general representation operators (Jeevanjee, 2011)) in quantum mechanics. The famous Wigner-Eckart theorem describes the general structure of these operators by Clebsch-Gordan coefficients, with the degrees of freedom given by reduced matrix elements. By generalizing this theorem, we find a general characterization and parameterization of $G$ -steerable kernel spaces. For specific examples, like $G = \\mathrm { S O ( 3 ) }$ or compact subgroups of $G = \\mathrm { O } ( 2 )$ , our kernel space solution specializes to earlier work, e.g., Worrall et al. (2016); Thomas et al. (2018); Weiler & Cesa (2019). Our main contributions are the following: ", + "bbox": [ + 174, + 208, + 823, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We present a generalized Wigner-Eckart theorem 4.1 for $G$ -steerable kernels. It describes the general structure of equivariant kernels in terms of 1) endomorphism bases, which generalize reduced matrix elements, 2) Clebsch-Gordan coefficients, and 3) harmonic basis functions on a suitable homogeneous space. In contrast to the usual formulation, we cover any compact group $G$ and both real and complex representations. • Corollary 4.2 explains how to parameterize $G$ -steerable kernels and thus GCNNs. • We apply the theorem exemplarily to solve for the kernel spaces for the symmetry groups SO(2) , $\\mathbb { Z } / 2$ , SO(3) and O(3) , considering both real and complex representations. Thereby, we demonstrate that the endomorphism bases, Clebsch-Gordan coefficients, and harmonic basis functions can usually be determined for practically relevant symmetry groups. ", + "bbox": [ + 196, + 325, + 825, + 474 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 SYMMETRY-CONSTRAINED OPERATORS AND THEIR MATRIX ELEMENTS", + "text_level": 1, + "bbox": [ + 174, + 493, + 802, + 510 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To motivate our generalized Wigner-Eckart theorem, we review quantum mechanical representation operators and $G$ -steerable kernels with an emphasis on the similarity of their underlying symmetry constraints. Due to their symmetries, the matrix elements of such operators and kernels are fully specified by a comparatively small number of reduced matrix elements or learnable parameters, respectively. This reduction is for representation operators described by the Wigner-Eckart theorem. For clarity, we discuss this theorem in its most popular form, i.e., for spherical tensor operators (SO(3)-representation operators transforming under irreducible representations). ", + "bbox": [ + 173, + 525, + 825, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The Representation Operator Constraint Consider a quantum mechanical system with symmetry under the action of some group $G$ , for instance rotations. The action of this symmetry group on quantum states is modeled by some unitary $G$ -representation1 $U : G \\to \\operatorname { U } ( { \\mathcal { H } } )$ on the Hilbert space $\\mathcal { H }$ . More specifically, $G$ acts on kets according to $\\left| \\psi \\right. \\mapsto \\left| \\psi ^ { \\prime } \\right. : = U ( g ) \\left| \\psi \\right.$ and on bras according to $\\langle \\psi | \\mapsto \\mathcal { \\bar { \\langle } } \\psi ^ { \\prime } | : = \\bar { \\langle } \\psi | U ( g ) ^ { \\dagger }$ , where $U ( g ) ^ { \\dagger }$ is the adjoint of $U ( g )$ . Observables of the system correspond to self-adjoint operators $A = A ^ { \\dagger }$ . The expectation value of such an observable in some quantum state $| \\psi \\rangle$ is given by $\\langle \\psi | A | \\psi \\rangle \\in \\mathbb { R }$ . ", + "bbox": [ + 174, + 628, + 825, + 728 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The transformation behaviors of states and observables need to be consistent with each other. As an example, consider a system consisting of a single, free particle in $\\mathbb { R } ^ { 3 }$ , which is (among other symmetries) symmetric under rotations $\\bar { G } = \\mathrm { S O } ( \\bar { 3 } )$ . The momentum of the particle in the direction of the three frame axes is measured by the three momentum operators $( P _ { 1 } , P _ { 2 } , P _ { 3 } )$ . Since the momentum of a classical particle transforms geometrically like a vector, one needs to demand the same for the momentum observable expectation values. If we denote by $p _ { i } : = \\langle \\psi | P _ { i } | \\psi \\rangle$ the expected momentum in $i$ -direction, this means that the expected momentum of a rotated system is given by $\\begin{array} { r } { p _ { i } ^ { \\prime } = \\sum _ { j } R _ { i j } p _ { j } = \\sum _ { j } R _ { i j } \\langle \\psi | P _ { j } | \\psi \\rangle } \\end{array}$ , where $\\bar { R } \\in \\mathrm { S O } ( 3 )$ is an element of the rotation group. This result should agree with the expectation values for rotated system states, that is, $p _ { i } ^ { \\prime } \\stackrel { } { = } \\langle \\psi ^ { \\prime } | P _ { i } | \\psi ^ { \\prime } \\rangle = \\langle \\psi | U ( R ) ^ { \\dagger } \\bar { P } _ { i } U ( R ) | \\psi \\rangle$ . As this argument is independent from the particular choice of state $| \\psi \\rangle$ , and making use of the linearity of the operations, this implies a consistency constraint $\\begin{array} { r } { \\sum _ { j } R _ { i j } P _ { j } \\ = \\ U ( R ) ^ { \\dagger } P _ { i } U ( R ) } \\end{array}$ , which identifies the collection $( P _ { 1 } , P _ { 2 } , P _ { 3 } )$ as a vector operator. Other geometric quantities are required to satisfy similar constraints: For instance, energy is a scalar (i.e., invariant) quantity and the Hamilton operator $H$ is a scalar operator, satisfying $\\overset { \\vartriangle } { \\boldsymbol { H } } = \\boldsymbol { U } ( \\boldsymbol { R } ) ^ { \\dagger } \\boldsymbol { H } \\boldsymbol { U } ( \\boldsymbol { R } )$ . Similarly, any matrix valued classical quantity corresponds to a rank $( 1 , 1 )$ Cartesian tensor operator $( M _ { i j } ) _ { i , j = 1 , 2 , 3 }$ subject to $\\begin{array} { r } { \\sum _ { k l } R _ { i k } \\hat { M _ { k l } } ( \\bar { R } ^ { - 1 } ) _ { l j } = \\hat { U } ( R ) ^ { \\dagger } M _ { i j } U ( R ) } \\end{array}$ . The overarching framework to study such situations is the notion of a representation operator, which we define as a family of operators $( A _ { 1 } , \\dotsc , A _ { N } )$ which are required to satisfy the constraint ", + "bbox": [ + 173, + 733, + 825, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 204 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6caf93adb572d2d0abab63401aecdf47d86d141b6f4b3cff372e064ae3d046f9.jpg", + "text": "$$\n{ \\sum } _ { j = 1 } ^ { N } { \\pi } ( g ) _ { i j } A _ { j } \\ = \\ U ( g ) ^ { \\dagger } A _ { i } U ( g ) \\qquad \\forall g \\in G ,\n$$", + "text_format": "latex", + "bbox": [ + 330, + 208, + 665, + 238 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\pi : G \\to \\mathrm { U } ( \\mathbb { C } ^ { N } )$ is some unitary representation of the symmetry group under consideration. The examples above correspond to specific choices of representations, namely the trivial representation $\\pi ( \\bar { R } ) = 1$ for scalars, the “standard” representation $\\pi ( R ) = R$ for vectors and the tensor product representation $\\pi ( R ) = R \\otimes ( R ^ { - 1 } ) ^ { \\top }$ for matrices. Spherical tensor operators, discussed below, correspond to the irreps (irreducible representations) of SO(3). ", + "bbox": [ + 174, + 243, + 825, + 314 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The Steerable Kernel Constraint Convolution kernels of group equivariant CNNs are required to satisfy a very similar constraint to that in Eq. (1). Before coming to such GCNNs, consider the case of conventional CNNs, processing image-like signals on a Euclidean space $\\mathbb { R } ^ { d }$ . Such signals are formalized as $c$ -channel feature maps $\\overline { { f } } : \\mathbb { R } ^ { d } \\ : \\ : \\mathbb { K } ^ { c }$ that assign a $c$ -dimensional feature vector $f ( x ) \\in \\mathbb { K } ^ { c }$ to each point $\\boldsymbol { x } \\in \\mathbb { R } ^ { d }$ , where we allow for K being either of the real or complex numbers $\\mathbb { R }$ or C. Each CNN layer maps its input feature map $f _ { \\mathrm { i n } } : \\mathbb { R } ^ { \\tilde { d } } \\to \\mathbb { K } ^ { c _ { \\mathrm { i n } } }$ via a convolution to an output feature map $f _ { \\mathrm { o u t } } : = \\dot { K } \\star f _ { \\mathrm { i n } } \\dot { : } \\mathbb { R } ^ { d } \\dot { } \\mathbb { K } ^ { c _ { \\mathrm { o u t } } }$ . Since the convolution maps $c _ { \\mathrm { i n } }$ input channels to $c _ { \\mathrm { o u t } }$ output channels, the kernel $K : \\mathbb { R } ^ { d } \\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ is matrix-valued. ", + "bbox": [ + 173, + 328, + 825, + 440 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Conventional CNNs are translation equivariant, however it is often desirable that the convolution is equivariant w.r.t. a larger symmetry group, for instance the isometries $\\operatorname { E } ( d )$ of $\\mathbb { R } ^ { d }$ (Weiler & Cesa, 2019). For simplicity, we consider semidirect product groups of the form $( \\ln ^ { d } , + ) \\rtimes G$ , where $G \\leq \\mathrm { G L } ( d )$ is any compact group. Group elements $t g \\in ( \\mathbb { R } ^ { d } , + ) \\rtimes G$ are uniquely split into a translation $t \\in ( \\mathbb { R } ^ { \\dot { d } } , + )$ and an element $g \\in G$ , stabilizing the origin. They act on $\\mathbf { \\bar { \\mathbb { R } } } ^ { d }$ according to $x \\mapsto ( t g ) \\cdot x : = g x + t$ . The equivariance of a GCNN – which is the analog to the symmetry of a quantum mechanical system – requires the feature spaces to be endowed with a group action of the symmetry group. A natural choice is to model the feature spaces as spaces of feature fields, for instance scalar, vector or tensor fields (Cohen & Welling, 2016b). ", + "bbox": [ + 173, + 446, + 825, + 573 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Such feature fields are defined as functions $f : \\mathbb { R } ^ { d } V$ , where the difference to conventional feature maps is that the space $V \\cong \\mathbb { K } ^ { c }$ of feature vectors is equipped with a group representation $\\rho : G \\to { \\mathrm { G L } } ( V )$ of the stabilizer $G$ . The full symmetry group acts on feature fields according to $f \\mapsto ( t g ) \\cdot \\dot { f } : = \\rho ( g ) \\circ f \\circ ( t g ) ^ { - 1 }$ , which is known as the induced representation of $\\rho$ . As proven in (Weiler et al., 2018a), the most general linear and equivariant map from an input field $f _ { \\mathrm { i n } } : \\mathbb { R } ^ { d } \\to V _ { \\mathrm { i n } }$ to an output field $f _ { \\mathrm { o u t } } : \\bar { \\mathbb { R } } ^ { d } \\to V _ { \\mathrm { o u t } }$ is a convolution with a $G$ -steerable kernel $K : \\mathbb { R } ^ { d } \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\cong \\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ . Such kernels take values in the space of linear operators from $V _ { \\mathrm { i n } }$ to $V _ { \\mathrm { o u t } }$ and are required to satisfy the $G$ -steerability (equivariance) constraint ", + "bbox": [ + 173, + 578, + 825, + 691 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/41f21ee5150bc7930f98286cfb1ef86c9e837dd31a0a60e6d0d51c9b8f98be48.jpg", + "text": "$$\nK ( g x ) = \\rho _ { \\mathsf { o u t } } ( g ) \\circ K ( x ) \\circ \\rho _ { \\mathsf { i n } } ( g ) ^ { - 1 } \\qquad \\forall g \\in G , x \\in \\mathbb { R } ^ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 295, + 695, + 700, + 715 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "One can easily check that a convolution with a $G$ -steerable kernel $K$ is indeed equivariant, i.e., satisfies $K \\star ( { \\dot { ( } } t g ) \\cdot f ) = ( t g ) \\cdot ( K \\star f )$ for any $t g \\in ( \\mathbb { R } ^ { d } , + ) \\rtimes G$ . This result was later generalized to feature fields on homogeneous spaces $H / G$ of unimodular locally compact groups $H$ (Cohen et al., 2019b) and on Riemannian manifolds with structure group $G$ (Cohen et al., 2019a). That the equivariance of the convolutional network requires $G$ -steerable kernels in any of these settings underlines the great practical relevance of our results. ", + "bbox": [ + 173, + 719, + 825, + 804 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The two constraints, Eq. (1) and Eq. (2), are remarkably similar: the left-hand-sides are in both cases given by a $G$ -transformation of the operator or kernel itself while the right-hand-sides are given by pre- and postcomposition of the operator or kernel with unitary representations. More details on this comparison can be found in Appendix C.1.3. ", + "bbox": [ + 174, + 809, + 825, + 867 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The Wigner-Eckart Theorem for Spherical Tensor Operators All information about a linear operator $A : { \\mathcal { H } } \\to { \\mathcal { H } }$ is encoded by its matrix elements $A _ { \\mu \\nu } ~ : = ~ \\langle \\mu | A | \\nu \\rangle ~ \\in ~ \\mathbb { C }$ relative to a given basis, where $\\vert \\nu \\rangle \\in { \\mathcal { H } }$ and $\\langle \\mu | \\in { \\mathcal { H } } ^ { * }$ denote basis elements of the Hilbert space and its dual. Similarly, all information about a convolution kernel $K$ is encoded by its matrix elements $K _ { \\mu \\nu } ( x ) : = \\langle \\mu | K ( x ) | \\nu \\rangle \\in \\mathbb { K }$ , where $| \\nu \\rangle \\in V _ { \\mathrm { i n } }$ and $\\langle \\mu \\vert \\in V _ { \\mathrm { o u t } } ^ { * }$ are elements of chosen bases for the input representation and dual output representation. Considering general operators and kernels, i.e., ignoring the symmetry constraints in Eqs. (1) and (2), all matrix elements are independent degrees of freedom. In the case of convolution kernels, they correspond directly to the $c _ { \\mathrm { o u t } } \\cdot c _ { \\mathrm { i n } }$ learnable parameters for every point of the kernel. However, if $A$ is a representation operator – or if $K$ is a $G$ -steerable kernel – the symmetry constraints couple the matrix elements to each other such that they can not be chosen freely anymore. For representation operators, this statement is made precise by the Wigner-Eckart theorem. ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Wigner-Eckart theorem is best known in its classical form, which applies specifically to spherical tensor operators. These operators are the representation operators for the irreps of SO(3), i.e., the Wigner D-matrices $D _ { j } : \\dot { \\mathrm { S O } } ( 3 ) \\mathrm { U } ( \\mathbb { C } ^ { 2 j + \\hat { 1 } } )$ . As such, spherical tensor operators of rank $j$ are defined as families $\\pmb { T } _ { j } = ( T _ { j } ^ { - j } , \\dots , T _ { j } ^ { j } ) ^ { \\top }$ of $2 j + 1$ operators $T _ { j } ^ { m }$ that satisfy the constraint $\\begin{array} { r } { \\sum _ { n = - j } ^ { j } D _ { j } ^ { m n } ( g ) T _ { j } ^ { n } = U ( g ) ^ { \\dagger } T _ { j } ^ { \\phantom { \\dagger } } U ( g ) } \\end{array}$ for any $g \\in \\mathrm { S O } ( 3 )$ . ", + "bbox": [ + 174, + 236, + 825, + 315 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to express the operators $T _ { j } ^ { m }$ in terms of matrix elements, we need to fix a basis of $\\mathcal { H }$ . Due to the SO(3)-symmetry of $\\mathbf { \\delta } _ { T _ { j } }$ , a natural choice are the angular momentum eigenstates2 $| l n \\rangle$ , where $l \\in { \\mathbb { N } } _ { \\geq 0 }$ and $n = - l , \\ldots , l$ . For fixed quantum numbers $j , l$ , and $J$ , there are $2 j + 1$ components $T _ { j } ^ { m }$ of $\\mathbf { \\delta } _ { T _ { j } }$ , $2 l + 1$ basis kets $| l n \\rangle$ , and $2 J + 1$ basis bras $\\langle J M \\vert$ . This implies that there are $( 2 J + 1 ) ( 2 j + 1 ) ( 2 l + 1 )$ different matrix elements $\\left. J M | T _ { j } ^ { m } | l n \\right. \\in \\mathbb { C }$ for these quantum numbers. According to the Wigner-Eckart theorem, all of these matrix elements are fully specified by one single number (Jeevanjee, 2011): ", + "bbox": [ + 173, + 320, + 825, + 424 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 2.1 (Wigner-Eckart theorem for Spherical Tensor Operators). Let $j , l , J \\ \\in \\ \\mathbb { N } _ { \\ge 0 }$ and let $\\mathbf { \\delta } _ { T _ { j } }$ be a spherical tensor operator of rank $j$ . Then there is a unique complex number, the reduced matrix element $\\lambda \\in \\mathbb { C }$ (often written $\\langle J \\| T _ { j } \\| l \\rangle \\in \\mathbb { C } )$ , that completely determines any of the $( 2 J + 1 ) ( 2 j + 1 ) ( 2 l + 1 )$ matrix elements $\\langle J M | T _ { j } ^ { m } | l n \\rangle$ by the relation ", + "bbox": [ + 174, + 428, + 825, + 484 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e68f3cc7b7c8dcbe03b64571a0bb1d2d645d673f65b9bc9ff5738c2fca31efd2.jpg", + "text": "$$\n\\langle J M | T _ { j } ^ { m } | l n \\rangle = \\lambda \\cdot \\langle J M | j m ; l n \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 489, + 619, + 508 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The coupling coefficients $\\langle J M | j m ; l n \\rangle$ , known as Clebsch-Gordan coefficients, are given by the projection of the tensor product basis $| j m ; l n \\rangle : = | j m \\rangle \\otimes | l n \\rangle$ on $\\vert J M \\rangle$ . They are purely algebraic and therefore independent of the spherical tensor operator $\\mathbf { \\delta } _ { T _ { j } }$ . ", + "bbox": [ + 174, + 513, + 825, + 556 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This result generalizes to arbitrary representation operators of the form in Eq. (1) (Agrawala, 1980). The similarities between representation operators and $G$ -steerable kernels suggests that a similar statement might hold for the matrix elements of $G$ -steerable kernels as well. As proven below, this is indeed the case: our generalized Wigner-Eckart theorem separates their independent degrees of freedom from purely algebraic relations between mutually dependent matrix elements. It does therefore give an explicit parametrization of the space of $G$ -steerable kernels. ", + "bbox": [ + 173, + 566, + 825, + 650 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 BUILDING BLOCKS OF STEERABLE KERNELS ", + "text_level": 1, + "bbox": [ + 174, + 670, + 583, + 686 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This chapter gives a brief introduction to the mathematical concepts that are required to formulate our Wigner-Eckart theorem for $G$ -steerable kernels. The first two of the following paragraphs explain why it is w.l.o.g. possible to restrict attention to steerable kernels on homogeneous spaces and to irreducible representations. The following three paragraphs discuss the building blocks of steerable kernels, which are endomorphisms, harmonic basis functions described by the Peter-Weyl theorem, and tensor product representations and their Clebsch-Gordan decomposition. An illustration of the concepts introduced in this chapter is given in Appendix A. ", + "bbox": [ + 173, + 700, + 825, + 799 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Restriction to Homogeneous Spaces Convolution kernels are usually defined on a Euclidean space $\\mathbb { R } ^ { d }$ , i.e., they are functions $\\hat { K ^ { \\mathbf { \\alpha } } } \\colon { \\mathbb { R } ^ { d } } \\to \\operatorname { H o m } _ { { \\mathbb { K } } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . The $G$ -steerability constraint in Eq. (2) relates kernel values $K ( x )$ at $x$ to kernel values $K ( g x )$ at all other points $g x$ on the orbit $G { \\bar { x } } : = \\{ g x \\vert g \\in G \\}$ of $x$ . To solve the constraint, it is therefore w.l.o.g. sufficient to consider restrictions of kernels to the individual orbits, from which the full solution on $\\mathbb { R } ^ { d }$ can be assembled (Weiler et al., 2018a). By construction, the orbits have the structure of a homogeneous space: ", + "bbox": [ + 174, + 806, + 825, + 890 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 3.1 (Homogeneous Space, Transitive Action). Let $\\cdot : G { \\times } X \\to X$ be a continuous action of a compact group $G$ on a topological space $X$ . Then $X$ is called a homogeneous space w.r.t. $G$ if $\\varnothing \\neq X$ and if for all $x , y \\in X$ there is a $g \\in G$ such that $g x = y$ . The action is then called transitive. ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We will in the following w.l.o.g. consider steerable kernels $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ on such homogeneous spaces $X$ . ", + "bbox": [ + 176, + 156, + 823, + 185 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Restriction to Irreducible Unitary Representations The theorems below apply specifically to unitary representations, that is, representations for which the automorphisms $\\rho ( g )$ preserve distances (Knapp, 2002). As asserted by Theorem B.20, this is not really a restriction as every finite-dimensional linear representation can be considered as being unitary. Thus, we assume $\\rho : G \\to \\operatorname { U } ( V )$ , where $\\mathrm { U } ( V )$ is the unitary group, i.e., the group of distance-preserving linear functions on $V$ . In the case of $\\mathbb { K } = \\mathbb { R }$ we say orthogonal instead of unitary and write $\\mathrm { O } ( V )$ . ", + "bbox": [ + 173, + 200, + 825, + 285 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Additionally, prior research has shown that it is sufficient to solve the kernel constraint in Eq. (2) for irreducible (unitary) input- and output representations instead of arbitrary finite-dimensional representations (Weiler & Cesa, 2019). This is possible due to the linearity of the constraint and the fact that any finite-dimensional unitary representation decomposes by Proposition B.38 into an orthogonal direct sum of irreps. The solution for general representations can thus be recovered from the solutions for irreps. More details on these considerations can be found in Section D.1.3. ", + "bbox": [ + 173, + 291, + 825, + 375 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "If two unitary irreps are related by an isometric intertwiner, they are isomorphic; see Definition B.18. The set of isomorphism classes of unitary irreps of $G$ is denoted by $\\widehat { G }$ . We assume that for each isomorphism class $j \\in { \\widehat { G } }$ we have picked a representative irrep $\\rho _ { j } : G \\to \\operatorname { U } ( V _ { j } )$ . We denote by $d _ { j }$ the dimension of $V _ { j }$ , so that we have $V _ { j } \\cong \\mathbb { K } ^ { d _ { j } }$ . ", + "bbox": [ + 174, + 382, + 825, + 445 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Overall, we can w.l.o.g. replace $\\mathbb { R } ^ { d }$ with $X$ and $\\rho _ { \\mathrm { i n } }$ and $\\rho _ { \\mathrm { o u t } }$ by $\\rho _ { l } : G \\to { \\mathrm { U } } ( V _ { l } )$ and $\\rho _ { J } : G \\to$ $\\mathrm { U } ( V _ { J } )$ , where $X$ is a homogeneous space and $\\rho _ { l }$ and $\\rho _ { J }$ are (representatives of isomorphism classes of) irreducible unitary representations of $G$ . This leads to our working definition of steerable kernels, to which we restrict from now on: ", + "bbox": [ + 174, + 452, + 825, + 508 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 3.2 (Steerable Kernel on a Homogeneous Space w.r.t. Unitary Irreps). Let $X$ be a homogeneous space of $G$ and $\\rho _ { l } : G \\to \\operatorname { U } ( V _ { l } )$ and $\\rho _ { J } : G \\to \\operatorname { U } ( V _ { J } )$ be representatives of isomorphism classes of irreducible unitary representations of $G$ . A $G$ -steerable kernel (on a homogeneous space and w.r.t. unitary irreps) is any function $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ such that the following $G$ - steerability constraint holds: ", + "bbox": [ + 173, + 512, + 825, + 583 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/68f900c6bf66d49091d715382373e0473e1ec49ded7ad11f2304bcb2b61198a2.jpg", + "text": "$$\nK ( g x ) = \\rho _ { J } ( g ) \\circ K ( x ) \\circ \\rho _ { l } ( g ) ^ { - 1 } \\qquad \\forall g \\in G , x \\in X .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 587, + 691, + 606 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Endomorphisms An important concept, underlying the reduced matrix elements in the WignerEckart theorem for spherical tensor operators, is that of endomorphisms of linear representations. ", + "bbox": [ + 171, + 619, + 821, + 647 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 3.3 (Endomorphism of a of Linear Representation). Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ be a linear representation. An endomorphism of $\\rho$ is a linear map $c : V V$ which satisfies $\\dot { c } \\circ \\dot { \\rho } ( g ) = \\rho ( g ) \\dot { \\circ } c$ for all $g \\in G$ . The space of all endomorphisms of $\\rho$ is written $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V )$ . ", + "bbox": [ + 176, + 651, + 821, + 694 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Endomorphisms play a central role in our generalized Wigner-Eckart theorem for steerable kernels. To get an insight why this is the case, consider a given steerable kernel $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ . The post-composition $( c \\circ K ) ( x ) : = c \\circ ( K ( \\bar { x } ) )$ of this kernel with any endomorphism $c \\in$ $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ is obviously still steerable, i.e., satisfies Eq. (3). A basis of the space of steerable kernels is therefore partly explained by bases of the endomorphism spaces, and thus occurs in our general solution. In the following, we write $\\{ c _ { r } \\mid r = 1 , \\ldots , E _ { J } \\}$ for the basis of $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ , where $E _ { J } : = \\dim ( \\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) )$ is the dimension of the endomorphism space.3 ", + "bbox": [ + 173, + 704, + 825, + 805 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The Peter-Weyl Theorem and Harmonic Basis Functions A cornerstone in our proof of the Wigner-Eckart theorem for steerable kernels is Theorem C.7. It states that the space of steerable kernels, which are $G$ -equivariant maps $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ , is isomorphic to the space of linear $G$ -equivariant maps of the form $\\widehat { K } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ . We are therefore interested in the representation theory of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ , which is described by the Peter-Weyl theorem.4 ", + "bbox": [ + 174, + 819, + 825, + 862 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 101, + 823, + 133 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.4 (Peter-Weyl Theorem, Existence of Harmonic Basis Functions). Let $G$ be a compact group and $X$ a homogeneous space. Let $\\widehat { G }$ be the set of isomorphism classes of irreducible representations. For $j \\in { \\widehat { G } }$ , let $\\rho _ { j } : G \\to \\operatorname { U } ( V _ { j } )$ be a representative with dimension $d _ { j } = \\dim ( V _ { j } )$ . Then there are multiplicities $m _ { j } \\in \\mathbb { N } _ { \\geq 0 }$ with $\\bar { m } _ { j } \\leq d _ { j }$ , and for each $i = 1 , \\ldots , m _ { j }$ there are harmonic basis functions $Y _ { j i } ^ { m } : X \\mathbb { K }$ , $m = 1 , \\ldots , d _ { j }$ , such that the following three properties hold: ", + "bbox": [ + 174, + 136, + 825, + 212 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1. The $Y _ { j i } ^ { m }$ , for fixed $j$ and $i$ , are steerable (Freeman & Adelson, 1991; Hel-Or & Teo, 1998), i.e., transformation via $g \\in G$ can be expressed by shifting basis coefficients with $\\rho _ { j }$ : $\\begin{array} { r } { Y _ { j i } ^ { m } ( g ^ { - 1 } x ) = \\Big ( \\sum _ { m ^ { \\prime } = 1 } ^ { d _ { j } } \\rho _ { j } ^ { m ^ { \\prime } m } ( g ) Y _ { j i } ^ { m ^ { \\prime } } \\Big ) ( x ) . } \\end{array}$ ", + "bbox": [ + 212, + 219, + 825, + 275 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2. Any square-integrable function $f : X \\to \\mathbb { K }$ can be uniquely expanded in terms of harmonic basis functions, i.e., $\\begin{array} { r } { \\mathrm { \\Delta } f \\ = \\ \\sum _ { j \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { m = 1 } ^ { d _ { j } } \\lambda _ { j i m } Y _ { j i } ^ { m } } \\end{array}$ with coefficients $\\lambda _ { j i m } \\in \\mathbb { K }$ . ", + "bbox": [ + 209, + 280, + 825, + 316 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3. The $Y _ { j i } ^ { m }$ are an orthonormal system with respect to the scalar product given by integration: $\\begin{array} { r } { \\int _ { X } \\overline { { Y _ { j i } ^ { m } ( x ) } } Y _ { j ^ { \\prime } i ^ { \\prime } } ^ { m ^ { \\prime } } ( x ) d x = \\delta _ { j j ^ { \\prime } } \\delta _ { i i ^ { \\prime } } \\delta _ { m m ^ { \\prime } } . } \\end{array}$ 5 ", + "bbox": [ + 210, + 321, + 821, + 358 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note the similarity of these properties to those encountered in usual Fourier analysis. Indeed, the Peter-Weyl theorem can be viewed as describing the harmonic analysis on arbitrary compact groups and their homogeneous spaces. ", + "bbox": [ + 176, + 364, + 823, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Tensor Products and Clebsch-Gordan Coefficients The last ingredients that we need to discuss are tensor product representations and Clebsch-Gordan coefficients. They appear, roughly speaking, in the following way: the kernel $K$ can be thought of as being built from harmonic basis functions $Y _ { j i } ^ { m }$ whype transform according to the correspondacts on an input feature field of type g irrep , the c $\\rho _ { j }$ . When a harmonic kernel componentbination will transform according to $\\rho _ { j }$ $\\rho _ { l }$ \ntheir tensor product $\\rho _ { j } \\otimes \\rho _ { l }$ . If the convolution should map to an output field of type $\\rho _ { J }$ , not any harmonic component $Y _ { j i } ^ { m }$ is admissible, but only those for which $\\rho _ { J }$ appears as a subrepresentation in the tensor product $\\rho _ { j } \\otimes \\rho _ { l }$ . The Clebsch-Gordan coefficients encode whether $\\rho _ { j } \\otimes \\rho _ { l }$ contains $\\rho _ { J }$ , and, if it does, in which way and how often $\\rho _ { J }$ is embedded in the tensor product. For more details on the definitions in this section see Appendix D.1. ", + "bbox": [ + 173, + 421, + 825, + 560 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition 3.5 (Tensor product representation). Let $\\rho : G \\to \\operatorname { U } ( V )$ and $\\tilde { \\rho } : G \\to \\mathrm { U } ( \\tilde { V } )$ be unitary representations. Then their tensor product $\\rho \\otimes { \\tilde { \\rho } } : G \\to \\operatorname { U } ( V \\otimes { \\tilde { V } } )$ is defined by: $\\left[ ( \\boldsymbol { \\rho } \\otimes \\tilde { \\boldsymbol { \\rho } } ) ( \\boldsymbol { g } ) \\right] ( \\boldsymbol { v } \\otimes \\boldsymbol { \\rho }$ $\\tilde { v } ) \\ = \\ \\left[ \\rho ( g ) \\right] ( v ) \\otimes \\left[ \\tilde { \\rho } ( g ) \\right] ( \\tilde { v } )$ . ", + "bbox": [ + 173, + 564, + 825, + 614 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The tensor product $\\rho _ { j } \\otimes \\rho _ { l }$ of two irreps is itself in general not irreducible anymore. However, as it is again a unitary representation, it splits by Proposition B.38 into a direct sum of irreducible unitary subrepresentations. Thus, there is an equivariant isomorphism ", + "bbox": [ + 173, + 621, + 825, + 664 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9b270359341f298d6a12ac541979c59e0d67281c61efb894f898132df2de12c9.jpg", + "text": "$$\n\\mathrm { C G } _ { j l } : V _ { j } \\otimes V _ { l } \\to \\bigoplus _ { J \\in \\widehat { G } } \\bigoplus _ { s = 1 } ^ { [ J ( j l ) ] } V _ { J } .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 665, + 635, + 694 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The integer $\\left[ J ( j l ) \\right]$ is the multiplicity of $V _ { J }$ in $V _ { j } \\otimes V _ { l }$ , which is zero for all but finitely many $J$ . ", + "bbox": [ + 169, + 695, + 802, + 712 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The matrix elements of $\\mathrm { C G } _ { j l }$ are denoted as Clebsch-Gordan coefficients: ", + "bbox": [ + 176, + 717, + 660, + 732 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition 3.6 (Clebsch-Gordan Coefficients). Let $Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n }$ be the basis tensors in $V _ { j } \\otimes V _ { l }$ and let the basis element $Y _ { J s } ^ { M }$ be the copy of $Y _ { J } ^ { M }$ with index $s$ in $\\oplus _ { J \\in { \\widehat { G } } } \\oplus _ { s = 1 } ^ { [ J ( j l ) ] } V _ { J }$ L[J(jl)]s=1 VJ . Then the ClebschGordan coefficients are the matrix elements of $\\mathrm { C G } _ { j l }$ brelative to these bases, ", + "bbox": [ + 173, + 734, + 825, + 781 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f4d9d6c7e3bf74ecf80265be76e62e60be1987f3dce794fe3d6cd07ddbba0e26.jpg", + "text": "$$\n s , J M | j m ; l n : = Y _ { J s } ^ { M } | \\mathrm { C G } _ { j l } | Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n } ,\n$$", + "text_format": "latex", + "bbox": [ + 341, + 784, + 653, + 804 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "i.e., the scalar product of $\\underline { { \\mathrm { C G } _ { j l } } } \\left( Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n } \\right)$ and $Y _ { J s } ^ { M }$ . ", + "bbox": [ + 173, + 805, + 524, + 824 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4Usually, the Peter-Weyl theorem uses $G$ itself as the homogeneous space and is formulated for complex representations (Knapp, 2002). However, generalizations to arbitrary homogeneous spaces and real representations are possible, as we explain in Appendix B.2 ", + "bbox": [ + 176, + 827, + 823, + 864 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5From a representation theoretic viewpoint, the functions $Y _ { j i } ^ { m }$ for fixed $j$ and $_ { i }$ span an irreducible subrepresentation $V _ { j i }$ of the unitary representation $\\lambda : G \\to \\operatorname { U } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) )$ given by $\\left[ \\lambda ( g ) f \\right] ( x ) : = f ( g ^ { - 1 } x ) . \\ L _ { \\mathbb { K } } ^ { 2 } ( X )$ then splits into an orthogonal direct sum $L _ { \\mathbb { K } } ^ { 2 } ( X ) = { \\widehat { \\bigoplus } } _ { j \\in { \\widehat { G } } } \\bigoplus _ { i = 1 } ^ { m _ { j } } V _ { j i }$ . This viewpoint is explained in the equivalent, more representation theoretic formulation of the Peter-Weyl theorem in Theorem B.22. ", + "bbox": [ + 174, + 866, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 A WIGNER-ECKART THEOREM FOR $G$ -STEERABLE KERNELS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 710, + 118 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Now that we have discussed all of the required ingredients, we are ready for stating our main theorem. Intuitively, our Wigner-Eckart theorem identifies exactly those combinations of harmonics, Clebsch-Gordan coefficients and endomorphisms that, when being assembled together, yield a $G$ - steerable kernel $K : X \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ . The kernel will thereby comprise all those harmonics $Y _ { j i } ^ { m }$ for which the tensor product $V _ { j } \\otimes V _ { l }$ contains $V _ { J }$ as a factor. The number of possible combinations depends therefore on the number of different isomorphism classes $j \\in { \\widehat { G } }$ for which $V _ { J }$ appears as a factor in the tensor product, the multiplicity $\\left[ J ( j l ) \\right]$ with which it occurs, and the multiplicities $m _ { j }$ of harmonics $Y _ { j i } ^ { m }$ in the Peter-Weyl decomposition that transform according to $\\rho _ { j }$ . In addition, each individual combination can subsequently be composed with an endomorphism in $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ , which increases the number of combinations by a factor of $E _ { J } = \\dim ( \\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) )$ to a total of $\\begin{array} { r } { \\Lambda _ { J l } : = E _ { J } \\cdot \\sum _ { j \\in \\widehat { G } } \\left[ J ( j l ) \\right] \\cdot m _ { j } } \\end{array}$ . This number is finite, as we explain in Remark D.18. ", + "bbox": [ + 173, + 132, + 825, + 294 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "How are such assembled steerable kernels parameterized? The learnable parameters correspond to the degrees of freedom in the individual components from which the kernel is built. While the Clebsch-Gordan coefficients and harmonic basis functions are fixed, the endomorphisms are elements of the $E _ { J }$ -dimensional vector spaces $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ . The degrees of freedom of a $G$ -steerable kernel are therefore identified with the choice of endomorphisms.6 This gives a total of $\\Lambda _ { J l }$ parameters which take values in $\\mathbb { K }$ . Note that the choice of endomorphisms corresponds directly to the choice of reduced matrix elements of spherical tensor operators. ", + "bbox": [ + 173, + 299, + 825, + 400 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For a kernel $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ , we write $\\langle J M | K ( x ) | l n \\rangle$ for the matrix elements of $K ( x ) \\in \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ with indices $n \\leq d _ { l }$ and $M \\leq d _ { J }$ , see also Definition D.9. Similarly, endomorphisms $c \\in \\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ have matrix elements $\\langle J M | c | J M ^ { \\prime } \\rangle$ with $M , M ^ { \\prime } \\ \\leq \\ d _ { J }$ . We furthermore write $\\langle i , j m | x \\rangle : = \\overline { { Y _ { j i } ^ { m } ( x ) } }$ . Finally, we denote the space of $G$ -steerable kernels by ${ \\mathrm { H o m } } _ { G } ( X , { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )$ . ", + "bbox": [ + 173, + 405, + 825, + 483 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our main result is the following Wigner-Eckart theorem for $G \\mathrm { \\Omega }$ -steerable kernels. Other versions at different levels of abstraction can be found in Theorems D.13 and D.16. ", + "bbox": [ + 174, + 487, + 823, + 517 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 4.1 (Wigner-Eckart Theorem for Steerable Kernels). There is a vector space isomorphism ", + "bbox": [ + 171, + 522, + 825, + 537 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/2a1db0566a234d7706ca882fc6bc691fa217839814d969c6be3f0dcd452cca30.jpg", + "text": "$$\n\\mathrm { G K e r : } \\bigoplus _ { \\substack { j \\in \\widehat { G } } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\bigoplus _ { s = 1 } ^ { \\lfloor J ( j l ) \\rfloor } \\mathrm { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) \\ \\ \\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 544, + 766, + 575 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A general steerable kernel $K = \\mathrm { G K e r } ( ( c _ { j i s } ) _ { j i s } )$ with $c _ { j i s } \\in \\operatorname { E n d } _ { G , \\operatorname { K } } ( V _ { J } )$ has matrix elements ", + "bbox": [ + 174, + 580, + 795, + 598 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/de3a46c9d8047f5ac3ffc6aba985094c284c934a8a0400abf26a30a18c6031f9.jpg", + "text": "$$\n\\underbrace { \\langle J M | K ( x ) | l n \\rangle } _ { \\mathrm { k e m e l ~ m a t i x ~ e l e m e n t s } } = \\sum _ { j \\in \\hat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { \\left[ J ( j l ) \\right] } \\sum _ { m = 1 } ^ { d _ { j } } \\sum _ { M ^ { \\prime } = 1 } ^ { d _ { J } } \\underbrace { \\langle J M | c _ { j i s } | J M ^ { \\prime } \\rangle } _ { \\mathrm { e n d o m o r p h i s m s } } \\cdot \\underbrace { \\langle s , J M ^ { \\prime } | j m ; l n \\rangle } _ { \\mathrm { C l e b s c h - G o r d a n } } \\cdot \\underbrace { \\langle i , j m | x \\rangle } _ { \\mathrm { h a m m i c s } } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 603, + 797, + 656 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof. We shortly sketch a proof of this theorem. We use the notation $\\mathrm { H o m } _ { G , \\mathbb { K } }$ to denote linear equivariant maps. The space of steerable kernels can be progressively transformed as follows: ", + "bbox": [ + 174, + 671, + 823, + 702 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/065bfb338e93080a3e3d7f378e5cf4c18c0dc32f6e09ce9a380043c7df856f58.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathrm { H o m } _ { G } \\big ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\big ) \\stackrel { ( 1 ) } { \\cong } \\mathrm { H o m } _ { G , \\mathbb { K } } \\big ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\big ) } \\\\ & { \\stackrel { \\mathrm { \\scriptsize 2 ) } } { \\cong } \\mathrm { H o m } _ { G , \\mathbb { K } } \\big ( \\widehat { \\bigoplus } _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } V _ { j i } , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\big ) \\stackrel { ( 3 ) } { \\cong } \\bigoplus _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\mathrm { H o m } _ { G , \\mathbb { K } } \\big ( V _ { j } , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\big ) } \\\\ & { \\stackrel { \\mathrm { \\scriptsize 4 ) } } { \\cong } \\bigoplus _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\mathrm { H o m } _ { G , \\mathbb { K } } \\big ( V _ { j } \\otimes V _ { l } , V _ { J } \\big ) \\stackrel { ( 5 ) } { \\cong } \\bigoplus _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\mathrm { H o m } _ { G , \\mathbb { K } } \\big ( \\bigoplus _ { J ^ { \\prime } \\in \\widehat { G } } \\bigoplus _ { s = 1 } ^ { [ J ^ { \\prime } ( j l ) ] } V _ { J ^ { \\prime } } , V _ { J } \\big ) } \\\\ & { \\stackrel { \\mathrm { \\scriptsize 6 ) } } { \\cong } \\bigoplus _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\bigoplus _ { s = 1 } ^ { [ J ( j l ) ] } \\mathrm { H o m } _ { G , \\mathbb { K } } ( V _ { J } , V _ { J } ) \\stackrel { ( 7 ) } { \\cong } \\bigoplus _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\bigoplus _ { s = 1 } ^ { [ J ( j l ) ] } \\mathrm { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 707, + 825, + 840 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In (1), we linearize the kernels such that they become representation operators, as detailed in Theorem C.7. Step (2) applies the representation-theoretic version of the Peter-Weyl Theorem B.22 to decompose $L _ { \\mathbb { K } } ^ { 2 } { \\bar { ( X ) } }$ in harmonic basis functions. Step (3) makes use of the well-known fact that linear maps can be described on each direct summand individually. Topological details are explained in Lemma D.20. In (4), we use the hom-tensor adjunction Proposition D.23. In (5), we use the Clebsch-Gordan decomposition Eq. (4), which provides us with Clebsch-Gordan coefficients. In (6), we use that nontrivial linear equivariant maps from $V _ { J ^ { \\prime } }$ to $V _ { J }$ exist by Schur’s Lemma B.29 only for $J ^ { \\prime } = J$ and, once again, that we can describe linear maps on each direct summand individually. Finally, in (7) we note that ${ \\mathrm { H o m } } _ { G , { \\mathbb K } } ( V _ { J } , V _ { J } ) = { \\mathrm { E n d } } _ { G , { \\mathbb K } } ( V _ { J } )$ is the space of endomorphisms. The formula of the matrix elements Eq. (6) is fully proven in Theorem D.13 by carefully tracing back all the isomorphisms above.7 ", + "bbox": [ + 174, + 843, + 825, + 900 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 200 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Technically, step (1) is the main gap that we had to bridge: it establishes that non-linear kernels on $X$ can be seen as linear representation operators on $L _ { \\mathbb { K } } ^ { 2 } ( X )$ . Steps (2) to (7) orient at the proof of the Wigner-Eckart theorem for representation operators by Agrawala (1980). However, it differs non-trivially from the reference by a) allowing the operator to be non-injective, b) topological considerations, since $L _ { \\mathbb { K } } ^ { 2 } ( X )$ is not simply a direct sum of irreps but its topological closure, and c) the possibility to allow for real representations, which is why we end up with endomorphisms. □ ", + "bbox": [ + 174, + 208, + 825, + 292 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We obtain the following corollary, which clarifies how steerable kernels can be parameterized: ", + "bbox": [ + 173, + 306, + 790, + 321 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Corollary 4.2. The space ${ \\mathrm { H o m } } _ { G } ( X , { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )$ of steerable kernels is spanned by basis kernels $\\{ K _ { j i s r } : X \\to \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\mid j \\in { \\widehat { G } } , \\ i \\leq m _ { j } ,$ $s \\leq [ J ( j l ) ]$ , $r \\leq E _ { J } \\}$ with matrix elements ", + "bbox": [ + 171, + 325, + 823, + 359 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/9ac3aae5b223313c2c41ae425955f89ae494bc49cd6b8fae9acc04f02e4798d1.jpg", + "text": "$$\n\\langle J M | K _ { j i s r } ( x ) | l n \\rangle = \\sum _ { m = 1 } ^ { d _ { j } } \\sum _ { M ^ { \\prime } = 1 } ^ { d _ { J } } \\langle J M | c _ { r } | J M ^ { \\prime } \\rangle \\cdot \\langle s , J M ^ { \\prime } | j m ; l n \\rangle \\cdot \\langle i , j m | x \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 197, + 373, + 782, + 404 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$A$ matrix-expression of the basis kernels from Eq. (7) is given in Eq. (22). Here, $c _ { r }$ is one of the $E _ { J }$ basis endomorphisms of $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ . This means that a general steerable kernel $K : X $ HomK(Vl, VJ ) is of the form K = Pj∈G Pmji=1 P[J(jl)s=1 $\\begin{array} { r } { K = \\sum _ { j \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { [ J ( j l ) ] } \\sum _ { r = 1 } ^ { E _ { J } } \\lambda _ { j i s r } \\cdot K _ { j i s r } } \\end{array}$ ] PEJr=1 λjisr · Kjisr with a total of ΛJ l = $E _ { J } \\cdot \\textstyle \\sum _ { j \\in { \\widehat { G } } } [ J ( j l ) ] \\cdot m _ { j }$ learnable parameters $\\lambda _ { j i s r } \\in \\mathbb { K } .$ . Overall, the kernel space can therefore be parameterized with an isomorphism $\\overline { { \\mathrm { G K e r } } } : \\mathbb { K } ^ { \\Lambda _ { J l } } \\to \\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) ) .$ . ", + "bbox": [ + 173, + 406, + 825, + 489 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Proof. We simply choose $K _ { j i s r } : = \\mathrm { G K e r } ( ( c _ { j ^ { \\prime } i ^ { \\prime } s ^ { \\prime } } ^ { j i s r } ) _ { j ^ { \\prime } i ^ { \\prime } s ^ { \\prime } } )$ with $c _ { j ^ { \\prime } i ^ { \\prime } s ^ { \\prime } } ^ { j i s r } = \\delta _ { j ^ { \\prime } j } \\cdot \\delta _ { i ^ { \\prime } i } \\cdot \\delta _ { s ^ { \\prime } s } \\cdot c _ { r }$ . Clearly, the $c ^ { j i s r }$ are a basis of $\\begin{array} { r l } { } & { \\bigoplus _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\bigoplus _ { s = 1 } ^ { [ J ( j l ) ] } \\mathrm { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) } \\end{array}$ L[J(jl)]s=1 EndG,K(VJ ), and since GKer is an isomorphism, the $K _ { j i s r }$ form a basis of steerable kernels. □ ", + "bbox": [ + 174, + 503, + 825, + 556 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Remark 4.3. The matrix elements $\\langle J M | c _ { j i s } | J M ^ { \\prime } \\rangle$ relate to the reduced matrix elements $\\lambda \\in \\mathbb { C }$ of spherical tensor operators as follows: in the case of spherical tensor operators one deals with complex irreps, whose endomorphism spaces are according to Schur’s Lemma D.8 generated by the identity. Consequently, such endomorphisms $c$ have matrix-elements $\\langle J M | c | J M ^ { \\prime } \\rangle = \\lambda \\delta _ { M M ^ { \\prime } }$ for some scaling factor $\\lambda \\in \\mathbb { C }$ . $\\lambda$ is denoted as the reduced matrix element of the spherical tensor operator. The analog to $\\lambda$ in our Wigner-Eckart theorem are the learnable parameters $\\lambda _ { j i s r } \\in \\mathbb { K }$ . ", + "bbox": [ + 173, + 563, + 825, + 648 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 667, + 344, + 684 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Harmonic convolution kernels date back to at least the early ’80s (Hsu & Arsenault, 1982; Rosen & Shamir, 1988). The term steerable filter was coined in Freeman & Adelson (1991). Hel-Or & Teo (1998) generalized steerable filters to Lie groups. Reisert & Burkhardt (2007) proposed matrix valued steerable kernels between representation spaces, which are similar to our $G$ -steerable kernels. ", + "bbox": [ + 174, + 699, + 825, + 756 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Steerable CNNs formulate GCNNs in the language of representation theory and feature fields. This design was proposed by Cohen & Welling (2016b), who specifically considered finite groups, for which the kernel constraint can be solved numerically. Weiler et al. (2018a) introduced the $G$ - steerability constraint in the form in Eq. (2) for $G = \\mathrm { S O ( 3 ) }$ . The authors choose a slightly different approach to solve the constraint in which they decompose the space $\\operatorname { H o m } _ { \\mathbb { R } } ( V _ { l } , V _ { J } ) \\stackrel { \\sim } { = } V _ { l } ^ { * } \\otimes V _ { J }$ instead of $V _ { j } \\otimes V _ { l }$ via Clebsch-Gordan coefficients. An essentially equivalent design was simultaneously proposed by Thomas et al. (2018), who decomposed $V _ { j } \\otimes V _ { l }$ as in the present work, see Appendix E.5. The case of complex valued irreps of $\\mathrm { S O } ( 2 )$ was investigated by Worrall et al. (2016) and Wiersma et al. (2020); see Appendix E.1. Weiler & Cesa (2019) solve the constraint for any, not necessarily irreducible, representation of the groups O(2), SO(2), $\\mathrm Ḋ \\mathrm Ḋ \\mathrm Ḋ \\ Ḍ Ḍ _ { N }$ and $\\mathrm { C } _ { N }$ . Their solution strategy is based on an expansion of the kernel in the Fourier basis of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ and solving for the Fourier coefficients satisfying the constraint. This is a special case of the strategy that we employ in the proof of our Wigner-Eckart theorem. de Haan et al. (2020) solve for SO(2)-steerable kernels by viewing them as invariants of the tensor product representation $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } ) ^ { * } \\otimes V _ { l } ^ { * } \\otimes V _ { J }$ . As they use real valued irreps, they can use that the duals are isomorphic to their original counterparts. Our Wigner-Eckart theorem unifies all of these results in one general framework. ", + "bbox": [ + 174, + 761, + 825, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To which use cases does the proposed kernel space solution apply? As argued by Cohen et al. (2019b), any $H$ -equivariant convolutional network on a homogeneous space $H / G$ needs to satisfy a $G$ -steerability constraint — if $H$ is locally compact and unimodular. While these works proved the necessity of steerable kernels, they did not solve the constraint – a gap which is filled by our WignerEckart theorem for compact groups $G$ , see also Remark D.15. This framework includes in particular the popular group convolutions on flat spaces (Cohen & Welling, 2016a) and homogeneous spaces of compact groups (Kondor & Trivedi, 2018) and Lie groups (Bekkers, 2020), including for instance the sphere (Cohen et al., 2018). Specifically, if $\\rho _ { \\mathrm { i n } }$ and $\\rho _ { \\mathrm { o u t } }$ are chosen to be regular representations $L _ { \\mathbb { K } } ^ { 2 } ( \\dot { G } )$ , steerable convolutions are equivalent to group convolutions (Weiler & Cesa, 2019). ", + "bbox": [ + 174, + 208, + 825, + 333 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A related line of work are Clebsch-Gordan Networks (Kondor et al., 2018; Kondor, 2018; Anderson et al., 2019; Bogatskiy et al., 2020). They apply bilinear equivariant nonlinearities which compute the tensor products of global irrep features. A subsequent Clebsch-Gordan decomposition disentangles the product features back into irrep features. Note that in this network design, the ClebschGordan coefficients are used in the nonlinear part, which differs from our use of these coefficients in the construction of steerable basis kernels, i.e. in the linear part of the network. ", + "bbox": [ + 174, + 339, + 825, + 424 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 EXAMPLE APPLICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 433, + 418, + 449 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Cohen et al. (2019b) showed in a fairly general setting that every GCNN is based on $G$ -steerable kernels. In practice, a basis for the space of $G \\mathrm { \\Omega }$ -steerable kernels needs to be determined for parameterizing GCNNs. This work determines the general structure of these basis kernels for compact (point-)symmetry groups $G$ and their homogeneous spaces $X$ : Corollary 4.2 explains that one needs to determine 1) the irreps $\\rho _ { l }$ of $G , 2$ ) harmonic basis functions $Y _ { i i } ^ { m }$ in $L _ { \\mathbb { K } } ^ { 2 } ( { \\dot { X } } )$ according to the Peter-Weyl Theorem 3.4, 3) the Clebsch-Gordan decomposition of $\\dot { V } _ { j } \\otimes V _ { l }$ , given by the ClebschGordan coefficients $\\langle s , J M | j m ; l n \\rangle$ , and 4) a basis of endomorphisms $c _ { r } \\in \\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ for any $J \\in { \\widehat { G } }$ . Given these ingredients, they can in a fifth step be put together according to Eq. (7) to obtain a complete, $\\Lambda _ { J l }$ -dimensional basis of $G$ -steerable kernels $K _ { j i s r } : X \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ . ", + "bbox": [ + 173, + 463, + 825, + 593 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Appendix E demonstrates this for the examples of $G$ being SO(2), SO(3), O(3), and $\\mathbb { Z } / 2$ , considering both real and complex irreps. In any of these cases, we derive the kernel bases following exactly the five steps outlined above. This procedure can easily be applied to further compact groups, for instance SU(2) or SU(3), which play an important role in physics applications of deep learning. ", + "bbox": [ + 174, + 599, + 825, + 656 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSIONS AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 665, + 504, + 681 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Prior work revealed that group equivariant convolutions generally rely on $G$ -steerable kernels. Our Wigner-Eckart theorem for $G$ -steerable kernels characterizes them for the practically relevant case of $G$ being any compact group. The degrees of freedom – or learnable parameters – correspond thereby precisely to the choice of endomorphisms. This mirrors the situation in quantum mechanics, where the degrees of freedom of spherical tensor operators are given by reduced matrix elements. ", + "bbox": [ + 174, + 695, + 825, + 766 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "It would be desirable to extend this result to non-compact groups, where the Peter-Weyl Theorem does not hold anymore. One alternative might be Pontryagin duality (Reiter, 1968), which describes the Fourier transform on locally compact abelian groups. Furthermore, for many non-compact, nonabelian groups, one can often find a direct integral decomposition of $L _ { \\mathbb { C } } ^ { 2 } ( G )$ . This generalization of the Peter-Weyl theorem can be found in Segal (1950) and Mautner (1955). Such generalizations of our Wigner-Eckart Theorem might lead to a better theoretical understanding of several recent work (Worrall & Welling, 2019; Bekkers, 2020; Sosnovik et al., 2020; Shutty & Wierzynski, 2020). ", + "bbox": [ + 174, + 772, + 825, + 871 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, we hope that the analogies between steerable kernels and representation operators appearing in physics inspire further research in this fascinating crossdisciplinary domain. This could lead to applications of GCNNs for learning tasks with physical symmetries. ", + "bbox": [ + 174, + 877, + 821, + 920 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 326, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Lucas Lang for discussions on the Wigner-Eckart Theorem and observables in physics and Patrick Forre for discussions on the link between steerable kernels and representation operators. ´ Additionally, we are greatful for discussions with Gabriele Cesa on the connection between real and complex representations of compact groups. Furthermore, we thank Stefan Dawydiak and Terrence Tao for online discussions on aspects surrounding a real version of the Peter-Weyl theorem. Finally, we thank Roberto Bondesan, Miranda Cheng, Tom Lieberum, and Rupert McCallum for feedback on different aspects of our work. 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", + "bbox": [ + 173, + 497, + 825, + 541 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "E.P. Wigner. Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Die Wissenschaft. J.W. Edwards, 1944. ", + "bbox": [ + 174, + 549, + 823, + 579 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Dana P. Williams. The Peter-Weyl Theorem for Compact Groups, 1991. URL https://math.d artmouth.edu/˜dana/bookspapers/pw.pdf. ", + "bbox": [ + 174, + 587, + 823, + 617 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Daniel Worrall and Max Welling. Deep Scale-spaces: Equivariance Over Scale. In Conference on Neural Information Processing Systems (NeurIPS). 2019. ", + "bbox": [ + 174, + 625, + 825, + 655 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Daniel E. Worrall, Stephan J. Garbin, Daniyar Turmukhambetov, and Gabriel J. Brostow. Harmonic Networks: Deep Translation and Rotation Equivariance. In Conference on Computer Vision and Pattern Recognition (CVPR), volume abs/1612.04642, 2016. ", + "bbox": [ + 176, + 662, + 823, + 705 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 101, + 302, + 121 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This appendix contains a detailed and rigorous treatment of the Wigner-Eckart theorem for steerable kernels, including background knowledge, proofs, and many example applications. ", + "bbox": [ + 174, + 127, + 823, + 155 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Chapter A, we shortly look at the simple example SO(2) for motivating the concepts and results in Section 3. ", + "bbox": [ + 176, + 161, + 821, + 190 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Everything afterwards, starting with Chapter B, can be read independently of the main paper and is a self-contained treatment of our investigations. In Chapter B, we start with the foundations of the representation theory of compact groups. We formulate the Peter-Weyl Theorem B.22, which tells us how to decompose the space of square-integrable functions on a homogeneous space into irreducible representations, leading to harmonic basis functions. In the second half, we include a proof of the more algebraic parts of this theorem. We do this since the theorem is usually only proven for complex representations in the literature, but we need it for real representation as well. ", + "bbox": [ + 174, + 196, + 825, + 295 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Chapter C we investigate steerable kernels and show their similarities to representation operators from physics and representation theory. In Theorem C.7 we will then proof a precise isomorphism between steerable kernels and representation operators on the space of square-integrable functions on a homogeneous space. We call these kernel operators. ", + "bbox": [ + 174, + 303, + 823, + 358 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Chapter D we will then formulate and prove the Wigner-Eckart theorem for steerable kernels of general compact groups D.13. The proof makes in essential parts use of the Peter-Weyl Theorem and Theorem C.7, and additionally of Schur’s Lemma B.29, the hom-tensor adjunction Proposition D.23, and the Clebsch-Gordan decomposition of tensor products. ", + "bbox": [ + 174, + 364, + 825, + 420 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Chapter E, we then look at specific example applications of our theory. In these examples, we look at specific compact transformation groups $G$ , specific, relevant homogeneous spaces $X$ of the group and one of the fields $\\mathbb { R }$ or C. For this combination we derive a basis for the space of steerable kernels between arbitrary irreducible input- and output representations of the group. Specifically, we look at harmonic networks (Worrall et al., 2016), SO(2)-equivariant networks for real representations (Weiler & Cesa, 2019), $\\mathbb { Z } _ { 2 }$ -equivariant networks for real representations, SO(3)-equivariant networks for both real and complex representations (Weiler et al., 2018a; Thomas et al., 2018), and O(3)-equivariant networks for both real and complex representations. The investigation of $\\mathbb { Z } _ { 2 }$ - equivariant CNNs will additionally show that our result is consistent with group convolutional CNNs for the regular representation (Cohen & Welling, 2016a). ", + "bbox": [ + 174, + 428, + 825, + 566 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Chapter F, we summarize some important notions and results from the theory of topological spaces, metric spaces, normed vector spaces, and (pre-)Hilbert spaces that we use throughout this appendix. ", + "bbox": [ + 174, + 574, + 825, + 616 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Chapters B, C, and D contain the bulk of the theoretical work. We recommend the reader to first only read the first halves of these chapters, Sections B.1, C.1 and D.1, since they contain the formulation of the most important results and the main intuitions, whereas the second halves of these chapters, i.e., Sections B.2, C.2 and D.2, mainly contain detailed proofs that can be skipped when going over the material for the first time. ", + "bbox": [ + 174, + 622, + 825, + 693 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "CONTENTS OF THE APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 715, + 426, + 732 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A Building Blocks of SO(2)-Steerable Kernels – Running Example for Section 3 1 9 \nB Representation Theory of Compact Groups 22 \nB.1 Foundations of Representation Theory and the Peter-Weyl Theorem 22 \nB.2 A Proof of the Peter-Weyl Theorem 29 \nC The Correspondence between Steerable Kernels and Representation Operators 4 0 \nC.1 Fundamentals of the Correspondence . 40 \nC.2 A Proof of the Correspondence between Steerable Kernels and Kernel Operators . 47 \nD A Wigner-Eckart Theorem for Steerable Kernels of General Compact Groups 51 \nD.1 A Wigner-Eckart Theorem for Steerable Kernels and their Kernel Bases 52 \nD.2 Proof of the Wigner-Eckart Theorem for Kernel Operators 63 ", + "bbox": [ + 163, + 750, + 831, + 926 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 161 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "E Example Applications 67 ", + "text_level": 1, + "bbox": [ + 174, + 180, + 826, + 195 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "E.1 SO(2)-Steerable Kernels for Complex Representations – Harmonic Networks . . . 68 \nE.2 SO(2)-Steerable Kernels for Real Representations . 70 \nE.3 $\\mathbb { Z } _ { 2 }$ -Steerable Kernels for Real Representations 75 \nE.4 SO(3)-Steerable Kernels for Complex Representations. 79 \nE.5 SO(3)-Steerable Kernels for Real Representations . . 81 \nE.6 O(3)-Steerable Kernels for Complex Representations 86 \nE.7 O(3)-Steerable Kernels for Real Representations 90 ", + "bbox": [ + 196, + 200, + 826, + 345 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F Mathematical Preliminaries 90 ", + "text_level": 1, + "bbox": [ + 174, + 363, + 825, + 378 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F.1 Topological Spaces, Normed Spaces, and Metric Spaces 90 \nF.2 Limits of nets and approximated Dirac delta functions . 94 \nF.3 Pre-Hilbert Spaces and Hilbert Spaces 95 ", + "bbox": [ + 197, + 385, + 826, + 444 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "LIST OF SYMBOLS ", + "bbox": [ + 174, + 463, + 330, + 479 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "GENERAL SET THEORY AND FUNCTIONS ", + "bbox": [ + 173, + 494, + 460, + 508 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$A \\cap B$ intersection of sets $A$ and $B$ $A \\cup B$ union of sets $A$ and $B$ \n$\\cap _ { i \\in I } A _ { i }$ intersection of sets $A _ { i }$ \nSi∈I Ai union of sets $A _ { i }$ \n$\\textstyle { \\bigsqcup } _ { i \\in I } A _ { i }$ union of sets $A _ { i }$ which are disjoint from each other \n$A \\subseteq B$ $A$ is a subset of $B$ \n$A \\subsetneq B$ $A$ is a strict subset of $B$ $A \\backslash B$ set of all elements in $A$ which are not in $B$ \n$A \\times B$ Cartesian product of sets or structures (e.g., groups) $A , B$ $\\varnothing$ empty set \n$X : = Y$ $X$ is defined as $Y$ $\\sim$ often an equivalence relation $[ x ]$ equivalence class with respect to an equivalence relation $\\mathbf { 1 } _ { A }$ indicator function of set $A$ $f \\circ g$ composition of two composable functions $f$ and $g$ f − 1 either the inverse of function $f$ or the preimage function $f | _ { A }$ restriction of a function $f$ to a subset $A$ ", + "bbox": [ + 183, + 523, + 635, + 762 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "NUMBERS AND COLLECTIONS OF NUMBERS ", + "text_level": 1, + "bbox": [ + 173, + 787, + 483, + 801 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "N natural numbers including 0 \n$\\mathbb { Z }$ integers \n$\\mathbb { R }$ field of real numbers \n$\\mathbb { C }$ field of complex numbers \n$\\mathbb { H }$ skew-field of quaternions \n$\\mathbb { K }$ one of the two fields $\\mathbb { R }$ and $\\mathbb { C }$ \n$\\mathbb { K } ^ { n }$ $n$ -dimensional canonical vector space over K \n$\\textstyle { \\overline { { x } } }$ complex conjugate of $x$ ", + "bbox": [ + 184, + 816, + 524, + 929 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "GROUPS ", + "text_level": 1, + "bbox": [ + 174, + 104, + 235, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$G$ a compact topological group $1 , e$ neutral element of a group with multiplication as operation 0 neutral element of an additive group $G \\rtimes H$ semidirect product of two groups $G$ and $H$ $\\mathrm { C } _ { N }$ group of planar rotations of a regular $N$ -gon $\\mathrm Ḋ \\mathrm Ḋ \\mathrm Ḋ \\ Ḍ Ḍ _ { N }$ group of planar rotations and reflections of a regular $N$ -gon $\\mathrm { S O } ( d )$ special orthogonal group in $d$ real dimensions $\\mathrm { O } ( d )$ orthogonal group in $d$ real dimensions $\\mathrm { O } ( V )$ orthogonal group of a real Hilbert space $V$ $\\mathrm { S U } ( d )$ special unitary group in $d$ complex dimensions $\\mathrm { U } ( d )$ unitary group in $d$ complex dimensions $\\mathrm { U } ( V )$ unitary group of a complex Hilbert space $V$ $\\operatorname { E } ( d )$ Euclidean motion group in $d$ dimensions ", + "bbox": [ + 184, + 133, + 643, + 314 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "BASIC REPRESENTATION THEORY ", + "text_level": 1, + "bbox": [ + 176, + 340, + 410, + 353 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\rho$ \n$\\rho ^ { v }$ \nρuv \n$\\rho ^ { \\mathrm { i n } } , \\rho ^ { \\mathrm { o u t } }$ \n$\\rho _ { \\mathrm { H o m } }$ \n$\\rho \\otimes \\rho ^ { \\prime }$ \n$\\operatorname { I n d } _ { G } ^ { H } \\rho$ \n$\\widehat { G }$ \n$l$ \n$\\rho _ { l }$ \n$V _ { l }$ \n$v _ { l } ^ { i }$ or $Y _ { l } ^ { n }$ \na linear representation of a group \nThe function $G V$ , $g \\mapsto \\rho ( g ) ( v )$ \nmatrix coefficient of the unitary representation $\\rho$ \nrepresentations of the in-field and out-field, respectively \nHom-representation on $\\mathrm { H o m } _ { \\mathrm { K } } ( V , V ^ { \\prime } )$ of representations $\\rho$ \nand $\\rho ^ { \\prime }$ \ntensor product representation on $V \\otimes V ^ { \\prime }$ of representations \n$\\rho$ and $\\rho ^ { \\prime }$ \ninduced representation on $H$ or a representation $\\rho$ on $G$ \nset of isomorphism classes of unitary representations on $G$ \nan isomorphism class of unitary representations \na representative of isomorphism class $l$ \nvector space on which $\\rho _ { l }$ acts \nfixed chosen orthonormal basis vector of $V _ { l }$ ", + "bbox": [ + 181, + 371, + 245, + 573 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 212, + 369, + 653, + 570 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "VECTOR SPACES AND HILBERT SPACES ", + "bbox": [ + 174, + 595, + 449, + 609 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\dim ( V )$ dimension of $\\mathbb { K }$ -vectorspace $V$ \n$V \\perp W$ $V$ and $W$ are perpendicular \n$V \\cong W$ $V$ and $W$ are isomorphic with respect to their structures \n$V \\not \\cong W$ $V$ and $W$ are not isomorphic with respect to their structures $\\langle f \\vert g \\rangle$ bra-ket notation of a scalar product on a Hilbert space \n$\\langle y | f | x \\rangle$ equivalent to $\\langle y | f ( x ) \\rangle$ for a function $f$ \n$\\operatorname { n u l l } ( f )$ null space of $f$ \n$\\operatorname { i m } ( f )$ image of $f$ $f ^ { * }$ adjoint of the operator $f$ $\\operatorname { i d } _ { V }$ identity function on $V$ ", + "bbox": [ + 183, + 625, + 648, + 765 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(HILBERT) SPACE CONSTRUCTIONS FROM OTHER SPACES ", + "text_level": 1, + "bbox": [ + 173, + 790, + 578, + 805 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "${ \\mathrm { H o m } } _ { \\mathrm { K } } ( V , W )$ space of K-linear functions from $V$ to $W$ $\\operatorname { G L } ( V )$ space of invertible $\\mathbb { K }$ -linear functions from $V$ to itself, sometimes written ${ \\mathrm { G L } } ( V , \\mathbb { K } )$ in the literature \n${ \\mathrm { H o m } } _ { G , \\mathbb { K } } ( V , W )$ space of intertwiners from $V$ to $W$ \n${ \\mathrm { H o m } } _ { G } ( X , W )$ space of $G$ -equivariant continuous maps from $X$ to $W$ , for a homogeneous space $X$ $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V )$ space of endomorphisms of $V$ , i.e., intertwiners from $V$ to $V$ ", + "bbox": [ + 181, + 820, + 702, + 931 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "V ⊗ W ", + "bbox": [ + 212, + 103, + 264, + 117 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "tensor product of two vector spaces over their common field. Also denotes the tensor product of pre-Hilbert spaces (orthogonal) direct sum of all $V _ { i }$ \ntopological closure of the (orthogonal) direct sum of all $V _ { i }$ vector subspace of a $\\mathbb { K }$ -vector space spanned by $M$ \northogonal complement of $V$ \neigenspace of $\\varphi$ for eigenvalue $\\lambda$ ", + "bbox": [ + 310, + 103, + 697, + 207 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "TOPOLOGICAL SPACES, METRIC SPACES, NORMED SPACES ", + "bbox": [ + 173, + 231, + 588, + 246 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$\\tau$ topology $U _ { x }$ open neighborhood of $x \\in X$ $\\mathcal { U } _ { x }$ set of all open neighborhoods of $x \\in X$ \nlimU∈Ux limit over the directed set of open neighborhoods of $x$ \n$\\scriptstyle \\operatorname* { l i m } _ { k \\to \\infty } x _ { k }$ limit of the sequence $( x _ { k } ) _ { k }$ $\\overline { { A } }$ topological closure of $A \\subseteq X$ $\\lVert x \\rVert$ norm of $x$ $| x |$ absolute value of $x$ $d ( { \\dot { x } } , { \\dot { x } } ^ { \\prime } )$ distance of $x , x ^ { \\prime }$ according to metric $d$ $\\mathrm { B } _ { \\epsilon } ( x )$ $\\epsilon$ -ball around $x$ according to some metric $d$ ", + "bbox": [ + 181, + 262, + 635, + 402 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "HOMOGENEOUS SPACES AND THE PETER-WEYL THEOREM ", + "bbox": [ + 173, + 426, + 586, + 440 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "X a homogeneous space of $G$ $x ^ { * } \\in X$ arbitrary point $S ^ { n }$ $n$ -dimensional sphere in $( n + 1 )$ -dimensional space $\\mu$ a measure on a compact group $G$ or its Homogeneous Space $X$ RX integral on a space $X$ with respect to its measure L2K(X), L2K(G) Hilbert space of square-integrable functions on $X$ and $G$ with values in $\\mathbb { K }$ λ unitary representation on $L _ { \\mathbb { K } } ^ { 2 } ( X )$ or $L _ { \\mathbb { K } } ^ { 2 } ( G )$ g(x) arbitrary lift of $x$ with respect to projection $\\pi : G $ $X$ , $g \\mapsto g x ^ { * }$ $\\operatorname { a v } ( f )$ average of $f : G \\to \\mathbb { K }$ along cosets $\\pi ^ { * }$ lift of functions $L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to { \\bar { L } } _ { \\mathbb { K } } ^ { 2 } ( G )$ $\\delta _ { x }$ Dirac delta function at point $x$ $\\delta _ { U }$ approximated Dirac delta function for nonempty open set $U$ $\\rho _ { l } ^ { i j }$ abbreviation for $\\rho _ { l } ^ { v ^ { i } v ^ { j } }$ for orthonormal basis vectors $v ^ { i } , v ^ { j } \\in$ $V _ { l }$ $\\mathcal { E }$ linear span of all matrix coefficients of irreducible unitary representations $\\mathcal { E } _ { l }$ linear span of all matrix coefficients of $\\rho _ { l }$ $\\mathcal { E } _ { l } ^ { j }$ linear span of all matrix coefficients $\\rho _ { l } ^ { i j }$ with varying $i$ but fixed $j$ $n \\llangle { \\mathbf { \\phi } } m _ { l }$ multiplicity of $l$ in orthogonal decomposition of $L _ { \\mathbb { K } } ^ { 2 } ( G )$ and $L _ { \\mathbb { K } } ^ { 2 } ( { \\bar { X } } )$ , respectively $V _ { l i }$ copy of $V _ { l }$ appearing in the Peter-Weyl decomposition of $L _ { \\mathbb { K } } ^ { 2 } { \\bar { ( X ) } }$ pli canonical projection $p _ { l i } \\colon L _ { \\mathbb { K } } ^ { 2 } ( X ) \\ \\to \\ V _ { l i }$ and $\\begin{array} { r l } { p _ { l i } } & { { } : } \\end{array}$ $\\oplus _ { l ^ { \\prime } i ^ { \\prime } } V _ { l ^ { \\prime } i ^ { \\prime } } \\to V _ { l i }$ $\\sin _ { m } , \\cos _ { m }$ the functions $x \\mapsto \\sin ( m x )$ and $x \\mapsto \\cos ( m x )$ $Y _ { l } ^ { n } , Y Y _ { l } ^ { n }$ complex- and real-valued version of a spherical harmonic $D _ { l } , \\mathrm { \\Delta } ^ { r } D _ { l }$ complex- and real-valued version of Wigner D-matrix ", + "bbox": [ + 183, + 454, + 700, + 893 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$K$ kernel $K : X \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ \n$K \\star f$ convolution of kernel $K$ with input $f$ $\\kappa$ kernel operator or (more generally) representation operator ${ \\mathcal { K } } : T \\to \\operatorname { H o m } _ { \\mathbb { K } } ( U , V )$ $\\widehat { K }$ kernel operator ${ \\widehat K } \\ : \\ L _ { \\mathbb K } ^ { 2 } ( X ) \\ \\to \\ \\operatorname { H o m } _ { \\mathbb K } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ corresponding to a kernel $K$ \n$\\kappa | _ { X }$ kernel $\\mathcal { K } | _ { X } : X \\to \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ corresponding to a kernel operator $\\kappa$ $\\tilde { \\kappa }$ for a representation operator ${ \\mathcal { K } } : T \\to \\operatorname { H o m } _ { \\mathbb { K } } ( U , V )$ , this denotes the corresponding map ${ \\tilde { \\mathcal { K } } } : T \\otimes U V$ under the hom-tensor adjunction ", + "bbox": [ + 184, + 132, + 637, + 292 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "THE WIGNER-ECKART THEOREM ", + "text_level": 1, + "bbox": [ + 176, + 319, + 408, + 333 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "input- and output representations on the spaces $V _ { l }$ and $V _ { J }$ fixed chosen orthonormal basis vectors of the abstract irreducible representations $V _ { j }$ , $V _ { l }$ , $V _ { J }$ \nmatrix element of $K ( x )$ for a kernel $K$ and $x \\in X$ \ndimension of $l ^ { \\star }$ th irrep $V _ { l }$ as $\\mathbb { K }$ -vector space \nnumber of times $V _ { j }$ is in the Peter-Weyl decomposition of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ \nnumber of times $V _ { J }$ is in the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ \nendomorphisms, mostly on $V _ { J }$ . $c _ { j i s }$ are endomorphisms appearing in the Wigner-Eckart theorem for steerable kernels basis endomorphism of $\\rho _ { J }$ , indexed with index set $r =$ $1 , \\ldots , E _ { J }$ \nmatrix element at indices $M , M ^ { \\prime }$ for endomorphism $c$ linear equivariant isometric embeddings $l _ { s } : V _ { J } \\to V _ { j } \\otimes V _ { l }$ and $l _ { j i s } : V _ { J } V _ { j i } \\otimes V _ { l }$ \nprojection $p _ { j i s } : V _ { j i } \\otimes V _ { l } V _ { J }$ corresponding to (i.e.: adjoint to) the embedding $l _ { j i s }$ \nClebsch-Gordan coefficient corresponding to $l _ { s }$ \n3-dimensional matrix of Clebsch-Gordan coefficients \nharmonic basis function, for example, spherical harmonic. Element of $V _ { j i } \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( X )$ \nshorthand notation for $\\mathrm { l i m } _ { U \\in \\mathcal { U } _ { x } } \\left. Y _ { j i } ^ { m } \\middle | \\delta _ { U } \\right.$ . Equal to $\\overline { { Y _ { j i } ^ { m } ( x ) } }$ row vector with entries $\\langle i , j m \\vert x \\rangle$ \nisomorphism between tuples of endomorphisms and kernel operators \nisomorphism between tuples of endomorphisms and steerable kernels \nbasis kernel \nlearnable parameter ", + "bbox": [ + 285, + 347, + 699, + 771 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "$\\begin{array} { r l r } & { } & { \\rho _ { l } , \\rho _ { J } } \\\\ & { } & { Y _ { j } ^ { m } , Y _ { l } ^ { n } , Y _ { J } ^ { M } } \\\\ & { } & \\\\ & { } & { \\langle J M | K ( x ) | l n \\rangle } \\\\ & { } & { d _ { l } } \\\\ & { } & { m _ { j } } \\\\ & { } & { \\left[ J ( j l ) \\right] } \\\\ & { } & \\\\ & { } & { c , c _ { j i s } } \\end{array}$ $c _ { r }$ $\\langle J M | c | J M ^ { \\prime } \\rangle$ pjis $\\begin{array} { c } { { \\langle s , J M | j m ; l n \\rangle } } \\\\ { { \\mathrm { C G } _ { J ( j l ) s } } } \\\\ { { Y _ { j i } ^ { m } } } \\end{array}$ $\\langle i , j m \\vert x \\rangle$ $\\langle i , j | x \\rangle$ Rep GKer $K _ { j i s r }$ $w _ { j i s r }$ ", + "bbox": [ + 183, + 340, + 294, + 771 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "LIST OF THEOREMS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 341, + 118 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.22 Theorem (Peter-Weyl Theorem) 28 \nB.27 Theorem (Density of Matrix Coefficients) 30 \nC.7 Theorem (Kernel-Operator-Correspondence) . . . 46 \nC.8 Theorem (Kernel-Operator-Correspondence, Restated) 47 \nD.11 Theorem (Wigner-Eckart Theorem) 56 \nD.13 Theorem (Wigner-Eckart Theorem for Steerable Kernels) . . 58 \nD.16 Theorem (Steerable Kernel Bases) 61 \nF.24 Theorem (Heine-Borel Theorem) . . . 93 ", + "bbox": [ + 196, + 132, + 826, + 300 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "LIST OF DEFINITIONS ", + "text_level": 1, + "bbox": [ + 173, + 319, + 357, + 337 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.1 Definition (Group, Abelian Group) . . 22 \nB.2 Definition (Subgroup) . . 23 \nB.3 Definition (Group Homomorphism) 23 \nB.4 Definition (Topological Group, Compact Group) . . 23 \nB.5 Definition (Group Action) 23 \nB.6 Definition (Orbit) . 23 \nB.7 Definition (Transitive Action, Homogeneous Space) . 23 \nB.8 Definition (Stabilizer Subgroup) 24 \nB.10 Definition (Linear Representation) 24 \nB.11 Definition (Intertwiner) . 24 \nB.12 Definition (Equivalent Representations) 24 \nB.13 Definition (Invariant Subspace, Subrepresentation, Closed Subrepresentation) 25 \nB.14 Definition (Irreducible Representation) . 25 \nB.15 Definition (Unitary Group) 25 \nB.16 Definition (Unitary Transformation) 25 \nB.17 Definition (Unitary Representation) 25 \nB.18 Definition (Isomorphism of Unitary Representations) 25 \nB.24 Definition (Matrix Coefficients) 29 \nC.1 Definition (Hom-Representation) . . 42 \nC.3 Definition (Steerable Kernel) . . 43 \nC.4 Definition (Representation Operator) . . 44 \nC.6 Definition (Kernel Operator) 45 \nD.1 Definition (Tensor Product) . . 52 \nD.2 Definition (Tensor Product of pre-Hilbert spaces) 52 \nD.3 Definition (Tensor Product Representation) 53 \nD.6 Definition (Clebsch-Gordan Coefficients) 54 \nD.7 Definition (Endomorphism) . . . 55 \nD.9 Definition (Matrix Element) 55 \nD.12 Definition (Reduced Matrix Element) 56 \nE.11 Definition (Real, Complex, and Quaternionic Type Irreducible Representations) . . 83 \nE.12 Definition (Restriction and Extension) . 84 \nE.14 Definition (Real Type Complex Representation) 84 \nE.21 Definition (Tensor Product Representation) 87 \nF.1 Definition (Topological Space, Open Sets, Closed Sets) 90 \nF.2 Definition (Open Neighborhood) 91 \nF.3 Definition (Hausdorff Space) 91 \nF.4 Definition (Subspace) . . 91 \nF.5 Definition (Closure, Density) 91 \nF.6 Definition (Continuous Function, Homeomorphism) . 91 \nF.7 Definition (Open Cover, Compact Space) 91 \nF.10 Definition (Product Topology) 91 \nF.11 Definition (Quotient Map, Quotient Space) . 91 \nF.13 Definition (Norm) . . . 92 \nF.14 Definition (Metric) 92 \nF.15 Definition (Convergent Sequence) 92 \nF.16 Definition (Continuity in Metric Spaces) . 92 \nF.17 Definition (Uniform Continuity) 92 \nF.19 Definition (Cauchy Sequence) 93 \nF.20 Definition (Complete Metric Space) 93 \nF.21 Definition (Completion) . . 93 \nF.23 Definition (Boundedness) . . 93 \nF.26 Definition (Partially Ordered Set, Directed Set) 94 \nF.28 Definition (Net) . 94 \nF.29 Definition (Convergence of Nets) . . 94 \nF.30 Definition (Approximated Dirac Delta) . . 94 \nF.32 Definition (pre-Hilbert Space, Hilbert space) . 95 \nF.35 Definition (Orthogonality) 95 \nF.36 Definition (Orthogonal Complement) . 95 \nF.39 Definition (Orthonormal System) . 96 \nF.40 Definition (Orthonormal Basis) . 96 \nF.42 Definition (Adjoint of an Operator) . . 96 ", + "bbox": [ + 196, + 342, + 828, + 930 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 196, + 102, + 828, + 829 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A BUILDING BLOCKS OF SO(2)-STEERABLE KERNELS – RUNNING EXAMPLE FOR SECTION 3 ", + "text_level": 1, + "bbox": [ + 174, + 845, + 748, + 880 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this short chapter, we briefly explain the components of steerable kernels at the specific example of real valued irreps of the circle group SO(2). While this example is quite simple, it shows some non-trivial properties like 2-dimensional endomorphism spaces $\\operatorname { E n d } _ { \\operatorname { S O ( 2 ) } , \\operatorname { R } } ( V _ { J } )$ for $J > 0$ and a Clebsch-Gordan decomposition in which the multiplicity $\\left[ J ( j l ) \\right]$ can differ from 0 or 1. To give a quick overview: Example A.1 considers the circle as an orbit and homogeneous space of SO(2) while Example A.2 introduces the real valued irreps. Their endomorphisms are stated in Example A.3. As discussed in Example A.4, the Peter-Weyl theorem corresponds here to the usual Fourier series on $S ^ { 1 }$ . The Clebsch-Gordan decomposition of tensor products of the irreps are discussed in Example A.5. With these ingredients, we are ready to instantiate the kernel spaces as described by our Wigner-Eckart Theorem 4.1 for steerable kernels, for which we refer, including proofs, to Section E.2. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 231 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "SO(2)-steerable kernels $K : \\mathbb { R } ^ { 2 } \\mathrm { H o m } _ { \\mathrm { R } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\cong \\mathbb { R } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ allow for rotation equivariant convolutions. For instance, a convolution with an SO(2)-steerable kernel on $\\mathbb { R } ^ { 2 }$ is guaranteed to be $\\mathrm { S E } ( 2 ) = ( \\mathbb { R } ^ { 2 } , + ) \\rtimes \\mathrm { S O } ( 2 )$ equivariant while a convolution with an SO(2)-steerable kernel on $S ^ { 2 }$ will be $\\mathrm { S O ( 3 ) }$ -equivariant. ", + "bbox": [ + 174, + 236, + 825, + 292 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Homogeneous Spaces $\\mathrm { S O } ( 2 )$ acts on the kernel’s domain $\\mathbb { R } ^ { 2 }$ by rotating it. The orbits of the action are therefore given by 1) the origin $\\{ 0 \\}$ and 2) circles of arbitrary radius. We know that the kernel constraint can be solved on each orbit individually, and so we can restrict to looking at those. Since $\\{ 0 \\}$ is rather trivial, we specifically consider the circle $S ^ { 1 }$ as a more interesting homogeneous space. ", + "bbox": [ + 173, + 310, + 825, + 382 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Example A.1. Consider the circle $S ^ { 1 }$ and the rotation group SO(2). For convenience, we reparameterize both: we view SO(2) as the group of angles $\\phi \\in \\mathbb { R } / 2 \\pi \\mathbb { Z } \\cong [ 0 , 2 \\pi ] / \\scriptscriptstyle \\mathrm { 0 } . . 2 \\pi$ and $S ^ { 1 }$ as the space $\\mathbb { R } / 2 \\pi \\mathbb { Z }$ as well. Then the action of SO(2) on $S ^ { 1 }$ is given by $\\phi \\cdot x : = ( \\phi + x )$ mod $2 \\pi$ . It is easy to see that this action is transitive, which makes the circle a homogeneous space of SO(2). ", + "bbox": [ + 174, + 387, + 825, + 445 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Irreducible Representations As it is sufficient to solve the kernel constraint for irreducible orthogonal input- and output representations, we now state a classification of those up to isomorphism. ", + "bbox": [ + 171, + 463, + 823, + 492 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Example A.2. The irreducible orthogonal representations $\\rho _ { l } : \\mathrm { S O } ( 2 ) \\to \\mathrm { O } ( V _ { l } )$ of SO(2) are labeled by indices (“quantum numbers”) $l \\in \\mathbb { N } _ { \\geq 0 }$ . For $l = 0$ , one has the trivial representation with $V _ { 0 } = \\mathbb { R }$ and $\\rho _ { 0 } ( \\phi ) = \\mathrm { \\bar { i d } _ { R } }$ . For $l \\geq 1$ , one has $\\bar { V _ { l } = \\mathbb { R } ^ { 2 } }$ and ", + "bbox": [ + 173, + 497, + 825, + 541 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ef6fd5f6e72711c4e52528bb67e45b1fc05ebbf47b652d4e1e57a4d91e830cf8.jpg", + "text": "$$\n\\rho _ { l } ( \\phi ) = \\binom { \\cos ( l \\phi ) } { \\sin ( l \\phi ) } - \\frac { \\sin ( l \\phi ) } { \\cos ( l \\phi ) } \\Biggr ) ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 551, + 614, + 585 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "i.e., rotation matrices of “frequency $l ^ { \\dag }$ . The isomorphism classes of irreducible orthogonal representations are then given by $\\widehat { \\mathrm { S O } ( 2 ) } \\cong \\mathbb { N } _ { \\ge 0 }$ . ", + "bbox": [ + 173, + 594, + 823, + 630 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We are thus in the following considering SO(2)-steerable kernels of the form ", + "bbox": [ + 173, + 641, + 679, + 656 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/0209aae7e2db4a82b4ecfda80b2414e5418b4fc27770e0f55c34b6c032919c46.jpg", + "text": "$$\nK : S ^ { 1 } \\to \\operatorname { H o m } _ { \\mathbb { R } } ( V _ { l } , V _ { J } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 408, + 666, + 588, + 685 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $l , J \\geq 0$ ", + "bbox": [ + 174, + 695, + 276, + 710 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Endomorphisms Remember that if $c : V _ { J } \\to V _ { J }$ is an endomorphism, i.e., commutes with $\\rho _ { J }$ , that $c \\circ K$ is then steerable as well. Thus, we now look at a classification of the endomorphisms of the irreducible orthogonal representations: ", + "bbox": [ + 174, + 728, + 823, + 772 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Example A.3. Let $G = \\mathrm { S O ( 2 ) }$ with the irreducible representations $\\rho _ { J }$ as in Example A.2. Clearly, the endomorphism space $\\mathrm { E n d } _ { \\mathrm { S O ( 2 ) , R } } ( V _ { 0 } )$ is 1-dimensional, i.e., $E _ { 0 } = 1$ . For all $J \\geq 1$ , the endomorphism space is two-dimensional $E _ { J } = 2$ ) and given by combinations of scalings and rotations8 on $\\bar { V } _ { J } = \\mathbb { R } ^ { \\bar { 2 } }$ . A basis of this space is given by the following two matrices: ", + "bbox": [ + 173, + 776, + 825, + 837 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/9efcb04d7de7e8f9f9ea7618d2b3c57b9e1957facc1611e41e4b0887a7da01cb.jpg", + "text": "$$\nc _ { 1 } = \\mathrm { i d } _ { \\mathrm { R } ^ { 2 } } = \\left( { \\begin{array} { c c } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\end{array} } \\right) , \\quad c _ { 2 } = \\left( { \\begin{array} { c c } { 0 } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} } \\right) , \\quad \\operatorname { s p a n } _ { \\mathrm { R } } \\{ c _ { 1 } , c _ { 2 } \\} = \\mathrm { E n d } _ { \\mathrm { S O } ( 2 ) , \\mathrm { K } } ( V _ { J } ) .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 845, + 764, + 881 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "That $c _ { 1 }$ is an endomorphism of $\\rho _ { J }$ for $J \\geq 1$ is immediately clear. That the same holds for $c _ { 2 }$ is checked by the following simple calculation: ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/9556f3f58e3906f332b94dfcae3c3e25e2c349c6c0239d3ca6762f50b53f26df.jpg", + "text": "$$\n{ \\begin{array} { r l } { c _ { 2 } \\rho _ { J } ( \\phi ) \\ = \\ { \\binom { 0 } { 1 } } \\ - 1 ( { \\begin{array} { c c } { \\cos ( J \\phi ) } & { - \\sin ( J \\phi ) } \\\\ { \\sin ( J \\phi ) } & { \\cos ( J \\phi ) } \\end{array} } ) \\ = \\ { \\binom { - \\sin ( J \\phi ) } { \\cos ( J \\phi ) } } \\ - \\ \\cos ( J \\phi ) \\ ) } \\\\ { = \\ ( { \\begin{array} { c c } { \\cos ( J \\phi ) } & { - \\sin ( J \\phi ) } \\\\ { \\sin ( J \\phi ) } & { \\cos ( J \\phi ) } \\end{array} } ) \\ { \\binom { 0 } { 1 } } \\ - 1 ) \\ = \\ \\rho _ { J } ( \\phi ) \\ c _ { 2 } } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 228, + 140, + 766, + 213 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The proof that there are no other endomorphisms is sketched in Proposition E.5. ", + "bbox": [ + 173, + 218, + 697, + 234 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Peter-Weyl and Harmonic Basis Functions Another ingredient that we need to construct SO(2)- steerable kernels on $S ^ { 1 }$ is the decomposition of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ into its irreducible subrepresentations $V _ { j i }$ , which the Peter-Weyl theorems guarantees to exist. Less abstractly, we are interested in an orthonormal set of harmonic (steerable) basis functions on $S ^ { 1 }$ that span $\\bar { L _ { \\mathbb { R } } ^ { 2 } } ( S ^ { 1 } )$ – which corresponds to the usual Fourier series on $S ^ { 1 }$ . ", + "bbox": [ + 173, + 250, + 825, + 320 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Example A.4. As in Example A.1, we assume $G = \\mathrm { S O } ( 2 )$ and $X \\ : = \\ : S ^ { 1 }$ . A standard result in harmonic analysis says that square-integrable functions $f : S ^ { 1 } \\to \\mathbb { R }$ , i.e., $f \\in L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ , can be uniquely written as an infinite sum of sine and cosine terms, ", + "bbox": [ + 173, + 325, + 825, + 368 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/e24a3778807d39353fc99a80a4ea7ea7261e7e5927dc8a2cd9eaa2df45f20175.jpg", + "text": "$$\nf ( x ) = a _ { 0 } + \\sum _ { j = 1 } ^ { \\infty } a _ { j } \\cos ( j x ) + b _ { j } \\sin ( j x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 346, + 377, + 650, + 405 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $a _ { 0 }$ and $a _ { j } , b _ { j } , j \\geq 1$ are real-valued expansion coefficients. ", + "bbox": [ + 176, + 411, + 606, + 428 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "How does this result relate to the harmonic basis functions in the Peter-Weyl theorem 3.4? As stated above, we have isomorphism classes ${ \\widehat { G } } = { \\widehat { \\mathrm { S O } ( 2 ) } } \\cong \\mathbb { N } _ { \\geq 0 }$ of irreps with representatives $\\rho _ { j }$ . A comparison of the Fourier series in Eq. (8) with property 2 in the Peter-Weyl theorem 3.4 suggests the following identification of harmonic basis functions $Y _ { j } ^ { m }$ and coefficients $\\lambda _ { j m }$ , ", + "bbox": [ + 174, + 433, + 826, + 496 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/8a4bf8794a509e153f262a340f8845701860b0b4ff8e6f3203ce0914720a9659.jpg", + "text": "$$\n\\begin{array} { r l } { Y _ { 0 } ^ { 1 } = \\cos _ { 0 } = 1 , \\quad } & { \\lambda _ { 0 1 } = a _ { 0 } \\quad \\quad \\mathrm { f o r } \\quad \\quad j = 0 } \\\\ { Y _ { j } ^ { 1 } = \\cos _ { j } \\quad \\quad , \\quad } & { \\lambda _ { j 1 } = a _ { j } \\quad \\quad \\mathrm { f o r } \\quad \\quad j \\geq 1 } \\\\ { Y _ { j } ^ { 2 } = \\sin _ { j } \\quad \\quad , \\quad } & { \\lambda _ { j 2 } = b _ { j } \\quad \\quad \\mathrm { f o r } \\quad \\quad j \\geq 1 , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 330, + 503, + 666, + 568 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where we introduced the shorthand notations $\\cos _ { j } ( x ) : = \\cos ( j x )$ and $\\sin _ { j } ( x ) : = \\sin ( j x )$ . Note that we dropped the index $i = 1 , \\dotsc , m _ { j }$ since $m _ { j } ~ = ~ 1$ for any $j \\in \\widehat { \\mathrm { S O } ( 2 ) }$ . As expected, we have indices $m = 1$ for $j = 0$ with $d _ { j } \\ \\bar { = } \\ \\dim ( \\bar { V _ { 0 } } ) = 1$ and indices $m = 1 , 2$ for $j \\geq 1$ with $d _ { j } = \\dim ( V _ { j } ) = 2$ . The orthogonality relations in property 3 of the Peter-Weyl theorem hold up to a simple normalization of these basis functions and are easily checked by explicitly computing the scalar products. Property 1, i.e., the SO(2)-steerability of the harmonic bases, is trivial for $j = 0$ . For $j \\geq 1$ , the standard angle summation formulas for cosines and sines lead to the following expressions for harmonics that are translated by $\\phi \\in \\mathrm { S O } ( 2 )$ : ", + "bbox": [ + 173, + 573, + 826, + 691 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/a8f616ef83033c8619fd93a543d39caa3ef1ee3b30ca210a91185f2ad6048967.jpg", + "text": "$$\n\\begin{array} { r l } & { \\cos _ { j } ( x - \\phi ) = \\cos _ { j } ( x ) \\cos _ { j } ( - \\phi ) - \\sin _ { j } ( x ) \\sin _ { j } ( - \\phi ) = \\left( \\rho _ { j } ^ { 1 1 } ( \\phi ) \\cos _ { j } + \\rho _ { j } ^ { 2 1 } ( \\phi ) \\sin _ { j } \\right) ( x ) } \\\\ & { \\sin _ { j } ( x - \\phi ) = \\cos _ { j } ( x ) \\sin _ { j } ( - \\phi ) + \\sin _ { j } ( x ) \\cos _ { j } ( - \\phi ) = \\left( \\rho _ { j } ^ { 1 2 } ( \\phi ) \\cos _ { j } + \\rho _ { j } ^ { 2 2 } ( \\phi ) \\sin _ { j } \\right) ( x ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 199, + 699, + 800, + 743 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "which is just property 1 in the Peter-Weyl theorem. This is concisely summarized by ", + "bbox": [ + 174, + 747, + 728, + 763 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/e8c4c9a28d292af386c9682a3b70b3b024c17462a085030a7c9debf90b10ba0d.jpg", + "text": "$$\n\\left( \\cos _ { j } \\atop \\sin _ { j } \\right) ( \\phi ^ { - 1 } \\cdot x ) = \\left( \\cos _ { j } \\atop \\sin _ { j } \\right) ( x - \\phi ) = \\rho _ { j } ( \\phi ) ^ { \\top } \\left( \\cos _ { j } \\atop \\sin _ { j } \\right) ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 285, + 770, + 710, + 806 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "which shows that the basis functions $Y _ { j } ^ { 1 } = \\cos _ { j }$ and $Y _ { j } ^ { 2 } = \\sin _ { j }$ span an invariant subspace $V _ { j }$ of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ under rotations. From a more abstr splits into the orthogonal direct sum , the Peter-Weyl theorem just states that. $L _ { \\mathbb { R } } ^ { 2 ^ { \\circ } } ( S ^ { 1 } )$ $\\bigoplus _ { j \\in { \\widehat { \\mathrm { S O } ( 2 ) } } } V _ { j }$ ", + "bbox": [ + 174, + 814, + 825, + 864 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Tensor Products and Clebsch-Gordan Coefficients Finally, we need to investigate the tensor products of irreducible representations and their decomposition via Clebsch-Gordan coefficients. They will be used to correctly assemble harmonic basis functions to steerable kernels. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Example A.5. Remember the irreducible representations $\\rho _ { l } : \\mathrm { S O } ( 2 ) \\to \\mathrm { O } ( V _ { l } )$ given in Example A.2. As we prove in Proposition E.4, including a description of the Clebsch-Gordan coefficients, the tensor products decompose as follows: ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/80ef5e5e857775c4b530e8cd3321ab15b8b86fd75c9f328f6d1160df1f3a7f33.jpg", + "text": "$$\nV _ { 0 } \\otimes V _ { 0 } \\cong V _ { 0 } , \\quad V _ { j } \\otimes V _ { 0 } \\cong V _ { j } , \\quad V _ { 0 } \\otimes V _ { l } \\cong V _ { l } , \\quad V _ { j } \\otimes V _ { l } \\cong V _ { | j - l | } \\oplus V _ { j + l } ,\n$$", + "text_format": "latex", + "bbox": [ + 240, + 152, + 756, + 170 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where the last isomorphism only holds if $j , l \\geq 1$ and $j \\neq l$ . If $j , l \\geq 1$ and $j = l$ , then we obtain ", + "bbox": [ + 176, + 174, + 803, + 191 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/a2575abc4ac7c733ec78f6084990d3128a01abb1990447289ba4bba69c8252d8.jpg", + "text": "$$\nV _ { l } \\otimes V _ { l } \\cong ( V _ { 0 } ) ^ { 2 } \\oplus V _ { 2 l } ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 194, + 576, + 214 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "i.e., $V _ { 0 }$ here appears twice in the decomposition of a tensor product of irreducible representations. We therefore have multiplicities $\\left[ J ( j l ) \\right]$ which are 1 for [0(00 $) ] , [ j ( j 0 ) ] , [ l ( 0 l ) ] , \\bar { [ } | j - l | ( j l ) ]$ , $[ j + l ( j l ) ]$ and $[ 2 l ( l l ) ]$ while $[ 0 ( l l ) ] = 2$ . Any other multiplicity is zero. ", + "bbox": [ + 174, + 219, + 825, + 262 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Wigner-Eckart theorem for SO(2)-steerable kernels With these ingredients one can then determine all SO(2)-steerable kernels. This is explained in Proposition E.6. ", + "bbox": [ + 171, + 276, + 823, + 306 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B REPRESENTATION THEORY OF COMPACT GROUPS", + "text_level": 1, + "bbox": [ + 174, + 325, + 624, + 342 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In this chapter, we outline the main ingredients of the representation theory of compact groups that we need for our applications to steerable CNNs. Usually, this theory is only developed for representations over the complex numbers. However, since we want to apply it also to steerable CNNs using real representations, we need to be a bit more careful. In particular, we need to make sure that the Peter-Weyl theorem is correctly stated and proven. ", + "bbox": [ + 174, + 356, + 825, + 426 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The outline is as follows: In Section B.1, we start by stating all the important definitions and concepts from group theory and representation theory of (unitary) representations that are needed for formulating the Peter-Weyl theorem. After defining Haar measures both for compact groups and their homogeneous spaces and shortly discussing their square-integrable functions, we formulate the Peter-Weyl Theorem B.22. In Section B.2, then, we give a proof of this version of the Peter-Weyl theorem, carefully making sure to not use properties that are only true over C. In some essential steps, mainly the density of the matrix coefficients in the regular representation, we refer to the literature, since the proof clearly does not make use of C per se. While we initially only give the proof for the regular representation, i.e., the space of square-integrable functions on the group itself, we end this section with a discussion of general unitary representations and, in particular, the space of square-integrable functions for an arbitrary homogeneous space. ", + "bbox": [ + 174, + 433, + 825, + 587 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In the whole chapter, let K be the field of real or complex numbers. ", + "bbox": [ + 174, + 593, + 614, + 608 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B.1 FOUNDATIONS OF REPRESENTATION THEORY AND THE PETER-WEYL THEOREM ", + "bbox": [ + 176, + 625, + 774, + 640 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B.1.1 PRELIMINARIES OF TOPOLOGICAL GROUPS AND THEIR ACTIONS ", + "bbox": [ + 176, + 650, + 684, + 665 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In this section, we define preliminary concepts from topological groups and their actions. This material can, for example, be found in detail in Arkhangel’skii & Tkachenko (2008). For the topological concepts that we use, we refer to Chapter F.1. ", + "bbox": [ + 174, + 674, + 825, + 717 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Definition B.1 (Group, Abelian Group). A group $G = ( G , \\cdot , ( \\cdot ) ^ { - 1 } , e )$ , most often simply written $G$ , consists of the following data: ", + "bbox": [ + 171, + 720, + 823, + 751 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "1. A set $G$ of group elements $g \\in G$ ", + "bbox": [ + 214, + 761, + 455, + 776 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "2. A multiplication $\\cdot : G \\times G \\to G , ( g , h ) \\mapsto g \\cdot h .$ ", + "bbox": [ + 212, + 784, + 553, + 801 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "3. An inversion $( \\cdot ) ^ { - 1 } : G \\to G , g \\mapsto g ^ { - 1 }$ . ", + "bbox": [ + 212, + 808, + 493, + 825 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "4. A distinguished unit element $e \\in G$ . It is also called neutral element. ", + "bbox": [ + 210, + 833, + 683, + 848 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "They are assumed to have the following properties for all $g , h , k \\in G$ : ", + "bbox": [ + 173, + 858, + 632, + 875 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "1. The multiplication is associative: $g \\cdot ( h \\cdot k ) = ( g \\cdot h ) \\cdot k$ . ", + "bbox": [ + 212, + 885, + 607, + 901 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "2. The unit element is neutral with respect to multiplication: $e \\cdot g = g = g \\cdot e$ ", + "bbox": [ + 209, + 909, + 725, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "3. The inversion of an element multiplied with itself is the neutral element: $g \\cdot g ^ { - 1 } = g ^ { - 1 } \\cdot g =$ e. ", + "bbox": [ + 204, + 102, + 823, + 132 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "A group is called abelian if, additionally, the multiplication is commutative: $g \\cdot h = h \\cdot g$ for all $g , h \\in G$ . If this is the case, a group is often written as $G = ( G , + , - ( \\cdot ) , 0 )$ . ", + "bbox": [ + 174, + 143, + 823, + 174 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "If we consider several groups at once, say $G$ and $H$ , then we often do not distinguish their multiplication, inversion, and neutral elements in notation. It will be clear from the context which group the operation belongs to. ", + "bbox": [ + 174, + 184, + 825, + 227 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition B.2 (Subgroup). Let $G$ be a group and $H \\subseteq G$ a subset. $H$ is called a subgroup if: ", + "bbox": [ + 168, + 231, + 789, + 247 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "1. For all $h , h ^ { \\prime } \\in H$ we have $\\boldsymbol { h } \\cdot \\boldsymbol { h } ^ { \\prime } \\in H$ . \n2. For all $h \\in H$ we have $h ^ { - 1 } \\in H$ . \n3. The neutral element $e \\in G$ is in $H$ . ", + "bbox": [ + 210, + 256, + 482, + 321 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Consequently, $H$ is also a group with the restrictions of the multiplication and inversion of $G$ to $H$ . ", + "bbox": [ + 171, + 333, + 823, + 348 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition B.3 (Group Homomorphism). Let $G$ and $H$ be groups. A function $f : G \\to H$ is called a group homomorphism if it respects the multiplication, inversion, and neutral element, i.e., for all $g , h \\in G$ : ", + "bbox": [ + 173, + 352, + 825, + 395 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "1. $f ( g \\cdot h ) = f ( g ) \\cdot f ( h )$ . \n2. $f ( g ^ { - 1 } ) = f ( g ) ^ { - 1 }$ . \n3. $f ( e ) = e$ . ", + "bbox": [ + 210, + 405, + 387, + 472 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "The second and third properties automatically follow from the first and so do not need to be verified in order to prove that a certain function is a group homomorphism. ", + "bbox": [ + 173, + 482, + 821, + 511 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition B.4 (Topological Group, Compact Group). Let $G$ be a group and $\\tau$ be a topology of the underlying set of $G$ . Then $G = ( \\bar { G } , \\mathcal { T } )$ is called a topological group (Arkhangel’skii $\\&$ Tkachenko, 2008) if both multiplication $G \\times G \\to G$ , $( x , y ) \\mapsto x \\cdot y$ and inversion $G G$ , $x \\mapsto x ^ { - 1 }$ are continuous maps. Additionally, we always assume the topology to be Hausdorff. ", + "bbox": [ + 173, + 515, + 825, + 571 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "A topological group is called compact if the underlying topological space is compact. ", + "bbox": [ + 176, + 578, + 732, + 593 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "From now on, all groups considered are compact topological groups. Furthermore, whenever $G$ is a finite group, we assume that it is a topological group with the discrete topology, i.e., the topology with respect to which all subsets of $G$ are open. ", + "bbox": [ + 173, + 604, + 826, + 647 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We will need the following definition in order to define homogeneous spaces: ", + "bbox": [ + 174, + 654, + 683, + 669 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition B.5 (Group Action). Let $G$ be a compact group and $X$ a topological space. Then a group action of $G$ on $X$ is a continuous function $\\cdot : G \\times X \\to X$ with the following properties: ", + "bbox": [ + 173, + 672, + 826, + 702 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "1. $( g \\cdot h ) \\cdot x = g \\cdot ( h \\cdot x )$ for all $g , h , \\in G$ and $x \\in X$ . \n2. $e \\cdot x = x$ for all $x \\in X$ . ", + "bbox": [ + 210, + 713, + 570, + 753 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We will often simply write $g x$ instead of $g \\cdot x$ . Also, note that the multiplication within $G$ is denoted by the same symbol as the group action on the space $X$ . ", + "bbox": [ + 173, + 765, + 823, + 794 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition B.6 (Orbit). Let $\\cdot : G \\times X \\to X$ be a group action. Let $x \\in X$ . Then it’s orbit, denoted $G \\cdot x$ , is given by the set ", + "bbox": [ + 173, + 797, + 823, + 825 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/c9e593aac8f69b7b5b8aae1d7fdca56485df583b49787f540038e57ead7caf3f.jpg", + "text": "$$\nG \\cdot x : = \\{ g \\cdot x \\mid g \\in G \\} \\subseteq X .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 825, + 599, + 842 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition B.7 (Transitive Action, Homogeneous Space). Let $\\cdot : G \\times X \\to X$ be a group action. This action is called transitive if for all $x , y \\in X$ there exists $g \\in G$ such that $g x = y$ . Equivalently, each orbit is equal to $X$ , that is: For all $x \\in X$ we have $G \\cdot x = X$ . ", + "bbox": [ + 174, + 845, + 825, + 888 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "$X$ is called a homogeneous space (with respect to the action) if the action is transitive, $X$ is Hausdorff and $X \\neq \\emptyset$ . ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "The Hausdorff condition and non-emptiness in the definition of homogeneous spaces is needed for Lemma B.21, which is necessary to even define a normalized Haar measure on a homogeneous space. Some texts in the literature may define homogeneous spaces without these conditions. ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Definition B.8 (Stabilizer Subgroup). Let · : $G \\times X \\to X$ be a group action. Let $x \\in X$ . The stabilizer subgroup $G _ { x }$ is the subgroup of $G$ given by ", + "bbox": [ + 171, + 150, + 823, + 179 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/7438fef34ec440ac4268fec9c403823a534df4f08a89d2378847feaa1e219820.jpg", + "text": "$$\nG _ { x } : = \\{ g \\in G \\mid g x = x \\} \\subseteq G .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 185, + 602, + 204 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Example B.9. The multiplication of the group $G$ is a group action of $G$ on itself. $G$ is a homogeneous space with this action. Furthermore, for each $g \\in G$ the stabilizers $G _ { g }$ are the trivial subgroup $e$ . ", + "bbox": [ + 173, + 210, + 825, + 253 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In general, homogeneous spaces with the property that all stabilizers are trivial are called torsors or principal homogeneous spaces. Principal homogeneous spaces are topologically indistinguishable from the group itself. ", + "bbox": [ + 174, + 260, + 825, + 303 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "B.1.2 LINEAR AND UNITARY REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 318, + 527, + 333 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In this section, we define many of the foundational concepts about linear and unitary representations (Knapp, 2002; Kowalski, 2014). ", + "bbox": [ + 173, + 343, + 823, + 371 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Whenever we will consider linear or unitary representations of compact groups, we want those representations to be continuous. This requires that the vector spaces on which our groups act carry themselves a topology. Prototypical examples of such vector spaces are (pre-)Hilbert spaces. They are the main examples of vector spaces considered in this work. Foundational concepts about (pre)Hilbert spaces can be found in Chapter F.3. The most important difference between how we view pre-Hilbert spaces and how it can often be found in the literature is that in this work, scalar products are antilinear in the first component and linear in the second. This is the convention usually chosen in physics. ", + "bbox": [ + 173, + 377, + 825, + 489 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For a vector space $V$ over $\\mathbb { K }$ let $\\operatorname { G L } ( V )$ be the group of invertible linear functions from $V$ to $V$ . Sometimes in the literature, this is also written ${ \\mathrm { G L } } ( V , \\mathbb { K } )$ . The multiplication is given by function composition and the neutral element by the identity function $\\operatorname { i d } _ { V }$ on $V$ . ", + "bbox": [ + 173, + 496, + 825, + 539 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Definition B.10 (Linear Representation). Let $G$ be a compact group and $V$ be a $\\mathbb { K }$ -vector space carrying a topology, for example, a (pre)-Hilbert space. Then a linear representation of $G$ on $V$ is a group homomorphism $\\rho : G \\to { \\mathrm { G L } } ( V )$ which is continuous in the following sense: for all $v \\in V$ , the function ", + "bbox": [ + 173, + 542, + 825, + 598 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/2a7a603f448bb88b6a0d40bd5271cae7e3757018e2a65e38f88a937e31ae889d.jpg", + "text": "$$\n\\rho ^ { v } : G \\to V , g \\mapsto \\rho ^ { v } ( g ) : = \\rho ( g ) ( v )\n$$", + "text_format": "latex", + "bbox": [ + 377, + 598, + 620, + 616 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "is continuous. From the definition we obtain $\\rho ( e ) = \\mathrm { i d } _ { V }$ , $\\rho ( g \\cdot h ) = \\rho ( g ) \\circ \\rho ( h )$ and $\\rho ( g ^ { - 1 } ) =$ $\\rho ( g ) ^ { - 1 }$ for all $g , h \\in G$ . For simplicity, we also just say representation or $G$ -representation instead of linear representation. Instead of denoting the representation by $\\rho$ , we often denote it by $V$ if the function $\\rho$ is clear from the context. ", + "bbox": [ + 173, + 619, + 825, + 676 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Note that in this definition, $V$ can be any abstract topological $\\mathbb { K }$ -vector space with a topology and does not need to be a space $\\mathbb { K } ^ { n }$ or something similar. Consequently, we usually do not view the functions $\\rho ( g )$ as matrices, but as abstract linear automorphisms from $V$ to $V$ . ", + "bbox": [ + 174, + 688, + 825, + 729 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Definition B.11 (Intertwiner). Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ and $\\rho ^ { \\prime } : G \\to { \\mathrm { G L } } ( V ^ { \\prime } )$ be two representations over the same group $G$ . An intertwiner between them is a linear function $f : V \\ { \\stackrel { } { \\to } } \\ V ^ { \\prime }$ that is additionally equivariant with respect to $\\rho$ and $\\rho ^ { \\prime }$ and continuous. Equivariance means that for all $g \\in G$ one has $f \\circ \\rho ( g ) = \\rho ^ { \\prime } ( g ) \\circ f$ , which means the following diagram commutes: ", + "bbox": [ + 174, + 733, + 825, + 791 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/9e74190799347ae43f416d1e394b0aabd4c33850d52c7f2df3402f6e074eb6c2.jpg", + "text": "$$\n\\begin{array} { c } { { V \\xrightarrow { f } V ^ { \\prime } } } \\\\ { { \\rho ( g ) \\downarrow } } \\\\ { { V \\xrightarrow [ f ] { } \\nu ^ { \\prime } } } \\end{array} \\downarrow \\rho ^ { \\prime } ( g )\n$$", + "text_format": "latex", + "bbox": [ + 431, + 800, + 566, + 871 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Definition B.12 (Equivalent Representations). Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ and $\\rho ^ { \\prime } : G \\to { \\mathrm { G L } } ( V ^ { \\prime } )$ be two representations. They are called equivalent if there is an intertwiner $f : V \\to V ^ { \\prime }$ that has an inverse. That is, there exists an intertwiner ${ \\tilde { f } } : V ^ { \\prime } \\to V$ such that $\\tilde { f } \\circ f = \\operatorname { i d } _ { V }$ and $f \\circ { \\tilde { f } } = \\operatorname { i d } _ { V ^ { \\prime } }$ . ", + "bbox": [ + 174, + 878, + 823, + 924 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In categorical terms, equivalent representations are isomorphic in the category of linear representations. The reason we do not call them isomorphic is that there is a stronger notion of isomorphism between representations which we will later use, namely isomorphisms of unitary representations. ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition B.13 (Invariant Subspace, Subrepresentation, Closed Subrepresentation). Let $\\rho : G $ $\\operatorname { G L } ( V )$ be a representation. An invariant subspace $W \\subseteq V$ is a linear subspace of $V$ such that $\\rho ( g ) ( w ) \\in W$ for all $g \\in G$ and $w \\in W$ . Consequently, the restriction $\\rho | _ { W } : G \\to { \\mathrm { G L } } ( W )$ , $g \\mapsto \\rho ( g ) | _ { W } : W \\to W$ is a representation as well, called subrepresentation of $\\rho$ . ", + "bbox": [ + 173, + 150, + 825, + 207 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "A subrepresentation is called closed if $W$ is closed in the topology of $V$ ", + "bbox": [ + 174, + 213, + 648, + 228 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition B.14 (Irreducible Representation). A representation $\\rho : G \\to { \\mathrm { G L } } ( V )$ is called irreducible if $V \\neq 0$ and if the only closed subrepresentations of $V$ are 0 and $V$ itself. An irreducible representation is also shortly called irrep. ", + "bbox": [ + 174, + 232, + 825, + 275 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition B.15 (Unitary Group). Let $V$ be a pre-Hilbert space. The unitary group $\\mathrm { U } ( V )$ of $V$ is defined as the group of all linear invertible maps $f : V \\to V$ that respect the inner product, i.e., $\\langle f ( x ) | f ( y ) \\rangle = \\langle x | \\bar { y } \\rangle$ for all $x , y \\in V$ . It is a group with respect to the usual composition and inversion of invertible linear maps. ", + "bbox": [ + 174, + 279, + 825, + 335 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Note that if the field $\\mathbb { K }$ is the real numbers, then what we call “unitary” is actually called orthogonal, and the group would be denoted $\\mathrm { O } ( V )$ . However, the mathematical properties are essentially the same, and since the term “unitary” is more widely used (as normally, representations over the complex numbers are considered) we stick with “unitary”. ", + "bbox": [ + 173, + 347, + 825, + 404 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "More generally, we have the following: ", + "bbox": [ + 176, + 410, + 431, + 425 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition B.16 (Unitary Transformation). Let $V , V ^ { \\prime }$ be two pre-Hilbert spaces. A unitary transformation $f : V \\to V ^ { \\prime }$ is a bijective linear function such that ${ \\bar { \\langle f ( x ) | } } f ( y ) \\rangle = \\langle x | y \\rangle$ for all $x , y \\in V$ . These can be regarded as isomorphisms between pre-Hilbert spaces. ", + "bbox": [ + 176, + 429, + 823, + 472 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Note that unitary transformations are in particular isometries, i.e., they keep the distances of vectors with respect to the metric defined by the scalar product. For the definition of this metric, see the discussion before and after Definition F.14. ", + "bbox": [ + 174, + 484, + 825, + 526 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition B.17 (Unitary Representation). Let $V$ be a pre-Hilbert space and $G$ a group. Then a representation $\\rho : G \\to { \\mathrm { G L } } ( V )$ is called a unitary representation if $\\rho ( g ) \\in \\operatorname { U } ( V )$ for all $g \\in G$ . We then write $\\rho : G \\to \\operatorname { U } ( V )$ . ", + "bbox": [ + 174, + 530, + 825, + 574 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In this whole chapter, the space $V$ of a unitary representation is supposed to be a Hilbert space, instead of just a pre-Hilbert space. Only in chapter $\\mathrm { D }$ will we consider unitary representations on pre-Hilbert spaces. Note that all finite-dimensional pre-Hilbert spaces are already complete by Proposition F.47, so in these cases, there is no difference. The same proposition also shows that for finite-dimensional unitary representations, we can ignore the topological closedness condition in order to check whether it is irreducible. It will later turn out that all irreducible representations of a compact group are automatically finite-dimensional anyway, see Proposition B.31, so this further simplifies our considerations. ", + "bbox": [ + 173, + 585, + 825, + 696 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "As before with the unitary group, a unitary representation is actually called “orthogonal representation” when the field is the real numbers $\\mathbb { R }$ . ${ \\bar { \\operatorname { U } } } ( V )$ is then replaced by $\\mathrm { O } ( V )$ . We again stick with $\\mathrm { U } ( V )$ whenever the field $\\mathbb { K }$ is not specified. ", + "bbox": [ + 174, + 704, + 820, + 746 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition B.18 (Isomorphism of Unitary Representations). Let $\\rho : G \\to \\operatorname { U } ( V )$ , $\\rho ^ { \\prime } : G \\to \\operatorname { U } ( V ^ { \\prime } )$ be unitary representations and $f : V \\to { \\bar { V } } ^ { \\prime }$ an intertwiner. $f$ is called an isomorphism (of unitary representations) if, additionally, $f$ is a unitary transformation. The representations are then called isomorphic. For this, we write $\\rho \\cong \\rho ^ { \\prime }$ or $V \\cong V ^ { \\prime }$ depending on whether we want to emphasize the representations or the underlying vector spaces. ", + "bbox": [ + 173, + 750, + 825, + 820 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We note the following, which we will frequently use: due to the unitarity of $\\rho ( g )$ for a unitary representation $\\rho$ , we have $\\rho ( g ) ^ { * } = \\rho ( g ) ^ { - 1 }$ , i.e., the adjoint is the inverse. Adjoints are defined in Definition F.42 and this statement is proven more generally in Proposition F.44. Overall, this means that $\\langle \\rho ( g ) ( v ) | w \\rangle = \\langle v | \\rho ( g ) ^ { - 1 } ( w ) \\rangle$ for all $v , w$ and $g$ . ", + "bbox": [ + 174, + 832, + 823, + 890 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In the end, it will turn out that the Peter-Weyl theorem which we aim at is exclusively a statement about unitary representations. One may then wonder whether this is too restrictive. After all, the representations that we consider for steerable CNNs (with precise definitions given in Section C.1) are not necessarily unitary, and so it is not immediately obvious how the Peter-Weyl theorem will be able to help for those. However, as it turns out, all linear representations on finite-dimensional spaces can be considered as unitary, and so the theory applies. We will discuss this in Proposition B.20 once we understand Haar measures on compact groups. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "B.1.3 THE HAAR MEASURE, THE REGULAR REPRESENTATION AND THE PETER-WEYL THEOREM ", + "bbox": [ + 173, + 188, + 787, + 215 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Now that we have introduced many notions in the representation theory of compact groups, we can formulate the most important result, the Peter-Weyl theorem that we will use throughout this work. In the next section, we will then go through a step-by-step proof of this theorem. The material in this section is based on Nachbin & Bechtolsheim (1965); Kowalski (2014) and Knapp (2002). We thank Stefan Dawydiak for a discussion about the Peter-Weyl theorem over the real numbers (Dawydiak, 2020). ", + "bbox": [ + 173, + 227, + 825, + 310 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We assume that the reader knows what a measure is (Tao, 2013). Let $G$ be a compact group. A standard result is that there exists a measure $\\mu$ on $G$ , called a Haar measure that, among other properties, fulfills the following: ", + "bbox": [ + 174, + 316, + 825, + 359 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "1. $\\mu ( S )$ can be evaluated for all Borel sets $S \\subseteq G$ . Here, the Borel sets form the smallest \nso-called $\\sigma$ -algebra that contains all the open sets. \n2. In particular, we can evaluate $\\mu ( S )$ for all open or closed sets $S \\subseteq G$ . \n3. The Haar measure is normalized: $\\mu ( G ) = 1$ . \n4. $\\mu$ is left and right invariant: $\\mu ( g S ) = \\mu ( S ) = \\mu ( S g )$ for all $g \\in G$ and $S$ measurable. \n5. $\\mu$ is inversion invariant: $\\mu ( S ^ { - 1 } ) = \\mu ( S )$ for all $S$ measurable. ", + "bbox": [ + 210, + 369, + 821, + 474 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "These properties then translate into properties of the associated Haar integral: let $f : G \\to \\mathbb { K }$ be integrable with respect to $\\mu$ , then we obtain: ", + "bbox": [ + 173, + 483, + 823, + 513 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "1. $\\textstyle \\int _ { G } 1 d g = 1$ for the constant function with value 1. \n2. $\\begin{array} { r } { \\int _ { G } f ( h g ) d g = \\int _ { G } f ( g ) d g = \\int _ { G } f ( g h ) d g } \\end{array}$ for all $h \\in G$ . \n3. $\\textstyle \\int _ { G } f ( g ^ { - 1 } ) d g = \\int _ { G } f ( g ) d g$ . ", + "bbox": [ + 210, + 522, + 599, + 582 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Example B.19 (Finite Groups). If $G$ is a finite group with $n$ elements, then the Haar measure is just the normalized counting measure which assigns $\\textstyle \\mu ( g ) = { \\frac { 1 } { n } }$ for all $g \\in G$ . Each function $f : G \\to \\mathbb { K }$ is then integrable, and its integral is just given by ", + "bbox": [ + 174, + 583, + 825, + 626 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/d84f3dc99c79bc077443d4903cf3b36d825f541b81b18042be415dc5e15ecbf3.jpg", + "text": "$$\n\\int _ { G } f ( g ) d g = { \\frac { 1 } { n } } \\sum _ { g \\in G } f ( g ) .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 630, + 586, + 670 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In this special case, one can easily verify all properties of Haar measures and Haar integrals stated above. ", + "bbox": [ + 174, + 674, + 825, + 702 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "With this measure defined, we can already understand why all linear representations on finitedimensional spaces can be considered as unitary: ", + "bbox": [ + 174, + 713, + 823, + 742 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proposition B.20. Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ be a linear representation on a finite-dimensional space $V$ . Then there exists a scalar product $\\langle \\cdot | \\cdot \\rangle _ { \\rho } : V \\times V \\to \\mathbb { K }$ that makes $\\left( V , \\langle \\cdot | \\cdot \\rangle \\right)$ a Hilbert space and such that $\\rho$ becomes a unitary representation with respect to this scalar product. ", + "bbox": [ + 173, + 744, + 825, + 790 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. Since $V$ is finite-dimensional, there is an isomorphism of vector spaces to some $\\mathbb { K } ^ { n }$ . Consequently, there is some scalar product $\\langle \\cdot | \\cdot \\rangle : V \\times V \\to \\mathbb { K }$ that makes $V$ a Hilbert space. However, this scalar product does not necessarily make $\\rho$ a unitary representation. However, we can define $\\langle \\cdot | \\cdot \\rangle _ { \\rho } : V \\times V \\to \\mathbb { K }$ by ", + "bbox": [ + 173, + 804, + 825, + 861 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/0f2a7cd26058ca7503645dce75fbcf18d1099de749268fd4c6f5192acad2009d.jpg", + "text": "$$\n\\langle v | w \\rangle _ { \\rho } : = \\int _ { G } \\langle \\rho ( g ) ( v ) | \\rho ( g ) ( w ) \\rangle d g .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 861, + 620, + 895 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "That this integral exists is due to the continuity of linear representations and since also the scalar product is continuous by Proposition F.38. It can easily be checked that this construction makes $V$ a ", + "bbox": [ + 173, + 896, + 823, + 924 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Hilbert space. And due to the right invariance of the Haar measure, we can check that $\\rho$ is a unitary representation with respect to this scalar product. Namely, for arbitrary $g ^ { \\prime } \\in G$ we have: ", + "bbox": [ + 169, + 103, + 821, + 133 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/aa86cfdbf10cbe80d5cdddc4254caf964cc32c56fe1747d08ce40b8b38637da5.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\left. \\rho ( g ^ { \\prime } ) ( v ) \\big | \\rho ( g ^ { \\prime } ) ( w ) \\right. _ { \\rho } = \\int _ { G } \\left. \\rho ( g ) \\rho ( g ^ { \\prime } ) v \\big | \\rho ( g ) \\rho ( g ^ { \\prime } ) w \\right. d g } } \\\\ { { \\displaystyle \\qquad = \\int _ { G } \\left. \\rho ( g g ^ { \\prime } ) ( v ) \\big | \\rho ( g g ^ { \\prime } ) ( w ) \\right. d g } } \\\\ { { \\displaystyle \\qquad = \\int _ { G } \\left. \\rho ( g ) ( v ) \\big | \\rho ( g ) ( w ) \\right. d g } } \\\\ { { \\displaystyle \\qquad = \\left. v \\big | w \\right. _ { \\rho } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 137, + 687, + 257 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Now, for a measure space $Y$ with corresponding measure $\\mu$ , we can consider the space of squareintegrable functions on $Y$ with values in $\\mathbb { K }$ , denoted $L _ { \\mathbb { K } } ^ { 2 } ( Y )$ (the measure is omitted in the notation since there is usually no ambiguity). In these spaces, functions are identified if they coincide on a set with measure 0. $L _ { \\mathrm { K } } ^ { 2 } ( Y )$ is clearly a vector space over $\\mathbb { K }$ , but it turns out that it can even be considered to be a Hilbert space as follows: ", + "bbox": [ + 173, + 286, + 825, + 357 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/0b90ca2db477e301885768f46813155624873da2c861e483dce06427a728fb07.jpg", + "text": "$$\n\\langle f | g \\rangle : = \\int _ { Y } { \\overline { { f ( y ) } } } g ( y ) d y .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 359, + 583, + 393 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Here, the overline means complex conjugation. The Hilbert space properties are easily verified. ", + "bbox": [ + 173, + 395, + 797, + 411 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In particular, one can consider the space $L _ { \\mathbb { K } } ^ { 2 } ( G )$ of square-integrable functions on the group $G$ itself. Now the claim is that $L _ { \\mathbb { K } } ^ { 2 } ( G )$ can actually be equipped with a prototypical structure as a unitary representation over $G$ which makes this space, in some sense, “universal among unitary representations”. This works with the following canonical representation, called the regular representation: ", + "bbox": [ + 173, + 416, + 825, + 473 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/7c5ff47988b50ee0f0027c170dd47a7acd8dc98acfa7fa08f1373f7565fb7f81.jpg", + "text": "$$\n\\lambda : G \\to \\operatorname { U } ( L _ { \\mathbb { K } } ^ { 2 } ( G ) ) , \\left[ \\lambda ( g ) ( f ) \\right] ( g ^ { \\prime } ) : = f ( g ^ { - 1 } g ^ { \\prime } ) .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 476, + 661, + 494 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "continuity of this map is non-trivial and is, for example, shown in Knapp (2002). However, the more algebraic properties of being a unitary representation are easy to appreciate. First of all, we clearly see that $\\lambda$ is a group homomorphism mapping each group element to a linear automorphism. And finally, the unitarity of this representation can be understood as a direct consequence of the properties of the Haar measure, where we notably make only use of the left-invariance: ", + "bbox": [ + 174, + 497, + 825, + 568 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/bb83fe24b82d6581e12323baf3db5b9ac912924716344a2336562380e7025828.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\langle \\lambda ( g ) ( f ) | \\lambda ( g ) ( h ) \\rangle = \\int _ { G } \\overline { { \\left[ \\lambda ( g ) ( f ) \\right] ( g ^ { \\prime } ) } } \\cdot \\left[ \\lambda ( g ) ( h ) \\right] ( g ^ { \\prime } ) d g ^ { \\prime } } \\\\ { \\displaystyle = \\int _ { G } \\overline { { f ( g ^ { - 1 } g ^ { \\prime } ) } } \\cdot h ( g ^ { - 1 } g ^ { \\prime } ) d g ^ { \\prime } } \\\\ { \\displaystyle = \\int _ { G } \\overline { { f ( g ^ { \\prime } ) } } h ( g ^ { \\prime } ) d g ^ { \\prime } } \\\\ { \\displaystyle = \\langle f | h \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 571, + 692, + 691 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We saw in Example B.9 that $G$ is a homogeneous space with respect to the action on itself. We can now ask whether these constructions can also work if $X$ is an arbitrary homogeneous space of $G$ . This requires us to define a suitable measure on $X$ . This is indeed possible. For a fixed element $x ^ { * } \\in X$ , denote the stabilizer subgroup by $H = G _ { x ^ { * } } \\ \\subseteq { \\cal { G } }$ . Then the Hausdorff property of $X$ allows to write down a homeomorphism between $X$ and $G / H$ , which in turn will allow us to use a canonical measure on $G / H$ that we study below. We denote cosets $g H \\in G / H$ by $[ g ]$ . ", + "bbox": [ + 173, + 693, + 826, + 776 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma B.21. Let $X$ be a homogeneous space of the compact group $G$ and $H$ the stabilizer subgroup of a fixed element $x ^ { * } \\in X$ . Then the map ", + "bbox": [ + 169, + 780, + 823, + 808 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/0111143d99f4a317328c39360320f9aaae79807f433f953bc49ec370e01fda1e.jpg", + "text": "$$\n\\varphi : G / H \\to X , [ g ] \\mapsto g x ^ { * }\n$$", + "text_format": "latex", + "bbox": [ + 406, + 811, + 589, + 829 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "is a homeomorphism. Furthermore, $H$ is topologically closed. ", + "bbox": [ + 173, + 832, + 581, + 848 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. Let ${ \\tilde { \\varphi } } : G \\to X$ , $g \\mapsto g x ^ { * }$ . This map is equal to the composition of the maps $G G \\times X$ , $g \\mapsto \\left( g , x ^ { * } \\right)$ and $G \\times X \\to X$ , $( g , x ) \\mapsto g x$ . Both these are continuous, and thus $\\tilde { \\varphi }$ is continuous as well. Furthermore, note that if $g ^ { - 1 } g ^ { \\prime } \\in H$ , then there is $h \\in H$ such that $g ^ { \\prime } = g h$ , and thus ", + "bbox": [ + 173, + 861, + 826, + 905 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/b84f0e92f2ada9f5eb2fcf1d76b684c4a30e92193d7be4e2b4f270d397e0bdbe.jpg", + "text": "$$\n\\tilde { \\varphi } ( g ^ { \\prime } ) = \\tilde { \\varphi } ( g h ) = ( g h ) x ^ { * } = g ( h x ^ { * } ) = g x ^ { * } = \\tilde { \\varphi } ( g )\n$$", + "text_format": "latex", + "bbox": [ + 328, + 909, + 669, + 925 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "which means that by Proposition F.12, the map $\\varphi : G / H \\to X , [ g ] \\mapsto g x ^ { * }$ is a well-defined continuous map. It is surjective since the action is transitive by definition of a homogeneous space. Furthermore, it is injective since if $g x ^ { * } = g ^ { \\prime } x ^ { * }$ then $x ^ { * } = ( g ^ { - 1 } g ^ { \\prime } ) x ^ { * }$ and thus $g ^ { - 1 } { \\bar { g } } ^ { \\prime } \\in H$ , which means $[ g ] = [ g ^ { \\prime } ]$ . ", + "bbox": [ + 174, + 102, + 825, + 161 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Overall, $\\varphi$ is a continuous bijective map from $G / H$ to $X$ . Furthermore, $G / H$ is compact since it is the continuous image of the compact group $G$ under the projection $G G / H$ , see Proposition F.8. Since $X$ is Hausdorff by definition of homogeneous spaces, $\\varphi$ is a homeomorphism according to Proposition F.9. ", + "bbox": [ + 173, + 165, + 825, + 223 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Now, since $X$ is Hausdorff and $\\varphi$ is a homeomorphism, it follows that $G / H$ is Hausdorff as well. Then, necessarily, $H$ is a topologically closed subgroup of $G$ , see Bourbaki (1998), Chapter III, Section 2.5, Proposition 13. □ ", + "bbox": [ + 173, + 228, + 825, + 272 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Every space $G / H$ where $H$ is topologically closed allows a measure $\\mu$ with similar properties to those of $G$ (Nachbin $\\&$ Bechtolsheim, 1965). Since the stabilizer $H$ is closed and $X \\cong { \\overline { { G } } } / H$ by Lemma B.21, we can do these constructions for $X$ as well, as we outline now. The only properties that we now miss are the right-invariance and inversion-invariance: We simply can’t ask for them since $G$ does not naturally act on $X$ from the right and since we cannot invert elements in $X$ . But left-invariance does hold and this means that ", + "bbox": [ + 173, + 286, + 825, + 371 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/d9c60121ec902c3af8161d7fc9b19637cdc1bd4c1f762700147de119431a3363.jpg", + "text": "$$\n\\lambda : G \\to L _ { \\mathbb { K } } ^ { 2 } ( X ) , [ \\lambda ( g ) ( f ) ] ( x ) : = f ( g ^ { - 1 } x )\n$$", + "text_format": "latex", + "bbox": [ + 351, + 376, + 647, + 393 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "makes $L _ { \\mathbb { K } } ^ { 2 } ( X )$ a unitary representation over $G$ , as can be shown in the exact same way as for $L _ { \\mathbb { K } } ^ { 2 } ( G )$ ", + "bbox": [ + 171, + 398, + 820, + 415 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Let $\\widehat { G }$ be the set of isomorphism classes of irreducible unitary representations over $G$ . Furthermore, let $\\rho _ { l } : G \\to V _ { l }$ be a fixed representative of such an isomorphism class $l \\in \\widehat { G }$ . We write isomorphism classes as $\" l \"$ (and later also $j$ and $J$ ) in order to bring to mind quantum numbers used in quantum mechanics. Recall from linear algebra that a countable sum of subspaces of a vector space is called direct if no nontrivial subspace of any of the considered spaces is contained in the sum of all the other considered spaces.9 Furthermore, recall that two subspaces $U , W \\subseteq V$ of a Hilbert space $V$ are called perpendicular or orthogonal if $\\langle u \\vert w \\rangle = 0$ for all $u \\in U$ and $w \\in W$ . We then write $U \\perp W$ . We can now formulate the Peter-Weyl theorem. Intuitively, it says that $L _ { \\mathbb { K } } ^ { 2 } ( X )$ splits into an orthogonal direct sum of the irreducible unitary representations, where each irreducible unitary representation appears maximally as often as its own dimension (and may not appear at all): ", + "bbox": [ + 173, + 422, + 825, + 565 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Theorem B.22 (Peter-Weyl Theorem). Let $G$ be a compact group. Let $X$ be a homogeneous space. There are numbers $m _ { l } \\in \\mathbb { N } _ { \\geq 0 }$ for all $l \\in \\widehat { G }$ and closed-invariant subspaces $V _ { l i } \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( X )$ for all $l \\in \\widehat { G }$ and $i \\in \\{ 1 , \\ldots , m _ { l } \\}$ such that the following hold: ", + "bbox": [ + 174, + 569, + 826, + 618 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "1. $V _ { l i } \\cong V _ { l }$ as unitary representations for all i and $l$ ", + "bbox": [ + 214, + 628, + 557, + 643 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "2. $m _ { l } \\leq \\dim ( V _ { l } ) < \\infty$ for all $l$ . ", + "bbox": [ + 212, + 651, + 426, + 667 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "3. $V _ { l i } \\perp V _ { l ^ { \\prime } j }$ whenever ${ \\mathit { l } } \\neq { \\mathit { l } } ^ { \\prime }$ or $i \\neq j$ ", + "bbox": [ + 210, + 675, + 468, + 693 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "4. $\\textstyle \\bigoplus _ { l \\in { \\widehat { G } } } \\bigoplus _ { i = 1 } ^ { m _ { l } } V _ { l i }$ is topologically dense in $L _ { \\mathbb { K } } ^ { 2 } ( X )$ , written $L _ { \\mathbb { K } } ^ { 2 } ( X ) = { \\widehat { \\bigoplus } } _ { l \\in { \\widehat { G } } } \\bigoplus _ { i = 1 } ^ { m _ { l } } V _ { l i }$ ", + "bbox": [ + 210, + 702, + 797, + 722 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Now additionally consider $G$ as a homogeneous space of itself. Then the same holds for $L _ { \\mathbb { K } } ^ { 2 } ( G )$ as well, with numbers $n _ { l } \\le \\dim ( V _ { l } ) < \\infty$ . We additionally have the following: ", + "bbox": [ + 173, + 733, + 823, + 762 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "1. $m _ { l } \\le n _ { l }$ . \n2. If $\\mathbb { K } = \\mathbb { C } ,$ , then $n _ { l } = \\dim ( V _ { l } )$ . ", + "bbox": [ + 210, + 772, + 431, + 813 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Note that the representative $V _ { l }$ is not assumed to be embedded in $L _ { \\mathbb { K } } ^ { 2 } ( X )$ . It is just isomorphic, as a unitary representation, to each of the $V _ { l i } \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( X )$ . ", + "bbox": [ + 174, + 821, + 823, + 852 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Example B.23. For $G = \\mathrm { S O } ( 2 )$ and $\\mathbb { K } = \\mathbb { C }$ we have $L _ { \\mathbb { C } } ^ { 2 } ( \\mathrm { S O ( 2 ) } ) = \\widehat { \\oplus } _ { l \\in \\mathbb { Z } } V _ { l 1 }$ and all irreducible representations $V _ { l }$ are 1-dimensional. ", + "bbox": [ + 176, + 856, + 820, + 888 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "For $G = \\mathrm { S O ( 2 ) }$ and $\\mathbb { K } = \\mathbb { R }$ , we obtain $L _ { \\mathbb { R } } ^ { 2 } ( \\mathrm { S O } ( 2 ) ) = \\widehat { \\bigoplus } _ { l \\geq 0 } V _ { l 1 }$ , and all irreducible representations $V _ { l }$ with $l \\geq 1$ are two-dimensional, whereas $V _ { 0 }$ is one-dimensional. Thus, here we see an example where the multiplicity of most irreducible representations in the regular representation is 1 and therefore smaller than their dimension, which cannot happen for representations over the complex numbers. ", + "bbox": [ + 173, + 101, + 825, + 175 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Both of these results are standard results in harmonic analysis. These examples are discussed in more detail, especially with respect to their applications in deep learning, in Section E.1 and E.2. ", + "bbox": [ + 176, + 183, + 823, + 212 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "B.2 A PROOF OF THE PETER-WEYL THEOREM ", + "text_level": 1, + "bbox": [ + 174, + 227, + 511, + 242 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "This section presents a proof of the Peter-Weyl theorem, as formulated in Theorem B.22. We mostly skip the analytical parts of the proof,10 since they are well-presented in the literature and clearly work over both the real and complex numbers. However, the more algebraic parts of the proof usually make use of the property of the complex numbers to be algebraically closed, which does not hold for the real numbers. This is invoked usually both in the proof of a version of Schur’s lemma, as well as in proving Schur’s orthogonality. We therefore carefully adapt the proof of the Peter-Weyl theorem in the literature so that it also works over the real numbers, and formulate and prove versions of Schur’s Lemma B.29 and Schur’s orthogonality B.30 that work in general. ", + "bbox": [ + 174, + 253, + 825, + 366 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "This section can be skipped if the interest is mainly in the applications of the Peter-Weyl theorem. \nIn this case, the reader is advised to directly move on to Chapter C. ", + "bbox": [ + 176, + 372, + 820, + 400 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "We note the following convention that applies to this section: for all unitary representations $\\rho : G $ $\\mathrm { U } ( V )$ that we consider here, $V$ is a Hilbert space (instead of just a pre-Hilbert space). ", + "bbox": [ + 176, + 406, + 820, + 436 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "B.2.1 DENSITY OF MATRIX COEFFICIENTS ", + "text_level": 1, + "bbox": [ + 176, + 450, + 483, + 465 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "An important ingredient in the construction of the spaces $V _ { l i }$ that appear in the formulation of the Peter-Weyl Theorem B.22 are matrix coefficients, which together generate those spaces in case that one considers the regular representation on $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . ", + "bbox": [ + 174, + 474, + 826, + 518 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Definition B.24 (Matrix Coefficients). Let $\\rho : G \\to \\operatorname { U } ( V )$ be a unitary representation. A matrix coefficient is any function of the form ", + "bbox": [ + 173, + 520, + 823, + 549 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/e7723529c9e28f39d1c9609e3c2bf4cf53cdd3693b70286a675bf381c3b02e03.jpg", + "text": "$$\n\\rho ^ { u v } : G \\to \\mathbb { K } , g \\mapsto { \\overline { { \\langle u | \\rho ( g ) ( v ) \\rangle } } }\n$$", + "text_format": "latex", + "bbox": [ + 390, + 556, + 607, + 575 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "for arbitrary $u , v \\in V$ ", + "bbox": [ + 174, + 580, + 320, + 595 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "The term “matrix coefficient” comes from the analogy to matrix elements of linear maps between pre-Hilbert spaces of which orthonormal bases are fixed. Later, in Definition D.9 we will also define the notion of “matrix elements” separately. The term “matrix coefficient” only applies to unitary representations. ", + "bbox": [ + 173, + 606, + 825, + 662 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Remark B.25. By definition of linear representations, the function $g \\mapsto \\rho ( g ) ( v )$ is continuous. Thus, since scalar products of Hilbert spaces are also continuous as functions on $V \\times V$ , see Proposition F.38, every matrix coefficient $\\rho ^ { u v } : G \\to \\mathbb { K }$ is continuous. As a continuous function on a compact space, it is of course also square-integrable, i.e., $\\rho ^ { u v } \\in L _ { \\mathbb { K } } ^ { 2 } ( G )$ . The Peter-Weyl theorem basically asserts that these matrix coefficients can be considered as the building blocks of all square-integrable functions. ", + "bbox": [ + 173, + 665, + 825, + 747 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Furthermore, one may wonder why there is a complex conjugation in the definition. The reason for this is that, otherwise, the isomorphism that we will construct in Proposition B.35 is not linear but conjugate linear. The reason why this can nevertheless be called a matrix coefficient is that this actually is the matrix coefficient (without complex conjugation) on a conjugate Hilbert space, as explained in the next Proposition, which we took from Williams (1991). ", + "bbox": [ + 174, + 755, + 825, + 825 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proposition B.26. Let $\\rho : G \\to \\operatorname { U } ( V )$ be a unitary representation on a Hilbert space $V$ with scalar multiplication $\\cdot _ { V }$ and scalar product $\\langle \\cdot | \\cdot \\rangle _ { V }$ . We have the following: ", + "bbox": [ + 171, + 828, + 823, + 858 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "1. ${ \\tilde { V } } : = V$ (equality as abelian groups) with $\\alpha \\cdot _ { \\tilde { V } } v : = \\overline { { \\alpha } } \\cdot _ { V } \\ i$ v and $\\langle u | v \\rangle _ { \\tilde { V } } : = \\overline { { \\langle u | v \\rangle } }$ is again $a$ Hilbert space, the so-called conjugate Hilbert space of $V$ . ", + "bbox": [ + 209, + 871, + 823, + 900 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "2. ${ \\tilde { \\rho } } : G \\to \\operatorname { U } ( { \\tilde { V } } )$ with $\\tilde { \\rho } ( g ) : = \\rho ( g )$ is again a unitary representation. ", + "bbox": [ + 210, + 102, + 673, + 119 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "3. For the matrix coefficients, we have $\\tilde { \\rho } ^ { u v } ( g ) = \\overline { { \\rho ^ { u v } ( g ) } }$ . ", + "bbox": [ + 210, + 128, + 586, + 146 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. All these assertions are easy to check. As a demonstration, we do 3: ", + "bbox": [ + 173, + 162, + 668, + 178 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/2990ecc36b7f97fc73c5313c40b2de62abfc07dfef6596ec3cb2ece6ff90987c.jpg", + "text": "$$\n\\widetilde { \\rho } ^ { u v } ( g ) = \\overline { { \\langle u | \\widetilde { \\rho } ( g ) ( v ) \\rangle } } _ { \\widetilde { V } } = \\overline { { \\overline { { \\langle u | \\rho ( g ) ( v ) \\rangle } } } } _ { V } = \\overline { { \\rho ^ { u v } ( g ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 185, + 668, + 208 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "That’s what we wanted to show. ", + "bbox": [ + 174, + 214, + 385, + 229 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "As a consequence of this proposition, the matrix coefficient $\\rho ^ { u v } ( g )$ is equal to $\\overline { { \\tilde { \\rho } ^ { u v } ( g ) } }$ , thus being a “matrix coefficient without complex conjugation above the scalar product” of the conjugate unitary representation. ", + "bbox": [ + 174, + 246, + 823, + 290 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Theorem B.27 (Density of Matrix Coefficients). The linear span of the matrix-coefficients of finitedimensional, unitary, irreducible representations of $G$ are dense in $L _ { \\mathbb { K } } ^ { 2 } ( G )$ for all compact groups $G$ . ", + "bbox": [ + 173, + 292, + 825, + 335 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. For $\\mathbb { K } = \\mathbb { C }$ , this is shown in Knapp (2002). The same proof, without adaptions, also works for $\\mathbb { K } = \\mathbb { R }$ . Note that the cited proof uses a definition of matrix coefficients without the complex conjugation. However, Proposition B.26 shows those span the same space, and thus we can apply it to our situation. □ ", + "bbox": [ + 173, + 353, + 825, + 410 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "B.2.2 SCHUR’S LEMMA, SCHUR’S ORTHOGONALITY AND CONSEQUENCES ", + "text_level": 1, + "bbox": [ + 176, + 428, + 707, + 444 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "In this section, we state and prove versions of Schur’s lemma and Schur’s Orthogonality (Knapp, 2002) that are valid for both $\\mathbb { K } = \\mathbb { R }$ and $\\mathbb { K } = \\mathbb { C }$ . ", + "bbox": [ + 173, + 452, + 823, + 481 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Lemma B.28. Let $\\rho : G \\to \\operatorname { U } ( V )$ and $\\rho ^ { \\prime } : G \\to \\operatorname { U } ( V ^ { \\prime } )$ be unitary representations. Furthermore, let $f : V \\to V ^ { \\prime }$ be an intertwiner. Then the adjoint $f ^ { * } : V ^ { \\prime } \\to V$ is also an intertwiner. ", + "bbox": [ + 173, + 486, + 823, + 515 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. The adjoint $f ^ { * } : V ^ { \\prime } \\to V$ is the unique continuous linear function from $V ^ { \\prime }$ to $V$ such that, for all $v \\in V$ and $v ^ { \\prime } \\in V ^ { \\prime }$ , we have ", + "bbox": [ + 173, + 531, + 823, + 560 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/d0d11a5ec3e05b43ea776f90767969fe723ea8d7ea79f2fee89d3284ffb77812.jpg", + "text": "$$\n\\langle f ( v ) | v ^ { \\prime } \\rangle = \\langle v | f ^ { * } ( v ^ { \\prime } ) \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 416, + 568, + 580, + 587 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "This always exists according to Definition F.42. Note that with $f$ being an intertwiner and using the unitarity of the representations, we obtain for all $g \\in G , v \\in V$ and $v ^ { \\prime } \\in V ^ { \\prime }$ : ", + "bbox": [ + 171, + 593, + 825, + 622 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/028739d5286e798af141634239d1cc5b2a0543483c7b3db00d320cec423c85f7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\langle v | \\rho ( g ) f ^ { * } ( v ^ { \\prime } ) \\rangle = \\langle \\rho ( g ^ { - 1 } ) ( v ) | f ^ { * } ( v ^ { \\prime } ) \\rangle } \\\\ & { \\qquad = \\langle f \\rho ( g ^ { - 1 } ) ( v ) | v ^ { \\prime } \\rangle } \\\\ & { \\qquad = \\langle \\rho ^ { \\prime } ( g ^ { - 1 } ) f ( v ) | v ^ { \\prime } \\rangle } \\\\ & { \\qquad = \\langle f ( v ) | \\rho ^ { \\prime } ( g ) ( v ^ { \\prime } ) \\rangle } \\\\ & { \\qquad = \\langle v | f ^ { * } \\rho ^ { \\prime } ( g ) ( v ^ { \\prime } ) \\rangle } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 370, + 627, + 625, + 728 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "from which we deduce $\\rho ( g ) f ^ { * } = f ^ { * } \\rho ^ { \\prime } ( g )$ from Proposition F.45 for all $g \\in G$ , i.e., $f ^ { * }$ is an intertwiner. ", + "bbox": [ + 174, + 732, + 820, + 762 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Lemma B.29 (Schur’s Lemma for unitary Representations). Assume $\\rho : G \\to \\operatorname { U } ( V )$ and $\\rho ^ { \\prime } : \\mathbf { \\tau }$ $G \\mathrm { U } ( V ^ { \\prime } )$ are irreducible unitary representations with $V$ finite-dimensional. Also assume that $f : V \\to V ^ { \\prime }$ is an intertwiner. Then either $f ~ = ~ 0$ or there is $\\mu \\in \\mathbb { R } _ { > 0 }$ such that $\\mu f$ is an isomorphism. ", + "bbox": [ + 173, + 771, + 825, + 829 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. For this proof, we follow the exposition of Tao (2011). We thank Terrence Tao for confirming in the discussion below his blogpost that this lemma can also be proven over the real numbers. ", + "bbox": [ + 173, + 845, + 823, + 875 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Let $f ^ { * } : V ^ { \\prime } \\to V$ be the adjoint of $f$ , which is also an intertwiner by Lemma B.28. Now, set $\\varphi : = f ^ { * } \\circ f : V \\to V$ . As a composition of intertwiners, $\\varphi$ is also an intertwiner. Furthermore, for arbitrary composable continuous linear functions between Hilbert spaces one always has $( g \\circ$ $h ) ^ { * } = h ^ { * } \\circ g ^ { * }$ and $\\left( g ^ { * } \\right) ^ { * } = g$ , which easily follows from the definition and uniqueness of adjoints. Consequently, we have ", + "bbox": [ + 173, + 881, + 825, + 924 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 102, + 823, + 132 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/498920e939f1d7e2c60d174ded97c26b0c3610a8e0dc2f3cf04894dc203f9c00.jpg", + "text": "$$\n\\varphi ^ { \\ast } = ( f ^ { \\ast } \\circ f ) ^ { \\ast } = f ^ { \\ast } \\circ ( f ^ { \\ast } ) ^ { \\ast } = f ^ { \\ast } \\circ f = \\varphi ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 135, + 647, + 152 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "and so $\\varphi$ is self-adjoint. Thus, $\\langle \\varphi ( v ) | w \\rangle = \\langle v | \\varphi ( w ) \\rangle$ for all $v , w \\in V$ , from which we conclude that the matrix of $\\varphi$ corresponding to any orthonormal basis of $V$ is Hermitian or, if $\\mathbb { K } = \\mathbb { R }$ , even symmetric. Such an orthonormal basis exists by Proposition F.41. From the Spectral Theorem for Hermitian or symmetric matrices (Horn & Johnson, 2012) we conclude that $\\varphi$ is unitarily (or for real matrices: orthogonally) diagonalizable with only real eigenvalues. Thus, there is an orthogonal decomposition of $V$ into eigenspaces: $V = \\oplus _ { \\lambda }$ eigenvalue $E _ { \\lambda } ( \\varphi )$ . ", + "bbox": [ + 173, + 155, + 825, + 242 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Let $E _ { \\lambda } ( \\varphi )$ be any eigenspace. We now claim that it is an invariant subspace of $\\rho$ . Indeed, for all $g \\in G$ and $v \\in E _ { \\lambda } ( \\varphi )$ we have since $\\varphi$ is an intertwiner: ", + "bbox": [ + 171, + 248, + 825, + 277 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/66a9781df153ffef2b1624c4c456a299fc02ca6549bc0fb2572963453867fc57.jpg", + "text": "$$\n\\varphi ( \\rho ( g ) ( v ) ) = \\rho ( g ) ( \\varphi ( v ) ) = \\rho ( g ) ( \\lambda v ) = \\lambda \\rho ( g ) ( v ) .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 280, + 668, + 297 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Since $V$ is finite-dimensional, $E _ { \\lambda } ( \\varphi )$ is topologically closed by Proposition F.47, and since $V$ is irreducible, we necessarily have $E _ { \\lambda } ( \\varphi ) = 0$ or $\\bar { E } _ { \\lambda } ( \\varphi ) = V$ . Since not all eigenspaces can be zero, we conclude that there is an eigenvalue $\\lambda$ with $E _ { \\lambda } ( \\varphi ) = V$ , meaning $\\varphi = \\lambda \\mathrm { i d } _ { V }$ . ", + "bbox": [ + 176, + 301, + 823, + 344 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Assume $f \\neq 0$ . We now claim that $\\lambda > 0$ . Indeed, note that for all $v \\in V$ we have ", + "bbox": [ + 173, + 349, + 712, + 364 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/27da126f4846184ce9769c18b8ce2d17766c935116ea2efbf94b5cbbe03ae1e0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda \\| v \\| ^ { 2 } = \\langle \\varphi ( v ) | v \\rangle } \\\\ & { \\qquad = \\langle f ^ { * } \\circ f ( v ) | v \\rangle } \\\\ & { \\qquad = \\langle f ( v ) | f ( v ) \\rangle } \\\\ & { \\qquad = \\| f ( v ) \\| ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 366, + 576, + 441 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Thus, if $v \\in V$ is any vector with $f ( v ) \\neq 0$ , then we obtain $\\begin{array} { r } { \\lambda = \\left( \\frac { \\| f ( v ) \\| } { \\| v \\| } \\right) ^ { 2 } > 0 } \\end{array}$ ", + "bbox": [ + 171, + 444, + 696, + 465 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Now define $g : V \\to V ^ { \\prime }$ as $g = \\lambda ^ { - \\frac { 1 } { 2 } } f$ . $g$ is clearly still an intertwiner. We can also show it is an isometry: ", + "bbox": [ + 176, + 473, + 823, + 503 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/0fdddbf5a00b62670f6dd8cebf072ae9768e360fd8e834e7247d5e55e1af4bce.jpg", + "text": "$$\n\\begin{array} { l } { { \\langle g ( v ) | g ( w ) \\rangle = \\lambda ^ { - 1 } \\langle f ( v ) | f ( w ) \\rangle } } \\\\ { { \\ = \\lambda ^ { - 1 } \\langle \\varphi ( v ) | w \\rangle } } \\\\ { { \\ = \\lambda ^ { - 1 } \\lambda \\langle v | w \\rangle } } \\\\ { { \\ = \\langle v | w \\rangle . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 390, + 503, + 607, + 582 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Note that since $V ^ { \\prime }$ is irreducible and $f ( V ) ~ \\subseteq ~ V ^ { \\prime }$ topologically closed due to $V$ being finitedimensional, we necessarily have that $f$ is surjective. Thus, we have shown that $\\mu f$ with $\\mu : = \\lambda ^ { - \\frac { 1 } { 2 } }$ is an isomorphism of unitary representations. □ ", + "bbox": [ + 173, + 583, + 825, + 630 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proposition B.30 (Schur’s Orthogonality). Let $\\rho : G \\to \\operatorname { U } ( V )$ and $\\rho ^ { \\prime } : G \\to \\operatorname { U } ( V ^ { \\prime } )$ be nonisomorphic irreducible unitary representations of the compact group $G$ , of which at least one is finite-dimensional. Let $\\rho ^ { u v }$ and $\\rho ^ { \\prime } { } ^ { \\bar { u } ^ { \\prime } v ^ { \\prime } }$ be matrix coefficients of them, which are functions in $L _ { \\mathbb { K } } ^ { 2 } ( G )$ due to their continuity. Then they are orthogonal, i.e., $\\left. \\rho ^ { u v } \\Big | \\rho ^ { \\prime u ^ { \\prime } v ^ { \\prime } } \\right. = 0 .$ . ", + "bbox": [ + 173, + 636, + 825, + 707 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proof. Without loss of generality, we can assume $V ^ { \\prime }$ to be finite-dimensional. Assume that $l : V ^ { \\prime } $ $V$ is any linear function. We can associate to it the function $f : V ^ { \\prime } \\to V$ given by ", + "bbox": [ + 173, + 719, + 821, + 750 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/beb3eeb87f6358f68b026b36d89ca499d4ad2cd617ee323ef2e266e1cb900221.jpg", + "text": "$$\nf ( w ^ { \\prime } ) : = \\int _ { G } \\rho ( g ) l \\rho ^ { \\prime } ( g ) ^ { - 1 } w ^ { \\prime } d g .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 752, + 606, + 785 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "For all $h \\in G$ we have ", + "bbox": [ + 173, + 787, + 323, + 801 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/17c7af7ce1a0df66fa1562f1c2fa13836dd6ddaa362788ffbd50fc438bebc6b5.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { \\rho ( h ) f \\rho ^ { \\prime } ( h ) ^ { - 1 } = \\int _ { G } \\rho ( h ) \\rho ( g ) l \\rho ^ { \\prime } ( g ) ^ { - 1 } \\rho ^ { \\prime } ( h ) ^ { - 1 } d g } } \\\\ { \\displaystyle { \\quad = \\int _ { G } \\rho ( h g ) l \\rho ^ { \\prime } ( h g ) ^ { - 1 } d g } } \\\\ { \\displaystyle { \\quad = \\int _ { G } \\rho ( g ) l \\rho ^ { \\prime } ( g ) ^ { - 1 } d g } } \\\\ { \\displaystyle { \\quad = f , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 804, + 663, + 922 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "and thus $\\rho ( h ) f = f \\rho ^ { \\prime } ( h )$ , which means that $f$ is an intertwiner. In this derivation, $\\rho ( h )$ could be put insight the integral since $\\rho ( h )$ is continuous and an integral is a limit over finite sums, which commutes with the continuous $\\rho ( h )$ . By Schur’s Lemma B.29, we necessarily have $f = 0$ . Now look at the specific linear function $l : V ^ { \\prime } \\to V$ given by $l ( w ^ { \\prime } ) : = \\langle v ^ { \\prime } | w ^ { \\prime } \\rangle v$ with the fixed vectors $v , v ^ { \\prime }$ corresponding to the matrix coefficients. We obtain $f = 0$ , for $f$ defined as before, and thus: ", + "bbox": [ + 174, + 102, + 823, + 174 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/9d015937f92827238ee88e903a06b0357d6d162b72e3e12cf7caba6d29207f88.jpg", + "text": "$$\n\\begin{array} { r l } { 0 = \\langle u | f ( u ^ { \\prime } ) \\rangle = u \\Big \\vert \\int _ { G } \\varphi ^ { ( i _ { 2 } ) / i _ { 2 } \\beta ^ { \\prime } } ( g ) ^ { - 1 } ( u ^ { \\prime } ) d g } & { } \\\\ & { = \\int _ { G } u | \\rho ( g ) | i \\rho ^ { \\prime } ( g ) ^ { - 1 } ( u ^ { \\prime } ) \\rangle d g } \\\\ & { = \\int _ { G } u \\Big \\vert \\rho ( g ) [ \\langle v ^ { \\prime } | \\rho ^ { \\prime } ( g ) ^ { - 1 } ( u ^ { \\prime } ) \\rangle v ] d g } \\\\ & { = \\int _ { G } u | \\rho ( g ) ( v ) \\rangle \\cdot v ^ { \\prime } | \\rho ( g ) ^ { - 1 } ( u ^ { \\prime } ) \\rangle d g } \\\\ & { = \\int _ { G } \\frac { 1 } { \\langle u | \\rho ( g ) ( v ) \\rangle } \\cdot \\frac { 1 } { \\langle u ^ { \\prime } | \\rho ( g ) ( v ^ { \\prime } ) \\rangle / d g } } \\\\ & { = \\int _ { G } \\frac { 1 } { \\rho ^ { \\prime \\prime \\prime } ( g ) \\rho ( g ) \\rho ^ { \\prime \\prime } ( g ) d g } } \\\\ & { = \\int _ { G } \\frac { 1 } { \\rho ^ { \\prime \\prime \\prime } ( g ) \\rho ^ { \\prime \\prime \\prime } ( g ) d g } } \\\\ & { = \\langle \\rho ^ { \\mathrm { a r } } | \\rho ^ { \\mathrm { a r } } \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 180, + 681, + 404 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "In this derivation, the integral could be put out of the scalar product since the scalar product is continuous, see Proposition F.38, and since integrals are certain limits over finite sums, with which the scalar product commutes. □ ", + "bbox": [ + 173, + 409, + 825, + 450 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Note that there are more general Schur’s orthogonality relations in the case that $\\mathbb { K } = \\mathbb { C }$ , see Knapp (2002), Corollary 4.10. These then engage with the matrix coefficients of one and the same representation. This, together with a version of Schur’s lemma that only holds over $\\mathbb { C }$ leads to the strengthening of the Peter-Weyl theorem that shows that the multiplicities $n _ { l }$ are given by $\\dim ( V _ { l } )$ . ", + "bbox": [ + 174, + 465, + 825, + 522 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proposition B.31. All irreducible unitary representations of a compact group $G$ are finitedimensional. ", + "bbox": [ + 169, + 526, + 823, + 554 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof. Assume $\\rho : G \\to \\operatorname { U } ( V )$ was an irreducible unitary representation on an infinite-dimensional space $V$ . Let $\\rho ^ { u v }$ be any of its matrix coefficients. By Proposition B.30, and since an infinitedimensional representation can never be isomorphic to a finite-dimensional representation, $\\rho ^ { u v }$ is perpendicular to all matrix coefficients of finite-dimensional irreducible unitary representations. Due to the linearity of the scalar product, $\\rho ^ { u v }$ is perpendicular to the whole linear span of these matrix coefficients and thus to the topological closure of this span. The last step follows from the continuity of the scalar product, see Proposition F.38. By Theorem B.27 this closure is the whole space $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . Therefore, $\\rho ^ { u v }$ is even perpendicular to itself, and thus $\\rho ^ { u v } = 0$ . ", + "bbox": [ + 173, + 569, + 825, + 683 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Overall, for arbitrary $u , v \\in V$ and $g \\in G$ we obtain $0 = \\rho ^ { u v } ( g ) = \\overline { { \\langle u | \\rho ( g ) ( v ) \\rangle } }$ and thus (by setting $u = \\rho ( g ) ( v ) ) \\rho ( g ) ( v ) = 0$ and consequently $\\rho ( g ) = 0$ . We obtain $\\rho = 0$ , a contradiction. Thus infinite-dimensional irreducible unitary representations cannot exist. □ ", + "bbox": [ + 174, + 689, + 825, + 733 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "As a consequence, we mention that the finiteness conditions in Schur’s lemma and Schur’s Orthogonality were not necessary to state since all irreducible unitary representations are finite-dimensional anyway. We obtain from this and from Schur’s Lemma B.29 that isomorphism classes and equivalence classes of irreducible unitary representations are one and the same. ", + "bbox": [ + 174, + 748, + 825, + 804 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "B.2.3 A PROOF OF THE PETER-WEYL THEOREM FOR THE REGULAR REPRESENTATION ", + "bbox": [ + 173, + 819, + 790, + 834 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "In this section, we engage with the Peter-Weyl theorem for the regular representation on $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . The case of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ for a homogeneous space $X$ will be dealt with in Section B.2.4. The core arguments in the proofs of this section are adapted from Williams (1991). ", + "bbox": [ + 174, + 843, + 825, + 887 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "As before, let $\\widehat { G }$ be the set of isomorphism classes of irreducible representations of $G$ . For $l \\in \\widehat { G }$ let $\\rho _ { l }$ bbe a representative for the isomorphism class $l$ . Furthermore, for each $\\rho _ { l } : G \\to \\operatorname { U } ( V _ { l } )$ ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "let $v _ { l } ^ { 1 } , \\ldots , v _ { l } ^ { \\dim ( V _ { l } ) }$ be an arbitrary orthonormal basis, which exists due to Proposition F.41 (mostly written without the superscript, i.e., as $v ^ { 1 } , v ^ { 2 } , \\ldots$ , if the corresponding isomorphism class is clear). Denote $\\rho _ { l } ^ { i j } : = \\rho _ { l } ^ { v ^ { i } v ^ { j } }$ . Remember that matrix coefficients of unitary representations are continuous by Remark B.25, and thus functions in $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . Then, let ${ \\mathcal { E } } \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( G )$ be the linear span of the matrix coefficients of all irreducible unitary representations. In the next Lemma, we want to show that $\\mathcal { E }$ is already spanned by the matrix coefficients corresponding to representatives of isomorphism classes and their orthonormal bases: ", + "bbox": [ + 173, + 99, + 826, + 208 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma B.32. We have ", + "bbox": [ + 174, + 214, + 333, + 229 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/6e08327ba1691863c309e95819f61d92521d3bf829976e1005d0df8279e85a83.jpg", + "text": "$$\n\\mathcal { E } = \\operatorname { s p a n } _ { \\mathbb { K } } \\left\\{ \\rho _ { l } ^ { i j } \\vert l \\in \\widehat { G } , i , j \\in \\{ 1 , \\ldots , \\dim ( V _ { l } ) \\} \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 241, + 671, + 268 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. First, we show that isomorphic representations don’t add distinct matrix coefficients. Thus, let $\\rho \\cong \\rho _ { l }$ and let $f : V \\to V _ { l }$ be the corresponding isomorphism. Then we have $\\rho _ { l } ( g ) \\circ f = f \\circ \\rho ( g )$ and thus, since $f$ is a unitary transformation, $\\rho ( g ) = f ^ { * } \\circ \\bar { \\rho _ { l } } ( g ) \\circ f$ , for all $g \\in G$ , see Proposition F.44. Now let $u , v \\in V$ be arbitrary. We obtain ", + "bbox": [ + 173, + 305, + 826, + 363 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/f1bab2ebb78f2a0b32288985d096fbf400efdeb484f90461dc1666ed9829d512.jpg", + "text": "$$\n\\begin{array} { l } { \\rho ^ { u v } ( g ) = \\overline { { \\langle u | \\rho ( g ) ( v ) \\rangle } } } \\\\ { = \\overline { { \\langle u | f ^ { * } \\rho _ { l } ( g ) f ( v ) \\rangle } } } \\\\ { = \\overline { { \\langle f ( u ) | \\rho _ { l } ( g ) ( f ( v ) ) \\rangle } } } \\\\ { = \\rho _ { l } ^ { f ( u ) f ( v ) } ( g ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 398, + 376, + 599, + 463 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "which proves the first claim. Now we want to show that we only need to consider the $\\rho _ { l } ^ { i j }$ . Thus, let $u , v \\in V _ { l }$ be arbitrary. They allow for linear combinations ", + "bbox": [ + 173, + 477, + 825, + 506 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/cbc4b62ba52764c5b343af9412be385708fee42364f90cd93a05379663abde64.jpg", + "text": "$$\nu = \\sum _ { i } \\lambda ^ { i } v ^ { i } , v = \\sum _ { i } \\mu ^ { i } v ^ { i }\n$$", + "text_format": "latex", + "bbox": [ + 400, + 520, + 598, + 546 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "with coefficients $\\lambda ^ { i } , \\mu ^ { i } \\in \\mathbb { K }$ . We obtain: ", + "bbox": [ + 174, + 559, + 439, + 575 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/857131aff9ce9549358ae5f38775abac808794b8025577d339fd6e8835b8bdc2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\rho _ { l } ^ { u v } ( g ) = \\overline { { \\langle u | \\rho _ { l } ( g ) ( v ) \\rangle } } } \\\\ & { \\qquad = \\sum _ { i } \\sum _ { j } \\lambda ^ { i } \\overline { { \\mu ^ { j } } } \\cdot \\overline { { \\langle v ^ { i } | \\rho _ { l } ( g ) ( v ^ { j } ) \\rangle } } } \\\\ & { \\qquad = \\left( \\sum _ { i } \\sum _ { j } \\lambda ^ { i } \\overline { { \\mu ^ { j } } } \\rho _ { l } ^ { i j } \\right) ( g ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 588, + 635, + 667 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "thus showing that $\\rho _ { l } ^ { u v }$ is in the linear span of the matrix coefficients corresponding to the orthonormal basis. This concludes the proof. □ ", + "bbox": [ + 171, + 678, + 825, + 708 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "For an isomorphism class $l \\in { \\widehat { G } }$ , let $\\mathcal { E } _ { l } : = \\operatorname { s p a n } \\left\\{ \\rho _ { l } ^ { i j } \\mid i , j \\in \\{ 1 , \\dots , \\dim ( V _ { l } ) \\} \\right\\} \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( G )$ be the linear subspace of $\\mathcal { E }$ generated by matrix coefficients corresponding to $l$ . Let furthermore for all $j$ the space $\\mathcal { E } _ { l } ^ { j } \\subseteq \\mathcal { E } _ { l }$ be the subspace generated by all $\\rho _ { l } ^ { i j }$ for $i \\in \\{ 1 , \\dots , \\dim ( V _ { l } ) \\}$ . In the next lemma, we prove that these are actually closed subrepresentations of the regular representation. ", + "bbox": [ + 173, + 746, + 825, + 814 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma B.33. For $j \\in \\{ 1 , \\dots , \\dim ( V _ { l } ) \\}$ , $\\mathcal { E } _ { l } ^ { j }$ is a closed invariant subspace of $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . In particular, $\\mathcal { E } _ { l }$ is a closed invariant subspace of $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . ", + "bbox": [ + 173, + 821, + 823, + 856 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. Closedness follows immediately since this space is finite-dimensional and thus complete, see Proposition F.47. We need to show that $\\lambda ( g ) \\rho _ { l } ^ { i j } \\in \\mathcal { E } _ { l } ^ { j }$ for all $g \\in G$ and all $i , j$ . We can compute ", + "bbox": [ + 173, + 892, + 825, + 925 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "this directly: ", + "bbox": [ + 173, + 103, + 258, + 118 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/8a7556fbba64d725d0c7d98d476998a3471c885d79dd55784e3cebd44b4e1270.jpg", + "text": "$$\n\\begin{array} { r l } { \\left( \\lambda ( g ) \\rho _ { i } ^ { i j } \\right) ( g ^ { \\prime } ) = \\rho _ { i } ^ { i j } ( g ^ { - 1 } g ^ { \\prime } ) } \\\\ & { = \\overline { { \\langle v ^ { \\downarrow } | \\rho _ { i } ( g ^ { - 1 } g ^ { \\prime } ) ( v ^ { i j } ) \\rangle } } } \\\\ & { = \\overline { { \\langle \\rho _ { i } ( g ) ( v ^ { i } ) | \\rho _ { i } ( g ^ { \\prime } ) ( v ^ { j } ) \\rangle } } } \\\\ & { = \\overline { { \\langle \\sum _ { i ^ { \\prime } } \\langle v ^ { i ^ { \\prime } } | \\rho _ { i } ( g ) ( v ^ { i } ) \\rangle } } \\overline { { v ^ { i ^ { \\prime } } | \\rho _ { i } ( g ^ { \\prime } ) ( v ^ { i } ) \\rangle } } } \\\\ & { = \\sum _ { i ^ { \\prime } } \\overline { { \\langle v ^ { \\downarrow } | \\rho _ { i } ( g ) ( v ^ { i } ) \\rangle } } \\cdot \\overline { { \\langle v ^ { i ^ { \\prime } } | \\rho _ { i } ( g ^ { \\prime } ) ( v ^ { j } ) \\rangle } } } \\\\ & { = \\sum _ { i ^ { \\prime } } \\overline { { \\langle v ^ { \\downarrow } | \\rho _ { i } ( g ^ { - 1 } ) ( v ^ { i ^ { \\prime } } ) \\rangle } } \\overline { { \\rho _ { i } ^ { \\prime } \\langle j ^ { \\prime } \\rangle } } } \\\\ & { = \\left( \\sum _ { i ^ { \\prime } } \\rho _ { i ^ { \\prime } } ^ { i i ^ { \\prime } } ( g ^ { - 1 } ) \\rho _ { i ^ { \\prime } } ^ { i ^ { \\prime } } \\right) ( g ^ { \\prime } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 125, + 686, + 303 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "where the coefficients $\\rho _ { l } ^ { i i ^ { \\prime } } ( g ^ { - 1 } )$ do not depend on $g ^ { \\prime }$ . Consequently, $\\lambda ( g ) \\rho _ { l } ^ { i j } \\in \\mathcal { E } _ { l } ^ { j }$ . ", + "bbox": [ + 173, + 309, + 715, + 327 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Lemma B.34. Let $\\rho : G \\to \\operatorname { U } ( V )$ and $\\rho ^ { \\prime } : G \\to \\operatorname { U } ( V ^ { \\prime } )$ be unitary representations, $\\rho$ being irreducible and $V ^ { \\prime } \\ne 0$ . Furthermore, assume that $f : V \\to V ^ { \\prime }$ is a surjective intertwiner. Then $V ^ { \\prime }$ is also irreducible and $f$ an equivalence. ", + "bbox": [ + 173, + 334, + 825, + 378 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof. Assume by contradiction that $V ^ { \\prime }$ is reducible. Thus, there is a nontrivial closed invariant subspace $0 \\subsetneq W \\subsetneq V ^ { \\prime }$ . Now the following can easily be checked: ", + "bbox": [ + 173, + 395, + 823, + 426 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "1. $0 \\subsetneq f ^ { - 1 } ( W ) \\subsetneq V$ . \n2. $f ^ { - 1 } ( W )$ is an invariant subspace of $V$ . \n3. $f ^ { - 1 } ( W )$ is a closed subset of $V$ . ", + "bbox": [ + 210, + 439, + 490, + 506 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Once we have this, we have a contradiction to the fact that $V$ is irreducible. ", + "bbox": [ + 174, + 520, + 666, + 536 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "1 and 2 can be checked by the reader, and 3 follows since $V$ is, as an irreducible representation, finite-dimensional by Proposition B.31 and thus every subspace is closed by Proposition F.47. ", + "bbox": [ + 173, + 541, + 823, + 571 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Therefore, we know that $V ^ { \\prime }$ is irreducible. Now use Schur’s Lemma B.29 to conclude that $f$ , being nonzero, necessarily is an equivalence. □ ", + "bbox": [ + 173, + 577, + 823, + 607 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proposition B.35. There is an equivalence of representations $f _ { l } ^ { j } : V _ { l } \\mathcal { E } _ { l } ^ { j }$ given on the orthonormal basis by $f _ { l } ^ { j } ( v ^ { i } ) = \\rho _ { l } ^ { i j }$ . Consequently, there is an isomorphism $V _ { l } \\cong \\mathcal { E } _ { l } ^ { j }$ of unitary representations. ", + "bbox": [ + 173, + 617, + 825, + 665 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof. We need to show that $f _ { l } ^ { j }$ is equivariant. Using the result of the derivation of Lemma B.33, we compute ", + "bbox": [ + 173, + 680, + 821, + 712 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/11d24fde898cc6c3e236a16b138020b3cd5c8c05683ee5d1ebb4d77ff02b5736.jpg", + "text": "$$\n\\begin{array} { r l } & { f _ { l } ^ { j } ( \\rho _ { l } ( g ) ( v ^ { i } ) ) = f _ { l } ^ { j } \\left( \\sum _ { i ^ { \\prime } } \\langle v ^ { i } | \\rho _ { l } ( g ) ( v ^ { i } ) \\rangle v ^ { i ^ { \\prime } } \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 718, + 651, + 873 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "so $f _ { l } ^ { j } \\circ \\rho _ { l } ( g ) = \\lambda ( g ) \\circ f _ { l } ^ { j }$ for all $g \\in G$ , which is what we wanted to show. That $f$ is an intertwiner also requires it to be continuous: this follows since $V _ { l }$ is finite-dimensional, and so all linear functions on it are continuous. ", + "bbox": [ + 173, + 880, + 825, + 924 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Now, that $f _ { l } ^ { j }$ is even an equivalence follows from Lemma B.34 by noting that $\\mathcal { E } _ { l } ^ { j } \\neq 0$ . Indeed, if it was zero then we would have $\\rho _ { l } ^ { i j } ( g ) = 0$ for all $i$ , and thus $\\rho ( g )$ would not be invertible, in contrast that it is a unitary automorphism. ", + "bbox": [ + 173, + 102, + 825, + 150 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Thus, there is even an isomorphism $V _ { l } \\cong \\mathcal { E } _ { l } ^ { j }$ by Schur’s Lemma B.29. ", + "bbox": [ + 174, + 156, + 630, + 174 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma B.36. Let $\\rho : G \\to \\operatorname { U } ( V )$ be a unitary representation. Let $V _ { 1 } \\subseteq V$ be a subrepresentation. \nThen the orthogonal complement $V _ { 1 } ^ { \\perp }$ is a subrepresentation as well. ", + "bbox": [ + 169, + 180, + 823, + 210 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof. We have $\\langle v \\vert v _ { 1 } \\rangle = 0$ for all $v \\in V _ { 1 } ^ { \\bot }$ and all $v _ { 1 } \\in V _ { 1 }$ . Now, let $g \\in G$ be arbitrary. From the unitarity of $\\rho$ we obtain ", + "bbox": [ + 171, + 223, + 825, + 252 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/2a151138903d41d40e549356aa9b8c5e88b37d0f8e2f7562901c7346f3cf46c9.jpg", + "text": "$$\n\\langle \\rho ( g ) ( v ) | v _ { 1 } \\rangle = \\bigl \\langle v \\bigl | \\rho ( g ^ { - 1 } ) ( v _ { 1 } ) \\bigr \\rangle = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 251, + 622, + 270 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "The last step follows from $\\rho ( g ^ { - 1 } ) ( v _ { 1 } ) \\in V _ { 1 }$ , which holds since $V _ { 1 }$ is a subrepresentation. Overall, this shows $\\hat { \\rho } ( g ) ( v ) \\in V _ { 1 } ^ { \\bot }$ as well, and so this is a subrepresentation. □ ", + "bbox": [ + 173, + 273, + 823, + 303 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma B.37. Let $\\rho : G \\to \\operatorname { U } ( V )$ be a finite-dimensional unitary representation. Furthermore, assume that $W _ { 1 } , W _ { 2 }$ are irreducible subrepresentations. If they are not isomorphic, then they are perpendicular, i.e., $\\langle w _ { 1 } | w _ { 2 } \\rangle = 0$ for all $w _ { 1 } \\in W _ { 1 }$ and $w _ { 2 } \\in W _ { 2 }$ . ", + "bbox": [ + 173, + 310, + 825, + 354 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof. Let $P : V \\to W _ { 1 }$ be the orthogonal projection from $V$ to $W _ { 1 }$ , defined as the adjoint of the canonical inclusion $i : W _ { 1 } \\to V$ , i.e., defined by the property ", + "bbox": [ + 169, + 368, + 823, + 397 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/8a98097ff5dd1d67f8d72848a3fcb319eaa9ceedd15dfaf44af08c7832ef852e.jpg", + "text": "$$\n\\langle w _ { 1 } | P ( v ) \\rangle = \\langle i ( w _ { 1 } ) | v \\rangle = \\langle w _ { 1 } | v \\rangle\n$$", + "text_format": "latex", + "bbox": [ + 385, + 404, + 611, + 421 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "for all $v \\in V$ and $w _ { 1 } \\in W _ { 1 }$ , see also Proposition F.46. We now show that $P$ is equivariant. For all $g \\in G$ , $v \\in V$ and $w _ { 1 } \\in W _ { 1 }$ we have: ", + "bbox": [ + 173, + 428, + 823, + 455 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/4f5dc6dd89f44c500044c6b2b167217477bbb763e255673daf03a0dbf4caf5a0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\langle w _ { 1 } \\vert P ( \\rho ( g ) ( v ) ) \\rangle = \\langle w _ { 1 } \\vert \\rho ( g ) ( v ) \\rangle } \\\\ & { \\qquad = \\langle \\rho ( g ^ { - 1 } ) ( w _ { 1 } ) \\vert v \\rangle } \\\\ & { \\qquad = \\langle \\rho ( g ^ { - 1 } ) ( w _ { 1 } ) \\vert P ( v ) \\rangle } \\\\ & { \\qquad = \\langle w _ { 1 } \\vert \\rho ( g ) ( P ( v ) ) \\rangle , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 361, + 460, + 635, + 545 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "where we used in the third step that $W _ { 1 }$ is a subrepresentation. Since this holds for all $w _ { 1 } \\in W _ { 1 }$ , we obtain $P ( \\rho ( g ) ( v ) ) = \\rho ( g ) ( P \\bar { ( } v ) )$ by Proposition F.45 and overall that $P$ is equivariant. ", + "bbox": [ + 174, + 547, + 825, + 577 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "In particular, also the restriction $P | _ { W _ { 2 } } : W _ { 2 } \\to W _ { 1 }$ is equivariant. Since $W _ { 1 }$ and $W _ { 2 }$ are not isomorphic, we obtain by Schur’s Lemma B.29 that $P | _ { W _ { 2 } } = 0$ , i.e., for all $w _ { 1 } \\in W _ { 1 }$ and $w _ { 2 } \\in W _ { 2 }$ we have $\\langle w _ { 1 } | w _ { 2 } \\rangle = \\langle \\dot { w } _ { 1 } | P | _ { W _ { 2 } } ( w _ { 2 } ) \\rangle = \\langle w _ { 1 } | 0 \\rangle = 0$ . Thus, $W _ { 1 }$ and $W _ { 2 }$ are perpendicular as claimed. □ ", + "bbox": [ + 173, + 583, + 825, + 640 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proposition B.38. Let $\\rho : G \\to \\operatorname { U } ( V )$ be any finite-dimensional unitary representation. Then $V$ decomposes into an orthogonal direct sum ", + "bbox": [ + 173, + 647, + 825, + 676 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/4485812893e2c03933435e610c6bd8bba8f08bca92044a859be297dbaf5f620d.jpg", + "text": "$$\nV = \\bigoplus _ { i = 1 } ^ { n } V _ { i }\n$$", + "text_format": "latex", + "bbox": [ + 457, + 683, + 540, + 724 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "such that $V _ { i } \\subseteq V$ are irreducible subrepresentations of $\\rho$ ", + "bbox": [ + 173, + 729, + 549, + 744 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof. Let $V _ { 1 }$ be any irreducible subrepresentation of $V$ : This can be obtained by noting that if $V$ is not already irreducible (in which case $V _ { 1 } = V ,$ ), then we find a nontrivial subrepresentation $0 \\subsetneq W \\subsetneq V$ . By iteratively proceeding with $W$ , we eventually need to reach an irreducible representation since $V$ is finite-dimensional. ", + "bbox": [ + 173, + 760, + 825, + 816 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Now, let $V _ { 1 } ^ { \\perp }$ be the orthogonal complement of $V _ { 1 }$ . From Lemma B.36 we know that this is a subrepresentation of $V$ . By induction on the dimension of $V$ , and since $V _ { 1 } ^ { \\perp }$ has strictly smaller dimension, we can assume that ${ \\bf \\dot { \\cal V } } _ { 1 } ^ { \\perp }$ already splits into an orthogonal direct sum of irreducible subrepresentations $V _ { 1 } ^ { \\perp } = \\textstyle \\bigoplus _ { i = 2 } ^ { n } V _ { i }$ , and overall, ${ \\bar { V } } { \\bar { = } } \\bigoplus _ { i = 1 } ^ { n } V _ { i }$ is the decomposition we were looking for. □ ", + "bbox": [ + 173, + 823, + 825, + 881 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "The following proposition will not be used now, but we make use of it later when showing that there are only finitely many basis kernels in a steerable CNN for a compact group: ", + "bbox": [ + 173, + 895, + 826, + 924 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proposition B.39 (Krull-Remak-Schmidt Theorem). In the situation of Proposition B.38, the orthogonal direct sum decomposition is essentially unique. That is, the type and multiplicities of the irreducible direct summands is always the same. ", + "bbox": [ + 174, + 103, + 821, + 146 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Proof. If one has one decomposition of $V$ in which an irreducible representation $U$ does not appear, then it cannot appear in any decomposition since $U$ would be perpendicular to all the irreps in the decomposition of $V$ by Lemma B.37 and thus zero. Therefore, the types of irreducible representations is always the same. That the multiplicities are always the same follows by the same argument and for dimension-reasons. □ ", + "bbox": [ + 174, + 162, + 825, + 232 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "We can now finally prove The Peter-Weyl Theorem B.22 for the case that $X = G$ ", + "bbox": [ + 174, + 250, + 710, + 265 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Proof. By Proposition B.38 and Lemma B.33 there is some orthogonal decomposition $\\begin{array} { r l } { \\mathcal { E } _ { l } } & { { } = } \\end{array}$ $\\textstyle \\bigoplus _ { i = 1 } ^ { n _ { i } } V _ { l i }$ into irreducible invariant subspaces. Now assume that there is an $i$ such that $V _ { l i } \\not \\cong V _ { l }$ . By Proposition B.35 this means that $V _ { l i } \\not \\approx \\mathcal { E } _ { l } ^ { j }$ for all $j = 1 , \\dots , \\dim ( V _ { l } )$ . By Lemma B.37 we obtain $V _ { l i } \\perp \\mathcal { E } _ { l } ^ { j }$ for all $j$ and thus, since $\\textstyle \\sum _ { j } \\mathcal { E } _ { l } ^ { j } = \\mathcal { E } _ { l }$ , we obtain $V _ { l i } \\perp \\mathcal { E } _ { l }$ and overall $V _ { l i } = 0$ , a contradiction. ", + "bbox": [ + 173, + 281, + 826, + 359 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Thus, the assumption was wrong and all $V _ { l i }$ in the orthogonal direct sum are isomorphic to $V _ { l }$ . ", + "bbox": [ + 173, + 366, + 790, + 382 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Now let ${ \\mathit { l } } \\neq { \\mathit { l } } ^ { \\prime }$ and $i , j$ be arbitrary. We have $\\mathcal { E } _ { l } \\perp \\mathcal { E } _ { l ^ { \\prime } }$ by Proposition B.30, and thus in particular Vli ⊥ Vl0j . Furthermore, we have nl ≤ dim(Vl) since El = Pdim(j=1 $\\begin{array} { r } { \\mathcal { E } _ { l } \\stackrel { \\cdot } { = } \\sum _ { j = 1 } ^ { \\dim ( V _ { l } ) } \\mathcal { E } _ { l } ^ { j } = \\bigoplus _ { i = 1 } ^ { n _ { l } } \\bar { V } _ { l i } } \\end{array}$ , and $\\mathrm { d i m } ( V _ { l } ) < \\infty$ by Proposition B.31. ", + "bbox": [ + 174, + 387, + 825, + 435 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Moreover, we have $\\begin{array} { r } { \\bigoplus _ { l \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { n _ { l } } V _ { l i } = \\bigoplus _ { l \\in \\widehat { G } } \\mathcal { E } _ { l } = \\mathcal { E } } \\end{array}$ , which is topologically dense in $L _ { \\mathbb { K } } ^ { 2 } ( G )$ by Theorem B.27. ", + "bbox": [ + 174, + 440, + 825, + 470 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Finally, that $n _ { l } = \\dim ( V _ { l } )$ if $\\mathbb { K } = \\mathbb { C }$ follows by invoking a stronger version of Schur’s orthogonality than we have developed, and which works only over the complex numbers (Knapp, 2002). □ ", + "bbox": [ + 173, + 477, + 825, + 506 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "B.2.4 A PROOF OF THE PETER-WEYL THEOREM FOR GENERAL $L _ { \\mathbb { K } } ^ { 2 } ( X )$ ", + "bbox": [ + 173, + 522, + 684, + 537 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Now let $X$ be a homogeneous space of $G$ . Then, as mentioned in Section B.1.3, there is a measure $\\mu$ on $X$ which is left- $G$ -invariant (Nachbin & Bechtolsheim, 1965) in the sense that we have for all $g \\in G$ and all square-integrable functions $f \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ : ", + "bbox": [ + 174, + 546, + 825, + 590 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/7821f645002201c5e2cd1ab7110751fd693fbda308a6b87ab598922dfb8d1f9e.jpg", + "text": "$$\n\\int _ { X } f ( g \\cdot x ) d x = \\int _ { X } f ( x ) d x .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 597, + 598, + 631 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Furthermore, let $\\pi : G \\to X$ be the projection given by $g \\mapsto g x ^ { * }$ for a fixed element $x ^ { * } \\in X$ . One important result is that there is a Fubini-like theorem for evaluation of integrals on $G$ using the invariant measure on $X$ . Namely, for arbitrary $x \\in X$ , let $g ( x ) \\in G$ be any lift, i.e., any element in $G$ with $\\pi ( g ( x ) ) = x$ . This exists since the action is transitive. Let $H : = G _ { x ^ { * } } \\subseteq G$ be the stabilizer subgroup. For a square-integrable function $f : G \\to \\mathbb { K }$ , we can then construct the average $\\operatorname { a v } ( f ) : X \\to \\mathbb { K }$ by ", + "bbox": [ + 173, + 636, + 825, + 719 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/b30b3dd517781a211342d685d075b4dc4af22ae8411e13e82ab5a735a4deb856.jpg", + "text": "$$\n\\operatorname { a v } ( f ) ( x ) : = \\int _ { H } f ( g ( x ) h ) d h ,\n$$", + "text_format": "latex", + "bbox": [ + 398, + 718, + 596, + 752 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "where we integrate using the Haar-measure on $H$ .11 If it is hard to understand why this is called an average, note that $X \\cong G / H$ , i.e., points in $X$ can be interpreted as cosets of $G$ , and then the average just averages over cosets.12 ", + "bbox": [ + 174, + 756, + 825, + 799 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "This construction is well-defined, i.e., does not depend on the specific choice of the lift $g ( x )$ . Indeed, let $g ( x ) ^ { \\prime }$ be another lift of $x$ . Then $g ( x ) ^ { \\prime } = g ( x ) h ^ { \\prime }$ for some $h ^ { \\prime } \\in H$ , since $H$ is the stabilizer ", + "bbox": [ + 176, + 805, + 823, + 835 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "subgroup. Consequently, using the invariance of the Haar measure, we see: ", + "bbox": [ + 174, + 103, + 665, + 118 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/0821100b48209805f065ea9ea4211c75288b7f5e98e771d60d911bd95b1fa662.jpg", + "text": "$$\n\\int _ { H } f ( g ( x ) ^ { \\prime } h ) d h = \\int _ { H } f ( g ( x ) h ^ { \\prime } h ) d h = \\int _ { H } f ( g ( x ) h ) d h ,\n$$", + "text_format": "latex", + "bbox": [ + 305, + 121, + 687, + 154 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "and thus the well-definedness of the average $\\operatorname { a v } ( f ) : X \\to \\mathbb { K }$ . Integration of $f$ on the whole of $G$ is a “complete” average, and thus we can hope that averaging $\\operatorname { a v } ( f )$ leads to this complete integral. This is indeed the case, i.e., $\\operatorname { a v } ( f )$ is square-integrable on $X$ and one has (Nachbin $\\&$ Bechtolsheim, 1965) ", + "bbox": [ + 174, + 156, + 825, + 212 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/39712bad956e3f34448e443caae0be00d26bc5afd9fe40bd71bf88e594654f0e.jpg", + "text": "$$\n\\int _ { G } f ( g ) d g = \\int _ { X } \\operatorname { a v } ( f ) ( x ) d x .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 210, + 599, + 243 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "We will use this important result later in order to see that $L _ { \\mathbb { K } } ^ { 2 } ( X )$ embeds with good properties into $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . ", + "bbox": [ + 174, + 244, + 823, + 275 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "We now want to prove the Peter-Weyl theorem for $L _ { \\mathbb { K } } ^ { 2 } ( X )$ . We first present a general argument showing an orthogonal decomposition of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ into irreducible subspaces, and then use a specific argument to deduce that the multiplicities of irreducible subrepresentations are necessarily bounded by the multiplicities in $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . ", + "bbox": [ + 174, + 280, + 825, + 338 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proposition B.40. Let $\\rho : G \\to \\operatorname { U } ( V )$ be any unitary representation. Then there is a dense subrepresentation which splits as an orthogonal direct sum of irreducible subrepresentations. ", + "bbox": [ + 173, + 339, + 821, + 368 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proof. We sketch the proof in Kowalski (2014), Corollary 5.4.2. In this book, the proof is done only for the complex numbers C, but it is obvious that each step carries over without any changes to arbitrary $\\mathbb { K } \\in \\{ \\mathbb { R } , \\mathbb { C } \\}$ . The rough steps are as follows: ", + "bbox": [ + 176, + 382, + 825, + 426 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "1. From $\\rho$ one builds a function $\\overline { { { \\rho } } } ~ : ~ L _ { \\mathbb { K } } ^ { 2 } ( G ) ~ \\to ~ \\operatorname { H o m } _ { \\mathbb { K } } ( V , V )$ , given by $\\overline { { \\rho } } ( \\varphi ) ( v ) \\ =$ $\\textstyle \\int _ { G } \\varphi ( { \\dot { g } } ) \\rho ( g ) ( v ) d g$ . This is analogous to our construction of kernel operators (special representation operators) from kernels, which we will handle in the next chapter, See Theorem C.7. \n2. Given $v \\in V$ fixed, one obtains the function ${ \\overline { { \\rho } } } ^ { v } : L _ { \\mathbb { K } } ^ { 2 } ( G ) \\to V$ , $\\varphi \\mapsto { \\overline { { \\rho } } } ( \\varphi ) ( v )$ . One can check easily that this is an intertwiner. \n3. For each finite-dimensional subrepresentation $E \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( G )$ , the image ${ \\overline { { \\rho } } } ^ { v } ( E ) \\subseteq V$ is a finite-dimensional subrepresentation of $V$ . \n4. For $v \\ne 0$ , using analytical arguments and the Peter-Weyl theorem for $L _ { \\mathbb { K } } ^ { 2 } ( G )$ , one can prove that there is an $E$ such that ${ \\overline { { \\rho } } } ^ { v } ( E ) \\subseteq V$ is not zero. ", + "bbox": [ + 212, + 439, + 825, + 609 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Having that, one can use Proposition B.38 in order to deduce that $\\overline { { \\rho } } ^ { v } ( E )$ contains an irreducible subrepresentation, and so does $V$ . ", + "bbox": [ + 176, + 622, + 823, + 651 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "With this at hand, one can proceed inductively as follows: Given an irreducible subrepresentation $V _ { 1 } ~ \\subseteq ~ V$ , one can consider the orthogonal complement $V _ { 1 } ^ { \\perp }$ , which is by Lemma B.36 again a subrepresentation of $V$ . Thus, this also has, by the same argument as above, an irreducible subrepresentation $V _ { 2 }$ and so on. By induction (or better: using Zorn’s Lemma), one can then “fill up” $V$ with orthogonal irreducible subrepresentations, deducing the result. □ ", + "bbox": [ + 173, + 657, + 825, + 728 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Consequently, since $L _ { \\mathbb { K } } ^ { 2 } ( X )$ carries a unitary representation of $G$ by $\\left[ \\lambda ( g ) ( \\varphi ) \\right] ( x ) : = \\varphi ( g ^ { - 1 } x )$ , we can deduce that it contains a dense subrepresentation which splits as an orthogonal direct sum of irreducible subrepresentations. But we would like to know more details about this, in particular the multiplicities of the irreps. For this to work, we want to embed $L _ { \\mathbb { K } } ^ { 2 } ( X )$ into $L _ { \\mathbb { K } } ^ { 2 } ( G )$ and thus deduce a more specific result from the decomposition of $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . ", + "bbox": [ + 173, + 742, + 825, + 814 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Let as before $x ^ { * } \\in X$ be an arbitrary point and let $\\pi : G \\to X$ be the projection given by $\\pi ( g ) : =$ $g x ^ { * }$ . Consider the function $\\pi ^ { * } : L _ { \\mathbb { K } } ^ { 2 } { \\dot { ( X ) } } L _ { \\mathbb { K } } ^ { 2 } ( G )$ given by $\\pi ^ { * } ( \\varphi ) : = \\varphi \\circ \\pi$ . It is unclear a priori whether this is well-defined: For example, it might be that an $f : X \\to \\mathbb { K }$ which is zero outside a measure 0 set gets lifted to $\\pi ^ { * } ( f ) : G \\to \\mathbb { K }$ which does not have this property, and thus $\\pi ^ { * }$ would not be an actual function.13 Thus, we need some lemmas: ", + "bbox": [ + 173, + 819, + 825, + 890 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Lemma B.41. Let $f : X \\to \\mathbb { K }$ be square-integrable. Then we have $\\operatorname { a v } ( \\pi ^ { * } ( f ) ) = f$ . ", + "bbox": [ + 171, + 102, + 725, + 119 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Using Eq. (9) and that $H$ is the stabilizer subgroup we compute: ", + "bbox": [ + 173, + 136, + 643, + 152 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/34eb67582d561a8686d9692e0c96317b0517bda3e6f118a4cd63041dc772bdc6.jpg", + "text": "$$\n\\begin{array} { r l } { { \\operatorname { a v } ( \\pi ^ { * } ( f ) ) ( x ) - \\int _ { H } \\pi ^ { * } ( f ) ( g ( x ) h ) d h } } \\\\ & { = \\int _ { H } f ( \\pi ( g ( x ) h ) ) d h } \\\\ & { = \\int _ { H } f ( \\pi ( g ( x ) ) ) d h } \\\\ & { = \\int _ { H } f ( x ) d h } \\\\ & { = \\int _ { H } f ( x ) d h } \\\\ & { = f ( x ) \\int _ { H } 1 d h } \\\\ & { = f ( x ) \\mu ( H ) } \\\\ & { = f ( x ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 374, + 160, + 620, + 363 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma B.42. Let $A \\subseteq X$ be any measurable set. Let $\\mathbf { 1 } _ { A } : X \\to \\{ 0 , 1 \\} \\subseteq \\mathbb { K }$ be its indicator function. Then $\\pi ^ { * } ( \\mathbf { 1 } _ { A } ) = \\mathbf { 1 } _ { \\pi ^ { - 1 } ( A ) }$ . ", + "bbox": [ + 171, + 395, + 826, + 426 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. This can easily be checked. ", + "bbox": [ + 174, + 443, + 403, + 458 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma B.43. Let $\\varphi : X \\to \\mathbb { K }$ be zero outside a measure zero set $A$ . Then $\\pi ^ { * } ( \\varphi )$ is zero outside $\\pi ^ { - 1 } ( A )$ which is also a measure zero set. ", + "bbox": [ + 173, + 469, + 825, + 498 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. If $g \\not \\in \\pi ^ { - 1 } ( A )$ then $\\pi ( g ) \\not \\in A$ and thus: ", + "bbox": [ + 173, + 516, + 483, + 532 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/fb258c2cfdc68304bb1959c1c652221f24990070a5ea1ba64e1db8b6ffda3af8.jpg", + "text": "$$\n0 = \\varphi ( \\pi ( g ) ) = \\pi ^ { * } ( \\varphi ) ( g )\n$$", + "text_format": "latex", + "bbox": [ + 413, + 540, + 584, + 558 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "which proves the first statement. The second is shown as follows using both Lemmas B.41 and B.42 and Eq. (9): ", + "bbox": [ + 171, + 564, + 823, + 593 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/493f53a31e2800b7a9258984c38f6ff9dbe1e8c1f82d6781022c61637abb0849.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mu ( \\pi ^ { - 1 } ( A ) ) = \\displaystyle \\int _ { G } \\mathbf { 1 } _ { \\pi ^ { - 1 } ( A ) } ( g ) d g } \\\\ & { \\quad \\quad \\quad \\quad = \\displaystyle \\int _ { G } \\pi ^ { - } ( \\mathbf { 1 } _ { A } ) ( g ) d g } \\\\ & { \\quad \\quad \\quad = \\displaystyle \\int _ { X } \\arcsin ( \\pi ^ { * } ( \\mathbf { 1 } _ { A } ) ) ( x ) d x } \\\\ & { \\quad \\quad \\quad = \\displaystyle \\int _ { X } \\mathbf { 1 } _ { A } ( x ) d x } \\\\ & { \\quad \\quad \\quad = \\mu ( A ) } \\\\ & { \\quad \\quad = 0 , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 377, + 601, + 619, + 772 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "thus showing what was claimed. ", + "bbox": [ + 173, + 777, + 387, + 792 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Thus, our concern about well-definedness as a function is invalid and we can now prove an embedding result: ", + "bbox": [ + 174, + 810, + 823, + 839 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proposition B.44. $\\pi ^ { * } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to L _ { \\mathbb { K } } ^ { 2 } ( G )$ is a well-defined intertwiner and a unitary transformation, i.e., for all ϕ, ψ ∈ L2K(X ) we have hπ∗(ϕ)|π∗(ψ)iL2K(G) = hϕ|ψiL2K(X). ", + "bbox": [ + 174, + 843, + 820, + 877 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. For well-definedness, we still need to show that $\\pi ^ { * } ( \\varphi )$ is again square-integrable for squareintegrable $\\varphi : X \\to \\mathbb { K }$ . This is indeed the case due to Eq. (9). Namely, let $| \\pi ^ { * } ( \\varphi ) | ^ { 2 } : G \\to { \\bar { \\mathbb { K } } }$ and ", + "bbox": [ + 173, + 895, + 825, + 925 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "consider its average $\\operatorname { a v } ( | \\pi ^ { * } ( \\varphi ) | ^ { 2 } )$ . Clearly, we have $| \\pi ^ { * } ( \\varphi ) | ^ { 2 } = \\pi ^ { * } ( | \\varphi | ^ { 2 } )$ and thus, using Lemma B.41, $\\mathrm { a v } ( | \\pi ^ { * } ( \\varphi ) | ^ { 2 } ) = | \\ddot { \\varphi } | ^ { 2 }$ . We obtain: ", + "bbox": [ + 171, + 102, + 823, + 133 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/7327525bdc79ae0426c1d8f28cd8193c3b2d67b259c2816bd593619c573f3fab.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\int _ { G } | \\pi ^ { * } ( \\varphi ) | ^ { 2 } ( g ) d g = \\int _ { X } \\mathrm { a v } ( | \\pi ^ { * } ( \\varphi ) | ^ { 2 } ) ( x ) d x } \\\\ { \\displaystyle \\qquad = \\int _ { X } | \\varphi ( x ) | ^ { 2 } d x } \\\\ { \\displaystyle \\qquad < \\infty . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 137, + 647, + 223 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Thus, $\\pi ^ { * }$ is not only well-defined but even fulfills $\\| \\pi ^ { * } ( \\varphi ) \\| _ { L _ { \\mathbb { K } } ^ { 2 } ( G ) } = \\| \\varphi \\| _ { L _ { \\mathbb { K } } ^ { 2 } ( X ) }$ , which also shows the continuity of $\\pi ^ { * }$ . With similar arguments, we show that $\\pi ^ { * }$ respects the whole scalar product, i.e., is a uniform transformation: ", + "bbox": [ + 174, + 228, + 825, + 273 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/d5b8bf66253ec63119e49d06dfc65e1bf8c243f511386933f88e812d5a6c6cd7.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\langle \\pi ^ { * } ( \\varphi ) | \\pi ^ { * } ( \\psi ) \\rangle _ { L _ { \\mathrm { R } } ^ { 2 } ( G ) } = \\int _ { G } \\left( \\overline { { \\pi ^ { * } ( \\varphi ) } } \\cdot \\pi ^ { * } ( \\psi ) \\right) ( g ) d g } \\ ~ } \\\\ { { \\displaystyle ~ = \\int _ { X } \\mathrm { a v } \\big ( \\overline { { \\pi ^ { * } ( \\varphi ) } } \\cdot \\pi ^ { * } ( \\psi ) \\big ) ( x ) d x } \\ ~ } \\\\ { { \\displaystyle ~ = \\int _ { X } \\overline { { \\varphi ( x ) } } \\psi ( x ) d x } \\ ~ } \\\\ { { \\displaystyle ~ = \\langle \\varphi | \\psi \\rangle _ { L _ { \\mathrm { R } } ^ { 2 } ( X ) } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 279, + 674, + 401 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "The step from the second to the third line follows as before by noting that $\\overline { { \\pi ^ { * } ( \\varphi ) } } \\cdot \\pi ^ { * } ( \\psi ) = \\pi ^ { * } ( \\overline { { \\varphi } } \\cdot \\psi )$ and invoking Lemma B.41 again. ", + "bbox": [ + 174, + 407, + 821, + 438 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "The linearity of $\\pi ^ { * }$ is obvious, and the equivariance is done as follows: note that for arbitrary $g , g ^ { \\prime } \\in G$ we have $\\pi ( g ^ { - 1 } g ^ { \\prime } ) = ( g ^ { - 1 } g ^ { \\prime } ) x ^ { * } \\stackrel { \\cdot } { = } g ^ { - 1 } ( g ^ { \\prime } x ^ { * } ) = g ^ { - 1 } \\pi ( g ^ { \\prime } )$ and therefore: ", + "bbox": [ + 171, + 443, + 821, + 473 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/58aed321edfc5b64f3120e570dae18912808fb99a31d9575c9f0f2ceac2778e2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left[ \\pi ^ { * } ( \\lambda ( g ) \\varphi ) \\right] ( g ^ { \\prime } ) = ( \\lambda ( g ) \\varphi ) ( \\pi ( g ^ { \\prime } ) ) } \\\\ & { \\qquad = \\varphi ( g ^ { - 1 } \\pi ( g ^ { \\prime } ) ) } \\\\ & { \\qquad = \\varphi ( \\pi ( g ^ { - 1 } g ^ { \\prime } ) ) } \\\\ & { \\qquad = \\pi ^ { * } ( \\varphi ) ( g ^ { - 1 } g ^ { \\prime } ) } \\\\ & { \\qquad = [ \\lambda ( g ) \\pi ^ { * } ( \\varphi ) ] ( g ^ { \\prime } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 372, + 477, + 622, + 577 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Thus, we shown everything which was to show. ", + "bbox": [ + 174, + 580, + 485, + 595 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Thus, $\\pi ^ { * } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to L _ { \\mathbb { K } } ^ { 2 } ( G )$ is an embedding which even preserves the scalar product. We can therefore view $L _ { \\mathbb { K } } ^ { 2 } ( X )$ as a subspace: $L _ { \\mathbb { K } } ^ { 2 } ( X ) \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( G )$ . 14 ", + "bbox": [ + 173, + 609, + 823, + 642 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "We can finally complete the proof of the Peter-Weyl Theorem B.22: ", + "bbox": [ + 173, + 647, + 619, + 662 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Proof of Theorem B.22. Assume that ", + "bbox": [ + 174, + 678, + 419, + 693 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/b76effe371e07e2973e16c98fd5aaa34df65e951063289f843a3b3cba708e1bd.jpg", + "text": "$$\n\\bigoplus _ { l \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { l } } V _ { l i } \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( X ) \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( G )\n$$", + "text_format": "latex", + "bbox": [ + 387, + 698, + 607, + 742 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "is a dense subspace such that the direct sum is orthogonal, where $V _ { l i } \\cong V _ { l }$ for all $l , i$ . This exists by Proposition B.40. ", + "bbox": [ + 173, + 748, + 825, + 776 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Remember that $n _ { l }$ denotes the multiplicity of $V _ { l }$ as a subrepresentation in $L _ { \\mathbb { K } } ^ { 2 } ( G )$ . We now want to show that $m _ { l } \\le n _ { l }$ . Since $V _ { l i }$ is perpendicular to all $\\mathcal { E } _ { l ^ { \\prime } }$ with ${ \\mathit { l } } ^ { \\prime } \\neq { \\mathit { l } }$ by Lemma B.37, $V _ { l i }$ must be contained in the orthogonal complement of $\\oplus _ { l ^ { \\prime } \\neq l } \\mathcal { E } _ { l ^ { \\prime } }$ . This is exactly $\\mathcal { E } _ { l }$ , which we show in a final lemma after this proof. So $V _ { l i } \\subseteq \\mathcal { E } _ { l }$ for all $i$ . Thus, we obtain the result $m _ { l } \\le n _ { l }$ by dimension reasons. This was all there was left to show. □ ", + "bbox": [ + 173, + 782, + 825, + 856 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Proof. We already know $\\mathcal { E } _ { l } \\subseteq \\big ( \\bigoplus _ { l \\neq l ^ { \\prime } \\in \\widehat { G } } \\mathcal { E } _ { l ^ { \\prime } } \\big ) ^ { \\perp }$ from Proposition B.30. Now, assume this inclusion is not an equality. Then there is $v \\notin \\mathcal { E } _ { l }$ such that $v \\in \\left( \\bigoplus _ { l \\neq l ^ { \\prime } \\in \\widehat { G } } \\mathcal { E } _ { l ^ { \\prime } } \\right) ^ { \\perp }$ . The space spanK $( v , \\mathcal { E } _ { l } )$ does contain an orthonormal basis by Proposition F.41, where the procedure of Gram-Schmidt orthonormalization allows starting with an orthonormal basis of $\\mathcal { E } _ { l }$ and to fill it up to one of the whole space spanK $( v , \\mathcal { E } _ { l } )$ . Thus, we can assume $v \\in \\mathcal { E } _ { l } ^ { \\bot }$ as well. Overall, $v \\in \\left( \\bigoplus _ { l ^ { \\prime } \\in \\widehat { G } } \\mathcal { E } _ { l ^ { \\prime } } \\right) ^ { \\perp }$ , and by taking topological closure and using that the scalar product is continuous by Proposition F.38, obtain $v \\in \\bigl ( \\widehat { \\mathbb { O } } _ { l ^ { \\prime } \\in \\widehat { G } } \\mathcal { E } _ { l ^ { \\prime } } \\bigr ) ^ { \\perp } = ( L _ { \\mathbb { K } } ^ { 2 } ( G ) ) ^ { \\perp }$ by the Peter-Weyl theorem for the regular representation. This means $v = 0 \\in \\mathcal { E } _ { l }$ , a contradiction to $v \\notin \\mathcal { E } _ { l }$ . ", + "bbox": [ + 173, + 101, + 825, + 236 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Thus, our assumption is wrong and such a vector $v$ cannot exist. We obtain the equality as desired. □ ", + "bbox": [ + 171, + 242, + 823, + 265 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "C THE CORRESPONDENCE BETWEEN STEERABLE KERNELS ANDREPRESENTATION OPERATORS", + "text_level": 1, + "bbox": [ + 173, + 301, + 725, + 335 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "In this chapter, we formulate and prove Theorem C.7, which gives a precise one-to-one correspondence between steerable kernels on the one hand, and certain representation operators which we call kernel operators on the other hand. Representation operators are a representation-theoretic abstraction of the scalar, vector and tensor operators from physics, that were explained in Section 2. The correspondence will allow us to prove a Wigner-Eckart theorem for steerable kernels in Chapter D and, ultimately, to obtain a complete description of steerable kernel bases. We formulate the correspondence in Section C.1, while Section C.2 gives a detailed and rigorous proof of it. ", + "bbox": [ + 174, + 356, + 825, + 454 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "As in Chapter B, K is either of the two fields R or C. ", + "bbox": [ + 176, + 460, + 522, + 476 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "C.1 FUNDAMENTALS OF THE CORRESPONDENCE ", + "text_level": 1, + "bbox": [ + 178, + 503, + 524, + 517 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "In Section C.1, we formulate the correspondence between steerable kernels and special representation operators that we name kernel operators. We do this by first studying steerable CNNs and the kernel constraint in Section C.1.1, which progressively leads us to consider steerable kernels on homogeneous spaces of general compact groups in Section C.1.2. This abstract formulation of steerable kernels will show apparent similarities to the concept of representation operators in Section C.1.3. We study them in purely representation-theoretic terms in Section C.1.4. However, they importantly differ in the fact that steerable kernels are not linear, whereas representation operators are – this is a difference that we need to bridge. Finally, after defining kernel operators as special representation operators, we give the formulation of the correspondence in Theorem C.7 in Section C.1.5 and shortly give some intuitions about why it is true. ", + "bbox": [ + 173, + 534, + 825, + 672 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "C.1.1 STEERABLE KERNELS AND THE RESTRICTION TO HOMOGENEOUS SPACES ", + "bbox": [ + 176, + 698, + 745, + 713 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "The concept of steerable CNNs outlined here follows (Weiler et al., 2018a; Weiler & Cesa, 2019). In a nutshell, they work as follows: ", + "bbox": [ + 173, + 727, + 823, + 755 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "The network is supposed to process feature fields $f : \\mathbb { R } ^ { d } \\mathbb { K } ^ { c }$ with $d \\in \\mathbb { N }$ . $c$ is the dimension of the features themselves, i.e., the number of channels. For example, planar RGB-images correspond to the case $d = 2$ and $c = 3$ . ", + "bbox": [ + 174, + 761, + 825, + 804 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Furthermore, a compact group $G$ (Definition B.4) is considered that acts on $\\mathbb { R } ^ { d }$ , for example, the special orthogonal group $\\mathrm { S O } ( d )$ , the orthogonal group $\\mathrm { O } ( d )$ or the finite groups $\\mathrm { C } _ { N }$ or $\\mathrm Ḋ \\mathrm Ḋ \\mathrm Ḋ \\ Ḍ Ḍ _ { N }$ if $d = 2$ . 15 Then for each layer, the input and output features have a certain type, i.e., representation, which may differ from layer to layer. That is, the input (and output as well) consists of a function $f : \\mathbb { R } ^ { d } \\mathbb { K } ^ { \\dot { c } }$ , and $G$ acts on $\\mathbb { K } ^ { c }$ with a linear representation $\\rho$ , see Definition B.10. This action induces an action of the semi-direct product $( \\mathbb { R } ^ { d } , + ) \\rtimes G$ on the space of all signals,16 where $t \\in ( { \\mathbb { R } } ^ { d } , + )$ and $g \\in G$ : ", + "bbox": [ + 174, + 810, + 825, + 881 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 102, + 823, + 119 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/a2d03d504cabd58558ae54b1f5d204b88c174639e448745e1ccd680ad4cd54f6.jpg", + "text": "$$\n\\left( \\left[ \\operatorname { I n d } _ { G } ^ { \\mathbb { R } ^ { d } \\times G } \\rho \\right] ( t g ) \\cdot f \\right) ( x ) : = \\rho ( g ) \\cdot f ( g ^ { - 1 } ( x - t ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 126, + 673, + 151 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Let the kernel that “maps” between the layers by convolution17 be given by a function ", + "bbox": [ + 174, + 159, + 736, + 174 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/a5c625c6ab36c4f6ec3b23c60c559acfdf475f9174a1ff4885f954436c5f4788.jpg", + "text": "$$\nK : \\mathbb { R } ^ { d } \\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } } .\n$$", + "text_format": "latex", + "bbox": [ + 426, + 181, + 573, + 198 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "That is, for an input $f _ { \\mathrm { i n } } : \\mathbb { R } ^ { d } \\mathbb { K } ^ { c _ { \\mathrm { i n } } }$ , the output $f _ { \\mathrm { o u t } } : \\mathbb { R } ^ { d } \\mathbb { K } ^ { c _ { \\mathrm { o u t } } }$ is given by ", + "bbox": [ + 173, + 207, + 696, + 223 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/8bc42004becefc654b417740374b0152ad5cf897af3ad0bbd9853470b97f94a8.jpg", + "text": "$$\nf _ { \\mathrm { o u t } } ( x ) = \\left[ K \\star f _ { \\mathrm { i n } } \\right] ( x ) = \\int _ { \\mathbb { R } ^ { d } } K ( y ) f _ { \\mathrm { i n } } ( x + y ) d y ,\n$$", + "text_format": "latex", + "bbox": [ + 330, + 229, + 666, + 262 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "where $K ( y ) \\in \\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ acts for any $\\boldsymbol { y } \\in \\mathbb { R } ^ { d }$ as a linear transformation from $\\mathbb { K } ^ { c _ { \\mathrm { i n } } }$ to $\\mathbb { K } ^ { c _ { \\mathrm { o u t } } }$ ", + "bbox": [ + 169, + 270, + 772, + 286 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "The goal is now to find kernels $K$ such that convolution with these kernels commutes with the induced actions on the input and output fields. That is, for all input fields $f _ { \\mathrm { i n } }$ and for all $t \\in \\mathbb { R } ^ { d }$ and $g \\in G$ we want the following property: ", + "bbox": [ + 173, + 292, + 825, + 335 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/74cc9f90188bb8b01d3a5725cfd56ac391662d90875d9bf88f91469d7a04896c.jpg", + "text": "$$\n\\begin{array} { r } { K \\star \\left( \\left[ \\mathrm { I n d } _ { G } ^ { \\mathrm { R } ^ { d } \\rtimes G } \\rho _ { \\mathrm { i n } } \\right] ( t g ) \\cdot f _ { \\mathrm { i n } } \\right) = \\left[ \\mathrm { I n d } _ { G } ^ { \\mathrm { R } ^ { d } \\rtimes G } \\rho _ { \\mathrm { o u t } } \\right] ( t g ) \\cdot \\left( K \\star f _ { \\mathrm { i n } } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 342, + 725, + 367 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "It was shown in Weiler et al. (2018a) that a kernel $K$ has this equivariance property if and only if the kernel satisfies a certain constraint. We are rederiving it here for convenience. ", + "bbox": [ + 173, + 373, + 823, + 402 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Writing out both sides we obtain the following equality that needs to hold for all $f _ { \\mathrm { i n } }$ and all $x , t \\in$ $\\mathbb { R } ^ { d }$ : ", + "bbox": [ + 173, + 407, + 823, + 436 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/a4dbaf5abc89637166524c44ae82359b55c5985bc1e7e8795d8e4b45b5170958.jpg", + "text": "$$\n\\int _ { \\mathbb { R } ^ { d } } K ( y ) \\rho _ { \\mathrm { i n } } ( g ) f _ { \\mathrm { i n } } \\big ( g ^ { - 1 } ( x + y - t ) \\big ) d y = \\rho _ { \\mathrm { o u t } } ( g ) \\int _ { \\mathbb { R } ^ { d } } K ( y ) f _ { \\mathrm { i n } } \\big ( g ^ { - 1 } ( x - t ) + y \\big ) d y .\n$$", + "text_format": "latex", + "bbox": [ + 217, + 441, + 779, + 476 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Substituting $y = g ^ { - 1 } y$ on the left side and using $| \\operatorname* { d e t } g | = 1$ due to the compactness of $G$ , and putting $\\rho _ { \\mathrm { o u t } } ( g )$ inside the integral on the right side, which is possible due to linearity, we obtain: ", + "bbox": [ + 173, + 482, + 823, + 512 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/36dd758e0fdb1d7d725ba882c12e85af628451c3b349ae9fb6b02e7796c4d8b2.jpg", + "text": "$$\n\\int _ { \\mathbb { R } ^ { d } } \\left[ K ( g y ) \\rho _ { \\mathrm { i n } } ( g ) \\right] f _ { \\mathrm { i n } } ( g ^ { - 1 } x - g ^ { - 1 } t + y ) d y = \\int _ { \\mathbb { R } ^ { d } } \\left[ \\rho _ { \\mathrm { o u t } } ( g ) K ( y ) \\right] f _ { \\mathrm { i n } } ( g ^ { - 1 } x - g ^ { - 1 } t + y ) d y .\n$$", + "text_format": "latex", + "bbox": [ + 186, + 518, + 808, + 553 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Since this needs to hold for all fields $f _ { \\mathrm { i n } }$ , we necessarily have $K ( g x ) \\rho _ { \\mathrm { i n } } ( g ) = \\rho _ { \\mathrm { o u t } } ( g ) K ( x )$ for all $\\boldsymbol { x } \\in \\mathbb { R } ^ { d }$ and all $g \\in G$ and obtain the kernel constraint ", + "bbox": [ + 173, + 559, + 823, + 588 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/cf25847bc790b015bb9222eb7920905cd2b297b6b87d75efaf340088c02a85eb.jpg", + "text": "$$\nK ( g x ) = \\rho _ { \\mathrm { o u t } } ( g ) \\circ K ( x ) \\circ \\rho _ { \\mathrm { i n } } ( g ) ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 594, + 625, + 613 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "This work will create a general theory for how to solve this kernel constraint, which means to find a parameterization for the space of all kernels that fulfill this constraint. We now explain how to make this problem more tractable: formally, the action of $G$ on $\\mathbb { R } ^ { d }$ is a group action as in Definition B.5. However, it cannot be transitive as in Definition B.7 since $G$ is compact and $\\mathbb { R } ^ { d }$ is not. Thus $\\mathbb { R } ^ { d }$ splits into a disjoint union of orbits (Definition B.6), of the action: ", + "bbox": [ + 173, + 619, + 825, + 690 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/0a2ae15dc9978b77553e44799ed0a53cb57c222611bde3ad7d7e0d18839ba41b.jpg", + "text": "$$\n\\mathbb { R } ^ { d } = \\bigcup _ { k \\in K } X _ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 447, + 698, + 550, + 733 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "That this is a disjoint union can be explained as follows: define the relation $\\sim$ on $\\mathbb { R } ^ { d }$ by $x \\sim x ^ { \\prime }$ if $g x = x ^ { \\prime }$ for some $g \\in G$ . This is then an equivalence relation, and so $\\mathbb { R } ^ { d }$ splits into a disjoint union of equivalence classes. One then can show that these equivalence classes are precisely the orbits of the group action. For example, such orbits take the form of spheres $S ^ { d - 1 }$ if $G = { \\mathrm { S O } } ( d )$ or $G = \\mathrm { O } ( d )$ and the form of a finite set of points if $G = \\mathrm { C } _ { N }$ or $G = \\mathrm { D } _ { N }$ . ", + "bbox": [ + 173, + 741, + 825, + 813 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "The idea is now that the kernel constraint 10 only constrains the behavior of the kernel at each orbit individually, and thus a solution on each orbit can be “patched together” to a solution on the whole of $\\mathbb { R } ^ { d }$ . Indeed, assume that $K _ { k } : X _ { k } \\to \\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ individually fulfill the kernel constraint, which means that for all $x _ { k } \\in X _ { k }$ and $g \\in G$ we have ", + "bbox": [ + 174, + 818, + 825, + 847 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/4272149630aa5566a90829e5717ee9a6d46b764d0047765e11ee74931a018ac3.jpg", + "text": "$$\nK _ { k } ( g x _ { k } ) = \\rho _ { \\mathrm { o u t } } ( g ) \\circ K _ { k } ( x _ { k } ) \\circ \\rho _ { \\mathrm { i n } } ( g ) ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 145, + 638, + 164 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Then, define the patch of these orbit-kernels by $K : \\mathbb { R } ^ { d } \\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ as $K ( x ) = K _ { k } ( x )$ if $x \\in X _ { k }$ . This is well-defined since each $x$ is in precisely one orbit. Then clearly, $K$ satisfies the kernel constraint 10. Moreover, each kernel $K$ which fulfills the kernel constraint emerges from such a construction, since we can simply set $K _ { k } : = K | _ { X _ { k } }$ . Overall, we see that we can restrict our attention to orbits. In Weiler et al. (2018b) and later Weiler et al. (2018a), a discretized implementation is done where the kernel is discretized into finitely many orbits with a smooth Gaussian radial profile. We will come back to these practical questions of parameterization in Remark D.19, once we have fully developed the theory of steerable CNNs. ", + "bbox": [ + 173, + 176, + 825, + 290 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "C.1.2 AN ABSTRACT DEFINITION OF STEERABLE KERNELS ", + "text_level": 1, + "bbox": [ + 173, + 313, + 602, + 327 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Motivated by the discussion in the last section, we now define steerable kernels in precise terms and will stick to that definition throughout this work. The definition will be more abstract than usual in the deep learning community, but we are rewarded since such an abstract definition makes it easier to apply representation-theoretic results. ", + "bbox": [ + 174, + 339, + 825, + 396 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Without loss of generality, we will in the rest of this work only consider kernels on orbits. Thus, let $X : = G \\cdot x$ be an arbitrary orbit. We consider steerable kernels $K : X \\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ . Note that the restriction of the action $\\dot { G } \\times \\mathbb { R } ^ { d } \\to \\mathbb { R } ^ { d }$ to $X$ , written $G \\times X \\to X$ , makes $X$ to a homogeneous space of $G$ , see Definition B.7. Thus, instead of viewing $X$ as a subset of $\\mathbb { R } ^ { d }$ , we view $X$ as an arbitrary homogeneous space of an arbitrary compact group $G$ . Notably, this framework is more general than usually studied in the context of steerable CNNs on $\\mathbb { R } ^ { d }$ , since we allow also groups that are not Lie groups and homogeneous spaces which are not naturally embedded in an $\\mathbf { \\bar { \\mathbb { R } } } ^ { d }$ , as well as finite homogeneous spaces of finite groups all at the same time. ", + "bbox": [ + 173, + 402, + 825, + 515 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Furthermore, we replace $\\mathbb { K } ^ { c _ { \\mathrm { i n } } }$ and $\\mathbb { K } ^ { c _ { \\mathrm { o u t } } }$ by coordinate-independent $\\mathbb { K }$ -vector spaces $V _ { \\mathrm { i n } }$ and $V _ { \\mathrm { o u t } }$ , and therefore $\\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ by the space of linear functions from $V _ { \\mathrm { i n } }$ to $V _ { \\mathrm { o u t } }$ , written $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . We assume there are linear representations $\\rho _ { \\mathrm { i n } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { o u t } } )$ . ", + "bbox": [ + 174, + 521, + 825, + 564 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Overall, this means that steerable kernels are certain maps $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . The only property they need to fulfill is the kernel constraint $K ( g x ) = \\rho _ { \\mathrm { o u t } } ( g ) \\circ K ( x ) \\circ \\rho _ { \\mathrm { i n } } ( g ) ^ { - 1 }$ for all $g \\in G$ and $x \\in X$ . This can be viewed in representation-theoretic terms by defining the Homrepresentation: ", + "bbox": [ + 174, + 570, + 825, + 626 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Definition C.1 (Hom-Representation). Let $\\rho _ { \\mathrm { i n } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { o u t } } )$ be two finite-dimensional $G$ -representations over the field K. The space $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ of $\\mathbb { K }$ -linear (not necessarily $G$ -equivariant) functions from $V _ { \\mathrm { i n } }$ to $V _ { \\mathrm { o u t } }$ also carries an induced $G$ -representation, with action ", + "bbox": [ + 174, + 633, + 825, + 690 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/d6fb13230aa7f1c26716420ac2bb28488594e1de1fc32623384f732a23a827ab.jpg", + "text": "$$\n\\left[ \\rho _ { \\mathrm { H o m } } ( g ) \\right] ( f ) : = \\rho _ { \\mathrm { o u t } } ( g ) \\circ f \\circ \\rho _ { \\mathrm { i n } } ( g ) ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 698, + 633, + 718 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "We call this the Hom-representation. ", + "bbox": [ + 176, + 731, + 415, + 746 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Remark C.2. Of course, one needs to check that this is indeed a linear representation. Continuity follows from the continuity of $\\rho _ { \\mathrm { i n } }$ and $\\rho _ { \\mathrm { o u t } }$ as follows: the topology on $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ is just the Euclidean topology of $\\mathbb { K } ^ { c _ { \\mathrm { o u t } } \\times c _ { \\mathrm { i n } } }$ coming from a basis of $V _ { \\mathrm { i n } }$ and $V _ { \\mathrm { o u t } }$ . In these bases, $\\rho _ { \\mathrm { i n } } ( g )$ and $\\rho _ { \\mathrm { o u t } } ( g )$ are given by matrices. All matrix coefficients are continuous by Remark B.25. Now, in order to show that $\\rho _ { \\mathrm { H o m } }$ is continuous, pick a fixed element $f \\in \\mathbb { K } ^ { c _ { \\mathrm { i n } } \\times c _ { \\mathrm { o u t } } }$ . One needs to show that the map ", + "bbox": [ + 173, + 752, + 825, + 838 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/6cee25340259130a6319ea871c87a5b3a96c40bd64ab1802a1ace1ceff9b7e4d.jpg", + "text": "$$\n\\rho _ { \\mathrm { H o m } } ^ { f } : G \\to \\mathbb { K } ^ { c _ { \\mathrm { i n } } \\times c _ { \\mathrm { o u t } } } , g \\mapsto \\rho _ { \\mathrm { o u t } } ( g ) \\circ f \\circ \\rho _ { \\mathrm { i n } } ( g ^ { - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 323, + 845, + 673, + 867 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "is continuous. Since all matrix coefficients are continuous and since also the inversion $G \\to G , g \\mapsto$ $g ^ { - 1 }$ is continuous by the definition of a topological group, the map $\\rho _ { \\mathrm { H o m } } ^ { f }$ is basically just a stacked linear combination of continuous functions and thus continuous itself. ", + "bbox": [ + 174, + 878, + 825, + 924 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "The linearity of each $\\rho _ { \\mathrm { H o m } } ( g )$ is also clear. So what needs to be checked is that $\\rho _ { \\mathrm { H o m } }$ is a group homomorphism. And indeed, it is, exploiting the corresponding property of $\\rho _ { \\mathrm { i n } }$ and $\\rho _ { \\mathrm { o u t } }$ : ", + "bbox": [ + 171, + 102, + 823, + 133 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/12ef860cdef2162bcadbe5205198d6d41911bd85395a36632aae9ac95b48207b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left[ \\rho _ { \\mathrm { H o m } } ( g g ^ { \\prime } ) \\right] ( f ) = \\rho _ { \\mathrm { o u t } } ( g g ^ { \\prime } ) \\circ f \\circ \\rho _ { \\mathrm { i n } } ( g g ^ { \\prime } ) ^ { - 1 } } \\\\ & { \\qquad = \\rho _ { \\mathrm { o u t } } ( g ) \\circ \\left( \\rho _ { \\mathrm { o u t } } ( g ^ { \\prime } ) \\circ f \\circ \\rho _ { \\mathrm { i n } } ( g ^ { \\prime } ) ^ { - 1 } \\right) \\circ \\rho _ { \\mathrm { i n } } ( g ) ^ { - 1 } } \\\\ & { \\qquad = \\left[ \\rho _ { \\mathrm { H o m } } ( g ) \\right] \\left( \\left[ \\rho _ { \\mathrm { H o m } } ( g ^ { \\prime } ) \\right] ( f ) \\right) } \\\\ & { \\qquad = \\left[ \\rho _ { \\mathrm { H o m } } ( g ) \\circ \\rho _ { \\mathrm { H o m } } ( g ^ { \\prime } ) \\right] ( f ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 140, + 720, + 223 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "and so the claim follows. ", + "bbox": [ + 174, + 231, + 338, + 246 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "With this definition in mind, steerable kernels $K : X \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ are just functions with the property $K ( g x ) = [ \\rho _ { \\mathrm { H o m } } ( g ) ] \\left( K ( x ) \\right)$ . Summarizing, we have the following abstract definition of steerable kernels (different from Definition 3.2, we here allow also input- and output representations that are not irreducible and make explicit reference to the Hom-representation): ", + "bbox": [ + 174, + 256, + 825, + 313 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Definition C.3 (Steerable Kernel). Let $G$ be any compact group and $X$ be any homogeneous space of $G$ . Furthermore, let $\\rho _ { \\mathrm { i n } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { o u t } } )$ be finite-dimensional representations of $G$ . We assume that $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ is equipped with the Hom-representation $\\rho _ { \\mathrm { H o m } }$ . A $G$ -steerable kernel is an equivariant function $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ , i.e., a function such that ", + "bbox": [ + 173, + 318, + 825, + 388 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/a682e028e3a045eb56b3b417cf75265dbf2b3b59cbd2079d309f4f95e31b4cba.jpg", + "text": "$$\nK ( g x ) = \\left[ \\rho _ { \\mathrm { H o m } } ( g ) \\right] ( K ( x ) )\n$$", + "text_format": "latex", + "bbox": [ + 403, + 393, + 594, + 411 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "for all $g \\in G$ and $x \\in X$ . We denote the vector-space of all these kernels by ", + "bbox": [ + 173, + 420, + 673, + 435 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/1e72383e4f80402628d4624a65e39967d80897a04558f43e15c0ad815873205c.jpg", + "text": "$$\n\\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) = \\{ K : X \\to \\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\mid K { \\mathrm { ~ i s ~ s t e e r a b l e ~ } } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 225, + 445, + 772, + 463 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Notably, steerable kernels are not linear in a meaningful sense with respect to their input. ", + "bbox": [ + 173, + 473, + 758, + 489 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "That the space of steerable kernels forms a vector space, as claimed in this definition, can easily be checked. ", + "bbox": [ + 173, + 501, + 825, + 531 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "C.1.3 MORE DETAILS ON THE COMPARISON OF REPRESENTATION OPERATORS ANDSTEERABLE KERNELS", + "text_level": 1, + "bbox": [ + 176, + 550, + 769, + 579 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Steerable kernels satisfy the constraint ", + "bbox": [ + 176, + 590, + 428, + 606 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/733e2e3dce1049642e6b315ef7f8ad2a3f4f6e9436db43f2b22f1070f2bd0819.jpg", + "text": "$$\nK ( g x ) = \\rho _ { \\mathrm { o u t } } ( g ) \\circ K ( x ) \\circ \\rho _ { \\mathrm { i n } } ( g ) ^ { - 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 372, + 614, + 624, + 635 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "whereas, as we saw in Section 2, representation operators are collections $( A _ { 1 } , \\dotsc , A _ { N } )$ of operators $A _ { i } : \\mathcal { H } \\to \\mathcal { H }$ that satisfy the constraint ", + "bbox": [ + 174, + 645, + 823, + 674 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/4981d3486166949993122f21452fc3ea28b08026ed3986abe54a40cdc065fbd0.jpg", + "text": "$$\n\\sum _ { j = 1 } ^ { N } \\pi ( g ) _ { i j } A _ { j } \\ = \\ U ( g ) ^ { \\dagger } A _ { i } U ( g ) \\qquad \\forall g \\in G .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 684, + 665, + 715 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Hereby, $U : G \\to \\operatorname { U } ( { \\mathcal { H } } )$ and $\\pi : G \\to \\operatorname { U } ( \\mathbb { C } ^ { N } )$ are unitary representations. Unfortunately, these equations still look somewhat different from each other. We can make them more similar by inverting $g$ and using the unitarity of $\\pi$ (note the swap of $j$ and $i$ and the complex conjugation): ", + "bbox": [ + 173, + 724, + 825, + 770 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/ec67ad2f43b5b08a6d72f4558845b853a6ccfdf56abde91223736151ee8036cd.jpg", + "text": "$$\n\\sum _ { j = 1 } ^ { N } \\overline { { { \\pi } } } ( g ) _ { j i } A _ { j } \\ = \\ U ( g ) A _ { i } U ( g ) ^ { \\dagger } \\qquad \\forall g \\in G .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 779, + 665, + 810 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "In order to make the analogy to steerable kernels stronger, we would like to interpret a representation operator as one object $\\pmb { A }$ instead of separate operators $A _ { i }$ , in the same way as a kernel $K$ is one single object and not just a disjoint collection of linear functions in $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . For this, we interpret $\\pmb { A }$ as a function that assigns to arbitrary vectors in $\\mathbb { C } ^ { N }$ an operator. Namely, let $\\left\\{ \\boldsymbol { e } _ { i } \\right\\}$ be the standard basis of $\\mathbb { C } ^ { N }$ . We then define $\\pmb { A }$ as the unique linear map which is given on basis elements as follows: ", + "bbox": [ + 173, + 819, + 825, + 904 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/5030091c86dc576dbdf1197961534c87ac019ca5950cccf7146dcd71bf00fb0f.jpg", + "text": "$$\nA : e _ { i } \\mapsto A _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 454, + 909, + 544, + 924 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "We can then deduce the following, where we use the linearity of $\\pmb { A }$ in the second step, the definition of $A _ { j }$ in the third and fifth step, and Eq. (13) in the fourth step: ", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/95f2f035b33da435ef0dab5b9667007ead8edb5d3b90f4eadd663391925374b5.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { \\pmb { A } \\big ( \\overline { { \\pi } } ( g ) ( e _ { i } ) \\big ) = \\pmb { A } \\bigg ( \\sum _ { j } \\overline { { \\pi } } ( g ) _ { j i } e _ { j } \\bigg ) } } \\\\ { \\displaystyle = \\sum _ { j } \\overline { { \\pi } } ( g ) _ { j i } \\pmb { A } ( e _ { j } ) } \\\\ { \\displaystyle = \\sum _ { j } \\overline { { \\pi } } ( g ) _ { j i } \\pmb { A } _ { j } } \\\\ { \\displaystyle = { U } ( g ) { A } _ { i } { U } ( g ) ^ { \\dagger } } \\\\ { \\displaystyle = { U } ( g ) { A } ( e _ { i } ) { U } ( g ) ^ { \\dagger } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 379, + 140, + 616, + 284 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "If now $v = \\textstyle \\sum _ { i } \\lambda _ { i } e _ { i }$ is an arbitrary vector in $\\mathbb { C } ^ { N }$ , not necessarily a standard basis vector, then from the linearity of $\\pmb { A }$ and Eq. (14) we obtain ", + "bbox": [ + 173, + 289, + 823, + 318 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/1c99679a7b4b2de56feba16c6bdd6ac724d2e46a1e4d4023f1603c89c5b5ec09.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { A } \\big ( \\mpb { \\pi } ( g ) ( v ) \\big ) = \\pmb { U } ( g ) \\pmb { A } ( v ) \\pmb { U } ( g ) ^ { - 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 324, + 616, + 344 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "This equation is essentially the starting point for the definition of a representation operator as it can be found in Jeevanjee [2011]. ", + "bbox": [ + 174, + 351, + 826, + 380 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "This, finally, really looks like Eq. (12). In this comparison, the action of the group $G$ on $\\mathbb { R } ^ { d }$ in deep learning is replaced by the action of $G$ via $\\overline { { \\pi } }$ on the space $\\mathbb { C } ^ { N }$ . The main difference is that steerable kernels are not necessarily linear. This difference will be bridged in Theorem C.7. ", + "bbox": [ + 174, + 386, + 825, + 428 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "C.1.4 REPRESENTATION OPERATORS AND KERNEL OPERATORS", + "text_level": 1, + "bbox": [ + 176, + 444, + 629, + 459 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Now that we have a clear abstract idea of what steerable kernels are and saw strong analogies to representation operators, we can begin to formulate precise theoretical connections. In this section, we therefore begin with formulating a purely representation-theoretic and more abstract working definition of representation operators and will then formulate the main theorem of this chapter, Theorem C.7. ", + "bbox": [ + 174, + 468, + 825, + 539 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "We come to the main definition, which is directly motivated from Eq. (15). It differs from (Jeevanjee, 2011) by allowing the input- and output representations to differ. We furthermore restrict to finitedimensional input- and output representations due to our specific applications. As explained in Section C.1.3, this new definition furthermore somewhat differs from the one given in Section 2 since now we view representation operators as one object instead of viewing it as a collection of several linear operators. ", + "bbox": [ + 173, + 545, + 825, + 630 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Definition C.4 (Representation Operator). Let $\\rho _ { \\mathrm { i n } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G \\to \\mathrm { G L } ( V _ { \\mathrm { o u t } } )$ be finite-dimensional $G$ -representations. Let $\\lambda : G \\ \\to \\ \\mathrm { G L } ( T )$ be a third $G$ -representation, not necessarily finite-dimensional. Then a representation operator is an intertwiner $\\kappa : T $ $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ , where the right space is equipped with the Hom-representation as in Definition C.1. We denote the vector space of all these representation operators by ", + "bbox": [ + 173, + 633, + 825, + 704 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/d6f12ad9167f0789321084c4f0eeeb4bc171ca18beec7e5ad10312f4b8806ee6.jpg", + "text": "$$\n\\mathrm { H o m } _ { G , \\mathbb { K } } ( T , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) = \\{ K : T \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\mid K \\mathrm { i s ~ a n ~ i n t e r t w i n e r } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 214, + 712, + 782, + 729 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Note that representation operators are by definition linear, which is a requirement that needs to be satisfied for the standard Wigner-Eckart theorem. We clearly see strong similarities between this definition and the formalization of steerable kernels in Definition C.3. The main difference is that we assume representation operators to be linear. This is in notation captured by the subscript K that we put in the corresponding Hom-space. One may think that there is another difference, namely coming from the fact that intertwiners are by definition continuous with respect to the topologies involved. Two things need to be said about this: ", + "bbox": [ + 174, + 743, + 825, + 842 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "1. First of all, one may wonder what continuity for representation operators actually means. This can be clarified as follows: By assumption, $G$ -representations are always on vector spaces with topologies, and thus $T$ has a topology. Furthermore, in Remark C.2 we clarified the topology on $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . Then, being continuous just means, as always, to be continuous with respect to the topologies of these two spaces. ", + "bbox": [ + 215, + 853, + 825, + 924 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "2. The second remark is that this apparent difference in the requirement of continuity for steerable kernels and representation operators is actually non-existent. This is explained by the following Proposition which says that steerable kernels are automatically continuous. Note that this is not true for steerable kernels that are defined on the domain $\\mathbb { R } ^ { d }$ – in that case, continuity is only guaranteed when restricting to orbits. ", + "bbox": [ + 210, + 103, + 823, + 174 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Proposition C.5. Let $K : X \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ be a steerable kernel. Then $K$ is continuous. ", + "bbox": [ + 173, + 176, + 799, + 193 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Proof. For brevity, denote $V : = \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ and $\\rho : = \\rho _ { \\mathrm { H o m } }$ . Let $x ^ { * } \\in X$ be any point and $G _ { x ^ { * } }$ the stabilizer corresponding to the action of $G$ on $X$ . Remember the homeomorphism $\\varphi : G / H \\to X , [ g ] \\mapsto g x ^ { * }$ from Lemma B.21. Since this is a homeomorphism, the kernel $K$ is continuous if and only if the composition $K \\circ \\varphi$ is continuous, since then $K = ( K \\circ \\varphi ) \\circ \\varphi ^ { - 1 }$ is a composition of continuous functions. Thus, we evaluate $K \\circ \\varphi$ : ", + "bbox": [ + 173, + 205, + 825, + 277 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/97fee34c9c0b996d8549d5cdd86cd1dd06e359bd60289f5618813c6ed262acb0.jpg", + "text": "$$\n( K \\circ \\varphi ) ( [ g ] ) = K ( \\varphi ( [ g ] ) ) = K ( g x ^ { * } ) = \\rho ( g ) ( K ( x ^ { * } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 281, + 679, + 299 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "where in the last step we have used the equivariance of $K$ . Thus, if we set $v ^ { * } : = K ( x ^ { * } ) \\in V$ , then we obtain the simple relation $( K \\circ \\varphi ) ( [ g ] ) = \\rho ( g ) ( v ^ { \\ast } ) .$ . This is by definition just the unique map on the quotient, $G / H V$ , coming from $\\rho ^ { v ^ { * } } : G \\to V$ , $g \\mapsto \\rho ( g ) ( v ^ { * } )$ . This last map is continuous by definition of a linear representation. The universal property of quotients Proposition F.12 then shows that $K \\circ \\varphi$ is continuous as well, and so we are done. All of this is visualized in the following commutative diagram, where $q : G \\to G / H$ , $g \\mapsto [ g ]$ is the canonical projection: ", + "bbox": [ + 174, + 303, + 825, + 390 + ], + "page_idx": 44 + }, + { + "type": "image", + "img_path": "images/12025ef1c9575e91d1a62ae6faf58eb2a16e846dd383a7a534dc5ae9b73ae9bc.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 370, + 397, + 624, + 481 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Thus, the only difference between steerable kernels and representation operators is indeed the linearity. We now look at special representation operators that play the main role in this work: ", + "bbox": [ + 173, + 520, + 825, + 550 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Definition C.6 (Kernel Operator). Let $\\rho _ { \\mathrm { i n } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { o u t } } )$ be finitedimensional $G$ -representations. Let $\\lambda : G \\to \\operatorname { U } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) )$ be the standard unitary representation on the space of square-integrable functions of a homogeneous space $X$ , given, as in Section B.1.3, by ", + "bbox": [ + 173, + 553, + 823, + 595 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/9cf15e8f069e3b16ebaabcd2113ef17934f787bd8162a031eb0b7690b2ef6003.jpg", + "text": "$$\n\\left[ \\lambda ( g ) ( \\varphi ) \\right] ( g ^ { \\prime } ) = \\varphi ( g ^ { - 1 } g ^ { \\prime } ) .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 598, + 588, + 617 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "A kernel operator is a representation operator ${ \\mathcal { K } } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . We denote the space of these by ", + "bbox": [ + 173, + 621, + 823, + 650 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/5192e01c706b62beff4fb1e1b61699e61441a27f659005f2f97c6bdbdd679ec6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { H o m } _ { G , \\mathrm { K } } ( L _ { \\mathrm { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) } \\\\ & { \\qquad = \\{ K : L _ { \\mathrm { K } } ^ { 2 } ( X ) \\to \\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\mid K \\mathrm { ~ i s ~ a n ~ i n t e r t w i n e r ~ } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 651, + 759, + 694 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Notably, kernel operators are $\\mathbb { K }$ -linear in their input. ", + "bbox": [ + 174, + 695, + 516, + 710 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "C.1.5 FORMULATION OF THE CORRESPONDENCE BETWEEN STEERABLE KERNELS ANDKERNEL OPERATORS", + "text_level": 1, + "bbox": [ + 176, + 724, + 794, + 753 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "The following Theorem lies at the heart of our investigations and establishes that steerable kernels can be considered as kernel operators, which we defined as special representation operators. More precisely, we will give an explicit isomorphism between the space of steerable kernels and the space of kernel operators. ", + "bbox": [ + 174, + 762, + 825, + 819 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "We shortly explain why the theorem is useful. First of all, using a Wigner-Eckart theorem for kernel operators that we prove in Theorem D.13, one can explicitly describe a basis $B$ of the space of kernel operators $\\operatorname { H o m } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) )$ . Then, since we have an isomorphism of vector spaces to the space of steerable kernels, one can “carry over” this basis to a basis for the space of steerable kernels, namely $\\operatorname { H o m } _ { G } ( X , \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) )$ . This basis will then have a convenient explicit form that we establish in Theorem D.16 and is exactly what we need in order to parameterize an equivariant neural network layer. We now come to a precise formulation of the theorem: ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Theorem C.7 (Kernel-Operator-Correspondence). Let $\\rho _ { \\mathrm { i n } } : G \\to \\mathrm { G L } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G \\to$ $\\mathrm { G L } ( V _ { \\mathrm { o u t } } )$ be finite-dimensional $G$ -representations and $X$ be a homogeneous space of $G$ . Then there is an isomorphism ", + "bbox": [ + 174, + 102, + 825, + 146 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/3ec95c97449344c172ffac7869450acaad6be94a7bc6d6be34480da859b21a52.jpg", + "text": "$$\n\\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , \\overbrace { V _ { \\mathrm { o u t } } ) ) } ^ { \\widehat { ( \\cdot ) } } \\overbrace { \\mathrm { H o m } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) } ^ { \\widehat { ( \\cdot ) } }\n$$", + "text_format": "latex", + "bbox": [ + 230, + 148, + 766, + 231 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "between the space of steerable kernels on the left and the space of kernel operators on the right. The two maps are defined as follows: ", + "bbox": [ + 173, + 233, + 825, + 262 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "1. For a steerable kernel $K : X \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ , the extension ${ \\widehat K } \\ : \\ L _ { \\mathbb { K } } ^ { 2 } ( X ) \\ \\to$ $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ is given by ", + "bbox": [ + 212, + 272, + 823, + 304 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/cc949f6c821dac590a83d0f58a9b39b755ae084b339538bf9ba3d744cad6c31a.jpg", + "text": "$$\n{ \\widehat { K } } ( f ) : = \\int _ { X } f ( x ) K ( x ) d x .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 306, + 619, + 339 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "2. For a kernel operator ${ \\mathcal K } : L _ { \\mathbb K } ^ { 2 } ( X ) \\to \\operatorname { H o m } _ { \\mathbb K } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ,$ , the restriction $\\mathcal { K } | _ { X } : X \\ $ $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ is given by ", + "bbox": [ + 212, + 348, + 825, + 380 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/31237aeeaaf0e1d3a5a24bdb050483ec0f30a33e3ae056a4ace3eccbe41dc32e.jpg", + "text": "$$\nK | _ { X } ( x ) : = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\mathcal { K } ( \\delta _ { U } ) .\n$$", + "text_format": "latex", + "bbox": [ + 446, + 382, + 609, + 406 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Hereby, $\\mathcal { U } _ { x }$ is the directed set of open neighborhoods of $x$ , see Example F.27. $\\delta _ { U } : X \\to \\mathbb { K }$ \nis the approximated Dirac delta function with $\\begin{array} { r } { \\delta _ { U } ( y ) = \\frac { 1 } { \\mu ( U ) } i f y \\in \\dot { U } } \\end{array}$ and $\\delta _ { U } ( y ) = 0$ , else. \nThe limit is a limit of nets as in Definition $F . 2 9$ . ", + "bbox": [ + 230, + 409, + 825, + 454 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "This theorem requires some explanation. First of all, $\\widehat { K }$ is supposed to be a kernel operator, i.e., a map $L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . Thus, $\\widehat K ( f )$ should be a linear function $V _ { \\mathrm { i n } } \\to V _ { \\mathrm { o u t } }$ . The formal expression of it can indeed be considered as such: ", + "bbox": [ + 178, + 464, + 825, + 511 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/cb6105e8f89a653b00a9222c5a2d9814f8bccf22a6bef99d93b7dbb663d1b7cc.jpg", + "text": "$$\n{ \\widehat { K } } ( f ) = \\int _ { X } f ( x ) K ( x ) d x : v _ { \\mathrm { i n } } \\mapsto \\int _ { X } f ( x ) \\left[ K ( x ) \\right] ( v _ { \\mathrm { i n } } ) d x \\in V _ { \\mathrm { o u t } } .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 513, + 720, + 546 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Due to the continuity of $K$ proven in Proposition ${ \\mathrm { C } } . 5 ^ { 1 8 }$ and the integrability of $f$ , the function $X V _ { \\mathrm { o u t } }$ , $x \\mapsto f ( x ) \\left[ K ( x ) \\right] ( v _ { \\mathrm { i n } } )$ is also integrable, meaning the expression in Eq. (16) can be evaluated. This explains the meaning of the map $\\widehat { ( \\cdot ) }$ in Theorem C.7. ", + "bbox": [ + 173, + 550, + 825, + 598 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "For the map $( \\cdot ) | _ { X }$ in the other direction, we want to shortly explain the intuitions in a more informal way. For this, we consider Dirac delta functions $\\delta _ { x }$ for $x \\in X$ . Such a “function” $\\delta _ { x } : X \\to \\mathbb { K }$ for a point $x \\in X$ can be imagined as a function taking value infinity at $x$ and zero elsewhere. It is characterized by the property that $\\begin{array} { r } { \\int _ { X } \\delta _ { x } ( x ^ { \\prime } ) f ( x ^ { \\prime } ) d x ^ { \\prime } = f ( x ) } \\end{array}$ for any function $f \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ . We think of $\\delta _ { x }$ as being a function in $L _ { \\mathbb { K } } ^ { 2 } ( X )$ , even though technically, it is not in this space. This is since $\\propto \\notin \\mathbb { K }$ . ", + "bbox": [ + 173, + 603, + 825, + 691 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Now, informally, we can think of the limit $\\begin{array} { r } { K | _ { X } ( x ) = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } K ( \\delta _ { U } ) } \\end{array}$ as being given by $\\mathcal { K } ( \\delta _ { x } )$ , the value that $\\kappa$ takes at the Dirac delta function $\\delta _ { x }$ . This is since the limit of nets progressively “shrinks down” the open neighborhood $U$ of $x$ . Of course, $\\kappa ( \\delta _ { x } )$ is not really well-defined since $\\delta _ { x } \\notin L _ { \\mathbb { K } } ^ { 2 } ( X )$ , but we can pretend that it is for gaining intuitions. ", + "bbox": [ + 173, + 695, + 825, + 753 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Now that we have understood the formulation of the theorem, we might wonder, why should such a theorem be true? A first intuition comes from an analogy with linear algebra: namely, assume $B$ is a basis of a $\\mathbb { K }$ -vector space $V$ and $W$ any other vector space. Then linear maps ${ \\hat { f } } : V \\to W$ are in one-to-one correspondence with (not assumed to be linear) functions $f : B \\to W$ , and this isomorphism is given by restriction and linear extension: ", + "bbox": [ + 173, + 758, + 825, + 833 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/fbb42f2250e81803b3539a584f3d3f7301891f034e5b7f01552283d628f90faf.jpg", + "text": "$$\n\\mathrm { H o m } ( B , W ) \\xrightarrow [ ( \\cdot ) ] { \\widehat { ( \\cdot ) } } \\mathrm { H o m } _ { \\mathbb { K } } ( V , W ) .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 835, + 638, + 901 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Thus, we can think of the homogeneous space $X$ as a “continuous basis” of the space of squareintegrable functions. Sums are then replaced by integrals, and evaluations at a basis element by evaluations at Dirac delta functions of elements in $X$ . ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "For the actual proof of Theorem C.7, informally, one direction seems pretty clear from the properties of the Dirac delta: ", + "bbox": [ + 171, + 152, + 823, + 180 + ], + "page_idx": 46 + }, + { + "type": "equation", + "img_path": "images/dbb200d2793471923b049981f3dc454acfff9bf3b2718415dc68b756003d86eb.jpg", + "text": "$$\n\\widehat K \\vert _ { X } ( x ) = \\widehat K ( \\delta _ { x } ) = \\int _ { X } \\delta _ { x } ( x ^ { \\prime } ) K ( x ^ { \\prime } ) d x ^ { \\prime } = K ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 185, + 668, + 218 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "But the other direction is less obvious: it seems like the space of kernel operators is considerably larger than the space of steerable kernels, since kernel operators are defined on a larger space. Therefore it is hard to believe that the construction is also inverse in the other direction. However, it pays off to ponder a bit more over what the Dirac delta construction does: Basically, we “embed” $X$ into $L _ { \\mathbb { K } } ^ { \\dot { 2 } } ( X )$ by means of the Dirac delta functions, i.e., $x \\mapsto \\delta _ { x }$ and, as such, view $X$ as a subset of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ (albeit a subset that is only in approximation in that space). Steerable kernels are then “partial” kernel operators in the sense that they are only defined on this subset $X \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( X )$ . What then needs to be understood is why there is only one unique extension of each steerable kernel $K$ to a kernel operator $\\kappa$ on the whole of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ : if this is understood, then the space of kernel operators cannot be larger than the space of steerable kernels. And indeed, if there is an extension of $K$ to $\\kappa$ on $L _ { \\mathbb { K } } ^ { 2 } ( X )$ , it has to be unique: each $f \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ can be approximated by finite linear combinations of scaled indicator functions. Then by linearity of the kernel operator $\\kappa$ , we can evaluate $\\kappa ( f )$ by knowing $\\kappa ( \\delta \\sigma )$ for scaled indicator functions $\\delta _ { U }$ on small measurable sets $U$ . And these approximate $K ( x ) \\stackrel { \\cdot } { = } \\dot { \\mathcal { K } } ( \\delta _ { x } )$ for $x \\in U$ arbitrarily well by construction. This determines the behavior of $\\kappa$ . The details of all of this can be found in the next section. ", + "bbox": [ + 173, + 223, + 825, + 433 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "C.2 A PROOF OF THE CORRESPONDENCE BETWEEN STEERABLE KERNELS AND KERNELOPERATORS", + "text_level": 1, + "bbox": [ + 176, + 449, + 802, + 477 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Here, we give a step-by-step proof of Theorem C.7. The details of this investigation will not be needed later, and so a reader who is mainly interested in the applications to steerable CNNs can safely skip reading this section and go on reading Chapter D. ", + "bbox": [ + 174, + 488, + 823, + 531 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "C.2.1 A REDUCTION TO UNITARY IRREDUCIBLE REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 546, + 661, + 560 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "In this section, we make the proof more manageable by reducing $\\mathrm { H o m _ { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ to an irreducible representation. First, remember that Proposition B.20 shows that there is a scalar product on $\\mathrm { H o m } _ { \\mathrm { K } } \\mathrm { \\bar { ( } } V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ such that it’s Hom-representation becomes unitary. Since all norms on finitedimensional spaces are equivalent, as is well known, this will not change the topology. Then, we can decompose $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ into an orthogonal direct sum of irreducible unitary representations by Proposition B.38. Let ${ \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\cong \\bigoplus _ { i = 1 } ^ { n } V _ { i }$ be such a decomposition. We get canonical19 isomorphisms ", + "bbox": [ + 173, + 569, + 825, + 667 + ], + "page_idx": 46 + }, + { + "type": "equation", + "img_path": "images/cc924729865affc145457a253367da09abac707caf1affc8361cf2ade7cfac76.jpg", + "text": "$$\n\\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) \\cong \\bigoplus _ { i = 1 } ^ { n } \\mathrm { H o m } _ { G } ( X , V _ { i } )\n$$", + "text_format": "latex", + "bbox": [ + 328, + 665, + 669, + 707 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 710, + 200, + 723 + ], + "page_idx": 46 + }, + { + "type": "equation", + "img_path": "images/e6639467151a56788d2d75780ee438c24e2812a89608f0fb740e4e3482d347e5.jpg", + "text": "$$\n\\mathrm { H o m } _ { G , \\mathrm { K } } ( L _ { \\mathrm { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) \\cong \\bigoplus _ { i = 1 } ^ { n } \\mathrm { H o m } _ { G , \\mathrm { K } } ( L _ { \\mathrm { K } } ^ { 2 } ( X ) , V _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 276, + 718, + 720, + 760 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Thus, we can show Theorem C.7 by showing it for irreducible unitary representations instead of $\\mathrm { H o m } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } )$ . Overall, we have reduced our Theorem to the following, simpler statement: ", + "bbox": [ + 174, + 762, + 825, + 791 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Theorem C.8 (Kernel-Operator-Correspondence, Restated). Let $\\rho : G \\to \\operatorname { U } ( V )$ be an irreducible unitary representation and $X$ a homogeneous space of $G$ . Then there is an isomorphism ", + "bbox": [ + 173, + 795, + 825, + 824 + ], + "page_idx": 46 + }, + { + "type": "equation", + "img_path": "images/cabdca10dd0582b99519bdecd2951bf35ff0ec79f096c70292d6473948af5618.jpg", + "text": "$$\n\\mathrm { H o m } _ { G } ( X , V ) \\overset { \\widehat { ( \\cdot ) } } { \\longleftarrow } \\overset { \\widehat { ( \\cdot ) } } { \\mathrm { H o m } _ { G , \\mathbb { K } } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , V )\n$$", + "text_format": "latex", + "bbox": [ + 331, + 830, + 665, + 900 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "which is given as follows: for $K \\in \\mathrm { H o m } _ { G } ( X , V )$ we set $\\begin{array} { r } { \\widehat { K } ( f ) = \\int _ { X } f ( x ) K ( x ) d x } \\end{array}$ and for $\\kappa \\in$ ${ \\mathrm { H o m } } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , V )$ we set $K | _ { X } ( x ) = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\mathcal { K } ( \\delta _ { U } )$ , with $\\delta _ { U }$ being an approximated Dirac delta function as before. ", + "bbox": [ + 173, + 101, + 825, + 148 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "From now on, we assume that $X$ and $\\rho : G \\to \\operatorname { U } ( V )$ is fixed as in the formulation of Theorem C.8. ", + "bbox": [ + 173, + 160, + 820, + 176 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "C.2.2 WELL-DEFINEDNESS OF $\\widehat { ( \\cdot ) }$ ", + "text_level": 1, + "bbox": [ + 176, + 194, + 421, + 212 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Lemma C.9. The function ${ \\widehat { ( \\cdot ) } } : \\operatorname { H o m } _ { G } ( X , V ) \\to \\operatorname { H o m } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , V )$ is well-defined, i.e.: for an equivariant function $K : X V$ , the function ${ \\widehat { K } } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to V$ is linear, equivariant and continuous. ", + "bbox": [ + 173, + 223, + 826, + 272 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Proof. Linearity of $\\widehat { K }$ is clear. Equivariance can be proven using the equivariance of $K$ and the left invariance of the Haar measure on the homogeneous space $X$ : ", + "bbox": [ + 169, + 292, + 823, + 323 + ], + "page_idx": 47 + }, + { + "type": "equation", + "img_path": "images/e7954c0061f1fee2fed4a2eff2c5121b03e3a9b42da68453c7b851da382f589c.jpg", + "text": "$$\n\\begin{array} { r l } { { \\widehat { R } ( \\lambda ( g ) f ) = \\int _ { X } ( \\lambda ( g ) f ) ( x ) K ( x ) d x } } \\\\ & { = \\int _ { X } f ( g ^ { - 1 } \\cdot x ) K ( x ) d x } \\\\ & { = \\int _ { X } f ( x ) K ( g \\cdot x ) d x } \\\\ & { = \\int _ { X } f ( x ) [ \\rho ( g ) ( K ( x ) ) ] d x } \\\\ & { = \\rho ( g ) [ \\int _ { X } f ( x ) K ( x ) d x ] } \\\\ & { = \\rho ( g ) [ \\widehat { R } ( f ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 330, + 632, + 530 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "The action by $\\rho ( g )$ could be put out of the integral since $\\rho ( g )$ it is linear and continuous, and since integrals can be approximated by finite sums. ", + "bbox": [ + 173, + 536, + 821, + 565 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Now about continuity: By Proposition F.18, we only need to show continuity in 0. Thus, let $( f _ { k } ) _ { k }$ be a sequence of functions $f _ { k } \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ with $\\begin{array} { r } { \\operatorname* { l i m } _ { k \\to \\infty } \\| f _ { k } \\| _ { L ^ { 2 } } = 0 } \\end{array}$ . Then we obtain ", + "bbox": [ + 173, + 571, + 823, + 602 + ], + "page_idx": 47 + }, + { + "type": "equation", + "img_path": "images/dca6f1578432107a4a1b1f2baa06d4f23072c495136500bc200437e25eb4f557.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\| \\widehat K ( f _ { k } ) \\| _ { V } = \\| \\int _ { X } f _ { k } ( x ) K ( x ) d x \\| _ { V } } } \\\\ & { } & { \\leq \\int _ { X } | f _ { k } ( x ) | \\cdot \\| K ( x ) \\| _ { V } d x } \\\\ & { } & { \\leq \\operatorname* { m a x } _ { x ^ { \\prime } } \\| K ( x ^ { \\prime } ) \\| _ { V } \\cdot \\int _ { X } | f _ { k } ( x ) | d x , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 608, + 651, + 713 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "where the continuity of $K$ proven in Proposition C.5 was used.20 For the right expression, using the Cauchy-Schwarz inequality Proposition F.34 we obtain ", + "bbox": [ + 173, + 720, + 821, + 751 + ], + "page_idx": 47 + }, + { + "type": "equation", + "img_path": "images/cac3ff319f075239e8cbbf6dfa74abadb31ab18fcb808ab360a13f2d32ef770f.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\int _ { X } | f _ { k } ( x ) | d x = \\int _ { X } | f _ { k } ( x ) | \\cdot 1 d x } \\\\ { \\displaystyle } & { = | \\langle | f _ { k } | | 1 \\rangle | } \\\\ { \\displaystyle } & { \\le \\| f _ { k } \\| _ { L ^ { 2 } } \\cdot \\| 1 \\| _ { L ^ { 2 } } } \\\\ { \\displaystyle } & { = \\| f _ { k } \\| _ { L ^ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 385, + 758, + 611, + 847 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "So, overall, if $\\begin{array} { r } { \\operatorname* { l i m } _ { k \\to \\infty } \\| f _ { k } \\| _ { L ^ { 2 } } = 0 } \\end{array}$ , then $\\begin{array} { r } { \\operatorname* { l i m } _ { k \\infty } \\| \\widehat { K } ( f _ { k } ) \\| _ { V } = 0 } \\end{array}$ as well, which proves continuity. ", + "bbox": [ + 171, + 856, + 821, + 875 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "C.2.3 WELL-DEFINEDNESS OF $( \\cdot ) | _ { X }$ ", + "text_level": 1, + "bbox": [ + 174, + 102, + 437, + 119 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "While it is clear that the limit $\\mathrm { l i m } _ { U \\in \\mathcal { U } _ { x } } \\mathcal { K } ( \\delta _ { U } )$ from Theorem C.8 is unique if it exists (Conway, 2014), it is somewhat unclear why it exists in the first place. For this, we need to better understand the properties of the (approximated) Dirac delta. The most important one is the following, which we hinted at already in the intuitions we gave before this section: basically, Dirac deltas help for evaluating continuous functions at specific points: ", + "bbox": [ + 173, + 126, + 826, + 198 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Lemma C.10. For each $x \\in X$ and $Y : X \\mathbb { K }$ continuous we have $\\begin{array} { r } { \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } { \\langle \\delta _ { U } | Y \\rangle } = Y ( x ) } \\end{array}$ . ", + "bbox": [ + 171, + 199, + 802, + 217 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Proof. We have ", + "bbox": [ + 173, + 229, + 281, + 244 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/b7b86c92ad150c33a45275fa3726046fdfaa5c5deefbc15e2999374b15c884d2.jpg", + "text": "$$\n\\begin{array} { r l } { { \\big | \\delta _ { U } | Y - Y ( x ) \\big | = \\bigg | \\int _ { X } \\delta _ { U } ( x ^ { \\prime } ) Y ( x ^ { \\prime } ) d x ^ { \\prime } - \\mu ( U ) \\cdot \\frac { 1 } { \\mu ( U ) } Y ( x ) \\bigg | } } \\\\ & { = \\bigg | \\int _ { U } \\frac { 1 } { \\mu ( U ) } Y ( x ^ { \\prime } ) d x ^ { \\prime } - \\int _ { U } \\frac { 1 } { \\mu ( U ) } Y ( x ) d x ^ { \\prime } \\bigg | } \\\\ & { = \\bigg | \\int _ { U } \\frac { 1 } { \\mu ( U ) } ( Y ( x ^ { \\prime } ) - Y ( x ) ) d x ^ { \\prime } \\bigg | } \\\\ & { \\leq \\int _ { U } \\frac { 1 } { \\mu ( U ) } | Y ( x ^ { \\prime } ) - Y ( x ) | d x ^ { \\prime } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 247, + 712, + 387 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Let $\\epsilon > 0$ . Since $Y$ is continuous in $x$ , there is $U _ { \\epsilon } \\in \\mathcal { U } _ { x }$ such that $Y ( x ^ { \\prime } ) \\in \\mathrm { B } _ { \\epsilon } ( Y ( x ) )$ for all $x ^ { \\prime } \\in U _ { \\epsilon }$ or, equivalently, $| Y ( x ^ { \\prime } ) - Y ( x ) | < \\epsilon$ . Thus, for all $U _ { \\epsilon } \\supseteq U$ , i.e., all $U _ { \\epsilon } \\leq U$ in $\\mathcal { U } _ { x }$ we obtain ", + "bbox": [ + 171, + 388, + 821, + 417 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/40be291b04f4b97b071122ed60728c7cdfe673910dad5d23c57ea76426012851.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\left| ~ \\left. \\delta _ { U } | Y \\right. - Y ( x ) \\right| \\le \\int _ { U } \\frac 1 { \\mu ( U ) } | Y ( x ^ { \\prime } ) - Y ( x ) | d x ^ { \\prime } } \\\\ { \\displaystyle \\qquad \\le \\int _ { U } \\frac 1 { \\mu ( U ) } \\epsilon d x ^ { \\prime } } \\\\ { \\displaystyle \\qquad = \\epsilon \\cdot \\mu ( U ) \\cdot \\frac 1 { \\mu ( U ) } } \\\\ { \\displaystyle \\qquad = \\epsilon } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 421, + 665, + 537 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "and consequently $\\begin{array} { r } { \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } { \\langle \\delta _ { U } | Y \\rangle } = Y ( x ) } \\end{array}$ . ", + "bbox": [ + 176, + 541, + 464, + 558 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Before we can show the well-definedness of $\\kappa | _ { X }$ , we first want to get a better description of $\\kappa$ . For this, recall from the Peter-Weyl theorem that $L _ { \\mathbb { K } } ^ { 2 } ( X ) = { \\widehat { \\bigoplus } } _ { l \\in { \\widehat { G } } } \\bigoplus _ { i = 1 } ^ { m _ { l } } V _ { l i }$ . With this at our disposal, bwe can formulate the following Lemma on the form of intertwiners on $L _ { \\mathbb { K } } ^ { 2 } ( X )$ : ", + "bbox": [ + 174, + 570, + 826, + 621 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Lemma C.11. Let ${ \\mathcal { K } } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to V$ be an intertwiner. Let $l \\in \\widehat { G }$ be the unique index such that $V \\cong V _ { l i }$ for all $i = 1 , \\dots , m _ { l }$ . Let $Y _ { l i } ^ { n }$ , $n = 1 , \\ldots , d _ { l }$ be an orthonormal basis of $V _ { l i }$ where $d _ { l } = \\dim ( V _ { l } )$ . Then ", + "bbox": [ + 173, + 625, + 825, + 667 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/644812e12b2851269ad2a706099054ea4883b5e3d203ba5d32799f1e513fed43.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { K } ( f ) = et { } { ' } \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { l } } \\langle Y _ { l i } ^ { n } | f \\rangle \\mathcal { K } ( Y _ { l i } ^ { n } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 666, + 629, + 695 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "for all $f \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ ", + "bbox": [ + 171, + 694, + 302, + 712 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Proof. We can write $f \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ according to the discussion after Definition F.40 as ", + "bbox": [ + 174, + 724, + 728, + 741 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/a4453e0086c061db85aa930c1c5601a224c39f80b3f32e296d79e19de1266a62.jpg", + "text": "$$\nf = \\sum _ { l ^ { \\prime } \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { l ^ { \\prime } } } \\sum _ { n = 1 } ^ { [ l ^ { \\prime } ] } \\langle Y _ { l ^ { \\prime } i } ^ { n } | f \\rangle Y _ { l ^ { \\prime } i } ^ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 742, + 633, + 773 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Note that $\\mathcal { K } \\vert _ { V _ { l ^ { \\prime } i } } : V _ { l ^ { \\prime } i } V$ is an intertwiner as well, and so by Schur’s Lemma B.29 it is necessarily zero unless ${ \\mathit { l } } ^ { \\prime } = l$ is the unique index such that $V _ { l i } \\cong V$ . Due to its continuity and linearity, $\\kappa$ commutes with infinite sums and we obtain ", + "bbox": [ + 176, + 776, + 821, + 818 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/987b2984b1b212e09ee4abfe24b717feafe760738662085b2adf9e877e32e389.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { K } ( f ) = \\sum _ { l ^ { \\prime } \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { l ^ { \\prime } } } \\sum _ { n = 1 } ^ { [ l ^ { \\prime } ] } \\left. Y _ { l ^ { \\prime } i } ^ { n } | f \\right. K \\left( Y _ { l ^ { \\prime } i } ^ { n } \\right) } \\\\ & { \\qquad = \\sum _ { l ^ { \\prime } \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { l ^ { \\prime } } } \\sum _ { n = 1 } ^ { [ l ^ { \\prime } ] } \\left. Y _ { l ^ { \\prime } i } ^ { n } | f \\right. K | _ { V _ { l ^ { \\prime } i } } \\left( Y _ { l ^ { \\prime } i } ^ { n } \\right) } \\\\ & { \\qquad = \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { l } } \\left. Y _ { l i } ^ { n } | f \\right. K ( Y _ { l i } ^ { n } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 818, + 673, + 911 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Corollary C.12. We have $\\begin{array} { r } { \\mathcal { K } | _ { X } ( x ) = \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { l } } \\overline { { Y _ { l i } ^ { n } ( x ) } } \\mathcal { K } ( Y _ { l i } ^ { n } ) } \\end{array}$ . In particular, the defining limit exists. ", + "bbox": [ + 173, + 99, + 825, + 133 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Proof. Since the $Y _ { l i } ^ { n }$ are by the proof of the Peter-Weyl theorem in the finite-dimensional space $\\mathcal { E } _ { l }$ spanned by matrix coefficients of the irreducible representation $\\rho _ { l } : G \\to \\operatorname { U } ( V _ { l } )$ and since these matrix coefficients are continuous by Remark B.25, the $Y _ { l i } ^ { n }$ are as finite linear combinations of them also continuous functions. Thus, from Lemma C.10 and C.11 together we obtain: ", + "bbox": [ + 174, + 147, + 825, + 204 + ], + "page_idx": 49 + }, + { + "type": "equation", + "img_path": "images/ae88804f510194dd697dd08d3a82bb35453e055056aa203d372cf001aaecf946.jpg", + "text": "$$\n\\begin{array} { r l } { { K | \\boldsymbol { x } ( \\boldsymbol { x } ) = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } K ( \\delta _ { U } ) } } \\\\ & { = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { l } } \\langle Y _ { l i } ^ { n } | \\delta _ { U } \\rangle K ( Y _ { l i } ^ { n } ) } \\\\ & { = \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { l } } [ \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\langle Y _ { l i } ^ { n } | \\delta _ { U } \\rangle ] K ( Y _ { l i } ^ { n } ) } \\\\ & { = \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { l } } \\frac { Y _ { l i } ^ { n } ( x ) } { Y _ { l i } ^ { n } ( x ) } K ( Y _ { l i } ^ { n } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 208, + 669, + 332 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "The complex conjugation came into play since the order in the scalar product is swapped compared to Lemma C.10. □ ", + "bbox": [ + 171, + 334, + 823, + 363 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Thus, since we now know that $\\kappa | _ { X }$ as a function makes sense, we can finally prove the welldefinedness of $\\kappa \\mapsto \\mathcal { K } | _ { X }$ , ", + "bbox": [ + 171, + 377, + 823, + 407 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Lemma C.13. The function $( \\cdot ) | _ { X } : \\mathrm { H o m } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , V ) \\to \\mathrm { H o m } _ { G } ( X , V )$ is well-defined, that is: for a linear, equivariant and continuous function ${ \\mathcal { K } } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to V$ , the restriction $\\mathcal { K } | _ { X } : X \\to V$ is equivariant. ", + "bbox": [ + 173, + 409, + 825, + 454 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Proof. We have ", + "bbox": [ + 174, + 468, + 281, + 483 + ], + "page_idx": 49 + }, + { + "type": "equation", + "img_path": "images/579584778d3ef31e6d785555e0e94b28914386a8c4c9a2cb5eee7f9601376c0b.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle K | { \\boldsymbol x } ( { \\boldsymbol g } \\cdot { \\boldsymbol x } ) = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } K ( \\delta { \\boldsymbol U } ) } } \\\\ { ~ = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } K ( \\delta _ { { \\boldsymbol U } { \\boldsymbol U } } ) } \\\\ { ~ = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } K ( \\lambda ( { \\boldsymbol g } ) \\delta _ { \\boldsymbol U } ) } \\\\ { ~ = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\rho ( { \\boldsymbol g } ) [ K ( \\delta _ { U } ) ] } \\\\ { ~ = \\rho ( \\boldsymbol { g } ) \\big [ \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } K ( \\delta _ { U } ) \\big ] } \\\\ { ~ = \\rho ( { \\boldsymbol g } ) [ K | x ( x ) ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 380, + 488, + 614, + 640 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "where the steps are justified as follows: The first step is just the definition of $\\kappa | _ { X }$ . The second step uses that the open neighborhood of $g x$ are precisely the $g$ -translated open neighborhoods of $x$ since $g : X \\to X$ is a homeomorphism. The third step is easy to check. The fourth step uses the equivariance of $\\kappa$ . The fifth step uses the continuity of $\\rho ( g )$ , which follows since $\\rho ( g )$ is a unitary transformation. The last step is again the definition of $\\kappa | _ { X }$ . □ ", + "bbox": [ + 174, + 642, + 826, + 713 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "C.2.4 $\\widehat { ( \\cdot ) }$ AND $( \\cdot ) | _ { X }$ ARE INVERSE TO EACH OTHER ", + "text_level": 1, + "bbox": [ + 173, + 728, + 544, + 747 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "We can now finish the proof of Theorem C.8 and consequently of Theorem C.7: ", + "bbox": [ + 174, + 755, + 696, + 771 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Proof of Theorem C.8. After all the preparation, we only need to still show that the maps $\\widehat { ( \\cdot ) }$ and $( \\cdot ) | _ { X }$ are inverse to each other. For ${ \\hat { K } } | _ { X } = K$ , i.e., the injectivity of the function $K \\mapsto { \\widehat { K } }$ and surjectivity of the function $\\kappa \\mapsto \\mathcal { K } | _ { X }$ , we compute: ", + "bbox": [ + 173, + 787, + 825, + 837 + ], + "page_idx": 49 + }, + { + "type": "equation", + "img_path": "images/33f3ed56f5523cedf70a7a20bceb7b8172cba25f8c636a50fcadc364a152f27e.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\left. \\widehat { K } \\right| _ { X } ( x ) = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\widehat { K } \\left( \\delta _ { U } \\right) } } \\\\ { ~ } \\\\ { { \\displaystyle ~ = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\int _ { X } \\delta _ { U } ( x ^ { \\prime } ) K ( x ^ { \\prime } ) d x ^ { \\prime } } } \\\\ { ~ } \\\\ { { \\displaystyle ~ = K ( x ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 372, + 842, + 624, + 922 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "The last step follows from Lemma C.10 by identifying $V = V _ { l }$ with $\\mathbb { K } ^ { d _ { l } }$ and viewing $K$ as consisting of continuous component functions $K ^ { n } : X \\mathbb { K }$ , $n \\in \\{ 1 , \\ldots , d _ { l } \\}$ . The continuity of $K$ was shown in Proposition C.5. ", + "bbox": [ + 176, + 102, + 823, + 146 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "For showing ${ \\widehat { \\ K | _ { X } } } = \\kappa$ we do a computation using the description of $\\kappa$ from Lemma C.11 and the description of $\\kappa | _ { X }$ from Corollary C.12: ", + "bbox": [ + 169, + 155, + 823, + 185 + ], + "page_idx": 50 + }, + { + "type": "equation", + "img_path": "images/0af451e0739914d8b98b71ee243bc7b6b5579f351a904420215c48b6b6502b49.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\widehat { K | _ { X } } ( f ) = \\int _ { X } f ( x ) \\mathcal { K } | _ { X } ( x ) d x } } \\\\ { { \\displaystyle = \\int _ { X } f ( x ) \\Big ( \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { l } } \\overline { { { Y _ { l i } ^ { n } ( x ) } } } \\mathcal { K } ( Y _ { l i } ^ { n } ) \\Big ) d x } } \\\\ { { \\displaystyle ~ = \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { i } } \\Big ( \\int _ { X } f ( x ) \\overline { { { Y _ { l i } ^ { n } ( x ) } } } d x \\Big ) \\mathcal { K } ( Y _ { l i } ^ { n } ) } } \\\\ { { \\displaystyle ~ = \\sum _ { i = 1 } ^ { m _ { l } } \\sum _ { n = 1 } ^ { d _ { i } } \\langle Y _ { l i } ^ { n } | f \\rangle \\mathcal { K } ( Y _ { l i } ^ { n } ) } } \\\\ { { \\displaystyle ~ = \\mathcal { K } ( f ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 209, + 681, + 359 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "This finally finishes the proof. ", + "bbox": [ + 174, + 382, + 372, + 397 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "D A WIGNER-ECKART THEOREM FOR STEERABLE KERNELS OF GENERAL COMPACT GROUPS ", + "text_level": 1, + "bbox": [ + 176, + 436, + 810, + 470 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "In Chapter C we have seen the most important theoretical insight of this work: steerable kernels on a homogeneous space $X$ correspond one-to-one to kernel operators (certain representation operators) on the space of square-integrable functions $L _ { \\mathbb { K } } ^ { 2 } ( X )$ . In this chapter, we will develop the most important consequence of this correspondence: a Wigner-Eckart theorem for steerable kernels and consequently a description of a basis for steerable kernels. This works for both fields $\\mathbb { R }$ and C, for an arbitrary compact group $G$ , an arbitrary homogeneous space $X$ and arbitrary finite-dimensional input- and output fields. Additionally, it covers the general theory of equivariant CNNs on homogeneous spaces developed in (Cohen et al., 2019b). ", + "bbox": [ + 174, + 498, + 825, + 611 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "In Section D.1 we will work towards formulating the most important theorems. Since these will involve tensor products, we will start with defining and studying tensor products of pre-Hilbert spaces and (unitary) representations. Afterward, we will define the Clebsch-Gordan coefficients, which relate a tensor product of irreducible representations to the irreducible subrepresentations of this tensor product. This will lead to a formulation of the original Wigner-Eckart theorem similar as it appears in quantum mechanics, including a proof. The original Wigner-Eckart theorem is a statement about representation operators on irreducible representations. However, we consider kernel operators on $\\dot { L } _ { \\mathbb { K } } ^ { 2 } ( X )$ which is not irreducible. Also, different from the original Theorem, we also consider representations over the real numbers, which leads to a replacement of reduced matrix elements by endomorphisms. Therefore we then formulate a generalization of the original theorem. Then, using the correspondence between kernel operators and steerable kernels from Theorem C.7, we can transform this into a Wigner-Eckart theorem for steerable kernels and ultimately a statement about a basis of the space of steerable kernels. We conclude with some remarks about how to use the basis kernels in practice. ", + "bbox": [ + 174, + 617, + 825, + 811 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Afterward, in Section D.2, we give the remaining proof of the Wigner-Eckart theorem for kernel operators, which we omit in the section before. First, we reduce the statement to the dense subspace of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ which is a direct sum of all irreducible subrepresentations. We then describe a correspondence between representation operators and intertwiners on a certain tensor product, the so-called hom-tensor adjunction. Finally, we finish with the full proof of the Wigner-Eckart theorem. ", + "bbox": [ + 174, + 818, + 825, + 888 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "As always, let $\\mathbb { K }$ be either of the two fields $\\mathbb { R }$ and $\\mathbb { C }$ and $G$ be a compact topological group. $X$ is any homogeneous space of $G$ . ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "D.1 A WIGNER-ECKART THEOREM FOR STEERABLE KERNELS AND THEIR KERNEL BASES ", + "bbox": [ + 173, + 103, + 818, + 118 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "D.1.1 TENSOR PRODUCTS OF PRE-HILBERT SPACES AND UNITARY REPRESENTATIONS ", + "bbox": [ + 173, + 128, + 789, + 145 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "In order to state the Wigner-Eckart theorem, we need the notion of representations on tensor products. This is defined similarly to Hom-representations, see Definition C.1. For this, we first need to discuss the notion of a tensor product of vector spaces: ", + "bbox": [ + 176, + 154, + 823, + 196 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Definition D.1 (Tensor Product). Let $V$ and $V ^ { \\prime }$ be two vector spaces over $\\mathbb { K }$ . Then $V \\otimes V ^ { \\prime }$ , the tensor product of $V$ and $V ^ { \\prime }$ , is a vector space over $\\mathbb { K }$ with the following properties: ", + "bbox": [ + 171, + 200, + 823, + 231 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "1. There is a bilinear function $\\otimes : V \\times V ^ { \\prime } \\to V \\otimes V ^ { \\prime } , ( v , v ^ { \\prime } ) \\mapsto v \\otimes v ^ { \\prime } . V \\otimes V ^ { \\prime }$ is generated by elements of the form $v \\otimes v ^ { \\prime }$ . ", + "bbox": [ + 204, + 241, + 825, + 271 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "2. It has the following universal property: for any bilinear function $\\beta : V \\times V ^ { \\prime } \\to P$ into a vector space $P$ , there is a unique linear function ${ \\overline { { \\beta } } } : V \\otimes V ^ { \\prime } \\to P$ given on elements of the form $v \\otimes v ^ { \\prime }$ by $\\overline { { \\beta } } ( v \\otimes v ^ { \\prime } ) = \\beta ( v , v ^ { \\prime } )$ . In other words, the following diagram commutes: ", + "bbox": [ + 212, + 279, + 825, + 327 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/3c876d25ad074686f589b61f9123db4fcbc095c8f27031ebc7e6cf0fd432635b.jpg", + "text": "$$\n\\begin{array} { l } { { V \\times V ^ { \\prime } \\xrightarrow [ ] { \\beta } P } } \\\\ { { \\otimes } } \\\\ { { V \\otimes V ^ { \\prime } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 465, + 335, + 589, + 400 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "3. If $V$ and $V ^ { \\prime }$ are finite-dimensional with bases $\\{ v _ { 1 } , \\ldots , v _ { n } \\} \\subseteq V$ and $\\{ v _ { 1 } ^ { \\prime } , \\ldots , v _ { m } ^ { \\prime } \\} \\subseteq V ^ { \\prime }$ , then $\\{ v _ { i } \\otimes v _ { j } ^ { \\prime } \\} _ { i , j } \\subseteq V \\otimes V ^ { \\prime }$ is a basis of $V \\otimes V ^ { \\prime }$ . In particular, the dimension of $V \\otimes V ^ { \\prime }$ is $n \\cdot m$ . ", + "bbox": [ + 210, + 417, + 826, + 460 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Property 3 follows from 1 and 2 and would therefore not necessarily be needed in the definition. The explicit construction of tensor products shall not matter for our purposes since the properties above characterize it up to isomorphism. The second property stated in the definition is of large importance since it tells us how we can define linear functions on $V \\otimes V ^ { \\prime }$ : if we have a guess for such a function $\\varphi : V \\otimes V ^ { \\prime } \\to P$ (of which we don’t yet know whether its “assignment rule” is well-defined), then we just need to test whether the function $\\tilde { \\varphi } : V \\times V ^ { \\prime } \\to P$ given by $\\tilde { \\varphi } ( v , v ^ { \\prime } ) : = \\varphi ( v \\otimes v ^ { \\prime } )$ is bilinear. If it is, then $\\varphi$ is a well-defined linear function. We will use this soon in the following context: Assume $f : V \\to V$ and $g : V ^ { \\prime } \\to V ^ { \\prime }$ are linear functions. Then we would like to define a function $f \\otimes g : V \\otimes V ^ { \\prime } \\to V \\otimes V ^ { \\prime }$ by $( f \\otimes g ) ( v \\otimes v ^ { \\prime } ) = f ( v ) \\otimes g ( v ^ { \\prime } )$ . For this to work, we need to test whether the assignment $( v , v ^ { \\prime } ) \\mapsto f ( v ) \\otimes g ( v ^ { \\prime } )$ is a bilinear function $V \\times V ^ { \\prime } \\to V \\otimes V ^ { \\prime }$ . Clearly, it is, and so $f \\otimes g$ is a well-defined linear function! We use this in Definition D.3 in order to define the tensor product of representations. ", + "bbox": [ + 173, + 472, + 825, + 640 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Since we actually deal with Hilbert spaces most of the time, we would like to build tensor products of Hilbert spaces. However, their definition is not completely straightforward since one cannot just take the tensor product of the underlying vector spaces but needs to additionally build the completion of the resulting space (Kadison & Ringrose, 1997). Since this complicates the considerations related to a correspondence we later formulate in Proposition D.23, we go a slightly different route. Instead of describing the tensor product of Hilbert spaces, we describe the tensor product of pre-Hilbert spaces, which does not require a completion step. Recall from Definition F.3 that a pre-Hilbert space is basically a Hilbert space that is not necessarily complete. ", + "bbox": [ + 173, + 646, + 825, + 758 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Definition D.2 (Tensor Product of pre-Hilbert spaces). Let $V , V ^ { \\prime }$ be two pre-Hilbert spaces with scalar products $\\langle \\cdot | \\cdot \\rangle$ and $\\langle \\cdot | \\cdot \\rangle ^ { \\prime }$ . Then the tensor product of vector spaces $V \\otimes V ^ { \\prime }$ can be made into a pre-Hilbert space using the scalar product which is given on generators by ", + "bbox": [ + 173, + 762, + 825, + 806 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/00df4f3662c0bbf1ae58bfc744152bd2f7fc89076384dfe6b579f0b187d074f8.jpg", + "text": "$$\n\\left. v \\otimes v ^ { \\prime } | w \\otimes w ^ { \\prime } \\right. _ { \\otimes } : = \\left. v | w \\right. \\cdot \\left. v ^ { \\prime } | w ^ { \\prime } \\right. ^ { \\prime } .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 814, + 627, + 835 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "This is then anti-linearly extended in the first (i.e., “Bra”), and linearly extended in the second (i.e., “Ket”) component. ", + "bbox": [ + 171, + 840, + 825, + 869 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "One can show that this makes $V \\otimes V ^ { \\prime }$ a pre-Hilbert space. For simplicity, we will from now on not notationally distinguish the different scalar products involved. With this preparation, we can come to the notion of tensor product representations: ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Definition D.3 (Tensor Product Representation). Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ and $\\rho ^ { \\prime } : G \\to { \\mathrm { G L } } ( V ^ { \\prime } )$ be two linear representations, where $V$ and $V ^ { \\prime }$ are pre-Hilbert spaces. Then on the tensor product $V \\otimes V ^ { \\prime }$ of pre-Hilbert spaces, we can define the tensor product representation $\\rho \\otimes \\rho ^ { \\prime }$ by ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 52 + }, + { + "type": "equation", + "img_path": "images/2bbecbf736218406ac2737cf35d064fd534d8c8e18d30e57d1280bc8c603a5ec.jpg", + "text": "$$\n\\rho \\otimes \\rho ^ { \\prime } : G \\to { \\mathrm { G L } } ( V \\otimes V ^ { \\prime } ) , g \\mapsto \\rho ( g ) \\otimes \\rho ^ { \\prime } ( g ) ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 155, + 655, + 174 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "where $\\rho ( g ) \\otimes \\rho ^ { \\prime } ( g ) : V \\otimes V ^ { \\prime } \\to V \\otimes V ^ { \\prime }$ is given on generators by ", + "bbox": [ + 174, + 183, + 612, + 200 + ], + "page_idx": 52 + }, + { + "type": "equation", + "img_path": "images/1f31bb4ee1e2b6e70ece621e4c1312c1014fbd1c7047912e51742951ffe84774.jpg", + "text": "$$\n\\left( \\rho ( g ) \\otimes \\rho ^ { \\prime } ( g ) \\right) ( v \\otimes v ^ { \\prime } ) : = \\rho ( g ) ( v ) \\otimes \\rho ^ { \\prime } ( g ) ( v ^ { \\prime } ) .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 209, + 656, + 228 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Lemma D.4. The map $\\rho \\otimes \\rho ^ { \\prime } : G \\to { \\mathrm { G L } } ( V \\otimes V ^ { \\prime } )$ defined above is a linear representation. ", + "bbox": [ + 171, + 237, + 772, + 253 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Proof. Clearly, each $( \\boldsymbol { \\rho } \\otimes \\boldsymbol { \\rho } ^ { \\prime } ) ( \\boldsymbol { g } )$ is linear and we have $( \\rho \\otimes \\rho ^ { \\prime } ) ( g g ^ { \\prime } ) = ( \\rho \\otimes \\rho ^ { \\prime } ) ( g ) \\circ ( \\rho \\otimes \\rho ^ { \\prime } ) ( g ^ { \\prime } )$ . Thus, for showing that it is a linear representation, we need to show it is continuous. Assume we already knew continuity of all maps $( \\bar { \\rho \\otimes } \\rho ^ { \\prime } ) ^ { v \\otimes v ^ { \\prime } } : G \\to V \\otimes V ^ { \\prime }$ , $g \\mapsto \\left[ ( \\rho \\otimes \\rho ^ { \\prime } ) ( g ) \\right] ( v \\otimes v ^ { \\prime } )$ . Then for linear combinations $\\begin{array} { r } { \\xi = \\sum _ { i = 1 } ^ { n } \\lambda _ { i } \\left( v _ { i } \\otimes v _ { i } ^ { \\prime } \\right) } \\end{array}$ we obtain using the linearity of $( \\boldsymbol { \\rho } \\otimes \\boldsymbol { \\rho } ^ { \\prime } ) ( \\boldsymbol { g } )$ : ", + "bbox": [ + 173, + 280, + 825, + 343 + ], + "page_idx": 52 + }, + { + "type": "equation", + "img_path": "images/8f2dfc0e17b71da6878e3b03195c6003894d24cfea9f78ce7a3a4a3d725256be.jpg", + "text": "$$\n\\begin{array} { l } { { ( \\rho \\otimes \\rho ^ { \\prime } ) ^ { \\xi } ( g ) = \\left[ ( \\rho \\otimes \\rho ^ { \\prime } ) ( g ) \\right] ( \\xi ) } } \\\\ { { \\ = \\left[ ( \\rho \\otimes \\rho ^ { \\prime } ) ( g ) \\right] \\left( \\sum _ { i = 1 } ^ { n } \\lambda _ { i } \\left( v _ { i } \\otimes v _ { i } ^ { \\prime } \\right) \\right) } } \\\\ { { \\ = \\sum _ { i = 1 } ^ { n } \\lambda _ { i } \\left[ ( \\rho \\otimes \\rho ^ { \\prime } ) ( g ) \\right] ( v _ { i } \\otimes v _ { i } ^ { \\prime } ) } } \\\\ { { \\ = \\left( \\sum _ { i = 1 } ^ { n } \\lambda _ { i } ( \\rho \\otimes \\rho ^ { \\prime } ) ^ { v _ { i } \\otimes v _ { i } ^ { \\prime } } \\right) ( g ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 352, + 671, + 458 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Now, since scalar multiplication and addition in topological vector spaces is continuous, and since pre-Hilbert spaces are special topological vector spaces, the continuity of $( \\boldsymbol { \\rho } \\otimes \\boldsymbol { \\rho } ^ { \\prime } ) ^ { \\xi }$ follows from that of all $( \\rho \\otimes \\rho ^ { \\prime } ) ^ { v \\otimes v ^ { \\prime } }$ . ", + "bbox": [ + 176, + 464, + 823, + 510 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "What’s left is proving the continuity of functions of the form $( \\rho \\otimes \\rho ^ { \\prime } ) ^ { v \\otimes v ^ { \\prime } }$ . For notational simplicity, write $f = \\rho ^ { v } : G \\to V$ and $f ^ { \\prime } : \\rho ^ { \\prime } { } ^ { v ^ { \\prime } }$ , which are both continuous since $\\rho$ and $\\rho ^ { \\prime }$ are linear representations. We want to show that also $f \\otimes f ^ { \\prime } : G \\to V \\otimes V ^ { \\prime }$ is continuous. We can test continuity in each point $g _ { 0 } \\in G$ separately by Definition F.6. For each $g \\in G$ we then obtain, with $\\mathrm { R e }$ being the real part of a complex number: ", + "bbox": [ + 173, + 517, + 825, + 592 + ], + "page_idx": 52 + }, + { + "type": "equation", + "img_path": "images/d492ecb42451edc60766e319b6b972e9a4839e664a26c9f536c18bbd9e6aee4a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| ( f \\otimes f ^ { \\prime } ) ( g ) - ( f \\otimes f ^ { \\prime } ) ( g _ { 0 } ) \\| ^ { 2 } } \\\\ & { \\qquad = \\| [ f ( g ) \\otimes f ^ { \\prime } ( g ) - f ( g ) \\otimes f ^ { \\prime } ( g _ { 0 } ) ] + [ f ( g ) \\otimes f ^ { \\prime } ( g _ { 0 } ) - f ( g _ { 0 } ) \\otimes f ^ { \\prime } ( g _ { 0 } ) ] \\| ^ { 2 } } \\\\ & { \\qquad = \\| f ( g ) \\otimes [ f ^ { \\prime } ( g ) - f ^ { \\prime } ( g _ { 0 } ) ] + [ f ( g ) - f ( g _ { 0 } ) ] \\otimes f ^ { \\prime } ( g _ { 0 } ) \\| ^ { 2 } } \\\\ & { \\qquad = \\| f ( g ) \\otimes [ f ^ { \\prime } ( g ) - f ^ { \\prime } ( g _ { 0 } ) ] \\| ^ { 2 } + \\| [ f ( g ) - f ( g _ { 0 } ) ] \\otimes f ^ { \\prime } ( g _ { 0 } ) \\| ^ { 2 } } \\\\ & { \\qquad + 2 \\operatorname { R e } f ( g ) \\otimes [ f ^ { \\prime } ( g ) - f ^ { \\prime } ( g _ { 0 } ) ] | [ f ( g ) - f ( g _ { 0 } ) ] \\otimes f ^ { \\prime } ( g _ { 0 } ) } \\\\ & { \\qquad = \\| f ( g ) \\| ^ { 2 } \\cdot \\| f ^ { \\prime } ( g ) - f ^ { \\prime } ( g _ { 0 } ) \\| ^ { 2 } + \\| f ( g ) - f ( g _ { 0 } ) \\| ^ { 2 } \\cdot \\| f ^ { \\prime } ( g _ { 0 } ) \\| ^ { 2 } } \\\\ & { \\qquad + 2 \\operatorname { R e } ( \\langle f ( g ) | f ( g ) - f ^ { \\prime } ( g _ { 0 } ) \\| ^ { 2 } + \\| f ( g ) - f ( g _ { 0 } ) \\| ^ { 2 } \\cdot \\| f ^ { \\prime } ( g _ { 0 } ) \\| ^ { 2 } } \\\\ & { \\qquad + 2 \\operatorname { R e } ( \\langle f ( g ) | f ( g ) - f ( g _ { 0 } ) \\rangle \\cdot \\langle f ^ { \\prime } ( g ) - f ^ { \\prime } ( g _ { 0 } ) | f ^ { \\prime } ( g _ { 0 } ) \\rangle ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 598, + 784, + 766 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "All in all we see the following: If $g$ is sufficiently close to $g _ { 0 }$ , then due to the continuity of $f , f ^ { \\prime }$ , the scalar product, multiplication in $\\mathbb { K }$ and the real part, $\\| ( f \\otimes f ^ { \\prime } ) ( g ) - ( f \\otimes f ^ { \\prime } ) ( g _ { 0 } ) \\| ^ { 2 }$ gets arbitrarily close to 0. This shows the continuity of $f \\otimes f ^ { \\prime }$ and we are done. □ ", + "bbox": [ + 173, + 773, + 826, + 818 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Lemma D.5. Let $\\rho : G \\to \\operatorname { U } ( V )$ and $\\rho ^ { \\prime } : G \\to \\operatorname { U } ( V ^ { \\prime } )$ be unitary representations on pre-Hilbert spaces. Then also $\\rho \\otimes \\rho ^ { \\prime } : G \\to \\operatorname { U } ( V \\otimes V ^ { \\prime } )$ is a well-defined unitary representation. ", + "bbox": [ + 173, + 837, + 823, + 868 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Proof. According to Lemma D.4 we only need to check whether all $\\rho ( g ) \\otimes \\rho ^ { \\prime } ( g )$ are unitary transformations. This follows immediately from the unitarity of $\\rho ( g )$ and $\\rho ^ { \\prime } ( g )$ . ", + "bbox": [ + 173, + 895, + 825, + 925 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "D.1.2 THE CLEBSCH-GORDAN COEFFICIENTS AND THE ORIGINAL WIGNER-ECKART THEOREM ", + "bbox": [ + 169, + 103, + 776, + 132 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "In this section, we describe the Clebsch-Gordan coefficients and the original Wigner-Eckart theorem. Except for the proof, we roughly follow Jeevanjee (2011). For the proof, we follow the more general treatment in Agrawala (1980).21 ", + "bbox": [ + 174, + 141, + 825, + 184 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "For our aims, let $\\rho _ { j } : G \\to \\operatorname { U } ( V _ { j } )$ and $\\rho _ { l } : G \\to \\operatorname { U } ( V _ { l } )$ be representatives of isomorphism classes of irreducible unitary representations.22 Then consider their tensor product representation ", + "bbox": [ + 169, + 190, + 825, + 222 + ], + "page_idx": 53 + }, + { + "type": "equation", + "img_path": "images/484f4e58ce77b2f8dedf567f4c3956bba3c830d978b527ca39fd46c06a5b7fac.jpg", + "text": "$$\n\\rho _ { j } \\otimes \\rho _ { l } : G \\to \\operatorname { U } ( V _ { j } \\otimes V _ { l } )\n$$", + "text_format": "latex", + "bbox": [ + 408, + 228, + 589, + 247 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "which is again a unitary representation according to Lemma D.5. If $V _ { j }$ and $V _ { l }$ are of dimension $d _ { j }$ and $d _ { l }$ , respectively, then $V _ { j } \\otimes V _ { l }$ is of dimension $d _ { j } \\cdot d _ { l }$ . Since it is a finite-dimensional unitary representation, it is itself an orthogonal direct sum of finitely many irreducible unitary representations by Proposition B.38: ", + "bbox": [ + 173, + 253, + 825, + 309 + ], + "page_idx": 53 + }, + { + "type": "equation", + "img_path": "images/63e53b0109dfe6985d7ed799f98fb01eb1d0f8d025ade18a006642a84e0859a7.jpg", + "text": "$$\nV _ { j } \\otimes V _ { l } \\cong \\bigoplus _ { J \\in \\widehat { G } } \\bigoplus _ { s = 1 } ^ { [ J ( j l ) ] } V _ { J } .\n$$", + "text_format": "latex", + "bbox": [ + 415, + 308, + 583, + 358 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Here $\\widehat { G }$ is, as before, the set of isomorphism classes of irreducible unitary representations and $\\left[ J ( j l ) \\right]$ is the number of times that $\\rho _ { J } : G \\to \\operatorname { U } ( V _ { J } )$ appears in the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ . Note that for most $J$ we have $[ J ( j l ) ] = 0$ , and for some $J$ we may have $[ J ( j l ) ] > 1$ , see Section E.2, where it turns out that $\\rho _ { 0 }$ is contained twice in $\\rho _ { m } \\otimes \\rho _ { m }$ . ", + "bbox": [ + 173, + 363, + 825, + 421 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Now, choose – once and for all – orthonormal bases of all involved irreps, which exists according to Proposition F.41: ", + "bbox": [ + 171, + 426, + 823, + 455 + ], + "page_idx": 53 + }, + { + "type": "equation", + "img_path": "images/2bb493214f56ed22810abb60de5e3a1efaf576f6a43528fa0418136dd728b963.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\left\\{ Y _ { j } ^ { m } \\mid m = 1 , \\ldots , d _ { j } \\right\\} \\subseteq V _ { j } , } \\\\ & { } & { \\left\\{ Y _ { l } ^ { n } \\mid n = 1 , \\ldots , d _ { l } \\right\\} \\subseteq V _ { l } , } \\\\ & { } & { \\left\\{ Y _ { J } ^ { M } \\mid M = 1 , \\ldots , d _ { J } \\right\\} \\subseteq V _ { J } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 393, + 462, + 602, + 525 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "This notation is supposed to remind about spherical harmonics since they form a basis for irreducible representations of the group SO(3). But as mentioned in the footnote, we do not consider these basis elements to be functions here. ", + "bbox": [ + 174, + 529, + 823, + 571 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Furthermore, let $l _ { s } : V _ { J } V _ { j } \\otimes V _ { l }$ be the linear, equivariant and isometric (i.e., scalar product preserving) embeddings that correspond to the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ into irreps, where $s$ ranges in $\\{ 1 , \\ldots , [ \\bar { J ( j l ) } ] \\}$ . With this in mind, we can define the Clebsch-Gordan coefficients: ", + "bbox": [ + 176, + 577, + 820, + 621 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Definition D.6 (Clebsch-Gordan Coefficients). The Clebsch-Gordan Coefficients are given by ", + "bbox": [ + 173, + 625, + 790, + 640 + ], + "page_idx": 53 + }, + { + "type": "equation", + "img_path": "images/e6102f2dbec4dbde818b68959c5c9a5826d968e5cc7b7ddf88e14a31117cfba8.jpg", + "text": "$$\n\\langle s , J M | j m ; l n \\rangle : = \\Big \\langle l _ { s } ( Y _ { J } ^ { M } ) \\big | Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n } \\Big \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 647, + 637, + 670 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Note that in the literature, people usually only consider Clebsch-Gordan coefficients of the specific groups SO(3), SU(2), SU(3) or similar groups appearing in physics. Also note that in the physics context, there is only one linear, equivariant, isometric embedding $l _ { s }$ , which follows directly from Schur’s Lemma D.8. Therefore, it is sensible that the embedding is usually not part of the notation of these coefficients. In our case, however, when considering real representations, there can be several such embeddings $l _ { s }$ . This happens if the endomorphism space of $V _ { J }$ is nontrivial. An example is given by the two-dimensional irreducible representations of SO(2) over the real numbers which we discuss in Section E.2. Since, however, we do not want to depart too much from the notation usually considered in physics, we also omit the embedding from the notation. The index $s$ however needs to be present in order to index the possibly different appearances of $V _ { J }$ in $V _ { j } \\otimes V _ { l }$ . ", + "bbox": [ + 173, + 684, + 825, + 824 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "With this preparation, we can explain the Wigner-Eckart theorem the way it is usually considered in physics, as a prelude for the generalization that we consider in the next section. ", + "bbox": [ + 174, + 830, + 823, + 859 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "In this (and only this!) section, we assume that our field is $\\mathbb { C }$ , since this is the case considered in physics. The Wigner-Eckart theorem aims to obtain a description for all possible representation operators ${ \\mathcal { K } } : { \\bar { V _ { j } } } \\operatorname { H o m } _ { \\mathbb { C } } ( V _ { l } , V _ { J } )$ . This is, for example, useful for describing state transitions in the electrons of hydrogen atoms. To motivate the generalization in the next section, we shortly explain the derivation: we can consider the equivalent function ${ \\mathcal { K } } : V _ { j } \\otimes V _ { l } \\to V _ { J }$ given by $\\tilde { \\mathcal { K } } ( v _ { j } \\otimes v _ { l } ) : = \\left[ \\mathcal { K } ( v _ { j } ) \\right] ( v _ { l } )$ on the tensor product. As one can compute, and as we will see in more generality in Proposition D.23, ${ \\tilde { \\cal K } } : V _ { j } \\otimes V _ { l } V _ { J }$ is an intertwiner, where on the left we consider the tensor product representation. We assume, as is the case for $G = \\mathrm { S O ( 3 ) }$ or $G = { \\mathrm { S U } } ( 2 )$ for usual applications in physics, that $V _ { J }$ is exactly once a direct summand of $V _ { j } \\otimes V _ { l }$ . Then, since by Schur’s Lemma B.29 there cannot be nontrivial equivariant linear maps between nonisomorphic irreps, $\\tilde { \\mathcal { K } }$ restricted to each direct summand of $V _ { j } \\otimes V _ { l }$ vanishes, except the one isomorphic to $V _ { J }$ . More precisely, assume that ", + "bbox": [ + 173, + 103, + 825, + 276 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/c6120224aabee95a65f8120a79b00e52e941d97dac7af2582d8741df58dcbad6.jpg", + "text": "$$\nV _ { j } \\otimes V _ { l } \\cong V _ { J } \\oplus \\bigoplus _ { l ^ { \\prime } } V _ { l ^ { \\prime } }\n$$", + "text_format": "latex", + "bbox": [ + 418, + 275, + 578, + 308 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "is a decomposition of $V _ { j } \\otimes V _ { l }$ into copies of irreducible representations, where each $V _ { l ^ { \\prime } }$ is nonisomorphic to $V _ { J }$ . Then the information contained in $\\tilde { \\kappa }$ is equal to the information contained in the restriction ${ \\tilde { \\cal K } } | _ { V _ { J } } : V _ { J } V _ { J }$ . Since it is an intertwiner from a representation to itself, it deserves a special name. We state the following definition for arbitrary $\\mathbb { K } \\in \\{ \\mathbb { R } , \\mathbb { C } \\}$ , since it will be of crucial importance in our generalization of the Wigner-Eckart theorem: ", + "bbox": [ + 173, + 309, + 825, + 383 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Definition D.7 (Endomorphism). Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ be a linear representation. An intertwiner from $V$ to $V$ is called endomorphism. The vector space of endomorphisms is written as ", + "bbox": [ + 171, + 386, + 823, + 415 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/d58770c109a20434dba5c76ffd1f555e630781bcbea2e7d1079334392b97f92c.jpg", + "text": "$$\n\\operatorname { E n d } _ { G , \\mathbb { K } } ( V ) : = \\operatorname { H o m } _ { G , \\mathbb { K } } ( V , V ) .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 417, + 606, + 435 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "A version of Schur’s lemma gives a simple description for endomorphisms of irreducible representations in the case that the underlying field is the complex numbers $\\mathbb { C }$ . It makes use of the property of the complex numbers to be algebraically closed: ", + "bbox": [ + 173, + 445, + 825, + 488 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Lemma D.8 (Schur’s Lemma). Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ be an irreducible representation. If the underlying field is the complex numbers $\\mathbb { C }$ , then the set of endomorphisms, i.e., intertwiners from $V$ to $V$ , only consists of the complex multiples of the identity: ", + "bbox": [ + 173, + 491, + 823, + 534 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/367bf499a3f822ff7b63cbce34145686b6b33c6f5b985e9596b479b0683a526b.jpg", + "text": "$$\n\\operatorname { E n d } _ { G , \\mathbb { C } } ( V ) = \\{ c \\cdot { \\mathrm { i d } } _ { V } \\mid c \\in \\mathbb { C } \\} \\cong \\mathbb { C } .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 536, + 625, + 554 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Proof. See Jeevanjee (2011). ", + "bbox": [ + 174, + 568, + 366, + 582 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "This means that ${ \\tilde { \\mathcal { K } } } | _ { V J } = c \\cdot \\operatorname { i d } _ { V J }$ for some complex number $c \\in \\mathbb { C }$ . Now if we let $p : V _ { j } \\otimes V _ { l } \\to V _ { J }$ be the projection corresponding to the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ , then we obtain ", + "bbox": [ + 173, + 598, + 825, + 628 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/177cc3edafea6431e5e44bb52a6ce4de2ee43e8c6efc584cb310ddeb6c6dac89.jpg", + "text": "$$\n\\tilde { \\cal K } = \\tilde { \\cal K } | _ { V _ { J } } \\circ p = ( c \\cdot \\mathrm { i d } _ { V _ { J } } ) \\circ p = c \\cdot p .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 632, + 625, + 651 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "That is, we have just found out that one complex number, $c$ , is able to completely characterize $\\tilde { \\kappa }$ and consequently $\\kappa !$ This is basically already the Wigner-Eckart theorem. However, it is useful to find a formulation that describes $\\kappa$ with respect to bases of the different irreducible representations. For this, we define matrix elements of representation operators. Before we come to the definition, we introduce some notation: If $f : V \\to V ^ { \\prime }$ is a linear continuous map between Hilbert spaces, we set ", + "bbox": [ + 173, + 655, + 826, + 738 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/9a6454b077cc92efa5e13d6eb4e65f63155c2d3d32f4d55e25cab7b0c18591f9.jpg", + "text": "$$\n\\langle y | f | x \\rangle : = \\langle y | f ( x ) \\rangle\n$$", + "text_format": "latex", + "bbox": [ + 431, + 738, + 566, + 756 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "for each $x \\in V$ and $y \\in V ^ { \\prime }$ . The symmetry in this notation is supposed to remind about the fact that $f$ has an adjoint, see Definition F.42, and thus can be applied to $y$ just as well as to $x$ , but we will not make use of this fact. ", + "bbox": [ + 174, + 756, + 825, + 797 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Definition D.9 (Matrix Element). Let $T , V _ { l }$ and $V _ { J }$ be unitary representations with orthonormal bases $\\{ Y _ { j } ^ { m } \\} \\subseteq T$ (with $j$ possibly also varying), $\\{ Y _ { l } ^ { n } \\} \\subseteq V _ { l }$ and $\\left\\{ Y _ { J } ^ { M } \\right\\} \\subseteq V _ { J }$ , respectively. Let ${ \\mathcal { K } } : T \\to { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ be a representation operator. Then it’s matrix elements are given by the scalars ", + "bbox": [ + 173, + 800, + 825, + 857 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/36fe667989f35d4559236f3b432e5f32b34d68190bd044e1cad792e3cb05217c.jpg", + "text": "$$\n\\big < J M | \\mathcal { K } _ { j } ^ { m } \\big | l n \\big > : = \\big < Y _ { J } ^ { M } \\big | \\mathcal { K } ( Y _ { j } ^ { m } ) \\big | Y _ { l } ^ { n } \\big > .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 856, + 625, + 876 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "In the same way, if $f : V _ { l } \\to V _ { J }$ is any linear (not necessarily equivariant) map, then its matrix elements are given by the scalars ", + "bbox": [ + 171, + 876, + 825, + 904 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/94038cfd626d64f94e037fa0f0cf16012c855700ad9fcb80c0e6014434a263e8.jpg", + "text": "$$\n\\langle J M | f | l n \\rangle : = \\left. Y _ { J } ^ { M } \\big | f \\big | Y _ { l } ^ { n } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 906, + 598, + 926 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Remark D.10. We shortly explain this term. Usually, in linear algebra, one has to do with linear functions $f : V \\to V ^ { \\prime }$ between vector spaces carrying bases $\\{ v _ { j } \\} \\subseteq V$ and $\\{ v _ { i } ^ { \\prime } \\} \\subseteq V ^ { \\prime }$ . For each basis element $v _ { j } \\in V$ one can then find coefficients $A _ { i j } \\in \\mathbb { K }$ such that ", + "bbox": [ + 173, + 103, + 825, + 147 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/81e10038932007cb28bdefe1cdb2bd86df0bf9ed43b27b43a7c8e70dad0e920d.jpg", + "text": "$$\nf ( v _ { j } ) = \\sum _ { i } A _ { i j } v _ { i } ^ { \\prime } .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 154, + 566, + 180 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "The $A _ { i j }$ are called the matrix elements of $f$ and characterize $f$ completely. Now if the bases are orthonormal bases as in Definition F.40, then the coefficients are given by ", + "bbox": [ + 173, + 186, + 823, + 215 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/96db351404cdb3d382f651f496c85318772365fab4b8e93f08c1bb985fc140f3.jpg", + "text": "$$\nA _ { i j } = \\langle v _ { i } ^ { \\prime } | f ( v _ { j } ) \\rangle = \\langle v _ { i } ^ { \\prime } | f | v _ { j } \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 395, + 222, + 601, + 241 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "In a similar way we can understand the matrix elements of a representation operator, only that the linear function itself depends on a chosen basis vector of $V _ { j }$ . As for linear functions, the matrix elements of a representation operator completely characterize it. ", + "bbox": [ + 174, + 247, + 825, + 290 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "One last remark: since in this section, $V _ { J }$ appears only once as a direct summand in $V _ { j } \\otimes V _ { l }$ , we omit the additional “quantum number” $s$ in the notation for the Clebsch-Gordan coefficients. With this preparation, we can formulate and prove the original version of the Wigner-Eckart theorem. Remember that there is a unique complex number $c$ such that $\\tilde { \\kappa }$ is given by $\\tilde { \\mathcal { K } } = c \\cdot p$ for a projection $p : V _ { j } \\otimes V _ { l } \\to V _ { J }$ . We now denote this by $\\langle J \\| \\mathcal { K } \\| l \\rangle : = c$ . ", + "bbox": [ + 173, + 297, + 825, + 371 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Theorem D.11 (Wigner-Eckart Theorem). The matrix elements of the representation operator $\\kappa$ : $V _ { j } \\to { \\mathrm { H o m } } _ { \\mathbb { C } } ( V _ { l } , V _ { J } )$ are given by ", + "bbox": [ + 173, + 375, + 823, + 404 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/7324d55c47c62484a633353411020bb50f3336b5f963de5b71dee6e349923997.jpg", + "text": "$$\n\\big \\langle J M \\big | K _ { j } ^ { m } \\big | l n \\big \\rangle = \\big \\langle J \\| K \\| l \\big \\rangle \\cdot \\big \\langle J M \\big | j m ; l n \\big \\rangle ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 411, + 640, + 433 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "with the $\\langle J M | j m ; l n \\rangle$ being the Clebsch-Gordan coefficients (which are independent from the representation operator $\\mathcal { K }$ ). ", + "bbox": [ + 169, + 440, + 821, + 469 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Proof. Let $i : V _ { J } \\to V _ { j } \\otimes V _ { l }$ be the embedding corresponding to the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ . It is an adjoint of the projection $p : V _ { j } \\otimes V _ { l } \\to V _ { J }$ according to the proof of Proposition F.46. By what we’ve argued above, there exists some $c \\in \\mathbb { C }$ such that: ", + "bbox": [ + 174, + 486, + 825, + 530 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/72db206744ff4a6440782d73a7062d01f40831d6982df0d5a97eec709aab3229.jpg", + "text": "$$\n\\begin{array} { r l } & { \\langle J M | K _ { j } ^ { m } | l n \\rangle = \\langle Y _ { J } ^ { M } | K ( Y _ { j } ^ { m } ) | Y _ { l } ^ { n } \\rangle } \\\\ & { \\quad \\quad \\quad \\quad = \\langle Y _ { J } ^ { M } | \\tilde { K } ( Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n } ) \\rangle } \\\\ & { \\quad \\quad \\quad = \\langle Y _ { J } ^ { M } | c \\cdot p ( Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n } ) \\rangle } \\\\ & { \\quad \\quad \\quad = c \\cdot \\big \\langle Y _ { J } ^ { M } \\big | p ( Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n } ) \\big \\rangle } \\\\ & { \\quad \\quad \\quad = c \\cdot \\big \\langle i ( Y _ { J } ^ { M } ) \\big | Y _ { j } ^ { m } \\otimes Y _ { l } ^ { n } \\big \\rangle } \\\\ & { \\quad \\quad \\quad = \\langle J \\| K \\| l \\rangle \\cdot \\langle J M | j m ; l n \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 536, + 640, + 665 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "As a short explanation: in the fifth step it was used that $i$ and $p$ are adjoint to each other, and consequently, we move from considering the tensor product in $V _ { J }$ to that one in $V _ { j } \\otimes V _ { l }$ . In the last step, the definition of the Clebsch-Gordan coefficients was used, and additionally, the notation $\\langle J \\| \\kappa \\| \\bar { l } \\rangle : = c$ that we mentioned before the theorem. The index $s$ is everywhere missing since $V _ { J }$ appears only once in $V _ { j } \\otimes V _ { l }$ . This finishes the proof. □ ", + "bbox": [ + 174, + 667, + 825, + 739 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Definition D.12 (Reduced Matrix Element). The unique number $c = \\langle J \\| K \\| l \\rangle \\in \\mathbb { C }$ in this theorem is called the reduced matrix element. To reiterate, it characterizes the representation operator completely. ", + "bbox": [ + 173, + 748, + 825, + 792 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "D.1.3 REDUCTION TO IRREDUCIBLE UNITARY REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 806, + 647, + 823 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Let $G$ be any compact group and $X$ any homogeneous space of $G$ . Before we state the WignerEckart Theorem for steerable kernels in the next section, we first want to explain why we can restrict to the case of irreducible unitary input- and output representations. Our explanations are adapted from Weiler & Cesa (2019). ", + "bbox": [ + 174, + 832, + 825, + 888 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Thus, let $\\rho _ { \\mathrm { i n } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G \\to { \\mathrm { G L } } ( V _ { \\mathrm { o u t } } )$ be general finite-dimensional input- and output representations. We consider the task of finding a basis for the space of steerable kernels $\\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) )$ . By Theorem B.20 and Proposition B.38, there are equivalences of representations (i.e., linear isomorphisms that intertwine between the representations) ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 102, + 825, + 133 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/73dead773729f7dac2e87ecffa1b6769f952a6810fa76cf0d9e8ab0eaef9d655.jpg", + "text": "$$\nQ _ { \\mathrm { i n } } : V _ { \\mathrm { i n } } \\to \\bigoplus _ { \\mu \\in I _ { \\mathrm { i n } } } V _ { \\mu } , \\qquad Q _ { \\mathrm { o u t } } : V _ { \\mathrm { o u t } } \\to \\bigoplus _ { \\nu \\in I _ { \\mathrm { o u t } } } V _ { \\nu } ,\n$$", + "text_format": "latex", + "bbox": [ + 325, + 138, + 671, + 174 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "where $\\rho _ { \\mu } : G \\to \\operatorname { U } ( V _ { \\mu } )$ and $\\rho _ { \\nu } : G \\to \\operatorname { U } ( V _ { \\nu } )$ are irreducible unitary representations. Both for the input- and the output representation, the same irrep can appear several times, e.g., there can be $\\mu \\neq \\bar { \\mu ^ { \\prime } }$ such that $\\rho _ { \\mu } \\cong \\rho _ { \\mu ^ { \\prime } }$ . Now, notice that the map ", + "bbox": [ + 174, + 180, + 825, + 226 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/04c7e841f7ee32178ec4c29f4d34243c95bb467081f546d78c05c1d6111541f1.jpg", + "text": "$$\n\\Phi _ { Q _ { \\mathrm { o u t } } , Q _ { \\mathrm { i n } } } : { \\mathrm { H o m } } _ { G } \\left( X , { \\mathrm { H o m } } _ { \\mathrm { K } } \\left( \\bigoplus _ { \\mu \\in I _ { \\mathrm { i n } } } V _ { \\mu } , \\bigoplus _ { \\nu \\in I _ { \\mathrm { o u t } } } V _ { \\nu } \\right) \\right) \\to { \\mathrm { H o m } } _ { G } ( X , { \\mathrm { H o m } } _ { \\mathrm { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) )\n$$", + "text_format": "latex", + "bbox": [ + 204, + 231, + 792, + 272 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "given for all $x \\in X$ by ", + "bbox": [ + 173, + 279, + 325, + 295 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/125f034143cd1c5e7efe2524f1c307a8d1be02daad709b71c347a74e3ce2ee22.jpg", + "text": "$$\n\\left[ \\Phi _ { Q _ { \\mathrm { o u t } } , Q _ { \\mathrm { i n } } } ( K ) \\right] ( x ) : = Q _ { \\mathrm { o u t } } ^ { - 1 } \\circ K ( x ) \\circ Q _ { \\mathrm { i n } }\n$$", + "text_format": "latex", + "bbox": [ + 357, + 291, + 640, + 311 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "is clearly an isomorphism. Thus, once a basis for the first kernel space is known, we just need to postcompose and precompose each basis kernel with $Q _ { \\mathrm { o u t } } ^ { - 1 }$ and $Q _ { \\mathrm { i n } }$ , respectively, in order to get a basis for the space we actually care about. Furthermore, the map ", + "bbox": [ + 176, + 320, + 823, + 363 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/53933547a6f4c34b22621f0cc68544d21c98f081732f8e9b52fcabbcb2ffa589.jpg", + "text": "$$\n\\Psi : \\bigoplus _ { \\nu \\in I _ { \\mathrm { o n t } } } \\bigoplus _ { \\mu \\in I _ { \\mathrm { i n } } } \\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { \\mu } , V _ { \\nu } ) ) \\to \\mathrm { H o m } _ { G } \\left( X , \\mathrm { H o m } _ { \\mathbb { K } } \\left( \\bigoplus _ { \\mu \\in I _ { \\mathrm { i n } } } V _ { \\mu } , \\bigoplus _ { \\nu \\in I _ { \\mathrm { o n t } } } V _ { \\nu } \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 197, + 369, + 799, + 411 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "given by ", + "bbox": [ + 173, + 417, + 232, + 433 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/7172e311a157a672740ae8846b6bd0fda40688e10644a40ea1fe830f77f066e5.jpg", + "text": "$$\n\\left[ \\Psi ( ( K ^ { \\nu \\mu } ) _ { \\nu , \\mu } ) ( x ) \\right] \\left( ( v _ { \\mu } ) _ { \\mu } \\right) : = \\Bigg ( \\sum _ { \\mu \\in I _ { \\mathrm { i n } } } K ^ { \\nu \\mu } ( x ) ( v _ { \\mu } ) \\Bigg ) _ { \\nu } \\in \\bigoplus _ { \\nu \\in I _ { \\mathrm { o u t } } } V _ { \\nu } ,\n$$", + "text_format": "latex", + "bbox": [ + 271, + 430, + 725, + 470 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "where $x \\in X$ and $( v _ { \\mu } ) _ { \\mu } \\in \\oplus _ { \\mu \\in I _ { \\mathrm { i n } } } V _ { \\mu }$ are arbitrary, is also clearly an isomorphism. It expresses that we can take a collection of steerable kernels $( K ^ { \\nu \\mu } ) _ { \\nu , \\mu }$ and build with it a block-matrix, which is steerable again, as can easily be checked. Accordingly, if we have basis kernels for a space ${ \\mathrm { H o m } } _ { G } ( X , { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { \\mu } , V _ { \\nu } ) )$ for some $\\mu , \\nu$ , then we can, by applying $\\Psi$ , map it to block basis kernels which are zero outside the block with indices $\\nu$ and $\\mu$ . Overall, by doing this for all $\\mu , \\nu$ , we thus recover a full basis for the space ${ \\mathrm { H o m } } _ { G } \\left( X , { \\mathrm { H o m } } _ { \\mathbb { K } } \\left( \\bigoplus _ { \\mu \\in I _ { \\mathrm { i n } } } V _ { \\mu } , \\bigoplus _ { \\nu \\in I _ { \\mathrm { o u t } } } V _ { \\nu } \\right) \\right)$ . By applying the base change $\\Phi _ { Q _ { \\mathrm { o u t } } , Q _ { \\mathrm { i n } } }$ from above, we thus get a basis for $\\operatorname { H o m } _ { G } ( X , \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) )$ . In summary, knowing a basis of steerable kernels for irreducible unitary input- and output representations gives us one for all finite-dimensional input- and output representations. Finally, note that the transformation of basis kernels using $\\Phi _ { Q _ { \\mathrm { o u t } } , Q _ { \\mathrm { i n } } }$ and $\\Psi$ can be done in the network initialization process and does not need to be performed in each forward pass. ", + "bbox": [ + 173, + 473, + 825, + 640 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "D.1.4 THE WIGNER-ECKART THEOREM FOR STEERABLE KERNELS ", + "text_level": 1, + "bbox": [ + 174, + 655, + 655, + 670 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "Now that we have seen the Wigner-Eckart theorem in a version similar to how it usually appears in physics, it is time to state the version which we will need in this work for applications in deep learning. The treatment is similar to the formulation in Agrawala (1980), which presents a generalization of the Wigner-Eckart theorem to the case that $V _ { J }$ may appear several times as a direct summand in the direct sum decomposition of the tensor product. However, this paper still only considers the Wigner-Eckart theorem for the case of the complex numbers C. If we allow the real numbers as well, we cannot be sure that endomorphisms of irreducible representations are just given by one number. This is a complication we will deal with below by allowing matrix elements of general endomorphisms. Furthermore, we will deal with topological considerations that did not play a role in Agrawala (1980). And lastly, we transport the theorem over into the nonlinear realm of steerable kernels. ", + "bbox": [ + 173, + 680, + 825, + 833 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "As discussed in the last section, we can restrict the considerations to (representatives of isomorphism classes of) irreducible unitary input- and output representations. Thus, assume the inputrepresentation to be the irrep $\\rho _ { l } : \\dot { G } { \\bf \\overrightarrow { \\bf { \\sigma } } } { \\bf { \\bf { U } } } ( V _ { l } )$ and the output-representation to be the irrep $\\rho _ { J } : G \\to \\operatorname { U } ( V _ { J } )$ . The idea is now that kernel operators ${ \\mathcal { K } } : { \\dot { L } } _ { \\mathbb { K } } ^ { 2 } ( { \\dot { X } } ) \\to { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ can be described on each direct summand of the domain individually, and that on each of these summands, arguments similar to those for the original Wigner-Eckart theorem apply. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "According to the Peter-Weyl Theorem B.22 the space $L _ { \\mathbb { K } } ^ { 2 } ( X )$ has a dense subset which is a direct sum of irreducible unitary representations: ", + "bbox": [ + 173, + 102, + 825, + 132 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/86849a3ae274804a43a067be77bd8e88e0bbfb7efad39acd80024c147340f9bd.jpg", + "text": "$$\nL _ { \\mathbb { K } } ^ { 2 } ( X ) = { \\widehat { \\bigoplus _ { j \\in { \\widehat { G } } } } } \\bigoplus _ { i = 1 } ^ { m _ { j } } V _ { j i } .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 136, + 576, + 183 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "Each $V _ { j i }$ is, as a subrepresentation of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ , isomorphic to $V _ { j }$ . $V _ { j }$ is itself not assumed to be embedded in $L _ { \\mathbb { K } } ^ { 2 } ( X )$ . ", + "bbox": [ + 173, + 185, + 823, + 218 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "For arbitrary $j \\in { \\widehat { G } }$ , fix once and for all orthonormal bases $\\{ Y _ { j i } ^ { m } \\} \\subseteq V _ { j i }$ corresponding to the basis $\\{ Y _ { j _ { . } } ^ { m } \\}$ of $V _ { j }$ .23 Furthermore, assume that for all $s = 1 , \\ldots , [ J ( j l ) ]$ ], $p _ { j i s } : V _ { j i } \\otimes V _ { l } V _ { J }$ is a projection which is an adjoint of the linear equivariant isometric embedding $l _ { j i s } : V _ { J } V _ { j i } \\otimes V _ { l }$ . This is assumed to be aligned with the embeddings $V _ { J } \\to V _ { j } \\otimes V _ { l }$ with respect to the isomorphisms $V _ { j } \\cong V _ { j i }$ that underlie the correspondence of basis elements $Y _ { j } ^ { m } \\sim Y _ { j i } ^ { m }$ . What this means is that the Clebsch-Gordan coefficients with respect to all of these embeddings, for all $i$ , are equal: ", + "bbox": [ + 173, + 224, + 825, + 314 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/6cb9fdaa5a398ceeef14ae4b418eaef8cdae8c9f14d89fa355cfa875a0bd9e71.jpg", + "text": "$$\n l _ { j i s } ( Y _ { J } ^ { M } ) | Y _ { j i } ^ { m } \\otimes Y _ { l } ^ { n } = s , J M | j m ; l n .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 316, + 642, + 337 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "Now we state and prove the Wigner-Eckart theorem, which gives an explicit description of representation operators ${ \\mathcal { K } } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ in terms of endomorphisms of $V _ { J }$ and then transfers this statement over to a statement about steerable kernels $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ . Before we state the theorem, we want to shortly explain what to expect: in the derivation of the original Wigner-Eckart theorem in Section D.1.2, we saw that a kernel operator could be expressed as ${ \\bar { \\mathcal { K } } } : V _ { j } \\otimes { \\bar { \\mathcal { V } } } _ { l } V _ { J }$ . This was in turn equal to $\\tilde { \\mathcal { K } } = c \\circ p$ for an endomorphism $c : V _ { J } \\to V _ { J }$ and the projection $p$ corresponding to the appearance of $V _ { J }$ in the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ . This time, however, $V _ { J }$ can be found often in $L _ { \\mathbb { K } } ^ { 2 } ( X ) \\otimes V _ { l }$ , namely: ", + "bbox": [ + 173, + 338, + 826, + 454 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "1. For each isomorphism class of irreps $j \\in { \\widehat { G } }$ , \n2. For each appearance $i = 1 , \\ldots , m _ { j }$ of the irrep $V _ { j }$ in $L _ { \\mathbb { K } } ^ { 2 } ( X )$ and \n3. For each appearance $s = 1 , \\ldots , \\left[ J ( j l ) \\right]$ of the irrep $V _ { J }$ in the tensor product representation $V _ { j } \\otimes V _ { l }$ . $\\left[ J ( j l ) \\right]$ can be zero, which means that $j$ does not contribute. ", + "bbox": [ + 210, + 463, + 825, + 532 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "We therefore expect $\\tilde { \\kappa }$ to be a whole sum of compositions of endomorphisms with projections, for each combination of valid $j , i$ and $s$ . Furthermore, the specific structure of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ will be exploited as well by using orthogonal projections from $L _ { \\mathbb { K } } ^ { 2 } ( X )$ to summands $V _ { j i }$ . Overall, we hope this sufficiently motivates the theorem: ", + "bbox": [ + 174, + 544, + 825, + 601 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "Theorem D.13 (Wigner-Eckart Theorem for Steerable Kernels). We state the theorem in three parts: ", + "bbox": [ + 178, + 603, + 823, + 619 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "1. (Basis-independent Wigner-Eckart for Kernel Operators) There is an isomorphism of vector spaces ", + "bbox": [ + 202, + 627, + 823, + 659 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/f33be5076d9751d7bba09c259cb04385190c0746e5c44dd9e9d782468754cece.jpg", + "text": "$$\n\\mathrm { R e p } : \\bigoplus _ { j \\in \\hat { G } } ^ { m _ { j } } \\bigoplus _ { i = 1 } ^ { [ J ( j l ) ] } \\mathrm { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) \\mathrm { H o m } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )\n$$", + "text_format": "latex", + "bbox": [ + 289, + 660, + 766, + 709 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "which is given by ", + "bbox": [ + 232, + 713, + 348, + 728 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/9ff9610ff9734b91aef2f9616ef2713ba9ec6fd9b0529df336434865353d66cb.jpg", + "text": "$$\n\\left[ \\mathrm { R e p } ( ( c _ { j i s } ) _ { j i s } ) ( \\varphi ) \\right] ( v _ { l } ) : = \\sum _ { j \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { \\left[ J ( j l ) \\right] } \\sum _ { m = 1 } ^ { d _ { j } } \\left. Y _ { j i } ^ { m } \\middle \\vert \\varphi \\right. \\cdot c _ { j i s } \\left( p _ { j i s } ( Y _ { j i } ^ { m } \\otimes v _ { l } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 254, + 731, + 771, + 780 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "where $( c _ { j i s } ) _ { j i s }$ is a tuple of endomorphisms, $\\varphi : X \\to \\mathbb { K }$ is any square-integrable function and $v _ { l } \\in V _ { l }$ is any element. ", + "bbox": [ + 230, + 784, + 823, + 813 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "2. (Basis-independent Wigner-Eckart for Steerable Kernels) There is an isomorphism of vector spaces ", + "bbox": [ + 209, + 820, + 823, + 849 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/83703ab5a520ca767617006691afe2bbd5fb9353ad31208fb8455772dd9e8daf.jpg", + "text": "$$\n{ \\mathrm { G K e r : } } \\bigoplus _ { j \\in { \\widehat { G } } } ^ { m _ { j } } \\bigoplus _ { i = 1 } ^ { [ J ( j l ) ] } { \\mathrm { E n d } } _ { G , \\mathbb { K } } ( V _ { J } ) \\to { \\mathrm { H o m } } _ { G } ( X , { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )\n$$", + "text_format": "latex", + "bbox": [ + 308, + 852, + 746, + 901 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "which is given by ", + "bbox": [ + 232, + 103, + 348, + 118 + ], + "page_idx": 58 + }, + { + "type": "equation", + "img_path": "images/0c0beadbb8ef611add69bd70655f02bf604d56ba4676948caf14fda5f8f316db.jpg", + "text": "$$\n\\left[ \\mathrm { G K e r } ( ( c _ { j i s } ) _ { j i s } ) ( x ) \\right] ( v _ { l } ) : = \\sum _ { j \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { \\left[ J ( j l ) \\right] } \\sum _ { m = 1 } ^ { d _ { j } } \\left. i , j m | x \\right. \\cdot c _ { j i s } \\left( p _ { j i s } ( Y _ { j i } ^ { m } \\otimes v _ { l } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 261, + 121, + 790, + 170 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "where $\\left( { { c _ { j i s } } } \\right) _ { j i s }$ is a tuple of endomorphisms, $x \\in X$ is any point and $v _ { l } ~ \\in ~ V _ { l }$ is any element. Here, $\\langle i , j m | x \\rangle : = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\left. Y _ { j i } ^ { m } \\big | \\delta _ { U } \\right.$ , which is according to Proposition $C . I O$ equal to $\\overline { { Y _ { j i } ^ { m } ( x ) } }$ . ", + "bbox": [ + 230, + 172, + 825, + 224 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "3. (Basis-dependent Wigner-Eckart for Steerable Kernels) Let $K = \\mathrm { G K e r } ( ( c _ { j i s } ) _ { j i s } )$ be the steerable kernel corresponding to the tuple of endomorphisms $\\left( { { c _ { j i s } } } \\right) _ { j i s }$ according to the isomorphism above. Then the matrix elements of $K ( x ) \\in \\mathrm { H o m } _ { \\mathbb { K } } ( \\bar { V } _ { l } , V _ { J } )$ are explicitly given by ", + "bbox": [ + 215, + 232, + 823, + 289 + ], + "page_idx": 58 + }, + { + "type": "equation", + "img_path": "images/f6dbed6720968205fc7c218763da3712ea53a80960fa6947c05c9bfdcdf74531.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\langle J M | K ( x ) | l n \\rangle = } } \\\\ & { } & { \\displaystyle \\sum _ { j \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { [ J ( j l ) ] } \\sum _ { m = 1 } ^ { d _ { j } } \\sum _ { M ^ { \\prime } = 1 } ^ { d _ { J } } \\big \\langle J M \\big | c _ { j i s } \\big | J M ^ { \\prime } \\big \\rangle \\cdot \\big \\langle s , J M ^ { \\prime } \\big | j m ; l n \\big \\rangle \\cdot \\big \\langle i , j m \\big | x \\big \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 291, + 758, + 361 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "Remark D.14. Before we come to the proof, we have some remarks to make about this theorem: ", + "bbox": [ + 176, + 363, + 802, + 377 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "1. In line with the usual convention, we call the $\\left. J M \\middle | c _ { j i s } \\middle | J M ^ { \\prime } \\right.$ the generalized reduced matrix elements of the representation operator $\\kappa$ . Different from the situation in physics, these can depend nontrivially on the specific basis indices $M$ and $M ^ { \\prime }$ . If the space of endomorphisms is 1-dimensional, as is the case when considering representations over $\\mathbb { C }$ , then each $c _ { j i s }$ is a diagonal matrix, meaning that it is characterized by only one complex number, for simplicity with the same name $c _ { j i s }$ . Then one has $\\langle J M | { \\bar { c _ { j i s } } } | J \\bar { M } ^ { \\prime } \\rangle = \\delta _ { M \\bar { M } ^ { \\prime } } .$ · $c _ { j i s }$ and the sum over $M ^ { \\prime }$ disappears. What this means for the matrix form of basis kernels of steerable CNNs will be discussed in Corollary D.17. ", + "bbox": [ + 214, + 386, + 825, + 500 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "2. The coefficients $\\langle s , J M ^ { \\prime } | j m ; l n \\rangle$ are as before the Clebsch-Gordan coefficients. Note that the input $x$ of $K$ appears only in $\\langle i , j m \\vert x \\rangle$ . Those two parts of the right-hand side of the formula are always the same, independent of the kernel $K$ . ", + "bbox": [ + 214, + 507, + 825, + 553 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "3. The Clebsch-Gordan coefficients are traditionally defined with respect to isometric embeddings $l _ { j i s } : V _ { J } V _ { j } \\otimes V _ { l }$ since this makes them less ambiguous. However, we mention that the property of being isometric is no requirement for the construction of Clebsch-Gordan coefficients or the proof of the Wigner-Eckart theorem, being equivariant and linear is sufficient. This then means that the copies $l _ { s } ( Y _ { J } ^ { M } )$ do not anymore form an orthonormal basis. We will use this relaxation in the example in Section E.2, where we do not want to be bothered with obtaining isometric embeddings. ", + "bbox": [ + 214, + 561, + 825, + 660 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "4. The names for the isomorphisms in the theorem are meant as follows: Rep is the map that maps a tuple of endomorphisms to a kernel operator, which is a special representation operator. GKer maps a tuple of endomorphisms to a $\\mathbf { G }$ -steerable kernel. It is not meant as a notation for a kernel in the sense of a nullspace in linear algebra. ", + "bbox": [ + 214, + 669, + 825, + 724 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "5. Furthermore, a reader with a background in abstract algebra may wonder why we build the direct sum of spaces of endomorphisms instead of the direct product. The reason is that a posteriori, it turns out that only finitely many $j$ contribute nontrivially, and so the direct sum is equal to the direct product. For a proof of the finiteness, see Remark D.18 below. ", + "bbox": [ + 214, + 733, + 825, + 790 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "6. As a last remark, we want to mention that part 1 of the theorem is not the most general version we could do. We chose to formulate the Wigner-Eckart theorem for $L _ { \\mathbb { K } } ^ { 2 } { \\bar { ( X ) } }$ specifically since this is the space we use it for. However, an appropriate isomorphism can probably be formulated for any unitary representation instead of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ , only that we then need to take care that we replace direct sums by direct products if the index sets on the left side are infinite. Additionally, $V _ { l }$ and $V _ { J }$ could be replaced by arbitrary finite-dimensional representations, and an appropriate adaptation of the theorem would apply. Whether $V _ { l }$ and $V _ { J }$ could also be replaced by infinite-dimensional unitary representations would need to be explored, but an extension to such a case seems possible. ", + "bbox": [ + 212, + 797, + 825, + 924 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "Proof of Theorem D.13. The proof of 1 will be done in Section D.2 since it requires some work. However, the proofs of 2 and 3 are relatively straightforward once we believe 1 and so we do them here: ", + "bbox": [ + 174, + 102, + 825, + 146 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "From 1 we know that Rep is an isomorphism. Furthermore, from Theorem C.7 we know that ", + "bbox": [ + 171, + 151, + 784, + 167 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/36bb5c21f439bb2e656005ce07b6856731c732cc5755fb907e6f1a5c7034ae32.jpg", + "text": "$$\n( \\cdot ) | _ { X } : \\mathrm { H o m } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) ) \\to \\mathrm { H o m } _ { G , \\mathbb { K } } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )\n$$", + "text_format": "latex", + "bbox": [ + 251, + 172, + 745, + 193 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "is an isomorphism as well, and this is given by $\\begin{array} { r } { K | _ { X } ( x ) : = \\operatorname* { l i m } _ { U \\in \\mathcal { U } _ { x } } \\mathcal { K } ( \\delta _ { U } ) } \\end{array}$ , where we take the limit over the directed set of open neighborhoods of $x$ . We define the isomorphism GKer now simply as the composition, i.e., $\\mathrm { G K e r } : = ( \\bar { \\cdot } ) | _ { X } \\circ \\mathrm { R e p }$ . This isomorphism is then explicitly given by: ", + "bbox": [ + 173, + 198, + 826, + 241 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/c7a65dae5a5c96c4f99b15e07d6cea8b33cd675f2bfdb23d4a807ad0598fc99a.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\left[ \\operatorname { G K e r } \\left( \\left( c _ { j i s } \\right) _ { j i s } \\right) ( x ) \\right] \\left( v _ { l } \\right) = \\left[ \\operatorname { R e p } \\left( \\left( c _ { j i s } \\right) _ { j i s } \\right) \\right] \\left( x \\right) \\left( v _ { l } \\right) } } \\\\ { { \\displaystyle \\qquad = \\operatorname * { l i m } _ { v \\in \\mathbb { Z } _ { i } } \\left[ \\operatorname { R e p } \\left( \\left( c _ { j i s } \\right) _ { j i s } \\right) \\left( \\delta _ { U } \\right) \\right] \\left( v _ { l } \\right) } } \\\\ { { \\displaystyle \\qquad = \\operatorname * { l i m } _ { v \\in \\mathbb { Z } _ { i } } \\sum _ { j \\in \\widehat { G } ^ { \\delta } = i = 1 } \\sum _ { s = 1 } ^ { m _ { j } } \\sum _ { m = 1 } ^ { \\left\\lfloor T / \\delta \\right\\rceil } \\left( Y _ { j i } ^ { m } \\right) \\cdot c _ { j i s } \\left( p _ { j i s } \\left( Y _ { j i } ^ { m } \\otimes v _ { l } \\right) \\right) } } \\\\ { { \\displaystyle \\qquad = \\sum _ { j \\in \\widehat { G } ^ { \\delta + 1 } } \\sum _ { s = 1 } ^ { m _ { j } } \\sum _ { m = 1 } ^ { \\left\\lfloor T / \\delta \\right\\rfloor } \\sum _ { \\left[ \\operatorname { t i m } _ { v \\in \\mathcal { G } _ { i } } \\left. Y _ { j i } ^ { m } \\right. \\right] } \\cdot c _ { j i s } \\left( p _ { j i s } \\left( Y _ { j i } ^ { m } \\otimes v _ { l } \\right) \\right) } } \\\\ { { \\displaystyle \\qquad = \\sum _ { j \\in \\widehat { G } ^ { \\delta + 1 } } \\sum _ { s = 1 } ^ { m _ { j } } \\sum _ { m = 1 } ^ { \\left\\lfloor T / \\delta \\right\\rfloor } \\sum _ { \\left( i , j \\ m \\right) \\left\\lfloor x \\right. \\cdot c _ { j i s } \\left( p _ { j i s } \\left( Y _ { j i } ^ { m } \\otimes v _ { l } \\right) \\right) } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 246, + 792, + 443 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "This already proves 2. Now, in the following computation, we will use that $c _ { j i s } \\circ p _ { j i s } = c _ { j i s } \\circ$ ◦ $\\operatorname { i d } _ { V _ { J } } \\circ p _ { j i s }$ pired by notation in physics, we can write the identity on . For 3, we then compute $V _ { J }$ as $\\mathrm { i d } _ { V _ { J } } \\ =$ $\\sum _ { M ^ { \\prime } = 1 } ^ { d _ { J } } \\left| Y _ { J } ^ { M ^ { \\prime } } \\right. \\cdot \\left. Y _ { J } ^ { M ^ { \\prime } } \\right|$ ", + "bbox": [ + 173, + 446, + 825, + 494 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/f14436978f02e01e5c6434b8ed642026c74b08b57eeb673a29a5426c09c9f981.jpg", + "text": "$$\n\\begin{array} { r l } & { ( J M | K ( x ) | i n ) } \\\\ & { = Y _ { j } ^ { M } \\middle | K ( x ) \\middle | Y _ { n } ^ { n } } \\\\ & { = Y _ { j } ^ { M } \\middle | [ G K e ( ( \\sigma _ { j \\mathrm { s i } } ) j _ { \\mathrm { s i } } ) ( x ) ] ( Y _ { 1 } ^ { n } ) } \\\\ & { = \\displaystyle \\sum _ { j \\in \\widetilde { G } ^ { \\ast } = 1 } \\sum _ { s = 1 } ^ { m } \\sum _ { m = 1 } ^ { d } i , j m | x \\cdot Y _ { j } ^ { M } | c _ { j \\mathrm { s i } } \\circ p _ { j \\mathrm { s i } } | Y _ { j } ^ { m } \\otimes Y _ { 1 } ^ { m } } \\\\ & { \\quad \\mathrm { ~ } } \\\\ & { = \\displaystyle \\sum _ { j \\in \\widetilde { G } ^ { \\ast } = 1 } \\sum _ { s = 1 } ^ { | J ( \\widetilde { J } ) | } \\sum _ { m = 1 } ^ { d } \\sum _ { s = 1 } ^ { d } i , j m | x \\cdot Y _ { j } ^ { M } | c _ { j \\mathrm { s i } } | Y _ { j } ^ { M ^ { \\prime } } \\cdot Y _ { j } ^ { M ^ { \\prime } } \\middle | p _ { j \\mathrm { s i } } | Y _ { j } ^ { m } } \\\\ & { \\quad \\mathrm { ~ } } \\\\ & { \\quad \\mathrm { ~ } = \\displaystyle \\sum _ { j \\in \\widetilde { G } ^ { \\ast } = 1 } ^ { m } \\sum _ { s = 1 } ^ { | J ( \\widetilde { J } ) | } \\sum _ { m = 1 } ^ { d } \\sum _ { s = 1 } ^ { d } J M \\middle | c _ { j \\mathrm { s i } } | J M \\cdot X _ { j } ^ { M } \\middle | c _ { j \\mathrm { s i } } | Y _ { j } ^ { M ^ { \\prime } } \\cdot Y _ { j } ^ { M ^ { \\prime } } \\middle | p _ { j \\mathrm { s i } } | Y _ { j } ^ { m } } \\\\ & \\quad = \\displaystyle \\sum _ { j \\in \\widetilde { G } ^ { \\ast } = 1 } ^ { m } \\sum _ { s = 1 } ^ { | J ( \\widetilde { J } ) | } \\sum _ { m = 1 } ^ { d } \\sum _ { s = 1 } ^ { d } J M \\middle | c _ { j \\mathrm { s i } } | J M \\cdot s , J M \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 498, + 767, + 717 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "In the last step, we used the Clebsch-Gordan coefficients, see Definition D.6 and, as mentioned before, that $p _ { j i s }$ is adjoint to the embedding $l _ { j i s } : V _ { J } V _ { j i } \\otimes V _ { l }$ . □ ", + "bbox": [ + 174, + 720, + 821, + 751 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "Remark D.15. Here, we want to argue that our kernel space solution also covers that of general equivariant CNNs on homogeneous spaces (Cohen et al., 2019b). One definition of the kernel space in that setting is ", + "bbox": [ + 173, + 758, + 825, + 801 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/4ee295f58f226306dc14cefac0b20c4f1ec79dcc0fd31d48d9cdf502e1f07e34.jpg", + "text": "$$\n\\begin{array} { r l } & { \\operatorname { H o m } _ { G _ { \\mathrm { i n } } \\times G _ { \\mathrm { o u t } } } ( H , \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) } \\\\ & { = \\big \\{ K : H \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\mid K ( g _ { \\mathrm { o u t } } h g _ { \\mathrm { i n } } ) = \\rho _ { \\mathrm { o u t } } ( g _ { \\mathrm { o u t } } ) \\circ K ( h ) \\circ \\rho _ { \\mathrm { i n } } ( g _ { \\mathrm { i n } } ) \\big \\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 806, + 767, + 847 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "where $H$ is a loccally compact group and $G _ { \\mathrm { i n } } , G _ { \\mathrm { o u t } } \\subseteq H$ are subgroups with input- and output representations $\\rho _ { \\mathrm { i n } } : G _ { \\mathrm { i n } } \\to \\mathrm { G L } ( V _ { \\mathrm { i n } } )$ and $\\rho _ { \\mathrm { o u t } } : G _ { \\mathrm { o u t } } \\to \\mathrm { G L } ( V _ { \\mathrm { o u t } } )$ . For compact groups $G _ { \\mathrm { i n } }$ and $G _ { \\mathrm { o u t } }$ , this is covered by our setting as follows: we define $G : = G _ { \\mathrm { o u t } } \\times G _ { \\mathrm { i n } }$ and $\\pmb { g } : = ( g _ { \\mathrm { o u t } } , g _ { \\mathrm { i n } } )$ . We can define the left action of $G$ on $H$ by $g \\cdot h : = { { g } _ { \\mathrm { o u t } } } h { { g } _ { \\mathrm { i n } } ^ { - 1 } }$ . Furthermore, we can reformulate the representations of $G _ { \\mathrm { i n } }$ and $G _ { \\mathrm { o u t } }$ to representations of the group $G$ by setting $\\rho _ { \\mathrm { i n } } : G \\to \\mathrm { G L } ( V _ { \\mathrm { i n } } )$ ", + "bbox": [ + 173, + 852, + 825, + 925 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "with $\\rho _ { \\mathrm { i n } } ( g ) : = \\rho _ { \\mathrm { i n } } ( g _ { \\mathrm { i n } } )$ , and similarly for $\\pmb { \\rho _ { \\mathbf { o u t } } }$ . We furthermore notice that in Eq. (19) we could also have inverted $g _ { \\mathrm { i n } }$ since that constraint needs to apply to all elements of $G _ { \\mathrm { i n } }$ . Thus, we then see that the kernel space can be equivalently defined by ", + "bbox": [ + 173, + 103, + 826, + 147 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/143f1160bb103220d5ed932750f51e512976e32c1d3057672a0e194f49b45fe1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\operatorname { H o m } _ { G _ { \\mathrm { i n } } \\times G _ { \\mathrm { o u t } } } ( H , \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) ) } \\\\ & { = \\big \\{ K : H \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { \\mathrm { i n } } , V _ { \\mathrm { o u t } } ) \\mid K ( g \\cdot h ) = \\rho _ { \\mathrm { o u t } } ( g ) \\circ K ( h ) \\circ \\rho _ { \\mathrm { i n } } ( g ) ^ { - 1 } \\big \\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 150, + 754, + 193 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "which precisely is the kernel constraint of steerable CNNs in Eq. (2). Thus, if we restrict to a homogeneous space of the action of $G$ on $H$ , we recover steerable kernels as in Definition 3.2 and can apply Theorem 4.1. ", + "bbox": [ + 173, + 195, + 825, + 238 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "D.1.5 GENERAL STEERABLE KERNEL BASES ", + "text_level": 1, + "bbox": [ + 178, + 252, + 501, + 267 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Now that we have a Wigner-Eckart theorem for steerable kernels, which gives a one-to-one correspondence between steerable kernels and tuples of endomorphisms, we can finally describe what a basis of the space of steerable kernels looks like. For this, additionally to the notation in the last section, we assume that $\\{ c _ { r } \\mid r = 1 , \\ldots , E _ { J } \\}$ is a basis of $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ . ", + "bbox": [ + 173, + 276, + 825, + 334 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Theorem D.16 (Steerable Kernel Bases). $A$ basis of the space of steerable kernels ${ \\mathrm { H o m } } _ { G } ( X , { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )$ ) is given by ", + "bbox": [ + 173, + 337, + 821, + 367 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/c28322ee7799648b54a92b8f12d7f9d49d7fd01732aea1164daf83c3bbf64804.jpg", + "text": "$$\n\\{ K _ { j i s r } : X \\to \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\mid j \\in \\widehat { G } , i \\in \\{ 1 , \\ldots , m _ { j } \\} , s \\in \\{ 1 , \\ldots , [ J ( j l ) ] \\} , r = 1 , \\ldots , E _ { J } \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 186, + 372, + 808, + 393 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "where the basis kernels $K _ { j i s r }$ have matrix elements ", + "bbox": [ + 174, + 398, + 514, + 414 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/fd5dedb86f5736b0de58e03b572433d3caa4cc515de9bcc52f649463d1d7e98e.jpg", + "text": "$$\n\\langle J M | K _ { j i s r } ( x ) | l n \\rangle = \\sum _ { m = 1 } ^ { d _ { j } } \\sum _ { M ^ { \\prime } = 1 } ^ { d _ { J } } \\left. J M \\big | c _ { r } \\big | J M ^ { \\prime } \\right. \\cdot \\left. s , J M ^ { \\prime } \\big | j m ; l n \\right. \\cdot \\left. i , j m \\big | x \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 236, + 420, + 761, + 465 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Now, for each $M ^ { \\prime } \\in \\{ 1 , \\ldots , d _ { J } \\}$ , let $\\mathrm { C G } _ { J ( j l ) s } ^ { M ^ { \\prime } }$ be the $d _ { j } \\times d _ { l }$ -matrix of Clebsch-Gordan coefficients $\\langle s , J M ^ { \\prime } | j m ; l n \\rangle$ , with only $m$ and $n$ varying. Furthermore, let $\\langle i , j | x \\rangle$ be the row vector with entries $\\langle i , j m \\vert x \\rangle$ for $m = 1 , \\ldots , d _ { j }$ . In matrix-notation with respect to the bases $\\{ Y _ { J } ^ { M } \\} \\subseteq V _ { J }$ and $\\{ Y _ { l } ^ { n } \\} \\subseteq V _ { l }$ , we can then express the basis kernel $K _ { j i s r } ( x ) : V _ { l } V _ { J }$ as follows: ", + "bbox": [ + 173, + 472, + 826, + 535 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/a7721a0ee20841a2d78ba1818d786a643b015776f08bec80ff6dc25bf3d5c70c.jpg", + "text": "$$\nK _ { j i s r } ( x ) = c _ { r } \\cdot \\left( \\begin{array} { l } { \\langle i , j | x \\rangle \\cdot { \\mathrm { C G } } _ { J ( j l ) s } ^ { 1 } } \\\\ { \\vdots } \\\\ { \\langle i , j | x \\rangle \\cdot { \\mathrm { C G } } _ { J ( j l ) s } ^ { d _ { J } } } \\end{array} \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 540, + 632, + 603 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "In this formula, all “dots” mean conventional matrix multiplication and $c _ { r }$ is by abuse of notation the matrix of the endomorphism $c _ { r }$ . ", + "bbox": [ + 174, + 607, + 823, + 636 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Proof. For the first statement, note that a basis for Lj∈G Lmji=1 $\\begin{array} { r l } { } & { \\bigoplus _ { j \\in \\widehat { G } } \\bigoplus _ { i = 1 } ^ { m _ { j } } \\bigoplus _ { s = 1 } ^ { [ J ( j l ) ] } \\mathrm { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) } \\end{array}$ L[J(jl)]s=1 EndG,K(VJ ) is given by all the tuples $t _ { j i s r } : = ( 0 , \\ldots , c _ { r } , \\ldots , 0 )$ that have $c _ { r }$ at position $j i s$ , for all combinations of $j , i , s$ and $r$ . Thus, from the isomorphism GKer in the second part of Theorem D.13 we obtain that all $K _ { j i s r } : =$ $\\operatorname { G K e r } ( t _ { j i s r } )$ together form a basis for the space of steerable kernels $\\mathrm { H o m } _ { G } ( X , \\mathrm { H o m } _ { \\mathbb { K } } ( \\bar { V _ { l } } , V _ { J } ) )$ . When applying the basis-dependent form in part 3 of that theorem to $K _ { j i s r }$ , the first three sums in Eq. (18) just disappear since $t _ { j i s r }$ is zero almost everywhere. Furthermore, $c _ { j i s }$ is replaced by the basis endomorphism $c _ { r }$ . We obtain the claimed result. ", + "bbox": [ + 173, + 651, + 826, + 756 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "For the final statement on the matrix representation, note that ", + "bbox": [ + 173, + 761, + 575, + 776 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/586469485b4c2ce7a2d56e121c0714385356cbb0e1a7e7188eedd1b414b6a1f6.jpg", + "text": "$$\n\\begin{array} { l } { { \\langle J M | K _ { j i s r } ( x ) | l n \\rangle = \\sum _ { m = 1 } ^ { d _ { j } } \\sum _ { M ^ { \\prime } = 1 } ^ { d _ { J } } \\left. J M | c _ { r } \\big | J M ^ { \\prime } \\right. \\cdot \\left. s , J M ^ { \\prime } \\big | j m ; l n \\right. \\cdot \\left. i , j m \\big | x \\right. } } \\\\ { { = \\sum _ { M ^ { \\prime } = 1 } ^ { d _ { J } } \\left. J M \\big | c _ { r } \\big | J M ^ { \\prime } \\right. \\sum _ { m = 1 } ^ { d _ { j } } \\left. i , j m \\big | x \\right. \\cdot \\left. s , J M ^ { \\prime } \\big | j m ; l n \\right. } } \\\\ { { = c _ { r } ^ { M } \\cdot \\left( \\sum _ { m = 1 } ^ { d _ { j } } \\left. i , j m \\big | x \\right. \\cdot \\left. s , J M ^ { \\prime } \\big | j m ; l n \\right. \\right) _ { M ^ { \\prime } = 1 } ^ { d _ { J } } } } \\\\ { { = c _ { r } ^ { M } \\cdot \\left( \\left. i , j \\big | x \\right. \\cdot \\mathrm { C G } _ { J ( j l ) s } ^ { M ^ { \\prime } - n } \\right) _ { M ^ { \\prime } = 1 } ^ { d _ { J } } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 781, + 784, + 905 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Here, $c _ { r } ^ { M }$ is the $M ^ { \\prime }$ ’th row of the matrix $c _ { r }$ . The result follows by dropping the indices $M$ and $n$ . ", + "bbox": [ + 169, + 909, + 795, + 925 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "The next corollary means that endomorphisms can be ignored if the space of endomorphisms is 1-dimensional, which is in particular the case if $\\mathbb { K } = \\mathbb { C }$ . ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Corollary D.17. Assume that $\\dim ( \\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } ) ) = 1$ . Then a basis of steerable kernels $K : X $ ${ \\mathrm { H o m } } _ { \\mathrm { K } } ( V _ { l } , V _ { J } )$ is given by all $K _ { j i s }$ with matrices ", + "bbox": [ + 174, + 133, + 821, + 162 + ], + "page_idx": 61 + }, + { + "type": "equation", + "img_path": "images/e1ccf95d8267622d4a14b5aaaecaf553a7a526ec155cd2ca4a1cb0f21e059c88.jpg", + "text": "$$\nK _ { j i s } ( x ) = \\left( \\begin{array} { c } { \\langle i , j | x \\rangle \\cdot \\mathrm { C G } _ { J ( j l ) s } ^ { 1 } } \\\\ { \\vdots } \\\\ { \\langle i , j | x \\rangle \\cdot \\mathrm { C G } _ { J ( j l ) s } ^ { d _ { J } } } \\end{array} \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 165, + 616, + 227 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "In particular, this is the case $i f \\mathbb { K } = \\mathbb { C }$ . ", + "bbox": [ + 176, + 228, + 429, + 242 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Proof. In this case, a basis for the space of endomorphisms is given by the single endomorphism $c = \\operatorname { i d } _ { V _ { J } }$ . Postcomposition with the identity does not change the matrix, and so the result follows. ", + "bbox": [ + 174, + 257, + 823, + 286 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "For $\\mathbb { K } = \\mathbb { C }$ we have $\\dim ( \\operatorname { E n d } _ { G , \\mathbb { C } } ( V _ { J } ) ) = 1$ by Schur’s Lemma D.8, and thus the result follows. ", + "bbox": [ + 174, + 291, + 799, + 308 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "We end with two remarks regarding the parameterization of steerable CNNs. The first remark considers the case of steerable CNNs of the form $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ on a homogeneous space $X$ . The second remark connects this back to the case that $X$ is an orbit embedded in $\\bar { \\mathbb { R } } ^ { d }$ . ", + "bbox": [ + 173, + 321, + 825, + 364 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Remark D.18 (Parameterization in the abstract). First of all, we want to understand that there are only finitely many basis kernels $K _ { j i s r }$ . To this end, note that the index sets for $i , s ,$ , and $r$ are necessarily finite for all $j$ , and thus we need to understand the finite range of $j$ . A priori, $j$ can run over the whole set $\\widehat { G }$ , which can be infinite. But, as we argue now, for only finitely many $j \\in { \\widehat { G } }$ we can have $V _ { J }$ in a direct sum decomposition of $V _ { j } \\otimes V _ { l }$ , which rescues the finiteness: ", + "bbox": [ + 174, + 366, + 825, + 438 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Namely, $V _ { J }$ is in the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ if and only if the vector space ${ \\mathrm { H o m } } _ { G , \\mathbb { K } } ( V _ { j } \\otimes V _ { l } , V _ { J } )$ is nonzero by Schur’s Lemma B.29. By the hom-tensor adjunction that we will show in Proposition D.23 in more generality, this is the case if an only if ${ \\mathrm { H o m } } _ { G , \\mathbb { K } } ( V _ { j } , { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )$ is nonzero. And finally, this is the case if and only if $V _ { j }$ is in a direct sum decomposition of the representation ${ \\mathrm { H o m } } _ { \\mathrm { K } } ( V _ { l } , V _ { J } )$ , again by Schur’s lemma. Now, since ${ \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ is finite-dimensional, this can only be the case for finitely many $j$ , and so we are done.24 ", + "bbox": [ + 173, + 443, + 825, + 542 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Overall, this means the following: To parameterize an equivariant neural network, one needs arbitrary parameters $w _ { j i s r } \\in \\mathbb { K }$ for all combinations of $j \\in { \\widehat { G } }$ , $i \\in \\{ 1 , \\ldots , m _ { j } \\}$ , $s \\in \\{ 1 , \\ldots , [ J ( j l ) ] \\}$ and $r = 1 , \\dots , E _ { J }$ . A general steerable Kernel $K : X \\to \\operatorname { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ then takes the form ", + "bbox": [ + 173, + 547, + 825, + 593 + ], + "page_idx": 61 + }, + { + "type": "equation", + "img_path": "images/214f68ee23bdb4acf302fa2b49f934532ff48a4b1f695e3e5b16792a42d6cba8.jpg", + "text": "$$\n\\begin{array} { r } { K = \\sum _ { j \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { [ J ( j l ) ] } \\sum _ { r = 1 } ^ { E _ { J } } w _ { j i s r } K _ { j i s r } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 594, + 661, + 626 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "with the basis kernels $K _ { j i s r }$ as in Theorem D.16. ", + "bbox": [ + 176, + 626, + 496, + 641 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Remark D.19 (Parameterization in practice). Remember that our original motivation for the use of homogeneous spaces in Section C.1.1 was that $\\mathbb { R } ^ { d }$ splits as a disjoint union of homogeneous spaces, on which the kernel constraint acts separately. For simplicity, we assume that the compact group acting on $\\mathbb { R } ^ { d }$ is either $G = \\mathrm { S O } ( d )$ or $\\bar { G } = \\mathrm { O } \\bar { ( } d )$ , but the general ideas hold also for the finite transformation groups in $\\mathbb { R } ^ { d }$ – the only difference is that in these finite cases, the set of representatives of orbits becomes larger. ", + "bbox": [ + 173, + 641, + 826, + 726 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Thus, $\\mathbb { R } ^ { d }$ splits into orbits $\\begin{array} { r } { \\mathbb { R } ^ { d } = \\bigcup _ { r \\geq 0 } S ^ { n - 1 } ( r ) } \\end{array}$ , where $S ^ { n - 1 } ( r )$ is the sphere of radius $r$ (with $S ( 0 ) = \\{ 0 \\}$ being a single point). ", + "bbox": [ + 174, + 729, + 821, + 762 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "We’ll discuss the orbit $X _ { 0 } = \\{ 0 \\}$ , the origin, separately below. But note that all other orbits are necessarily homeomorphic to each other and thus can be treated on equal footing. Therefore, let $S ^ { n - 1 }$ be the standard sphere with radius 1 and $K _ { j i s r } : S ^ { n - 1 } \\to \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ be basis kernels for this choice. Then for a general steerable kernel ${ \\underset { \\cdot } { K } } : { \\mathbb { R } ^ { d } } { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ there are arbitrary functions $w _ { j i s r } : \\mathbb { R } _ { > 0 } \\to \\mathbb { K }$ such that, for all $x \\in \\mathbb { R } ^ { d } \\setminus \\{ 0 \\}$ , we have: ", + "bbox": [ + 173, + 768, + 825, + 842 + ], + "page_idx": 61 + }, + { + "type": "equation", + "img_path": "images/62c323f090151692284651f2dc1dca6578fe4fb70fe60c82349253a4b21aea98.jpg", + "text": "$$\nK ( \\boldsymbol { x } ) = \\sum _ { \\boldsymbol { j } \\in \\widehat { G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { [ J ( \\boldsymbol { j } \\boldsymbol { l } ) ] } \\sum _ { r = 1 } ^ { E _ { \\boldsymbol { j } } } w _ { j i s r } ( \\Vert \\boldsymbol { x } \\Vert ) \\cdot K _ { j i s r } \\left( \\frac { \\boldsymbol { x } } { \\Vert \\boldsymbol { x } \\Vert } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 266, + 844, + 730, + 877 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "For $x = 0$ , we might use our heavy theory to solve the kernel constraint, but it is more illuminating to do it from scratch since this case is so simple: we have $K ( 0 ) : V _ { l } V _ { J }$ , and the kernel constraint takes the form ", + "bbox": [ + 173, + 103, + 823, + 145 + ], + "page_idx": 62 + }, + { + "type": "equation", + "img_path": "images/4993fa989abaeb29404968c648da876739818d7d573277d94cc3deb75a1060c2.jpg", + "text": "$$\nK ( 0 ) = K ( g \\cdot 0 ) = \\rho _ { J } ( g ) \\circ K ( 0 ) \\circ \\rho _ { l } ( g ) ^ { - 1 }\n$$", + "text_format": "latex", + "bbox": [ + 349, + 143, + 647, + 162 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "for all $g \\in G$ , which is equivalent to $K ( 0 ) \\circ \\rho _ { l } ( g ) = \\rho _ { J } ( g ) \\circ K ( 0 )$ for all $g \\in G$ . This just means that $K ( 0 ) : V _ { l } V _ { J }$ is an intertwiner, and by Schur’s Lemma B.29 it is either $0$ if $l \\neq J$ or an arbitrary endomorphism $V _ { J } V _ { J }$ if $l ~ = ~ J$ . Thus, assuming $l ~ = ~ J$ and choosing basis-endomorphisms $c _ { r } : V _ { J } \\to V _ { J }$ , there are coefficients $w _ { r } \\in \\mathbb { K }$ such that ", + "bbox": [ + 173, + 166, + 825, + 223 + ], + "page_idx": 62 + }, + { + "type": "equation", + "img_path": "images/d7c39611606be0732dce4908ac1f07a73dcfaf85521a49f1ca7bf7e9ff28a2cf.jpg", + "text": "$$\nK ( 0 ) = \\sum _ { r = 1 } ^ { E _ { J } } w _ { r } \\cdot c _ { r } .\n$$", + "text_format": "latex", + "bbox": [ + 418, + 229, + 578, + 258 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "The reader may find it interesting to check that this solution is precisely what is also predicted by our theory using that $L _ { \\mathbb { K } } ^ { 2 } ( \\{ 0 \\} ) \\cong \\mathbb { K }$ is just isomorphic to the trivial representation of $G$ . ", + "bbox": [ + 171, + 265, + 823, + 294 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "All in all, we now know what the most general steerable kernels look like. In practice, one needs to choose the functions $w _ { j i s r } : \\mathbb { R } _ { > 0 } \\to \\mathbb { K }$ . For representations over the real numbers, i.e., with $\\mathbb { K } = \\mathbb { R }$ , one choice is to only consider finitely many radii and Gaussian radial profiles around them. Then instead of learning the whole function $w _ { j i s r }$ , one learns finitely many real parameters that choose “how activated” a basis kernel $K _ { j i s r }$ is for a certain radius. This is, for example, the route taken in Weiler et al. (2018b;a); Weiler & Cesa (2019). If one deals with complex representations, one usually goes the same route, only that the parameters that choose how “activated” the basis kernels are will then be complex numbers. One can either parameterize them as $a + i b$ with a real part $a$ and a complex part $b$ . This intuitively means that $a$ activates the standard version of the kernel $K _ { j i s r }$ , whereas $b$ activates the kernel $i K _ { j i s r }$ , which can be imagined as a version of the kernel turned by $9 0 °$ . One other possibility is to parameterize a complex number as $\\alpha \\cdot e ^ { i \\beta }$ with a scaling factor $\\alpha > 0$ and a phase shift $\\beta$ . This is the route chosen in Worrall et al. (2016). ", + "bbox": [ + 173, + 299, + 825, + 468 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "In Chapter E we will look at examples of determining the basis kernels $K _ { j i s r }$ , which will hopefully further illuminate the theorem. In the next section, we go back to the theory and prove the remaining parts of the Wigner-Eckart theorem. ", + "bbox": [ + 176, + 478, + 823, + 520 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "D.2 PROOF OF THE WIGNER-ECKART THEOREM FOR KERNEL OPERATORS", + "text_level": 1, + "bbox": [ + 174, + 537, + 705, + 553 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "In this section, we prove the first part of Theorem D.13, the Wigner-Eckart theorem for Kernel Operators, since we have skipped this in the last section. It is not necessary to read this section and the reader may wish to directly go to the chapter on examples E. We will make frequent use of topological concepts from Chapter F.1 in this section. ", + "bbox": [ + 174, + 564, + 825, + 621 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "The strategy is the following: in Section D.2.1, we show that ", + "bbox": [ + 173, + 627, + 573, + 642 + ], + "page_idx": 62 + }, + { + "type": "equation", + "img_path": "images/d14308c689b885d6ff522eb2d5645d86f4acfb793a8f88cb0bebabd45c4a15f0.jpg", + "text": "$$\n\\mathrm { H o m } _ { G , \\mathbb { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) ) \\cong \\mathrm { H o m } _ { G , \\mathbb { K } } \\bigg ( \\bigoplus _ { j \\in \\widehat { G } } ^ { m _ { j } } V _ { j i } , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\bigg ) ,\n$$", + "text_format": "latex", + "bbox": [ + 235, + 650, + 761, + 696 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "which basically means that we can ignore the “topological closure” of the direct sum which is dense in $L _ { \\mathbb { K } } ^ { 2 } ( X )$ . This works, intuitively, since kernel operators are continuous, and so they are determined by what they do on a dense subset. Then, in section D.2.2, we show that ", + "bbox": [ + 174, + 704, + 826, + 747 + ], + "page_idx": 62 + }, + { + "type": "equation", + "img_path": "images/29f8fe0ab6a295cb1f23b99c2b698e78202f081aa44666a97ffc313ee507dbb6.jpg", + "text": "$$\n\\mathrm { H o m } _ { G , \\mathrm { K } } \\left( \\bigoplus _ { j \\in \\widehat { G } } ^ { m _ { j } } V _ { j i } , \\mathrm { H o m } _ { \\mathrm { K } } ( V _ { l } , V _ { J } ) \\right) \\cong \\mathrm { H o m } _ { G , \\mathrm { K } } \\left( \\bigoplus _ { j \\in \\widehat { G } } ^ { m _ { j } } V _ { j i } \\otimes V _ { l } , V _ { J } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 236, + 752, + 759, + 800 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "which is the main step that we need in order to be able to make use of the Clebsch-Gordan coefficients, namely when we decompose the tensor product. Finally, in Section D.2.3, we finish the proof of Theorem D.13. ", + "bbox": [ + 176, + 808, + 823, + 849 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "D.2.1 REDUCTION TO A DENSE SUBSPACE OF $L _ { \\mathbb { K } } ^ { 2 } ( X )$ ", + "text_level": 1, + "bbox": [ + 173, + 864, + 560, + 882 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "In this section, we reduce the statement to representation operators on $\\textstyle \\bigoplus _ { j \\in { \\widehat { G } } } \\bigoplus _ { i = 1 } ^ { m _ { j } } V _ { j i }$ . For simplicity, we write the double direct sum from now on as $\\bigoplus _ { j i }$ . ", + "bbox": [ + 173, + 892, + 823, + 926 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "Furthermore, remember that $V _ { l }$ and $V _ { J }$ are finite-dimensional, and thus ${ \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ can be identified with matrices in $\\mathbb { K } ^ { d _ { J } \\times d _ { l } }$ . This space is a Euclidean space and thus has a scalar product and consequently also a norm, see Chapter F.1. Consequently, each kernel operator is a continuous map between normed vector spaces, which we’ll use in the following. ", + "bbox": [ + 174, + 102, + 825, + 160 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "A short terminological note: kernel operators are just representation operators on $L _ { \\mathbb { K } } ^ { 2 } ( X )$ and only have their name due to the relation to steerable kernels. Thus, the terminological difference to representation operators in the following reduction result has no further meaning: ", + "bbox": [ + 176, + 166, + 823, + 209 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Lemma D.20. The restriction map ", + "bbox": [ + 174, + 213, + 406, + 228 + ], + "page_idx": 63 + }, + { + "type": "equation", + "img_path": "images/1cd83c8444e393940a1a5304584caf622448633111943ee890bdf43e2306584a.jpg", + "text": "$$\n\\mathrm { H o m } _ { G , \\mathrm { K } } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) ) \\to \\mathrm { H o m } _ { G , \\mathrm { K } } \\left( \\bigoplus _ { j i } V _ { j i } , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 246, + 236, + 748, + 265 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "given by $K \\mapsto \\mathcal { K } | _ { \\oplus _ { j i } V _ { j i } }$ , between kernel operators on the left and representation operators on the right is an isomorphism. ", + "bbox": [ + 173, + 271, + 825, + 303 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Proof. First of all, the kernel operators on the left are actually uniformly continuous by Proposition F.18. Thus, by Lemma F.22, the restriction map is an injection into uniformly continuous representation operators on $\\bigoplus _ { j i } V _ { j i }$ . The set of all these maps is equal to the set of all representation operators by Proposition F.18 again. ", + "bbox": [ + 173, + 321, + 825, + 381 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Thus, in order to be finished, we only need to see that the unique extension of a representation operator ${ \\mathcal { K } } : \\oplus _ { j i } V _ { j i } \\to { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , { \\bar { V _ { J } } } )$ to a continuous function $\\overline { { \\mathcal { K } } } : L _ { \\mathbb { K } } ^ { 2 } ( X ) \\to \\operatorname { H o m } _ { K } ^ { \\bullet } ( V _ { l } , V _ { J } )$ is a kernel operator, which means it is linear and equivariant. ", + "bbox": [ + 173, + 387, + 825, + 431 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "For linearity, let $a \\in \\mathbb { K }$ and $f \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ . Let $( f _ { k } ) _ { k }$ be a sequence in $\\textstyle \\bigoplus _ { j i } V _ { j i }$ that converges to $f$ . Using the continuity of $\\overline { { \\mathcal { K } } }$ and the linearity of $\\kappa$ we obtain: ", + "bbox": [ + 173, + 436, + 823, + 470 + ], + "page_idx": 63 + }, + { + "type": "equation", + "img_path": "images/2ccbf6ee02f1539971050b9e8a45b72fbc12b19bbe85089822b5f9587c1dcafb.jpg", + "text": "$$\n\\begin{array} { r l } { K ( a \\cdot f ) = \\overline { { K } } \\big ( \\underset { k \\infty } { \\operatorname* { l i m } } \\big ( a \\cdot f _ { k } \\big ) \\big ) } & { } \\\\ & { = \\underset { k \\infty } { \\operatorname* { l i m } } \\overline { { K } } \\big ( a \\cdot f _ { k } \\big ) } \\\\ & { = \\underset { k \\infty } { \\operatorname* { l i m } } K \\big ( a \\cdot f _ { k } \\big ) } \\\\ & { = \\underset { k \\infty } { \\operatorname* { l i m } } a \\cdot K \\big ( f _ { k } \\big ) } \\\\ & { = \\underset { k \\infty } { \\operatorname* { l i m } } \\frac { K } { \\big ( f _ { k } \\big ) } } \\\\ & { = a \\cdot \\frac { 1 } { k \\infty } \\overline { { K } } \\big ( f _ { k } \\big ) } \\\\ & { = a \\cdot \\overline { { K } } \\big ( \\underset { k \\infty } { \\operatorname* { l i m } } f _ { k } \\big ) } \\\\ & { = a \\cdot \\overline { { K } } ( f ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 400, + 477, + 591, + 647 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Linearity with respect to addition can be shown similarly. For the equivariance we can argue in the same way, only that we additionally need to use the continuity of the representations $\\lambda : G \\to$ $\\mathrm { U } ( L _ { \\mathbb { K } } ^ { 2 } ( X ) )$ and $\\rho _ { \\mathrm { H o m } } : G \\to { \\mathrm { G L } } ( { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) )$ . □ ", + "bbox": [ + 174, + 655, + 825, + 699 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "D.2.2 THE HOM-TENSOR ADJUNCTION ", + "text_level": 1, + "bbox": [ + 174, + 717, + 462, + 732 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Lemma D.21. Let ${ \\mathcal { K } } : \\oplus _ { l i } V _ { l i } \\to V$ be linear and equivariant, where $V$ is an irrep. Then $\\kappa$ is continuous. ", + "bbox": [ + 171, + 742, + 823, + 771 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Proof. By Schur’s Lemma D.8,25 we know that $\\kappa$ factors through the irreducible representations that are isomorphic to $V$ . That is, let $V _ { j }$ be that irrep and $p _ { j i } : \\mathsf { \\bar { \\theta } } _ { l i } V _ { l i } \\to V _ { j i }$ be the canonical projections. Then there are intertwiners $c _ { i } : V _ { j i } \\to V$ such that $\\textstyle K = \\sum _ { i } c _ { i } \\circ p _ { j i }$ . Each $c _ { i }$ is continuous since it is a linear function between finite-dimensional normed vector spaces. Since also summation on normed vector spaces is continuous, we only need to show that the projections $p _ { j i }$ are continuous. ", + "bbox": [ + 173, + 790, + 825, + 875 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "This follows from the following fact on how the norm on $\\bigoplus _ { l i } V _ { l i }$ is composed from the norms on each $V _ { l i }$ : For an element $\\begin{array} { r } { f = \\bar { \\sum _ { l i } } f _ { l i } \\in \\bigoplus _ { l i } V _ { l i } } \\end{array}$ with $f _ { l i } \\in V _ { l i }$ , we have: ", + "bbox": [ + 169, + 102, + 825, + 133 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/85c839ffaf687c4172edb07a4b12b6d56348fec40447a7b89f38418080a17ab8.jpg", + "text": "$$\n\\begin{array} { r } { \\| f \\| ^ { 2 } = \\sum _ { l i } \\| f _ { l i } \\| ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 428, + 137, + 568, + 162 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "The reason for this is that the $V _ { l i }$ are perpendicular to each other. Consequently, if $( f ^ { k } ) _ { k }$ with $f ^ { k } \\in \\oplus _ { l i } V _ { l i }$ converges to 0, then also ${ \\bar { ( } } p _ { j i } ( f ^ { k } ) { \\big ) } _ { k } ~ = ~ ( f _ { j i } ^ { k } ) _ { k }$ converges to 0, which shows the continuity of $p _ { j i }$ in 0 and thus general continuity by Proposition F.18. □ ", + "bbox": [ + 173, + 166, + 826, + 209 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Remark D.22. Note the curious fact that we cannot get rid of the equivariance condition in the preceding Lemma. I.e., if we have a linear function $\\textstyle { \\mathcal { K } } : \\bigoplus _ { l } V _ { l } \\to V$ , then we cannot deduce that $\\kappa$ is continuous. We omit the index $i$ for simplicity. If equivariance is no requirement, then we only deal with vector spaces, which are in general isomorphic to spaces of (maybe infinite) tuples of elements in $\\mathbb { K }$ . Thus, let the function $\\mathcal { K } : \\bar { \\bigoplus } _ { l \\in \\mathbb { N } } \\mathbb { K } \\mathbb { K }$ given by ", + "bbox": [ + 173, + 215, + 825, + 287 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/7694beafc22904dc461bd106711a4108d37292c6887a5b045ce7c5d4bdfc3ffe.jpg", + "text": "$$\n\\begin{array} { r } { ( a _ { l } ) _ { l } \\mapsto \\sum _ { l } l \\cdot a _ { l } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 436, + 291, + 562, + 318 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "This is linear but not continuous in 0. The latter can be seen by considering the sequence $( a ^ { k } ) _ { k }$ with $a ^ { k } = ( 0 , \\ldots , 0 , \\frac { 1 } { k } , 0 , \\ldots )$ that has value $\\textstyle { \\frac { 1 } { k } }$ on position $k$ and otherwise only zeros. This sequence converges to the 0-sequence in norm. However, we have ${ \\mathcal { K } } ( a ^ { k } ) = 1$ for all $k$ , thus the images do not converge to $0 = \\kappa ( 0 )$ . \u0003 ", + "bbox": [ + 173, + 320, + 825, + 381 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "From the preceding lemma, we are able to obtain the following alternative description of representation operators: ", + "bbox": [ + 174, + 387, + 825, + 416 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Proposition D.23 (Hom-tensor Adjunction). The map ", + "bbox": [ + 174, + 420, + 532, + 434 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/f41efcd4702dba80b9ee77419a1e163149125582dd7b91d03b938867e3b28d1e.jpg", + "text": "$$\n\\quad \\widetilde { \\mathrm { ( \\cdot ) } } : \\mathrm { H o m } _ { G , \\mathbb { K } } \\left( \\bigoplus _ { j i } V _ { j i } , \\mathrm { H o m } _ { \\mathbb { K } } ( V _ { l } , V _ { J } ) \\right) \\to \\mathrm { H o m } _ { G , \\mathbb { K } } \\left( \\left( \\bigoplus _ { j i } V _ { j i } \\right) \\otimes V _ { l } , V _ { J } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 235, + 438, + 761, + 467 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "given by ", + "bbox": [ + 173, + 469, + 232, + 484 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/efc2e0085f460b214705b94fdced264e222012b9616dff1836255472177f775a.jpg", + "text": "$$\n\\tilde { \\mathcal { K } } ( v _ { j } \\otimes v _ { l } ) : = \\left[ \\mathcal { K } ( v _ { j } ) \\right] ( v _ { l } )\n$$", + "text_format": "latex", + "bbox": [ + 408, + 481, + 588, + 501 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "is an isomorphism. ", + "bbox": [ + 174, + 501, + 299, + 516 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Proof. For continuity, note the following: by straightforward extensions of Lemma D.21, all linear and equivariant maps $\\oplus _ { j i } V _ { j i } \\to { \\mathrm { H o m } } _ { \\mathbb { K } } ( V _ { l } , V _ { J } )$ and $( \\bigoplus _ { j i } V _ { j i } ) \\otimes V _ { l } V _ { J }$ are necessarily continuous, and thus we can ignore continuity altogether. The rest of the proof can be done as in Agrawala (1980). For illustrating the most important part, we show that $\\tilde { \\mathcal { K } }$ is actually equivariant: ", + "bbox": [ + 174, + 530, + 826, + 598 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/6c57bd558ec38c51c2b57d5e5efc6ab38b09f408eceabd6185e341bd6fa94d79.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { K } \\big ( [ ( \\rho _ { j } \\otimes \\rho _ { l } ) ( g ) ] ( v _ { j } \\otimes v _ { l } ) \\big ) = \\tilde { K } \\big ( [ \\rho _ { j } ( g ) ] ( v _ { j } ) \\otimes [ \\rho _ { l } ( g ) ] ( v _ { l } ) \\big ) } \\\\ & { \\qquad = [ K ( \\rho _ { j } ( g ) ( v _ { j } ) ) ] \\big ( \\rho _ { l } ( g ) ( v _ { l } ) \\big ) } \\\\ & { \\qquad = \\big [ \\rho _ { \\mathrm { H o m } } ( g ) ( K ( v _ { j } ) ) \\big ] ( \\rho _ { l } ( g ) ( v _ { l } ) ) } \\\\ & { \\qquad = \\big ( \\rho _ { J } ( g ) \\circ \\mathcal { K } ( v _ { j } ) \\circ \\rho _ { l } ( g ) ^ { - 1 } \\big ) \\big ( \\rho _ { l } ( g ) ( v _ { l } ) \\big ) } \\\\ & { \\qquad = \\rho _ { J } ( g ) ( \\mathcal { K } ( v _ { j } ) ( v _ { l } ) ) } \\\\ & { \\qquad = \\rho _ { J } ( g ) ( \\tilde { K } ( v _ { j } \\otimes v _ { l } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 601, + 730, + 723 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Remark D.24. Some readers may wonder why this is called an adjunction. With removing some of the notation in the Proposition, one has ", + "bbox": [ + 176, + 744, + 823, + 773 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/7f7fcaaca471630cbb611e2e78e26c48de21a2a9b0c82373611f832255bc9fab.jpg", + "text": "$$\n\\mathrm { H o m } _ { G , \\mathbb { K } } ( T , \\mathrm { H o m } _ { \\mathbb { K } } ( U , V ) ) \\cong \\mathrm { H o m } _ { G , \\mathbb { K } } ( T \\otimes U , V ) .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 777, + 669, + 795 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Now, for notational clarity, set $F : = \\operatorname { H o m } _ { \\mathbb { K } } ( U , \\cdot )$ and $\\begin{array} { r } { H : = ( \\cdot ) \\otimes U } \\end{array}$ and remove the subscripts. Then the formula can be written as ", + "bbox": [ + 171, + 797, + 826, + 827 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/d7241edc0637bc6871c03fc44e490625a48e7e270d8953e6a108b84295b98ebc.jpg", + "text": "$$\n\\mathrm { H o m } ( T , F ( V ) ) \\cong \\mathrm { H o m } ( H ( T ) , V ) .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 830, + 619, + 848 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "With replacing the notation if the Hom-spaces with a scalar product, and the isomorphism sign with equality, this reads as follows: ", + "bbox": [ + 173, + 851, + 825, + 878 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/dde4900ad5b0d63cd1c0c8f24f2f90fb7f25ff347936a56b5320981a0647456e.jpg", + "text": "$$\n\\langle T | F ( V ) \\rangle = \\langle H ( T ) | V \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 415, + 877, + 583, + 895 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Similar to adjoints in Hilbert spaces, we can then view $F$ and $H$ as adjoint to each other. In categorical terms, they are a pair of adjoint functors, see Lane et al. (1998). ", + "bbox": [ + 169, + 896, + 826, + 924 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "D.2.3 PROOF OF THEOREM D.13 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 416, + 118 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "After the work done in the prior sections, we are ready to complete the proof of Theorem D.13! ", + "bbox": [ + 169, + 127, + 799, + 143 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "Proof of Theorem D.13. Only the first part of that theorem still needs to be proven. We have the following string of isomorphisms, which we will explain below: ", + "bbox": [ + 171, + 157, + 823, + 186 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/a3c2121e8298e914565bbf95b30b9b81d3d363872941d245209037b9ea146ef9.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathrm { H o m } _ { \\mathcal { Q } , \\mathbb { R } } ( \\mathcal { L } _ { \\mathbb { R } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathbb { R } } ( V _ { i } , V _ { j } ) ) \\cong \\mathrm { H o m } _ { \\mathcal { Q } , \\mathbb { R } } \\left( \\bigoplus _ { j _ { i } } V _ { j _ { i } } , \\mathrm { H o m } _ { \\mathbb { R } } ( V _ { i } , V _ { j } ) \\right) } \\\\ & { \\cong \\mathrm { H o m } _ { \\mathcal { Q } , \\mathbb { R } } \\left( \\left( \\bigoplus _ { j _ { i } } V _ { j _ { i } } \\right) \\otimes V _ { i } , V _ { j } \\right) } \\\\ & { \\cong \\mathrm { H o m } _ { \\mathbb { Q } , \\mathbb { R } } \\left( \\bigoplus _ { j _ { i } } ( V _ { j _ { i } } \\otimes V _ { i } ) , V _ { j } \\right) } \\\\ & { \\cong \\underset { \\hat { \\mathcal { P } } } { \\bigoplus } \\mathrm { H o m } _ { \\mathcal { Q } , \\mathbb { R } } ( V _ { j _ { i } } \\otimes V _ { i } , V _ { j } ) } \\\\ & { \\cong \\underset { \\hat { \\mathcal { P } } } { \\bigoplus } \\underset { \\hat { \\mathcal { P } } } { \\bigoplus } \\mathrm { H o m } _ { \\mathcal { Q } , \\mathbb { R } } ( V _ { j _ { i } } , V _ { j } ) } \\\\ & { \\cong \\underset { \\hat { \\mathcal { P } } } { \\bigoplus } \\underset { \\hat { \\mathcal { P } } } { \\bigoplus } \\mathrm { H o m } _ { \\mathcal { Q } , \\mathbb { R } } ( V _ { j _ { i } } , V _ { j } ) } \\\\ & { = \\underset { \\hat { \\mathcal { P } } } { \\bigoplus } \\underset { \\hat { \\mathcal { P } } = 1 } { \\bigoplus } \\underset { \\hat { \\mathcal { P } } = 1 } { \\bigoplus } \\mathrm { E m } _ { \\mathbb { Q } , \\mathbb { R } } ( V _ { j } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 191, + 748, + 407 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "The steps are justified as follows: ", + "bbox": [ + 174, + 411, + 393, + 428 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "1. For the first step, use Lemma D.20. ", + "bbox": [ + 214, + 440, + 465, + 457 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "2. For the second step, use Proposition D.23. ", + "bbox": [ + 214, + 464, + 511, + 481 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "3. For the third step, use that there is a natural isomorphism $\\left( \\bigoplus _ { j i } V _ { j i } \\right) \\otimes V _ { l } \\cong \\bigoplus _ { j i } ( V _ { j i } \\otimes V _ { l } ) .$ ", + "bbox": [ + 210, + 489, + 821, + 508 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "4. For the fourth step, use that linear equivariant maps can be described on each direct summand individually (and that we do not need to worry about continuity due to Lemma D.21). ", + "bbox": [ + 205, + 517, + 825, + 546 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "5. For the fifth step, precompose with the linear equivariant isometric embeddings $l _ { j i s } : V _ { J } $ $V _ { j i } \\otimes V _ { l }$ and use, again, that linear equivariant maps can be described on each direct summand individually. Furthermore, use Schur’s Lemma B.29 in order to see that the other summands disappear. ", + "bbox": [ + 210, + 554, + 826, + 612 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "6. The last step is just a reformulation. ", + "bbox": [ + 210, + 621, + 468, + 636 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "Now, we call the string of isomorphisms from right to left ", + "bbox": [ + 173, + 648, + 553, + 665 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/56daa1b2df4b3ecbc632c21fcf13cda98e9ab668903705d012b2e6d615f0f255.jpg", + "text": "$$\n\\mathrm { R e p } : \\bigoplus _ { j \\in \\hat { G } } ^ { m _ { j } } \\bigoplus _ { i = 1 } ^ { [ J ( j l ) ] } \\mathrm { E n d } _ { G , \\mathrm { K } } ( V _ { J } ) \\mathrm { H o m } _ { G , \\mathrm { K } } ( L _ { \\mathrm { K } } ^ { 2 } ( X ) , \\mathrm { H o m } _ { \\mathrm { K } } ( V _ { l } , V _ { J } ) )\n$$", + "text_format": "latex", + "bbox": [ + 259, + 670, + 736, + 719 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "and are only left with understanding that it is actually given by Eq. (17). For this, we take a tuple $( c _ { j i s } ) _ { j i s }$ of endomorphisms and explicitly trace back “where it comes from”. As in Lemma D.21, let $p _ { j i } : \\oplus _ { j ^ { \\prime } i ^ { \\prime } } V _ { j ^ { \\prime } i ^ { \\prime } } \\to V _ { j i }$ be the canonical projection, which is by Proposition F.46 explicitly given by $\\begin{array} { r } { p _ { j i } ( \\varphi ) = \\sum _ { m = 1 } ^ { d _ { j } } \\left. Y _ { j i } ^ { m } \\middle | \\varphi \\right. Y _ { j i } ^ { m } } \\end{array}$ . Furthermore, let $p _ { j i s } : V _ { j i } \\otimes V _ { l } V _ { J }$ be the projections corresponding to the embeddings $l _ { j i s }$ . Then from bottom to top, $( c _ { j i s } ) _ { j i s }$ gets transformed as follows: ", + "bbox": [ + 171, + 723, + 828, + 803 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/b1860166b2b7a7fffadd4006d33409242c4d152f289be6c82195ef125d1678d7.jpg", + "text": "$$\n\\begin{array} { r l } \\displaystyle ( c _ { j i s } ) _ { j i s } \\mapsto \\left( \\begin{array} { l } { \\left[ { \\boldsymbol J } ( j l ) \\right] } \\\\ { \\displaystyle \\sum _ { s = 1 } ^ { \\boldsymbol J } c _ { j i s } \\circ p _ { j i s } \\right) _ { j i } } \\\\ { \\mapsto \\displaystyle \\sum _ { j \\in \\widehat { \\boldsymbol G } } \\sum _ { i = 1 } ^ { m _ { j } } \\sum _ { s = 1 } ^ { [ { \\boldsymbol J } ( j l ) ] } c _ { j i s } \\circ p _ { j i s } \\circ ( p _ { j i } \\otimes \\mathrm { i d } _ { V _ { i } } ) } \\\\ { \\mapsto \\mathrm { R e p } ( ( c _ { j i s } ) _ { j i s } ) } \\end{array} \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 808, + 668, + 924 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "In the last step, the hom-tensor adjunction Proposition D.23 is used, but in the other direction. As an illustration, the composition of functions over which we sum can be shown in the following commutative diagram: ", + "bbox": [ + 173, + 103, + 823, + 146 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/27da207f6375f54d965606464effff3b7f266cb5e0806837673c7354042b41b1.jpg", + "text": "$$\n\\displaystyle \\bigoplus _ { i ^ { \\prime } j ^ { \\prime } } V _ { j ^ { \\prime } i ^ { \\prime } } \\otimes V _ { l } \\xrightarrow [ ] { p _ { j i } \\otimes \\mathrm { i d } _ { V _ { l } } } V _ { j i } \\otimes V _ { l } \\xrightarrow [ ] { p _ { j i s } } V _ { J } \\xrightarrow [ ] { c _ { j i s } } V _ { J }\n$$", + "text_format": "latex", + "bbox": [ + 285, + 154, + 710, + 181 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/552b4c8e971678abe6e7ff852d2a9ee1a6444d7ff83aa7aead01454a9365b591.jpg", + "text": "$$\nc _ { j i s } \\circ p _ { j i s } \\circ ( p _ { j i } \\otimes \\mathrm { i d } _ { V _ { l } } )\n$$", + "text_format": "latex", + "bbox": [ + 467, + 215, + 589, + 229 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "We obtain: ", + "bbox": [ + 173, + 236, + 246, + 251 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/afcf23e922d6c930c9dc1659f5d5c81b2c2dfcd014d5adabfc7610054455a6ff.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\big [ \\mathbb { R e p } \\big ( ( c _ { j i s } ) _ { j i s } \\big ) ( \\varphi ) \\big ] ( v _ { l } ) = \\sum _ { j \\in \\widetilde { \\mathcal { O } } } \\displaystyle \\sum _ { s = 1 } ^ { m _ { \\beta } } \\sum _ { s = 1 } ^ { \\lfloor ( J / \\beta ) \\rfloor } \\Big [ c _ { j i s } \\circ p _ { j i s } \\circ ( p _ { j i s } \\tilde { \\varphi } ) \\mathrm { d } \\psi _ { l } \\big ) \\Big ] ( \\varphi \\otimes v _ { l } ) } & { } \\\\ & { = \\displaystyle \\sum _ { j \\in \\widetilde { \\mathcal { O } } } \\sum _ { i ^ { \\prime } = 1 } ^ { m _ { \\beta } } \\sum _ { s = 1 } ^ { \\lfloor ( J / \\beta ) \\rfloor } \\Big ( c _ { j i s } \\circ p _ { j j s } \\circ ( p _ { j i s } ( \\varphi ) \\otimes v _ { l } ) \\Big ) ( p _ { j i s } \\circ v _ { l } ) } \\\\ & { = \\displaystyle \\sum _ { j \\in \\widetilde { \\mathcal { O } } } \\sum _ { i ^ { \\prime } = 1 } ^ { m _ { \\beta } } \\sum _ { s = 1 } ^ { \\lfloor ( J / \\beta ) \\rfloor } \\Big ( c _ { j i s } \\circ p _ { j j s } \\circ \\Big ( \\displaystyle \\sum _ { m = 1 } ^ { d _ { \\beta } } \\big \\langle Y _ { j i } ^ { m } \\big | \\varphi \\rangle Y _ { j i } ^ { m } \\otimes v _ { l } \\Big ) } \\\\ & { = \\displaystyle \\sum _ { j \\in \\widetilde { \\mathcal { O } } } \\sum _ { i ^ { \\prime } = 1 } ^ { m _ { \\beta } } \\sum _ { s = 1 } ^ { \\lfloor ( J / \\beta ) \\rfloor } \\sum _ { j \\in \\widetilde { \\mathcal { O } } } \\Big \\langle Y _ { j i s } ^ { m } \\circ ( p _ { j i s } ( \\varphi ) \\cdot c _ { j i s } \\circ ( p _ { j i s } ( \\varphi ) \\wedge Y _ { j i } ^ { m } \\otimes v _ { l } ) \\Big \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 253, + 761, + 454 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "That, finally, finishes the proof. ", + "bbox": [ + 174, + 457, + 380, + 472 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "E EXAMPLE APPLICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 491, + 419, + 507 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "In this chapter, we develop some relevant examples of the theory outlined in prior chapters. All of these examples are applications of Theorem D.16 and Corollary D.17. These examples are concerned with the following question: Given a specific field $\\mathbb { K } \\in \\{ \\mathbb { R } , \\mathbb { C } \\}$ , compact transformation group $G$ and homogeneous space $X$ of $G$ , how can a basis of steerable kernels $K : X $ ${ \\mathrm { H o m } } _ { \\mathrm { K } } ( V _ { l } , V _ { J } )$ for given irreducible representations $\\rho _ { l } : G \\to \\operatorname { U } ( V _ { l } )$ and $\\rho _ { J } : G \\to \\operatorname { U } ( V _ { J } )$ be determined? The theorems give an outline for what needs to be done in order to succeed in this task, and the steps are always as follows: ", + "bbox": [ + 173, + 522, + 825, + 621 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "1. For each $l \\in \\widehat { G }$ , a representative for the isomorphism class of irreducible representations $l$ needs to be determined. That is, one needs to determine $\\rho _ { l } : G \\to \\operatorname { U } ( V _ { l } )$ and an orthonormal basis $\\{ Y _ { l } ^ { n } \\mid n \\in \\{ 1 , \\ldots , d _ { l } \\} \\}$ . We omit the index $n$ if there is only one basis element. Usually, we have $V _ { l } = \\mathbb { K } ^ { d _ { l } }$ and the orthonormal basis is just the standard basis. \n2. The Peter-Weyl Theorem B.22 gives the existence-statement for a decomposition of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ into irreducible subrepresentations. We need an explicit such decomposition, i.e.: we need to find multiplicities $m _ { j }$ , irreducible subrepresentations $V _ { j i } \\cong V _ { j }$ for $i \\in \\{ 1 , \\ldots , m _ { j } \\}$ and basis functions $Y _ { j i } ^ { m } \\in V _ { j i } \\subseteq L _ { \\mathbb { K } } ^ { 2 } ( X )$ corresponding to the $Y _ { j } ^ { m }$ such that $L _ { \\mathbb { K } } ^ { 2 } ( X ) =$ ${ \\widehat { \\bigoplus } } _ { j \\in { \\widehat { G } } } \\oplus _ { i = 1 } ^ { m _ { j } } V _ { j i } .$ \n3. For each combination of $j , l$ and $J$ in $\\widehat { G }$ , one needs to find the number of times $[ J ( j l ) ]$ that $V _ { J }$ appears in a direct sum decomposition of $V _ { j } \\otimes V _ { l }$ . Then, for each $s \\in \\{ 1 , \\ldots , [ J ( \\dot { j } l ) ] \\}$ , and for all basis-indices $M , m$ and $n$ , one needs to determine the Clebsch-Gordan coefficients $\\langle s , J M | j m ; l n \\rangle$ . We omit the index $s$ if $V _ { J }$ appears only once in the direct sum decomposition of $V _ { j } \\otimes V _ { l }$ . \n4. For each $J$ one needs to determine a basis $\\{ c _ { r } \\mid r = 1 , \\ldots , E _ { J } \\}$ of the space of endomorphisms of $V _ { J }$ , namely $\\operatorname { E n d } _ { G , \\mathbb { K } } ( V _ { J } )$ . ", + "bbox": [ + 210, + 628, + 825, + 886 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "Once all of this is done, one can then simply write down the basis kernels according to Eq. (22) or, in case that the space of endomorphisms is 1-dimensional, Eq. (23). The ingredients determined above are purely representation-theoretic information about the situation at hand, which hopefully makes the reader appreciate the results even more: we do not simply determine basis kernels; we understand in detail, along the way, the representation theory of the group and homogeneous space. ", + "bbox": [ + 176, + 895, + 821, + 924 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "Note that we are not concerned with practical considerations related to how fine-grained to do this in practice (for example, if the space on which the kernels operate splits into infinitely many orbits). For such questions, we refer back to Remark D.19. ", + "bbox": [ + 174, + 152, + 821, + 195 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "In the following sections, we discuss harmonic networks (SO(2)-equivariant CNNs with complex representations), SO(2)-equivariant CNNs with real representations, reflection-equivariant networks, SO(3)-equivariant CNNs with both complex and real representations, and O(3)-equivariant CNNs with both complex and real representations. For each of these examples, we go through the four steps outlined above. We recommend looking at the first example in detail: we conduct it in the greatest detail and it is the easiest to understand and thus serves as a nice introduction. ", + "bbox": [ + 174, + 202, + 825, + 286 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "E.1 SO(2)-STEERABLE KERNELS FOR COMPLEX REPRESENTATIONS – HARMONIC NETWORKS ", + "text_level": 1, + "bbox": [ + 173, + 301, + 759, + 330 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "Here, we explain how the kernel constraint for harmonic networks (Worrall et al., 2016) can be solved using our theory. In the case of harmonic networks, we have $\\mathbb { K } = \\mathbb { C }$ , $G = \\mathrm { S O ( 2 ) }$ , $\\dot { X } = S ^ { 1 }$ . As in most examples that follow, we ignore the solution of the kernel constraint in the origin, since it is usually easy to solve. For simplifying the formulas, we employ the isomorphism ", + "bbox": [ + 173, + 342, + 826, + 398 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/211e6b161f1af69b67551299329941a5aaedb9916dfb38c1dc9b7e58f18ae4e4.jpg", + "text": "$$\n\\mathrm { S O ( 2 ) } \\stackrel { \\sim } { \\longrightarrow } \\mathrm { U ( 1 ) } , \\qquad \\left( { a } \\quad - { b } \\right) \\mapsto a + i b\n$$", + "text_format": "latex", + "bbox": [ + 352, + 404, + 645, + 439 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "and always write $\\mathrm { U } ( 1 )$ instead of SO(2). Here, $\\mathrm { U } ( 1 )$ is the group of rotations of $\\mathbb { C }$ , i.e., the group of elements in $\\mathbb { C }$ with absolute value 1. It is also called the circle group since the group elements lie on a circle in the complex plane. Note that the change from SO(2) to the isomorphic group U(1) is done purely for convenience reasons, and SO(2) could be used just as well. ", + "bbox": [ + 174, + 444, + 825, + 501 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "We now go through the four steps outlined above. Our statements about the representation theory of the circle group can be found in Kowalski (2014), chapter 5. ", + "bbox": [ + 174, + 507, + 823, + 536 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "E.1.1 CONSTRUCTION OF THE IRREDUCIBLE REPRESENTATIONS OF U(1) ", + "bbox": [ + 174, + 550, + 694, + 565 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "We have ${ \\widehat { \\mathrm { U } ( 1 ) } } = \\mathbb { Z }$ , and for $l \\in \\mathbb { Z }$ we can construct a representative $\\rho _ { l } : \\mathrm { U } ( 1 ) \\to \\mathrm { U } ( V _ { l } )$ as follows: $V _ { l } = \\mathbb { C }$ is just the canonical 1-dimensional C-vector space, and $\\rho _ { l }$ is given by ", + "bbox": [ + 169, + 578, + 823, + 609 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/73e842e570be0a000a97af8467f645c9f309061caa0206f0f21a8b4602bcafc1.jpg", + "text": "$$\n\\left[ \\rho _ { l } ( g ) \\right] ( z ) : = g ^ { l } \\cdot z ,\n$$", + "text_format": "latex", + "bbox": [ + 431, + 614, + 565, + 633 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "where $g$ is regarded as an element in C. One can easily check that this is an irreducible representation. The orthonormal basis element for each such representation is just given by $1 \\in \\mathbb { C } = V _ { l }$ . This already answers step 1 of the outline above. ", + "bbox": [ + 176, + 638, + 825, + 681 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "E.1.2 THE PETER-WEYL THEOREM FOR $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 1 } )$ ", + "text_level": 1, + "bbox": [ + 174, + 695, + 524, + 712 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "For step 2, we need to determine the Peter-Weyl decomposition of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 1 } )$ , where we regard $S ^ { 1 }$ as a subset of C. Let $Y _ { l 1 } : S ^ { 1 } \\to \\mathbb { C }$ be given by $Y _ { l 1 } ( z ) = z ^ { - l }$ . Let $\\bar { V } _ { l 1 } \\subseteq L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 1 } )$ just be given by its span: $V _ { l 1 } = \\mathrm { s p a n } _ { \\mathbb { C } } ( Y _ { l 1 } )$ . We want to see that this is a subrepresentation of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 1 } )$ . To see this, remember that the unitary representation on $L _ { \\mathbb { C } } ^ { 2 } ( X )$ is given by $\\lambda : \\mathrm { U } ( 1 ) \\mathrm { U } ( \\bar { L } _ { \\mathbb { C } } ^ { 2 } ( \\dot { S } ^ { 1 } ) )$ with $\\left[ \\lambda ( g ) \\varphi \\right] ( z ) = \\varphi ( g ^ { - 1 } z )$ . We have ", + "bbox": [ + 173, + 719, + 825, + 792 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/87126d5e0f48368a30d1a1b5fc4a3c95395d87c7a581a2bc11be964b17b73345.jpg", + "text": "$$\n\\left[ \\lambda ( g ) Y _ { l 1 } \\right] ( z ) = Y _ { l 1 } ( g ^ { - 1 } z ) = ( g ^ { - 1 } z ) ^ { - l } = g ^ { l } \\cdot z ^ { - l } = \\left( g ^ { l } \\cdot Y _ { l 1 } \\right) ( z )\n$$", + "text_format": "latex", + "bbox": [ + 279, + 799, + 718, + 819 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "and thus $\\lambda ( g ) Y _ { l 1 } = g ^ { l } Y _ { l 1 } \\in V _ { l 1 }$ , which is what we claimed. Since the $V _ { l 1 }$ are 1-dimensional, they are necessarily irreducible for dimension reasons. Now, an important result from Fourier analysis is that the $Y _ { l 1 }$ for $l \\in \\mathbb { Z }$ actually form an orthonormal basis of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 1 } )$ and that, consequently, the Peter-Weyl decomposition of $L _ { \\mathbb { C } } ^ { \\frac { \\cdot } { 2 } } ( S ^ { 1 } )$ looks as follows: ", + "bbox": [ + 173, + 825, + 825, + 882 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/d066f96af429b4fa5be2c0ab0bd9287ac53a88f371fe9a587c65e35d6d1ad593.jpg", + "text": "$$\nL _ { \\mathbb { C } } ^ { 2 } ( S ^ { 1 } ) = \\widehat { \\bigoplus _ { l \\in \\mathbb { Z } } } V _ { l 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 434, + 887, + 562, + 928 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "From this we see that the multiplicities $m _ { l }$ are all given by 1. What is missing is the connection to the irreps $\\rho _ { l } : \\mathrm { U } ( 1 ) \\to \\mathrm { U } ( V _ { l } )$ , but we have already indicated this in the notation. Namely, the map $f _ { l } : V _ { l } V _ { l 1 }$ given by $z \\mapsto z \\cdot Y _ { l 1 }$ is clearly an isomorphism of vector spaces, and due to Eq. (24) even an isomorphism of representations: ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/18e2c0992256700796e02adb44f4978109ca02a6a7a7e0fcf08075605a669148.jpg", + "text": "$$\n\\begin{array} { r l } & { f _ { l } \\big ( \\rho _ { l } ( g ) ( z ) \\big ) = f _ { l } \\big ( g ^ { l } \\cdot z \\big ) } \\\\ & { \\quad \\quad = ( g ^ { l } \\cdot z ) \\cdot Y _ { l 1 } } \\\\ & { \\quad \\quad = z \\cdot ( g ^ { l } \\cdot Y _ { l 1 } ) } \\\\ & { \\quad \\quad = z \\cdot ( \\lambda ( g ) ( Y _ { l 1 } ) ) } \\\\ & { \\quad \\quad = \\lambda ( g ) \\big ( z \\cdot Y _ { l 1 } \\big ) } \\\\ & { \\quad \\quad = \\lambda ( g ) \\big ( f _ { l } ( z ) \\big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 393, + 165, + 601, + 287 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "Thus, $f _ { l } \\circ \\rho _ { l } ( g ) = \\lambda ( g ) \\circ f _ { l }$ for all $g \\in \\operatorname { U } ( 1 )$ and, as claimed, $f _ { l }$ turns out to be an isomorphism. \nThis finishes step 2 of the outline above. ", + "bbox": [ + 171, + 291, + 821, + 320 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "E.1.3 THE CLEBSCH-GORDAN DECOMPOSITION ", + "text_level": 1, + "bbox": [ + 173, + 334, + 524, + 349 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "For step 3, we proceed as follows: The map ", + "bbox": [ + 174, + 359, + 464, + 375 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/9d5e93efbdf417fb9137820078f779b97cff469b22e4f4e3d76222f81c55263a.jpg", + "text": "$$\nf : V _ { j } \\otimes V _ { l } \\to V _ { j + l } , z _ { j } \\otimes z _ { l } \\mapsto z _ { j } \\cdot z _ { l }\n$$", + "text_format": "latex", + "bbox": [ + 370, + 381, + 627, + 398 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "is clearly well-defined and linear by the universal property of tensor products, see Definition D.1. Furthermore, it is an isometry: namely, since the scalar product in $\\mathbb { C }$ is just the usual multiplication (with the left entry being complex conjugated), we obtain ", + "bbox": [ + 174, + 405, + 825, + 446 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/68b4b7fce646e101db778d3888ddb9dda1a76307c8be05ab0eecbbd5784c5483.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left. f ( z _ { j } \\otimes z _ { l } ) \\big | f ( z _ { j } ^ { \\prime } \\otimes z _ { l } ^ { \\prime } ) \\right. = \\left. z _ { j } z _ { l } \\big | z _ { j } ^ { \\prime } z _ { l } ^ { \\prime } \\right. } \\\\ & { \\qquad = \\overline { { z _ { j } z _ { l } } } \\cdot z _ { j } ^ { \\prime } z _ { l } ^ { \\prime } } \\\\ & { \\qquad = \\overline { { z _ { j } } } z _ { j } ^ { \\prime } \\cdot \\overline { { z _ { l } } } z _ { l } ^ { \\prime } } \\\\ & { \\qquad = \\left. z _ { j } \\big | z _ { j } ^ { \\prime } \\right. \\cdot \\left. z _ { l } | z _ { l } ^ { \\prime } \\right. } \\\\ & { \\qquad = \\left. z _ { j } \\otimes z _ { l } \\big | z _ { j } ^ { \\prime } \\otimes z _ { l } ^ { \\prime } \\right. . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 450, + 651, + 556 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "In the last step, we have used the definition of the scalar product on the tensor product, Definition D.2. Thus, $f$ is an isomorphism of Hilbert spaces. Finally, it also respects the representations since ", + "bbox": [ + 173, + 558, + 820, + 587 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/c16057bb5e223b5256946a379042d70b264690ca90f2da1683b8edf8e2700d3d.jpg", + "text": "$$\n\\begin{array} { r l } & { f \\big ( \\left[ \\left( \\rho _ { j } \\otimes \\rho _ { l } \\right) ( g ) \\right] ( z _ { j } \\otimes z _ { l } ) \\big ) = f \\big ( \\left[ \\rho _ { j } ( g ) \\right] ( z _ { j } ) \\otimes \\left[ \\rho _ { l } ( g ) \\right] ( z _ { l } ) \\big ) } \\\\ & { \\qquad = f \\big ( g ^ { j } z _ { j } \\otimes g ^ { l } z _ { l } \\big ) } \\\\ & { \\qquad = g ^ { j } z _ { j } \\cdot g ^ { l } z _ { l } } \\\\ & { \\qquad = g ^ { j + l } \\cdot \\left( z _ { j } z _ { l } \\right) } \\\\ & { \\qquad = \\left[ \\rho _ { j + l } ( g ) \\right] ( f ( z _ { j } \\otimes z _ { l } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 590, + 699, + 691 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "and thus $f \\circ ( \\rho _ { j } \\otimes \\rho _ { l } ) ( g ) = \\rho _ { j + l } ( g ) \\circ f$ for all $g \\in \\operatorname { U } ( 1 )$ . Finally, the basis vectors correspond in the simplest possible way since $f ( 1 \\otimes 1 ) = 1$ . ", + "bbox": [ + 174, + 695, + 821, + 726 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "Overall, what we’ve shown is the following: $V _ { J }$ is a direct summand of $V _ { j } \\otimes V _ { l }$ if and only if $J = j + l$ . If this is the case, we have $[ J ( j l ) ] = 1$ and can thus omit the index $s$ . The only ClebschGordan coefficient is then given by $\\langle J 1 | j 1 l 1 \\rangle = 1$ since the basis elements directly correspond. ", + "bbox": [ + 174, + 731, + 825, + 775 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "E.1.4 ENDOMORPHISMS OF $V _ { J }$ ", + "text_level": 1, + "bbox": [ + 176, + 789, + 400, + 804 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "This is the simplest part: Since we are considering representations over C, Schur’s Lemma D.8 tells us that $\\operatorname { E n d } _ { \\operatorname { U } ( 1 ) , \\operatorname { C } } ( V _ { J } )$ is 1-dimensional for each irrep $J$ , and thus we can ignore the endomorphisms altogether. ", + "bbox": [ + 174, + 813, + 825, + 856 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "E.1.5 BRINGING EVERYTHING TOGETHER ", + "text_level": 1, + "bbox": [ + 176, + 871, + 480, + 886 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "We now show that a basis of steerable kernels $K : S ^ { 1 } \\to \\operatorname { H o m } _ { \\mathbb { C } } ( V _ { l } , V _ { J } )$ of the group $\\mathrm { U } ( 1 )$ is given, when expressed as $1 \\times 1$ -matrix parameterized by $S ^ { 1 }$ , by the basis function $Y _ { l - J } : S ^ { 1 } \\to \\mathbb { C }$ . We ", + "bbox": [ + 176, + 893, + 823, + 924 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "remove the index “1” at the basis function to remove clutter. How can we see this result, using Eq. (23)? ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "Note that $V _ { J }$ can only appear as a direct summand of $V _ { j } \\otimes V _ { l }$ if $j = J - l$ by what we’ve shown above. The “matrix” of Clebsch-Gordan coefficients $\\mathrm { C G } _ { J ( ( J - l ) l ) }$ is then just the number 1. We can omit the vacuous indices $i$ and $s$ and obtain that the only basis kernel is given by ", + "bbox": [ + 173, + 138, + 823, + 181 + ], + "page_idx": 69 + }, + { + "type": "equation", + "img_path": "images/09de066bec3d61e78701f5315147124ba122ff8d722afd2cf8b064f28ad0ef84.jpg", + "text": "$$\n\\begin{array} { r } { K _ { J - l } ( x ) = \\langle Y _ { J - l } | x \\rangle = \\overline { { Y _ { J - l } ( x ) } } } \\\\ { = \\overline { { x ^ { - ( J - l ) } } } } \\\\ { = x ^ { - ( l - J ) } } \\\\ { = Y _ { l - J } ( x ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 385, + 188, + 609, + 270 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "This result is precisely equal to the one obtained in the original paper (Worrall et al., 2016). This concludes our investigations of harmonic networks. ", + "bbox": [ + 173, + 275, + 825, + 304 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "E.2 SO(2)-STEERABLE KERNELS FOR REAL REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 320, + 630, + 335 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "In this section, we look at the case $\\mathbb { K } = \\mathbb { R }$ , $G = \\mathrm { S O ( 2 ) }$ and $X = S ^ { 1 }$ . In the following sections, we again step by step determine the representation-theoretic ingredients that we need for the application of our theorem. Compared to Chapter A, which focuses more on the components themselves and how they relate to the general situation, this section has a stronger focus on actually determining the final kernels, which also involves the task of determining the Clebsch-Gordan coefficients explicitly. We remark that the resulting kernels are not new, since Weiler & Cesa (2019) have solved for this kernel basis already. However, we want to emphasize again that with our method, we learn more about the representation theory of SO(2) and thus get an overall better conceptual understanding of how the kernels arise. ", + "bbox": [ + 173, + 345, + 825, + 473 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "Since it will help the presentation of our results, we set $\\mathrm { S O ( 2 ) } = \\mathbb { R } / 2 \\pi \\mathbb { Z } .$ , i.e., we view ${ \\mathrm { S O } } ( 2 )$ as a group of angles. We also set $S ^ { 1 } = \\mathbb { R } / 2 \\pi \\mathbb { Z }$ , i.e., we take the interval $[ 0 , 2 \\pi ]$ as the space where our functions are defined. Consequently, since we want our Haar measure to be normalized, we have to put the fraction $\\scriptstyle { \\frac { 1 } { 2 \\pi } }$ before all of our integrals, different from what we did in our treatment of SO(2) over $\\mathbb { C }$ . ", + "bbox": [ + 173, + 478, + 825, + 550 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "Note that since we now consider representations over the real numbers, unitary representations become orthogonal and we write $\\mathrm { O } ( \\bar { V } )$ instead of $\\mathrm { U } ( V )$ . ", + "bbox": [ + 174, + 556, + 823, + 585 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "E.2.1 CONSTRUCTION OF THE IRREDUCIBLE REPRESENTATIONS OF SO(2) ", + "text_level": 1, + "bbox": [ + 173, + 599, + 702, + 614 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "The irreps of SO(2) over $\\mathbb { R }$ are given by $\\rho _ { l } : \\mathrm { S O } ( 2 ) \\to \\mathrm { O } ( V _ { l } )$ , $l \\in \\mathbb { N } _ { \\geq 0 }$ . For $l = 0$ , we have $V _ { 0 } = \\mathbb { R }$ and the action is trivial. For $l \\geq 1$ , $V _ { l } = \\mathbb { R } ^ { 2 }$ as a vector space. The action is given by ", + "bbox": [ + 174, + 623, + 825, + 654 + ], + "page_idx": 69 + }, + { + "type": "equation", + "img_path": "images/f08f4adbbfc7dc6f28c7c6eaea0d86c89c501a88f16a9cd9f2af76230c77476c.jpg", + "text": "$$\n\\left[ \\rho _ { l } ( \\phi ) \\right] ( v ) = \\left( \\begin{array} { c c } { \\cos ( l \\phi ) } & { - \\sin ( l \\phi ) } \\\\ { \\sin ( l \\phi ) } & { \\phantom { + } \\cos ( l \\phi ) } \\end{array} \\right) \\cdot v\n$$", + "text_format": "latex", + "bbox": [ + 364, + 660, + 633, + 695 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "for $\\phi \\in \\mathrm { S O } ( 2 ) = \\mathbb { R } / 2 \\pi \\mathbb { Z }$ . The orthonormal basis is in both cases just given by standard basis vectors. ", + "bbox": [ + 171, + 702, + 823, + 731 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "E.2.2 THE PETER-WEYL THEOREM FOR $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ ", + "text_level": 1, + "bbox": [ + 174, + 744, + 524, + 762 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "Now we look at square-integrable functions $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ that we now assume to take real values. As before, SO(2) acts on this space by $( \\lambda ( \\phi ) f ) ( x ) = f ( x - \\phi )$ .26 For notational simplicity, we write $\\mathrm { c o s } _ { l }$ for the function that maps $x$ to $\\cos ( l x )$ , and analogously for $\\mathrm { s i n } _ { l }$ . One then can show the following, which is a standard result in Fourier analysis: ", + "bbox": [ + 173, + 770, + 825, + 829 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "Proposition E.1. The functions $\\mathrm { c o s } _ { l }$ , $\\mathrm { s i n } _ { l }$ , $l \\geq 1$ span an irreducible invariant subspace of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ of dimension 2, explicitly given by ", + "bbox": [ + 173, + 833, + 823, + 862 + ], + "page_idx": 69 + }, + { + "type": "equation", + "img_path": "images/2042eb2ff3caf5b89bb7b95ca81b0925a7b84284498ebacbd47ddeadc009f272.jpg", + "text": "$$\n\\mathrm { s p a n } _ { \\mathbb { R } } ( \\cos _ { l } , \\sin _ { l } ) = \\big \\{ \\alpha \\cos _ { l } + \\beta \\sin _ { l } \\mid \\alpha , \\beta \\in \\mathbb { R } \\big \\}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 867, + 661, + 887 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "which is isomorphic as an orthogonal representation to $V _ { l }$ by ${ \\sqrt { 2 } } \\cos _ { l } \\mapsto { \\binom { 1 } { 0 } }$ and $\\sqrt { 2 } \\sin _ { l } \\mapsto$ $\\binom { 0 } { 1 }$ .27 Furthermore, $\\mathrm { { s i n } _ { 0 } = 0 }$ and $\\cos _ { 0 } = 1$ are constant functions and their span is 1-dimensional and equivariantly isomorphic to $V _ { 0 }$ by $\\cos _ { 0 } \\mapsto 1$ . ", + "bbox": [ + 173, + 99, + 826, + 178 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Finally, the functions ${ \\sqrt { 2 } } \\cdot \\cos _ { l }$ , $\\sqrt { 2 } \\cdot \\sin _ { l }$ form an orthonormal basis of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ , i.e., every function can be written uniquely as a (possibly infinite) linear combination of these basis functions. ", + "bbox": [ + 174, + 184, + 825, + 214 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "When setting $V _ { l 1 } = \\mathrm { s p a n } _ { \\mathbb { R } } ( \\cos _ { l } , \\sin _ { l } )$ , we thus obtain a decomposition ", + "bbox": [ + 174, + 222, + 640, + 238 + ], + "page_idx": 70 + }, + { + "type": "equation", + "img_path": "images/7b03e2534f5818dfc4955d3d43ea7a12f59d6e54f2d66f755d7d05c4bb124fef.jpg", + "text": "$$\nL _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } ) = { \\widehat { \\bigoplus _ { l \\geq 0 } } } V _ { l 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 434, + 239, + 562, + 279 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Thus, we have $m _ { l } = 1$ for all $l \\in \\mathbb N$ . All in all, we know everything there is to know about the Peter-Weyl theorem in our situation. ", + "bbox": [ + 173, + 279, + 825, + 309 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "E.2.3 THE CLEBSCH-GORDAN DECOMPOSITION ", + "text_level": 1, + "bbox": [ + 174, + 321, + 524, + 337 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "We now do the explicit decomposition of $V _ { j } \\otimes V _ { l }$ into irreps, which will give us the Clebsch-Gordan coefficients that we need. Instead of doing the decomposition in terms of $V _ { j }$ and $V _ { l }$ themselves, in the proofs we actually use the isomorphic images $V _ { j 1 }$ and $V _ { l 1 }$ in $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ . For doing so, we first need some trigonometric formulas in our disposal: ", + "bbox": [ + 173, + 345, + 826, + 404 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Lemma E.2. The sine and cosine functions fulfill the following rules: ", + "bbox": [ + 173, + 406, + 632, + 421 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "$l . \\ \\cos _ { j + l } = \\cos _ { j } \\cos _ { l } - \\sin _ { j } \\sin _ { l } .$ 2. $\\begin{array} { r } { \\sin _ { j + l } = \\sin _ { j } \\cos _ { l } + \\cos _ { j } \\sin _ { l } } \\end{array}$ . 3. $\\mathrm { c o s } _ { j - l } = \\mathrm { c o s } _ { j } \\mathrm { c o s } _ { l } + \\mathrm { s i n } _ { j } \\mathrm { s i n } _ { l } .$ . 4. $\\begin{array} { r } { \\sin _ { j - l } = \\sin _ { j } \\cos _ { l } - \\cos _ { j } \\sin _ { l } . } \\end{array}$ . ", + "bbox": [ + 210, + 429, + 434, + 516 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Proof. The first two are well-known and the last two follow directly from the first two using sin−j = $- \\sin _ { j }$ and $\\mathrm { c o s } _ { - j } = \\mathrm { c o s } _ { j }$ . ", + "bbox": [ + 173, + 529, + 825, + 559 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "We will need the following general lemma: ", + "bbox": [ + 174, + 573, + 459, + 588 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Lemma E.3. Let $f : V \\to V ^ { \\prime }$ be an intertwiner between representations $\\rho : G \\to { \\mathrm { G L } } ( V )$ and $\\rho ^ { \\prime } : G \\to { \\mathrm { G L } } ( V ^ { \\prime } )$ . Then ${ \\mathrm { n u l l } } ( f ) = \\{ v \\in V | f ( v ) = 0 \\}$ is an invariant linear subspace of $V .$ ", + "bbox": [ + 174, + 589, + 825, + 619 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Proof. This can easily be checked by the reader. ", + "bbox": [ + 173, + 632, + 493, + 648 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "As a remark on notation for the following proposition: We write the Clebsch-Gordan coefficients $\\mathrm { C G } _ { J ( j l ) s }$ of irreps $V _ { J }$ , $V _ { j }$ and $V _ { l }$ with dimensions $d _ { J } , d _ { j }$ and $d _ { l }$ as a $d _ { J } \\times ( d _ { j } \\times d _ { l } )$ -tensor. That is, it consists of $d _ { J }$ “rows”, each of which is a $d _ { j } \\times d _ { l }$ -matrix. If $V _ { J }$ appears only once in the tensor product, we omit the index $s$ as before. ", + "bbox": [ + 174, + 661, + 826, + 719 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Proposition E.4. We have the following decomposition results: ", + "bbox": [ + 173, + 720, + 591, + 736 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "1. For $j = l = 0$ we have $V _ { 0 } \\otimes V _ { 0 } \\cong V _ { 0 }$ and Clebsch-Gordan coefficients $\\mathrm { C G } _ { 0 ( 0 0 ) } = \\bigl ( [ 1 ] \\bigr )$ . ", + "bbox": [ + 207, + 743, + 813, + 761 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "2. For $j = 0 , l > 0$ we have $V _ { 0 } \\otimes V _ { l } \\cong V _ { l }$ and Clebsch-Gordan coefficients $\\mathrm { C G } _ { l ( 0 l ) } =$ $\\left( \\begin{array} { l l } { { [ 1 } } & { { 0 ] } } \\\\ { { [ 0 } } & { { 1 ] } } \\end{array} \\right)$ . ", + "bbox": [ + 209, + 767, + 825, + 816 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "3. For $j > 0$ , $l = 0$ , we get $V _ { j } \\otimes V _ { 0 } \\cong V _ { j }$ and Clebsch-Gordan coefficients $\\mathrm { C G } _ { j ( j 0 ) } =$ ", + "bbox": [ + 204, + 823, + 825, + 840 + ], + "page_idx": 70 + }, + { + "type": "equation", + "img_path": "images/007e3c5717a4eb21adace79e5083b966464dc21b7c0cb9e87daefb6601e8f90f.jpg", + "text": "$$\n\\begin{array} { r } { ( [ \\begin{array} { l } { 1 } \\\\ { 0 } \\\\ { [ 0 ^ { \\vphantom { \\frac { 1 } { 2 } } } ] } \\end{array} ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 838, + 294, + 904 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "4. For $j > l > 0$ we get $V _ { j } \\otimes V _ { l } \\cong V _ { j - l } \\oplus V _ { j + l }$ . The Clebsch-Gordan coefficients are given ", + "bbox": [ + 207, + 102, + 825, + 119 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/ef79b192bb91388aab9c2980d295b9450eb9e127d94a45ed9ce20fb4681c31ae.jpg", + "text": "$$\nb y \\mathrm { C G } _ { j - l , ( j l ) } = ( \\begin{array} { r r } { { [ 1 } } & { { 0 ] } } \\\\ { { 0 } } & { { 1 } } \\\\ { { [ 0 } } & { { - 1 ] } } \\\\ { { 1 } } & { { 0 ] } } \\end{array} ) a n d \\mathrm { C G } _ { j + l , ( j l ) } = ( \\begin{array} { r r } { { [ 1 } } & { { 0 ] } } \\\\ { { 0 } } & { { - 1 } } \\\\ { { [ 0 } } & { { 1 } } \\\\ { { 1 } } & { { 0 ] } } \\end{array} ) .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 118, + 656, + 183 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "5. For $l > j > 0$ we get $V _ { j } \\otimes V _ { l } \\cong V _ { l - j } \\oplus V _ { j + l }$ . The Clebsch-Gordan coefficients are given ", + "bbox": [ + 202, + 188, + 823, + 204 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/62f1279ed4f424aa769b7234047e5e62b1a542017b7c3a3f9f0a0fe7e85a1edb.jpg", + "text": "$$\nb y \\mathrm { C G } _ { ( l - j ) ( j l ) } = ( [ \\begin{array} { l l } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\\\ { 0 } & { 1 } \\\\ { - 1 } & { 0 } \\end{array} ] ) a n d \\mathrm { C G } _ { j + l , ( j l ) } = ( \\begin{array} { l l } { [ 1 } & { 0 } \\\\ { 0 } & { - 1 } \\\\ { [ 0 } & { 1 ] } \\\\ { 1 } & { 0 } \\end{array} ) .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 203, + 663, + 267 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "6. For Cleb $j ~ = ~ l ~ > ~ 0$ , we getefficients $V _ { l } \\otimes V _ { l } \\cong V _ { 0 } ^ { 2 } \\oplus V _ { 2 l }$ n the, and $\\mathrm { C G } _ { 0 ( l l ) 1 } = \\left( \\left[ \\begin{array} { c c } { { 1 } } & { { 0 } } \\\\ { { 0 } } & { { 1 } } \\end{array} \\right] \\right)$ $\\mathrm { C G } _ { 0 ( l l ) 2 } = \\left( \\left[ \\begin{array} { c c } { { 0 } } & { { \\mp 1 } } \\\\ { { \\pm 1 } } & { { 0 } } \\end{array} \\right] \\right)$ $\\begin{array} { r } { \\mathrm { C G } _ { 2 l , ( l l ) } = \\left( \\begin{array} { c c } { \\left[ 1 \\right]} & { 0 } \\\\ { 0 } & { - 1 } \\\\ { \\left[ 0 } & { 1 \\right] } \\\\ { 1 } & { 0 } \\end{array} \\right) , } \\end{array}$ the last one being the same as the Clebsch-Gordan coefficients $\\mathrm { C G } _ { j + l , ( j l ) }$ from above. In $\\mathrm { C G } _ { 0 ( l l ) 1 }$ and $\\mathrm { C G } _ { 0 ( l l ) 2 }$ , a fourth index is present, namely 1 and 2, respectively. This is the index “ $s$ ” that was missing in all the prior examples, since this is the first time an irrep appears more than once in a tensor product decomposition. Note that for $\\mathrm { C G } _ { 0 ( l l ) 2 }$ , we have exactly one positive and one negative entry present, but both are equally valid and mirror the lower halves in $\\mathrm { C G } _ { j - l , ( j l ) }$ from part 4 and $\\mathrm { C G } _ { l - j , ( j l ) }$ from part 5. ", + "bbox": [ + 212, + 273, + 826, + 467 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "Proof. In the proof, instead of working directly with the irreps $\\rho _ { j } : \\mathrm { S O } ( 2 ) \\mathrm { O } ( V _ { j } )$ , we use the isomorphic copies $V _ { j 1 }$ in $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ given in Proposition E.1. Since we think that it does not help understanding to carry the index “1” in all computations, we omit this index. ", + "bbox": [ + 174, + 479, + 825, + 525 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "The proof of 1, 2, and 3 is clear. ", + "bbox": [ + 174, + 530, + 383, + 545 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "For 4 and 5, consider the (unnormalized) basis $\\left\\{ \\cos _ { j } \\otimes \\cos _ { l } , \\cos _ { j } \\otimes \\sin _ { l } , \\sin _ { j } \\otimes \\cos _ { l } , \\sin _ { j } \\otimes \\sin _ { l } \\right\\}$ of $V _ { j } \\otimes V _ { l }$ . Our goal is to express these basis elements with respect to basis elements of invariant subspaces. We do this by explicitly constructing an isomorphism to a decomposition of irreps. To that end, let $p : V _ { j } \\otimes V _ { l } \\to ^ { \\cdot } L _ { \\mathbb { R } } ^ { 2 } ( \\hat { S } ^ { 1 } )$ be given by $f \\otimes g \\mapsto f \\cdot g$ , which is clearly a well-defined intertwiner. We get as image of $p$ the set ", + "bbox": [ + 173, + 551, + 826, + 622 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/6f25cdc874c3e860cc6cc5d2a6a4dc489b1517873f23920dd323f33f5d9a2087.jpg", + "text": "$$\n\\begin{array} { r l } & { \\operatorname { i m } ( p ) = \\operatorname { s p a n } _ { \\mathbb { R } } \\big ( p ( \\cos _ { j } \\otimes \\cos _ { l } ) , ~ p ( \\cos _ { j } \\otimes \\sin _ { l } ) , ~ p ( \\sin _ { j } \\otimes \\cos _ { l } ) , ~ p ( \\sin _ { j } \\otimes \\sin _ { l } ) \\big ) } \\\\ & { \\qquad = \\operatorname { s p a n } _ { \\mathbb { R } } \\big ( \\cos _ { j } \\cdot \\cos _ { l } , ~ \\cos _ { j } \\cdot \\sin _ { l } , ~ \\sin _ { j } \\cdot \\cos _ { l } , ~ \\sin _ { j } \\cdot \\sin _ { l } \\big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 227, + 626, + 767, + 667 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "From Lemma E.2 we obtain: ", + "bbox": [ + 174, + 670, + 362, + 684 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/cbefbd35e2edc072eaa65e0224fcd1f3c69dab823124c79e538ac2ec208483c9.jpg", + "text": "$$\n\\begin{array} { r l } { ( a ) } & { p ( \\cos _ { j } \\otimes \\cos _ { l } ) \\ - \\ p ( \\sin _ { j } \\otimes \\sin _ { l } ) \\ = \\ \\cos _ { j + l } , } \\\\ { ( b ) } & { p ( \\cos _ { j } \\otimes \\sin _ { l } ) \\ + \\ p ( \\sin _ { j } \\otimes \\cos _ { l } ) \\ = \\ \\sin _ { j + l } , } \\\\ { ( c ) } & { p ( \\cos _ { j } \\otimes \\cos _ { l } ) \\ + \\ p ( \\sin _ { j } \\otimes \\sin _ { l } ) \\ = \\ \\cos _ { j - l } , } \\\\ { ( d ) } & { p ( \\sin _ { j } \\otimes \\cos _ { l } ) \\ - \\ p ( \\cos _ { j } \\otimes \\sin _ { l } ) \\ = \\ \\sin _ { j - l } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 686, + 665, + 761 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "Since the right hand sides are linearly independent basis functions of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 1 } )$ , we obtain: ", + "bbox": [ + 173, + 763, + 758, + 780 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/6068ebbc6bfa7939383d368e30ffd106643f9101d761fd7043397150dff6caa3.jpg", + "text": "$$\n\\begin{array} { r } { \\operatorname { i m } ( p ) = \\operatorname { s p a n } _ { \\mathbb { R } } \\big ( \\cos _ { j + l } , ~ \\sin _ { j + l } , ~ \\cos _ { j - l } , ~ \\sin _ { j - l } \\big ) = V _ { | j - l | } \\oplus V _ { j + l } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 785, + 728, + 804 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "Note for the last step that due to symmetry, $\\cos _ { j - l } = \\cos _ { l - j }$ and $\\mathrm { s i n } _ { j - l } = - \\sin _ { l - j }$ ", + "bbox": [ + 174, + 809, + 725, + 824 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "We now specialize to the case of 4, i.e., $j > l > 0$ . In this case, $V _ { | j - l | } = V _ { j - l }$ , and the basis is given by $\\mathrm { c o s } _ { j - l }$ and $\\mathrm { s i n } _ { j - l }$ , as in the right hand sides of Eq. (25) (c) and (d). Consequently, Eq. (25) is already the expansion of the new basis elements with the old, and the coefficients are consequently the Clebsch-Gordan coefficients.28 More precisely, if we want to compute, for example, CGj−l,(jl), ", + "bbox": [ + 174, + 829, + 823, + 887 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "then we observe from (c) that ", + "bbox": [ + 174, + 103, + 369, + 117 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/0f55b7f46f32d082a74facfc9a75e7af4b03c159dc32c15614ed080fad59b69d.jpg", + "text": "$$\n\\begin{array} { r l r } { \\cos _ { j - l } = } & { + 1 \\cdot p ( \\cos _ { j } \\otimes \\cos _ { l } ) } & { + 0 \\cdot p ( \\cos _ { j } \\otimes \\sin _ { l } ) } \\\\ & { + 0 \\cdot p ( \\sin _ { j } \\otimes \\cos _ { l } ) } & { + 1 \\cdot p ( \\sin _ { j } \\otimes \\sin _ { l } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 123, + 673, + 161 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "from which we can already read the upper half of $\\mathrm { C G } _ { j - l , ( j l ) }$ as the coefficients in this equation (which we conveniently visually arranged in the right way). For the lower half, we do proceed the same for $\\mathrm { s i n } _ { j - l }$ , using (d). Then, for $\\mathrm { C G } _ { j + l , ( j l ) }$ , we proceed exactly the same, using parts (a) and (b). That proves 4. ", + "bbox": [ + 173, + 165, + 826, + 222 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "For 5, we have $l > j > 0$ . In this case, $V _ { | j - l | } = V _ { l - j }$ , i.e., the basis is given by $\\cos _ { l - j } = \\cos _ { j - l }$ and $\\mathrm { s i n } _ { l - j } = - \\mathrm { s i n } _ { j - l }$ . The latter means that in part (d) of Eq. (25), we need to replace $\\mathrm { s i n } _ { j - l }$ by $\\mathrm { s i n } _ { l - j }$ and thus change the signs on the left hand side. This change means that $\\mathrm { C G } _ { j + l , ( j l ) }$ will remain the same as in 4, the upper half of $\\mathrm { C G } _ { l - j , ( j l ) }$ will remain the same as the upper half of $\\mathrm { C G } _ { j - l , ( j l ) }$ from part 4 since the cosine in part (c) of Eq. (25) is symmetric, and the lower part will flip the signs. This fully proves 5. ", + "bbox": [ + 173, + 228, + 825, + 313 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "Finally, we prove 6. We have $j = l$ and still consider the same function $p$ . Note that $p ( \\cos _ { j } \\otimes \\cos _ { l } ) +$ $p ( \\sin _ { j } \\otimes \\sin _ { l } ) = 1$ and $p ( \\sin _ { j } \\otimes \\cos _ { l } ) - p ( \\cos _ { j } \\otimes \\sin _ { l } ) = 0$ are constant functions that span the 1- dimensional trivial representation. Thus, we see that $p$ is a surjection ", + "bbox": [ + 173, + 319, + 825, + 362 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/5a9c180a903e18e5a37d8f935580e3cf7185178e288d233bad7ff63b8aedae2b.jpg", + "text": "$$\np : V _ { l } \\otimes V _ { l } \\to V _ { 0 } \\oplus V _ { 2 l }\n$$", + "text_format": "latex", + "bbox": [ + 419, + 368, + 578, + 385 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "with null space spanned by $\\begin{array} { r } { \\sin _ { j } \\otimes \\cos _ { l } - \\cos _ { j } \\otimes \\sin _ { l } } \\end{array}$ . Such a null space is automatically an invariant subspace as well, and since it is one-dimensional, it also must be isomorphic to the trivial representation. Overall, this gives us an isomorphism ", + "bbox": [ + 174, + 392, + 825, + 434 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/7e6fd0020da353c85b3f6dddd5ad198ada8e4998fb95cb7b2c121bbf8afd812a.jpg", + "text": "$$\nV _ { l } \\otimes V _ { l } \\cong V _ { 0 } ^ { 2 } \\oplus V _ { 2 l } .\n$$", + "text_format": "latex", + "bbox": [ + 428, + 439, + 570, + 458 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "From this, we can as before read off the Clebsch-Gordan coefficients. The only thing that changes is that parts (c) and (d) of Eq. (25) now correspond to two different copies of $V _ { 0 }$ , which means that the Clebsch-Gordan coefficients $\\mathrm { C G } _ { 0 ( l l ) }$ now split up in two parts $\\mathrm { C G } _ { 0 ( l l ) 1 }$ and $\\mathrm { C G } _ { 0 ( l l ) 2 }$ . Note that in the trivial representation, the isomorphism that sends the basis vector to its negative is clearly equivariant, which means that both combinations of signs that we give in the final formula for $\\mathrm { C G } _ { 0 ( l l ) 2 }$ are valid. □ ", + "bbox": [ + 173, + 463, + 825, + 549 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "E.2.4 ENDOMORPHISMS OF $V _ { J }$ ", + "text_level": 1, + "bbox": [ + 174, + 564, + 401, + 579 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "We now describe the endomorphisms of the irreducible representations, our last ingredient: ", + "bbox": [ + 174, + 588, + 772, + 603 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "Proposition E.5. We have $\\operatorname { E n d } _ { \\mathrm { S O ( 2 ) } , \\mathbb { R } } ( V _ { 0 } ) \\cong \\mathbb { R } ,$ , i.e., multiplications with all real numbers are valid endomorphisms of $V _ { 0 }$ . For $l \\geq 1$ , we get ", + "bbox": [ + 173, + 606, + 820, + 636 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/26ad43ad213f7a6f6713bd82525ffd13795fb368bdf3f5a6e8d94b69f3f15375.jpg", + "text": "$$\n\\operatorname { E n d } _ { \\operatorname { S O } ( 2 ) , \\mathbb { R } } ( V _ { l } ) = \\{ \\binom { a } { b } \\ \\frac { - b } { a } ) \\Big | a , b \\in \\mathbb { R } \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 351, + 641, + 643, + 676 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "which is the set of all scaled rotations of $\\mathbb { R } ^ { 2 }$ . When identifying $ { \\mathbb { R } } ^ { 2 } \\cong \\mathbb { C }$ , we can also view these transformations as arbitrary multiplications with a complex number. ", + "bbox": [ + 173, + 683, + 820, + 713 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "As a consequence, $\\mathrm { i d _ { R } }$ is a basis for $\\operatorname { E n d } _ { \\mathrm { S O ( 2 ) } , \\mathrm { R } } ( V _ { 0 } )$ and $\\left\\{ { \\left( { \\begin{array} { c c } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\end{array} } \\right) } , { \\left( { \\begin{array} { c c } { 0 } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} } \\right) } \\right\\}$ a basis for $\\mathrm { E n d } _ { \\mathrm { S O ( 2 ) } , \\mathrm { R } } ( V _ { l } ) f o r l \\ge 1 .$ . ", + "bbox": [ + 173, + 718, + 825, + 767 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "Proof Sketch. For $l \\geq 1$ and an arbitrary matrix $E = { \\binom { a b } { c d } }$ that commutes with all rotation matrices $\\rho _ { l } ( \\phi )$ , i.e., $E \\circ \\rho _ { l } ( \\phi ) = \\rho _ { l } ( \\phi ) \\circ E$ , one can easily show the constraints $a = d$ and $b = - c ,$ from which the result follows. □ ", + "bbox": [ + 173, + 781, + 825, + 842 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "E.2.5 BRINGING EVERYTHING TOGETHER ", + "text_level": 1, + "bbox": [ + 174, + 103, + 480, + 118 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Now we have done all needed preparation and can solve the kernel constraint explicitly, using the matrix-form of the Wigner-Eckart theorem for steerable kernels, Theorem D.16. This is, as mentioned before, a new derivation of the results in Weiler & Cesa (2019). One can compare with table 8 in their appendix which only differs by (irrelevant) constants. ", + "bbox": [ + 173, + 127, + 825, + 184 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Proposition E.6. We consider steerable kernels $K : S ^ { 1 } \\mathrm { H o m } _ { \\mathbb { R } } ( V _ { l } , V _ { J } )$ , where $V _ { l }$ and $V _ { J }$ are irreducible representations of SO(2). Then the following holds: ", + "bbox": [ + 173, + 188, + 823, + 218 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "1. For $l = J = 0$ , we get $K ( x ) = a \\cdot ( 1 )$ for every $x \\in S ^ { 1 }$ and an arbitrary real number $a \\in \\mathbb { R }$ independent of $x$ . ", + "bbox": [ + 209, + 229, + 823, + 260 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "$l = 0$ , $J > 0$ , a basis for steerable kernels is given by $\\binom { \\cos _ { J } } { \\sin _ { J } }$ and $\\left( { - \\sin _ { J } } \\right) $ ", + "bbox": [ + 210, + 268, + 772, + 301 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "3. For $l > 0$ and $J = 0$ , a basis for steerable kernels is given by (cosl $\\mathrm { s i n } _ { l }$ ), $\\begin{array} { r l } { ( \\sin _ { l } } & { { } - \\cos _ { l } ) } \\end{array}$ $ { l } , J \\quad > \\quad 0$ , and \u0012 $\\begin{array} { r l } { \\left( \\begin{array} { c c } { \\cos _ { J - l } } & { - \\sin _ { J - l } } \\\\ { \\sin _ { J - l } } & { \\cos _ { J - l } } \\end{array} \\right) } \\end{array}$ , $\\left( { \\begin{array} { c c } { - \\sin _ { J - l } } & { - \\cos _ { J - l } } \\\\ { \\cos _ { J - l } } & { - \\sin _ { J - l } } \\end{array} } \\right) , \\left( { \\begin{array} { c c } { \\cos _ { J + l } } & { \\sin _ { J + l } } \\\\ { \\sin _ { J + l } } & { - \\cos _ { J + l } } \\end{array} } \\right) $ \u0013, and \u0012 $\\binom { - \\sin _ { J + l } } { \\cos _ { J + l } } \\quad \\cos _ { J + l } \\rangle$ ", + "bbox": [ + 202, + 309, + 821, + 327 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "", + "bbox": [ + 212, + 335, + 825, + 401 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Proof. The proof of 1 is clear. ", + "bbox": [ + 173, + 415, + 374, + 430 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "For 2, note that $V _ { J }$ can only appear in $V _ { j } \\otimes V _ { 0 }$ if $j = J$ . The relevant Clebsch-Gordan coefficients are by Proposition E.4 therefore $\\mathrm { C G } _ { J ( J 0 ) } = \\left( \\begin{array} { l } { { \\left[ 1 \\right] } } \\\\ { { 0 } } \\\\ { { \\left[ 0 \\right] } } \\\\ { { 1 } } \\end{array} \\right) .$ Furthermore, the orthonormal basis of $V _ { j 1 } = V _ { J 1 }$ is given by Proposition E.1 up to constants by $\\langle \\cos _ { J } , \\sin _ { J } \\rangle$ , which we have to write as a row-vector according to Theorem D.16. Thereby, we can ignore the complex conjugation since we work over the real numbers. Our final ingredient is the endomorphism basis of $V _ { J }$ , which is by Proposition E.5 given by $c _ { 1 } = \\mathrm { i d } _ { \\mathrm { R } ^ { 2 } }$ and $c _ { 2 } = { \\binom { 0 } { 1 } } \\quad { \\binom { - 1 } { 0 } }$ . Overall, the basis kernels are given by ", + "bbox": [ + 171, + 435, + 826, + 587 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/fdaf238d13379990043db2a1079f62cb405a73d2a6addde508d7a51cc9f07751.jpg", + "text": "$$\nc _ { i } \\cdot { \\sqrt { [ \\cos _ { J } \\quad \\sin _ { J } ] \\cdot { \\sqrt { 1 } } } } ) = c _ { i } \\cdot { \\binom { \\cos _ { J } } { \\sin _ { J } } } .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 593, + 643, + 659 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "The result follows. ", + "bbox": [ + 173, + 665, + 299, + 679 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "For 3, we find $V _ { 0 }$ only in $V _ { j } \\otimes V _ { l }$ if $j ~ = ~ l$ , and even twice so. The relevant Clebsch-Gordan coefficients are therefore by Proposition E.4 given by $\\mathrm { C G } _ { 0 ( l l ) 1 } = \\left( \\left[ 1 \\begin{array} { c c } { { 0 } } \\\\ { { 0 } } \\end{array} \\right] \\right)$ and $\\mathrm { C G } _ { 0 ( l l ) 2 } \\ =$ $\\left( { \\begin{array} { r r } { \\left[ 0 \\right. } & { \\left. - 1 \\right] } \\\\ { 1 } & { 0 } \\end{array} } \\right)$ . The basis-functions in $V _ { j 1 } = V _ { l 1 }$ are by Proposition E.1 up to constants $\\{ \\cos _ { l } , \\sin _ { l } \\}$ again written as a row-vector. Finally, $V _ { J } = V _ { 0 }$ has only $\\mathrm { i d _ { R } }$ as a basis-endomorphism by Proposition E.5, so this can be ignored altogether by Corollary D.17. We obtain the following basis for steerable kernels: ", + "bbox": [ + 171, + 684, + 826, + 805 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/1efe41c4a930db90fa480a76679ad6fdb431a94cb50af51c0a288ce9e1424bd4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left( \\left[ \\cos \\iota \\quad \\sin \\iota \\right] \\left[ \\begin{array} { c c } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\end{array} \\right] \\right) = ( 1 \\cos \\iota \\quad 1 \\sin \\iota ) } \\\\ & { \\quad \\left( \\left[ \\cos \\iota \\quad \\sin \\iota \\right] \\left[ \\begin{array} { c c } { 0 } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} \\right] \\right) = ( 1 \\sin \\iota \\quad - 1 \\cos \\iota ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 809, + 656, + 881 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "For 4, we consider only the case $l < J$ . By Proposition E.4 we have ", + "bbox": [ + 173, + 885, + 620, + 901 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/9f26b7beeda918829c53f3d56092d7a6dafca65070fd41c040f15f50ea6eeadd.jpg", + "text": "$$\nV _ { J - l } \\otimes V _ { l } \\cong V _ { \\left| 2 l - J \\right| } \\oplus V _ { J } , V _ { l + J } \\otimes V _ { l } \\cong V _ { J } \\oplus V _ { 2 l + J } ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 909, + 679, + 926 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "i.e., $j ~ = ~ J - l$ and $j = l + J$ leads to a tensor product decomposition containing $V _ { J }$ , but no other $j$ does. Thus, the relevant Clebsch-Gordan coefficients are by Proposition E.4 the matrices $\\mathrm { C G } _ { J , ( J - l , l ) }$ and $\\mathrm { C G } _ { J , ( l + J , l ) }$ . ", + "bbox": [ + 173, + 102, + 825, + 148 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "We now consider the first case, i.e., $j = J - l$ . The Clebsch-Gordan coefficients are $\\mathrm { C G } _ { J , ( J - l , l ) } =$ $\\mathrm { C G } _ { l + j , ( j l ) } = \\left( \\left[ 0 \\begin{array} { r r } { { 1 } } & { { 0 } } \\\\ { { 0 } } & { { - 1 } } \\\\ { { 0 } } & { { 1 } } \\\\ { { 1 } } & { { 0 } } \\end{array} \\right] \\right) .$ The basis functions of $V _ { ( J - l ) 1 }$ are by Proposition E.1 furthermore given by $\\{ \\cos _ { J - l } , \\sin _ { J - l } \\}$ . Finally, $V _ { J }$ has again the two basis endomorphisms $c _ { 1 } = \\mathrm { i d } _ { \\mathbb { R } ^ { 2 } }$ and $c _ { 2 }$ from above. Thus, we obtain the following basis kernel for $c _ { 1 }$ : ", + "bbox": [ + 171, + 151, + 826, + 258 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/32e18e73490b9e30332823510b4ca1c8c081c95f098a5753bbbf3020262c527c.jpg", + "text": "$$\nc _ { 1 } \\cdot { ( \\begin{array} { l l l } { [ \\cos _ { J - l } } & { \\sin _ { J - l } ] \\cdot } \\\\ { [ \\cos _ { J - l } } & { \\sin _ { J - l } ] \\cdot } \\\\ { [ \\cos _ { J - l } } & { \\sin _ { J - l } ] \\cdot } \\\\ { } & { \\sin _ { J - l } ] \\cdot } \\end{array} ) } = { ( \\begin{array} { l l l } { \\cos _ { J - l } } & { - \\sin _ { J - l } } \\\\ { \\sin _ { J - l } } & { \\cos _ { J - l } ) } \\end{array} ) } .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 265, + 710, + 330 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "Consequently, for $c _ { 2 }$ as the basis endomorphism we need to postcompose with $c _ { 2 }$ and get: ", + "bbox": [ + 169, + 337, + 761, + 352 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/1904e555716a4e187bee038ea38ab3995b74db8f2c28004acfe024c19d1eb071.jpg", + "text": "$$\n\\binom { 0 } { 1 } - 1 \\atop 0 \\sp { . } \\cdot \\binom { \\cos _ { J - l } } { \\sin _ { J - l } } - \\sin _ { J - l } \\sp { . } \\biggr ) = \\left( \\begin{array} { r c } { { - \\sin _ { J - l } } } & { { - \\cos _ { J - l } } } \\\\ { { \\cos _ { J - l } } } & { { - \\sin _ { J - l } } } \\end{array} \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 289, + 358, + 709, + 393 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "These are half of the basis kernels. For the other half, we need to look at the case $j = l + J$ . The Clebsch-Gordan coefficients are by part 4 of Proposition E.4 given by $\\mathrm { C G } _ { J , ( l + J , l ) } = \\mathrm { C G } _ { j - l , ( j l ) } =$ $\\left( { \\begin{array} { r r } { \\left[ 1 \\right. } & { \\left. 0 \\right] } \\\\ { 0 } & { 1 } \\\\ { \\left[ 0 \\right. } & { - 1 } \\\\ { 1 } & { \\left. 0 \\right] } \\end{array} } \\right) .$ The basis functions of $V _ { ( l + J ) 1 }$ are by Proposition E.1 furthermore given by $\\{ \\cos _ { J + l } , \\sin _ { J + l } \\}$ . For the basis endomorphism $c _ { 1 }$ we thus get the basis kernel ", + "bbox": [ + 173, + 400, + 826, + 506 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/af288440692005bc14675abdf2d45eaae5317a22cc5c7225445d303d43b2d0b0.jpg", + "text": "$$\n\\begin{array}{c} c _ { 1 } \\cdot { ( \\begin{array} { l l } { [ \\cos _ { J + l } } & { \\sin _ { J + l } ] \\cdot } \\\\ { [ \\cos _ { J + l } } & { \\sin _ { J + l } ] \\cdot } & { [ 0 } \\end{array} } 1 \\\\ { [ \\cos _ { J + l } } & { \\sin _ { J + l } ] \\cdot { [ \\begin{array} { l l } { 0 } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} ] } ) } & { = ( \\sin _ { J + l } \\quad \\cdot \\sin _ { J + l } ) . } \\end{array} ) \n$$", + "text_format": "latex", + "bbox": [ + 287, + 512, + 710, + 579 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "Consequently, for $c _ { 2 }$ as the basis endomorphism we need to postcompose with $c _ { 2 }$ and get: ", + "bbox": [ + 169, + 584, + 761, + 599 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/7006c156d0910e9e47a943ba94fe4b91acccefbad807d66286c3cb6a0af90bab.jpg", + "text": "$$\n\\left( \\begin{array} { c c } { { 0 } } & { { - 1 } } \\\\ { { 1 } } & { { 0 } } \\end{array} \\right) \\cdot \\left( \\begin{array} { c c } { { \\cos _ { J + l } } } & { { \\sin _ { J + l } } } \\\\ { { \\sin _ { J + l } } } & { { - \\cos _ { J + l } } } \\end{array} \\right) = \\left( \\begin{array} { c c } { { - \\sin _ { J + l } } } & { { \\cos _ { J + l } } } \\\\ { { \\cos _ { J + l } } } & { { \\sin _ { J + l } } } \\end{array} \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 606, + 700, + 642 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "Overall, for the case $l \\ < \\ J$ we have determined all four basis kernels in Eqs. (26), (27), (28), and (29). The cases $l = J$ and $l > J$ can be considered analogously, and in every case the correct Clebsch-Gordan coefficients have to be picked. By using $\\mathrm { c o s } _ { l - J } = \\mathrm { c o s } _ { J - l }$ and $\\mathrm { s i n } _ { l - J } = - \\mathrm { s i n } _ { J - l }$ , this will, in the end, always lead to the same final formulas. This result is consistent with Table 8 in Weiler & Cesa (2019). □ ", + "bbox": [ + 173, + 647, + 826, + 719 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "E.3 $\\mathbb { Z } _ { 2 }$ -STEERABLE KERNELS FOR REAL REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 736, + 607, + 751 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "In this section, we discuss steerable CNNs that use the finite group $\\mathbb { Z } _ { 2 }$ , which we identify with $( \\{ - 1 , + 1 \\} , \\cdot )$ , for their symmetries. We let this group act on the plane $\\mathbb { R } ^ { 2 }$ by vertical reflections, though other choices are possible as well: ", + "bbox": [ + 173, + 762, + 825, + 805 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/7c29a9ab9357b95f8d510c7ea7fc79885b61150a63fe9d5a4108db7a9bb96379.jpg", + "text": "$$\nx \\cdot { \\binom { a } { b } } = { \\binom { x a } { b } } .\n$$", + "text_format": "latex", + "bbox": [ + 434, + 811, + 563, + 848 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "This example is simple and one may see it as contrived to apply our relatively heavy theory to it. We include it mainly as a demonstration that our results can also be applied to non-smooth finite groups as instances of compact groups. Furthermore, we will fully recover the relationship to the original group convolutional CNNs from Cohen & Welling (2016a) and thereby demonstrate that all the different developed theories are consistent with each other. ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "E.3.1 THE IRREDUCIBLE REPRESENTATIONS OF $\\mathbb { Z } _ { 2 }$ OVER THE REAL NUMBERS ", + "bbox": [ + 174, + 103, + 736, + 118 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Let $\\rho : \\mathbb { Z } _ { 2 } \\to { \\mathrm { G L } } ( V )$ be an irreducible real representation. Note that ", + "bbox": [ + 173, + 127, + 627, + 142 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/f7bc686e4d7e8cb2609a39f368637a1304bcb342b1adeccd74a2b3a2e76a746e.jpg", + "text": "$$\n\\rho ( - 1 ) \\circ \\rho ( - 1 ) = \\rho ( ( - 1 ) \\cdot ( - 1 ) ) = \\rho ( 1 ) = \\mathrm { i d } _ { V } ,\n$$", + "text_format": "latex", + "bbox": [ + 334, + 148, + 661, + 166 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "and thus $\\rho ( - 1 )$ is an involution satisfying the equation $\\rho ( - 1 ) ^ { 2 } - \\mathrm { i d } _ { V } = 0$ . It is well-known from linear algebra that involutions are diagonalizable, and thus $\\rho ( - 1 )$ leaves 1-dimensional subspaces invariant. By irreducibility of $\\rho$ this means that $V$ itself needs to be 1-dimensional. Consequently, we can assume $V = \\mathbb { R }$ without loss of generality. Note that the computations above mean that we have ", + "bbox": [ + 173, + 171, + 826, + 239 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/961cfc9d1bcdeee995a2dba266110c2d85292571d6da3f07b83290c4cac2522f.jpg", + "text": "$$\n\\left( \\rho ( - 1 ) - \\mathrm { i d _ { \\mathbb { R } } } \\right) \\circ \\left( \\rho ( - 1 ) + \\mathrm { i d _ { \\mathbb { R } } } \\right) = 0\n$$", + "text_format": "latex", + "bbox": [ + 370, + 238, + 627, + 257 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "and thus we need to have $\\rho ( - 1 ) - \\mathrm { i d _ { \\mathrm { R } } } \\ = \\ 0$ or $\\rho ( - 1 ) + \\mathrm { i d _ { \\mathrm { R } } } \\ = \\ 0$ . It follows $\\rho ( - 1 ) \\ = \\ \\mathrm { i d _ { \\mathrm { R } } }$ or $\\rho ( - 1 ) = - \\operatorname { i d } _ { \\mathbb { R } }$ . Overall, all these investigations mean that we have precisely two irreducible representations of $\\mathbb { Z } _ { 2 }$ up to equivalence. We call them $\\rho _ { + } : \\mathbb { Z } _ { 2 } \\to \\operatorname { O } ( V _ { + } )$ and $\\rho _ { - } : \\mathbb { Z } _ { 2 } \\to \\operatorname { O } ( V _ { - } )$ , where $\\rho _ { + } ( - 1 ) = \\mathrm { i d } _ { \\mathrm { R } }$ and $\\rho _ { - } \\dot { ( } - 1 ) = - \\operatorname { i d } _ { \\mathbb { R } }$ and $V _ { + } = V _ { - } = \\mathbb { R }$ . ", + "bbox": [ + 174, + 260, + 825, + 318 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "E.3.2 THE PETER-WEYL THEOREM FOR $L _ { \\mathbb { R } } ^ { 2 } ( X )$ ", + "text_level": 1, + "bbox": [ + 173, + 329, + 521, + 345 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Here we do the Peter-Weyl decomposition for $L _ { \\mathbb { R } } ^ { 2 } ( X )$ , where $X$ is one of the two homogeneous spaces $X = \\{ - 1 , 1 \\}$ and $X = \\{ 0 \\}$ with the obvious actions coming from the groups $\\mathbb { Z } _ { 2 }$ . This time, we also discuss orbits with only one point since we later want to get a description of kernels on the whole of $\\mathbb { R } ^ { 2 }$ for comparisons with group convolutional CNNs. ", + "bbox": [ + 173, + 354, + 825, + 411 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "We start with $X = \\{ - 1 , 1 \\}$ . Note that the measure on $X$ is just the normalized counting measure, and thus all functions $f : X \\to \\mathbb { R }$ are square-integrable. We define the two functions ", + "bbox": [ + 171, + 417, + 823, + 446 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/8d673b713381feb81557afbaa30e0c7ebb9a2be78dfc0d67b46590322fb38e96.jpg", + "text": "$$\n\\begin{array} { r l } & { f _ { + } : X \\mathbb { R } , f _ { + } ( x ) = 1 \\mathrm { ~ f o r ~ a l l ~ } x \\in X = \\{ - 1 , 1 \\} , } \\\\ & { f _ { - } : X \\mathbb { R } , f _ { - } ( x ) = x \\mathrm { ~ f o r ~ a l l ~ } x \\in X = \\{ - 1 , 1 \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 452, + 665, + 489 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "We then define $V _ { + 1 } = \\operatorname { s p a n } _ { \\mathbb { R } } ( f _ { + } )$ and $V _ { - 1 } = \\operatorname { s p a n } _ { \\mathbb { R } } ( f _ { - } )$ . This gives a decomposition ", + "bbox": [ + 173, + 492, + 745, + 508 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/ab4b8731e60643b426d668d4849f4732216616394c4de4a206e37fe7475a47ed.jpg", + "text": "$$\nL _ { \\mathbb { R } } ^ { 2 } ( X ) = V _ { + 1 } \\oplus V _ { - 1 }\n$$", + "text_format": "latex", + "bbox": [ + 424, + 513, + 573, + 532 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "since we have for all $f \\in L _ { \\mathbb { R } } ^ { 2 } ( X )$ ", + "bbox": [ + 173, + 536, + 393, + 553 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/962cdc8fc22c5dfc908b781d1149ad47a592310625342dd997cb02a237f52179.jpg", + "text": "$$\nf = \\frac { f ( 1 ) + f ( - 1 ) } { 2 } \\cdot f _ { + } + \\frac { f ( 1 ) - f ( - 1 ) } { 2 } \\cdot f _ { - } .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 558, + 656, + 590 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Furthermore, the maps $1 \\mapsto f _ { + }$ and $1 \\mapsto f _ { - }$ give isomorphisms of representations $V _ { + } \\cong V _ { + 1 }$ and $V _ { - } \\cong V _ { - 1 }$ , respectively. ", + "bbox": [ + 176, + 594, + 823, + 623 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Now, assume that $X = \\{ 0 \\}$ with the trivial action coming from $\\mathbb { Z } _ { 2 }$ . Then $L _ { \\mathbb { R } } ^ { 2 } ( X ) = V _ { + 1 }$ generated from the function $f _ { + } : X \\to \\mathbb { R }$ , $f _ { + } ( 0 ) = 1$ . As before, $1 \\mapsto f _ { + }$ gives an isomorphism $V _ { + } \\cong V _ { + 1 }$ . This concludes the investigations of the Peter-Weyl theorem. ", + "bbox": [ + 174, + 628, + 825, + 672 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "E.3.3 THE CLEBSCH-GORDAN DECOMPOSITION ", + "bbox": [ + 173, + 686, + 524, + 702 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "We have the following four isomorphisms of representations: ", + "bbox": [ + 173, + 710, + 575, + 727 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/304e92c02beacc063ef1327cea7347ae343a3040edf6685ecb8fc24dde37c9e2.jpg", + "text": "$$\n\\begin{array} { l l } { { V _ { + } \\otimes V _ { + } \\cong V _ { + } , } } & { { V _ { + } \\otimes V _ { - } \\cong V _ { - } , } } \\\\ { { V _ { - } \\otimes V _ { + } \\cong V _ { - } , } } & { { V _ { - } \\otimes V _ { - } \\cong V _ { + } , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 377, + 731, + 619, + 768 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "each time simply given by $a \\otimes b \\mapsto a b$ . It can easily be checked that these are isomorphisms. In Section E.6.3 the reader can find a proof for similar, sign-dependent isomorphisms for the case that the group is O(3). For each such isomorphism, there is precisely one Clebsch-Gordan coefficient and it is just given by 1. Thus, as in the case of harmonic networks in Section E.1.5, we can just ignore the Clebsch-Gordan coefficients altogether in the final formulas for our basis kernels. ", + "bbox": [ + 173, + 772, + 825, + 843 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "E.3.4 ENDOMORPHISMS OF $V _ { + }$ AND $V _ { - }$ ", + "text_level": 1, + "bbox": [ + 176, + 856, + 459, + 872 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Since $V _ { + }$ and $V _ { - }$ are themselves only 1-dimensional, the endomorphism spaces are necessarily 1- dimensional as well and just given by arbitrary $1 \\times 1$ -matrices, i.e., arbitrary stretchings. As in the example of harmonic networks, we can therefore ignore the endomorphisms as well. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "E.3.5 BRINGING EVERYTHING TOGETHER ", + "text_level": 1, + "bbox": [ + 176, + 103, + 480, + 118 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Different from the other examples, we will in this section not only engage with the final steerable kernels on homogeneous spaces but also discuss how these assemble to kernels defined on the whole plane $\\mathbb { R } ^ { 2 }$ . In the end, we will then also discuss how kernels for the regular representation would look like. ", + "bbox": [ + 173, + 130, + 825, + 185 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "But first, we engage with the homogeneous spaces. We start with $X = \\{ - 1 , 1 \\}$ and consider steerable kernels $K \\colon \\bar { X } \\operatorname { H o m } _ { \\mathbb { R } } ( V _ { \\mathrm { i n } } , \\bar { V } _ { \\mathrm { o u t } } )$ for irreducible $V _ { \\mathrm { i n } }$ and $V _ { \\mathrm { o u t } }$ . There are four possibilities for the input and output representations: ", + "bbox": [ + 174, + 193, + 825, + 236 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : X \\to \\operatorname { H o m } _ { \\mathbb { R } } ( V _ { + } , V _ { + } )$ ", + "bbox": [ + 176, + 253, + 511, + 270 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "$V _ { + }$ can only be in a tensor product $V \\otimes V _ { + }$ if the sign of $V$ is positive as well. Such a space appears precisely once in $L _ { \\mathbb { R } } ^ { 2 } ( X )$ according to Section E.3.2. Since endomorphisms and Clebsch-Gordan coefficients do not appear by what we’ve shown before, and since complex conjugation doesn’t do anything over the real numbers, a basis for steerable kernels is just given by the one kernel $K _ { + } = f _ { + }$ itself. Here, we identify ${ \\mathrm { H o m } } _ { \\mathrm { R } } ( V _ { + } , V _ { + } )$ with $\\mathbb { R }$ since it only consists of $1 \\times 1$ -matrices. ", + "bbox": [ + 173, + 280, + 825, + 352 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : X \\to \\operatorname { H o m } _ { \\mathbb { R } } ( V _ { + } , V _ { - } )$ : ", + "bbox": [ + 174, + 371, + 514, + 386 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "By the same arguments, a basis is given by the one kernel $K _ { - } = f _ { - }$ . ", + "bbox": [ + 174, + 397, + 625, + 412 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : X \\to \\operatorname { H o m } _ { \\mathbb { R } } ( V _ { - } , V _ { + } )$ : ", + "bbox": [ + 174, + 431, + 514, + 448 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Again, a basis for steerable kernels is given by $K _ { - } = f _ { - }$ . ", + "bbox": [ + 174, + 458, + 552, + 473 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : X \\to \\operatorname { H o m } _ { \\mathbb { R } } ( V _ { - } , V _ { - } )$ : ", + "bbox": [ + 174, + 491, + 514, + 508 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "A basis is given by $K _ { + } = f _ { + }$ ", + "bbox": [ + 174, + 518, + 372, + 534 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Finally, we also need to engage with the case that $X = \\{ 0 \\}$ consists only of a single point. Similarly to above, in the “even” case that the signs of input- and output representations agree, a basis is given by $K _ { + } = f _ { + }$ with $f _ { + } ( 0 ) = 1$ . If, however, the signs do not agree, then only $K = 0$ fulfills the constrained and the basis is empty. ", + "bbox": [ + 173, + 540, + 825, + 597 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Now, we assemble this to kernels on the whole of $\\mathbb { R } ^ { 2 }$ . We saw above that we only need to distinguish two cases, namely (a) the case that the signs of input and output representation agree and (b) that they do not. ", + "bbox": [ + 176, + 602, + 823, + 646 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "For case (a), let $K : \\mathbb { R } ^ { 2 } \\mathbb { R }$ be a steerable kernel, where $\\mathbb { R }$ is isomorphic to the Hom-space between equal-sign representations. $\\mathbb { R } ^ { 2 }$ splits disjointly into orbits, namely $\\left\\{ \\left( { a \\atop b } \\right) , \\left( { - a \\atop b } \\right) \\right\\}$ for all $a \\in \\mathbb { R } _ { \\geq 0 }$ and $b \\in \\mathbb { R }$ . If $a = 0$ , then the orbit is just a single point, which means that we have a vertical line of single-point orbits. The solution above showed that on each orbit, the kernel needs to be constant (since $f _ { + }$ is constant) and overall this just translates to ", + "bbox": [ + 173, + 651, + 825, + 739 + ], + "page_idx": 76 + }, + { + "type": "equation", + "img_path": "images/e873a3d7971bf7ef1e562446cf6067d9c4fb1fb2d2a576eb3f488891ca9d32d9.jpg", + "text": "$$\nK \\left( { a atop b } \\right) = K \\left( { - a \\atop b } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 428, + 750, + 568, + 785 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "for all $a \\geq 0$ and $b \\in \\mathbb { R }$ . Consequently, $K$ is just an arbitrary left-right symmetric kernel. ", + "bbox": [ + 169, + 795, + 756, + 810 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "In the case that the input- and output representations do not share their sign, by the same arguments we see that $K : \\mathbb { R } ^ { 2 } \\mathbb { R }$ is an arbitrary left-right anti-symmetric kernel which is zero on the vertical line $\\binom { 0 } { b }$ for arbitrary $b \\in \\mathbb { R }$ . ", + "bbox": [ + 174, + 815, + 823, + 877 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Other than these left-right restrictions, the kernel can be freely learned. Overall, this means that we learn one “half” of the kernel and can recover the other half by the symmetry property derived above. ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "E.3.6 GROUP CONVOLUTIONAL CNNS FOR $\\mathbb { Z } _ { 2 }$ ", + "text_level": 1, + "bbox": [ + 173, + 102, + 514, + 118 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "We now investigate what all this means if we consider regular representations instead of irreducible representations, thus corresponding to group convolutional kernels as in (Cohen & Welling, 2016a). In this case, we will see an interesting “twist” in the kernel, which makes this example more interesting than one might initially think. The twist emerges as follows: For regular representations, we consider steerable kernels ", + "bbox": [ + 173, + 127, + 825, + 195 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/385733048616d6a196f03a335985eebf8ce0773fe541844a96072a2af48a805c.jpg", + "text": "$$\nK : \\mathbb { R } ^ { 2 } \\mathrm { H o m } _ { \\mathbb { R } } ( L _ { \\mathbb { R } } ^ { 2 } ( \\mathbb { Z } _ { 2 } ) , L _ { \\mathbb { R } } ^ { 2 } ( \\mathbb { Z } _ { 2 } ) )\n$$", + "text_format": "latex", + "bbox": [ + 374, + 194, + 624, + 213 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "Now, there are two relatively canonical bases we can choose in the left and the right space. We already know from above that $\\{ f _ { + } , f _ { - } \\}$ is the basis to choose if we want to express steerable kernels corresponding to irreducible representations. However, for vanilla group convolutional CNNs, the basis usually chosen is $\\{ e _ { + 1 } , e _ { - 1 } \\}$ where $e _ { + 1 } ( x ) = \\delta _ { + 1 , x }$ and $e _ { - 1 } ( x ) = \\delta _ { - 1 , x }$ . We then obtain the following four base change relations: ", + "bbox": [ + 173, + 215, + 825, + 286 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/b97961d5e1e08e7032b341a9979a3400fd6200e1907610d0f8ef97a4742b8d06.jpg", + "text": "$$\n\\begin{array} { l } { { f _ { + } = e _ { + 1 } + e _ { - 1 } , \\quad f _ { - } = e _ { + 1 } - e _ { - 1 } , } } \\\\ { { e _ { + 1 } = \\displaystyle \\frac { 1 } { 2 } f _ { + } + \\displaystyle \\frac { 1 } { 2 } f _ { - } , \\quad e _ { - 1 } = \\displaystyle \\frac { 1 } { 2 } f _ { + } - \\displaystyle \\frac { 1 } { 2 } f _ { - } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 291, + 645, + 344 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "Thus, the base change matrices are given by ", + "bbox": [ + 176, + 345, + 464, + 361 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/81ab53ac13fa7d610522a901cb289d65481268b67c963b344b135ed08531ae24.jpg", + "text": "$$\nB = \\left( { \\begin{array} { c c } { 1 } & { 1 } \\\\ { 1 } & { - 1 } \\end{array} } \\right) , \\quad B ^ { - 1 } = \\left( { \\begin{array} { c c } { { \\frac { 1 } { 2 } } } & { 1 } \\\\ { { \\frac { 1 } { 2 } } } & { - { \\frac { 1 } { 2 } } } \\end{array} } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 366, + 635, + 402 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "Now, assume that $K : \\mathbb { R } ^ { 2 } \\mathrm { H o m } _ { \\mathbb { R } } ( L _ { \\mathbb { R } } ^ { 2 } ( \\mathbb { Z } _ { 2 } ) , L _ { \\mathbb { R } } ^ { 2 } ( \\mathbb { Z } _ { 2 } ) ) \\cong \\mathbb { R } ^ { 2 \\times 2 }$ is expressed with respect to the basis $\\{ f _ { + } , f _ { - } \\}$ . If we write $K$ as a matrix ", + "bbox": [ + 174, + 407, + 820, + 438 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/f6978a2670729205ac20ac0dfeacedde31a1218d6ae45df275b9a3283caee4b1.jpg", + "text": "$$\nK = \\left( \\begin{array} { c c } { { K _ { 1 1 } } } & { { K _ { 1 2 } } } \\\\ { { K _ { 2 1 } } } & { { K _ { 2 2 } } } \\end{array} \\right)\n$$", + "text_format": "latex", + "bbox": [ + 429, + 443, + 566, + 478 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "then we know that $K _ { 1 1 }$ and $K _ { 2 2 }$ map between equal-sign representations and $K _ { 1 2 }$ and $K _ { 2 1 }$ between unequal-sign representations. Consequently, from what we’ve found above, $K _ { 1 1 }$ and $K _ { 2 2 }$ are symmetric, whereas $K _ { 1 2 }$ and $K _ { 2 1 }$ are antisymmetric. What we now want to figure out is how exactly this translates to a property of the kernel expressed in the basis $\\{ e _ { + } , e _ { - } \\}$ . ", + "bbox": [ + 174, + 483, + 825, + 541 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "Thus, let $K ^ { \\prime }$ be this corresponding kernel. Then using the base change matrices above we obtain ", + "bbox": [ + 169, + 546, + 803, + 563 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/e0817cdd5c545981e513c52992189d29d3be4793476a6adb723abc16bbd9bec7.jpg", + "text": "$$\n{ \\begin{array} { r l } { \\left( K _ { 1 1 } ^ { \\prime } } & { K _ { 1 2 } ^ { \\prime } \\right) = K ^ { \\prime } } \\\\ { \\left( K _ { 2 1 } ^ { \\prime } } & { K _ { 2 2 } ^ { \\prime } \\right) = K ^ { \\prime } } \\\\ & { \\qquad = B \\cdot K \\cdot B ^ { - 1 } } \\\\ & { \\qquad = \\left( 1 \\qquad 1 \\right) \\cdot \\left( K _ { 1 1 } \\quad K _ { 1 2 } \\right) \\cdot \\left( { \\frac { 1 } { 2 } } \\quad \\qquad { \\frac { 1 } { 2 } } \\right) } \\\\ & { \\qquad = \\left( { \\frac { 1 } { 2 } } \\left[ K _ { 1 1 } + K _ { 1 2 } + K _ { 2 1 } + K _ { 2 2 } \\right] \\quad { \\frac { 1 } { 2 } } \\left[ K _ { 1 1 } - K _ { 1 2 } + K _ { 2 1 } - K _ { 2 2 } \\right] \\right) . } \\\\ & { \\qquad = \\left( { \\frac { 1 } { 2 } } \\left[ K _ { 1 1 } + K _ { 1 2 } - K _ { 2 1 } - K _ { 2 2 } \\right] \\quad { \\frac { 1 } { 2 } } \\left[ K _ { 1 1 } - K _ { 1 2 } - K _ { 2 1 } + K _ { 2 2 } \\right] \\right) . } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 220, + 566, + 779, + 695 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "What symmetry properties does this kernel obey? In order to understand this, we use the following convention: for $\\boldsymbol { y } \\in \\mathbb { R } ^ { 2 }$ we set $- y = { \\binom { - y _ { 1 } } { y _ { 2 } } }$ , i.e., the vertically flipped image of $y$ . Then we have, using the symmetry and anti-symmetry of the entries of the original kernel $K$ : ", + "bbox": [ + 174, + 696, + 826, + 757 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/63eee08cbf4ecaad4b6d9b2ec87f1f47d4841a3804c5de6a5ab623b5de838354.jpg", + "text": "$$\n\\begin{array} { l } { { K _ { 2 2 } ^ { \\prime } ( - y ) = \\displaystyle \\frac { 1 } { 2 } \\big [ K _ { 1 1 } ( - y ) - K _ { 1 2 } ( - y ) - K _ { 2 1 } ( - y ) + K _ { 2 2 } ( - y ) \\big ] } } \\\\ { { \\ ~ } } \\\\ { { \\ ~ = \\displaystyle \\frac { 1 } { 2 } \\big [ K _ { 1 1 } ( y ) + K _ { 1 2 } ( y ) + K _ { 2 1 } ( y ) + K _ { 2 2 } ( y ) \\big ] } } \\\\ { { \\ ~ = K _ { 1 1 } ^ { \\prime } ( y ) , } } \\\\ { { \\ ~ K _ { 2 1 } ^ { \\prime } ( - y ) = \\displaystyle \\frac { 1 } { 2 } \\big [ K _ { 1 1 } ( - y ) + K _ { 1 2 } ( - y ) - K _ { 2 1 } ( - y ) - K _ { 2 2 } ( - y ) \\big ] } } \\\\ { { \\ ~ = \\displaystyle \\frac { 1 } { 2 } \\big [ K _ { 1 1 } ( y ) - K _ { 1 2 } ( y ) + K _ { 2 1 } ( y ) - K _ { 2 2 } ( y ) \\big ] } } \\\\ { { \\ ~ = K _ { 1 2 } ^ { \\prime } ( y ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 289, + 763, + 709, + 924 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "Thus the second row of $K ^ { \\prime }$ is basically the same as the first, only that the kernels swap with each other and are internally flipped. This is a special case of the outcome in Cohen & Welling (2016a), which is also described clearly in Weiler et al. (2018b): in group convolutional kernels which are steerable with respect to finite groups, the kernels get copied and applied in all orientations demanded by the group. ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "What we would still like to understand is if we can also reverse the direction: That is, assume that we start with a group convolutional kernel $K ^ { \\prime }$ of which we know that $K _ { 2 2 ^ { \\prime } } ( - y ) = K _ { 1 1 } ^ { \\prime } ( y )$ and $K _ { 2 1 } ^ { \\prime } ( - y ) = K _ { 1 2 } ^ { \\prime } ( y )$ for all $\\boldsymbol { y } \\in \\mathbb { R } ^ { 2 }$ . If we then do a base change, we would like to know if the resulting kernel consists of symmetric and antisymmetric entries. Namely, set ", + "bbox": [ + 176, + 180, + 825, + 236 + ], + "page_idx": 78 + }, + { + "type": "equation", + "img_path": "images/b0e0eb518e54dda230bdb16e3235bdecb63d2d664d248e2c07a99271489dfd98.jpg", + "text": "$$\n\\begin{array} { r l } { \\bigg ( K _ { 1 1 } } & { K _ { 1 2 } \\bigg ) = K } \\\\ { K _ { 2 1 } } & { K _ { 2 2 } \\bigg ) = K } \\\\ & { \\quad \\quad = B ^ { - 1 } \\cdot K ^ { \\prime } \\cdot B } \\\\ & { \\quad \\quad = \\bigg ( \\frac { 1 } { 2 } \\qquad \\frac { 1 } { 2 } \\bigg ) \\cdot \\bigg ( K _ { 1 1 } ^ { \\prime } \\quad K _ { 1 2 } ^ { \\prime } \\bigg ) \\cdot \\binom { 1 } { 1 } \\quad 1 \\bigg ) } \\\\ & { \\quad \\quad = \\bigg ( \\frac { 1 } { 2 } \\qquad - \\frac { 1 } { 2 } \\bigg ) \\cdot \\bigg ( K _ { 2 1 } ^ { \\prime } \\quad K _ { 2 2 } ^ { \\prime } \\bigg ) \\quad \\frac { 1 } { 2 } \\big [ K _ { 1 1 } ^ { \\prime } - K _ { 1 2 } ^ { \\prime } + K _ { 2 1 } ^ { \\prime } - K _ { 2 2 } ^ { \\prime } \\big ] } \\\\ & { \\quad \\quad = \\bigg ( \\frac { 1 } { 2 } \\big [ K _ { 1 1 } ^ { \\prime } + K _ { 1 2 } ^ { \\prime } - K _ { 2 1 } ^ { \\prime } - K _ { 2 2 } ^ { \\prime } \\big ] \\quad \\frac { 1 } { 2 } \\big [ K _ { 1 1 } ^ { \\prime } - K _ { 1 2 } ^ { \\prime } - K _ { 2 1 } ^ { \\prime } + K _ { 2 2 } ^ { \\prime } \\big ] \\bigg ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 220, + 242, + 779, + 371 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "The reader can easily check that we can deduce that $K _ { 1 1 }$ and $K _ { 2 2 }$ are symmetric and that $K _ { 1 2 }$ and $K _ { 2 1 }$ are anti-symmetric. We have thus fully shown the equivalence of the kernel solutions in the setting of steerable CNNs compared to the setting of group convolutional CNNs for the specific group $\\mathbb { Z } _ { 2 }$ . ", + "bbox": [ + 173, + 373, + 826, + 430 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "E.4 SO(3)-STEERABLE KERNELS FOR COMPLEX REPRESENTATIONS. ", + "text_level": 1, + "bbox": [ + 178, + 446, + 665, + 462 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "In the first two sections, we have discussed SO(2)-equivariant kernels (i.e., SE(2)-equivariant neural networks) both over $\\mathbb { C }$ and R. The situation over R was considerably more complicated and required new arguments. In this section, we will discuss SO(3)-equivariant kernels (i.e., SE(3)-equivariant neural networks) for complex representations. In Section E.5 we will then look at the real case, which will essentially give the exact same results, thus differing somewhat from the considerations about SO(2). Different from the earlier sections, we will from now on be less explicit and care more about the general properties of the different functions and coefficients we consider. SO(3)- equivariant networks with real representations have before been implemented in Weiler et al. (2018a) and Thomas et al. (2018), among others. ", + "bbox": [ + 173, + 472, + 825, + 598 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "E.4.1 THE IRREDUCIBLE REPRESENTATIONS OF SO(3) OVER THE COMPLEX NUMBERS ", + "text_level": 1, + "bbox": [ + 171, + 613, + 789, + 628 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "In this section, we state the complex irreducible representations of SO(3). We will not state the matrices explicitly since the matrix elements are considerably more complicated than in the earlier examples that we saw. For each $l \\in \\mathbb { N } _ { \\geq 0 }$ , there is one irreducible unitary representation ", + "bbox": [ + 174, + 637, + 825, + 680 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "The matrices $D _ { l } ( g )$ for $g \\in \\mathrm { S O } ( 3 )$ are called the Wigner $D$ -matrices.29 There are, up to equivalence, no other irreducible representations of SO(3) over $\\mathbb { C }$ . A reference for all this is the original work Wigner (1944). ", + "bbox": [ + 174, + 712, + 825, + 756 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "We note that the indices for the dimensions in $\\mathbb { C } ^ { 2 l + 1 }$ are $- l , - l + 1 , \\dots , l - 1 , l$ by general convention. ", + "bbox": [ + 173, + 760, + 821, + 777 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "E.4.2 THE PETER-WEYL THEOREM FOR $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ AS A REPRESENTATION OF SO(3) ", + "bbox": [ + 178, + 790, + 759, + 808 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "Here, we describe how $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ , considered as a unitary representation via $\\lambda ~ : ~ \\mathrm { S O ( 3 ) } ~ $ $\\mathrm { U } \\big ( L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } ) \\big )$ , with $[ \\lambda ( g ) \\varphi ] ( x ) = \\varphi ( g ^ { - 1 } x )$ , contains densely a direct sum of irreducible representations. For doing so, we proceed by first describing spherical harmonics without formulas and stating their orthonormality properties, and then stating how they transform under rotation. This will then yield the result. Note that we do not need to describe explicit formulas for the spherical harmonics, which are again somewhat complicated since we are more interested in their properties in relation to Hilbert space theory and representation theory. A reference for all this is MacRobert (1947). ", + "bbox": [ + 174, + 814, + 825, + 900 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 133 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "The spherical harmonics are continuous functions $Y _ { l } ^ { n } : S ^ { 2 } \\to \\mathbb { C }$ for $l \\in \\mathbb { N } _ { \\geq 0 }$ and $n = - l , \\ldots , l$ . Thus, they are elements of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ . They have the following properties: ", + "bbox": [ + 174, + 137, + 825, + 169 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "1. $\\left. Y _ { l } ^ { n } \\Big | Y _ { l ^ { \\prime } } ^ { n ^ { \\prime } } \\right. = \\delta _ { l l ^ { \\prime } } \\delta _ { n n ^ { \\prime } }$ for all $l , l ^ { \\prime } , n , n ^ { \\prime } .$ . \n2. The linear span of the spherical harmonics is dense in $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ . \n3. They transform as follows under rotation: $\\begin{array} { l } { { \\lambda ( g ) ( Y _ { l } ^ { n } ) ~ = ~ \\sum _ { n ^ { \\prime } = - l } ^ { l } D _ { l } ^ { n ^ { \\prime } n } ( g ) Y _ { l } ^ { n ^ { \\prime } } } } \\end{array}$ , where $D _ { l } ^ { n ^ { \\prime } n } ( g )$ are the matrix elements of the Wigner D-matrices defined in Section E.4.1. ", + "bbox": [ + 210, + 179, + 825, + 265 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Properties 1 and 2 together imply that the spherical harmonics form an orthonormal basis of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ , see Definition F.40. Let ", + "bbox": [ + 171, + 276, + 825, + 304 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/b6c2d811be3b8461a743ac5885bef4b3795beb3ba2310507b2225135863a3c49.jpg", + "text": "$$\nV _ { l 1 } : = \\operatorname { s p a n } _ { \\mathbb { C } } ( Y _ { l } ^ { n } \\mid n = - l , \\ldots , l ) .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 303, + 616, + 320 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Then we already obtain $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } ) = { \\widehat { \\bigoplus } } _ { l \\geq 0 } V _ { l 1 }$ . Now, let $e ^ { n } \\in \\mathbb { C } ^ { 2 l + 1 }$ be the $n$ ’th standard basis vector, for $n = - l , \\ldots , l .$ . Then property 3 means that the linear map given on basis vectors by ", + "bbox": [ + 174, + 324, + 826, + 358 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/d440473b51d08e900497866b2e4936d4889a44719db1b0a2beb5b7bebe83ae5f.jpg", + "text": "$$\nf : V _ { l } \\to V _ { l 1 } , e ^ { n } \\mapsto Y _ { l } ^ { n }\n$$", + "text_format": "latex", + "bbox": [ + 416, + 364, + 580, + 381 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "is an isomorphism of unitary representations. More precisely, $f$ is clearly a unitary transformation and a linear isomorphism, and it is furthermore equivariant on basis vectors since ", + "bbox": [ + 171, + 387, + 823, + 417 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/4e9612dd3c8b84abf1a296501ab6c4b62d6e4e98cb67a485a4e513bdccd9d40d.jpg", + "text": "$$\n\\begin{array} { l } { f \\left( D _ { l } ( g ) ( e ^ { n } ) \\right) = f \\left( \\sum _ { n ^ { \\prime } = - l } ^ { l } D _ { l } ^ { n ^ { \\prime } n } ( g ) e ^ { n ^ { \\prime } } \\right) } \\\\ { \\quad \\quad \\quad = \\sum _ { n ^ { \\prime } = - l } ^ { l } D _ { l } ^ { n ^ { \\prime } n } ( g ) f ( e ^ { n ^ { \\prime } } ) } \\\\ { \\quad \\quad \\quad = \\sum _ { n ^ { \\prime } = - l } ^ { l } D _ { l } ^ { n ^ { \\prime } n } ( g ) Y _ { l } ^ { n ^ { \\prime } } } \\\\ { \\quad \\quad = \\lambda ( g ) ( Y _ { l } ^ { n } ) } \\\\ { \\quad \\quad = \\lambda ( g ) ( f ( e ^ { n } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 422, + 645, + 556 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "General equivariance then follows from equivariance on basis vectors. This concludes this section. ", + "bbox": [ + 169, + 559, + 816, + 575 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "E.4.3 THE CLEBSCH-GORDAN DECOMPOSITION ", + "text_level": 1, + "bbox": [ + 173, + 589, + 524, + 604 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Explicit formulas for the Clebsch-Gordan coefficients of SO(3) are given in Bohm & Lowe (1993). ¨ The most important fact is the following: There is a decomposition ", + "bbox": [ + 173, + 613, + 823, + 643 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/e7b1a1bf6e21c60b2779f872f62a77de47d91fdbd5e555ddcfbe7b0a8b22f6be.jpg", + "text": "$$\nV _ { j } \\otimes V _ { l } \\cong \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J }\n$$", + "text_format": "latex", + "bbox": [ + 426, + 648, + 571, + 696 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "of representations. Furthermore, the Clebsch-Gordan coefficients $\\langle J M | j m ; l n \\rangle$ are all real numbers, a fact that we will use in Section E.5. ", + "bbox": [ + 171, + 703, + 823, + 733 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "E.4.4 ENDOMORPHISMS OF $V _ { J }$ ", + "text_level": 1, + "bbox": [ + 174, + 747, + 400, + 762 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "As in the case of harmonic networks, this is again simple: we are considering representations over C, and so Schur’s Lemma D.8 tells us that $\\mathrm { E n d } _ { \\mathrm { S O ( 3 ) } } ( V _ { J } )$ is 1-dimensional for each irrep $J$ . We can therefore ignore the endomorphisms once again. ", + "bbox": [ + 174, + 771, + 825, + 814 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "E.4.5 BRINGING EVERYTHING TOGETHER ", + "text_level": 1, + "bbox": [ + 176, + 829, + 480, + 844 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Now, with all this prior work, let us determine the equivariant kernels $K : S ^ { 2 } \\mathrm { H o m } _ { \\mathbb { C } } ( V _ { l } , V _ { J } )$ for the irreducible representations $D _ { l } : \\mathrm { S O ( 3 ) } \\mathrm { U } ( V _ { l } )$ and $D _ { J } : \\mathrm { S O ( 3 ) } \\mathrm { U } ( V _ { J } )$ . For this, we use Eq. (23). Since each $V _ { j }$ appears only once in the direct sum decomposition of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ according to Section E.4.2 and since $V _ { J }$ can only appear once in the direct sum decomposition of a tensor product $V _ { j } \\otimes V _ { l }$ according to Section E.4.3 , we do not need the indices $i$ and $s$ . Furthermore, as mentioned in the last section, the endomorphisms are trivial, which is why we also do not need the index $r$ . Overall, we see that we simply have basis kernels $K _ { j } : S ^ { 2 } \\to \\operatorname { H o m } _ { \\mathbb { C } } ( V _ { l } , V _ { J } )$ for all $j$ with $| l - J | \\leq j \\leq l + J$ .30 They are explicitly given by ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 147 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/fe64c4229f4516ae36c405ddb55460b201f00859c13a3c0ffa2eceef8e976da6.jpg", + "text": "$$\nK _ { j } ( x ) = \\left( \\begin{array} { c } { \\langle j | x \\rangle \\cdot \\mathrm { C G } _ { J ( j l ) } ^ { 1 } } \\\\ { \\vdots } \\\\ { \\langle j | x \\rangle \\cdot \\mathrm { C G } _ { J ( j l ) } ^ { d _ { J } } } \\end{array} \\right)\n$$", + "text_format": "latex", + "bbox": [ + 400, + 154, + 598, + 217 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "for all $x \\in S ^ { 2 }$ . Remembering that $\\langle j m | x \\rangle = \\overline { { Y _ { j } ^ { m } ( x ) } }$ , the individual matrix elements of $K _ { j } ( x )$ are then given by ", + "bbox": [ + 176, + 224, + 825, + 255 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/1dad0bba48eb75866cdcc01e2b1c43318c9b2ce17bf7ed4c28119296cd70dfd3.jpg", + "text": "$$\n\\langle J M | K _ { j } ( x ) | l n \\rangle = \\sum _ { m = - j } ^ { j } \\langle J M | j m ; l n \\rangle \\cdot \\overline { { Y _ { j } ^ { m } } } ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 252, + 661, + 299 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "This ends the discussion. ", + "bbox": [ + 174, + 300, + 338, + 315 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "E.5 SO(3)-STEERABLE KERNELS FOR REAL REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 332, + 632, + 347 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "In this section, we want to argue why the results in the last section transfer over to the real case as well. Most of the investigations in this section are probably well-known. However, we were not able to find sources that explicitly explain the representation theory of SO(3) over the real numbers, and so we develop lots of it here from scratch. We thereby make use of the theory over C, some results about real spherical harmonics, and the general theory of real and quaternionic representations outlined in Brocker & Dieck (2003). We need to somewhat turn the order around in ¨ this section in order to develop the results. Therefore we first investigate the Peter-Weyl theorem, then look at the endomorphism spaces of the appearing irreducible representations and afterward, as a consequence, show that the representations appearing in the decomposition of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } )$ are already exhaustive. ", + "bbox": [ + 173, + 358, + 825, + 498 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "E.5.1 THE PETER-WEYL THEOREM FOR $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } )$ AS A REPRESENTATION OF SO(3) ", + "bbox": [ + 169, + 511, + 758, + 529 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "The most important finding is the following, which is taken from Gallier & Quaintance (2020): One can do a base change for the spherical harmonics as follows to obtain real versions of them. Namely, let ", + "bbox": [ + 173, + 536, + 825, + 578 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/6020587ab03f0ade2d1915bb33cc4c2ee2b2fc31d9d51c4caf1e0a7138be1378.jpg", + "text": "$$\n{ } ^ { r } Y _ { l } ^ { n } = { \\left\\{ \\begin{array} { l l } { \\displaystyle { \\frac { i } { \\sqrt { 2 } } } \\left( Y _ { l } ^ { n } - ( - 1 ) ^ { n } Y _ { l } ^ { - n } \\right) } & { { \\mathrm { i f } } n < 0 , } \\\\ { \\displaystyle Y _ { l } ^ { 0 } } & { { \\mathrm { i f } } n = 0 , } \\\\ { \\displaystyle { \\frac { 1 } { \\sqrt { 2 } } } \\left( Y _ { l } ^ { - n } + ( - 1 ) ^ { n } Y _ { l } ^ { n } \\right) } & { { \\mathrm { i f } } n > 0 . } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 344, + 575, + 651, + 654 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "One can then show that these functions are real-valued continuous functions and therefore $^ { r } Y _ { l } ^ { n } \\in$ $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } )$ . Furthermore, they are an orthonormal basis of this space. We can then, as before, set $^ r V _ { l 1 }$ as the span of the $^ r Y _ { l } ^ { n } \\in \\dot { L } _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } )$ and obtain a decomposition ", + "bbox": [ + 173, + 655, + 823, + 699 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/c3eed9d274366a0b9d0ac82f9ca20f8b5146a41ce5055c2a2c97f930f2d9a4ba.jpg", + "text": "$$\nL _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } ) = { \\widehat { \\bigoplus } _ { l \\geq 0 } r _ { l 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 704, + 578, + 737 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "We need to understand the transformation properties of these real-valued spherical harmonics under rotation. To understand this explicitly, we set $B _ { l } \\in \\mathbb { C } ^ { ( 2 l + 1 ) \\times ( 2 l + 1 ) }$ as the (complex) base change matrix between the complex and real spherical harmonics. Its entries are given according to Eq. (31) such that the following relation holds for all $n = - l , \\ldots , l$ : ", + "bbox": [ + 173, + 742, + 825, + 800 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/ecb53a612260394a68a5721b8d2e17c1ba61bb0f1f93c13b821831bd596134ee.jpg", + "text": "$$\nr { Y _ { l } ^ { n } } = \\sum _ { n ^ { \\prime } = - l } ^ { l } B _ { l } ^ { n ^ { \\prime } n } \\cdot Y _ { l } ^ { n ^ { \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 806, + 584, + 851 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "Since for a given $l$ , both the complex and real spherical harmonics are linearly independent, the matrix $B _ { l }$ is invertible. Let $B _ { l } ^ { - 1 }$ be its inverse. Then it is generally known from linear algebra that ", + "bbox": [ + 174, + 857, + 823, + 887 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "we also obtain the inverse relation: ", + "bbox": [ + 174, + 103, + 403, + 117 + ], + "page_idx": 81 + }, + { + "type": "equation", + "img_path": "images/49b03c23063533775e212bb85ed89a346de0ce0d808bf0b6c091e6e3f135d42b.jpg", + "text": "$$\nY _ { l } ^ { n } = \\sum _ { n ^ { \\prime } = - l } ^ { l } ( B _ { l } ^ { - 1 } ) ^ { n ^ { \\prime } n } \\cdot { } ^ { r } Y _ { l } ^ { n ^ { \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 121, + 598, + 166 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Using both these relations and the rotation properties of the complex spherical harmonics from Section E.4.2 we obtain the following rotation property for the real spherical harmonics: ", + "bbox": [ + 173, + 171, + 825, + 202 + ], + "page_idx": 81 + }, + { + "type": "equation", + "img_path": "images/0cd1c43a66684f6e16b602540e0fa33ac5df38f0073db3d502ebc2557d03301b.jpg", + "text": "$$\n\\begin{array} { r l } { \\lambda ( g ) ( T Y _ { t } ^ { n } ) = \\displaystyle \\sum _ { n = - l } ^ { l } B _ { t } ^ { n , n } \\cdot \\lambda ( g ) ( Y _ { t } ^ { n , n } ) } \\\\ { = \\displaystyle \\sum _ { n = - l } ^ { l } B _ { t } ^ { n , n } \\sum _ { n = - l } ^ { l } D _ { t } ^ { n , n } ( g ) \\cdot Y _ { t } ^ { n , n } } \\\\ { = \\displaystyle \\sum _ { n = - l } ^ { l } B _ { t } ^ { n , n } \\sum _ { n = - l } ^ { l } D _ { t } ^ { n , n } ( g ) \\cdot Y _ { t } ^ { n , n } } \\\\ { = \\displaystyle \\sum _ { n = - l } ^ { l } B _ { t } ^ { n , n } \\sum _ { n = - l } ^ { l } D _ { t } ^ { n , n } ( g ) \\cdot \\sum _ { n = - l } ^ { l } ( B _ { t } ^ { - 1 } ) ^ { n , n } \\cdots \\gamma _ { t } ^ { n , n } } \\\\ { = \\displaystyle \\sum _ { n = - l } ^ { l } \\left( \\displaystyle \\sum _ { n = - l } ^ { l } \\displaystyle \\sum _ { n = - l - n - l } ^ { l } ( B _ { t } ^ { - 1 } ) ^ { n , n } \\cdots D _ { t } ^ { n , 2 n , 1 } ( g ) \\cdot B _ { t } ^ { n , 1 , n } \\right) \\cdot Y _ { t } ^ { n , n } } \\\\ { = \\displaystyle \\sum _ { n = - l } ^ { l } \\left( B _ { t } ^ { - 1 } \\cdot D _ { t } ( g ) \\cdot B _ { t } \\right) ^ { n , n } \\cdot \\gamma _ { t } ^ { n , n } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 207, + 738, + 435 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Now if we set ${ } ^ { r } D _ { l } ( g ) : = B _ { l } ^ { - 1 } \\cdot D _ { l } ( g ) \\cdot B _ { l }$ , then we obtain the transformation property ", + "bbox": [ + 174, + 441, + 738, + 458 + ], + "page_idx": 81 + }, + { + "type": "equation", + "img_path": "images/59ac32ef00546da76ae13face744459fc782d5912b7529498bfbd25dfa3548d7.jpg", + "text": "$$\n\\lambda ( g ) ( ^ { r } Y _ { l } ^ { n } ) = \\sum _ { n ^ { \\prime } = - l } ^ { l } { ^ { r } D _ { l } ( g ) ^ { n ^ { \\prime } n } \\cdot ^ { r } Y _ { l } ^ { n ^ { \\prime } } }\n$$", + "text_format": "latex", + "bbox": [ + 372, + 463, + 625, + 510 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "which is analogous to the one in Section E.4.2. ", + "bbox": [ + 176, + 515, + 478, + 530 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Proof. Note that since ${ } ^ { r } Y _ { l } ^ { n }$ is a real-valued function, the rotation $\\lambda ( g ) ( ^ { r } Y _ { l } ^ { n } )$ is real-valued as well. Thus, it is in the space $L _ { \\mathbb { R } } ^ { 2 ^ { \\circ } } ( S ^ { 2 } )$ . The real spherical harmonics are a basis of this space, which means that the coefficients when expanding $\\lambda ( g ) ( ^ { r } Y _ { l } ^ { n } )$ in this basis are necessarily real as well. These coefficients are precisely given by the ${ } ^ { r } D _ { l } ( g ) ^ { n ^ { \\prime } n }$ according to Eq. (32). □ ", + "bbox": [ + 173, + 565, + 825, + 626 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Now, we have the choice to view $^ r D _ { l }$ as either a real or a complex representation, but first we take the complex viewpoint and see it as a function $^ r D _ { l } : \\mathrm { S O ( 3 ) } \\bar { } \\mathrm { G } \\bar { \\mathrm { L } } ( \\mathbb { C } ^ { 2 l + 1 } )$ . Notationwise, the following is important: the $\\cdot _ { r } , \\cdot $ in ${ \\boldsymbol { r } } _ { D _ { l } }$ indicates that the elements in this matrix are real but does not tell us on which space it acts. This will always be clarified by the context. We have the following: ", + "bbox": [ + 173, + 640, + 826, + 696 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Lemma E.8. $^ { r } D _ { l } : \\mathrm { S O ( 3 ) } \\to \\mathrm { U } ( \\mathbb { C } ^ { 2 l + 1 } )$ is an irreducible unitary representation and isomorphic to $D _ { l }$ . ", + "bbox": [ + 173, + 699, + 825, + 729 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Proof. First of all, it is an actual linear representation since ", + "bbox": [ + 174, + 744, + 563, + 760 + ], + "page_idx": 81 + }, + { + "type": "equation", + "img_path": "images/48e07c25200ac09fb5a26a600e628b62ee7b8c333ae0a1d291fff8e78e969ece.jpg", + "text": "$$\n^ { r } D _ { l } ( g g ^ { \\prime } ) = B _ { l } ^ { - 1 } D _ { l } ( g g ^ { \\prime } ) B _ { l } = B _ { l } ^ { - 1 } D _ { l } ( g ) B _ { l } B _ { l } ^ { - 1 } D _ { l } ( g ^ { \\prime } ) B _ { l } = { } ^ { r } D _ { l } ( g ) \\cdot { } ^ { r } D _ { l } ( g ^ { \\prime } )\n$$", + "text_format": "latex", + "bbox": [ + 238, + 763, + 759, + 784 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "where we used that $D _ { l }$ is a linear representation. Now since $Y _ { l } ^ { n }$ and $^ { r } Y _ { l } ^ { n }$ are both orthonormal bases of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ , the base change matrix $B _ { l }$ needs to be a unitary matrix. Consequently, ${ } ^ { r } D _ { l } ( g ) =$ $B _ { l } ^ { - 1 } D _ { l } ( g ) \\dot { B } _ { l }$ is as a product of unitary transformations itself unitary, which means that ${ } ^ { r } D _ { l }$ is a unitary representation. Furthermore, we obtain $B _ { l } \\cdot { \\dot { \\boldsymbol { r } } } D _ { l } ( g ) = D _ { l } ( { \\dot { \\boldsymbol { g } } } ) \\cdot B _ { l }$ , which means that $B _ { l }$ gives an isomorphism $r _ { D _ { l } } \\cong D _ { l }$ of unitary representations. From the fact that $D _ { l }$ is irreducible, we obtain that ${ \\bf \\Delta } ^ { r } D _ { l }$ is irreducible as well. □ ", + "bbox": [ + 173, + 787, + 826, + 876 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Now we take the real viewpoint. Let ${ } ^ { r } V _ { l } = \\mathbb { R } ^ { 2 l + 1 }$ . ", + "bbox": [ + 173, + 888, + 506, + 905 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Lemma E.9. $^ r D _ { l } : \\mathrm { S O ( 3 ) } \\mathrm { O } ( ^ { r } V _ { l } )$ is an irreducible orthogonal representation. ", + "bbox": [ + 173, + 909, + 714, + 925 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Proof. ${ } ^ { r } D _ { l } ( g )$ is a unitary matrix for each $g \\in \\mathrm { S O } ( 3 )$ by Lemma E.8, and since its matrix elements are real by Lemma E.7, it automatically is an orthogonal matrix. If it was reducible, then there would be a real base change matrix that brings ${ } ^ { r } D _ { l }$ in a nontrivial block-diagonal shape. However, this base change would in particular be complex, meaning that we would conclude that the complex version of the representation ${ } ^ { r } D _ { l }$ is reducible. But it is not, due to Lemma E.8. □ ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Now, remember that $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } ) = { \\widehat { \\bigoplus } _ { l \\geq 0 } } ^ { r } V _ { l 1 }$ and that ${ } ^ { r } V _ { l 1 }$ is generated from the real spherical harmonics. Also, remember that the real spherical harmonics transform as in Eq. (32). Thus, with the same arguments as in Eq. (30) we obtain ${ \\bf { \\Lambda } } ^ { r } V _ { l 1 } \\cong { \\bf { \\Lambda } } ^ { r } V _ { l }$ , which is from the preceding lemmas an irreducible orthogonal representation. Thus, we have found the Peter-Weyl decomposition of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } )$ . ", + "bbox": [ + 174, + 190, + 825, + 267 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "E.5.2 ENDOMORPHISMS OF ${ } ^ { r } V _ { J }$ ", + "text_level": 1, + "bbox": [ + 176, + 280, + 408, + 296 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "In the next section, we will show that the ${ } ^ { r } D _ { J } : \\mathrm { S O } ( 3 ) \\mathrm { O } ( { } ^ { r } V _ { J } )$ already given an exhaustive list of the irreducible representations of SO(3) over the real numbers. In this section, we first describe their endomorphism spaces since this will help in showing that there cannot be any other irreducible representations. Fortunately, the situation is again simple: ", + "bbox": [ + 173, + 305, + 825, + 362 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Proposition E.10. $\\operatorname { E n d } _ { \\mathrm { S O ( 3 ) , R } } ( ^ { r } V _ { J } )$ is one-dimensional for each $J \\geq 0$ . ", + "bbox": [ + 174, + 366, + 653, + 382 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Proof. Let $f : { } ^ { r } V _ { J } { } ^ { r } V _ { J }$ be an endomorphism. Since ${ } ^ { r } V _ { J } = \\mathbb { R } ^ { 2 J + 1 }$ we can view $f$ as a matrix in $\\mathbb { R } ^ { ( 2 \\bar { J } + 1 ) \\times ( 2 \\bar { J } + 1 ) }$ . That $f$ is an endomorphism then means ", + "bbox": [ + 174, + 398, + 823, + 430 + ], + "page_idx": 82 + }, + { + "type": "equation", + "img_path": "images/5e2f73c3a9af88c08d048cad28e9a9f4825a86a55e5c489bfc6e29a71cbf8777.jpg", + "text": "$$\nf \\cdot { } ^ { r } D _ { J } ( g ) = { } ^ { r } D _ { J } ( g ) \\cdot f\n$$", + "text_format": "latex", + "bbox": [ + 413, + 436, + 584, + 455 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "for all $g \\in \\mathrm { S O } ( 3 )$ . Now note that as a real matrix, f is in particular a complex matrix, i.e., $f \\in$ $\\mathbb { C } ^ { ( 2 J + 1 ) \\times ( 2 J + 1 ) }$ . Also, remember that we can view ${ } ^ { r } D _ { J }$ also as a complex irreducible representation $^ r D _ { J } : \\mathrm { S O ( 3 ) } \\to \\mathrm { U } ( \\mathbb { C } ^ { 2 J + 1 } )$ by Lemma E.8. What this means is that $f \\in \\mathrm { E n d } _ { \\mathrm { S O ( 3 ) } , \\mathrm { C } } ( \\mathbb { C } ^ { 2 J + 1 } )$ , which is isomorphic to C by Schur’s Lemma D.8. Thus, $f$ is a complex multiple of the identity. Since $f$ is a real matrix, it is thus a real multiple of the identity. The result follows. □ ", + "bbox": [ + 174, + 460, + 825, + 534 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "E.5.3 GENERAL NOTES ON THE RELATION BETWEEN REAL AND COMPLEX REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 550, + 707, + 578 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "In the next section we show that there can, up to isomorphism, not be other irreducible representations than the ${ } ^ { r } D _ { l } : \\mathrm { S O ( 3 ) } \\mathrm { O } ( { } ^ { r } V _ { l } )$ . In order to do so, we first need to better understand the relationship between real and complex representations of compact groups. These investigations will carry over to the investigations for O(3) that we do in Section E.6 as well. ", + "bbox": [ + 174, + 588, + 825, + 645 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "The following definition of a classification of real irreducible representations of a compact group $G$ can be found in Brocker & Dieck (2003), Theorem II. ¨ 6.7. In this book, it is a theorem, since the authors give an independent but equivalent definition of these notions. ", + "bbox": [ + 176, + 651, + 821, + 694 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Definition E.11 (Real, Complex, and Quaternionic Type Irreducible Representations). Let $\\rho : G $ $\\mathrm { O } ( V )$ be a real irreducible representation of a compact group $G$ . Then $\\rho$ is said to be of ", + "bbox": [ + 174, + 698, + 825, + 727 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "1. real type if $\\operatorname { E n d } _ { G , \\mathbb { R } } ( V ) \\cong \\mathbb { R }$ , \n2. complex type if $\\operatorname { E n d } _ { G , \\mathbb { R } } ( V ) \\cong \\mathbb { C }$ and \n3. quaternionic type if $\\operatorname { E n d } _ { G , \\mathbb { R } } ( V ) \\cong \\mathbb { H }$ , where $\\mathbb { H }$ are the quaternions. ", + "bbox": [ + 210, + 737, + 679, + 803 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Here, these isomorphisms respect both addition and multiplication. The multiplication in the endomorphism spaces is thereby given by composition of functions. ", + "bbox": [ + 173, + 813, + 823, + 842 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Furthermore, Brocker & Dieck (2003) shows in Theorem II. ¨ 6.3 that there is no other possibility for an irreducible real representation, i.e., they can be completely categorized by being of real, complex or quaternionic type. Additionally, since R, C and $\\mathbb { H }$ already differ in their R-dimension, it is enough to check whether the R-dimension of an endomorphism space is 1, 2 or 4 in order to do the classification. ", + "bbox": [ + 174, + 853, + 823, + 924 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "In order to compare real and complex representations we need to define two functors between those:31 ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Definition E.12 (Restriction and Extension). Let ${ } ^ { c } \\rho : G \\to \\mathrm { G L } ( { } ^ { c } V )$ be a complex representation. Furthermore, let $r _ { \\rho } : G \\to { \\mathrm { G L } } ( { \\boldsymbol { \\mathbf { \\mathit { r } } } } _ { V } )$ be a real representation. Then we define their restriction and extension as follows: ", + "bbox": [ + 176, + 137, + 825, + 179 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "1. Set $r ( ^ { c } V )$ as the $\\mathbb { R }$ -vector space that has the same underlying abelian group as $^ c V$ and the scalar multiplication from $\\mathbb { R }$ which is the restriction of the multiplication from C. The restriction $r ( ^ { c } \\rho ) : G \\to { \\mathrm { G L } } ( r ( ^ { c } V ) )$ is defined as the exact same map as $^ c \\rho$ , only that $r ( { } ^ { c } \\rho ) ( g ) : r ( { } ^ { \\overset { } { c } } V ) \\to r ( { } ^ { c } V )$ is now viewed as an automorphism of real vector spaces. ", + "bbox": [ + 210, + 191, + 825, + 247 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "2. We define the extension by $e ( ^ { r } V ) : = \\mathbb { C } \\otimes _ { \\mathbb { R } } { } ^ { r } V$ , where $\\mathbb { C }$ is regarded as an $\\mathbb { R }$ -vector space. This construction becomes a C-vector space by scalar multiplication $z \\cdot ( z ^ { \\prime } \\otimes v ) : = ( z z ^ { \\prime } ) \\otimes v$ We can then define $e ( { } ^ { r } \\rho ) : G \\to { \\mathrm { G L } } ( e ( { \\bar { r } } V ) )$ by setting $e ( \\bar { r _ { \\rho } } ) ( g ) : = \\mathrm { i d } _ { \\mathbb { C } } \\otimes ( \\bar { r _ { \\rho } } ( g ) )$ . ", + "bbox": [ + 217, + 257, + 821, + 301 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Note that the extension operation doubles the $\\mathbb { R }$ -dimension, whereas for the restriction it stays equal. Therefore, we can not hope that these operations are inverse to each other. However, we have the following, almost as nice statement: ", + "bbox": [ + 176, + 313, + 825, + 354 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Proposition E.13. For each real representation $\\rho : G \\to { \\mathrm { G L } } ( V )$ there is a natural isomorphism $r ( e ( V ) ) \\cong V \\oplus V$ of R-representations. ", + "bbox": [ + 173, + 359, + 823, + 388 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Proof. This is the first statement in Brocker & Dieck (2003), Proposition II. ¨ 6.1. ", + "bbox": [ + 174, + 407, + 694, + 422 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "The following definition is actually not the definition that Brocker & Dieck (2003) formulate. How-¨ ever, it is an equivalent characterization that follows from their Proposition II.6.6 (vii), (viii) and (ix) and is more convenient for our needs: ", + "bbox": [ + 176, + 443, + 825, + 484 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Definition E.14 (Real Type Complex Representation). Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ be a complex irreducible representation. Then $\\rho$ is called of real type if there is an isomorphism of real representations $r ( V ) \\cong \\bar { U } \\oplus U$ where ", + "bbox": [ + 173, + 489, + 823, + 532 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "1. $\\rho _ { U } : G \\to { \\mathrm { G L } } ( U )$ is an irreducible real representation and ", + "bbox": [ + 212, + 544, + 617, + 560 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "2. $r ( \\rho ) : G \\to { \\mathrm { G L } } ( r ( V ) )$ is the restriction of $\\rho$ , as defined in Definition E.12. ", + "bbox": [ + 209, + 569, + 723, + 584 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Proposition E.15. Assume $G$ is a compact group such that all complex irreducible representations are of real type. Then also all real irreducible representations are of real type. ", + "bbox": [ + 173, + 589, + 821, + 618 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Proof. This follows from Brocker & Dieck (2003), Proposition II. ¨ 6.6 (ii) and (iii). ", + "bbox": [ + 173, + 637, + 710, + 652 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Proposition E.16. Let $\\rho : G \\to { \\mathrm { G L } } ( V )$ be an irreducible real representation of real type. Then its extension $e ( \\rho ) : G \\to { \\mathrm { G L } } ( e ( V ) )$ given as in Definition E.12 is an irreducible complex representation (also of real type). ", + "bbox": [ + 174, + 664, + 825, + 707 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Proof. This is precisely Brocker & Dieck (2003), Proposition II. ¨ 6.6(i). ", + "bbox": [ + 174, + 726, + 637, + 741 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "E.5.4 THE IRREDUCIBLE REPRESENTATIONS OF SO(3) OVER THE REAL NUMBERS ", + "bbox": [ + 173, + 761, + 758, + 776 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "The rough strategy is to use the fact that the ${ } ^ { r } D _ { l }$ , viewed as complex irreducible representations, are an exhaustive list of all the complex irreps. Then, using the restriction and extension operators $r$ and $e$ between real and complex representations, we can show that in the specific case of SO(3), there can not be any other real irreducible representations than the ${ } ^ { r } D _ { l }$ , viewed as real representations. ", + "bbox": [ + 174, + 786, + 825, + 843 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Lemma E.17. All complex irreducible representations of SO(3) are of real type. ", + "bbox": [ + 174, + 847, + 704, + 863 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Proof. From Section E.4.1 and Lemma E.8 we know that the $^ { r } D _ { l } : \\mathrm { S O ( 3 ) } \\to \\mathrm { U } ( \\mathbb { C } ^ { 2 l + 1 } )$ give us, up to equivalence, all the complex irreducible representations of SO(3). According to Definition E.14 we now need to understand that its restriction splits into the direct sum of twice the same irreducible real representation. We do this as follows: ", + "bbox": [ + 174, + 102, + 825, + 160 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "We can write $r ( \\mathbb { C } ^ { 2 l + 1 } ) = \\mathbb { R } ^ { 2 l + 1 } \\oplus ( i \\mathbb { R } ) ^ { 2 l + 1 } = { } ^ { r } V _ { l } \\oplus i ^ { r } V _ { l }$ , which is a decomposition of $\\mathbb { C } ^ { 2 l + 1 }$ when viewed as an $\\mathbb { R }$ -vector space. Then, we can note that both ", + "bbox": [ + 171, + 165, + 825, + 195 + ], + "page_idx": 84 + }, + { + "type": "equation", + "img_path": "images/2f142d4a50bf36bf2d8f6e0cb4d64ae4062517a2f6df1e891992ec59dd11e725.jpg", + "text": "$$\n\\begin{array} { r l } & { { ^ { r } D _ { l } } : \\mathrm { S O } ( 3 ) \\to \\mathrm { O } ( ^ { r } V _ { l } ) \\mathrm { a n d } } \\\\ & { { ^ { r } D _ { l } } : \\mathrm { S O } ( 3 ) \\to \\mathrm { O } ( i ^ { r } V _ { l } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 405, + 202, + 593, + 239 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "are well-defined $\\mathbb { R }$ -representations, which follows from the fact that the matrix elements are all real. Furthermore, the first map is actually an irreducible real representation by Lemma E.9. The second one is isomorphic to the first since one can show that ", + "bbox": [ + 173, + 244, + 825, + 286 + ], + "page_idx": 84 + }, + { + "type": "equation", + "img_path": "images/b7179c66a4884bb045b4d2e09d7280c13875e39f08d8743bc0191bf3259109fa.jpg", + "text": "$$\ni : { } ^ { r } V _ { l } \\to i ^ { r } V _ { l } , a \\mapsto i \\cdot a\n$$", + "text_format": "latex", + "bbox": [ + 413, + 295, + 583, + 310 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "is an isomorphism of real SO(3)-representations. This gives us precisely the splitting of $r ( \\mathbb { C } ^ { 2 l + 1 } )$ as a representation that we were looking for. □ ", + "bbox": [ + 173, + 318, + 825, + 348 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Corollary E.18. All irreducible real representations of SO(3) are of real type. ", + "bbox": [ + 174, + 356, + 689, + 372 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Proof. This follows directly from Lemma E.17 and Proposition E.15. ", + "bbox": [ + 173, + 388, + 629, + 405 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Proposition E.19. The ${ } ^ { r } D _ { l } : \\mathrm { S O } ( 3 ) \\mathrm { O } ( { } ^ { r } V _ { l } )$ are, up to equivalence, all real irreducible representations of SO(3). ", + "bbox": [ + 171, + 412, + 823, + 444 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Proof. Assume that $\\rho : \\mathrm { S O } ( 3 ) \\mathrm { G L } ( V )$ is an irreducible real representation of $\\mathrm { S O ( 3 ) }$ . It is of real type by Corollary E.18. By Proposition E.16, the extension ${ \\dot { e } } ( \\rho ) : G \\to { \\mathrm { G L } } ( { \\dot { e } } ( V ) )$ is an irreducible complex representation. Since the ${ \\boldsymbol { r } } _ { D _ { l } }$ give us all complex irreducible representations up to equivalence by Section E.4.1 and Lemma E.8, there is an equivalence of complex SO(3)- representations $e ( V ) \\cong \\mathbb { C } ^ { 2 l + 1 }$ for some $l$ . Since functors respect isomorphisms (and equivalences are isomorphisms in the categories of $G$ -representations) and the restriction operation is a functor,32 and using Proposition E.13 as well as the proof of Lemma E.17 we obtain: ", + "bbox": [ + 173, + 459, + 825, + 558 + ], + "page_idx": 84 + }, + { + "type": "equation", + "img_path": "images/17faf1140ff70b01cc3196f093d03ab3e90680a953c05976af6cf64d2470a5f4.jpg", + "text": "$$\nV \\oplus V \\cong r ( e ( V ) ) \\cong r ( \\mathbb { C } ^ { 2 l + 1 } ) \\cong { } ^ { r } V _ { l } \\oplus i ^ { r } V _ { l } = { } ^ { r } V _ { l } \\oplus { } ^ { r } V _ { l } .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 565, + 692, + 584 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Using the Krull-Remak-Schmidt Theorem B.39, we see that there is an isomorphism of SO(3)- representations $V \\cong { } ^ { r } V _ { l }$ . This finishes the proof. □ ", + "bbox": [ + 174, + 590, + 820, + 619 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "E.5.5 THE CLEBSCH-GORDAN DECOMPOSITION ", + "text_level": 1, + "bbox": [ + 173, + 636, + 524, + 651 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "We are almost there. The only thing left to understand is the Clebsch-Gordan decomposition. Remember the following from Section E.4.3: For the complex irreducible representations there are decompositions ", + "bbox": [ + 174, + 661, + 825, + 704 + ], + "page_idx": 84 + }, + { + "type": "equation", + "img_path": "images/87359d17a007850a06f79d0ce1ba1225592dce3d79b66d8296defe9d42554e23.jpg", + "text": "$$\nV _ { j } \\otimes V _ { l } \\cong \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J }\n$$", + "text_format": "latex", + "bbox": [ + 426, + 702, + 573, + 750 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "where on each space, the representations $D _ { j } , D _ { l }$ and $D _ { J }$ are given by the Wigner D-matrices. Furthermore, the Clebsch-Gordan coefficients are all real. Now, we know that ${ } ^ { r } D _ { l }$ is, as a complex representation, isomorphic to $D _ { l }$ by Lemma E.8, and such a representation then acts on $\\mathbb { C } ^ { 2 l + \\mathrm { \\hat { 1 } } }$ as well. Consequently, we also get the decomposition ", + "bbox": [ + 174, + 752, + 825, + 809 + ], + "page_idx": 84 + }, + { + "type": "equation", + "img_path": "images/851d89cee49ac3a1211b394e3b1dc869a65f9a76beccf0447c0cb16e24b117a3.jpg", + "text": "$$\n\\mathbb { C } ^ { 2 j + 1 } \\otimes \\mathbb { C } ^ { 2 l + 1 } \\cong \\bigoplus _ { J = | l - j | } ^ { l + j } \\mathbb { C } ^ { 2 J + 1 }\n$$", + "text_format": "latex", + "bbox": [ + 388, + 815, + 609, + 863 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "of the complex representations ${ } ^ { r } D _ { j }$ and ${ } ^ { r } D _ { l }$ . Obviously, the Clebsch-Gordan coefficients can be chosen to be exactly the same as before, and thus they are again real. ", + "bbox": [ + 176, + 869, + 823, + 900 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Let the above isomorphism be called $f$ . Now, we can view all involved vector spaces as $\\mathbb { R }$ -vector spaces as well. Furthermore, we have subspaces $^ r V _ { j } = \\mathbb { R } ^ { 2 j + 1 }$ , ${ } ^ { r } V _ { l } = \\mathbb { R } ^ { 2 l + 1 }$ and ${ } ^ { r } V _ { J } = \\mathbb { R } ^ { 2 J + 1 }$ which are also invariant under the representations ${ } ^ { r } \\bar { D } _ { j }$ , ${ \\bf \\Delta } ^ { r } D _ { l }$ and ${ } ^ { r } D _ { J }$ . Consequently, we can just restrict the isomorphism above to a map ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 85 + }, + { + "type": "equation", + "img_path": "images/cc15e3daeef289cb056b5036173147fda9c366753a92cd580cbdf5779bfd528b.jpg", + "text": "$$\nf | : { ^ { r } V _ { j } } \\otimes { ^ { r } V _ { l } } \\sp { \\bullet } \\bigoplus _ { J = | l - j | } ^ { l + j } { ^ { r } V _ { J } } .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 165, + 599, + 212 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "which is well-defined since the Clebsch-Gordan coefficients are real. It needs to be injective, since it is a restriction of an isomorphism. For dimension reasons, the restriction then needs to be an isomorphism, and obviously, it has the exact same Clebsch-Gordan coefficients as the original map $f$ .33 ", + "bbox": [ + 173, + 217, + 825, + 273 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "E.5.6 BRINGING EVERYTHING TOGETHER ", + "text_level": 1, + "bbox": [ + 178, + 287, + 480, + 303 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "By what we’ve shown in the last sections, we see that the situation is basically the same as in Section E.4.5. The only thing that changes is that we now use the real spherical harmonics, and therefore the complex conjugation disappears. What this overall means is the following: let ${ } ^ { r } D _ { l } :$ $\\mathrm { S O ( 3 ) } \\mathrm { O } ( ^ { r } V _ { l } ) \\bar $ and ${ } ^ { r } D _ { J } : \\mathrm { S O } ( 3 ) { } \\mathrm { O } ( { } ^ { r } V _ { J } )$ be the representations determining the input and output fields. Then a basis for steerable kernels $K : S ^ { 2 } \\ { \\overset { \\cdot } { \\to } } \\ \\operatorname { H o m } _ { \\mathbb { R } } ( { } ^ { r } V _ { l } , { } ^ { r } V _ { J } )$ is given by kernels $K _ { j } \\mathrm { : } S ^ { 2 } \\mathrm { H o m } _ { \\mathbb { R } } ( ^ { r } V _ { l } , ^ { r } V _ { J } )$ for all $| l - J | \\leq j \\leq l + J$ . The matrix elements are given by ", + "bbox": [ + 173, + 313, + 825, + 397 + ], + "page_idx": 85 + }, + { + "type": "equation", + "img_path": "images/757d52ebbdb931e81cd79531df690a6867ab5f75b7d9b99b0b89a747b5efc805.jpg", + "text": "$$\n\\langle J M | K _ { j } ( x ) | l n \\rangle = \\sum _ { m = - j } ^ { j } \\langle J M | j m ; l n \\rangle \\cdot { } ^ { r } Y _ { j } ^ { m } ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 402, + 666, + 449 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "E.6 O(3)-STEERABLE KERNELS FOR COMPLEX REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 463, + 653, + 479 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "In this section, we deal with O(3)-equivariant kernels for complex representations and then, in the next section, will transport the results over to real representations. In the earlier examples, we saw that the Peter-Weyl decomposition of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ always contained each irreducible representation of the symmetry group exactly once. The example of O(3) is the first in which this is not the case: parity will play a role in determining which irreducible representations make their way in the space of square-integrable functions and which do not. Overall, we hope that the example of O(3) is a sufficient justification for our use of the multiplicities $m _ { j }$ of irreducible representations that we considered in all our theorems. $\\mathrm { O ( 3 ) }$ -equivariant networks are to the best of our knowledge not described in any published work yet. ", + "bbox": [ + 173, + 488, + 826, + 616 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "E.6.1 THE IRREDUCIBLE REPRESENTATIONS OF O(3) ", + "text_level": 1, + "bbox": [ + 173, + 628, + 558, + 645 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "The most important observation is the following, after which we can deduce the irreducible representations of O(3) from those of SO(3): ", + "bbox": [ + 171, + 654, + 823, + 683 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "Lemma E.20. Let $\\mathbb { Z } _ { 2 } : = ( \\{ - 1 , + 1 \\} , \\cdot )$ be the group with two elements. Then the map ", + "bbox": [ + 178, + 686, + 746, + 702 + ], + "page_idx": 85 + }, + { + "type": "equation", + "img_path": "images/d7cb5453ff843fbafe28b4fbb11da28254a75c8aa019bda6704ee28cee578357.jpg", + "text": "$$\n\\cdot : \\mathbb { Z } _ { 2 } \\times \\operatorname { S O } ( 3 ) \\to \\operatorname { O } ( 3 ) , ( s , g ) \\mapsto s g\n$$", + "text_format": "latex", + "bbox": [ + 370, + 707, + 627, + 724 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "is an isomorphism of groups. ", + "bbox": [ + 174, + 729, + 364, + 744 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "Proof. It is a group homomorphism since $s \\in \\{ - 1 , + 1 \\}$ can be represented by a multiple of the identity matrix, and as such it commutes with every matrix $g$ . That $\\cdot$ is an isomorphism follows since all matrices in O(3) either have determinant 1 or $- 1$ . The matrices with determinant 1 form $\\mathrm { S O ( 3 ) }$ and are the image of $\\{ + 1 \\} \\times \\mathrm { S O } ( 3 )$ . The matrices with determinant $- 1$ are the image of $\\{ - 1 \\} \\times \\mathrm { S O } ( 3 )$ . □ ", + "bbox": [ + 173, + 758, + 825, + 832 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "Note the fact that for $g \\in \\mathrm { S O } ( 3 )$ , $- g$ has determinant $- 1$ , which we used in the proof. This does only hold for $g \\in \\mathrm { S O } ( d )$ with $d$ being odd. Therefore, the above lemma is not true for $d$ even. In the even case, we obtain a semidirect product and the story complicates somewhat. ", + "bbox": [ + 173, + 844, + 825, + 888 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "Earlier, we already considered tensor product representations of one and the same group. A related notion is that of tensor product representations of two different groups:34 ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Definition E.21 (Tensor Product Representation). Let $G$ and $H$ be two compact groups. Let $\\rho _ { G } :$ $G \\to { \\mathrm { G L } } ( V _ { G } )$ and $\\rho _ { H } : H \\to { \\mathrm { G L } } ( { \\bar { V } } _ { H } )$ be representations of the two groups $G$ and $H$ . Then the tensor product representation is given by ", + "bbox": [ + 174, + 136, + 825, + 179 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/80af19ed29e0d8cfd8bbcaba3a10db940407d428056670470ef078db44f484f3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\rho _ { G } \\otimes \\rho _ { H } : G \\times H \\to { \\mathrm { G L } } ( V _ { G } \\otimes V _ { H } ) , } \\\\ & { \\left[ \\left( \\rho _ { G } \\otimes \\rho _ { H } \\right) ( g , h ) \\right] ( v _ { G } \\otimes v _ { H } ) : = \\rho _ { G } ( g ) ( v _ { G } ) \\otimes \\rho _ { H } ( h ) ( v _ { H } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 184, + 700, + 224 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "This is again a linear representation. ", + "bbox": [ + 174, + 228, + 413, + 243 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Proposition E.22. Representatives of isomorphism classes of irreducible representations of $G \\times H$ are given precisely by all the $\\rho _ { G } \\otimes \\rho _ { H }$ , where $\\rho _ { G }$ and $\\rho _ { H }$ run through representatives of isomorphism classes of irreducible representations of $G$ and $H$ , respectively. ", + "bbox": [ + 173, + 247, + 826, + 290 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Proof. This is proven in chapter II, Proposition 4.14 and 4.15 of Brocker & Dieck (2003). ¨ ", + "bbox": [ + 169, + 304, + 764, + 320 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "It is important to note that the proof of the above proposition uses the property of the complex numbers to be algebraically closed in crucial steps, and therefore it is unclear how exactly a generalization to representations over the real numbers looks like. Therefore, we will not use the above proposition in our later considerations for real representations of O(3). ", + "bbox": [ + 173, + 334, + 825, + 391 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "However, in our current situation, we can apply it without problems. This proposition, together with Lemma E.20, suggests that we should understand the irreducible representations of $\\mathbb { Z } _ { 2 }$ . We already saw this for real representations before and essentially obtain the same result: ", + "bbox": [ + 173, + 397, + 825, + 440 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Lemma E.23. The irreducible representations of $\\mathbb { Z } _ { 2 }$ are up to equivalence precisely the following two, which we state for simplicity only on the generator: ", + "bbox": [ + 171, + 444, + 823, + 473 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/6fa0726b801303f73c1b454beda10ce8c91190e13ccc0ad23d525bc4b8735844.jpg", + "text": "$$\n\\begin{array} { r l } & { \\rho _ { + } : \\mathbb { Z } _ { 2 } \\to { \\mathrm { G L } } ( \\mathbb { C } ) , \\rho _ { + } ( - 1 ) = \\mathrm { i d } _ { \\mathbb { C } } } \\\\ & { \\rho _ { - } : \\mathbb { Z } _ { 2 } \\to { \\mathrm { G L } } ( \\mathbb { C } ) , \\rho _ { - } ( - 1 ) = - \\mathrm { i d } _ { \\mathbb { C } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 364, + 478, + 632, + 516 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Proof. This can be shown in exactly the same way as in Section E.3.1. ", + "bbox": [ + 174, + 526, + 635, + 542 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Thus we are ready to state our result about the irreducible representations of $\\mathrm { O ( 3 ) }$ : ", + "bbox": [ + 174, + 558, + 714, + 573 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Proposition E.24. The irreducible representations of O(3) are up to equivalence given as follows: for each $l \\in \\mathbb { N } _ { \\geq 0 }$ there are precisely two representations $\\overline { { D _ { l + } } } : \\mathrm { O } ( 3 ) \\to \\mathrm { U } ( V _ { l + } )$ and $D _ { l - } : \\mathrm { O } ( 3 ) \\to$ $\\mathrm { U } ( V _ { l - } )$ with $\\bar { V _ { l + } } = \\mathbb { C } ^ { 2 l + 1 } = V _ { l - }$ , given as follows: ", + "bbox": [ + 173, + 577, + 825, + 621 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/e1a23e21c4bd0fccb6dfc010cb951b0c2fcfdc012b1febab39335cc943bc15b0.jpg", + "text": "$$\n\\begin{array} { r l } & { D _ { l + } ( s g ) = D _ { l } ( g ) ~ f o r { a l l } s \\in \\mathbb { Z } _ { 2 } , ~ g \\in \\mathrm { S O ( 3 ) } . } \\\\ & { D _ { l - } ( s g ) = s D _ { l } ( g ) f o r { a l l } s \\in \\mathbb { Z } _ { 2 } , ~ g \\in \\mathrm { S O ( 3 ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 626, + 653, + 664 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Proof. Remember from Section E.4.1 that the irreducible representations of $\\mathrm { S O ( 3 ) }$ are given by the Wigner D-matrices $D _ { l }$ . From Lemma E.23 we know that the irreducible representations of $\\mathbb { Z } _ { 2 }$ are given by $\\rho _ { + }$ and $\\rho _ { - }$ . From the isomorphism ${ \\bf O } ( 3 ) \\cong { \\bf Z } _ { 2 } \\times \\mathrm { S O } ( 3 )$ from Lemma E.20 and from Proposition E.22 we thus obtain that the irreducible representations of O(3) are precisely given by all $\\rho _ { + } \\otimes D _ { l }$ and $\\rho _ { - } \\otimes D _ { l }$ . We now show that $\\rho _ { - } \\otimes D _ { l }$ is equivalent to $D _ { l - }$ : We have ", + "bbox": [ + 173, + 675, + 825, + 746 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/bb95ef859c31ba0a85696006bb6c607089b76d1bfe26568cf53d6b80238b2787.jpg", + "text": "$$\n\\rho _ { - } \\otimes D _ { l } : \\operatorname { O } ( 3 ) \\to \\operatorname { G L } ( \\mathbb { C } \\otimes V _ { l } ) , \\ \\left[ ( \\rho _ { - } \\otimes D _ { l } ) ( s g ) \\right] ( z \\otimes v ) = s z \\otimes \\left[ D _ { l } ( g ) \\right] ( v ) .\n$$", + "text_format": "latex", + "bbox": [ + 238, + 751, + 759, + 771 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Now, consider the linear isomorphism $f : \\mathbb { C } \\otimes V _ { l } V _ { l + }$ , $z \\otimes v \\mapsto z v$ . We only need to check that it is equivariant and are then done: ", + "bbox": [ + 174, + 775, + 823, + 804 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/dd29db71451d6384a9f818335fdbdb55b7bf0998c6945bb30a25f8412670e9ff.jpg", + "text": "$$\n\\begin{array} { r l } & { f \\big ( \\left[ ( \\rho _ { - } \\otimes D _ { l } ) ( s g ) \\right] ( z \\otimes v ) \\big ) = f \\big ( s z \\otimes \\left[ D _ { l } ( g ) \\right] ( v ) \\big ) } \\\\ & { \\qquad = s z \\cdot \\left[ D _ { l } ( g ) \\right] ( v ) } \\\\ & { \\qquad = [ s D _ { l } ( g ) ] ( z v ) } \\\\ & { \\qquad = [ D _ { l - } ( s g ) ] ( f ( z \\otimes v ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 808, + 676, + 883 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "The statement about $D _ { l + }$ can be shown using the exact same map $f$ . ", + "bbox": [ + 173, + 888, + 624, + 904 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "34It is not a direct generalization due to the presence of two different group elements being applied. ", + "bbox": [ + 184, + 910, + 772, + 924 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "E.6.2 THE PETER-WEYL THEOREM FOR $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ AS REPRESENTATION OF O(3) ", + "bbox": [ + 173, + 102, + 733, + 119 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "The considerations in this section follow almost entirely from Section E.4.2. There we saw that, as a representation over SO(3), we have a decomposition ", + "bbox": [ + 173, + 127, + 823, + 157 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/c4e22ab4daba662109a3a12a911b9274a1d1c64ddcc79ab2ccc83811b8ab96bd.jpg", + "text": "$$\nL _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } ) = { \\widehat { \\bigoplus _ { l \\geq 0 } } } V _ { l 1 }\n$$", + "text_format": "latex", + "bbox": [ + 437, + 164, + 560, + 205 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "with the spaces $V _ { l 1 }$ being spanned by the spherical harmonics $Y _ { l } ^ { n }$ , $n = - l , \\ldots , l$ . We immediately see that in $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ , viewed as a representation over O(3), there is not enough space for all the irreducible representations, since they appear in pairs as shown in Proposition E.24.35 Thus, we need to figure out which irreducible representations are present and which are not. The core of this question is answered by the following proposition: ", + "bbox": [ + 173, + 213, + 825, + 285 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Lemma E.25 (Parity in spherical harmonics). The spherical harmonics obey the following parity rules: ", + "bbox": [ + 171, + 290, + 823, + 318 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/71ee9bb31ac6ff1411a2342b1acdec51002045386c08297fffe818faa0cd3393.jpg", + "text": "$$\nY _ { l } ^ { n } ( s x ) = s ^ { l } \\cdot Y _ { l } ^ { n } ( x )\n$$", + "text_format": "latex", + "bbox": [ + 426, + 318, + 571, + 337 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "for all $l \\geq 0$ , $n = - l , \\ldots , l$ , $s \\in \\mathbb { Z } _ { 2 }$ and $x \\in S ^ { 2 }$ . ", + "bbox": [ + 171, + 340, + 493, + 357 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Proof. This is a well-known property of the spherical harmonics. ", + "bbox": [ + 173, + 376, + 601, + 392 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Thus, together with Section E.4.2 we get the following transformation behavior of spherical harmonics under the group O(3), where $s \\in \\mathbb { Z } _ { 2 }$ and $g \\in \\mathrm { S O } ( 3 )$ : ", + "bbox": [ + 174, + 412, + 823, + 443 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/e514a156af4a7479380f09cb1dd12cb0482906861cba8ecb95cb940079486a8e.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\lambda ( s g ) ( Y _ { l } ^ { n } ) = s ^ { l } \\lambda ( g ) ( Y _ { l } ^ { n } ) } } \\\\ { { \\displaystyle = s ^ { l } \\sum _ { n ^ { \\prime } = - l } ^ { l } D _ { l } ^ { n ^ { \\prime } n } ( g ) Y _ { l } ^ { n ^ { \\prime } } } } \\\\ { { \\displaystyle = \\sum _ { n ^ { \\prime } = - l } ^ { l } \\left( s ^ { l } D _ { l } ^ { n ^ { \\prime } n } ( g ) \\right) Y _ { l } ^ { n ^ { \\prime } } } } \\\\ { { \\displaystyle = \\left\\{ \\sum _ { n ^ { \\prime } = - l } ^ { l } D _ { l + } ^ { n ^ { \\prime } n } ( s g ) Y _ { l } ^ { n ^ { \\prime } } , l \\mathrm { ~ e v e n } \\right. } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 449, + 648, + 604 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Thus, we obtain the following decomposition of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ : ", + "bbox": [ + 173, + 612, + 549, + 628 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/f2ab4dce96ffdf8c22ce263f20238cc302fd096505f21dc88130eb19a31226d2.jpg", + "text": "$$\nL _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } ) = \\widehat { \\bigoplus _ { l \\geq 0 } } V _ { l 1 + } \\oplus \\widehat { \\bigoplus _ { l \\geq 0 \\atop l \\mathrm { o d d } } } V _ { l 1 - } .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 635, + 609, + 685 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Here, $V _ { l 1 + }$ and $V _ { l 1 - }$ are generated from the spherical harmonics of order $l$ and we have $V _ { l 1 + } \\cong V _ { l + }$ and $V _ { l 1 - } \\cong V _ { l - }$ as representations according to the transformation behavior we saw above. ", + "bbox": [ + 171, + 691, + 823, + 720 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "E.6.3 THE CLEBSCH-GORDAN DECOMPOSITION ", + "bbox": [ + 173, + 737, + 524, + 752 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Remember from Section E.4.3 that we have a decomposition of SO(3)-representations ", + "bbox": [ + 173, + 762, + 740, + 777 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/07c06d3747681daab6bd593e4d3718f110eeebd3431a90d68b931d76a07e0272.jpg", + "text": "$$\nV _ { j } \\otimes V _ { l } \\cong \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J }\n$$", + "text_format": "latex", + "bbox": [ + 426, + 785, + 571, + 833 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "given by real Clebsch-Gordan coefficients. Now for O(3), remember that as vector spaces we have for all $j$ (and equally for $l$ and $J$ ) equalities $V _ { j } = V _ { j - } = V _ { j + }$ , and so we guess that in the isomorphism above, we just need to figure out the correct signs in order to be compatible with the ", + "bbox": [ + 173, + 840, + 825, + 885 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "corresponding representations. The idea is that “multiplying the signs at the left” should lead to the “sign at the right”, and this paradigm leads us to believe that there are the following isomorphisms: ", + "bbox": [ + 169, + 103, + 825, + 133 + ], + "page_idx": 88 + }, + { + "type": "equation", + "img_path": "images/d7d610c1ff4d8e85115a980ee7ff290f7da8fb8c440d157bd8e0c0a6c1dfb6d7.jpg", + "text": "$$\n\\begin{array} { l l } { { V _ { j + } \\otimes V _ { l + } \\cong \\displaystyle \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J + } , } } & { { V _ { j + } \\otimes V _ { l - } \\cong \\displaystyle \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J - } , } } \\\\ { { V _ { j - } \\otimes V _ { l + } \\cong \\displaystyle \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J - } , } } & { { V _ { j - } \\otimes V _ { l - } \\cong \\displaystyle \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J + } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 138, + 687, + 234 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "Wethat $f : V _ { j } \\otimes V _ { l } \\to \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J }$ rphism since the arguments are always the same. So, assumeis an isomorphism and thus in particular intertwines the given representations. Now, we take the exact same map $f : V _ { j - } \\otimes V _ { l + } \\to \\bigoplus _ { J = | l - j | } ^ { l + j } V _ { J - }$ and only need to figure out that it is equivariant with respect to the given representations, using the same property for the original isomorphism we started with: ", + "bbox": [ + 173, + 239, + 826, + 320 + ], + "page_idx": 88 + }, + { + "type": "equation", + "img_path": "images/078e304bf13a7f5ce8ed69f380b993213de917132fdef883eb2bfb690adf0ae2.jpg", + "text": "$$\n\\begin{array} { l } { { { \\displaystyle f \\circ \\left[ D _ { j - } ( s g ) \\otimes D _ { l + } ( s g ) \\right] = f \\circ \\left[ s ( D _ { j } ( g ) \\otimes D _ { l } ( g ) ) \\right] } } } \\\\ { { { } } } \\\\ { { { } = s \\bigoplus _ { { \\scriptstyle { J = | l - j | } \\atop { { \\scriptstyle l + j } } \\atop { { \\scriptstyle J = | l - j | } \\atop { { \\scriptstyle J = | l - j | } \\atop { { \\scriptstyle J = | l - j | } \\atop { { \\scriptstyle J = | l - j | } \\atop { { \\scriptstyle J = | l - j | } \\atop { { \\scriptstyle J = | l - j | } } } } } } } } } } } \\\\ { { { { } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 325, + 679, + 443 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "This shows the claim. From these considerations, it also follows that the Clebsch-Gordan coefficients do not in any way depend on the signs of the spaces $V _ { j } , V _ { l } , V _ { J }$ . Thus, we write them generically as $\\langle J M | j m ; l n \\rangle$ . ", + "bbox": [ + 176, + 448, + 823, + 491 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "E.6.4 ENDOMORPHISMS OF $V _ { J }$ ", + "text_level": 1, + "bbox": [ + 176, + 505, + 400, + 520 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "As always over C, Schur’s Lemma D.8 shows that the endomorphism spaces are 1-dimensional, and thus we can ignore endomorphisms. ", + "bbox": [ + 173, + 529, + 825, + 559 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "E.6.5 BRINGING EVERYTHING TOGETHER ", + "text_level": 1, + "bbox": [ + 174, + 573, + 480, + 588 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "Now we can finally compute the basis for steerable kernels. The section on the Clebsch-Gordan decomposition suggests that we need to do a case distinction for this. Namely, the possible kernels depend on the signs of $V _ { l }$ and $V _ { J }$ . The results basically follow analogously to the results in Section E.4.5. ", + "bbox": [ + 173, + 597, + 826, + 654 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : S ^ { 2 } \\operatorname { H o m } _ { \\mathbb { C } } ( V _ { l + } , V _ { J + } )$ : ", + "bbox": [ + 174, + 667, + 529, + 685 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "$V _ { J + }$ can only be in a tensor product $V _ { j } \\otimes V _ { l + }$ if the sign of $j$ is positive. Spaces $V _ { j 1 + }$ appear in the tensor product decomposition of $L _ { \\mathbb { C } } ^ { 2 } ( S ^ { 2 } )$ precisely for even $j$ , according to Section E.6.2. Thus, a basis for steerable kernels is given by all $K _ { j }$ with even $j \\in \\left\\{ | l - J | , \\ldots , l + J \\right\\}$ . It has matrix elements ", + "bbox": [ + 173, + 693, + 826, + 752 + ], + "page_idx": 88 + }, + { + "type": "equation", + "img_path": "images/7b572adce8c91c933fc3ec51efd76eeab51b1555fc5477c57fac3fd47925cb70.jpg", + "text": "$$\n\\langle J M | K _ { j } ( x ) | l n \\rangle = \\sum _ { m = - j } ^ { j } \\langle J M | j m ; l n \\rangle \\cdot \\overline { { Y _ { j } ^ { m } } } ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 334, + 747, + 661, + 794 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "exactly as in Section E.4.5. ", + "bbox": [ + 174, + 795, + 352, + 810 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : S ^ { 2 } \\operatorname { H o m } _ { \\mathbb { C } } ( V _ { l + } , V _ { J - } )$ : ", + "bbox": [ + 173, + 824, + 529, + 842 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "Analogously, a basis for steerable kernels is given by all $K _ { j }$ , with odd $j \\in \\left\\{ | l - J | , \\dots , l + J \\right\\}$ . ", + "bbox": [ + 173, + 849, + 803, + 868 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : S ^ { 2 } \\operatorname { H o m } _ { \\mathbb { C } } ( V _ { l - } , V _ { J + } )$ : ", + "bbox": [ + 173, + 882, + 529, + 900 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "Again, a basis for steerable kernels is given by all $K _ { j }$ with odd $j \\in \\left\\{ | l - J | , \\dots , l + J \\right\\}$ . ", + "bbox": [ + 171, + 907, + 758, + 926 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "STEERABLE KERNELS $K : S ^ { 2 } \\operatorname { H o m } _ { \\mathbb { C } } ( V _ { l - } , V _ { J - } )$ : ", + "bbox": [ + 174, + 102, + 529, + 118 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "As in the first case, a basis for steerable kernels is given by all $K _ { j }$ with even $j \\in \\left\\{ | l - J | , \\dots , l + J \\right\\}$ . ", + "bbox": [ + 173, + 128, + 823, + 145 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Thus, we have determined all kernel bases for the group O(3) over the complex numbers. Compared to SO(3), we see that the kernel spaces get roughly halved. The reason for this is that with a bigger symmetry group, the kernel needs to obey more rules, which means that the kernel constraint has fewer solutions. ", + "bbox": [ + 174, + 151, + 825, + 208 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "E.7 O(3)-STEERABLE KERNELS FOR REAL REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 226, + 622, + 239 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Basically, we can argue exactly as in Section E.5.4 in order to transport the results for complex representations over to the real world. We shortly sketch the procedure and outcome. As we know from Section E.3.1, $\\rho _ { - } : \\mathbb { Z } _ { 2 } \\to \\operatorname { O } ( \\mathbb { R } )$ and $\\rho _ { + } : \\mathbb { Z } _ { 2 } \\to \\operatorname { O } ( \\mathbb { R } )$ are the only irreducible real representations of $\\mathbb { Z } _ { 2 }$ . Thus, for each $l \\geq 0$ we obtain two irreducible real representations $^ { r } D _ { l + } : \\mathrm { O } ( 3 ) \\to \\mathrm { O } ( ^ { r } V _ { l + } )$ and ${ } ^ { r } D _ { l - } : \\mathrm { O ( 3 ) } \\mathrm { O } ( { } ^ { r } V _ { l - } )$ . As before, they also act on complex vector spaces and are as such isomorphic to the complex irreducible representations of O(3). One can then show as in Lemma E.17 that all complex irreducible representations are of real type since they split into two copies of the real version of this representation. Thus, by Corollary E.18, all real irreducible representations are of real type, and this means that we can proceed exactly as in Proposition E.19 in order to show that the $^ r D _ { l + }$ and ${ } ^ { r } D _ { l - }$ are already all the irreducible real representations of O(3) up to equivalence. ", + "bbox": [ + 173, + 251, + 825, + 405 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "For the Peter-Weyl decomposition of $L _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } )$ , we only need to note that the real spherical harmonics emerge with a base change from the complex ones, as seen in Eq. (31), and thus fulfill the same parity rules as the complex spherical harmonics. This gives us a decomposition ", + "bbox": [ + 174, + 411, + 825, + 454 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/93367b89df56d2e3a22c8410771cba41c8fa2ef3e33c30e4e8f54445f71bca8b.jpg", + "text": "$$\nL _ { \\mathbb { R } } ^ { 2 } ( S ^ { 2 } ) = \\widehat { \\bigoplus _ { l \\geq 0 \\atop l \\mathrm { e v e n } } } ( ^ { r } V _ { l 1 + } ) \\oplus \\widehat { \\bigoplus _ { l \\geq 0 \\atop l \\mathrm { o d d } } } ( ^ { r } V _ { l 1 - } ) .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 460, + 632, + 508 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "For the Clebsch-Gordan coefficients, we again get decompositions ", + "bbox": [ + 171, + 515, + 609, + 530 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/7ad099196293254a393c9784235ff51d276e316f3010f71254aa1f407bc85624.jpg", + "text": "$$\n{ } ^ { r } V _ { j } \\otimes { } ^ { r } V _ { l } \\cong \\bigoplus _ { J = | l - j | } { } ^ { r } V _ { J }\n$$", + "text_format": "latex", + "bbox": [ + 415, + 536, + 583, + 584 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "where the signs on the left must “multiply to” the signs on the right, as in Section E.6.3. Finally, the endomorphism spaces must be 1-dimensional since the endomorphism spaces of the complex versions are 1-dimensional. ", + "bbox": [ + 174, + 590, + 823, + 632 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Overall, we obtain the same kernels as in Section E.6.5, only that we need to use the real spherical harmonics as our steerable filters and can get rid of the complex conjugation. ", + "bbox": [ + 173, + 638, + 823, + 669 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "F MATHEMATICAL PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 174, + 689, + 483, + 705 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "In this chapter, we state mathematical preliminaries that we use throughout the earlier chapters. In this whole chapter, K is one of the two fields $\\mathbb { R }$ or C. ", + "bbox": [ + 174, + 720, + 825, + 750 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "F.1 TOPOLOGICAL SPACES, NORMED SPACES, AND METRIC SPACES ", + "bbox": [ + 174, + 767, + 660, + 781 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Since in this work, we want to develop the theory of representations over compact groups, and since this is a topological property, we need to formulate some topological concepts (Conway, 2014). Additionally, the vector spaces on which our compact groups act also carry a topology, mostly coming from their Hilbert space structure. ", + "bbox": [ + 174, + 792, + 825, + 849 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Definition F.1 (Topological Space, Open Sets, Closed Sets). A topological space $( X , \\tau )$ consists of a set $X$ and a set $\\tau$ of subsets of $X$ , called the open sets, such that arbitrary unions and finite intersections of open sets are open. In particular, $X$ and the empty set $\\varnothing$ are open. Closed sets are the complements of open sets and fulfill dual axioms: arbitrary intersections and finite unions of closed sets are closed. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Let in the following $X$ and $Y$ be topological spaces. ", + "bbox": [ + 176, + 103, + 514, + 118 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.2 (Open Neighborhood). Let $x \\in X$ . An open set $U \\subseteq X$ is called open neighborhood of $x$ if $x \\in U$ . ", + "bbox": [ + 173, + 121, + 823, + 150 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.3 (Hausdorff Space). $X$ is called a Hausdorff space if two distinct points can always be separated by open sets, i.e., for all $x , y \\in X$ there exist $U _ { x } , U _ { y }$ open such that $x \\in U _ { x } , y \\in U _ { y }$ , and $U _ { x } \\cap U _ { y } = \\emptyset$ . ", + "bbox": [ + 174, + 154, + 823, + 196 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "In this work, all topological spaces are Hausdorff. ", + "bbox": [ + 173, + 207, + 501, + 222 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.4 (Subspace). Assume $A \\subseteq X$ is a subset. Then the set $\\mathcal { T } _ { A } : = \\{ U \\cap A | U \\in \\mathcal { T } \\}$ is a topology for $A$ and thus makes $A$ a topological space as well. It is called a subspace of $X$ . ", + "bbox": [ + 174, + 224, + 825, + 253 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Whenever we consider a subset of a topological space, it is viewed as a topological space with this construction. ", + "bbox": [ + 173, + 263, + 823, + 292 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.5 (Closure, Density). For $A \\subseteq X$ , its closure $\\overline { { A } }$ is defined as the smallest closed subset of $X$ that contains $A$ . Equivalently, it is the intersection of all closed subsets of $X$ containing $A$ , which is closed by the axioms of a topology. $A$ is called dense in $X$ if ${ \\overline { { A } } } = X$ . ", + "bbox": [ + 173, + 296, + 825, + 338 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.6 (Continuous Function, Homeomorphism). A function $f : X \\to Y$ is called continuous if preimages of open sets are always open. Equivalently, for each point $x _ { 0 } \\in X$ and each open neighborhood $V$ of $f { \\bar { ( x _ { 0 } ) } }$ there is an open neighborhood $U$ of $x _ { 0 }$ such that $f ( U ) \\subseteq V$ . ", + "bbox": [ + 173, + 342, + 825, + 385 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "A homeomorphism is a continuous bijective function with a continuous inverse. ", + "bbox": [ + 176, + 391, + 694, + 406 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Note that compositions of continuous functions are continuous as well. ", + "bbox": [ + 174, + 416, + 640, + 431 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.7 (Open Cover, Compact Space). An open cover of $X$ is a family of open sets $\\{ U _ { i } \\} _ { i \\in I }$ that cover $X$ , i.e., $\\textstyle X = \\bigcup _ { i \\in I } U _ { i }$ . $X$ is called compact if all open covers have a finite subcover, that is: For all open covers $\\{ U _ { i } \\} _ { i \\in I }$ there exists a finite subset $J \\subseteq I$ such that $\\{ U _ { i } \\} _ { i \\in J }$ is still an open cover of $X$ . ", + "bbox": [ + 173, + 435, + 825, + 492 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Proposition F.8. If $X$ is compact and $f : X \\to Y$ is continuous, then $f ( X ) \\subseteq Y$ is compact as well. In particular, if $f$ surjective, then $Y$ is compact. ", + "bbox": [ + 173, + 494, + 823, + 525 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Proof. See Sutherland (1975), Proposition 13.15. ", + "bbox": [ + 173, + 539, + 500, + 554 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Proposition F.9. Let $f : X \\to Y$ be a continuous bijection and assume that $X$ is compact and that $Y$ is Hausdorff. Then the inverse $f ^ { - 1 }$ is continuous as well and thus $f$ is a homeomorphism. ", + "bbox": [ + 174, + 561, + 826, + 590 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Proof. See Sutherland (1975), Proposition 13.26. ", + "bbox": [ + 174, + 604, + 500, + 621 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.10 (Product Topology). The product topology on $X \\times Y$ is the coarsest (i.e., smallest in terms of inclusion) topology that makes both projections $p _ { X } : X \\times Y \\to X$ and $p _ { Y } : X \\times Y Y$ continuous. ", + "bbox": [ + 174, + 627, + 825, + 670 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "If $Z$ is a third topological space and we have two continuous functions $f _ { X } : Z \\to X$ and $f _ { Y } : Z \\to$ $Y$ , then the function $f _ { X } \\times f _ { Y } : Z \\to X \\times Y$ , $z \\mapsto ( f _ { X } ( z ) , f _ { Y } ( z ) )$ is continuous as well. ", + "bbox": [ + 173, + 681, + 823, + 712 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Definition F.11 (Quotient Map, Quotient Space). A continuous function $f : X \\to Y$ is called a quotient map if $f$ is surjective and if $U \\subseteq Y$ is open if and only if $f ^ { - 1 } ( U ) \\subseteq X$ is open. ", + "bbox": [ + 171, + 713, + 823, + 743 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Let $\\sim$ be any equivalence relation on $X$ and $X / \\sim$ be the quotient set formed by identifying equivalent elements. Let $q : X \\to X / { \\sim }$ be the canonical function sending each element to its equivalence class. We define $U \\subseteq X / \\sim$ to be open if $q ^ { - 1 } ( U ) \\subseteq X$ is open. Then $q$ is a quotient map and $X / \\sim$ is called a quotient space. ", + "bbox": [ + 173, + 748, + 825, + 805 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Proposition F.12 (Universal property of Quotient Maps). Let $q : X \\to X / { \\sim }$ be a standard quotient map and $f : X \\to Y$ be any continuous function such that $f ( x ) = f ( x ^ { \\prime } )$ whenever $x \\sim x ^ { \\prime }$ . Then there is a unique continuous function ${ \\overline { { f } } } : X / { \\sim } \\to Y$ such that the following diagram commutes: ", + "bbox": [ + 174, + 808, + 825, + 853 + ], + "page_idx": 90 + }, + { + "type": "equation", + "img_path": "images/6a83dfaa6c9ac517881f55222544d14a0bcfb4fce873e7593df8606aa8764fcf.jpg", + "text": "$$\n \\begin{array} { c c c } { { X \\stackrel { f } { \\longrightarrow } Y } } \\\\ { { q \\downarrow \\stackrel { } { } \\stackrel { \\overline { { { f } } } } { f } } } \\end{array} \n$$", + "text_format": "latex", + "bbox": [ + 447, + 861, + 552, + 924 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "$\\overline { { f } }$ is given on equivalence classes by ${ \\overline { { f } } } ( [ x ] ) = f ( x )$ . ", + "bbox": [ + 173, + 102, + 514, + 119 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Proof. See Conway (2014), Proposition 2.8.7. ", + "bbox": [ + 174, + 132, + 478, + 148 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "It can be shown that all quotient maps are equivalent to a construction of the form $q : X \\to X / { \\sim }$ . Namely, for a quotient map $f : X \\to Y$ , define $\\sim$ by $x \\sim x ^ { \\prime }$ if $f ( x ) = f ( x ^ { \\prime } )$ . Then the map ${ \\overline { { f } } } : X / { \\sim } \\to Y$ , $[ x ] \\mapsto f ( x )$ is a well-defined continuous map by the universal property of quotient maps Proposition F.12. One can show that this is a homeomorphism. Thus for a quotient map $f : X \\to Y$ we also call $Y$ a quotient space. ", + "bbox": [ + 173, + 162, + 825, + 236 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Our route for defining concrete topologies is in most cases through the existence of inner products on Hilbert spaces, which will be defined in detail in Definition F.32. Namely, inner products define norms, which define metrics (Kaplansky, 2001), which in turn define topologies. For this, we need some definitions: ", + "bbox": [ + 173, + 242, + 825, + 297 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Definition F.13 (Norm). Let $V$ be a $\\mathbb { K }$ -vector space, A norm on $V$ is a map $\\| \\cdot \\| : V \\to \\mathbb { R } _ { \\ge 0 }$ with the following properties for all $\\lambda \\in \\mathbb { K }$ and $v , w \\in V$ : ", + "bbox": [ + 169, + 301, + 823, + 330 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "1. $\\| v \\| = 0$ if and only if $v = 0$ . \n2. $\\| \\lambda v \\| = | \\lambda | \\cdot \\| v \\|$ . \n3. Triangle inequality: $\\| v + w \\| \\leq \\| v \\| + \\| w \\|$ . ", + "bbox": [ + 210, + 340, + 522, + 406 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "If $\\langle \\cdot | \\cdot \\rangle : V \\times V \\to \\mathbb { K }$ is an inner product on a Hilbert space, then it defines a norm $\\| \\cdot \\| : V \\to \\mathbb { R } _ { \\ge 0 }$ by $\\| x \\| : = { \\sqrt { \\langle x | x \\rangle } }$ . ", + "bbox": [ + 168, + 415, + 823, + 446 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Definition F.14 (Metric). Let $Y$ be a set. A metric on $Y$ is a function $d : Y \\times Y \\to \\mathbb { R } _ { \\geq 0 }$ with the following properties for all $x , y , z \\in Y$ : ", + "bbox": [ + 173, + 450, + 825, + 479 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "1. $d ( x , y ) = 0$ if and only if $x = y$ . \n2. Symmetry: $d ( x , y ) = d ( y , x )$ . \n3. Triangle inequality: $d ( x , z ) \\leq d ( x , y ) + d ( y , z )$ . ", + "bbox": [ + 210, + 488, + 553, + 554 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "A norm $\\| \\cdot \\| : V \\to \\mathbb { R } _ { \\ge 0 }$ defines a metric $d : V \\times V \\to \\mathbb { R }$ by setting $d ( x , y ) : = \\| x - y \\|$ . In turn, a metric defines a topology as follows: open balls are given by all sets of the form $\\mathrm { B } _ { \\epsilon } ( x ) : = \\{ y \\in$ $V \\mid d ( x , y ) < \\epsilon \\}$ for all $x \\in V$ and $\\epsilon > 0$ . Open sets are then defined as arbitrary unions of arbitrary open balls. ", + "bbox": [ + 178, + 563, + 815, + 619 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Additionally, we need notions about convergence in this work. Since we will deal with them mostly in the context of metric spaces (with normed vector spaces and Hilbert spaces being special cases, as explained above), we focus on these notions for metric spaces. ", + "bbox": [ + 174, + 626, + 823, + 669 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Definition F.15 (Convergent Sequence). Let $Y$ be a metric space. Then a sequence $( y _ { k } ) _ { k }$ in $Y$ is said to converge to $y$ if for all $\\epsilon > 0$ there is a $k _ { \\epsilon } \\in \\mathbb { N }$ such that $y _ { k } \\in \\mathrm { B } _ { \\epsilon } ( y )$ for all $k \\geq k _ { \\epsilon }$ . ", + "bbox": [ + 173, + 672, + 821, + 702 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "With this in mind, one can give an equivalent definition of continuity that applies to metric spaces: ", + "bbox": [ + 171, + 712, + 816, + 727 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Definition F.16 (Continuity in Metric Spaces). A function $f : Y \\to Z$ between metric spaces is continuous in $y \\in Y$ if for each sequence $( y _ { k } ) _ { k }$ of points $y _ { k } \\in Y$ converging to a point $y \\in Y$ , we also have that the sequence $f ( y _ { k } )$ converges to $f ( y )$ . This can be understood in terms of the function “commuting with limits”: ", + "bbox": [ + 173, + 729, + 825, + 786 + ], + "page_idx": 91 + }, + { + "type": "equation", + "img_path": "images/d6948e5da9515e74c23421d200365ec524dce6b78ad704072ad9e962c1e5a994.jpg", + "text": "$$\n\\operatorname* { l i m } _ { k \\to \\infty } f ( y _ { k } ) = f { \\big ( } \\operatorname* { l i m } _ { k \\to \\infty } y _ { k } { \\big ) } .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 791, + 591, + 818 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Furthermore, $f : Y \\to Z$ is called continuous if it is continuous in all points $y \\in Y$ . ", + "bbox": [ + 173, + 823, + 722, + 839 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Equivalently, the following holds: $f : Y \\to Z$ is continuous in $y \\in Y$ if and only of for all $\\epsilon > 0$ there is a $\\delta > 0$ such that $f \\left( \\mathrm { B } _ { \\delta } ( y ) \\right) \\subseteq \\mathrm { B } _ { \\epsilon } ( f ( y ) )$ . ", + "bbox": [ + 173, + 848, + 825, + 878 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Definition F.17 (Uniform Continuity). A function $f : Y \\to Z$ between metric spaces is called uniformly continuous if for each $\\epsilon > 0$ there is a $\\delta > 0$ such that for all $y , y ^ { \\prime } \\in Y$ with $d _ { Y } ( y , y ^ { \\prime } ) < \\delta$ we obtain $d _ { Y } ( f ( y ) , f ( y ^ { \\prime } ) ) < \\epsilon$ . ", + "bbox": [ + 174, + 881, + 825, + 925 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "The following is a result we use several times in the main text: ", + "bbox": [ + 173, + 103, + 583, + 118 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proposition F.18. Let $f : V \\to V ^ { \\prime }$ be a linear function between normed vector spaces. Then the following are equivalent: ", + "bbox": [ + 173, + 121, + 825, + 150 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "1. $f$ is uniformly continuous. ", + "bbox": [ + 214, + 160, + 406, + 176 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "2. $f$ is continuous. ", + "bbox": [ + 212, + 183, + 338, + 199 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "3. $f$ is continuous in 0. ", + "bbox": [ + 212, + 207, + 367, + 222 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proof. Trivially, 1 implies 2, which in turn implies 3. Now assume 3, i.e., $f$ is continuous in 0. Let $\\epsilon > 0$ . Then by continuity in 0, there exists $\\delta > 0$ such that for all $v \\in V$ with $\\| v \\| = \\| v - 0 \\| < \\delta$ we obtain $\\| f ( v ) \\| = \\| f ( v ) - f ( 0 ) \\| < \\epsilon$ . Now let $v , v ^ { \\prime } \\in V$ be arbitrary with $\\| v - v ^ { \\prime } \\| < \\delta$ . Then by the linearity of $f$ we obtain: ", + "bbox": [ + 173, + 237, + 825, + 292 + ], + "page_idx": 92 + }, + { + "type": "equation", + "img_path": "images/545729a3a7306a5d4bc6ffd42b3339a74dcbeac9960ada1c646781f5d236d86b.jpg", + "text": "$$\n\\| f ( v ) - f ( v ^ { \\prime } ) \\| = \\| f ( v - v ^ { \\prime } ) \\| < \\epsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 296, + 619, + 314 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "which is exactly what we wanted to show. ", + "bbox": [ + 176, + 318, + 449, + 332 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Sometimes, sequences look like they converge since their elements get ever closer to each other. However, not all such sequences need to converge. Therefore, there is the following notion: ", + "bbox": [ + 174, + 347, + 823, + 376 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Definition F.19 (Cauchy Sequence). Let $Y$ be a metric space. A sequence $( y _ { k } ) _ { k }$ in $Y$ is a Cauchy Sequence if for all $\\epsilon > 0$ there is $k _ { \\epsilon } \\in \\mathbb { N }$ such that for all $\\bar { k } , k ^ { \\prime } > k _ { \\epsilon }$ we have $d ( y _ { k } , y _ { k ^ { \\prime } } ) < \\epsilon$ . ", + "bbox": [ + 173, + 378, + 823, + 409 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "For example, one can consider the metric space $\\mathbb { R } \\backslash \\{ 0 \\}$ together with the usual metric. Then the sequence $\\left( { \\frac { 1 } { k } } \\right) _ { k }$ is a Cauchy sequence but does not converge since the limit (in R!), which would be 0, is not in $\\mathbb { R } \\stackrel { \\cdot } { | } \\{ 0 \\}$ . Thus, the following notion is useful: ", + "bbox": [ + 174, + 416, + 825, + 463 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Definition F.20 (Complete Metric Space). A metric space $Y$ is called complete if every Cauchy sequence converges. ", + "bbox": [ + 174, + 465, + 823, + 494 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Definition F.21 (Completion). Let $Y$ be a metric space. A completion of $Y$ is a metric space $Y ^ { \\prime }$ which contains $Y$ as a dense subspace and such that $Y ^ { \\prime }$ is complete. ", + "bbox": [ + 173, + 497, + 821, + 526 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proposition F.22 (Universal Property of Completions). Assume that $Y \\subseteq Y ^ { \\prime }$ is a pair of metric spaces, where $Y ^ { \\prime }$ is a completion of $Y$ . Then the following universal property holds: ", + "bbox": [ + 171, + 529, + 823, + 559 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Let $Z$ be any complete metric space and $f : Y \\to Z$ be any uniformly continuous function. Then there is a unique continuous function $f ^ { \\prime } : Y ^ { \\prime } \\to Z$ that extends $f$ , i.e., such that $f ^ { \\prime } | _ { Y } = f$ . $f ^ { \\prime }$ furthermore is also uniformly continuous. This can be expressed by the following commutative diagram, where $i : Y \\to Y ^ { \\prime }$ is the canonical inclusion: ", + "bbox": [ + 173, + 564, + 825, + 621 + ], + "page_idx": 92 + }, + { + "type": "equation", + "img_path": "images/a6f293a9d7b25937a1d5c274dbc38a2e942a36ce6d60cbea91b42a446eacab17.jpg", + "text": "$$\n\\begin{array} { l } { { Y \\xrightarrow { \\quad f \\quad } Z } } \\\\ { { \\Biggl \\downarrow i { \\Biggl \\downarrow } ^ { \\quad } } } \\\\ { { Y ^ { \\prime } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 454, + 626, + 544, + 688 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proof. See, for example, Kaplansky (2001). ", + "bbox": [ + 174, + 704, + 464, + 719 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Definition F.23 (Boundedness). Let $Y$ be a metric space. A subset $A \\subseteq Y$ is called bounded if there is a constant $C > 0$ such that $d ( a , b ) \\leq C$ for all $a , b \\in A$ . ", + "bbox": [ + 171, + 727, + 825, + 756 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Theorem F.24 (Heine-Borel Theorem). A subset $A \\subseteq \\mathbb { K } ^ { d }$ is compact if and only if it is closed and bounded. ", + "bbox": [ + 176, + 757, + 823, + 786 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proof. See Conway (2014), Theorem 1.4.8. ", + "bbox": [ + 174, + 801, + 462, + 816 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Corollary F.25 (Extreme Value Theorem). Let $f : X \\to \\mathbb { R }$ be continuous, where $X$ is any nonempty compact topological space. Then $f$ has a maximum and a minimum. ", + "bbox": [ + 173, + 823, + 825, + 853 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proof. By Proposition F.8, $f ( X ) ~ \\subseteq \\mathbb { R }$ is compact. By Theorem F.24 this means that $f ( X )$ is closed and bounded. Boundedness means that the supremum is finite and closedness means that the supremum must lie in $f ( X )$ , and consequently it is a maximum. For the minimum, the same arguments apply. □ ", + "bbox": [ + 174, + 866, + 825, + 925 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "F.2 LIMITS OF NETS AND APPROXIMATED DIRAC DELTA FUNCTIONS ", + "text_level": 1, + "bbox": [ + 173, + 103, + 660, + 118 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "In this section, we discuss “limits of nets”, where a net can be imagined as a sequence over an index set which may be “too big to be handled as a sequence over the natural numbers”. They appear in the formulation of Theorem C.7. This material can, for example, be found in (Conway, 2014). ", + "bbox": [ + 176, + 128, + 825, + 171 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Definition F.26 (Partially Ordered Set, Directed Set). Let $I$ be an index set and $\\leq$ a relation on it. $I = ( I , \\leq )$ is a partially ordered set if: ", + "bbox": [ + 169, + 176, + 823, + 205 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "1. $\\leq$ is reflexive, i.e., $i \\leq i$ for all $i \\in I$ ", + "bbox": [ + 214, + 215, + 475, + 232 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "2. $\\leq$ is antisymmetric, that is: $i \\leq j$ and $j \\le i$ together imply $i = j$ . ", + "bbox": [ + 210, + 239, + 661, + 257 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "3. $\\leq$ is transitive, that is: $i \\leq j$ and $j \\le k$ together imply $i \\leq k$ . ", + "bbox": [ + 209, + 263, + 633, + 280 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "A partially ordered set $I$ is called directed if for all $i , j \\in I$ there exists $k \\in I$ such that $i \\leq k$ and $j \\le k$ . ", + "bbox": [ + 173, + 290, + 825, + 320 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Example F.27. Clearly, the natural numbers together with the standard order relation form a directed set. ", + "bbox": [ + 173, + 324, + 825, + 352 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "An important example for our purposes is the following: let $Z$ be any topological space (for example, a homogeneous space $X$ of a compact group $G$ ) and $x \\in Z$ be any point. Furthermore, define $\\mathcal { U } _ { x }$ as the set of open neighborhoods of $x$ , i.e., open sets $U \\subseteq Z$ such that $x \\in U$ . On this set, we define $U \\leq V$ if $U \\supseteq V$ , i.e., by reversed inclusion. Then $( { { \\mathcal { U } } _ { x } } , \\le )$ is a directed set: ", + "bbox": [ + 174, + 358, + 825, + 416 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "1. Reflexivity is clear since $V \\supseteq V$ for all $V$ . ", + "bbox": [ + 214, + 426, + 511, + 443 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "2. Antisymmetry is clear since $U \\supseteq V$ and $V \\supseteq U$ together clearly imply $U = V$ . ", + "bbox": [ + 209, + 450, + 751, + 467 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "3. Transitivity is clear since $U \\supseteq V$ and $V \\supseteq W$ together clearly imply $U \\supseteq W$ . ", + "bbox": [ + 209, + 474, + 745, + 491 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "4. For directedness, let $U , V \\in \\mathcal { U } _ { x }$ . Define $W = U \\cap V$ . Then $W \\in \\mathcal { U } _ { x }$ and clearly $U \\supseteq W$ and $V \\supseteq W$ , which is what was to show. ", + "bbox": [ + 210, + 498, + 825, + 529 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Note that $\\mathcal { U } _ { x }$ is usually not totally ordered, i.e., there are usually $U , V \\in \\mathcal { U } _ { x }$ such that neither $U \\supseteq V$ nor $V \\supseteq U$ . ", + "bbox": [ + 174, + 539, + 823, + 569 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Definition F.28 (Net). Let $Z$ be any topological space and $I$ a directed set. Then a net in $Z$ is a function $x : I Z$ . We write a net as $( x _ { i } ) _ { i \\in I }$ , in analogy to sequences. ", + "bbox": [ + 173, + 571, + 823, + 602 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Definition F.29 (Convergence of Nets). Let $( x _ { i } ) _ { i \\in I }$ be a net in a topological space $Z$ . Let $x \\in Z$ . We say that $( x _ { i } ) _ { i \\in I }$ converges to $x$ , written $\\operatorname* { l i m } _ { i \\in I } x _ { i } \\ = \\ x$ , if the following holds: for all open neighborhoods $U$ of $x$ there is an $i _ { 0 } \\in I$ such that for all $i \\geq i _ { 0 }$ we have $x _ { i } \\in U$ . ", + "bbox": [ + 174, + 604, + 825, + 648 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Now we define the approximated Dirac delta for the special case that $X$ is a homogeneous space of a compact group $G$ . Remember that there is a Haar measure $\\mu$ on $X$ . ", + "bbox": [ + 174, + 659, + 820, + 688 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Definition F.30 (Approximated Dirac Delta). For $\\varnothing \\neq U \\subseteq X$ open, we define the approximated Dirac delta by $\\delta _ { U } : X \\to \\mathbb { K }$ with ", + "bbox": [ + 171, + 691, + 823, + 720 + ], + "page_idx": 93 + }, + { + "type": "equation", + "img_path": "images/93a59ebaddfbc1d87066fb330ee809629ce5df0481822e15e6a32957cc5cc9c6.jpg", + "text": "$$\n\\delta _ { U } ( x ) = { \\frac { 1 } { \\mu ( U ) } } \\cdot \\mathbf { 1 } _ { U } ( x ) = { \\left\\{ { \\frac { 1 } { \\mu ( U ) } } , \\ x \\in U \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 349, + 727, + 630, + 770 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "We have $\\delta _ { U } \\in L _ { \\mathbb { K } } ^ { 2 } ( X )$ . ", + "bbox": [ + 174, + 776, + 328, + 792 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "A priori, it is unclear that open sets have positive measure, which is needed for the well-definedness of this construction, since otherwise we divide by zero. Thus, we need the following lemma: ", + "bbox": [ + 173, + 804, + 823, + 833 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Lemma F.31. Let $\\varnothing \\neq U \\subseteq X$ be an open set. Then $\\mu ( U ) > 0$ . ", + "bbox": [ + 173, + 837, + 591, + 852 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Proof. Consider the family of open sets $( g U ) _ { g \\in G }$ . That all of these sets are necessarily open follows since the action $G \\times X \\to X$ is continuous, and thus by the definition of a group action, each $g \\in G$ induces a homeomorphism $X \\to X , x \\mapsto g x$ . Now, since the action is transitive, $( g U ) _ { g \\in G }$ is an open cover of $X$ , and since $X$ is compact, see Definition F.7, it has an open subcover $( g _ { i } U ) _ { i = 1 } ^ { n }$ with $g _ { i } \\in G$ . Note that $\\mu ( g _ { i } U ) = \\mu ( U )$ for all $i$ since the measure $\\mu$ on $X$ is by definition left invariant under the action of $G$ . Overall, we obtain ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 94 + }, + { + "type": "equation", + "img_path": "images/6f6ab8f5fbafd8287576740c48e093e597f8bcd90885f550c6f56532c31745c2.jpg", + "text": "$$\n1 = \\mu ( X ) = \\mu { \\Biggr ( } \\bigcup _ { i = 1 } ^ { n } g _ { i } U { \\Biggr ) } \\leq \\sum _ { i = 1 } ^ { n } \\mu ( g _ { i } U ) = \\sum _ { i = 1 } ^ { n } \\mu ( U ) = n \\cdot \\mu ( U )\n$$", + "text_format": "latex", + "bbox": [ + 279, + 135, + 717, + 176 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "and thus $\\textstyle \\mu ( U ) \\geq { \\frac { 1 } { n } } > 0$ . ", + "bbox": [ + 174, + 180, + 338, + 198 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "F.3 PRE-HILBERT SPACES AND HILBERT SPACES ", + "bbox": [ + 173, + 212, + 527, + 227 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Here, we state foundational concepts in the theory of Hilbert spaces (Debnath & Mikusinski, 2005). Definition F.32 (pre-Hilbert Space, Hilbert space). A pre-Hilbert space $V = ( V , \\langle \\cdot | \\cdot \\rangle )$ consists of the following data: ", + "bbox": [ + 174, + 237, + 823, + 285 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "1. A vector space $V$ over K. ", + "bbox": [ + 214, + 295, + 401, + 310 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "2. An inner product $\\langle \\cdot | \\cdot \\rangle : V \\times V \\to \\mathbb { K }$ , $( x , y ) \\mapsto \\langle x | y \\rangle$ . ", + "bbox": [ + 210, + 316, + 588, + 335 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "It has the following properties that hold for all $x , x ^ { \\prime } , y , y ^ { \\prime } \\in V , \\lambda \\in \\mathbb { K }$ : ", + "bbox": [ + 173, + 343, + 637, + 359 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "1. The inner product is conjugate linear in the first component: $\\langle x + x ^ { \\prime } | y \\rangle = \\langle x | y \\rangle + \\langle x ^ { \\prime } | y \\rangle$ and $\\langle \\lambda x | y \\rangle = { \\overline { { \\lambda } } } \\langle x | y \\rangle$ , where $\\bar { \\lambda }$ is the complex conjugate of $\\lambda$ . ", + "bbox": [ + 209, + 368, + 821, + 401 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "2. The inner product is linear in the second component: $\\langle x | y + y ^ { \\prime } \\rangle = \\langle x | y \\rangle + \\langle x | y ^ { \\prime } \\rangle$ and $\\langle x | \\lambda y \\rangle = \\lambda \\langle x | y \\rangle$ . ", + "bbox": [ + 209, + 407, + 825, + 439 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "3. The inner product is conjugate symmetric: $\\langle y | x \\rangle = { \\overline { { \\langle x | y \\rangle } } }$ ", + "bbox": [ + 210, + 446, + 611, + 464 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "4. The inner product is positive definite: $\\langle x | x \\rangle > 0$ unless $x = 0$ . ", + "bbox": [ + 210, + 472, + 642, + 488 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "If additionally, the following statement holds, then $V$ is called a Hilbert Space: ", + "bbox": [ + 173, + 497, + 691, + 512 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "5. $V$ , together with the norm $\\| \\cdot \\| : V \\to V$ induced from the inner product by $\\| x \\| : = { \\sqrt { \\langle x | x \\rangle } }$ , and consequently the metric defined by $d ( x , y ) : = \\| x - y \\|$ , is a complete metric space as in Definition F.20. ", + "bbox": [ + 210, + 523, + 825, + 566 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Remark F.33. Of course, all Hilbert Spaces are pre-Hilbert spaces, and so all Propositions about pre-Hilbert spaces in the following apply to Hilbert spaces just as well. ", + "bbox": [ + 174, + 569, + 823, + 599 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Note that the first property follows from the second and third. We also mention that usually, inner products on Hilbert spaces are assumed to be linear in the first and conjugate linear in the second component, in contrast to how we view it. The reason for our choice is that our work is inspired by connections to physics where our convention is more common. It is basically the bra-ket convention. Furthermore, note that if $\\mathbb { K } = \\mathbb { R }$ , then conjugate linear maps are linear and thus the inner product will be linear in both components. Additionally, it will be symmetric instead of only conjugate symmetric. ", + "bbox": [ + 173, + 604, + 825, + 703 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Proposition F.34 (Cauchy-Schwartz Inequality). For any two elements $v , w$ in a pre-Hilbert space $V$ , we have ", + "bbox": [ + 171, + 705, + 821, + 733 + ], + "page_idx": 94 + }, + { + "type": "equation", + "img_path": "images/48b7f27dcd58fd1defe137f871e74a885aca7a4a6656d522305b265d64eadcb9.jpg", + "text": "$$\n| \\left. v | w \\right. | \\leq \\| v \\| \\cdot \\| w \\| .\n$$", + "text_format": "latex", + "bbox": [ + 423, + 732, + 571, + 750 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "We have equality if and only if v and w are linearly dependent. ", + "bbox": [ + 173, + 751, + 586, + 766 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Proof. See Debnath & Mikusinski (2005), Theorem 3.2.9. ", + "bbox": [ + 173, + 780, + 558, + 796 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Definition F.35 (Orthogonality). Two vectors $v , w$ in a pre-Hilbert space $V$ are called orthogonal, written $v \\perp w$ , i $\\mathbf { f } \\langle v | w \\rangle = 0$ . ", + "bbox": [ + 171, + 803, + 823, + 833 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Obviously, being orthogonal is a symmetric relation. ", + "bbox": [ + 176, + 842, + 516, + 857 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Definition F.36 (Orthogonal Complement). Let $V$ be a pre-Hilbert space and $W \\subseteq V$ a subset. \n$v \\in V$ is orthogonal to $W$ if $\\langle v \\mid w \\rangle = 0$ for all $w \\in W$ . ", + "bbox": [ + 173, + 859, + 820, + 890 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "The orthogonal complement of $W$ , denoted $W ^ { \\perp }$ , is the set of all vectors in $V$ that are orthogonal to $W$ . ", + "bbox": [ + 173, + 895, + 823, + 922 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Proposition F.37 (Closedness of Complements). Let $W \\subseteq V$ be a subset of a pre-Hilbert space $V$ Then $W ^ { \\perp }$ is a topologically closed linear subspace of $V$ . ", + "bbox": [ + 171, + 102, + 821, + 133 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proof. See Debnath & Mikusinski (2005), Theorem 3.6.2. ", + "bbox": [ + 173, + 146, + 558, + 162 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proposition F.38 (Continuity of Scalar Product). For any pre-Hilbert space $V$ , the scalar product $\\langle \\cdot | \\cdot \\rangle : V \\times V \\to \\mathbb { K }$ is continuous. ", + "bbox": [ + 173, + 169, + 826, + 199 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proof. See Debnath & Mikusinski (2005), Theorem 3.3.12. ", + "bbox": [ + 173, + 213, + 565, + 229 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Definition F.39 (Orthonormal System). A family $( v _ { i } ) _ { i \\in I }$ of elements in a pre-Hilbert space is called orthonormal system if $\\lVert \\boldsymbol { v } _ { i } \\rVert = 1$ for all $i \\in I$ and $v _ { i } \\perp v _ { j }$ for all $i \\neq j$ . ", + "bbox": [ + 173, + 236, + 825, + 266 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Definition F.40 (Orthonormal Basis). An orthonormal system $( v _ { i } ) _ { i \\in I }$ in a Hilbert space $V$ is called orthonormal basis if the linear span of all $\\{ v _ { i } \\} _ { i \\in I }$ is dense in $V$ . If this is the case, then each $v \\in V$ can be uniquely written as ", + "bbox": [ + 173, + 268, + 826, + 310 + ], + "page_idx": 95 + }, + { + "type": "equation", + "img_path": "images/3fcbda141e4a138a77490e722fe1afad2b8f3e04cf86003fd894feb87256b8e9.jpg", + "text": "$$\nv = \\sum _ { i \\in I } \\alpha _ { i } v _ { i }\n$$", + "text_format": "latex", + "bbox": [ + 454, + 309, + 544, + 343 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "with only countably many $\\alpha _ { i } \\in \\mathbb { K }$ being nonzero. The coefficients are given by $\\alpha _ { i } = \\langle v _ { i } | v \\rangle$ ", + "bbox": [ + 171, + 345, + 777, + 361 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "We stress that while the index set $I$ can be uncountably infinite, the sequence expansions of each element in $V$ only have countably many entries. It is obvious from the Peter-Weyl Theorem B.22 and this definition that the functions ", + "bbox": [ + 173, + 371, + 826, + 414 + ], + "page_idx": 95 + }, + { + "type": "equation", + "img_path": "images/18a56a08e9265560f864f599689dd86c24215791774edb78d82804c53fd95dcb.jpg", + "text": "$$\n\\left\\{ Y _ { l i } ^ { m } \\mid l \\in \\widehat { G } , i \\in \\{ 1 , \\ldots , m _ { l } \\} , m \\in \\{ 1 , \\ldots , d _ { l } \\} \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 417, + 663, + 444 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "form an orthonormal basis of $L _ { \\mathbb { K } } ^ { 2 } ( X )$ ", + "bbox": [ + 174, + 450, + 423, + 467 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proposition F.41 (Gram-Schmidt Orthonormalization). For every linearly independent sequence $( y _ { k } ) _ { k }$ in a pre-Hilbert space $V$ with $N \\in \\mathbb { N } \\cup \\{ \\infty \\}$ elements, one can find an orthonormal sequence $( v _ { k } ) _ { k }$ in $V$ such that the following holds: for all $n \\in \\mathbb { N }$ , $n \\leq N$ , the progressive linear span stays the same: ", + "bbox": [ + 173, + 469, + 825, + 523 + ], + "page_idx": 95 + }, + { + "type": "equation", + "img_path": "images/9845b01e82ed9016da3ef40e052d0310907531717af819d2db00981faa1a3ee7.jpg", + "text": "$$\n\\operatorname { s p a n } _ { \\mathbb { K } } ( v _ { 1 } , \\dots , v _ { n } ) = \\operatorname { s p a n } _ { \\mathbb { K } } ( y _ { 1 } , \\dots , y _ { n } ) .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 523, + 637, + 540 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "In particular, since every finite-dimensional Hilbert space has a vector space basis, it necessarily also has an orthonormal basis. ", + "bbox": [ + 174, + 542, + 825, + 571 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proof. See Debnath & Mikusinski (2005), page 110. ", + "bbox": [ + 173, + 585, + 521, + 602 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Definition F.42 (Adjoint of an Operator). Let $f : V \\to V ^ { \\prime }$ be a continuous linear function between Hilbert spaces. Then there is a unique continuous linear function $f ^ { * } : V ^ { \\prime } \\to V$ such that for all $v \\in V$ and $v ^ { \\prime } \\in V ^ { \\prime }$ one has: ", + "bbox": [ + 173, + 608, + 825, + 650 + ], + "page_idx": 95 + }, + { + "type": "equation", + "img_path": "images/29066d13a6f7756f7d8b2104c59a2242d3967991e2cebe2c1358637bd7a18d71.jpg", + "text": "$$\n\\langle f ( v ) | v ^ { \\prime } \\rangle _ { V ^ { \\prime } } = \\langle v | f ^ { * } ( v ^ { \\prime } ) \\rangle _ { V } .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 648, + 594, + 667 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "$f ^ { * }$ is called the adjoint of $f$ ", + "bbox": [ + 176, + 669, + 361, + 684 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "The existence of adjoints is, for example, discussed in Debnath & Mikusinski (2005), page 158. This book only considers the case of operators on a Hilbert space to itself, but these considerations generalize to the setting with two different Hilbert spaces. One has the following: ", + "bbox": [ + 174, + 695, + 825, + 738 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proposition F.43. Let $f : V \\to V ^ { \\prime }$ and $g : V ^ { \\prime } \\to V ^ { \\prime \\prime }$ be continuous linear functions between Hilbert spaces. Then: ", + "bbox": [ + 173, + 739, + 823, + 770 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "1. $( f ^ { * } ) ^ { * } = f .$ . \n2. $\\mathrm { i d } _ { V } ^ { * } = \\mathrm { i d } _ { V }$ . \n3. $( g \\circ f ) ^ { * } = f ^ { * } \\circ g ^ { * } .$ . ", + "bbox": [ + 210, + 780, + 362, + 845 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proof. All of these properties follow directly from the uniqueness of adjoints. ", + "bbox": [ + 174, + 858, + 684, + 875 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proposition F.44. Let $f : V \\to V ^ { \\prime }$ be a unitary transformation between Hilbert spaces, i.e., an invertible linear function such that $\\langle f ( v ) | f ( w ) \\rangle = \\langle v | \\bar { w } \\rangle$ for all $v , w \\in V$ . Then the adjoint is the inverse, i.e., $f ^ { * } = f ^ { - 1 }$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 95 + }, + { + "type": "text", + "text": "Proof. First of all, the inverse $f ^ { - 1 }$ is again continuous due to the unitarity of $f$ . Furthermore, due to the unitarity, we obtain ", + "bbox": [ + 173, + 102, + 825, + 132 + ], + "page_idx": 96 + }, + { + "type": "equation", + "img_path": "images/77e3c89a249908c73233dc48964ae94edcbd13a624a7f946d20bdda44672101c.jpg", + "text": "$$\n\\begin{array} { r } { \\langle f ( v ) | v ^ { \\prime } \\rangle = \\langle f ( v ) \\big | f ( f ^ { - 1 } ( v ^ { \\prime } ) ) \\rangle } \\\\ { = \\langle v \\big | f ^ { - 1 } ( v ^ { \\prime } ) \\rangle } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 392, + 132, + 606, + 175 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "for all $v \\in V$ and $v ^ { \\prime } \\in V ^ { \\prime }$ . Due to the uniqueness of adjoints, we obtain $f ^ { - 1 } = f ^ { * }$ ", + "bbox": [ + 173, + 176, + 714, + 193 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "The following proposition is sometimes used in the main text: ", + "bbox": [ + 174, + 207, + 580, + 222 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Proposition F.45. Let $v , w \\in V$ be two elements in a pre-Hilbert space such that $\\langle v | u \\rangle = \\langle w | u \\rangle$ for all $u \\in V$ . Then $v = w$ . ", + "bbox": [ + 176, + 224, + 825, + 252 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Proof. We have ", + "bbox": [ + 174, + 267, + 281, + 282 + ], + "page_idx": 96 + }, + { + "type": "equation", + "img_path": "images/eac47176193b0776fa52e4bfb11f8d9e5f0e3b4360c4b3cc25c0f8ec77d76684.jpg", + "text": "$$\n\\langle v - w | u \\rangle = \\langle v | u \\rangle - \\langle w | u \\rangle = 0\n$$", + "text_format": "latex", + "bbox": [ + 390, + 280, + 607, + 297 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "for all $u \\in V$ . In particular, when setting $u = v - w$ we obtain ", + "bbox": [ + 173, + 299, + 586, + 313 + ], + "page_idx": 96 + }, + { + "type": "equation", + "img_path": "images/443497c90b6e6fb27e2ceef497d10a4ae9b53ef0b75102f8abf875ade0503381.jpg", + "text": "$$\n\\langle v - w | v - w \\rangle = 0\n$$", + "text_format": "latex", + "bbox": [ + 433, + 316, + 565, + 333 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "and thus $v - w = 0$ , i.e., $v = w$ ", + "bbox": [ + 174, + 335, + 387, + 351 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Proposition F.46 (Orthogonal Projection Operators). Let $W \\subseteq V$ be a topologically closed subspace of a Hilbert space. Then there is a continuous linear function $P : V W$ such that for all $v \\in V$ and $w \\in W$ we have ", + "bbox": [ + 173, + 358, + 825, + 398 + ], + "page_idx": 96 + }, + { + "type": "equation", + "img_path": "images/9a0e57df60f215167690f1d6b13eb41908eb25642e0afdea2553169a8fdcabb8.jpg", + "text": "$$\n\\langle P ( v ) | w \\rangle = \\langle v | w \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 398, + 565, + 415 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Furthermore, if $W$ is finite-dimensional and $w _ { 1 } , \\ldots , w _ { n }$ and orthonormal basis, then $P$ is given explicitly by ", + "bbox": [ + 173, + 416, + 823, + 443 + ], + "page_idx": 96 + }, + { + "type": "equation", + "img_path": "images/8745b2c4398a83cad0658d4a7e57fc6ad2f268e8d87ad99a0b88f24c9ffd985b.jpg", + "text": "$$\nP ( v ) = \\sum _ { i = 1 } ^ { n } \\left. w _ { i } | v \\right. w _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 421, + 441, + 575, + 483 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Proof. That $W$ is topologically closed means that $W$ , with the scalar product inherited from $V$ , is a complete metric space. Thus, $W$ is a Hilbert space as well. Therefore, the continuous linear embedding $i : W \\to V$ given by $w \\mapsto w$ has an adjoint $i ^ { * } : V W$ by Definition F.42. Set $P : = i ^ { * }$ . For arbitrary $v \\in V$ and $w \\in W$ we obtain: ", + "bbox": [ + 173, + 496, + 826, + 553 + ], + "page_idx": 96 + }, + { + "type": "equation", + "img_path": "images/93fb59ea59b72a85b1eec72db383d21bc7bd179aef0c6369542ec25680892565.jpg", + "text": "$$\n\\begin{array} { c } { { \\langle P ( v ) | w \\rangle = \\langle i ^ { * } ( v ) | w \\rangle } } \\\\ { { { } } } \\\\ { { = \\langle v | i ( w ) \\rangle } } \\\\ { { { } } } \\\\ { { { } = \\langle v | w \\rangle . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 421, + 555, + 575, + 608 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "For the second statement, note that for all $j \\in \\{ 1 , \\ldots , n \\}$ we have, using that the $w _ { i }$ are orthonormal: ", + "bbox": [ + 173, + 609, + 823, + 626 + ], + "page_idx": 96 + }, + { + "type": "equation", + "img_path": "images/3979826e91a07512477782bb74ea7904cdbdbfd28078a9a444b23a17d75d5e92.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left. \\sum _ { i = 1 } ^ { n } \\left. w _ { i } | v \\right. w _ { i } \\middle | w _ { j } \\right. = \\sum _ { i = 1 } ^ { n } \\overline { { \\left. w _ { i } | v \\right. } } \\left. w _ { i } | w _ { j } \\right. } \\\\ & { \\qquad = \\langle v | w _ { j } \\rangle } \\\\ & { \\qquad = \\langle P ( v ) | w _ { j } \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 628, + 663, + 693 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "By Proposition F.45 and since the $w _ { j }$ generate $W$ we obtain $\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } \\left. w _ { i } | v \\right. w _ { i } = P ( v ) } \\end{array}$ as claimed. ", + "bbox": [ + 176, + 694, + 820, + 723 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Proposition F.47. Let $\\left( V , \\langle \\cdot | \\cdot \\rangle \\right)$ be a finite-dimensional pre-Hilbert space. Then this space is already complete and thus a Hilbert space. ", + "bbox": [ + 173, + 731, + 823, + 761 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "In particular, all finite-dimensional subspaces of Hilbert spaces are topologically closed. ", + "bbox": [ + 173, + 766, + 754, + 782 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Proof. The proof of the Gram-Schmidt orthonormalization in Proposition F.41 does not make use of the completeness of the Hilbert space, and thus it holds for pre-Hilbert spaces as well. Consequently, $V$ , being finite-dimensional, has an orthonormal basis. It is thus isomorphic to $\\mathbb { K } ^ { n }$ together with the standard scalar product, which is well-known to be complete. Thus, $V$ is a Hilbert space. ", + "bbox": [ + 173, + 795, + 825, + 853 + ], + "page_idx": 96 + }, + { + "type": "text", + "text": "Now, let $W \\subseteq V$ be a finite-dimensional subspace of a Hilbert space $V$ which may be infinitedimensional. Then $W$ is a pre-Hilbert space and by what was just shown a Hilbert space. Consequently, all sequences in $W$ which have a limit in $V$ need, by completeness, to have that limit already in $W$ . This shows that $W$ is topologically closed. □ ", + "bbox": [ + 174, + 859, + 825, + 916 + ], + "page_idx": 96 + } +] \ No newline at end of file diff --git a/parse/train/ajOrOhQOsYx/ajOrOhQOsYx_model.json b/parse/train/ajOrOhQOsYx/ajOrOhQOsYx_model.json new file mode 100644 index 0000000000000000000000000000000000000000..245c2854287bbabee2364b4e9fca6f9be37bc7c6 --- /dev/null +++ b/parse/train/ajOrOhQOsYx/ajOrOhQOsYx_model.json @@ -0,0 +1,200098 @@ +[ + { + "layout_dets": [ + { + "category_id": 1, + "poly": [ + 398, + 657, + 1301, + 657, + 1301, + 1113, + 398, + 1113 + ], + "score": 0.983 + }, + { + "category_id": 1, + "poly": [ + 298, + 1508, + 1403, + 1508, + 1403, + 1813, + 298, + 1813 + ], + "score": 0.981 + }, + { + "category_id": 1, + "poly": [ + 299, + 1247, + 1404, + 1247, + 1404, + 1492, + 299, + 1492 + ], + "score": 0.979 + }, + { + "category_id": 1, + "poly": [ + 299, + 1828, + 1401, + 1828, + 1401, + 1982, + 299, + 1982 + ], + "score": 0.969 + }, + { + "category_id": 0, + 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\\overline { { \\langle v ^ { i ^ { \\prime } } | \\rho _ { i } ( g ^ { \\prime } ) ( v ^ { j } ) \\rangle } } } \\\\ & { = \\sum _ { i ^ { \\prime } } \\overline { { \\langle v ^ { \\downarrow } | \\rho _ { i } ( g ^ { - 1 } ) ( v ^ { i ^ { \\prime } } ) \\rangle } } \\overline { { \\rho _ { i } ^ { \\prime } \\langle j ^ { \\prime } \\rangle } } } \\\\ & { = \\left( \\sum _ { i ^ { \\prime } } \\rho _ { i ^ { \\prime } } ^ { i i ^ { \\prime } } ( g ^ { - 1 } ) \\rho _ { i ^ { \\prime } } ^ { i ^ { \\prime } } \\right) ( g ^ { \\prime } ) } \\end{array}" + }, + { + "category_id": 14, + "poly": [ + 585, + 1581, + 1110, + 1581, + 1110, + 1924, + 585, + 1924 + ], + "score": 0.95, + "latex": "\\begin{array} { r l } & { f _ { l } ^ { j } ( \\rho _ { l } ( g ) ( v ^ { i } ) ) = f _ { l } ^ { j } \\left( \\sum _ { i ^ { \\prime } } \\langle v ^ { i } | \\rho _ { l } ( g ) ( v ^ { i } ) \\rangle v ^ { i ^ { \\prime } } \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad 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1525.0, + 1404.0, + 1574.0, + 1267.0, + 1574.0 + ], + "score": 1.0, + "text": "" + }, + { + "category_id": 15, + "poly": [ + 1370.0, + 1569.5, + 1405.0, + 1569.5, + 1405.0, + 1590.5, + 1370.0, + 1590.5 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 96, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/rygG4AVFvH/rygG4AVFvH.md b/parse/train/rygG4AVFvH/rygG4AVFvH.md new file mode 100644 index 0000000000000000000000000000000000000000..8305aa916dcc14188d21f510fb12d1e492d68933 --- /dev/null +++ b/parse/train/rygG4AVFvH/rygG4AVFvH.md @@ -0,0 +1,334 @@ +# CHAMELEON: ADAPTIVE CODE OPTIMIZATION FOR EXPEDITED DEEP NEURAL NETWORK COMPILATION + +Byung Hoon $\mathbf { A } \mathbf { h } \mathbf { n } ^ { 1 }$ , Prannoy Pilligundla1, Amir Yazdanbakhsh2, Hadi Esmaeilzadeh1 + +1 University of California, San Diego +2 Google Research +bhahn@eng.ucsd.edu, ppilligu@eng.ucsd.edu, ayazdan@google.com +hadi@eng.ucsd.edu + +# ABSTRACT + +Achieving faster execution with shorter compilation time can foster further diversity and innovation in neural networks. However, the current paradigm of executing neural networks either relies on hand-optimized libraries, traditional compilation heuristics, or very recently genetic algorithms and other stochastic methods. These methods suffer from frequent costly hardware measurements rendering them not only too time consuming but also suboptimal. As such, we devise a solution that can learn to quickly adapt to a previously unseen design space for code optimization, both accelerating the search and improving the output performance. This solution dubbed CHAMELEON leverages reinforcement learning whose solution takes fewer steps to converge, and develops an adaptive sampling algorithm that not only focuses on the costly samples (real hardware measurements) on representative points but also uses a domain-knowledge inspired logic to improve the samples itself. Experimentation with real hardware shows that CHAMELEON provides $4 . 4 5 \times$ speed up in optimization time over AutoTVM, while also improving inference time of the modern deep networks by $5 . 6 \%$ . + +# 1 INTRODUCTION + +The enormous computational intensity of Deep Neural Networks (DNNs) have resulted in developing either hand-optimized kernels, such as NVIDIA cuDNN or Intel MKL that serve as backend for a variety of programming environment such as TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2019). However, the complexity of the tensor operations in DNNs and the volatility of algorithms, which has led to unprecedented rate of innovation (LeCun, 2019), calls for developing automated compilation frameworks. To imitate or even surpass the success of hand-optimized libraries, recent research has developed stochastic optimization passes: for general code, STOKE (Schkufza et al., 2013), and neural network code, TVM (Chen et al., 2018a) and TensorComprehensions (Vasilache et al., 2018). TVM and TensorComprehensions are based on random or genetic algorithms to search the space of optimized code for neural networks. AutoTVM (Chen et al., 2018b) builds on top of TVM and leverage boosted trees (Chen & Guestrin, 2016) as part of the search cost model to avoid measuring the fitness of each solution (optimized candidate neural network code), and instead predict its fitness. However, even with these innovations the optimizing compilation time can be around 10 hours for ResNet-18 (He et al., 2016), and even more for deeper or wider networks. + +Since the general objective is to unleash new possibilities by developing automatic optimization passes, long compilation time hinders innovation and could put the current solutions in a position of questionable utility. To solve this problem, we first question the very statistical guarantees which the aforementioned optimization passes rely on. The current approaches are oblivious to the patterns in the design space of schedules that are available for exploitation, and causes inefficient search or even converges to solutions that may even be suboptimal. Also, we notice that current approaches rely on greedy sampling that neglects the distribution of the candidate solutions (configurations). While greedy sampling that passively filter samples based on the fitness estimations from the cost models work, many of their hardware measurements (required for optimization) tend to be redundant and wasteful. Moreover, we found that current solutions that rely on greedy sampling lead to significant fractions of the candidate configurations being redundant over iterations, and that any optimizing compiler are prone to invalid configurations which significantly prolongs the optimization time. As such, this work sets out to present an Adaptive approach dubbed CHAMELEON to significantly reduce the compilation time and offer automation while avoiding dependence to hand-optimization, enabling far more diverse tensor operations in the next generation DNNs. We tackle this challenge from two fronts with the following contributions: + +(1) Devising an Adaptive Exploration module that utilizes reinforcement learning to adapt to unseen design space of new networks to reduce search time yet achieve better performance. (2) Proposing an Adaptive Sampling algorithm that utilizes clustering to adaptively reduce the number of costly hardware measurements, and devising a domain-knowledge inspired Sample Synthesis to find configurations that would potentially yield better performance. + +Real hardware experimentation with modern DNNs (AlexNet, VGG-16, and ResNet-18) on a highend GPU (Titan $\mathrm { X p }$ ), shows that the combination of these two innovations, dubbed CHAMELEON, yields $4 . 4 5 \times$ speedup over the leading framework, AutoTVM. CHAMELEON is publicly available in the project page: https://bitbucket.org/act-lab/chameleon. + +# 2 CHALLENGES IN DEEP NEURAL NETWORK COMPILATION + +The general life-cycle of deep learning models from its birth to deployment comprises of two major stages. First stage is the designing and the training of a deep learning model by a research scientist, with the primary goal of achieving the highest feasible accuracy. Then, with a general demand to enable the intelligence on a wide range of devices (from mobile CPUs in the edge to cloud-scale GPUs), the second stage has emerged for the deployment of the pre-trained deep learning model to a target hardware by a deployment engineer. These stages are each iterative processes: research scientists iterate until it reaches the target performance in terms of accuracy whereas the deployment engineers iterate until the performance in terms of inference speed with a given hardware satisfies the given constraints. Importantly, these two stages are most often separate processes, and this paper mainly focuses on the second stage (deployment) of the cycle with an overarching goal of accelerating the overall deployment cycle by reducing the optimizing compilation time without compromising the performance of the output code. + +# 2.1 COMPILATION WORKFLOW FOR DEEP NEURAL NETWORKS + +![](images/4c1eda16cc68693383186b7adabf42830ab5f161819979c3ab152bb6ba67eb96.jpg) +Figure 1: Overview of our model compilation workflow, and highlighted is the scope of this work. + +Figure 1 illustrates how a compiler for DNNs takes an input model $\mathcal { M }$ and emits an optimized code $\tau ( \bar { \Theta } ^ { * } )$ that runs the model efficiently on a given hardware. This flow is commensurate with TensorComprehensions (Vasilache et al., 2018) and TVM (Chen et al., 2018a), using which we implement CHAMELEON that is available as a separate package for adoption in even other frameworks. The first phase of the workflow is the frontend compiler which performs the translation from the compiler and applies target-independent and white-box target-dependent optimizations that do not incorporate a measure of runtime. Target-independent passes transform the input DNN model without specificity to the target hardware. Operator fusion and data layout transformation in TVM are some examples of these passes, which lie in the same category as dead-code elimination or loop-invariant code motion in GCC (Stallman & DeveloperCommunity, 2009) or LLVM (Lattner & Adve, 2004). Target-dependent passes, on the other hand, the compiler takes the hardware architecture (target) into account while optimizing the program; however, this also does not actively leverage runtime measures. The last stage is a black-box optimization pass, called optimizing compiler, that given a measure of performance at runtime from the hardware can further optimize the code. CHAMELEON falls in this class by offering an optimizing compiler that adapts to different design space to be more swift in optimizing deep neural networks compared to conventional approaches. + +Table 1: Knobs in the design space to optimize convolution. + +
KNOBSDEFINITION
tile_f, tile-y, tile_xFactors for tiling and binding # of filters height,and width of feature maps.
tile_rc,tile_ry, tile_rxFactors for tiling reduction axis such as # of channels,height,and width of filters.
auto_unroll_max_stepThreshold of number of steps in the loop to be automatically unrolled.
unroll_explicitExplicitly unroll loop, this may let code generator to generate pragma unroll hint.
+ +![](images/a82336e7fd8660ed1af7676a9258e1f38a623d17fb06729397d666a7e6612433.jpg) +Figure 2: AutoTVM optimization time for ResNet-18 on Titan Xp. + +# 2.2 OPTIMIZING COMPILER FOR DEEP NEURAL NETWORKS + +Optimizing compilers (Kennedy & Allen, 2001) usually take a black-box approach and use hardware measurements to configure the optimization based on a measure of fitness $f$ of each solution. In order to make the problem tractable, the optimizing compilers for deep neural networks reduce the problem down to tuning the knobs $\theta$ for the output code template $\tau$ , and can be formulated as: + +$$ +\Theta ^ { * } = \operatorname * { a r g m a x } _ { \Theta } f ( \tau ( \Theta ) ) , \qquad \mathrm { f o r } \Theta \in \mathcal { D } _ { \Theta } . +$$ + +A combination of assignment to the knobs is said to be a configuration $\boldsymbol { \Theta } = ( \theta _ { 1 } , \theta _ { 2 } , . . . , \theta _ { n } )$ while the dimensions of the design space $\mathcal { D } _ { \Theta }$ is defined by the knobs. As such, in Equation 1, an optimizing compiler starts from a code template $\tau$ for each layer, and makes use of a search algorithm and real hardware measurements to efficiently find the best configuration $\Theta ^ { \ast } \in { \mathcal { D } } _ { \Theta }$ . In this context, there are three variables that determine the effectiveness of the optimizing compiler: (1) a large and diverse enough design space that covers a variety of transformations, (2) an effective search algorithm to adequately navigate this space, and (3) a mechanism to cut down the number of costly hardware measurements that check the fitness of a solution. Table 1 lists the knobs for performing convolution on a GPU, where it is crucial that the code (1) maximizes data reuse, (2) uses the shared memory wisely, and (3) minimizes bank conflicts. The knobs optimize various aspects of the execution, including tiling (e.g., tile x, tile y, . . . ), unrolling (e.g., auto unroll max step and unroll explicit), and these knobs define a design space with $1 0 ^ { 1 0 }$ possibilities. Given the vastness of the design space, the remaining challenges are designing an effective search algorithm and designing a mechanism that reduces the cost of each step in the search (i.e. reducing the need to measure the hardware). + +# 2.3 CHALLENGES IN DEEP NEURAL NETWORK COMPILATION + +As shown in Figure 2, optimizing compilation for DNNs may still take an eon even with the advances from prior works (Chen et al., 2018a;b; Vasilache et al., 2018) With active research (You et al., 2017; Goyal et al., 2017; Codreanu et al., 2017; Akiba et al., 2017; You et al., 2018; Mattson et al., 2019) that has been able to cut down the training time to only few hours (You et al., 2017; Goyal et al., 2017) and even minutes (You et al., 2018; Akiba et al., 2017) on big models (e.g., ResNet-50 (He et al., 2016)) for ImageNet, it renders the optimizing compilation time of the current solutions seem even more prominent. Especially, since the above-mentioned compilers have been integrated to the deep learning pipelines of major players in the industry (Liu et al., 2019; Rotem et al., 2018; Vasilache et al., 2018), many users of these pipelines including the deployment engineers must go through the compilation workflow depicted in Figure 1 numerous times. Therefore, current long compilation time can be a hindrance to deploying DNN in various hardware, hence a major bottleneck in enabling intelligence on wider range of target platforms. + +Furthermore, as we explore various neural topologies (Xie et al., 2019; Wortsman et al., 2019) for better performance as illustrated in Ahn et al. (2020), even deeper or wider networks (Szegedy et al., 2015; Zagoruyko & Komodakis, 2016), and new operations (Howard et al., 2017) to achieve higher performance (LeCun, 2019), we are forced to optimize the networks more frequently. The long optimization times are multiplied with such trend, leaving the practical utility of the current compiler solutions to question. As such, the primary goal of this work is reducing the optimizing compilation time to meet the immediate needs of the industry for expedited DNN compilation to foster further diversity and innovation in designing DNNs. + +![](images/29fcf8d4209155d1093cb560684815d3387661eb22015843e0cbad9b14a22354.jpg) +Figure 3: Overall design and compilation overview of the CHAMELEON. + +Such long optimization time results from the inefficiency of simulated annealing which (while it stochastically guarantees a reasonable solution after huge number of iterations) fails to capture the patterns in the design space that can be exploited during the search. On the other hand, we can see in the figure that majority of the optimization time is spent on reaching for measurements on real hardware that is used as a feedback for the aforementioned search. Also, current approach even suffers from numerous invalid configurations that not only wastes the limited hardware measurement budget that the compiler starts with, but also incurs serious overhead to reset the target hardware for subsequent hardware measurements. As such, it is important that a sampling mechanism that selects potential configurations for hardware measurements to be smarter to ensure that each measurement is maximizing the chances of achieving a good solution and that it evades the invalid configurations. However, the current approaches rely on greedy sampling that passively sample based on the estimations from the cost models. This not only has a tendency to overfit but also neglect that solutions are distributed non-uniformly and that there are numerous invalid configurations. + +# 3 CHAMELEON: ADAPTIVE CODE OPTIMIZATION FOR EXPEDITED DEEP NEURAL NETWORK COMPILATION + +As discussed in Section 2, current solutions fall short of providing a swift optimization framework for optimizing emergent deep neural networks, because of the futility of the search in adapting to the design space from a random walk based search algorithm and the inefficiency of the physical hardware measurements from the greedy sampling. Therefore, developing a new framework that can overcome current challenges to unfetter neural network innovation from a prolonged optimization times can be boiled down to two problems: $\textcircled{4}$ improving the the search algorithm to better adapt to the design space, and $\textcircled { \bullet }$ improving the sampling algorithm to both better adapt to the distribution of the solutions and decrease the possibility of running into invalid configurations. As such we make two innovations in the optimizing compiler for deep neural networks to develop CHAMELEON by applying reinforcement learning to the search that can adapt to new design spaces (Adaptive Exploration) and devising an Adaptive Sampling that replaces the current greedy sampling. + +# 3.1 OVERALL DESIGN OF CHAMELEON + +Figure 3 outlines the overall design of our optimizing compiler, dubbed CHAMELEON1, and gives an overview of the optimizing compilation process. CHAMELEON takes code template $\tau$ for each layer in the network and the corresponding design space $\mathcal { D } _ { \Theta }$ as its input, and iteratively optimizes the code for configuration $\Theta$ to finally output $\tau ( \Theta ^ { * } )$ . The proposed Adaptive Exploration maneuvers the design space while using a cost model as a proxy for hardware measurements to the output set of candidate configurations $S _ { \Theta }$ . These configurations are then sampled with Adaptive Sampling so that the sampled configurations $S _ { \Theta } ^ { \prime }$ subsume the initial candidate configurations while reducing its number significantly. The sampled configurations $S _ { \Theta } ^ { \prime }$ are then passed to the code generator which combines the input template $\tau$ and the configurations $S _ { \Theta } ^ { \prime }$ to create a set of $\tau ( \Theta )$ that are sent to real hardware for runtime measurements. Runtimes from the hardware are used as the measure of fitness $f$ and update the cost model to enhance the exploration of the subsequent iterations. After multiple iterations, $\tau ( \Theta ^ { * } )$ with the best fitness $f$ (shortest runtime) is selected as an output for the layer. + +3.2 ADAPTIVE EXPLORATION: LEARNING ABOUT THE UNSEEN DESIGN SPACE TO EXPEDITE CONVERGENCE OF OPTIMIZATION + +As stated in Section 2, the current state-of-the-art approach (Chen et al., 2018b) that leverages simulated annealing relies on the stochastic guarantees of its random walks. Therefore, the current approach requires numerous iterations of exploration to converge to a reasonable solution causing long compilation hours, thus insufficient to enable disruptive innovations in neural networks. We take an inspiring approach that avoids naive dependence on the stochastic guarantee of simulated annealing and leverage a technique that can learn to adapt to unseen design space to not only accelerate convergence but also bring some performance gains. As such, we develop Adaptive Exploration by leveraging Reinforcement Learning $( R L )$ , which is concerned with learning to maximize reward given an environment by making good exploration and exploitation tradeoffs, in our case maximizing fitness $f$ of the explored configurations $S _ { \Theta }$ . + +Reinforcement learning formulation. Our RL-based Adaptive Exploration module uses an actor-critic style $R L$ , where policy network learns to emit a set of directions (vector of increment/decrement/stay) for each knob in the design space that will increase $f$ of the next configuration and the value network learns the design space $\mathcal { D } _ { \Theta }$ to estimate the value of the action. The first layer of these networks that takes the current configuration $\Theta$ as input is shared to foster information sharing among the two networks, and its output is fed into the subsequent layers the networks. These networks not only learn the dependencies among the different knobs of the design space (which are interrelated) that helps our module navigate through the design space but also lean the potential gains of the modifications to the configurations. + +![](images/97d024af12ac3c4f745d8b752f71ea358c4b67e85da0a104058710f13b015c4e.jpg) +Figure 4: Adaptive Exploration Module of CHAMELEON in action. + +Learning procedure. Having formulated the RL-based Adaptive Exploration Module, an iteration of our optimization begins with a set of initial configurations and takes multiple search steps (episode) for each of the configurations. As shown in Figure 4, the agent makes an action and applies it to the configuration using configuration updater to get another configuration that potentially has better $f$ . After finishing multiple search steps in the episode, all configurations $S _ { \Theta }$ are evaluated using a cost model, which its return values are used as a surrogate reward to update our agent, to reduce the number of costly hardware measurements. By taking this approach, $f$ of $S _ { \Theta }$ improves as our module progresses through the episodes. In other words, by repeating multiple episodes and iterations, our Adaptive Exploration Module gradually learns to locate good configurations. + +# 3.3 ADAPTIVE SAMPLING: ADAPTING TO THE DISTRIBUTION TO REDUCE COSTLY HARDWARE MEASUREMENTS + +Reducing number of costly hardware measurements. After the exploration step (regardless of the exploration method), we observe that the candidate configurations are clustered in subregions of the design space and these clusters are non-uniformly distributed (Figure 5). We also find that, while the design space’s surface is discrete and un-smooth, a large fraction of configurations within each cluster achieve similar runtime (Figure 6). Utilizing these characteristics of the design space, we devise Adaptive Sampling that can sample a new set of candidates, by adapting to the shape of the design space and the non-uniformity of the distribution while leaving the performance of optimization intact. We first leverage clustering algorithm to find configurations that are representative of each cluster; the sampling module uses centroids as the representative configurations. Our Adaptive Sampling iterates over a different number of clusters for their respective centroids and the L2 loss. + +![](images/ed941247342c26d8e1fad3ef4dd55690ccc085092361400e5fd016f0b22e4c39.jpg) +Figure 5: Clusters of candidate configurations. + +![](images/d9a9b79e18493ca91b94c446e32aee7cb3ed27107e9f4a93ee28dc4354aa45ac.jpg) +Figure 6: Cumulative Distribution Function (CDF) of the difference in runtime among the configurations in the cluster. + +In the context of optimizing compiler, selecting the number of centroids for clustering entails making the important tradeoff between selecting more centroids for better performance or fewer centroids for a reduced number of hardware measurements. As such, we must devise a method that would automatically make the tradeoff in a reasonable manner. We take advantage of the decreasing trend in the aforementioned L2 loss as we increase the number of centroids, and devise a Threshold-based Swift Meta-Search to determine the number of clusters. By setting the threshold (hyperparameter) it allows the compiler to determine the point of diminishing return (knee of the curve), inflection point beyond which fewer centroids may lead to performance degradation and more clusters would prolong the optimization substantially. Overall, our sampling curtails the number of hardware measurements so that it is just enough to subsume the entire subspace of the candidate configurations. + +Improving candidate configurations using sample synthesis. While the above sampling algorithm significantly reduces the number of hardware measurements compared to the conventional greedy sampling, without impacting the performance of the output code, we are still left with a critical issue of redundancy among the candidate configurations. We find that the exploration algorithm (regardless of the type) combined with the greedy sampling frequently leads to redundancy among the candidate configurations over different iterations of optimization due to the overfitting of the cost model from the greediness of the sampling. Even though the exploration algorithm tries to explore unvisited regions of the design space, these explored (not exploited) configurations are discarded due to the greedy sampling which entirely depends on the cost model for its selections of the configurations. Therefore, the current greedy sampling algorithm has its limitation in focusing the hardware measurements to the same region over and over. + +On the other hand, we find that from a code optimization point of view, we know that many of the automated approaches for black-box optimization are prone to invalid configurations, which results from too large a tile that goes over the input feature map boundary or errors during memory accesses (cannot be solved analytically). These invalid configurations not only blow the chances for better exploration but also leads to an extra optimization time overhead to reset the physical hardware for the subsequent hardware measurement. We try to overcome both of these limitations by devising Sample Synthesis. When our compiler runs into redundant samples, the proposed synthesis method analyzes the candidate samples to determine the most probable (most frequent $=$ mode function) non-invalid choice for each knob to come up with a new configuration. This statistical combination of the most frequent knob settings yield configurations that combine the strengths of different knobs to converge to a better overall solution. In spirit, the recombination (crossover) operator in genetic algorithms also tries to combine the best features of the solutions with high fitness values. Algorithm 1 presents the integration of our Adaptive Sampling and the Sample Synthesis. + +# 3.4 IMPLEMENTATION DETAILS + +Architecture exploration for the adaptive exploration. We use Proximal Policy Optimization $( P P O )$ (Schulman et al., 2017), a policy gradient that has been shown to adapt to various problems and have good sample complexity, as our reinforcement learning algorithm. Since reinforcement learning could incur computational overhead that could prolong the optimization time, we optimize the actor-critic networks through architecture exploration to find good tradeoff for size of these networks (that determines the computational overhead) and the optimization performance. + +Algorithm 1 Adaptive Sampling and Sample Synthesis + +
1: procedure ADAPTIVESAMPLING(SΘ, UΘ)
2:V se: candidate configs, ve: visited configs new_candidates ← @,previous_loss ←0
3:for k in range(8, 64) do
4:new_candidates,clusters,L2_loss ← K-means.run(se,k)
5:if Threshold × L2_loss ≥ previous_loss then break>Exit loop at knee of loss curve
6:previous_loss ←L2_loss
7:end for
8:for candidate in new_candidates do Replace visited config with new config
9:if candidate in ve then new_candidates.replace(candidate, mode(s@))
10:end for
11:return new_candidates >Feed to Code Generator to make measurements on hardware end procedure
+ +Design choices for the adaptive sampling. We use a $\kappa$ -means Clustering to determine centroids of the configurations, because $\kappa$ -means has been shown effective in practice and it only requires $\kappa$ , over error $\epsilon$ or radius in other algorithms which are much more challenging to tune. For example, DBSCAN (Ester et al., 1996) or mean-shift clustering (Comaniciu & Meer, 2002) are very sensitive to the above hyperparameters. On the other hand, $\kappa$ can be framed as a lever to balance the performance and speed of optimizing compilation which abstracts away the aforementioned challenges, enabling the Threshold-based Swift Meta-Search that identifies the optimal number of clusters. + +Hyperparameter tuning. Hyperparameter tuning is a very important task in machine learningbased tools and models. As such, we present the hyperparameters we used for the evaluation in Table 7 (in appendix), which its tuning took several days. For the hyperparameters in Table 8 (in appendix), we used the same set of values that were used in the AutoTVM paper (Chen et al., 2018b) in order to conduct a fair comparison or CHAMELEON. Additionally, for parameters used in the Adaptive Exploration module, which is not present in AutoTVM, we have tuned the hyperparameters using the set of layers presented in Table 5 (in appendix). We emphasize, however, that the hyperparameters have been tuned offline before the deployment of CHAMELEON, and the hyperparameters are not changed during the use of the framework or the experimentation. So the tuning overhead is not part of the compilation after the Adaptive Exploration module is tuned once before releasing the compiler to the deployment practitioners. + +# 4 EVALUATION + +We integrate CHAMELEON into TVM (Chen et al., 2018a) to perform component evaluation and compare with AutoTVM (Chen et al., 2018b). We first evaluate components of CHAMELEON in Section 4.1 and Section 4.2 on set of convolution layers sampled from AlexNet (Krizhevsky et al., 2012), VGG-16 (Simonyan & Zisserman, 2015), and ResNet-18 (He et al., 2016). Then we provide end-to-end evaluation of CHAMELEON on both set of layers and end-to-end deep models, in Section 4.3. Due to space limitations, we present only the representative plots in the paper, and the complete set of results and the details of the parameters are provided in the appendix. + +4.1 ADAPTIVE EXPLORATION: IMPROVING EFFICACY OF SEARCH ALGORITHM + +In the previous approach (Chen et al., 2018b), authors have built a cost model to estimate fitness instead of performing costly measurements on real hardware, then used simulated annealing to find potentially optimal configurations. Figure 7(a) compares the number of search steps taken per iteration to reach or converge to the solution in simulated annealing and Adaptive Exploration, respectively. Overall, observation is that CHAMELEON’s Adaptive Exploration requires $2 . 8 8 \times$ less search steps compared to simulated annealing to find good solution. This comes from the ability of the reinforcement learning algorithm in Adaptive Exploration Module to (1) learn the correlation between different dimensions, and (2) reuse information across different iterations, instead of starting from scratch while naively relying on the stochastic guarantees of simulated annealing process. + +![](images/9ad0ad2777e161304155b61c563bcb6dbb86606a4be75504fda18bc6925fd81c.jpg) +Figure 7: Component evaluation of CHAMELEON. + +# 4.2 ADAPTIVE SAMPLING: REDUCING NUMBER OF COSTLY HARDWARE MEASUREMENTS + +Figure 7(b) summarizes the effect of applying CHAMELEON’s Adaptive Sampling module on simulated annealing and reinforcement learning based search. First, the results show that using Adaptive Sampling helps the framework to make less hardware measurements regardless of the search algorithm used. The Adaptive Sampling algorithm reduces the number of measurements by $1 . 9 8 \times$ when used with simulated annealing and $2 . 3 3 \times$ with reinforcement learning One observation is that the Adaptive Sampling is more effective with reinforcement learning search. This comes from the reinforcement learning agent’s capacity to better localize the search to meaningful samples (exploitation) while still aiming to find good solution by making diverse search (exploration). + +Diversity exploration of AutoTVM aims to spread out the candidate configurations with a regularizing effect that fosters uniform sampling. In contrast, our Adaptive Sampling uses a clustering algorithm to perform more measurements on the regions with higher likelihood of achieving better output performance, leading to a non-uniform sampling. While AutoTVM states that diversity-aware selection had no meaningful impact on + +![](images/80fe02a72091d19f2baf576de93c71118e759109eb64c4eccd2ecdfc04c1cc03.jpg) +Figure 8: Comparison to AutoTVM’s diversity exploration. + +most of the evaluated workloads, our Adaptive Sampling brings significant improvement as depicted in Figure 8. As shown, Adaptive Sampling brings an average of $1 3 . 5 \%$ and $1 9 . 0 \%$ improvement on simulated annealing and reinforcement learning, respectively. + +# 4.3 INTEGRATION: REDUCING OPTIMIZATION TIME AND OUTPUT INFERENCE TIME + +CHAMELEON integrates two components into the workflow: RL-based Adaptive Exploration (AE) and Adaptive Sampling (AS). This section compares the performance of CHAMELEON with AutoTVM (Chen et al., 2018b) that leverages Simulated Annealing (SA) for its exploration. + +Layer evaluation. Figure 9 shows the trend of output code performance of ResNet-18’s 11th layer over number of hardware measurements during optimization. The figure illustrates that our Adaptive Exploration finds better configurations than simulated annealing which results in better output code performance, and the Adaptive Sampling reduces number of hardware measurements significantly during optimization. Also, CHAMELEON’s Adaptive Exploration and Adaptive Sampling working in tandem emits better code with shorter optimization time than others. As such, Figure 10(a) compares optimization time and the performance of the output code in CHAMELEON and AutoTVM to confirm the observation. CHAMELEON achieved $1 . 1 7 \times$ better performance with $4 . 8 2 \times$ shorter optimization time compared to AutoTVM. Overall, the results suggest that our Adaptive Exploration effectively maneuvers the design space, and Adaptive Sampling reduces hardware measurements and the overall optimization time while even improving output performance. + +![](images/902919044cc53df38e045268760dae15b3b4e717f01247ba0a622d6db4bdd025.jpg) +Figure 9: Layer evaluation of output performance for ResNet-18’s 11th layer. + +HAMELEON significantly reduces number of hardware measurements (from 800 to 392) + +
NETWORKSA (AutoTVM)AESA + ASAE+ AS (CHAMELEON)
AlexNet4.31 Hours4.06 Hours1.25 Hours1.20 Hours
VGG-1611.18 Hours8.82 Hours2.57 Hours1.95 Hours
ResNet-189.13 Hours7.39 Hours2.14 Hours2.13 Hours
+ +End-to-end evaluation. Up until now, we have focused on evaluation with subset of layers. Now we continue our discussion to the applicability of CHAMELEON to optimization of end-to-end deep neural networks. Figure 10(b) shows that CHAMELEON spends $3 . 5 9 \times$ , $5 . 7 3 \times$ , and $4 . 2 8 \times$ less time than AutoTVM to optimize AlexNet, VGG-16, and ResNet-18, respectively. On average, our work shows $4 . 4 5 \times$ optimization time speedup while achieving up to $6 . 4 \%$ improvement in terms of performance of output code. Inference time in Figure 10(b) illustrates the speedup for optimized code. Raw numbers are available in Table 2 and Table 3. All in all, such improvements result from efficient Adaptive Exploration and the reduced number of hardware measurements from Adaptive Sampling. + +![](images/5bd4f96ac9884f3aa4fdc206e0be0d33f6da796de011e9711d54ecc3131f0c4e.jpg) +Figure 10: Layer and end-to-end evaluation. Dashed lines denote AutoTVM’s performance. + +Table 2: End-to-end evaluation of the optimization time for deep networks. + +
NETWORKSA (AutoTVM)AESA + ASAE +AS (CHAMELEON)
AlexNet1.0277 ms1.0207 ms0.9762 ms0.9673 ms
VGG-163.9829 ms3.9710 ms3.8733 ms3.8458 ms
ResNet-181.0258 ms0.9897 ms0.9897 ms0.9831 ms
+ +Table 3: End-to-end evaluation of the output performance for deep networks. + +# 5 RELATED WORKS + +CHAMELEON uniquely offers a solution that exclusively enables (i) Reinforcement Learning and (ii) Sampling in the context of (iii) Optimizing Compilers for neural networks. As such, we discuss the related work from each of the three independent research directions. + +Optimizing compilers. TensorComprehensions (Vasilache et al., 2018) and TVM (Chen et al., 2018a) use genetic algorithm and simulated annealing to choose parameters of polyhedral optimization for neural networks. In a more general context, some computing libraries (Whaley & Dongarra, 1998; Frigo & Johnson, 1998) make use of black box optimization and also profiling-based compilation passes (Chang et al., 1991; Novillo, 2014) utilize runtime information to generate optimized code. Later, AutoTVM (Chen et al., 2018b) incorporates learning with boosted trees within the cost model for TVM to reduce the number of real hardware measurements. While CHAMELEON is inspired and builds on these prior works, unlike them, it is based on reinforcement learning for Adaptive Exploration, and Adaptive Sampling that leverages clustering to reduce the number of measurements. + +Reinforcement learning for hyper-parameter optimization. There are a growing body of studies on using reinforcement learning to perform various optimizations (Gao et al., 2018; Mirhoseini et al., 2017; Nareyek, 2003; Mao et al., 2016; Xu et al., 2018; Mao et al., 2019) for a variety of objectives including hyper-parameter optimization for neural networks. For instance, DeepArchitect (Negrinho & Gordon, 2017) and NAS (Zoph & Le, 2017) use reinforcement learning to automate the process of designing deep neural network models and their associated parameters. HAQ (Wang et al., 2019) and ReLeQ (Elthakeb et al., 2018) use reinforcement learning to chose levels of quantization for the layers of a given deep neural network. AMC (He et al., 2018) formulates neural network compression as a RL problem. A most recent effort (Paliwal et al., 2020)–which will be published concurrent to ours in ICLR 2020–combined RL with graph neural networks and genetic algorithms to optimize DNN execution. Our work exclusively explores a different problem, that is optimizing compilers using reinforcement learning. + +Sampling algorithms for learning. Active learning is a broad field (Settles, 2009; Cohn et al., 1996; Sugiyama, 2006; Cai et al., 2013; Goetz et al., 2018; Wu et al., 2019) that uses a measure of the change in the model to decide which training data elements should be used to update the model. Passive learning (Yu & Kim, 2010; O’Neill et al., 2017) is an alternative view that independent of the model, analyze the distribution of the training data set and selects a subset. The Adaptive Sampling algorithm for CHAMELEON shares similarities with Passive learning but it differs in its context. The sampling is designed to reduce the number of samples (configuration) for hardware measurement from the exploration of the design space whilst performing an optimization to accelerate the process. + +# 6 CONCLUSION + +We present CHAMELEON to allow optimizing compilers to adapt to unseen design spaces of code schedules to reduce the optimization time. This paper is also an initial effort to bring reinforcement learning to the realm of optimizing compilers for neural networks, and we also develop an Adaptive Sampling with domain-knowledge inspired Sample Synthesis to not only reduce the number of samples required to navigate the design space but also augment its quality in terms of fitness. Experimentation with real-world deep models shows that CHAMELEON not only reduces the time for compilation significantly, but also improves the quality of the code. This encouraging result suggests a significant potential for various learning techniques to optimizing deep learning models. + +# ACKNOWLEDGEMENT + +We thank the anonymous reviewers for their insightful comments. We also thank Jinwon Lee and Jangho Kim for the fruitful discussions and feedbacks on the manuscript. This work was in part supported by generous gifts from Qualcomm, Google, Microsoft, and Xilinx as well as the Semiconductor Research Corporation (SRC) contract #2019-SD-2884, National Science Foundation (NSF) awards CNS#1703812, ECCS#1609823, CCF#1553192, Air Force Office of Scientific Research (AFOSR) Young Investigator Program (YIP) award #FA9550-17-1-0274, National Institute of Health (NIH) award #R01EB028350, and Air Force Research Laboratory (AFRL) and Defense Advanced Research Project Agency (DARPA) under agreement number #FA8650-20-2-7009. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. 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In NeurIPS, 2019. + +Dongrui Wu, Chin-Teng Lin, and Jian Huang. Active learning for regression using greedy sampling. Information Sciences, 2019. + +Saining Xie, Alexander Kirillov, Ross Girshick, and Kaiming He. Exploring randomly wired neural networks for image recognition. In ICCV, 2019. + +Zhongwen Xu, Hado P van Hasselt, and David Silver. Meta-gradient reinforcement learning. In NeurIPS, 2018. + +Yang You, Igor Gitman, and Boris Ginsburg. Large batch training of convolutional networks. arXiv, 2017. URL https://arxiv.org/pdf/1708.03888.pdf. + +Yang You, Zhao Zhang, Cho-Jui Hsieh, James Demmel, and Kurt Keutzer. ImageNet training in minutes. In ICPP, 2018. + +Hwanjo Yu and Sungchul Kim. Passive sampling for regression. In ICDM, 2010. + +Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. BMVC, 2016. + +Barret Zoph and Quoc V Le. Neural architecture search with reinforcement learning. ICLR, 2017. URL https://openreview.net/forum?id $=$ r1Ue8Hcxg. + +# APPENDIX + +A EXPERIMENTAL SETUP + +A.1 DNN MODELS AND LAYERS + +Table 4: Details of the DNN models used in evaluating CHAMELEON. + +
NETWORKDATASETNUMBER OF TASKS
AlexNetImageNet5
VGG-16ImageNet9
ResNet-18ImageNet12
+ +Table 5: Details of the layers used in evaluating CHAMELEON. + +
NAMEMODELLAYER TYPETASK INDEX
L1AlexNetconvolution1
L2AlexNetconvolution4
L3VGG-16convolution1
L4VGG-16convolution2
L5VGG-16convolution4
L6ResNet-18convolution6
L7ResNet-18convolution9
L8ResNet-18convolution11
+ +# A.2 HARDWARE SPECIFICATION + +Table 6: Details of the hardware used for evaluation of CHAMELEON. + +
SPECIFICATIONSDETAILS
GPUTitan Xp
Host CPU3.4G Hz Intel Core i7
Main Memory32GB 2400 MHz DDR3
+ +# A.3 HYPER-PARAMETERS + +Table 7: Hyper-parameters uses in CHAMELEON. + +
HYPERPARAMETERVALUEDESCRIPTION
iterationopt16number of iterations for optimization process (equivalent to 1Ooo hardware measurements)
modeGBT bGBTxgb-regtype of loss used for cost model
64maximum batch size of planning in GBT(Chen & Guestrin,2016) cost model per iteration of optimization process
episoderl128number of episodes for reinforcement learning
steprl500maximum steps of one reinforcement learning episode
thresholdmeta2.5threshold used for meta-search in sampling
+ +Table 8: Hyper-parameters uses in AutoTVM (Chen et al., 2018b). + +
HYPERPARAMETERVALUEDESCRIPTION
£(bGBT)1000totalnumber of hardwaremeasurements
modeGBTxgb-regtype of loss used for cost model
bGBT64batch size of planning in GBT(Chen& Guestrin,2016)
nsa128number of Markov chains in parallel simulated annealing
stepsa500maximum steps of one simulated annealing run
+ +Table 9: Hyper-parameters used in CHAMELEON’s PPO (Schulman et al., 2017) search agent. + +
HYPERPARAMETERVALUE
Adam Step Size Discount Factor1×10-3
GAE Parameter0.9
Number of Epochs0.99
Clipping Parameter3
Value Coefficient0.3
1.0
Entropy Coefficient0.1
+ +# B ADDITIONAL EXPERIMENTAL RESULTS + +# B.1 OPTIMIZATION TIME BREAKDOWN FOR DNN MODELS + +![](images/026a53d0d37fc394dbde9ee260ed92631908ad47f92de723d2f9d7a13438b6e1.jpg) +Figure 11: AutoTVM optimization time for AlexNet (Krizhevsky et al., 2012) and VGG-16 (Simonyan & Zisserman, 2015), and ResNet-18 (He et al., 2016) on Titan Xp. Numbers in bars denote fraction of time for measurements. + +![](images/3c8b2503e8f20893f004a618595f804ee4202bdce892dd2dd37cf5019a6c7a97.jpg) +Figure 12: Layer evaluations for AlexNet (Krizhevsky et al., 2012). + +![](images/01cae0b30b91a657d3523b35270fbc77354a18f41266d34fb3123c16b6bd7b20.jpg) +Figure 13: Layer evaluations for VGG-16 (Simonyan & Zisserman, 2015). + +![](images/8d074821723c328e9d090e41a9d4a947aea4d674ac66384eb9bb3c3baedf687b.jpg) +Figure 14: Layer evaluations for ResNet-18 (He et al., 2016). \ No newline at end of file diff --git a/parse/train/rygG4AVFvH/rygG4AVFvH_content_list.json b/parse/train/rygG4AVFvH/rygG4AVFvH_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..21b99a6581236f9f7ed07fbf83892e76bb067040 --- /dev/null +++ b/parse/train/rygG4AVFvH/rygG4AVFvH_content_list.json @@ -0,0 +1,1827 @@ +[ + { + "type": "text", + "text": "CHAMELEON: ADAPTIVE CODE OPTIMIZATION FOR EXPEDITED DEEP NEURAL NETWORK COMPILATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 810, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Byung Hoon $\\mathbf { A } \\mathbf { h } \\mathbf { n } ^ { 1 }$ , Prannoy Pilligundla1, Amir Yazdanbakhsh2, Hadi Esmaeilzadeh1 ", + "bbox": [ + 184, + 169, + 772, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 University of California, San Diego \n2 Google Research \nbhahn@eng.ucsd.edu, ppilligu@eng.ucsd.edu, ayazdan@google.com \nhadi@eng.ucsd.edu ", + "bbox": [ + 184, + 185, + 769, + 241 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 279, + 544, + 294 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Achieving faster execution with shorter compilation time can foster further diversity and innovation in neural networks. However, the current paradigm of executing neural networks either relies on hand-optimized libraries, traditional compilation heuristics, or very recently genetic algorithms and other stochastic methods. These methods suffer from frequent costly hardware measurements rendering them not only too time consuming but also suboptimal. As such, we devise a solution that can learn to quickly adapt to a previously unseen design space for code optimization, both accelerating the search and improving the output performance. This solution dubbed CHAMELEON leverages reinforcement learning whose solution takes fewer steps to converge, and develops an adaptive sampling algorithm that not only focuses on the costly samples (real hardware measurements) on representative points but also uses a domain-knowledge inspired logic to improve the samples itself. Experimentation with real hardware shows that CHAMELEON provides $4 . 4 5 \\times$ speed up in optimization time over AutoTVM, while also improving inference time of the modern deep networks by $5 . 6 \\%$ . ", + "bbox": [ + 233, + 308, + 764, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 539, + 336, + 554 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The enormous computational intensity of Deep Neural Networks (DNNs) have resulted in developing either hand-optimized kernels, such as NVIDIA cuDNN or Intel MKL that serve as backend for a variety of programming environment such as TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2019). However, the complexity of the tensor operations in DNNs and the volatility of algorithms, which has led to unprecedented rate of innovation (LeCun, 2019), calls for developing automated compilation frameworks. To imitate or even surpass the success of hand-optimized libraries, recent research has developed stochastic optimization passes: for general code, STOKE (Schkufza et al., 2013), and neural network code, TVM (Chen et al., 2018a) and TensorComprehensions (Vasilache et al., 2018). TVM and TensorComprehensions are based on random or genetic algorithms to search the space of optimized code for neural networks. AutoTVM (Chen et al., 2018b) builds on top of TVM and leverage boosted trees (Chen & Guestrin, 2016) as part of the search cost model to avoid measuring the fitness of each solution (optimized candidate neural network code), and instead predict its fitness. However, even with these innovations the optimizing compilation time can be around 10 hours for ResNet-18 (He et al., 2016), and even more for deeper or wider networks. ", + "bbox": [ + 173, + 569, + 825, + 763 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since the general objective is to unleash new possibilities by developing automatic optimization passes, long compilation time hinders innovation and could put the current solutions in a position of questionable utility. To solve this problem, we first question the very statistical guarantees which the aforementioned optimization passes rely on. The current approaches are oblivious to the patterns in the design space of schedules that are available for exploitation, and causes inefficient search or even converges to solutions that may even be suboptimal. Also, we notice that current approaches rely on greedy sampling that neglects the distribution of the candidate solutions (configurations). While greedy sampling that passively filter samples based on the fitness estimations from the cost models work, many of their hardware measurements (required for optimization) tend to be redundant and wasteful. Moreover, we found that current solutions that rely on greedy sampling lead to significant fractions of the candidate configurations being redundant over iterations, and that any optimizing compiler are prone to invalid configurations which significantly prolongs the optimization time. As such, this work sets out to present an Adaptive approach dubbed CHAMELEON to significantly reduce the compilation time and offer automation while avoiding dependence to hand-optimization, enabling far more diverse tensor operations in the next generation DNNs. We tackle this challenge from two fronts with the following contributions: ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(1) Devising an Adaptive Exploration module that utilizes reinforcement learning to adapt to unseen design space of new networks to reduce search time yet achieve better performance. (2) Proposing an Adaptive Sampling algorithm that utilizes clustering to adaptively reduce the number of costly hardware measurements, and devising a domain-knowledge inspired Sample Synthesis to find configurations that would potentially yield better performance. ", + "bbox": [ + 174, + 180, + 825, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Real hardware experimentation with modern DNNs (AlexNet, VGG-16, and ResNet-18) on a highend GPU (Titan $\\mathrm { X p }$ ), shows that the combination of these two innovations, dubbed CHAMELEON, yields $4 . 4 5 \\times$ speedup over the leading framework, AutoTVM. CHAMELEON is publicly available in the project page: https://bitbucket.org/act-lab/chameleon. ", + "bbox": [ + 174, + 257, + 825, + 314 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 CHALLENGES IN DEEP NEURAL NETWORK COMPILATION ", + "text_level": 1, + "bbox": [ + 176, + 340, + 687, + 358 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The general life-cycle of deep learning models from its birth to deployment comprises of two major stages. First stage is the designing and the training of a deep learning model by a research scientist, with the primary goal of achieving the highest feasible accuracy. Then, with a general demand to enable the intelligence on a wide range of devices (from mobile CPUs in the edge to cloud-scale GPUs), the second stage has emerged for the deployment of the pre-trained deep learning model to a target hardware by a deployment engineer. These stages are each iterative processes: research scientists iterate until it reaches the target performance in terms of accuracy whereas the deployment engineers iterate until the performance in terms of inference speed with a given hardware satisfies the given constraints. Importantly, these two stages are most often separate processes, and this paper mainly focuses on the second stage (deployment) of the cycle with an overarching goal of accelerating the overall deployment cycle by reducing the optimizing compilation time without compromising the performance of the output code. ", + "bbox": [ + 173, + 377, + 825, + 544 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 COMPILATION WORKFLOW FOR DEEP NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 173, + 569, + 627, + 583 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/4c1eda16cc68693383186b7adabf42830ab5f161819979c3ab152bb6ba67eb96.jpg", + "image_caption": [ + "Figure 1: Overview of our model compilation workflow, and highlighted is the scope of this work. " + ], + "image_footnote": [], + "bbox": [ + 243, + 607, + 753, + 657 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Figure 1 illustrates how a compiler for DNNs takes an input model $\\mathcal { M }$ and emits an optimized code $\\tau ( \\bar { \\Theta } ^ { * } )$ that runs the model efficiently on a given hardware. This flow is commensurate with TensorComprehensions (Vasilache et al., 2018) and TVM (Chen et al., 2018a), using which we implement CHAMELEON that is available as a separate package for adoption in even other frameworks. The first phase of the workflow is the frontend compiler which performs the translation from the compiler and applies target-independent and white-box target-dependent optimizations that do not incorporate a measure of runtime. Target-independent passes transform the input DNN model without specificity to the target hardware. Operator fusion and data layout transformation in TVM are some examples of these passes, which lie in the same category as dead-code elimination or loop-invariant code motion in GCC (Stallman & DeveloperCommunity, 2009) or LLVM (Lattner & Adve, 2004). Target-dependent passes, on the other hand, the compiler takes the hardware architecture (target) into account while optimizing the program; however, this also does not actively leverage runtime measures. The last stage is a black-box optimization pass, called optimizing compiler, that given a measure of performance at runtime from the hardware can further optimize the code. CHAMELEON falls in this class by offering an optimizing compiler that adapts to different design space to be more swift in optimizing deep neural networks compared to conventional approaches. ", + "bbox": [ + 173, + 702, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "table", + "img_path": "images/c2e5a4dd0b08feacfb84d10506f773673ba88a1cd59ac54e25896d4a354d2118.jpg", + "table_caption": [ + "Table 1: Knobs in the design space to optimize convolution. " + ], + "table_footnote": [], + "table_body": "
KNOBSDEFINITION
tile_f, tile-y, tile_xFactors for tiling and binding # of filters height,and width of feature maps.
tile_rc,tile_ry, tile_rxFactors for tiling reduction axis such as # of channels,height,and width of filters.
auto_unroll_max_stepThreshold of number of steps in the loop to be automatically unrolled.
unroll_explicitExplicitly unroll loop, this may let code generator to generate pragma unroll hint.
", + "bbox": [ + 194, + 101, + 566, + 231 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/a82336e7fd8660ed1af7676a9258e1f38a623d17fb06729397d666a7e6612433.jpg", + "image_caption": [ + "Figure 2: AutoTVM optimization time for ResNet-18 on Titan Xp. " + ], + "image_footnote": [], + "bbox": [ + 601, + 103, + 820, + 223 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 OPTIMIZING COMPILER FOR DEEP NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 173, + 276, + 602, + 291 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Optimizing compilers (Kennedy & Allen, 2001) usually take a black-box approach and use hardware measurements to configure the optimization based on a measure of fitness $f$ of each solution. In order to make the problem tractable, the optimizing compilers for deep neural networks reduce the problem down to tuning the knobs $\\theta$ for the output code template $\\tau$ , and can be formulated as: ", + "bbox": [ + 174, + 303, + 825, + 359 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6e38bda066d98c27abf8ecce1932f0e2c8e4e10202b43524d3b59987121af43c.jpg", + "text": "$$\n\\Theta ^ { * } = \\operatorname * { a r g m a x } _ { \\Theta } f ( \\tau ( \\Theta ) ) , \\qquad \\mathrm { f o r } \\Theta \\in \\mathcal { D } _ { \\Theta } .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 366, + 640, + 392 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A combination of assignment to the knobs is said to be a configuration $\\boldsymbol { \\Theta } = ( \\theta _ { 1 } , \\theta _ { 2 } , . . . , \\theta _ { n } )$ while the dimensions of the design space $\\mathcal { D } _ { \\Theta }$ is defined by the knobs. As such, in Equation 1, an optimizing compiler starts from a code template $\\tau$ for each layer, and makes use of a search algorithm and real hardware measurements to efficiently find the best configuration $\\Theta ^ { \\ast } \\in { \\mathcal { D } } _ { \\Theta }$ . In this context, there are three variables that determine the effectiveness of the optimizing compiler: (1) a large and diverse enough design space that covers a variety of transformations, (2) an effective search algorithm to adequately navigate this space, and (3) a mechanism to cut down the number of costly hardware measurements that check the fitness of a solution. Table 1 lists the knobs for performing convolution on a GPU, where it is crucial that the code (1) maximizes data reuse, (2) uses the shared memory wisely, and (3) minimizes bank conflicts. The knobs optimize various aspects of the execution, including tiling (e.g., tile x, tile y, . . . ), unrolling (e.g., auto unroll max step and unroll explicit), and these knobs define a design space with $1 0 ^ { 1 0 }$ possibilities. Given the vastness of the design space, the remaining challenges are designing an effective search algorithm and designing a mechanism that reduces the cost of each step in the search (i.e. reducing the need to measure the hardware). ", + "bbox": [ + 173, + 400, + 825, + 594 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 CHALLENGES IN DEEP NEURAL NETWORK COMPILATION ", + "text_level": 1, + "bbox": [ + 174, + 612, + 616, + 627 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As shown in Figure 2, optimizing compilation for DNNs may still take an eon even with the advances from prior works (Chen et al., 2018a;b; Vasilache et al., 2018) With active research (You et al., 2017; Goyal et al., 2017; Codreanu et al., 2017; Akiba et al., 2017; You et al., 2018; Mattson et al., 2019) that has been able to cut down the training time to only few hours (You et al., 2017; Goyal et al., 2017) and even minutes (You et al., 2018; Akiba et al., 2017) on big models (e.g., ResNet-50 (He et al., 2016)) for ImageNet, it renders the optimizing compilation time of the current solutions seem even more prominent. Especially, since the above-mentioned compilers have been integrated to the deep learning pipelines of major players in the industry (Liu et al., 2019; Rotem et al., 2018; Vasilache et al., 2018), many users of these pipelines including the deployment engineers must go through the compilation workflow depicted in Figure 1 numerous times. Therefore, current long compilation time can be a hindrance to deploying DNN in various hardware, hence a major bottleneck in enabling intelligence on wider range of target platforms. ", + "bbox": [ + 174, + 638, + 825, + 805 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Furthermore, as we explore various neural topologies (Xie et al., 2019; Wortsman et al., 2019) for better performance as illustrated in Ahn et al. (2020), even deeper or wider networks (Szegedy et al., 2015; Zagoruyko & Komodakis, 2016), and new operations (Howard et al., 2017) to achieve higher performance (LeCun, 2019), we are forced to optimize the networks more frequently. The long optimization times are multiplied with such trend, leaving the practical utility of the current compiler solutions to question. As such, the primary goal of this work is reducing the optimizing compilation time to meet the immediate needs of the industry for expedited DNN compilation to foster further diversity and innovation in designing DNNs. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/29fcf8d4209155d1093cb560684815d3387661eb22015843e0cbad9b14a22354.jpg", + "image_caption": [ + "Figure 3: Overall design and compilation overview of the CHAMELEON. " + ], + "image_footnote": [], + "bbox": [ + 236, + 88, + 758, + 243 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Such long optimization time results from the inefficiency of simulated annealing which (while it stochastically guarantees a reasonable solution after huge number of iterations) fails to capture the patterns in the design space that can be exploited during the search. On the other hand, we can see in the figure that majority of the optimization time is spent on reaching for measurements on real hardware that is used as a feedback for the aforementioned search. Also, current approach even suffers from numerous invalid configurations that not only wastes the limited hardware measurement budget that the compiler starts with, but also incurs serious overhead to reset the target hardware for subsequent hardware measurements. As such, it is important that a sampling mechanism that selects potential configurations for hardware measurements to be smarter to ensure that each measurement is maximizing the chances of achieving a good solution and that it evades the invalid configurations. However, the current approaches rely on greedy sampling that passively sample based on the estimations from the cost models. This not only has a tendency to overfit but also neglect that solutions are distributed non-uniformly and that there are numerous invalid configurations. ", + "bbox": [ + 173, + 287, + 825, + 468 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 CHAMELEON: ADAPTIVE CODE OPTIMIZATION FOR EXPEDITED DEEP NEURAL NETWORK COMPILATION ", + "text_level": 1, + "bbox": [ + 174, + 489, + 637, + 523 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As discussed in Section 2, current solutions fall short of providing a swift optimization framework for optimizing emergent deep neural networks, because of the futility of the search in adapting to the design space from a random walk based search algorithm and the inefficiency of the physical hardware measurements from the greedy sampling. Therefore, developing a new framework that can overcome current challenges to unfetter neural network innovation from a prolonged optimization times can be boiled down to two problems: $\\textcircled{4}$ improving the the search algorithm to better adapt to the design space, and $\\textcircled { \\bullet }$ improving the sampling algorithm to both better adapt to the distribution of the solutions and decrease the possibility of running into invalid configurations. As such we make two innovations in the optimizing compiler for deep neural networks to develop CHAMELEON by applying reinforcement learning to the search that can adapt to new design spaces (Adaptive Exploration) and devising an Adaptive Sampling that replaces the current greedy sampling. ", + "bbox": [ + 174, + 539, + 825, + 691 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 OVERALL DESIGN OF CHAMELEON ", + "text_level": 1, + "bbox": [ + 176, + 710, + 459, + 724 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 3 outlines the overall design of our optimizing compiler, dubbed CHAMELEON1, and gives an overview of the optimizing compilation process. CHAMELEON takes code template $\\tau$ for each layer in the network and the corresponding design space $\\mathcal { D } _ { \\Theta }$ as its input, and iteratively optimizes the code for configuration $\\Theta$ to finally output $\\tau ( \\Theta ^ { * } )$ . The proposed Adaptive Exploration maneuvers the design space while using a cost model as a proxy for hardware measurements to the output set of candidate configurations $S _ { \\Theta }$ . These configurations are then sampled with Adaptive Sampling so that the sampled configurations $S _ { \\Theta } ^ { \\prime }$ subsume the initial candidate configurations while reducing its number significantly. The sampled configurations $S _ { \\Theta } ^ { \\prime }$ are then passed to the code generator which combines the input template $\\tau$ and the configurations $S _ { \\Theta } ^ { \\prime }$ to create a set of $\\tau ( \\Theta )$ that are sent to real hardware for runtime measurements. Runtimes from the hardware are used as the measure of fitness $f$ and update the cost model to enhance the exploration of the subsequent iterations. After multiple iterations, $\\tau ( \\Theta ^ { * } )$ with the best fitness $f$ (shortest runtime) is selected as an output for the layer. ", + "bbox": [ + 173, + 736, + 825, + 876 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 ADAPTIVE EXPLORATION: LEARNING ABOUT THE UNSEEN DESIGN SPACE TO EXPEDITE CONVERGENCE OF OPTIMIZATION ", + "bbox": [ + 174, + 151, + 758, + 179 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As stated in Section 2, the current state-of-the-art approach (Chen et al., 2018b) that leverages simulated annealing relies on the stochastic guarantees of its random walks. Therefore, the current approach requires numerous iterations of exploration to converge to a reasonable solution causing long compilation hours, thus insufficient to enable disruptive innovations in neural networks. We take an inspiring approach that avoids naive dependence on the stochastic guarantee of simulated annealing and leverage a technique that can learn to adapt to unseen design space to not only accelerate convergence but also bring some performance gains. As such, we develop Adaptive Exploration by leveraging Reinforcement Learning $( R L )$ , which is concerned with learning to maximize reward given an environment by making good exploration and exploitation tradeoffs, in our case maximizing fitness $f$ of the explored configurations $S _ { \\Theta }$ . ", + "bbox": [ + 174, + 193, + 825, + 332 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Reinforcement learning formulation. Our RL-based Adaptive Exploration module uses an actor-critic style $R L$ , where policy network learns to emit a set of directions (vector of increment/decrement/stay) for each knob in the design space that will increase $f$ of the next configuration and the value network learns the design space $\\mathcal { D } _ { \\Theta }$ to estimate the value of the action. The first layer of these networks that takes the current configuration $\\Theta$ as input is shared to foster information sharing among the two networks, and its output is fed into the subsequent layers the networks. These networks not only learn the dependencies among the different knobs of the design space (which are interrelated) that helps our module navigate through the design space but also lean the potential gains of the modifications to the configurations. ", + "bbox": [ + 174, + 348, + 825, + 474 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/97d024af12ac3c4f745d8b752f71ea358c4b67e85da0a104058710f13b015c4e.jpg", + "image_caption": [ + "Figure 4: Adaptive Exploration Module of CHAMELEON in action. " + ], + "image_footnote": [], + "bbox": [ + 238, + 489, + 758, + 611 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Learning procedure. Having formulated the RL-based Adaptive Exploration Module, an iteration of our optimization begins with a set of initial configurations and takes multiple search steps (episode) for each of the configurations. As shown in Figure 4, the agent makes an action and applies it to the configuration using configuration updater to get another configuration that potentially has better $f$ . After finishing multiple search steps in the episode, all configurations $S _ { \\Theta }$ are evaluated using a cost model, which its return values are used as a surrogate reward to update our agent, to reduce the number of costly hardware measurements. By taking this approach, $f$ of $S _ { \\Theta }$ improves as our module progresses through the episodes. In other words, by repeating multiple episodes and iterations, our Adaptive Exploration Module gradually learns to locate good configurations. ", + "bbox": [ + 173, + 654, + 825, + 780 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 ADAPTIVE SAMPLING: ADAPTING TO THE DISTRIBUTION TO REDUCE COSTLY HARDWARE MEASUREMENTS ", + "text_level": 1, + "bbox": [ + 173, + 799, + 637, + 827 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Reducing number of costly hardware measurements. After the exploration step (regardless of the exploration method), we observe that the candidate configurations are clustered in subregions of the design space and these clusters are non-uniformly distributed (Figure 5). We also find that, while the design space’s surface is discrete and un-smooth, a large fraction of configurations within each cluster achieve similar runtime (Figure 6). Utilizing these characteristics of the design space, we devise Adaptive Sampling that can sample a new set of candidates, by adapting to the shape of the design space and the non-uniformity of the distribution while leaving the performance of optimization intact. We first leverage clustering algorithm to find configurations that are representative of each cluster; the sampling module uses centroids as the representative configurations. Our Adaptive Sampling iterates over a different number of clusters for their respective centroids and the L2 loss. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/ed941247342c26d8e1fad3ef4dd55690ccc085092361400e5fd016f0b22e4c39.jpg", + "image_caption": [ + "Figure 5: Clusters of candidate configurations. " + ], + "image_footnote": [], + "bbox": [ + 176, + 97, + 482, + 233 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/d9a9b79e18493ca91b94c446e32aee7cb3ed27107e9f4a93ee28dc4354aa45ac.jpg", + "image_caption": [ + "Figure 6: Cumulative Distribution Function (CDF) of the difference in runtime among the configurations in the cluster. " + ], + "image_footnote": [], + "bbox": [ + 509, + 94, + 807, + 208 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 275, + 825, + 330 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the context of optimizing compiler, selecting the number of centroids for clustering entails making the important tradeoff between selecting more centroids for better performance or fewer centroids for a reduced number of hardware measurements. As such, we must devise a method that would automatically make the tradeoff in a reasonable manner. We take advantage of the decreasing trend in the aforementioned L2 loss as we increase the number of centroids, and devise a Threshold-based Swift Meta-Search to determine the number of clusters. By setting the threshold (hyperparameter) it allows the compiler to determine the point of diminishing return (knee of the curve), inflection point beyond which fewer centroids may lead to performance degradation and more clusters would prolong the optimization substantially. Overall, our sampling curtails the number of hardware measurements so that it is just enough to subsume the entire subspace of the candidate configurations. ", + "bbox": [ + 174, + 338, + 825, + 477 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Improving candidate configurations using sample synthesis. While the above sampling algorithm significantly reduces the number of hardware measurements compared to the conventional greedy sampling, without impacting the performance of the output code, we are still left with a critical issue of redundancy among the candidate configurations. We find that the exploration algorithm (regardless of the type) combined with the greedy sampling frequently leads to redundancy among the candidate configurations over different iterations of optimization due to the overfitting of the cost model from the greediness of the sampling. Even though the exploration algorithm tries to explore unvisited regions of the design space, these explored (not exploited) configurations are discarded due to the greedy sampling which entirely depends on the cost model for its selections of the configurations. Therefore, the current greedy sampling algorithm has its limitation in focusing the hardware measurements to the same region over and over. ", + "bbox": [ + 174, + 494, + 825, + 647 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "On the other hand, we find that from a code optimization point of view, we know that many of the automated approaches for black-box optimization are prone to invalid configurations, which results from too large a tile that goes over the input feature map boundary or errors during memory accesses (cannot be solved analytically). These invalid configurations not only blow the chances for better exploration but also leads to an extra optimization time overhead to reset the physical hardware for the subsequent hardware measurement. We try to overcome both of these limitations by devising Sample Synthesis. When our compiler runs into redundant samples, the proposed synthesis method analyzes the candidate samples to determine the most probable (most frequent $=$ mode function) non-invalid choice for each knob to come up with a new configuration. This statistical combination of the most frequent knob settings yield configurations that combine the strengths of different knobs to converge to a better overall solution. In spirit, the recombination (crossover) operator in genetic algorithms also tries to combine the best features of the solutions with high fitness values. Algorithm 1 presents the integration of our Adaptive Sampling and the Sample Synthesis. ", + "bbox": [ + 174, + 655, + 825, + 835 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 854, + 405, + 868 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Architecture exploration for the adaptive exploration. We use Proximal Policy Optimization $( P P O )$ (Schulman et al., 2017), a policy gradient that has been shown to adapt to various problems and have good sample complexity, as our reinforcement learning algorithm. Since reinforcement learning could incur computational overhead that could prolong the optimization time, we optimize the actor-critic networks through architecture exploration to find good tradeoff for size of these networks (that determines the computational overhead) and the optimization performance. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/ba0f0c0e5452359556837de8f1e2bbe5e45df9f526c8ffd7fb4dd5761b9f0d1f.jpg", + "table_caption": [ + "Algorithm 1 Adaptive Sampling and Sample Synthesis " + ], + "table_footnote": [], + "table_body": "
1: procedure ADAPTIVESAMPLING(SΘ, UΘ)
2:V se: candidate configs, ve: visited configs new_candidates ← @,previous_loss ←0
3:for k in range(8, 64) do
4:new_candidates,clusters,L2_loss ← K-means.run(se,k)
5:if Threshold × L2_loss ≥ previous_loss then break>Exit loop at knee of loss curve
6:previous_loss ←L2_loss
7:end for
8:for candidate in new_candidates do Replace visited config with new config
9:if candidate in ve then new_candidates.replace(candidate, mode(s@))
10:end for
11:return new_candidates >Feed to Code Generator to make measurements on hardware end procedure
", + "bbox": [ + 178, + 114, + 826, + 291 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 319, + 825, + 361 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Design choices for the adaptive sampling. We use a $\\kappa$ -means Clustering to determine centroids of the configurations, because $\\kappa$ -means has been shown effective in practice and it only requires $\\kappa$ , over error $\\epsilon$ or radius in other algorithms which are much more challenging to tune. For example, DBSCAN (Ester et al., 1996) or mean-shift clustering (Comaniciu & Meer, 2002) are very sensitive to the above hyperparameters. On the other hand, $\\kappa$ can be framed as a lever to balance the performance and speed of optimizing compilation which abstracts away the aforementioned challenges, enabling the Threshold-based Swift Meta-Search that identifies the optimal number of clusters. ", + "bbox": [ + 174, + 377, + 823, + 474 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Hyperparameter tuning. Hyperparameter tuning is a very important task in machine learningbased tools and models. As such, we present the hyperparameters we used for the evaluation in Table 7 (in appendix), which its tuning took several days. For the hyperparameters in Table 8 (in appendix), we used the same set of values that were used in the AutoTVM paper (Chen et al., 2018b) in order to conduct a fair comparison or CHAMELEON. Additionally, for parameters used in the Adaptive Exploration module, which is not present in AutoTVM, we have tuned the hyperparameters using the set of layers presented in Table 5 (in appendix). We emphasize, however, that the hyperparameters have been tuned offline before the deployment of CHAMELEON, and the hyperparameters are not changed during the use of the framework or the experimentation. So the tuning overhead is not part of the compilation after the Adaptive Exploration module is tuned once before releasing the compiler to the deployment practitioners. ", + "bbox": [ + 174, + 492, + 825, + 643 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 666, + 313, + 681 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We integrate CHAMELEON into TVM (Chen et al., 2018a) to perform component evaluation and compare with AutoTVM (Chen et al., 2018b). We first evaluate components of CHAMELEON in Section 4.1 and Section 4.2 on set of convolution layers sampled from AlexNet (Krizhevsky et al., 2012), VGG-16 (Simonyan & Zisserman, 2015), and ResNet-18 (He et al., 2016). Then we provide end-to-end evaluation of CHAMELEON on both set of layers and end-to-end deep models, in Section 4.3. Due to space limitations, we present only the representative plots in the paper, and the complete set of results and the details of the parameters are provided in the appendix. ", + "bbox": [ + 174, + 698, + 825, + 795 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 ADAPTIVE EXPLORATION: IMPROVING EFFICACY OF SEARCH ALGORITHM ", + "bbox": [ + 176, + 813, + 733, + 828 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the previous approach (Chen et al., 2018b), authors have built a cost model to estimate fitness instead of performing costly measurements on real hardware, then used simulated annealing to find potentially optimal configurations. Figure 7(a) compares the number of search steps taken per iteration to reach or converge to the solution in simulated annealing and Adaptive Exploration, respectively. Overall, observation is that CHAMELEON’s Adaptive Exploration requires $2 . 8 8 \\times$ less search steps compared to simulated annealing to find good solution. This comes from the ability of the reinforcement learning algorithm in Adaptive Exploration Module to (1) learn the correlation between different dimensions, and (2) reuse information across different iterations, instead of starting from scratch while naively relying on the stochastic guarantees of simulated annealing process. ", + "bbox": [ + 174, + 840, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/9ad0ad2777e161304155b61c563bcb6dbb86606a4be75504fda18bc6925fd81c.jpg", + "image_caption": [ + "Figure 7: Component evaluation of CHAMELEON. " + ], + "image_footnote": [], + "bbox": [ + 174, + 98, + 821, + 267 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 300, + 825, + 342 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 ADAPTIVE SAMPLING: REDUCING NUMBER OF COSTLY HARDWARE MEASUREMENTS ", + "text_level": 1, + "bbox": [ + 173, + 358, + 810, + 373 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 7(b) summarizes the effect of applying CHAMELEON’s Adaptive Sampling module on simulated annealing and reinforcement learning based search. First, the results show that using Adaptive Sampling helps the framework to make less hardware measurements regardless of the search algorithm used. The Adaptive Sampling algorithm reduces the number of measurements by $1 . 9 8 \\times$ when used with simulated annealing and $2 . 3 3 \\times$ with reinforcement learning One observation is that the Adaptive Sampling is more effective with reinforcement learning search. This comes from the reinforcement learning agent’s capacity to better localize the search to meaningful samples (exploitation) while still aiming to find good solution by making diverse search (exploration). ", + "bbox": [ + 173, + 385, + 825, + 497 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Diversity exploration of AutoTVM aims to spread out the candidate configurations with a regularizing effect that fosters uniform sampling. In contrast, our Adaptive Sampling uses a clustering algorithm to perform more measurements on the regions with higher likelihood of achieving better output performance, leading to a non-uniform sampling. While AutoTVM states that diversity-aware selection had no meaningful impact on ", + "bbox": [ + 174, + 517, + 419, + 683 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/80fe02a72091d19f2baf576de93c71118e759109eb64c4eccd2ecdfc04c1cc03.jpg", + "image_caption": [ + "Figure 8: Comparison to AutoTVM’s diversity exploration. " + ], + "image_footnote": [], + "bbox": [ + 446, + 515, + 805, + 656 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "most of the evaluated workloads, our Adaptive Sampling brings significant improvement as depicted in Figure 8. As shown, Adaptive Sampling brings an average of $1 3 . 5 \\%$ and $1 9 . 0 \\%$ improvement on simulated annealing and reinforcement learning, respectively. ", + "bbox": [ + 174, + 684, + 825, + 726 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 INTEGRATION: REDUCING OPTIMIZATION TIME AND OUTPUT INFERENCE TIME ", + "text_level": 1, + "bbox": [ + 174, + 742, + 769, + 757 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "CHAMELEON integrates two components into the workflow: RL-based Adaptive Exploration (AE) and Adaptive Sampling (AS). This section compares the performance of CHAMELEON with AutoTVM (Chen et al., 2018b) that leverages Simulated Annealing (SA) for its exploration. ", + "bbox": [ + 174, + 768, + 825, + 810 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Layer evaluation. Figure 9 shows the trend of output code performance of ResNet-18’s 11th layer over number of hardware measurements during optimization. The figure illustrates that our Adaptive Exploration finds better configurations than simulated annealing which results in better output code performance, and the Adaptive Sampling reduces number of hardware measurements significantly during optimization. Also, CHAMELEON’s Adaptive Exploration and Adaptive Sampling working in tandem emits better code with shorter optimization time than others. As such, Figure 10(a) compares optimization time and the performance of the output code in CHAMELEON and AutoTVM to confirm the observation. CHAMELEON achieved $1 . 1 7 \\times$ better performance with $4 . 8 2 \\times$ shorter optimization time compared to AutoTVM. Overall, the results suggest that our Adaptive Exploration effectively maneuvers the design space, and Adaptive Sampling reduces hardware measurements and the overall optimization time while even improving output performance. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/902919044cc53df38e045268760dae15b3b4e717f01247ba0a622d6db4bdd025.jpg", + "image_caption": [ + "Figure 9: Layer evaluation of output performance for ResNet-18’s 11th layer. " + ], + "image_footnote": [], + "bbox": [ + 227, + 102, + 769, + 215 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 251, + 825, + 306 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/f49f6ccd90c211ab039138d0baab9e8213ca2e227b2b6a5c97fe54adf73f529c.jpg", + "table_caption": [ + "HAMELEON significantly reduces number of hardware measurements (from 800 to 392) " + ], + "table_footnote": [], + "table_body": "
NETWORKSA (AutoTVM)AESA + ASAE+ AS (CHAMELEON)
AlexNet4.31 Hours4.06 Hours1.25 Hours1.20 Hours
VGG-1611.18 Hours8.82 Hours2.57 Hours1.95 Hours
ResNet-189.13 Hours7.39 Hours2.14 Hours2.13 Hours
", + "bbox": [ + 274, + 611, + 715, + 690 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "End-to-end evaluation. Up until now, we have focused on evaluation with subset of layers. Now we continue our discussion to the applicability of CHAMELEON to optimization of end-to-end deep neural networks. Figure 10(b) shows that CHAMELEON spends $3 . 5 9 \\times$ , $5 . 7 3 \\times$ , and $4 . 2 8 \\times$ less time than AutoTVM to optimize AlexNet, VGG-16, and ResNet-18, respectively. On average, our work shows $4 . 4 5 \\times$ optimization time speedup while achieving up to $6 . 4 \\%$ improvement in terms of performance of output code. Inference time in Figure 10(b) illustrates the speedup for optimized code. Raw numbers are available in Table 2 and Table 3. All in all, such improvements result from efficient Adaptive Exploration and the reduced number of hardware measurements from Adaptive Sampling. ", + "bbox": [ + 173, + 321, + 825, + 434 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/5bd4f96ac9884f3aa4fdc206e0be0d33f6da796de011e9711d54ecc3131f0c4e.jpg", + "image_caption": [ + "Figure 10: Layer and end-to-end evaluation. Dashed lines denote AutoTVM’s performance. " + ], + "image_footnote": [], + "bbox": [ + 179, + 445, + 810, + 578 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/ca70579f4f7091141d0d39ecf57f6ae73c519db2a2fc82ce4327c505a2be72a1.jpg", + "table_caption": [ + "Table 2: End-to-end evaluation of the optimization time for deep networks. " + ], + "table_footnote": [], + "table_body": "
NETWORKSA (AutoTVM)AESA + ASAE +AS (CHAMELEON)
AlexNet1.0277 ms1.0207 ms0.9762 ms0.9673 ms
VGG-163.9829 ms3.9710 ms3.8733 ms3.8458 ms
ResNet-181.0258 ms0.9897 ms0.9897 ms0.9831 ms
", + "bbox": [ + 282, + 728, + 709, + 809 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 3: End-to-end evaluation of the output performance for deep networks. ", + "bbox": [ + 246, + 819, + 750, + 833 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 176, + 849, + 354, + 866 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "CHAMELEON uniquely offers a solution that exclusively enables (i) Reinforcement Learning and (ii) Sampling in the context of (iii) Optimizing Compilers for neural networks. As such, we discuss the related work from each of the three independent research directions. ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Optimizing compilers. TensorComprehensions (Vasilache et al., 2018) and TVM (Chen et al., 2018a) use genetic algorithm and simulated annealing to choose parameters of polyhedral optimization for neural networks. In a more general context, some computing libraries (Whaley & Dongarra, 1998; Frigo & Johnson, 1998) make use of black box optimization and also profiling-based compilation passes (Chang et al., 1991; Novillo, 2014) utilize runtime information to generate optimized code. Later, AutoTVM (Chen et al., 2018b) incorporates learning with boosted trees within the cost model for TVM to reduce the number of real hardware measurements. While CHAMELEON is inspired and builds on these prior works, unlike them, it is based on reinforcement learning for Adaptive Exploration, and Adaptive Sampling that leverages clustering to reduce the number of measurements. ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Reinforcement learning for hyper-parameter optimization. There are a growing body of studies on using reinforcement learning to perform various optimizations (Gao et al., 2018; Mirhoseini et al., 2017; Nareyek, 2003; Mao et al., 2016; Xu et al., 2018; Mao et al., 2019) for a variety of objectives including hyper-parameter optimization for neural networks. For instance, DeepArchitect (Negrinho & Gordon, 2017) and NAS (Zoph & Le, 2017) use reinforcement learning to automate the process of designing deep neural network models and their associated parameters. HAQ (Wang et al., 2019) and ReLeQ (Elthakeb et al., 2018) use reinforcement learning to chose levels of quantization for the layers of a given deep neural network. AMC (He et al., 2018) formulates neural network compression as a RL problem. A most recent effort (Paliwal et al., 2020)–which will be published concurrent to ours in ICLR 2020–combined RL with graph neural networks and genetic algorithms to optimize DNN execution. Our work exclusively explores a different problem, that is optimizing compilers using reinforcement learning. ", + "bbox": [ + 174, + 244, + 825, + 411 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Sampling algorithms for learning. Active learning is a broad field (Settles, 2009; Cohn et al., 1996; Sugiyama, 2006; Cai et al., 2013; Goetz et al., 2018; Wu et al., 2019) that uses a measure of the change in the model to decide which training data elements should be used to update the model. Passive learning (Yu & Kim, 2010; O’Neill et al., 2017) is an alternative view that independent of the model, analyze the distribution of the training data set and selects a subset. The Adaptive Sampling algorithm for CHAMELEON shares similarities with Passive learning but it differs in its context. The sampling is designed to reduce the number of samples (configuration) for hardware measurement from the exploration of the design space whilst performing an optimization to accelerate the process. ", + "bbox": [ + 174, + 428, + 825, + 540 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 560, + 318, + 577 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We present CHAMELEON to allow optimizing compilers to adapt to unseen design spaces of code schedules to reduce the optimization time. This paper is also an initial effort to bring reinforcement learning to the realm of optimizing compilers for neural networks, and we also develop an Adaptive Sampling with domain-knowledge inspired Sample Synthesis to not only reduce the number of samples required to navigate the design space but also augment its quality in terms of fitness. Experimentation with real-world deep models shows that CHAMELEON not only reduces the time for compilation significantly, but also improves the quality of the code. This encouraging result suggests a significant potential for various learning techniques to optimizing deep learning models. ", + "bbox": [ + 174, + 593, + 825, + 704 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENT ", + "text_level": 1, + "bbox": [ + 176, + 727, + 357, + 741 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank the anonymous reviewers for their insightful comments. We also thank Jinwon Lee and Jangho Kim for the fruitful discussions and feedbacks on the manuscript. This work was in part supported by generous gifts from Qualcomm, Google, Microsoft, and Xilinx as well as the Semiconductor Research Corporation (SRC) contract #2019-SD-2884, National Science Foundation (NSF) awards CNS#1703812, ECCS#1609823, CCF#1553192, Air Force Office of Scientific Research (AFOSR) Young Investigator Program (YIP) award #FA9550-17-1-0274, National Institute of Health (NIH) award #R01EB028350, and Air Force Research Laboratory (AFRL) and Defense Advanced Research Project Agency (DARPA) under agreement number #FA8650-20-2-7009. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. 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 AFRL, DARPA or the U.S. Government. 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", + "bbox": [ + 176, + 454, + 823, + 481 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 264, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A EXPERIMENTAL SETUP ", + "bbox": [ + 176, + 135, + 405, + 152 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 DNN MODELS AND LAYERS ", + "bbox": [ + 174, + 166, + 416, + 181 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/1f4b9e5e2139f8aea12a8b1c0cfa48d7fa165cc7ca2b1a98d173ffb34c503183.jpg", + "table_caption": [ + "Table 4: Details of the DNN models used in evaluating CHAMELEON. " + ], + "table_footnote": [], + "table_body": "
NETWORKDATASETNUMBER OF TASKS
AlexNetImageNet5
VGG-16ImageNet9
ResNet-18ImageNet12
", + "bbox": [ + 344, + 229, + 651, + 296 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/51615402d54a7b4125395cfc793ce077919a4bdac41bb9811ad6e61dc1f91fdb.jpg", + "table_caption": [ + "Table 5: Details of the layers used in evaluating CHAMELEON. " + ], + "table_footnote": [], + "table_body": "
NAMEMODELLAYER TYPETASK INDEX
L1AlexNetconvolution1
L2AlexNetconvolution4
L3VGG-16convolution1
L4VGG-16convolution2
L5VGG-16convolution4
L6ResNet-18convolution6
L7ResNet-18convolution9
L8ResNet-18convolution11
", + "bbox": [ + 326, + 363, + 669, + 492 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 HARDWARE SPECIFICATION", + "text_level": 1, + "bbox": [ + 176, + 522, + 411, + 537 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/39f0387a3536ecfabd68083e103df74299ebe51fca4f6e995f879cab9e1e0c17.jpg", + "table_caption": [ + "Table 6: Details of the hardware used for evaluation of CHAMELEON. " + ], + "table_footnote": [], + "table_body": "
SPECIFICATIONSDETAILS
GPUTitan Xp
Host CPU3.4G Hz Intel Core i7
Main Memory32GB 2400 MHz DDR3
", + "bbox": [ + 354, + 583, + 643, + 650 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 HYPER-PARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 367, + 117 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/eab548784d37517f4ceb800bdbdcb7b6747845330e0f86233dcd272f731a93cf.jpg", + "table_caption": [ + "Table 7: Hyper-parameters uses in CHAMELEON. " + ], + "table_footnote": [], + "table_body": "
HYPERPARAMETERVALUEDESCRIPTION
iterationopt16number of iterations for optimization process (equivalent to 1Ooo hardware measurements)
modeGBT bGBTxgb-regtype of loss used for cost model
64maximum batch size of planning in GBT(Chen & Guestrin,2016) cost model per iteration of optimization process
episoderl128number of episodes for reinforcement learning
steprl500maximum steps of one reinforcement learning episode
thresholdmeta2.5threshold used for meta-search in sampling
", + "bbox": [ + 191, + 165, + 807, + 296 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/b4cb30e55c7f5113d4da6353493ce0bd64e828363fb677a7294e655ec60e2b5c.jpg", + "table_caption": [ + "Table 8: Hyper-parameters uses in AutoTVM (Chen et al., 2018b). " + ], + "table_footnote": [], + "table_body": "
HYPERPARAMETERVALUEDESCRIPTION
£(bGBT)1000totalnumber of hardwaremeasurements
modeGBTxgb-regtype of loss used for cost model
bGBT64batch size of planning in GBT(Chen& Guestrin,2016)
nsa128number of Markov chains in parallel simulated annealing
stepsa500maximum steps of one simulated annealing run
", + "bbox": [ + 218, + 362, + 779, + 454 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/e887780cd91b6b8d0629642ce2ecdde3521610a9257416b939239f4a68ca4af4.jpg", + "table_caption": [ + "Table 9: Hyper-parameters used in CHAMELEON’s PPO (Schulman et al., 2017) search agent. " + ], + "table_footnote": [], + "table_body": "
HYPERPARAMETERVALUE
Adam Step Size Discount Factor1×10-3
GAE Parameter0.9
Number of Epochs0.99
Clipping Parameter3
Value Coefficient0.3
1.0
Entropy Coefficient0.1
", + "bbox": [ + 388, + 520, + 609, + 638 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B ADDITIONAL EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 537, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.1 OPTIMIZATION TIME BREAKDOWN FOR DNN MODELS ", + "text_level": 1, + "bbox": [ + 174, + 133, + 601, + 148 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/026a53d0d37fc394dbde9ee260ed92631908ad47f92de723d2f9d7a13438b6e1.jpg", + "image_caption": [ + "Figure 11: AutoTVM optimization time for AlexNet (Krizhevsky et al., 2012) and VGG-16 (Simonyan & Zisserman, 2015), and ResNet-18 (He et al., 2016) on Titan Xp. Numbers in bars denote fraction of time for measurements. " + ], + "image_footnote": [], + "bbox": [ + 174, + 165, + 818, + 305 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/3c8b2503e8f20893f004a618595f804ee4202bdce892dd2dd37cf5019a6c7a97.jpg", + "image_caption": [ + "Figure 12: Layer evaluations for AlexNet (Krizhevsky et al., 2012). " + ], + "image_footnote": [], + "bbox": [ + 181, + 468, + 812, + 655 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/01cae0b30b91a657d3523b35270fbc77354a18f41266d34fb3123c16b6bd7b20.jpg", + "image_caption": [ + "Figure 13: Layer evaluations for VGG-16 (Simonyan & Zisserman, 2015). " + ], + "image_footnote": [], + "bbox": [ + 181, + 99, + 813, + 369 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/8d074821723c328e9d090e41a9d4a947aea4d674ac66384eb9bb3c3baedf687b.jpg", + "image_caption": [ + "Figure 14: Layer evaluations for ResNet-18 (He et al., 2016). 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However, the complexity of the tensor operations in DNNs and the volatility of algo-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 494, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 505, + 509 + ], + "score": 1.0, + "content": "rithms, which has led to unprecedented rate of innovation (LeCun, 2019), calls for developing auto-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "mated compilation frameworks. 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TVM and TensorComprehensions are based on random or genetic algorithms to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "search the space of optimized code for neural networks. AutoTVM (Chen et al., 2018b) builds on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "top of TVM and leverage boosted trees (Chen & Guestrin, 2016) as part of the search cost model to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "avoid measuring the fitness of each solution (optimized candidate neural network code), and instead", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "predict its fitness. 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To solve this problem, we first question the very statistical guarantees which the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "aforementioned optimization passes rely on. The current approaches are oblivious to the patterns in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "the design space of schedules that are available for exploitation, and causes inefficient search or even", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "score": 1.0, + "content": "converges to solutions that may even be suboptimal. Also, we notice that current approaches rely", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "on greedy sampling that neglects the distribution of the candidate solutions (configurations). 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However, the current paradigm of exe-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "score": 1.0, + "content": "cuting neural networks either relies on hand-optimized libraries, traditional com-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "score": 1.0, + "content": "pilation heuristics, or very recently genetic algorithms and other stochastic meth-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "score": 1.0, + "content": "ods. These methods suffer from frequent costly hardware measurements rendering", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 469, + 311 + ], + "score": 1.0, + "content": "them not only too time consuming but also suboptimal. As such, we devise a so-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 470, + 322 + ], + "score": 1.0, + "content": "lution that can learn to quickly adapt to a previously unseen design space for code", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 322, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 142, + 322, + 469, + 333 + ], + "score": 1.0, + "content": "optimization, both accelerating the search and improving the output performance.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 469, + 344 + ], + "score": 1.0, + "content": "This solution dubbed CHAMELEON leverages reinforcement learning whose solu-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 355 + ], + "score": 1.0, + "content": "tion takes fewer steps to converge, and develops an adaptive sampling algorithm", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 469, + 367 + ], + "score": 1.0, + "content": "that not only focuses on the costly samples (real hardware measurements) on rep-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 365, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 469, + 376 + ], + "score": 1.0, + "content": "resentative points but also uses a domain-knowledge inspired logic to improve the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 470, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 470, + 388 + ], + "score": 1.0, + "content": "samples itself. Experimentation with real hardware shows that CHAMELEON pro-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 386, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 166, + 399 + ], + "score": 1.0, + "content": "vides", + "type": "text" + }, + { + "bbox": [ + 166, + 387, + 192, + 397 + ], + "score": 0.84, + "content": "4 . 4 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 386, + 469, + 399 + ], + "score": 1.0, + "content": "speed up in optimization time over AutoTVM, while also improving", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 357, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 333, + 410 + ], + "score": 1.0, + "content": "inference time of the modern deep networks by", + "type": "text" + }, + { + "bbox": [ + 333, + 398, + 354, + 408 + ], + "score": 0.85, + "content": "5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 397, + 357, + 410 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15, + "bbox_fs": [ + 141, + 244, + 470, + 410 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 206, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 208, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 208, + 442 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 504, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 504, + 464 + ], + "score": 1.0, + "content": "The enormous computational intensity of Deep Neural Networks (DNNs) have resulted in develop-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "ing either hand-optimized kernels, such as NVIDIA cuDNN or Intel MKL that serve as backend for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "a variety of programming environment such as TensorFlow (Abadi et al., 2016) and PyTorch (Paszke", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 483, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 505, + 498 + ], + "score": 1.0, + "content": "et al., 2019). However, the complexity of the tensor operations in DNNs and the volatility of algo-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 494, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 505, + 509 + ], + "score": 1.0, + "content": "rithms, which has led to unprecedented rate of innovation (LeCun, 2019), calls for developing auto-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "mated compilation frameworks. To imitate or even surpass the success of hand-optimized libraries,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "recent research has developed stochastic optimization passes: for general code, STOKE (Schkufza", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "et al., 2013), and neural network code, TVM (Chen et al., 2018a) and TensorComprehensions (Vasi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "lache et al., 2018). TVM and TensorComprehensions are based on random or genetic algorithms to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "search the space of optimized code for neural networks. AutoTVM (Chen et al., 2018b) builds on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "top of TVM and leverage boosted trees (Chen & Guestrin, 2016) as part of the search cost model to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "avoid measuring the fitness of each solution (optimized candidate neural network code), and instead", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "predict its fitness. However, even with these innovations the optimizing compilation time can be", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 484, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 484, + 606 + ], + "score": 1.0, + "content": "around 10 hours for ResNet-18 (He et al., 2016), and even more for deeper or wider networks.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 451, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Since the general objective is to unleash new possibilities by developing automatic optimization", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "passes, long compilation time hinders innovation and could put the current solutions in a position of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "questionable utility. To solve this problem, we first question the very statistical guarantees which the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "aforementioned optimization passes rely on. The current approaches are oblivious to the patterns in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "the design space of schedules that are available for exploitation, and causes inefficient search or even", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "score": 1.0, + "content": "converges to solutions that may even be suboptimal. Also, we notice that current approaches rely", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "on greedy sampling that neglects the distribution of the candidate solutions (configurations). While", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "greedy sampling that passively filter samples based on the fitness estimations from the cost models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "work, many of their hardware measurements (required for optimization) tend to be redundant and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "wasteful. 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KNOBSDEFINITION
tile_f, tile-y, tile_xFactors for tiling and binding # of filters height,and width of feature maps.
tile_rc,tile_ry, tile_rxFactors for tiling reduction axis such as # of channels,height,and width of filters.
auto_unroll_max_stepThreshold of number of steps in the loop to be automatically unrolled.
unroll_explicitExplicitly unroll loop, this may let code generator to generate pragma unroll hint.
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KNOBSDEFINITION
tile_f, tile-y, tile_xFactors for tiling and binding # of filters height,and width of feature maps.
tile_rc,tile_ry, tile_rxFactors for tiling reduction axis such as # of channels,height,and width of filters.
auto_unroll_max_stepThreshold of number of steps in the loop to be automatically unrolled.
unroll_explicitExplicitly unroll loop, this may let code generator to generate pragma unroll hint.
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Therefore, current", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "long compilation time can be a hindrance to deploying DNN in various hardware, hence a major", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 388, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 388, + 640 + ], + "score": 1.0, + "content": "bottleneck in enabling intelligence on wider range of target platforms.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 506, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "Furthermore, as we explore various neural topologies (Xie et al., 2019; Wortsman et al., 2019) for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "better performance as illustrated in Ahn et al. 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The", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "long optimization times are multiplied with such trend, leaving the practical utility of the current", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 697, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 505, + 713 + ], + "score": 1.0, + "content": "compiler solutions to question. 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Having formulated the RL-based Adaptive Exploration Module, an itera-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "score": 1.0, + "content": "tion of our optimization begins with a set of initial configurations and takes multiple search steps", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "(episode) for each of the configurations. 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In other words, by repeating multiple episodes and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 606, + 473, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 473, + 620 + ], + "score": 1.0, + "content": "iterations, our Adaptive Exploration Module gradually learns to locate good configurations.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 106, + 633, + 390, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 391, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 391, + 646 + ], + "score": 1.0, + "content": "3.3 ADAPTIVE SAMPLING: ADAPTING TO THE DISTRIBUTION TO", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 310, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 310, + 657 + ], + "score": 1.0, + "content": "REDUCE COSTLY HARDWARE MEASUREMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Reducing number of costly hardware measurements. 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We also find that, while", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "the design space’s surface is discrete and un-smooth, a large fraction of configurations within each", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "cluster achieve similar runtime (Figure 6). 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We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "take an inspiring approach that avoids naive dependence on the stochastic guarantee of simulated", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "annealing and leverage a technique that can learn to adapt to unseen design space to not only ac-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 504, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 504, + 231 + ], + "score": 1.0, + "content": "celerate convergence but also bring some performance gains. 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Having formulated the RL-based Adaptive Exploration Module, an itera-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "score": 1.0, + "content": "tion of our optimization begins with a set of initial configurations and takes multiple search steps", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "(episode) for each of the configurations. As shown in Figure 4, the agent makes an action and ap-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "plies it to the configuration using configuration updater to get another configuration that potentially", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 147, + 576 + ], + "score": 1.0, + "content": "has better", + "type": "text" + }, + { + "bbox": [ + 147, + 563, + 154, + 574 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 562, + 435, + 576 + ], + "score": 1.0, + "content": ". After finishing multiple search steps in the episode, all configurations", + "type": "text" + }, + { + "bbox": [ + 435, + 563, + 449, + 574 + ], + "score": 0.88, + "content": "S _ { \\Theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "are evaluated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "using a cost model, which its return values are used as a surrogate reward to update our agent, to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 584, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 429, + 598 + ], + "score": 1.0, + "content": "reduce the number of costly hardware measurements. By taking this approach,", + "type": "text" + }, + { + "bbox": [ + 430, + 585, + 437, + 596 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 584, + 450, + 598 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 450, + 585, + 464, + 596 + ], + "score": 0.89, + "content": "S _ { \\Theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 584, + 506, + 598 + ], + "score": 1.0, + "content": "improves", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "as our module progresses through the episodes. In other words, by repeating multiple episodes and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 606, + 473, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 473, + 620 + ], + "score": 1.0, + "content": "iterations, our Adaptive Exploration Module gradually learns to locate good configurations.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 519, + 506, + 620 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 633, + 390, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 391, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 391, + 646 + ], + "score": 1.0, + "content": "3.3 ADAPTIVE SAMPLING: ADAPTING TO THE DISTRIBUTION TO", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 310, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 310, + 657 + ], + "score": 1.0, + "content": "REDUCE COSTLY HARDWARE MEASUREMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Reducing number of costly hardware measurements. After the exploration step (regardless of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "the exploration method), we observe that the candidate configurations are clustered in subregions of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "the design space and these clusters are non-uniformly distributed (Figure 5). We also find that, while", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "the design space’s surface is discrete and un-smooth, a large fraction of configurations within each", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "cluster achieve similar runtime (Figure 6). Utilizing these characteristics of the design space, we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "devise Adaptive Sampling that can sample a new set of candidates, by adapting to the shape of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 218, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 504, + 230 + ], + "score": 1.0, + "content": "design space and the non-uniformity of the distribution while leaving the performance of optimiza-", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "score": 1.0, + "content": "tion intact. We first leverage clustering algorithm to find configurations that are representative of", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "each cluster; the sampling module uses centroids as the representative configurations. 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We first leverage clustering algorithm to find configurations that are representative of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "each cluster; the sampling module uses centroids as the representative configurations. Our Adaptive", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 252, + 501, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 501, + 264 + ], + "score": 1.0, + "content": "Sampling iterates over a different number of clusters for their respective centroids and the L2 loss.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "In the context of optimizing compiler, selecting the number of centroids for clustering entails making", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "the important tradeoff between selecting more centroids for better performance or fewer centroids", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "for a reduced number of hardware measurements. As such, we must devise a method that would", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "automatically make the tradeoff in a reasonable manner. We take advantage of the decreasing trend", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "in the aforementioned L2 loss as we increase the number of centroids, and devise a Threshold-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "Swift Meta-Search to determine the number of clusters. By setting the threshold (hyperparameter)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "it allows the compiler to determine the point of diminishing return (knee of the curve), inflection", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "point beyond which fewer centroids may lead to performance degradation and more clusters would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "prolong the optimization substantially. Overall, our sampling curtails the number of hardware mea-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 367, + 499, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 499, + 379 + ], + "score": 1.0, + "content": "surements so that it is just enough to subsume the entire subspace of the candidate configurations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 504, + 405 + ], + "score": 1.0, + "content": "Improving candidate configurations using sample synthesis. While the above sampling algo-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "rithm significantly reduces the number of hardware measurements compared to the conventional", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "greedy sampling, without impacting the performance of the output code, we are still left with a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "critical issue of redundancy among the candidate configurations. We find that the exploration al-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 435, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 450 + ], + "score": 1.0, + "content": "gorithm (regardless of the type) combined with the greedy sampling frequently leads to redundancy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "among the candidate configurations over different iterations of optimization due to the overfitting", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "of the cost model from the greediness of the sampling. Even though the exploration algorithm tries", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "score": 1.0, + "content": "to explore unvisited regions of the design space, these explored (not exploited) configurations are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "discarded due to the greedy sampling which entirely depends on the cost model for its selections of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "the configurations. Therefore, the current greedy sampling algorithm has its limitation in focusing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 501, + 355, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 355, + 515 + ], + "score": 1.0, + "content": "the hardware measurements to the same region over and over.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "On the other hand, we find that from a code optimization point of view, we know that many of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "automated approaches for black-box optimization are prone to invalid configurations, which results", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "from too large a tile that goes over the input feature map boundary or errors during memory accesses", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "(cannot be solved analytically). 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When our compiler runs into redundant samples, the proposed synthesis method", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 432, + 608 + ], + "score": 1.0, + "content": "analyzes the candidate samples to determine the most probable (most frequent", + "type": "text" + }, + { + "bbox": [ + 433, + 597, + 442, + 605 + ], + "score": 0.79, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "mode function)", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "non-invalid choice for each knob to come up with a new configuration. This statistical combina-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "tion of the most frequent knob settings yield configurations that combine the strengths of different", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "knobs to converge to a better overall solution. In spirit, the recombination (crossover) operator in", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "genetic algorithms also tries to combine the best features of the solutions with high fitness values.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 650, + 466, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 466, + 663 + ], + "score": 1.0, + "content": "Algorithm 1 presents the integration of our Adaptive Sampling and the Sample Synthesis.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49 + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 248, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 250, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 250, + 690 + ], + "score": 1.0, + "content": "3.4 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 56 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Architecture exploration for the adaptive exploration. We use Proximal Policy Optimization", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 107, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 134, + 721 + ], + "score": 0.26, + "content": "( P P O )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(Schulman et al., 2017), a policy gradient that has been shown to adapt to various problems", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "and have good sample complexity, as our reinforcement learning algorithm. Since reinforcement", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 58 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 77, + 295, + 185 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 77, + 295, + 185 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 77, + 295, + 185 + ], + "spans": [ 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"configurations in the cluster.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + } + ], + "index": 13.75 + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 262 + ], + "lines": [], + "index": 19.5, + "bbox_fs": [ + 105, + 218, + 505, + 264 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "In the context of optimizing compiler, selecting the number of centroids for clustering entails making", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "the important tradeoff between selecting more centroids for better performance or fewer centroids", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "for a reduced number of hardware measurements. As such, we must devise a method that would", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "automatically make the tradeoff in a reasonable manner. We take advantage of the decreasing trend", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "in the aforementioned L2 loss as we increase the number of centroids, and devise a Threshold-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "Swift Meta-Search to determine the number of clusters. By setting the threshold (hyperparameter)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "it allows the compiler to determine the point of diminishing return (knee of the curve), inflection", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "point beyond which fewer centroids may lead to performance degradation and more clusters would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "prolong the optimization substantially. Overall, our sampling curtails the number of hardware mea-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 367, + 499, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 499, + 379 + ], + "score": 1.0, + "content": "surements so that it is just enough to subsume the entire subspace of the candidate configurations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 267, + 506, + 379 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 504, + 405 + ], + "score": 1.0, + "content": "Improving candidate configurations using sample synthesis. While the above sampling algo-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "rithm significantly reduces the number of hardware measurements compared to the conventional", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "greedy sampling, without impacting the performance of the output code, we are still left with a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "critical issue of redundancy among the candidate configurations. We find that the exploration al-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 435, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 450 + ], + "score": 1.0, + "content": "gorithm (regardless of the type) combined with the greedy sampling frequently leads to redundancy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "among the candidate configurations over different iterations of optimization due to the overfitting", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "of the cost model from the greediness of the sampling. 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When our compiler runs into redundant samples, the proposed synthesis method", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 432, + 608 + ], + "score": 1.0, + "content": "analyzes the candidate samples to determine the most probable (most frequent", + "type": "text" + }, + { + "bbox": [ + 433, + 597, + 442, + 605 + ], + "score": 0.79, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "mode function)", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "non-invalid choice for each knob to come up with a new configuration. This statistical combina-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "tion of the most frequent knob settings yield configurations that combine the strengths of different", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "knobs to converge to a better overall solution. In spirit, the recombination (crossover) operator in", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "genetic algorithms also tries to combine the best features of the solutions with high fitness values.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 650, + 466, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 466, + 663 + ], + "score": 1.0, + "content": "Algorithm 1 presents the integration of our Adaptive Sampling and the Sample Synthesis.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 519, + 506, + 663 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 248, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 250, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 250, + 690 + ], + "score": 1.0, + "content": "3.4 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 56 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Architecture exploration for the adaptive exploration. We use Proximal Policy Optimization", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 107, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 134, + 721 + ], + "score": 0.26, + "content": "( P P O )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(Schulman et al., 2017), a policy gradient that has been shown to adapt to various problems", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "and have good sample complexity, as our reinforcement learning algorithm. Since reinforcement", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "learning could incur computational overhead that could prolong the optimization time, we optimize", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the actor-critic networks through architecture exploration to find good tradeoff for size of these", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 274, + 468, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 468, + 289 + ], + "score": 1.0, + "content": "networks (that determines the computational overhead) and the optimization performance.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 58, + "bbox_fs": [ + 106, + 699, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 91, + 506, + 231 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 82, + 329, + 94 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 330, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 330, + 96 + ], + "score": 1.0, + "content": "Algorithm 1 Adaptive Sampling and Sample Synthesis", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 109, + 91, + 506, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 91, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 109, + 91, + 506, + 231 + ], + "score": 0.738, + "html": "
1: procedure ADAPTIVESAMPLING(SΘ, UΘ)
2:V se: candidate configs, ve: visited configs new_candidates ← @,previous_loss ←0
3:for k in range(8, 64) do
4:new_candidates,clusters,L2_loss ← K-means.run(se,k)
5:if Threshold × L2_loss ≥ previous_loss then break>Exit loop at knee of loss curve
6:previous_loss ←L2_loss
7:end for
8:for candidate in new_candidates do Replace visited config with new config
9:if candidate in ve then new_candidates.replace(candidate, mode(s@))
10:end for
11:return new_candidates >Feed to Code Generator to make measurements on hardware end procedure
", + "type": "table", + "image_path": "ba0f0c0e5452359556837de8f1e2bbe5e45df9f526c8ffd7fb4dd5761b9f0d1f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 109, + 91, + 506, + 137.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 137.66666666666666, + 506, + 184.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 109, + 184.33333333333331, + 506, + 230.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "learning could incur computational overhead that could prolong the optimization time, we optimize", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the actor-critic networks through architecture exploration to find good tradeoff for size of these", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 274, + 468, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 468, + 289 + ], + "score": 1.0, + "content": "networks (that determines the computational overhead) and the optimization performance.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 329, + 312 + ], + "score": 1.0, + "content": "Design choices for the adaptive sampling. We use a", + "type": "text" + }, + { + "bbox": [ + 330, + 300, + 339, + 309 + ], + "score": 0.65, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "-means Clustering to determine centroids", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 311, + 504, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 227, + 323 + ], + "score": 1.0, + "content": "of the configurations, because", + "type": "text" + }, + { + "bbox": [ + 228, + 311, + 236, + 320 + ], + "score": 0.82, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 311, + 492, + 323 + ], + "score": 1.0, + "content": "-means has been shown effective in practice and it only requires", + "type": "text" + }, + { + "bbox": [ + 492, + 311, + 501, + 321 + ], + "score": 0.68, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 311, + 504, + 323 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 149, + 334 + ], + "score": 1.0, + "content": "over error", + "type": "text" + }, + { + "bbox": [ + 149, + 324, + 155, + 331 + ], + "score": 0.63, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "or radius in other algorithms which are much more challenging to tune. For example,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "DBSCAN (Ester et al., 1996) or mean-shift clustering (Comaniciu & Meer, 2002) are very sensitive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 306, + 356 + ], + "score": 1.0, + "content": "to the above hyperparameters. On the other hand,", + "type": "text" + }, + { + "bbox": [ + 307, + 344, + 316, + 353 + ], + "score": 0.78, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "can be framed as a lever to balance the perfor-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "mance and speed of optimizing compilation which abstracts away the aforementioned challenges,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 365, + 488, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 488, + 379 + ], + "score": 1.0, + "content": "enabling the Threshold-based Swift Meta-Search that identifies the optimal number of clusters.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "Hyperparameter tuning. Hyperparameter tuning is a very important task in machine learning-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "based tools and models. As such, we present the hyperparameters we used for the evaluation in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 410, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 426 + ], + "score": 1.0, + "content": "Table 7 (in appendix), which its tuning took several days. For the hyperparameters in Table 8 (in ap-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "pendix), we used the same set of values that were used in the AutoTVM paper (Chen et al., 2018b) in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "score": 1.0, + "content": "order to conduct a fair comparison or CHAMELEON. Additionally, for parameters used in the Adap-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "tive Exploration module, which is not present in AutoTVM, we have tuned the hyperparameters", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "using the set of layers presented in Table 5 (in appendix). We emphasize, however, that the hyper-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "parameters have been tuned offline before the deployment of CHAMELEON, and the hyperparameters", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "are not changed during the use of the framework or the experimentation. So the tuning overhead is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "not part of the compilation after the Adaptive Exploration module is tuned once before releasing the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 500, + 273, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 273, + 513 + ], + "score": 1.0, + "content": "compiler to the deployment practitioners.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 108, + 528, + 192, + 540 + ], + "lines": [ + { + "bbox": [ + 104, + 526, + 195, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 195, + 543 + ], + "score": 1.0, + "content": "4 EVALUATION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "We integrate CHAMELEON into TVM (Chen et al., 2018a) to perform component evaluation and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "compare with AutoTVM (Chen et al., 2018b). We first evaluate components of CHAMELEON in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "Section 4.1 and Section 4.2 on set of convolution layers sampled from AlexNet (Krizhevsky et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "2012), VGG-16 (Simonyan & Zisserman, 2015), and ResNet-18 (He et al., 2016). Then we pro-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "vide end-to-end evaluation of CHAMELEON on both set of layers and end-to-end deep models, in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "Section 4.3. Due to space limitations, we present only the representative plots in the paper, and the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 619, + 448, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 448, + 631 + ], + "score": 1.0, + "content": "complete set of results and the details of the parameters are provided in the appendix.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 108, + 644, + 449, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 451, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 451, + 657 + ], + "score": 1.0, + "content": "4.1 ADAPTIVE EXPLORATION: IMPROVING EFFICACY OF SEARCH ALGORITHM", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "In the previous approach (Chen et al., 2018b), authors have built a cost model to estimate fitness", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "instead of performing costly measurements on real hardware, then used simulated annealing to find", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "potentially optimal configurations. Figure 7(a) compares the number of search steps taken per iter-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "ation to reach or converge to the solution in simulated annealing and Adaptive Exploration, respec-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 434, + 722 + ], + "score": 1.0, + "content": "tively. Overall, observation is that CHAMELEON’s Adaptive Exploration requires", + "type": "text" + }, + { + "bbox": [ + 434, + 710, + 460, + 720 + ], + "score": 0.69, + "content": "2 . 8 8 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "less search", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "steps compared to simulated annealing to find good solution. This comes from the ability of the re-", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 91, + 506, + 231 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 82, + 329, + 94 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 330, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 330, + 96 + ], + "score": 1.0, + "content": "Algorithm 1 Adaptive Sampling and Sample Synthesis", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 109, + 91, + 506, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 91, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 109, + 91, + 506, + 231 + ], + "score": 0.738, + "html": "
1: procedure ADAPTIVESAMPLING(SΘ, UΘ)
2:V se: candidate configs, ve: visited configs new_candidates ← @,previous_loss ←0
3:for k in range(8, 64) do
4:new_candidates,clusters,L2_loss ← K-means.run(se,k)
5:if Threshold × L2_loss ≥ previous_loss then break>Exit loop at knee of loss curve
6:previous_loss ←L2_loss
7:end for
8:for candidate in new_candidates do Replace visited config with new config
9:if candidate in ve then new_candidates.replace(candidate, mode(s@))
10:end for
11:return new_candidates >Feed to Code Generator to make measurements on hardware end procedure
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We use a", + "type": "text" + }, + { + "bbox": [ + 330, + 300, + 339, + 309 + ], + "score": 0.65, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "-means Clustering to determine centroids", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 311, + 504, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 227, + 323 + ], + "score": 1.0, + "content": "of the configurations, because", + "type": "text" + }, + { + "bbox": [ + 228, + 311, + 236, + 320 + ], + "score": 0.82, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 311, + 492, + 323 + ], + "score": 1.0, + "content": "-means has been shown effective in practice and it only requires", + "type": "text" + }, + { + "bbox": [ + 492, + 311, + 501, + 321 + ], + "score": 0.68, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 311, + 504, + 323 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 149, + 334 + ], + "score": 1.0, + "content": "over error", + "type": "text" + }, + { + "bbox": [ + 149, + 324, + 155, + 331 + ], + "score": 0.63, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "or radius in other algorithms which are much more challenging to tune. For example,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "DBSCAN (Ester et al., 1996) or mean-shift clustering (Comaniciu & Meer, 2002) are very sensitive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 306, + 356 + ], + "score": 1.0, + "content": "to the above hyperparameters. 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Hyperparameter tuning is a very important task in machine learning-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "based tools and models. As such, we present the hyperparameters we used for the evaluation in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 410, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 426 + ], + "score": 1.0, + "content": "Table 7 (in appendix), which its tuning took several days. For the hyperparameters in Table 8 (in ap-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "pendix), we used the same set of values that were used in the AutoTVM paper (Chen et al., 2018b) in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "score": 1.0, + "content": "order to conduct a fair comparison or CHAMELEON. Additionally, for parameters used in the Adap-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "tive Exploration module, which is not present in AutoTVM, we have tuned the hyperparameters", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "using the set of layers presented in Table 5 (in appendix). We emphasize, however, that the hyper-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "parameters have been tuned offline before the deployment of CHAMELEON, and the hyperparameters", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "are not changed during the use of the framework or the experimentation. So the tuning overhead is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "not part of the compilation after the Adaptive Exploration module is tuned once before releasing the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 500, + 273, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 273, + 513 + ], + "score": 1.0, + "content": "compiler to the deployment practitioners.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 389, + 506, + 513 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 528, + 192, + 540 + ], + "lines": [ + { + "bbox": [ + 104, + 526, + 195, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 195, + 543 + ], + "score": 1.0, + "content": "4 EVALUATION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "We integrate CHAMELEON into TVM (Chen et al., 2018a) to perform component evaluation and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "compare with AutoTVM (Chen et al., 2018b). We first evaluate components of CHAMELEON in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "Section 4.1 and Section 4.2 on set of convolution layers sampled from AlexNet (Krizhevsky et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "2012), VGG-16 (Simonyan & Zisserman, 2015), and ResNet-18 (He et al., 2016). 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Due to space limitations, we present only the representative plots in the paper, and the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 619, + 448, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 448, + 631 + ], + "score": 1.0, + "content": "complete set of results and the details of the parameters are provided in the appendix.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 554, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 644, + 449, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 451, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 451, + 657 + ], + "score": 1.0, + "content": "4.1 ADAPTIVE EXPLORATION: IMPROVING EFFICACY OF SEARCH ALGORITHM", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 644, + 451, + 657 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "In the previous approach (Chen et al., 2018b), authors have built a cost model to estimate fitness", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "instead of performing costly measurements on real hardware, then used simulated annealing to find", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "potentially optimal configurations. Figure 7(a) compares the number of search steps taken per iter-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "ation to reach or converge to the solution in simulated annealing and Adaptive Exploration, respec-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 434, + 722 + ], + "score": 1.0, + "content": "tively. 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NETWORKSA (AutoTVM)AESA + ASAE+ AS (CHAMELEON)
AlexNet4.31 Hours4.06 Hours1.25 Hours1.20 Hours
VGG-1611.18 Hours8.82 Hours2.57 Hours1.95 Hours
ResNet-189.13 Hours7.39 Hours2.14 Hours2.13 Hours
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NETWORKSA (AutoTVM)AESA + ASAE +AS (CHAMELEON)
AlexNet1.0277 ms1.0207 ms0.9762 ms0.9673 ms
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NETWORKSA (AutoTVM)AESA + ASAE+ AS (CHAMELEON)
AlexNet4.31 Hours4.06 Hours1.25 Hours1.20 Hours
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NETWORKSA (AutoTVM)AESA + ASAE +AS (CHAMELEON)
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SPECIFICATIONSDETAILS
GPUTitan Xp
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NAMEMODELLAYER TYPETASK INDEX
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L2AlexNetconvolution4
L3VGG-16convolution1
L4VGG-16convolution2
L5VGG-16convolution4
L6ResNet-18convolution6
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SPECIFICATIONSDETAILS
GPUTitan Xp
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HYPERPARAMETERVALUEDESCRIPTION
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HYPERPARAMETERVALUE
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HYPERPARAMETERVALUEDESCRIPTION
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HYPERPARAMETERVALUE
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KNOBSDEFINITION
tile_f, tile-y, tile_xFactors for tiling and binding # of filters height,and width of feature maps.
tile_rc,tile_ry, tile_rxFactors for tiling reduction axis such as # of channels,height,and width of filters.
auto_unroll_max_stepThreshold of number of steps in the loop to be automatically unrolled.
unroll_explicitExplicitly unroll loop, this may let code generator to generate pragma unroll hint.
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1: procedure ADAPTIVESAMPLING(SΘ, UΘ)
2:V se: candidate configs, ve: visited configs new_candidates ← @,previous_loss ←0
3:for k in range(8, 64) do
4:new_candidates,clusters,L2_loss ← K-means.run(se,k)
5:if Threshold × L2_loss ≥ previous_loss then break>Exit loop at knee of loss curve
6:previous_loss ←L2_loss
7:end for
8:for candidate in new_candidates do Replace visited config with new config
9:if candidate in ve then new_candidates.replace(candidate, mode(s@))
10:end for
11:return new_candidates >Feed to Code Generator to make measurements on hardware end procedure
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NAMEMODELLAYER TYPETASK INDEX
L1AlexNetconvolution1
L2AlexNetconvolution4
L3VGG-16convolution1
L4VGG-16convolution2
L5VGG-16convolution4
L6ResNet-18convolution6
L7ResNet-18convolution9
L8ResNet-18convolution11
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NETWORKDATASETNUMBER OF TASKS
AlexNetImageNet5
VGG-16ImageNet9
ResNet-18ImageNet12
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SPECIFICATIONSDETAILS
GPUTitan Xp
Host CPU3.4G Hz Intel Core i7
Main Memory32GB 2400 MHz DDR3
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HYPERPARAMETERVALUEDESCRIPTION
iterationopt16number of iterations for optimization process (equivalent to 1Ooo hardware measurements)
modeGBT bGBTxgb-regtype of loss used for cost model
64maximum batch size of planning in GBT(Chen & Guestrin,2016) cost model per iteration of optimization process
episoderl128number of episodes for reinforcement learning
steprl500maximum steps of one reinforcement learning episode
thresholdmeta2.5threshold used for meta-search in sampling
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HYPERPARAMETERVALUEDESCRIPTION
£(bGBT)1000totalnumber of hardwaremeasurements
modeGBTxgb-regtype of loss used for cost model
bGBT64batch size of planning in GBT(Chen& Guestrin,2016)
nsa128number of Markov chains in parallel simulated annealing
stepsa500maximum steps of one simulated annealing run
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HYPERPARAMETERVALUE
Adam Step Size Discount Factor1×10-3
GAE Parameter0.9
Number of Epochs0.99
Clipping Parameter3
Value Coefficient0.3
1.0
Entropy Coefficient0.1
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