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+# A SPECTRAL APPROACH TO GENERALIZATION ANDOPTIMIZATION IN NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The recent success of deep neural networks stems from their ability to generalize well on real data; however, Zhang et al. (Zhang et al., 2016) have observed that neural networks can easily overfit random labels. This observation demonstrates that with the existing theory, we cannot adequately explain why gradient methods can find generalizable solutions for neural networks. In this work, we use a Fourierbased approach to study the generalization properties of gradient-based methods over 2-layer neural networks with sinusoidal activation functions. We prove that if the underlying distribution of data has nice spectral properties such as bandlimitedness, then the gradient descent method will converge to generalizable local minima. We also establish a Fourier-based generalization bound for bandlimited spaces, which generalizes to other activation functions. Our generalization bound motivates a grouped version of path norms for measuring the complexity of 2-layer neural networks with ReLU activation functions. We demonstrate numerically that regularization of this group path norm results in neural network solutions that can fit true labels without losing test accuracy while not overfitting random labels.
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs) have achieved state-of-the-art performance on a wide array of diverse tasks (LeCun et al., 2015). A given DNN architecture represents a highly rich space of hypotheses. However, numerous empirical results have demonstrated that a simple stochastic gradient descent (SGD) learner can efficiently search over this space to find a solution that achieves high performance on both training and test data. Despite many successful applications of DNNs to practical tasks such as computer vision (Krizhevsky et al., 2012), natural language processing (Collobert & Weston, 2008) and speech recognition (Hinton et al., 2012), our basic understanding of the factors that drive DNN generalization is still lacking.
+
+Addressing generalization for DNNs is hard for two fundamental reasons: 1) Empirical risk minimization for neural networks is a non-convex optimization problem with possibly many local minima, and 2) Two different local minima with the same training performance can achieve significantly different performance on test data. For these reasons, the neural network optimization method plays an important role in the generalizability of the local minima found. For example, SGD has been empirically shown to outperform large-batch gradient descent (Keskar et al., 2016). Also, the performance of gradient methods can be improved upon by incorporating the geometry of observed data (Duchi et al., 2011; Neyshabur et al., 2015a).
+
+For DNNs, however, a good optimization method is not sufficient for guaranteeing good generalization. Zhang et al. (Zhang et al., 2016) empirically demonstrate that a neural network trained by SGD can easily overfit random labels on the CIFAR-10 (Krizhevsky & Hinton, 2009) data. Yet, the same neural network fitted by the same SGD algorithm achieves good generalization performance for the original CIFAR-10 labels. This observation challenges the ability of traditional learning theory to explain why SGD learns generalizable hypotheses over neural networks. To shed light on this phenomenon, two recent works have developed generalization bounds and complexity measures for neural networks which can distinguish the local minima found for true and random labels. (Bartlett et al., 2017) proves a margin-based generalization bound and shows how it correlates with the generalization risk of DNNs when fitting true and random labels. (Neyshabur et al., 2017) explores different complexity scores for DNNs and how they behave differently for true and random labels. The complexity measures investigated in these works can effectively distinguish generalizable from poorly-generalizable local minima. They do not explain, however, why SGD converges to generalizable local minima when there exist poorly-generalizable local minima which can also perfectly fit the training set.
+
+![](images/eae1e91408a7d3edf31f6a8267c65fa3430e2b118f63f5b6bee8650d1cc10a30.jpg)  
+Figure 1: (a) A 2-layer neural network with activation function $\phi$ , (b) Training and test accuracy on CIFAR10 with true and random labels on a 2-layer neural network with 512 ReLU hidden units, regularized with an additive penalty: (b1) no penalty, (b2) $\ell _ { 2 }$ -norm, (b3) $\chi _ { 2 }$ -group path norm, (b4) $\ell _ { 1 }$ -path norm. The $\chi _ { 2 }$ -group path norm and $\ell _ { 1 }$ -path norm were successful to close the generalization gap for both true and random labels.
+
+To approach this question, one needs to understand the key characteristic of CIFAR-10’s original labeling which differentiates it from random labels and how it is exploited by SGD to achieve good generalization performance. In this work, we approach this problem in the Fourier domain where non-random labeling schemes behave completely differently from random labeling schemes. While signals recoverable from few measurements possess nice spectral properties such as bandlimitedness, fully random stochastic processes are not bandlimited and not recoverable from any finite number of measurements.
+
+Using spectral analysis, we focus on characterizing spectral properties of an underlying distribution which can be exploited by gradient-based methods to converge to generalizable local minima. We address this problem for 2-layer neural networks (see Figure 1a) with sinusoidal activation functions, where we show that if the underlying labeling scheme has limited bandwidth and Fourier $\ell _ { 1 }$ -norm (i.e. "nice" Fourier properties), we expect a gradient-based method to achieve good generalization performance. To arrive at this result, we first develop a Fourier-based generalization bound for 2-layer neural networks in terms of bandwidth and Fourier $\ell _ { 1 }$ -norm. Next, we prove that the local minima found by the gradient descent method over a 2-layer neural network with sine activation have bandwidth and Fourier $\ell _ { 1 }$ -norm bounded in terms of the spectral properties of the underlying labeling scheme.
+
+As a byproduct of our Fourier analysis, we derive generalization bounds for 2-layer neural networks with general activation functions. For bandlimited activation functions with finite Fourier $\ell _ { 1 }$ -norm, such as sinusoidal or Gaussian activation1, our bound is tighter than the generalization bound obtained using only the Lipschitz constant of the activation function. For ReLU-type activation functions, our generalization bound is comparable to Lipschitz-based bounds; however, it leads to a grouped version of the path norms developed in (Neyshabur et al., 2015a). We therefore call this capacity norm group path norm which can be used as an additive penalty to regularize 2-layer neural networks with ReLU activation. Our numerical experiments suggest that the generalization gap can be effectively tightened by regularizing the group path norm. Figure 1b demonstrates how group path norm regularization can help close the generalization gap for both true and random labels.
+
+# 2 RELATED WORK
+
+Generalization has been a topic of central interest in statistical learning theory (Vapnik, 1999; ShalevShwartz & Ben-David, 2014). Generalization bounds have been derived using the stability of a learning algorithm (Bousquet & Elisseeff, 2002) and various complexity measures of a function space such as VC-dimension (Vapnik, 2013) and Rademacher complexity (Bartlett & Mendelson, 2002). (Hardt et al., 2015) develops a stability-based generalization result for SGD as the learning algorithm, which holds for both convex and non-convex loss functions.
+
+We note that spectral analysis has provided a powerful framework for studying neural networks. (Barron, 1993) uses a Fourier-based approach to prove the universal approximation theorem for 2-layer neural networks. Similarly, (Lee et al., 2017) applies Fourier analysis to extend Barron’s result to a general feedforward neural network. (Rippel et al., 2015) uses a spectral approach to model and analyze convolutional neural networks (CNNs) and introduce the spectral pooling scheme for CNNs. Also, our Fourier-based approach to analyze SGD’s performance for 2-layer neural networks follows the same prinicples as the analysis performed in (Shamir, 2016) to prove the hardness of fitting periodic labeling schemes via gradient-based methods. We should note that in this work we use only periodic activation functions and not periodic labeling schemes. Therefore, the hardness result shown in (Shamir, 2016) does not affect our numerical experiments.
+
+In general, theoretical studies of neural networks can be categorized into three main categories: 1) Approximation: Neural networks have been proven to be powerful in expressing a very rich class of functions (Cybenko, 1989) and in general deeper networks need fewer neurons to express the same class of functions (Eldan & Shamir, 2016; Liang & Srikant, 2016). 2) Generalization: Tight bounds have been shown on the VC dimesnion of feedforward neural networks (Anthony & Bartlett, 2009; Harvey et al., 2017). Also, norm-based Rademacher complexity bounds have been developed at (Bartlett & Mendelson, 2002; Neyshabur et al., 2015b). Sharpness of local minima and its connection to their generalizibility have been the focus of several recent works (Keskar et al., 2016; Dinh et al., 2017; Neyshabur et al., 2017) 3) Optimization: theoretical studies have shown both positive (Andoni et al., 2014; Daniely, 2017) and negative (Shalev-Shwartz et al., 2017) results about the performance of gradient-based methods in training neural networks.
+
+# 3 PRELIMINARIES
+
+# 3.1 SUPERVISED LEARNING AND GENERALIZATION
+
+Suppose that we are given $n$ samples $( { \bf x } _ { i } , y _ { i } ) _ { i = 1 } ^ { n }$ drawn i.i.d. from a population distribution $P _ { \mathbf { X } , Y }$ Here $\mathbf { X }$ denotes the random vector of features and $Y$ denotes the target variable. Using these $n$ samples, the goal of a supervised learner is to find a prediction rule $f$ from a function space $\mathcal { F }$ which can predict $Y$ for an unseen test sample $\mathbf { X }$ . Therefore, given loss function $\ell$ the supervised learner wants to find $f ^ { * } \in { \mathcal { F } }$ minimizing the population risk, defined as $\mathbb { E } \big [ \ell \big ( f ( \mathbf { X } ) , Y \big ) \big ]$ averaged under the population distribution.
+
+However, the supervised learner does not know the population distribution $P _ { \mathbf { X } , Y }$ and has only access to the $n$ training samples. The supervised learner can minimize the empirical risk, defined as $\textstyle 1 / n \sum _ { i = 1 } ^ { n } \ell \bigl ( f ( \mathbf { x } _ { i } ) , \mathbf { \bar { y } } _ { i } \bigr )$ and find $f _ { n } ^ { \mathrm { e m p } }$ . Since we only observe a limited number of samples, the empirical risk would be different from the population risk. The generalization risk, defined for $f \in { \mathcal { F } }$ as $\begin{array} { r l } { } & { { \mathbb E } [ \ell ( f ( \mathbf { X } ) , Y ) ] - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( f ( \mathbf { x } _ { i } ) , y _ { i } ) } \end{array}$ , is the difference among the population risk and empirical risk for $f$ . Studying the behavior of $f _ { n } ^ { \mathrm { e m p } }$ ’s generalization risk for different function spaces and learning algorithms is a topic of central interest in statistical learning theory.
+
+# 3.2 FOURIER TRANSFORM AND BANDLIMITED FUNCTIONS
+
+Consider a real-valued function $f : \mathbb { R } ^ { k }  \mathbb { R }$ . The Fourier transform of this function, which we denote by $\hat { f }$ , is defined as
+
+$$
+{ \widehat { f } } ( \pmb { \xi } ) = \int f ( \mathbf { x } ) \exp \left( - 2 \pi i \pmb { \xi } ^ { T } \mathbf { x } \right) \mathrm { d } \mathbf { x } .
+$$
+
+Some important examples of Fourier transform are:
+
+• Sinusoidal function: $f ( \mathbf { x } ) = \exp ( 2 \pi i \omega ^ { T } \mathbf { x } )$ , then ${ \widehat { f } } ( \pmb { \xi } ) = \delta ( \pmb { \xi } - \omega )$ where $\delta$ denotes the Dirac delta function, which also implies
+
+• Gaussian function: $f ( \mathbf { x } ) = ( \sqrt { 2 \pi } \sigma ) ^ { k } \exp \bigl ( - \| \mathbf { x } \| _ { 2 } ^ { 2 } / 2 \sigma ^ { 2 } \bigr )$ , then $\widehat { f } ( \pmb { \xi } ) = \exp \left( - \sigma ^ { 2 } \| \pmb { \xi } \| _ { 2 } ^ { 2 } / 2 \right)$ . Thus, the Fourier transform of a Gaussian function preserves the Gaussian shape.
+
+A function $f$ is called $B$ -bandlimited if ${ \widehat { f } } ( \pmb { \xi } ) = 0$ for every $\boldsymbol { \xi }$ where $\| { \pmb \xi } \| _ { 2 } > B$ . The smallest $B$ for which this property holds is called the bandwidth of $f$ . We use $B ( f )$ to denote the bandwidth of function $f$ . We also use $\| { \widehat { f } } \| _ { 1 }$ to denote the $\ell _ { 1 }$ -norm of $f$ ’s Fourier transform,
+
+$$
+\| { \widehat { f } } \| _ { 1 } = \int | { \widehat { f } } ( \pmb { \xi } ) | \mathrm { d } \pmb { \xi }
+$$
+
+which we call the Fourier $\ell _ { 1 }$ -norm of $f$ . Fourier $\ell _ { 1 }$ -norm can be interpreted as the absolute volume under $f$ ’s Fourier transform, and is an approximate measure of $\widehat { f }$ ’s sparsity. Fourier $\ell _ { 1 }$ -norm is both scale and shift invariant, i.e. if we define $g ( \mathbf { x } ) = f ( \mathbf { W } \mathbf { x } + \mathbf { b } )$ for a real-valued $f$ and $\mathbf { W } \in \mathbb { R } ^ { r \times k }$ and $\mathbf { b } \in \mathbb { R } ^ { r }$ for some $r \leq k$ , then $\| { \widehat { g } } \| _ { 1 } = \| { \widehat { f } } \| _ { 1 }$ . Some other useful properties of Fourier transform are:
+
+• Synthesis: $\begin{array} { r } { f ( \mathbf { x } ) = \int \widehat { f } ( \pmb { \xi } ) \exp \left( 2 \pi i \pmb { \xi } ^ { T } \mathbf { x } \right) \mathrm { d } \pmb { \xi } } \end{array}$ , which also implies $\| { \widehat { f } } \| _ { 1 } = f ( 0 )$ if $\widehat { f }$ is real and non-negative.   
+• Shift: $\widehat { f } _ { \mathbf { b } } ( \pmb { \xi } ) = \exp ( 2 \pi i \mathbf { b } ^ { T } \pmb { \xi } ) \widehat { f } ( \pmb { \xi } )$ where $f _ { \mathbf { b } } ( \mathbf { x } ) : = f ( \mathbf { x } - \mathbf { b } )$ , which implies $\| { \widehat { f _ { \mathbf { b } } } } \| _ { 1 } = \| { \widehat { f } } \| _ { 1 }$ and $B ( f _ { \mathbf { b } } ) = B ( f )$ .   
+• Derivative: ${ \widehat { \nabla f } } ( \pmb { \xi } ) = 2 \pi i ~ { \widehat { f } } ( \pmb { \xi } ) \pmb { \xi }$ , where $\nabla f$ denotes the gradient of $f$ .   
+• Isometry: $\begin{array} { r } { \int f ( \mathbf { x } ) \overline { { g ( \mathbf { x } ) } } \mathrm { d } \mathbf { x } = \int \widehat { f } ( \pmb { \xi } ) \overline { { \widehat { g } ( \pmb { \xi } ) } } \mathrm { d } \pmb { \xi } } \end{array}$ where $\overline { z }$ denotes the complex conjugate of $z$ .   
+• Convolution: $\widehat { f g } = \widehat { f } \star \widehat { g }$ where $\star$ denotes the convolution operator i.e. ${ \widehat { f } } \star { \widehat { g } } ( \xi ) : = $ $\begin{array} { r } { \int \widehat { f } ( \pmb { \eta } ) \widehat { g } ( \pmb { \xi } - \pmb { \eta } ) \mathrm { d } \pmb { \eta } } \end{array}$ . Therefore, $B ( f g ) \le B ( f ) + B ( g )$ and $\| \widehat { f g } \| _ { 1 } \leq \| \widehat { f } \| _ { 1 } \| \widehat { g } \| _ { 1 }$ .
+
+# 4 A FOURIER-BASED GENERALIZATION BOUND
+
+Consider a supervised learning task with n training samples xi, yi
ni=1 and function space $\mathcal { F }$ . We are interested in uniform convergence bounds on the generalization risk. A standard approach to bound the generalization risk is based on the notion of Rademacher complexity. Given samples $\left( \mathbf { x } _ { i } , y _ { i } \right) _ { i = 1 } ^ { n }$ , the empirical Rademacher complexity of $\mathcal { F }$ is defined as
+
+$$
+\mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } ) : = \mathbb { E } _ { \pmb { \sigma } } \bigg [ \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sigma _ { i } f ( \mathbf { x } _ { i } ) \bigg ]
+$$
+
+where $\sigma _ { i }$ ’s are i.i.d. random variables uniformly distributed over $\{ - 1 , + 1 \}$ . In fact, the Rademacher complexity of $\mathcal { F }$ measures how well $\mathcal { F }$ can fit some random labels over input $\mathbf { x } _ { i }$ ’s. The following result shows how to bound the generalization risk over $\mathcal { F }$ through its Rademacher complexity.
+
+Theorem 1 (Bartlett & Mendelson (2002)). Consider a $\rho$ -Lipschitz loss function $\ell ( f ( \mathbf { x } ) , y )$ bounded as $| \ell ( z , y ) | \leq c .$ . Then, for any $\delta > 0$ , with probability at least $1 - \delta$
+
+$$
+\forall f \in \mathcal { F } : \quad \mathbb { E } \big [ \ell ( f ( \mathbf { X } ) , Y ) \big ] - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( f ( \mathbf { x } _ { i } ) , y _ { i } ) \leq 2 \rho \mathcal { R } _ { n } ^ { \mathrm { e n p } } ( \mathcal { F } ) + 4 c \sqrt { \frac { 2 \log ( 4 / \delta ) } { n } } .
+$$
+
+Since the Rademacher complexity of norm-bounded linear functions can be appropriately bounded (Kakade et al., 2009), one can effectively apply Theorem 1 to bound generalization risk over normbounded linear functions. To use Theorem 1 in the Fourier domain, here we provide a Rademacher complexity bound for bandlimited functions with bounded Fourier $\ell _ { 1 }$ -norm. We apply the following Rademacher complexity bound to bound generalization risk for 2-layer neural networks in Section 5, and also to analyze the performance of gradient-based methods with sinusoidal activation functions in Section 6.
+
+Theorem 2. Consider function space $\mathcal { F } = \{ \boldsymbol { f } : \mathbb { R } ^ { k }  \mathbb { R } $ s.t. $B ( f ) \leq B , \| { \widehat { f } } \| _ { 1 } \leq V \}$ of $B$ - bandlimited functions with $V$ -bounded Fourier $\mathbf { \dot { \ell } } _ { 1 }$ -norm. Then, the empirical Rademacher complexity for samples $( { \bf x } _ { i } , y _ { i } ) _ { i = 1 } ^ { n }$ is bounded as
+
+$$
+\mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } ) \leq V \sqrt { \frac { 4 k \log \left( 6 4 n B \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } \right) } { n } } .
+$$
+
+Proof. We defer the proof to the Appendix.
+
+Corollary 1. Assume that $\| \mathbf { X } \| _ { 2 } \leq C$ holds almost surely and the loss function $\ell$ is $\rho$ -Lipschitz. Then, for any $\delta > 0$ with probability at least $1 - \delta$ the following generalization bound holds for any $B$ -bandlimited function $f$ with $V$ -bounded Fourier $\ell _ { 1 }$ -norm:
+
+$$
+\mathbb { E } \big [ \ell ( f ( \mathbf { X } ) , Y ) \big ] - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( f ( \mathbf { x } _ { i } ) , y _ { i } ) \leq O \bigg ( \rho V \sqrt { \frac { k \log ( n B C / \delta ) } { n } } \bigg ) .
+$$
+
+Proof. The corrollary is a direct result of applying the bound in Theorem 2 to Theorem 1.
+
+The above corollary bounds the generalization risk uniformly over all bandlimited $f$ ’s such that $B ( f ) \le B$ and $\| { \widehat { f } } \| _ { 1 } \leq V$ . Next, we apply the above results to 2-layer neural networks.
+
+# 5 APPLICATION OF THEOREM 2 TO 2-LAYER NEURAL NETWORKS
+
+Consider a 2-layer neural network including $d$ neurons with activation function $\phi$ in the hidden layer (See Figure 1a). The output of this neural network is
+
+$$
+f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { x } ) = \mathbf { a } ^ { T } \phi ( \mathbf { W } \mathbf { x } + \mathbf { b } ) .
+$$
+
+If $\phi$ has bounded bandwidth and Fourier $\ell _ { 1 }$ -norm, we can apply Theorem 2 to bound the Rademacher complexity and hence generalization risk over the 2-layer neural network. Here, we use $\| \mathbf { W } \| _ { 2 , \infty }$ to denote the maximum $\ell _ { 2 }$ -norm $\left\| \mathbf { w } _ { i } \right\| _ { 2 }$ among all rows of $\mathbf { W }$ .
+
+Corollary 2. Let $\mathcal { F } _ { \phi } = \left\{ f ( \mathbf { x } ) = \mathbf { a } ^ { T } \phi ( \mathbf { W } \mathbf { x } + \mathbf { b } ) : ~ \| \mathbf { W } \| _ { 2 , \infty } \leq W , ~ \| \mathbf { a } \| _ { 1 } \leq A \right\}$ be the class of 2-layer neural networks where $B ( \phi ) = B$ and $\| \widehat { \phi } \| _ { 1 } = V$ . Then, the empirical Rademacher complexity of $\mathcal { F } _ { \phi }$ for samples $( { \bf x } _ { i } , y _ { i } ) _ { i = 1 } ^ { n }$ is bounded as follows
+
+$$
+\mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } _ { \phi } ) \leq O \biggl ( A V \sqrt { \frac { k \log \bigl ( n B W \operatorname* { m a x } \| \mathbf { x } _ { i } \| _ { 2 } \bigr ) } { n } } \biggr ) .
+$$
+
+Proof. We defer the proof to the Appendix.
+
+Notice that for bandlimited activation functions with bounded Fourier $\ell _ { 1 }$ -norm, the above generalization bound is increasing logarithmically with $\| \mathbf { W } \| _ { 2 , \infty }$ . For example, this result holds for sinusoidal activation $\phi ( x ) = \sin ( 2 \pi x )$ where $\| \widehat { \phi } \| _ { 1 } = 1$ , $B ( \phi ) = 1$ . On the other hand, the existing Rademacher complexity bounds which use only the Lipschitz constant of the activation function are linear in W’s norm (Bartlett & Mendelson, 2002). Therefore, by exploiting the spectral properties of $\phi$ , Corollary 2 results in a tighter generalization bound than the bounds using only the Lipschitz constant of $\phi$ .
+
+However, an unbounded function such as ReLU $\phi ( x ) = \operatorname* { m a x } ( x , 0 )$ has an infinite Fourier $\ell _ { 1 }$ -norm. Therefore, Corollary 2 does not directly apply to these functions. The following theorem uses a boundedness assumption on input $\mathbf { X }$ to apply Theorem 2 to ReLU-type activation functions. Although the following bound is growing faster than logarithmically with W’s norm, it introduces new capacity norms for 2-layer ReLU-based networks.
+
+Theorem 3. Suppose that $\phi _ { \alpha } ( x ) = \operatorname* { m a x } \{ x , \alpha x \}$ where $\alpha \in [ 0 , 1 ]$ is an arbitrary constant. Consider the pair of dual norms $( \| \cdot \| _ { p } , \| \cdot \| _ { q } )$ where $1 \leq p , q \leq \infty$ and $1 / p + 1 / q = 1$ . Assume that $\| \mathbf { x } _ { i } \| _ { p } \leq C$ holds for all $\mathbf { x } _ { i } \mathbf { \ ' } _ { s . }$ . Then, for $\begin{array} { r } { \mathcal { F } _ { \phi _ { \alpha } } = \left\{ f _ { \mathbf { a } , \mathbf { W } } ( \mathbf { x } ) = \mathbf { a } ^ { T } \phi _ { \alpha } ( \mathbf { W } \mathbf { x } ) : \sum _ { i = 1 } ^ { d } | a _ { i } | \| \mathbf { w } _ { i } \| _ { q } \leq V \right\} } \end{array}$
+
+$$
+\mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } _ { \phi _ { \alpha } } ) \leq O \bigg ( V C \sqrt { \frac { k \log ( n k C ) } { n } } \bigg ) .
+$$
+
+Proof. We relegate the proof to the Appendix.
+
+The above bound uses the complexity score $\begin{array} { r } { \sum _ { i = 1 } ^ { d } | a _ { i } | \| \mathbf { w } _ { i } \| _ { q } } \end{array}$ for each $f _ { \mathbf { a } , \mathbf { W } } ( \mathbf { x } ) = \mathbf { a } ^ { T } \phi _ { \alpha } ( \mathbf { W } \mathbf { x } )$ . We can rewrite this complexity score in the following way, which is an $\ell _ { 1 , q }$ -group norm on the product of weights for each path from the input nodes to the output node of the 2-layer neural network,
+
+$$
+\chi _ { q } \big ( f _ { \mathbf { a } , \mathbf { W } } \big ) = \sum _ { i = 1 } ^ { d } \biggl ( \sum _ { j = 1 } ^ { k } \bigl ( | a _ { i } | | w _ { i , j } | \bigr ) ^ { q } \biggr ) ^ { 1 / q } .
+$$
+
+Here $w _ { i , j }$ denotes the weight on the link from the $j$ th node of the input layer to the ith node of the hidden layer. Based on the path-norm function defined at (Neyshabur et al., 2015a), we call $\chi _ { q } \left( f _ { \mathbf { a } , \mathbf { w } } \right)$ the group path norm. For $q = 1$ , $\chi _ { 1 }$ -group path norm leads to the $\ell _ { 1 }$ -path norm for 2-layer neural networks. We can use group path norms as an additive regularization penalty to learn over 2-layer neural networks. In our numerical experiments, we test the performance of $\chi _ { 2 }$ -group path norm and $\ell _ { 1 }$ -path norm regularization to control the generalization risk over 2-layer neural networks.
+
+# 6 FOURIER ANALYSIS OF GRADIENT-BASED METHODS FOR 2-LAYER NEURAL NETWORKS WITH SINE ACTIVATION
+
+In this section, we apply Fourier analysis for a 2-layer neural network with sinusoidal activation. We aim to understand the connection between generalizibility of local minima found by gradient-based methods and spectral properties of the population distribution $P _ { \mathbf { X } , Y }$ . As a simplifying assumption, let’s assume that target variable $Y$ is a deterministic function $Y ( \mathbf { \dot { x } } )$ of input $\mathbf { X }$ , which we call the labeling scheme. In our analysis, we consider the squared-error loss $\ell ( y , y ^ { \prime } ) = ( y - y ^ { \prime } ) ^ { 2 }$ .
+
+We specifically ask this question: how can spectral properties of labeling scheme $Y ( \mathbf { x } )$ and population density function $P _ { \mathbf { X } } ( \mathbf { x } )$ affect the generalization performance of a gradient-based method? To address this question, we use a similar strategy to the analysis performed in (Mei et al., 2016) by establishing generalization results for both the empirical risk and the gradient of empirical risk. First, we show that the bandwidth and Fourier $\ell _ { 1 }$ -norm for the local minima of the population risk can be bounded in terms of the bandwidth and Fourier $\ell _ { 1 }$ -norm of $Y ( \mathbf { x } )$ and $P _ { \mathbf { X } } ( \mathbf { x } )$ . Next, we establish a generalization result for the gradient of the empirical risk, proving that the gradient of empirical risk would stay close to the gradient of population risk given that $Y ( \mathbf { x } )$ has limited bandwidth and Fourier $\ell _ { 1 }$ -norm. These two results show that by assuming a labeling scheme with limited bandwidth and Fourier $\ell _ { 1 }$ -norm, the local minima found by the gradient descent (in general large-batch gradient descent) method will generalize well.
+
+# 6.1 POPULATION RISK WITH SINUSOIDAL ACTIVATION
+
+Consider sinusoida $\begin{array} { r } { f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { x } ) = \sum _ { j = 1 } ^ { d } a _ { j } \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { x } + b _ { j } ) } \end{array}$ ing from a 2-layer neural network with the population risk will be $d$ $Y ( \mathbf { x } )$
+
+$$
+\mathbb { E } _ { P _ { \mathbf { X } } } \left[ \ell \big ( f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { x } ) , Y ( \mathbf { x } ) \big ) \right] = \mathbb { E } _ { P _ { \mathbf { X } } } \big [ \big ( Y ( \mathbf { x } ) - \sum _ { j = 1 } ^ { d } a _ { j } \mathrm { s i n } ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { x } + b _ { j } ) \big ) ^ { 2 } \big ] ,
+$$
+
+where the expectation is according to the population density function $P _ { \mathbf { X } } ( \mathbf { x } )$ .
+
+Lemma 1. Consider the population risk in (11). Assume $\mathbf { w } _ { j }$ satisfies $\forall i \neq j : \operatorname* { m i n } \{ \| \mathbf { w } _ { i } - { }$ $\mathbf { w } _ { j } \| _ { 2 } , \| \mathbf { w } _ { i } + \mathbf { w } _ { j } \| _ { 2 } \big \} > B ( P _ { \mathbf { X } } )$ . Then, $i f ( \mathbf { a } , \mathbf { W } , \mathbf { b } )$ is assumed to be a local minimum of the population risk,
+
+$$
+\big | a _ { j } \big | \leq 2 \big | \widehat { Y } \star \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { j } ) \big | .
+$$
+
+Proof. We defer the proof to the Appendix.
+
+Lemma 1 says that if the component $a _ { j } \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { x } )$ becomes isolated for a local minimum, by which we mean there are no other component $a _ { i } \sin ( 2 \pi \mathbf { w } _ { i } ^ { T } \mathbf { x } )$ with $\operatorname* { m i n } \{ \| \mathbf { w } _ { i } - \mathbf { w } _ { j } \| _ { 2 } , \| \mathbf { w } _ { i } + \mathbf { w } _ { j } \| _ { 2 } \}$ less than $P _ { \mathbf { X } }$ ’s bandwidth, then the value of $a _ { j }$ for that local minimum is nicely bounded in terms of the population distribution. This result leads to the following Theorem which describes the Fourier properties of the local minima of the population risk.
+
+Theorem 4. Consider the minimization problem of the population risk (11). If a local minimum $( \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } )$ satisfies the isolated components condition, i.e. for any two different $i , j$ we have $\operatorname* { m i n } \bigl \{ \| \mathbf { w } _ { i } ^ { * } - \mathbf { w } _ { j } ^ { * } \| _ { 2 } , \| \mathbf { w } _ { i } ^ { * } + \mathbf { w } _ { j } ^ { * } \| _ { 2 } \bigr \} > 2 B ( P _ { \mathbf { X } } )$ , then for the local minimum function $f _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } }$
+
+$$
+\begin{array} { r l } & { \bullet \mathcal { B } ( f _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } ) \leq \mathcal { B } ( Y ) + \mathcal { B } ( P _ { \mathbf { X } } ) , } \\ & { \bullet \| \widehat { f } _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } \| _ { 1 } \leq 2 \| \widehat { Y } \| _ { 1 } . } \end{array}
+$$
+
+Proof. We defer the proof to the Appendix.
+
+Theorem 4 implies that the bandwidth of the local minima of the population risk is less than the sum of bandwidths for $Y$ and $P _ { \mathbf { X } }$ . Also, the Fourier $\ell _ { 1 }$ -norm for the local minima of the population distribution is bounded by twice the Fourier $\ell _ { 1 }$ -norm of $Y$ .
+
+Remark 1. To apply Theorem 4, the bandwidth of $P _ { \mathbf { X } }$ needs to be smaller than half the distance among $\mathbf { w } _ { i } ^ { * }$ ’s. For example, suppose that $\mathbf { X } \sim { \mathcal { N } } ( { \pmb { \mu } } , \sigma ^ { 2 } \mathbf { I } _ { k \times k } )$ has a multivariate Gaussian distribution with mean $\pmb { \mu }$ and diagonal covariance matrix with standard deviation $\sigma$ . Then, the above theorem shows that if for any $i , j$ we have $\operatorname* { m i n } \bigr \{ \| \mathbf { w } _ { i } ^ { * } - \mathbf { w } _ { j } ^ { * } \| _ { 2 } , \| \mathbf { w } _ { i } ^ { * } + \mathbf { w } _ { j } ^ { * } \| _ { 2 } \bigr \} > 2 C / \sigma$ for some constant $C$ then
+
+$$
+\begin{array} { r l } & { \bullet \mathcal { B } ( f _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } ) \leq \mathcal { B } ( Y ) + O \big ( \sqrt { k } / \sigma \big ) , } \\ & { \bullet \| \widehat { f } _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } \| _ { 1 } \leq 2 ( 1 + d \exp ( - C ^ { 2 } / 2 ) ) \| \widehat { Y } \| _ { 1 } . } \end{array}
+$$
+
+Proof. See the proof of Theorem 4 in the Appendix.
+
+# 6.2 GENERALIZATION TO THE EMPIRICAL RISK
+
+Theorem 4 characterizes the Fourier properties of the local minima for the population risk. However, we want to investigate the generalization performance of the local minima of the empirical risk defined for training samples $\mathbf { \bar { \rho } } ( \mathbf { x } _ { i } , Y ( \mathbf { x } _ { i } ) ) _ { i = 1 } ^ { n }$ as
+
+$$
+\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell \big ( f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { x } _ { i } ) , Y ( \mathbf { x } _ { i } ) \big ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \bigg ( Y ( \mathbf { x } _ { i } ) - \sum _ { j = 1 } ^ { d } a _ { j } \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { x } _ { i } + b _ { j } ) \bigg ) ^ { 2 } .
+$$
+
+To address this question, note that the bandwidth and Fourier $\ell _ { 1 }$ -norm of the loss’s gradient with respect to each $a _ { j }$ are bounded in terms of the bandwidth and Fourier $\ell _ { 1 }$ -norm of $Y ( \mathbf { x } )$ as
+
+$$
+\begin{array} { r l } & { \quad \big \| \nabla _ { a _ { j } } \ell \big ( \widehat { f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } } ( \mathbf { x } ) , Y ( \mathbf { x } ) \big ) \big \| _ { 1 } \leq \| \widehat { Y } \| _ { 1 } + \| \mathbf { a } \| _ { 1 } , } \\ & { \mathcal { B } \big ( \nabla _ { a _ { j } } \ell \big ( f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { x } ) , Y ( \mathbf { x } ) \big ) \big ) \leq \mathcal { B } ( Y ) + 2 \| \mathbf { W } \| _ { 2 , \infty } . } \end{array}
+$$
+
+We can apply Corollary 1 to show that not only the empirical risk uniformly converges to the population risk but also the gradient of the empirical risk will stay close to the gradient of the population risk.
+
+Corollary 3. Consider $\begin{array} { r } { f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { x } ) = \sum _ { j = 1 } ^ { d } a _ { j } \sin ( \mathbf { w } _ { j } ^ { T } \mathbf { x } + b _ { j } ) } \end{array}$ and squared error loss \`. Then, given that $\| \mathbf { X } \| _ { 2 } \leq C$ , for any $\delta > 0$ with probability at least $1 - \delta$ we have
+
+$\forall j , \mathbf { a } , \mathbf { W } , \mathbf { b }$ s.t. $\| \mathbf { a } \| _ { 1 } + \| { \widehat { Y } } \| _ { 1 } \leq V , \ 2 \| \mathbf { W } \| _ { 2 , \infty } + B ( Y ) \leq B \ :$
+
+$$
+\begin{array} { r l } & { \displaystyle \mathbb { E } [ \nabla _ { a _ { j } } \ell ( f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { X } ) , Y ( \mathbf { X } ) ) ) ] - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } [ \nabla _ { a _ { j } } \ell ( f _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { x } _ { i } ) , Y ( \mathbf { x } _ { i } ) ) ) ] \Big \vert \leq O \big ( V \sqrt { \frac { k \log ( n B C / \delta ) } { n } } \big ) . } \end{array}
+$$
+
+Proof. The corollary is a direct result of Corollary (1) given (14) and (15). Note that the generalization bound holds with probability $1 - \delta$ for the derivative with respect to all $a _ { j }$ ’s, since the bounds in (14) and (15) hold for all $j$ ’s. □
+
+We emphasize that to prove Theorem 4 we need to analyze the risk function’s derivative only with respect to $a _ { j }$ ’s. Hence, generalization of the empirical risk’s gradient with respect to $a _ { j }$ ’s, which is shown in the above corollary under certain assumptions, is sufficient to apply an approximate version of Theorem 4 in section 8.6 to a local minimum $( { \bf a } ^ { * } , { \bf W } ^ { * } , { \bf b } ^ { * } )$ satisfying the isolated components assumption and found by the gradient descent approach initialized at a low $\left\| \mathbf { a } \right\| _ { 1 }$ and $\lVert \mathbf { W } \rVert _ { 2 , \infty } ^ { - }$ . We can conclude that with probability at least $1 - \delta$ the $\ell _ { 1 }$ -norm of $f _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } }$ ’s Fourier transform outside the bandwidth $\mathcal { B } ( Y ) + \mathcal { B } ( P _ { \mathbf { X } } )$ is bounded by $O \big ( d V \sqrt { \frac { k \log \big ( n B C / \delta \big ) } { n } } \big )$ , and also
+
+![](images/914686f543f146ed0aa239713cbfcd289a4969ea062bc0a533bafbbb1b15a0f0.jpg)  
+Figure 2: Training an test performance on cat and airplane CIFAR10 images with true and random labels. Sine activation and mean-squared-error loss were used.
+
+![](images/3a0eb3a058ec45fed9e212dca7302bb4d59e8eba4b3a5b5bba3855e8b79b0e81.jpg)  
+Figure 3: Training and test performance on cat and airplane CIFAR10 images with true and random labels. ReLU activation and cross-entropy loss were used.
+
+$$
+\| \widehat { f } _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } \| _ { 1 } \leq 2 \| \widehat { Y } \| _ { 1 } + O \big ( d V \sqrt { \frac { k \log ( n B C / \delta } { n } } \big ) .
+$$
+
+Based on the above discussion, if a large-batch gradient descent method starts learning from $f _ { \mathbf { a } , \mathbf { w } , \mathbf { b } }$ with low $\lVert \mathbf { a } \rVert _ { 1 }$ and $\| \mathbf { W } \| _ { 2 , \infty }$ and also we assume that the bandwidth and the Fourier $\ell _ { 1 }$ -norm for $Y ( \mathbf { x } )$ are properly bounded, Theorem 4 combined with Corollary 1 will guarantee good generalization performance for the local minima found by the gradient descent method.
+
+# 7 NUMERICAL EXPERIMENTS
+
+For all experiments described in this section, we implemented and trained the two-layer neural network described in Figure 1a using TensorFlow 1.3.0. We used SGD to train the model for 2000 epochs with an initial learning rate of 0.01. The learning rate decayed slightly each epoch at a rate of 0.95 every 390 epochs. We used $h = 5 1 2$ hidden units and a batch size of 128. When working with CIFAR10 data, we preprocessed the data as described in (Zhang et al., 2016), resulting in each training sample having dimension $d = 2 3 5 2$ . Initial weights from the first layer were sampled from $\mathcal { N } ( 0 , \bar { 0 . 0 1 } / \bar { d } )$ and initial weights from the second layer were sampled from $\dot { \mathcal { N } } ( 0 , 0 . 0 1 / h )$ .
+
+7.1 SGD GRADUALLY LEARNS HIGHER FOURIER $\ell _ { 1 }$ -NORM, BANDWIDTH HYPOTHESES
+
+We first numerically demonstrate that how Fourier $\ell _ { 1 }$ -norm and bandwidth both increases during training via SGD. Motivated by the analysis from Section 6, we use the squared-error as our loss function and sine as our activation function. Our samples consist of cats and airplanes from the CIFAR10 dataset with the labels mapped to $- 1$ and 1. We use 5000 and 2000 samples from each category for training and test, respectively. We arbitrarily chose two of the ten classes to accommodate our choice of loss function. We evaluate the network’s performance for both random and true labels.
+
+Figure 2a shows that without regularization, SGD learns to perfectly fit both the true and random labels, which is consistent with the results from Zhang et al. (2016). Additionally, the random labels are harder to learn, requiring more epochs before achieving a perfect fit. Figures 2b and 2c confirm that both Fourier $\ell _ { 1 }$ -norm and bandwidth consistently increase with training, highlighting how SGD gradually finds more complex hypotheses in order to fit the data. Finally, we see in figures 2d and 2e how both Fourier $\ell _ { 1 }$ -norm and bandwidth increase with generalization risk (the difference between test mean squared-error (MSE) and training MSE) with almost perfect correlation. This suggests that, as implied by the theory above, regularizing Fourier $\ell _ { 1 }$ -norm and bandwidth could improve generalizability of the final learned model.
+
+# 7.2 GROUP PATH NORM REGULARIZATION FOR RELU ACTIVATION
+
+We regularize group path norm for ReLU activation as motivated by Theorem 3. Although $\chi _ { 2 }$ -group path norm is not convex, it is differentiable and we can use it as an additive penalty and find a local minimum via SGD. Using the same experimental setup as from section 7.1, we swap sine for ReLU and test the network’s performance for both random and true labels.
+
+Figure 3a confirms that, like before, the network can fit both true and random labels. The generalization gap, however, remains large for random labels. By regularizing the $\ell _ { 2 }$ -norm of all the weights, we see that the generalization gap closes for both the true labels and the random labels without compromising test accuracy significantly (Figure 3b). This result is further improved when we use the $\chi _ { 2 }$ -group path norm and $\ell _ { 1 }$ -path norm (Figure 3c and 3d), demonstrating that direct regularization of Fourier $\ell _ { 1 }$ -norm leads to better generalization.
+
+We cross-validated the value of $\lambda$ for each regularization technique, and we chose the $\lambda$ that resulted in the smallest generalization gap with comparable validation performance. To fairly compare different regularization strategies, we tested five lambda values for each strategy and then reported the performance on the test set for the lambda value that resulted in the best performance on the validation set.
+
+We repeated the experiment using all 50000 CIFAR10 training samples (and 10000 test samples). We included all 10 classes and switched to cross-entropy loss. The results are shown in Figure 1b. Again, we see that while all regularization techniques give similar test performance, the generalization gap is closed significantly for the $\chi _ { 2 }$ -group path norm and $\ell _ { 1 }$ -path norm.
+
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+
+# 8 APPENDIX
+
+# 8.1 PROOF OF THEOREM 2
+
+We use a high-dimensional grid in the Fourier domain to approximate the Fourier transform of a $B$ -bandlimited function. Consider the ball $\displaystyle \mathbf { \bar { \boldsymbol { \{ \xi } } }  \colon \| \pmb { \xi } \| _ { 2 } \leq B \}$ . Using the bounds on the covering number for $\ell _ { 2 }$ -norm, for any $0 < \epsilon < B$ we can find a set of points $\{ \pmb { \xi } _ { j } : 1 \le j \le ( 3 B / \epsilon ) ^ { k } \}$ such that for any $\pmb { \xi }$ with $\| { \pmb { \xi } } \| _ { 2 } \le B$ , there exists some $\xi _ { j }$ with $\| \pmb { \xi } - \pmb { \xi } _ { j } \| _ { 2 } \le \epsilon$ .
+
+Let $S _ { j } = \{ \pmb { \xi } : \| \pmb { \xi } - \pmb { \xi } _ { j } \| _ { 2 } \leq \epsilon \}$ for each $1 \le j \le ( 3 B / \epsilon ) ^ { k }$ . Note that $\{ \pmb { \xi } : \ \| \pmb { \xi } \| _ { 2 } \le B \} \subset \cup _ { j } S _ { j }$ . We then define $S _ { j } ^ { \prime } = S _ { j } \setminus \cup _ { t = 1 } ^ { j - 1 } S _ { t }$ to have a group of disjoint sets $S _ { j } ^ { \prime }$ covering $\{ \pmb { \xi } : \ \| \pmb { \xi } \| _ { 2 } \leq B \}$ . Since any $f \in { \mathcal { F } }$ is assumed to be $B$ -bandlimited, for $f \in { \mathcal { F } }$
+
+$$
+\begin{array} { l } { f ( \mathbf { x } ) = \displaystyle \int \widehat { f } ( \pmb { \xi } ) \exp ( 2 \pi i \pmb { \xi } ^ { T } \mathbf { x } ) \mathrm { d } \pmb { \xi } } \\ { = \displaystyle \sum _ { j = 1 } ^ { ( 3 B / \epsilon ) ^ { k } } \int _ { \pmb { \xi } \in S _ { j } ^ { \prime } } \widehat { f } ( \pmb { \xi } ) \exp ( 2 \pi i \pmb { \xi } ^ { T } \mathbf { x } ) \mathrm { d } \pmb { \xi } . } \end{array}
+$$
+
+Then, for any $f \in { \mathcal { F } } = \{ f : { \mathcal { B } } ( f ) \leq B , \| { \widehat { f } } \| _ { 1 } \leq V \}$ we have
+
+$$
+\begin{array} { r l }    | | \begin{array} { l l } { f ( \Phi ) } & { - \frac { \lambda ( \xi ) \xi } { 2 } [ \exp ( \xi ) \log ( \xi ) ] | _ { \xi = \xi } | \langle \Phi \rangle | } \\ & { + | \frac { \lambda ( \xi ) \xi } { 2 } [ \exp ( \xi ) \log ( \xi ) ] | _ { \xi = \xi } | \langle \Phi \rangle | _ { \xi = \xi } | } \\ & { - | \frac { \lambda ( \xi ) \xi } { 2 } [ \frac { \lambda ( \xi ) } { 2 } [ \xi \xi ] [ \log ( \xi ) ) + \frac { \lambda ( \xi ) } { 2 } [ \xi ] [ \xi ] ] | _ { \xi = \xi } ^ { \xi } } \\ & { \le \frac { \lambda ( \xi ) } { 2 } [ \frac { \lambda ( \xi ) } { 2 } [ \xi ] [ \xi ] [ \xi ] [ \lambda ( \xi ) ) [ \xi ] [ \xi ] ] ] | _ { \xi = \xi } ^ { \xi } [ \xi ] [ \xi ] | _ { \xi = \xi } ^ { \xi } } \\ & { \overset { \mathrm { B ( a ) } } { \le } \frac { \lambda ( \xi ) } { 2 } \int _ { \xi \in \xi } [ \frac { \lambda ( \xi ) } { 2 } [ \xi ] [ \xi ] [ \xi ] \xi ] [ \xi - \xi ] \xi ] \le \frac { \lambda ( \xi ) } { 2 } [ \xi ] [ \xi ] \frac { \lambda ( \xi ) } { 2 } } \\ & { \le \frac { \lambda ( \xi ) } { 2 } [ \frac { \lambda ( \xi ) } { 2 } [ \xi ] [ \xi ] \xi ] [ \xi ] \xi [ \xi ] \xi [ \xi ] \xi [ \xi ] \xi ] \ } \\ & { \overset { \mathrm { C ( I ) } } { \le } \frac { \lambda ( \xi ) } { 2 } [ \frac { \lambda ( \xi ) } { 2 } \int _ { \xi \in \xi } [ \frac { \lambda ( \xi ) } { 2 } [ \xi ] \xi ] [ \xi ] \xi [ \xi ] \xi ] \xi } \\ &  \overset { \mathrm { B ( a ) } } { \le } \frac { \lambda ( \xi ) } { 2 } \int _ { \xi \in \xi } \frac { \lambda ( \xi ) } { 2 } \int _  \xi \ \end{array} \end{array}
+$$
+
+Here, (a) is a direct application of (17). (b) holds as $\exp ( i b z ) = \cos ( b z ) + i \sin ( b z )$ is $^ { b }$ -Lipschitz as a function of $z \in \mathbb { R }$ for any real number $b > 0$ . (c) holds because according to our definitions $\bar { S _ { j } ^ { \prime } } \subseteq S _ { j }$ and $S _ { j } = \{ \pmb { \xi } : \ \| \pmb { \xi } - \pmb { \xi } _ { j } \| _ { 2 } \leq \epsilon \}$ .
+
+Therefore, the following function space $\mathcal { F } _ { \epsilon }$ can approximate any $f \in { \mathcal { F } } = \left\{ f : { \mathcal { B } } ( f ) \leq B , \| { \widehat { f } } \| _ { 1 } \leq V \right\}$ within $2 \pi \epsilon C V$ accuracy for any $\| \mathbf { x } \| _ { 2 } \leq C$ . Here a is, in general, a vector of complex numbers, and $\begin{array} { r } { \| \dot { \bf a } \| _ { 1 } : = \sum _ { j } | a _ { j } | } \end{array}$
+
+where $| z |$ denotes the absolute value of complex number $z$
+
+$$
+\mathcal { F } _ { \epsilon } = \Bigg \{ f ( \mathbf { x } ) = \sum _ { j = 1 } ^ { ( 3 B / \epsilon ) ^ { k } } a _ { j } \exp ( 2 \pi i \pmb { \xi } _ { j } ^ { T } \mathbf { x } ) : \| \mathbf { a } \| _ { 1 } \leq V \Bigg \} .
+$$
+
+Then, $\mathcal { F } _ { \epsilon }$ is the space of $\ell _ { 1 }$ -norm bounded linear functions in terms of the input vector $\left[ \exp ( 2 \pi i \pmb { \xi } _ { j } ^ { T } \mathbf { x } ) \right] _ { j }$ . Now, we can apply a well-known bound (Shalev-Shwartz & Ben-David, 2014) on the Rademacher complexity of $\ell _ { 1 }$ -norm bounded linear space $\mathcal { F } _ { \mathrm { l i n } , 1 } = \{ \boldsymbol { f } : \mathbb { R } ^ { k }  \mathbb { R } $ s.t. $f ( \mathbf { x } ) = \mathbf { a } ^ { T } \mathbf { x }$ , $\| \mathbf { a } \| _ { 1 } \leq A \}$ as
+
+$$
+\mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } _ { \mathrm { l i n } , 1 } ) \leq A \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { \infty } \sqrt { \frac { 2 \log ( 2 k ) } { n } } .
+$$
+
+Applying the above bound, we can bound the Rademacher complexity of $\mathcal { F } _ { \epsilon }$ as
+
+$$
+\mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } _ { \epsilon } ) \leq V \sqrt { \frac { 2 k \log ( 6 B / \epsilon ) } { n } } .
+$$
+
+Since for each $f \in { \mathcal { F } }$ there exists $\tilde { f } \in \mathcal { F } _ { \epsilon }$ such that $\forall \| \mathbf { x } \| _ { 2 } \leq C : | f ( \mathbf { x } ) - \tilde { f } ( \mathbf { x } ) | \leq 2 \pi \epsilon C V$ ,
+
+$$
+\begin{array} { r l } {  { \mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } ) = \mathbb { E } _ { \sigma } \bigg [ \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sigma _ { i } f ( \mathbf { x } _ { i } ) \bigg ] } \quad } & { } \\ & { \leq \mathbb { E } _ { \sigma } \bigg [ \operatorname* { s u p } _ { \tilde { f } \in \mathcal { F } _ { \epsilon } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sigma _ { i } \tilde { f } ( \mathbf { x } _ { i } ) \bigg ] + 2 \pi \epsilon V \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } } \\ & { = \mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } _ { \epsilon } ) + 2 \pi \epsilon V \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } . } \end{array}
+$$
+
+Finally, combining (20) and (21) we obtain:
+
+$$
+\forall \epsilon > 0 : \quad \mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } ) \leq V \sqrt { \frac { 2 k \log ( 6 B / \epsilon ) } { n } } + 2 \pi \epsilon V \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } .
+$$
+
+If we choose the value $\begin{array} { r } { \epsilon = \frac { 1 } { 2 \pi n \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } } } \end{array}$ , then we get
+
+$$
+\begin{array} { r l } & { \mathcal { R } _ { n } ^ { \mathrm { e m p } } ( \mathcal { F } ) \leq V \bigg ( \sqrt { \frac { 2 k \log \left( 1 2 \pi n B \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } \right) } { n } } + \frac { 1 } { n } \bigg ) } \\ & { \qquad \leq V \sqrt { \frac { 4 k \log \left( 1 2 \pi n B \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } \right) + 2 / n } { n } } } \\ & { \qquad \leq V \sqrt { \frac { 4 k \log \left( 6 4 n B \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } \right) } { n } } , } \end{array}
+$$
+
+where the last inequality follows from the fact that $1 \leq k , n$ . Therefore, the proof is complete.
+
+# 8.2 PROOF OF COROLLARY 2
+
+First, we prove the following lemma.
+
+Lemma 2. Given function $f : \mathbb { R } ^ { k }  \mathbb { R }$ and matrix $\mathbf { W } \in \mathbb { R } ^ { k \times k }$ , we define $g ( \mathbf { x } ) = f ( \mathbf { W } \mathbf { x } )$ . Then,
+
+• $B ( g ) \leq \| \mathbf { W } \| _ { 2 } B ( f )$ with $\| \mathbf { W } \| _ { 2 }$ denoting the spectral norm of $\mathbf { W }$ , • $\| { \widehat { g } } \| _ { 1 } = \| { \widehat { f } } \| _ { 1 }$ .
+
+Proof. From the properties of the Fourier transform we know
+
+$$
+\widehat { g } ( \pmb { \xi } ) = \frac { 1 } { | \operatorname* { d e t } ( \mathbf { W } ) | } \widehat { f } \big ( \mathbf { W } ^ { - T } \pmb { \xi } \big ) .
+$$
+
+Therefore, $\begin{array} { r } { \widehat { g } ( \mathbf { W } ^ { T } \pmb { \xi } ^ { \prime } ) = \frac { 1 } { | \operatorname* { d e t } ( \mathbf { W } ) | } \widehat { f } \big ( \pmb { \xi } ^ { \prime } \big ) } \end{array}$ and if $\| \pmb { \xi } ^ { \prime } \| _ { 2 } \le B ( f )$ , then $\| \mathbf { W } ^ { T } \pmb { \xi } ^ { \prime } \| _ { 2 } \le \| \mathbf { W } \| _ { 2 } B ( f )$ gives an upperbound on $B ( g )$ . Also,
+
+$$
+\begin{array} { r l } & { \| \hat { g } \| _ { 1 } = \int \big | \hat { g } ( \xi ) \big | \mathrm { d } \xi } \\ & { \quad = \int \frac { 1 } { \big | \operatorname* { d e t } ( \mathbf { W } ) \big | } \big | \widehat { f } ( \mathbf { W } ^ { - T } \xi ) \big ) \big | \mathrm { d } \xi } \\ & { \quad = \frac { 1 } { \big | \operatorname* { d e t } ( \mathbf { W } ) \big | } \int \big | \widehat { f } ( \mathbf { W } ^ { - T } \xi ) \big | \mathrm { d } \xi } \\ & { \quad = \frac { 1 } { \big | \operatorname* { d e t } ( \mathbf { W } ) \big | } \int \big | \widehat { f } ( \xi ^ { \prime } ) \big ) \big | \frac { 1 } { \big | \operatorname* { d e t } ( \mathbf { W } ^ { - T } \setminus j ) \big | } \mathrm { d } \xi ^ { \prime } } \\ & { \quad = \int \big | \widehat { f } ( \xi ^ { \prime } ) \big | \mathrm { d } \xi ^ { \prime } } \\ & { \quad = \| \widehat { f } \| _ { 1 } . } \end{array}
+$$
+
+It can be seen that this result remains valid even if $\mathbf { W }$ is not an invertible matrix, which will complete the proof for Corollary 2. However, we continue proving Corollary 2 without using this fact.
+
+As shown in the above lemma, Fourier $\ell _ { 1 }$ -norm and bandwidth are invariant to an orthonormal transformation $\mathbf { W }$ . Given $f _ { i } ( \mathbf { x } ) = a _ { i } \phi ( \mathbf { w } _ { i } ^ { T } \mathbf { x } )$ , we define $g _ { i } ( { \bf x } ) = f _ { i } ( { \bf A } _ { i } { \bf x } )$ where $A _ { i }$ is an orthonormal matrix with ${ \bf w } _ { i }$ as an eigenvector. Note that $\lVert \widehat { f } _ { i } \rVert _ { 1 } = \lVert \widehat { g _ { i } } \rVert _ { 1 }$ and $B ( f _ { i } ) \ : = \ : B ( g _ { i } )$ . However, $g _ { i } ( \mathbf { x } )$ is a function of only bone of the coordinates, which we can assume, without loss of generality, to be the first coordinate. Hence, $g _ { i } ( \mathbf { x } ) = a _ { i } \phi ( \| \mathbf { w } _ { i } \| _ { 2 } x _ { 1 } )$ for the first coordinate $x _ { 1 }$ , implying $\begin{array} { r } { \widehat { g _ { i } } ( \pmb { \xi } ) = \frac { a _ { i } } { \| \mathbf { w } _ { i } \| _ { 2 } } \widehat { \phi } ( \frac { \xi _ { 1 } } { \| \mathbf { w } _ { i } \| _ { 2 } } ) . \delta _ { 2 } ( \xi _ { 2 } ) \ldots \delta _ { k } ( \xi _ { k } ) } \end{array}$ $\delta _ { j }$ is the Dirac delta function across the $j$ th dimension. Hence, we can use the above lemma in the 1-dimensional case to show $\| \widehat { g _ { i } } \| _ { 1 } = | a _ { i } | \| \widehat { \phi } \| _ { 1 }$ and $B ( g _ { i } ) = \| \mathbf { w } _ { i } \| _ { 2 } B ( \phi )$ . As a result,
+
+$$
+\| \widehat { f } _ { i } \| _ { 1 } = | a _ { i } | \| \widehat { \phi } \| _ { 1 } , \quad \mathcal { B } ( f _ { i } ) \leq \| \mathbf { w } _ { i } \| _ { 2 } \mathcal { B } ( \phi ) .
+$$
+
+Hence, for $\begin{array} { r } { f ( \mathbf { x } ) = \mathbf { a } ^ { T } \boldsymbol { \phi } ( \mathbf { W } \mathbf { x } + \mathbf { b } ) = \sum _ { i = 1 } ^ { d } a _ { i } \boldsymbol { \phi } ( \mathbf { w } _ { i } ^ { T } \mathbf { x } + b _ { i } ) } \end{array}$ we have
+
+$$
+\| \widehat { f } \| _ { 1 } \leq \| \mathbf { a } \| _ { 1 } \| \widehat { \phi } \| _ { 1 } , \quad \mathcal { B } ( f ) \leq \| \mathbf { W } \| _ { 2 , \infty } \mathcal { B } ( \phi ) .
+$$
+
+The corollary is then a direct application of Theorem 2.
+
+# 8.3 PROOF OF THEOREM 3
+
+Given a ReLU-type activation function $\phi _ { \alpha } ( z ) = \operatorname* { m a x } \{ z , \alpha z \}$ ,
+
+$$
+\phi _ { \alpha } \big ( \mathbf { w } ^ { T } \mathbf { x } \big ) = \| \mathbf { w } \| _ { q } \phi _ { \alpha } \big ( ( \frac { \mathbf { w } } { \| \mathbf { w } \| _ { q } } ) ^ { T } \mathbf { x } \big ) .
+$$
+
+Since ${ \left\| \frac { \mathbf { w } } { \left\| \mathbf { w } \right\| _ { q } } \right\| } _ { q } = 1$ , if $\| \mathbf { x } \| _ { p } \leq C$ , then ${ \Big | } { \big ( } { \frac { \mathbf { w } } { \| \mathbf { w } \| _ { q } } } { \big ) } ^ { T } \mathbf { x } { \Big | } \leq C$ and hence the input to $\phi _ { \alpha }$ in the R.H.S. of (28) is always between $- C$ and $C$ .
+
+Suppose that function $\psi _ { \alpha }$ satisfies $\psi _ { \alpha } ( z ) = \phi _ { \alpha } ( z )$ for $z \in [ - C , C ]$ . Then, based on the above discussion, we can bound the Rademacher complexity of ${ \mathcal F } _ { \phi _ { \alpha } }$ by finding a bound on the Rademacher complexity of $\mathcal { F } _ { \psi _ { \alpha } } = \left\{ f _ { \mathbf { v } , \mathbf { U } } ( \mathbf { x } ) = \mathbf { v } ^ { T } \psi _ { \alpha } ( \mathbf { U } \mathbf { x } ) : ~ \| \mathbf { v } \| _ { 1 } \leq V , \forall i : ~ \| \mathbf { u } _ { i } \| _ { q } = 1 \right\}$ .
+
+To find a good candidate for $\psi _ { \alpha }$ , we use a symmetrization trick to define
+
+$$
+\psi _ { \alpha } ( z ) = \left\{ \begin{array} { c l } { - \alpha C } & { \mathrm { i f } z < - C , } \\ { \phi _ { \alpha } ( z ) } & { \mathrm { i f } - C \le z < C , } \\ { \phi _ { \alpha } ( 2 C - z ) } & { \mathrm { i f } C \le z < 3 C , } \\ { - \alpha C } & { \mathrm { i f } 3 C \le z . } \end{array} \right.
+$$
+
+Note that $\begin{array} { r } { \psi _ { \alpha } ( z ) = ( 1 - \alpha ) C h ( \frac { z - C } { C } ) + 2 \alpha C h ( \frac { z - C } { 2 C } ) - \alpha C } \end{array}$ where $h ( z ) = \operatorname* { m a x } \{ 0 , 1 - | z | \} .$ . It can be seen that $\begin{array} { r } { { \widehat { h } } ( \xi ) = \big ( \frac { \sin ( \pi \xi ) } { \pi \xi } \big ) ^ { 2 } } \end{array}$ which is real and positive everywhere. Therefore, $\| \widehat { h } \| _ { 1 } = h ( 0 ) = 1$ which means that $\| \widehat { \psi _ { \alpha } } \| _ { 1 } \leq C ( 1 + 2 \alpha ) \leq 3 C .$ .
+
+Since $\vert \widehat { h } ( \xi ) \vert \le \frac { 1 } { \xi ^ { 2 } }$ , we have $\widehat { | \psi _ { \alpha } ( \xi ) | } \le \frac { 1 } { \xi ^ { 2 } }$ . For $B > 0$ , we let the $B$ -filtered $\psi _ { \alpha , B }$ be a function with the following Fourier transform:
+
+$$
+\widehat { \psi _ { \alpha , B } } ( \xi ) = \left\{ \begin{array} { l l } { \widehat { \psi _ { \alpha } } ( \xi ) } & { \mathrm { i f ~ } | \xi | \leq B } \\ { 0 } & { \mathrm { o t h e r w i s e } . } \end{array} \right.
+$$
+
+Then, since $\bigl | \widehat { \psi _ { \alpha } } ( \xi ) \bigr | \le \frac { 1 } { \xi ^ { 2 } }$ we have
+
+$$
+\forall z \in \mathbb { R } : \quad \left| \psi _ { \alpha } ( z ) - \psi _ { \alpha , B } ( z ) \right| \leq \int _ { | \xi | \geq B } \left| \widehat { \psi _ { \alpha } } ( \xi ) \right| \mathrm { d } \xi \leq \frac { 2 } { B } .
+$$
+
+Thus, for any $B > 0$ the defined $\psi _ { \alpha , B }$ approximates $\phi _ { \alpha }$ with a maximum error of $\frac { 2 } { B }$ uniformly over $[ - C , C ]$ . $\psi _ { \alpha , B }$ also satisfies $\| \widehat { \psi _ { \alpha , B } } \| _ { 1 } \leq 3 C$ and $\begin{array} { r } { B ( \psi _ { \alpha , B } ) = B } \end{array}$ . Applying Corollary 2, we get
+
+$$
+\forall B > 0 : \quad \mathcal { R } _ { n } ^ { \mathrm { e m p } } \big ( \mathcal { F } _ { \phi _ { \alpha } } \big ) \leq O \Bigg ( V C \sqrt { \frac { k \log \left( n B \operatorname* { m a x } \| \mathbf { x } _ { i } \| _ { 2 } \right) } { n } } \Bigg ) + \frac { 2 } { B } .
+$$
+
+Here we can bound m $\begin{array} { r } { \mathrm { a x } _ { i } \| \mathbf { x } _ { i } \| _ { 2 } \leq \sqrt { k } \operatorname* { m a x } _ { i } \| \mathbf { x } _ { i } \| _ { \infty } \leq \sqrt { k } C } \end{array}$ , and choose $B = n$ to get
+
+$$
+\mathcal { R } _ { n } ^ { \mathrm { e m p } } \big ( \mathcal { F } _ { \phi _ { \alpha } } \big ) \leq O \bigg ( V C \sqrt { \frac { k \log \big ( n k C \big ) } { n } } + \frac { 1 } { n } \bigg ) ,
+$$
+
+which completes the proof.
+
+# 8.4 PROOF OF LEMMA 1
+
+Note that
+
+$$
+\begin{array} { r l } { \overset { \_ } { \nabla } _ { \mathbf { X } _ { 3 } } \left[ \ell \big ( f _ { \mathrm { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { X } ) , Y ( \mathbf { X } ) \big ) \right] = \mathbb { E } _ { P _ { \mathbf { X } } } \left[ \nabla _ { a _ { j } } \ell \big ( f _ { \mathrm { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { X } ) , Y ( \mathbf { X } ) \big ) \right] } & { } \\ & { = \mathbb { E } _ { P _ { \mathbf { X } } } \left[ \nabla _ { a _ { j } } \big ( f _ { \mathrm { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { X } ) - Y ( \mathbf { X } ) \big ) ^ { 2 } \right] } \\ & { = \mathbb { E } _ { P _ { \mathbf { Y } } } \left[ 2 \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { X } + b _ { j } ) \big ( \underset { t = 1 } { \overset { d } { \sum } } a _ { t } \sin ( 2 \pi \mathbf { w } _ { t } ^ { T } \mathbf { X } + b _ { t } ) - Y ( \mathbf { X } ) \big ) \right] } \\ & { = \mathbb { E } _ { P _ { \mathbf { X } } } \left[ 2 a _ { j } \sin ^ { 2 } ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { X } + b _ { j } ) \right] - \mathbb { E } _ { P _ { \mathbf { X } } } \left[ 2 \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { X } + b _ { j } ) Y ( \mathbf { X } ) \right] } \\ & { \ + \mathbb { E } _ { P _ { \mathbf { X } } } \left[ \underset { t \neq j } { \sum } 2 a _ { t } \sin ( 2 \pi \mathbf { w } _ { t } ^ { T } \mathbf { X } + b _ { t } ) \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { X } + b _ { j } ) \right] } \\ & { = a _ { j } - 2 \left[ \cos ( b _ { j } ) \ln \big \{ \hat { Y } * \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { j } ) \big \} + \sin ( b _ { j } ) \mathrm { R e } \big \{ \hat { Y } * \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { j } ) \big \} \right] . } \end{array}
+$$
+
+To show the last equality, we use the isolatedness assumption for $\mathbf { w } _ { j }$ , i.e. $\forall t \neq j : \operatorname* { m i n } \{ \| \mathbf { w } _ { t } - \mathbf { w } _ { j } \| _ { 2 } , \| \mathbf { w } _ { t } +$ $\mathbf { w } _ { j } \| _ { 2 } \big \} > B ( P \mathbf { x } )$ , and also $\| \mathbf { w } _ { j } \| _ { 2 } \geq B ( P \mathbf { x } ) / 2$ . Then, for each $t$
+
+$$
+\begin{array} { r l } { \mathbb { E } _ { P \mathbf x } \left[ 2 \sin ( 2 \pi \mathbf w _ { t } ^ { T } \mathbf X + b _ { t } ) \sin ( 2 \pi \mathbf w _ { j } ^ { T } \mathbf X + b _ { t } ) \right] = 2 \int P \mathbf x ( \mathbf x ) \sin ( 2 \pi \mathbf w _ { t } ^ { T } \mathbf x + b _ { t } ) \sin ( 2 \pi \mathbf w _ { j } ^ { T } \mathbf x + b _ { t } ) \mathrm { d } \mathbf x } & { } \\ { = } & { \int P \mathbf x ( \mathbf x ) \biggl [ \cos ( 2 \pi ( \mathbf w _ { t } - \mathbf w _ { s } ) ^ { T } \mathbf x + b _ { t } - b _ { s } ) } \\ & { \phantom { 2 } - \cos ( 2 \pi ( \mathbf w _ { t } + \mathbf w _ { s } ) ^ { T } \mathbf x + b _ { t } + b _ { s } ) \biggr ] \mathrm { d } \mathbf x } \\ { = } & { 0 . 5 \exp ( j ( b _ { t } - b _ { s } ) ) \widehat { P } \widetilde { \mathbf x } ( \mathbf w _ { 1 } - \mathbf w _ { s } ) } \\ { + 0 . 5 \exp ( j ( b _ { t } - b _ { t } ) ) \widehat { P } \widetilde { \mathbf x } ( \mathbf w _ { j } - \mathbf w _ { t } ) } \\ & { \phantom { 2 } - 0 . 5 \exp ( j ( b _ { t } + b _ { s } ) ) \widehat { P } \widetilde { \mathbf x } ( \mathbf w _ { 1 } + \mathbf w _ { s } ) } \\ { - 0 . 5 \exp ( j ( b _ { t } + b _ { s } ) ) \widehat { P } \widetilde { \mathbf x } ( \mathbf w _ { 1 } + \mathbf w _ { s } ) } \\ & { \phantom { 2 } - 0 . 5 \exp ( j ( b _ { t } + b _ { t } ) ) \widehat { P } \widetilde { \mathbf x } ( - \mathbf w _ { t } - \mathbf w _ { j } ) } \\ { = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } t < j _ { s } } \\ { 1 } & { \mathrm { i f ~ } t = j _ { s } } \end{array} \right. } \end{array}
+$$
+
+Also, by applying the convolution property of Fourier transform we can show
+
+$$
+\begin{array} { r l } { \mathbb { E } _ { P _ { \mathbf { X } } } \left[ \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { X } + b _ { j } ) Y ( \mathbf { X } ) \right] = \displaystyle \int P _ { \mathbf { X } } ( \mathbf { x } ) Y ( \mathbf { x } ) \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { x } + b _ { j } ) \mathrm { d } \mathbf { x } } & { } \\ { = \displaystyle \int ( P _ { \mathbf { X } } \times Y ) ( \mathbf { x } ) \left[ \cos ( b _ { j } ) \sin ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { X } ) + \sin ( b _ { j } ) \cos ( 2 \pi \mathbf { w } _ { j } ^ { T } \mathbf { X } ) \right] \mathrm { d } \mathbf { x } } & { } \\ { = \cos ( b _ { j } ) \mathrm { I m } \{ \widehat { Y } \star \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { j } ) \} + \sin ( b _ { j } ) \mathrm { R e } \{ \widehat { Y } \star \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { j } ) \} . } & { } \end{array}
+$$
+
+Finally if $( \mathbf { a } , \mathbf { W } , \mathbf { b } )$ is a local minimum for the population risk, for all $t$ ’s we have $\nabla _ { a _ { t } } \bar { \mathbb { E } } _ { P _ { \mathbf { X } } } \big [ \ell \big ( \dot { f } _ { \mathbf { a } , \mathbf { W } , \mathbf { b } } ( \mathbf { X } ) , Y ( \mathbf { X } ) \big ) \big ] \ = \ 0$ . Therefore, due to the isolatedness assumption of $\mathbf { w } _ { j }$ we have
+
+$$
+| a _ { j } | = 2 \left| \cos ( b _ { j } ) \operatorname { I m } \bigl \{ \widehat { Y } \star \widehat { P } _ { \mathbf { X } } ( { \mathbf { w } } _ { j } ) \bigr \} + \sin ( b _ { j } ) \operatorname { R e } \bigl \{ \widehat { Y } \star \widehat { P } _ { \mathbf { X } } ( { \mathbf { w } } _ { j } ) \bigr \} \right| \leq 2 \bigl | \widehat { Y } \star \widehat { P } _ { \mathbf { X } } ( { \mathbf { w } } _ { j } ) \bigr | .
+$$
+
+# 8.5 PROOF OF THEOREM 4
+
+Since the isolatedness assumption holds for all $j$ ’s, by Lemma 1,
+
+$$
+\forall j : \quad | a _ { j } ^ { * } | \leq 2 \bigl | \widehat { Y } \star \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { j } ^ { * } ) \bigr | .
+$$
+
+If $\| \mathbf { w } _ { t } ^ { * } \| _ { 2 } > B ( Y ) + B ( P _ { \mathbf { X } } )$ holds for some $t$ , (37) implies
+
+$$
+| a _ { t } ^ { * } | \leq 2 \bigl | \widehat { Y } \star \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { t } ^ { * } ) \bigr | = 0 .
+$$
+
+Hence, $a _ { t } ^ { * }$ will be 0, implying there will be no component in $f _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } }$ with $\| \mathbf { w } _ { t } ^ { * } \| _ { 2 } > B ( Y ) + B ( P _ { \mathbf { X } } )$ . This discussion proves the first part of Theorem, i.e. $B ( f _ { \mathbf { a } ^ { * } } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } ) \leq B ( Y ) + B ( P _ { \mathbf { X } } )$ .
+
+To show the second part, note that
+
+$$
+\begin{array} { r l } { \| A _ { \varepsilon } ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 2 } } & { = | \alpha | ^ { 3 } \| _ { \infty } ^ { 3 } } \\ & { \stackrel { \mathrm { i . e . } } { = } 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 3 } } \\ & { = 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( s ) \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 3 } } \\ & { \stackrel { \mathrm { i . e . } } { = } 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) } \\ & { = 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 3 } } \\ & { = 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 3 } \mathrm { d } u \varepsilon } \\ & { = 2 \displaystyle \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 3 } \mathrm { d } u \varepsilon } \\ & { \stackrel { \mathrm { i . e . } } { = } 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 4 } } \\ & { = 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 3 } } \\ & { = 2 \displaystyle \sum _ { u = 1 } ^ { 3 } \int _ { u } ^ { 1 } | \hat { \rho } _ { u } ^ { \varepsilon } ( u ) ^ { \varepsilon } ( s ) \| _ { \infty } ^ { 3 } } \\ &  - 2 \displaystyle \int _ { u } ^ { 1 } \hat { \rho } _ { u } ^  \ \end{array}
+$$
+
+Here, (a) comes from Lemma 1. Also, since $\operatorname* { m i n } \Bigl \{ \| \mathbf { w } _ { t } ^ { * } - \mathbf { w } _ { r } ^ { * } \| _ { 2 } , \| \mathbf { w } _ { t } ^ { * } + \mathbf { w } _ { r } ^ { * } \| _ { 2 } \Bigr \} > 2 B ( P _ { \mathbf { X } } )$ is assumed for any $t \neq r$ , for any $\boldsymbol { \xi }$ at most one element in $\left[ \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { t } ^ { * } - \pmb { \xi } ) \right] _ { t = 1 } ^ { d }$ can be nonzero. Because if both $\widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { t } ^ { * } - \pmb { \xi } )$ and $\widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { r } ^ { * } - \pmb { \xi } )$ are nonzero for $r \neq t$ , then $\| \mathbf { w } _ { t } ^ { * } - \pmb { \xi } \| \leq B ( P _ { \mathbf { X } } )$ and also $\| \mathbf { w } _ { r } ^ { * } - \pmb { \xi } \| \leq B ( P _ { \mathbf { X } } )$ which results in $\| \mathbf { w } _ { t } ^ { * } - \mathbf { w } _ { r } ^ { * } \| \leq 2 B ( P _ { \mathbf { X } } )$ which is a contradiction. Hence,
+
+$$
+\sum _ { t = 1 } ^ { d } \left| \widehat { P } \mathbf { x } ( \mathbf { w } _ { t } ^ { * } - \pmb { \xi } ) \right| \leq \operatorname* { m a x } _ { \pmb { \xi } ^ { \prime } } \left| \widehat { P } \mathbf { x } ( \pmb { \xi } ^ { \prime } ) \right| \leq \int \left| P \mathbf { x } ( \mathbf { x } ) \right| \mathrm { d } \mathbf { x } = 1 ,
+$$
+
+which proves (b) and completes the proof.
+
+# 8.5.1 APPLYING THEOREM 4 TO MULTIVARIATE GAUSSIAN X
+
+Assume $\mathbf { X } \sim { \mathcal { N } } ( \mu , \sigma ^ { 2 } \mathbf { I } _ { k \times k } )$ has a multivariate Gaussian distribution with mean $\pmb { \mu }$ and standard deviation $\sigma$ . Then, the Fourier transform $\hat { P } _ { \mathbf { X } }$ has a Gaussian shape with mean 0 and standard deviation $1 / \sigma$ . Hence, if for any $i , j$ we have $\operatorname* { m i n } \bigr \{ \| \mathbf { w } _ { i } ^ { * } - \mathbf { w } _ { j } ^ { * } \| _ { 2 } , \| \mathbf { w } _ { i } ^ { * } + \mathbf { w } _ { j } ^ { * } \| _ { 2 } \bigr \} \stackrel {  } { > } 2 C / \sigma$ for some constant $C$ , the approximation error term which should be added to the upperbound in Equation (39) is $2 \| \widehat { Y } \| _ { 1 } d \exp ( - C ^ { 2 } / 2 )$ . Also, given any $\epsilon > 0$ the Fourier $\ell _ { 1 }$ norm outside the bandwidth $O ( \sqrt { k } \log ( 1 / \epsilon ) / \sigma )$ is at most $\epsilon$ . Therefore, Theorem 4 implies
+
+$$
+\begin{array} { r l } & { \bullet \mathcal { B } ( f _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } ) \leq \mathcal { B } ( Y ) + O \big ( \sqrt { k } / \sigma \big ) , } \\ & { \bullet \| \widehat { f } _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } \| _ { 1 } \leq 2 ( 1 + d \exp ( - C ^ { 2 } / 2 ) ) \| \widehat { Y } \| _ { 1 } . } \end{array}
+$$
+
+# 8.6 APPROXIMATE VERSION OF THEOREM 4
+
+Here we show an approximate version of Theorem 4 which applies to approximate population local minima.
+
+Theorem 5. Consider minimizing the population risk (11). Consider an approximate local minimum $( \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } )$ where $| \nabla _ { a _ { j } } \mathbb { E } \big [ \ell \big ( f _ { { \mathbf a } ^ { * } , { \mathbf W } ^ { * } , { \mathbf b } ^ { * } } ( { \mathbf { \dot { X } } } ) , Y ( { \mathbf { X } } ) \big ) \big ) \big ] | \ \leq \ \epsilon$ for all $j$ ’s. If for any two different $i , j$ we
+
+have $\operatorname* { m i n } \bigl \{ \| \mathbf { w } _ { i } ^ { * } - \mathbf { w } _ { j } ^ { * } \| _ { 2 } , \| \mathbf { w } _ { i } ^ { * } + \mathbf { w } _ { j } ^ { * } \| _ { 2 } \bigr \} > 2 B ( P _ { \mathbf { X } } )$ , then the Fourier $\ell _ { 1 }$ -norm of $f _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } }$ outside the bandwidth $\mathcal { B } ( Y ) + \mathcal { B } ( P _ { \mathbf { X } } )$ is bounded by $d \epsilon$ and
+
+$$
+\begin{array} { r } { \| \widehat { f } _ { \mathbf { a } ^ { * } , \mathbf { W } ^ { * } , \mathbf { b } ^ { * } } \| _ { 1 } \leq 2 \| \widehat { Y } \| _ { 1 } + d \epsilon . } \end{array}
+$$
+
+Proof. Since the isolated components condition holds, we can apply Lemma 1’s proof to show under the above assumptions
+
+$$
+\forall j : \quad | a _ { j } ^ { * } | \leq 2 \big | \hat { Y } \star \hat { P } _ { \mathbf { X } } ( \mathbf { w } _ { j } ^ { * } ) \big | + \epsilon .
+$$
+
+Then, a simple modification of Theorem 4’s proof according to the above inequality proves the above theorem. □
+
+# 8.7 PROOF OF THEOREM 4 WITHOUT THE ISOLATED COMPONENTS ASSUMPTION
+
+What happens if a component $\mathbf { w } _ { i }$ is not isolated from the other components which has been assumed in Theorem 4? As a simplifying assumption, we assume that $\mathbf b = \mathbf 0$ and $\widehat { P _ { \mathbf { X } } }$ is real. We can write
+
+$$
+\begin{array} { r l } { \displaystyle \forall i : \nabla _ { a _ { i } } \mathbb { E } _ { P _ { \mathbf { X } } } \left[ \ell \Big ( f _ { \mathbf { a } , \mathbf { w } } ( \mathbf { x } ) , Y ( \mathbf { x } ) \Big ) \right] = \sum _ { j = 1 } ^ { d } \left[ a _ { j } \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } - \mathbf { w } _ { j } ) \right] - \operatorname { I m } \left\{ \widehat { Y } \star \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } ) \right\} } & { } \\ { \displaystyle } & { \approx \sum _ { j = 1 } ^ { d } \left[ \left( a _ { j } - \int _ { \xi \in S _ { \mathbf { w } _ { j } } } \operatorname { I m } \{ \widehat { Y } ( \xi ) \} { \mathrm { d } } \xi \right) \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } - \mathbf { w } _ { j } ) \right] } \end{array}
+$$
+
+Here $S _ { \mathbf { w } _ { i } }$ ’s, which are centered around $\mathbf { w } _ { j }$ ’s, are disjoint sets covering the bandwidth region for $\widehat { Y }$ , i.e. $\{ \pmb { \xi } : \| \pmb { \xi } \| _ { 2 } ^ { \prime } \leq B ( Y ) \} \subseteq \bigcup _ { j } S _ { \mathbf { w } _ { j } }$ . Note that in (41) we have approximated the convolution integral as
+
+$$
+\begin{array} { r l } { \widehat { Y } \star \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } ) = \displaystyle \int _ { \pm : \| \xi \| _ { 2 } \leq B ( Y ) } \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } - \pmb { \xi } ) \widehat { Y } ( \pmb { \xi } ) \mathrm { d } \xi } & { } \\ { = \displaystyle \sum _ { j = 1 } ^ { d } \displaystyle \int _ { \pmb { \xi } \in S _ { \mathbf { w } _ { j } } } \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } - \pmb { \xi } ) \widehat { Y } ( \pmb { \xi } ) \mathrm { d } \xi } & { } \\ { \approx \displaystyle \sum _ { j = 1 } ^ { d } \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } - \mathbf { w } _ { j } ) \displaystyle \int _ { \pmb { \xi } \in S _ { \mathbf { w } _ { j } } } \widehat { Y } ( \pmb { \xi } ) \mathrm { d } \xi . } \end{array}
+$$
+
+Letting the gradient element in (41) be zero for all $a _ { i }$ ’s at a local minimum $( \mathbf { a } ^ { * } , \mathbf { W } ^ { * } )$ of the population risk, the following approximation holds in general case:
+
+$$
+\forall j : a _ { j } ^ { * } \approx \int _ { \pmb { \xi } \in S _ { \mathbf { w } _ { j } } } \mathrm { I m } \{ \widehat { Y } ( \pmb { \xi } ) \} \mathrm { d } \pmb { \xi } ,
+$$
+
+Here, the matrix $\left[ \widehat { P _ { \mathbf { X } } } ( \mathbf { w } _ { i } - \mathbf { w } _ { j } ) \right] _ { 1 \leq i , j \leq d }$ is positive-definite and hence invertible, because $\widehat { P \mathbf { x } }$ is the Fourier transform of $P \mathbf { x }$ and due to Bochner’s theorem a positive-definite kernel function. Therefore, the system of linear equations -PcX(w∗i − w∗j ) -a∗j − Rξ∈Sw $\begin{array} { r } { \left[ \widehat { P _ { { \bf X } } } ( { \bf w } _ { i } ^ { * } - { \bf w } _ { j } ^ { * } ) \right] \left[ a _ { j } ^ { * } - \int _ { \pmb { \xi } \in S _ { { \bf w } _ { j } } } \mathrm { I m } \{ \widehat { Y } ( \pmb { \xi } ) \} \mathrm { d } \pmb { \xi } \right] \approx \mathbf { 0 } } \end{array}$ would imply (42). This discussion indicates that the result of Theorem 4 would remain valid even if the isolated components condition does not hold.

parse/train/SJxstlHFPH/SJxstlHFPH.md

@@ -0,0 +1,333 @@
+# DIFFERENTIABLE REASONING OVER A VIRTUAL KNOWLEDGE BASE
+
+Bhuwan Dhingra1∗, Manzil Zaheer2, Vidhisha Balachandran1, Graham Neubig1, Ruslan Salakhutdinov1, William W. Cohen2
+
+1 School of Computer Science, Carnegie Mellon University 2 Google Research {bdhingra, vbalacha, gneubig, rsalakhu}@cs.cmu.edu {manzilzaheer, wcohen}@google.com
+
+# ABSTRACT
+
+We consider the task of answering complex multi-hop questions using a corpus as a virtual knowledge base $( K B )$ . In particular, we describe a neural module, DrKIT, that traverses textual data like a KB, softly following paths of relations between mentions of entities in the corpus. At each step the module uses a combination of sparse-matrix TFIDF indices and a maximum inner product search (MIPS) on a special index of contextual representations of the mentions. This module is differentiable, so the full system can be trained end-to-end using gradient based methods, starting from natural language inputs. We also describe a pretraining scheme for the contextual representation encoder by generating hard negative examples using existing knowledge bases. We show that DrKIT improves accuracy by 9 points on 3-hop questions in the MetaQA dataset, cutting the gap between text-based and KB-based state-of-the-art by $7 0 \%$ . On HotpotQA, DrKIT leads to a $1 0 \%$ improvement over a BERT-based re-ranking approach to retrieving the relevant passages required to answer a question. DrKIT is also very efficient, processing 10-100x more queries per second than existing multi-hop systems.1
+
+# 1 INTRODUCTION
+
+Large knowledge bases (KBs), such as Freebase and WikiData, organize information around entities, which makes it easy to reason over their contents. For example, given a query like “When was the Grateful Dead’s lead singer born?”, one can identify the entity Grateful Dead and the path of relations LeadSinger, BirthDate to efficiently extract the answer—provided that this information is present in the KB. Unfortunately, KBs are often incomplete (Min et al., 2013). While relation extraction methods can be used to populate KBs, this process is inherently error-prone, expensive and slow.
+
+Advances in open-domain QA (Moldovan et al., 2002; Yang et al., 2019) suggest an alternative— instead of performing relation extraction, one could treat a large corpus as a virtual KB by answering queries with spans from the corpus. This ensures facts are not lost in the relation extraction process, but also poses challenges. One challenge is that it is relatively expensive to answer questions using QA models which encode each document in a query-dependent fashion (Chen et al., 2017; Devlin et al., 2019)—even with modern hardware (Strubell et al., 2019; Schwartz et al., 2019). The cost of QA is especially problematic for certain complex questions, such as the example question above. If the passages stating that “Jerry Garcia was the lead singer of the Grateful Dead” and “Jerry Garcia was born in 1942” are far apart in the corpus, it is difficult for systems that retrieve and read a single passage to find an answer—even though in this example, it might be easy to answer the question after the relations were explicitly extracted into a KB. More generally, complex questions involving sets of entities or paths of relations may require aggregating information from multiple documents, which is expensive.
+
+One step towards efficient QA is the recent work of Seo et al. (2018; 2019) on phrase-indexed question answering (PIQA), in which spans in the text corpus are associated with question-independent contextual representations and then indexed for fast retrieval. Natural language questions are then answered by converting them into vectors that are used to perform maximum inner product search (MIPS) against the index. This can be done efficiently using approximate algorithms (Shrivastava & Li, 2014). However, this approach cannot be directly used to answer complex queries, since by construction, the information stored in the index is about the local context around a span—it can only be used for questions where the answer can be derived by reading a single passage.
+
+This paper addresses this limitation of phrase-indexed question answering. We introduce an efficient, end-to-end differentiable framework for doing complex QA over a large text corpus that has been encoded in a query-independent manner. Specifically, we consider “multi-hop” complex queries which can be answered by repeatedly executing a “soft” version of the operation below, defined over a set of entities $X$ and a relation $R$ :
+
+$$
+Y = X . { \mathrm { f o l l o w } } ( R ) = \{ x ^ { \prime } : \exists x \in X { \mathrm { ~ s . t . ~ } } R ( x , x ^ { \prime } ) { \mathrm { ~ h o l d s } } \} 
+$$
+
+In past work soft, differentiable versions of this operation were used to answer multi-hop questions against an explicit KB (Cohen et al., 2019). Here we propose a more powerful neural module which approximates this operation against an indexed corpus (a virtual KB). In our module, the input $X$ is a sparse-vector representing a weighted set of entities, and the relation $R$ is a dense feature vector, e.g. a vector derived from a neural network over a natural language query. $X$ and $R$ are used to construct a MIPS query used for retrieving the top- $K$ spans from the index. The output $Y$ is another sparse-vector representing the weighted set of entities, aggregated over entity mentions in the top- $K$ spans. We discuss pretraining schemes for the index in $\bar { \ S } \bar { 2 . 3 }$ .
+
+For multi-hop queries, the output entities $Y$ can be recursively passed as input to the next iteration of the same module. The weights of the entities in $Y$ are differentiable w.r.t the MIPS queries, which allows end-to-end learning without any intermediate supervision. We discuss an implementation based on sparse-matrix-vector products, whose runtime and memory depend only on the number of spans $K$ retrieved from the index. This is crucial for scaling up to large corpora, providing up to $1 5 \mathrm { x }$ faster inference than existing state-of-the-art multi-hop and open-domain QA systems. The system we introduce is called DrKIT (for Differentiable Reasoning over a Knowledge base of Indexed Text). We test DrKIT on the MetaQA benchmark for complex question answering, and show that it improves on prior text-based systems by 5 points on 2-hop and 9 points on 3-hop questions, reducing the gap between text-based and KB-based systems by $3 0 \%$ and $7 0 \%$ , respectively. We also test DrKIT on a new dataset of multi-hop slot-filling over Wikipedia articles, and show that it outperforms DrQA (Chen et al., 2017) and PIQA (Seo et al., 2019) adapted to this task. Finally, we apply DrKIT to multi-hop information retrieval on the HotpotQA dataset (Yang et al., 2018), and show that it significantly improves over a BERT-based reranking approach, while being $1 0 \mathrm { x }$ faster.
+
+# 2 DIFFERENTIABLE REASONING OVER A KB OF INDEXED TEXT
+
+We want to answer a question $q$ using a text corpus as if it were a KB. We start with the set of entities $z$ in the question $q$ , and would ideally want to follow relevant outgoing relation edges in the KB to arrive at the answer. To simulate this behaviour on text, we first expand $z$ to set of cooccurring mentions $m$ (say using TFIDF). Not all of these co-occurring mentions are relevant for the question $q$ , so we train a neural network which filters the mentions based on a relevance score of $q$ to $m$ . Then we can aggregate the resulting set of mentions $m$ to the entities they refer to, ending up with an ordered set $z ^ { \prime }$ of entities which are answer candidates, very similar to traversing the KB. Furthermore, if the question requires more than one hop to answer, we can repeat the above procedure starting with $z ^ { \prime }$ . This is depicted pictorially in Figure 1.
+
+We begin by first formalizing this idea in a probabilistic framework in $\ S 2 . 1$ . In $\ S 2 . 2$ , we describe how the expansion of entities to mentions and the filtering of mentions can be performed efficiently, using sparse-matrix products and MIPS algorithms (Johnson et al., 2017). Lastly we discuss a pretraining scheme for constructing the mention representations in $\ S 2 . 3$ .
+
+Notation: We denote the given corpus as $\mathcal { D } = \{ d _ { 1 } , d _ { 2 } , . . . \}$ , where each $d _ { k } = ( d _ { k } ^ { 1 } , \dots , d _ { k } ^ { L _ { k } } )$ is a sequence of tokens. We start by running an entity linker over the corpus to identify mentions of a fixed set of entities $\mathcal { E }$ . Each mention $m$ is a tuple $( e _ { m } , k _ { m } , i _ { m } , j _ { m } )$ denoting that the text span $d _ { k _ { m } } ^ { i _ { m } } , \ldots , d _ { k _ { m } } ^ { j _ { m } }$ in doc ent $k _ { m }$ mentions the $e _ { m } \in \mathcal { E }$ , and the collection of all mentions in the corpus is denoted as $\mathcal { M }$ . Note that typically $| { \mathcal { M } } | \gg | { \mathcal { E } } |$ .
+
+![](images/374a456e6fc1cc77a02aef3190f3252081d7c8bc178ae0bbe71a04ea91b08de3.jpg)  
+Figure 1: DrKIT answers multi-hop questions by iteratively mapping an input set of entities $X$ (The Grateful Dead, Bob Dylan) to an output set of entities $Y$ (Dylan & the Dead, American beauty, ...) which are related to any input entity by some relation $R$ (album by).
+
+# 2.1 DIFFERENTIABLE MULTI-HOP REASONING
+
+We assume a weakly supervised setting where during training we only know the final answer entities $a \in { \mathcal { E } }$ for a $T$ -hop question. We denote the latent sequence of entities which answer each of the intermediate hops as $z _ { 0 } , z _ { 1 } , \dotsc , z _ { T } \in \mathcal { E }$ , where $z _ { 0 }$ is mentioned in the question, and $z _ { T } = a$ . We can recursively write the probability of an intermediate answer as:
+
+$$
+\operatorname* { P r } ( z _ { t } | q ) = \sum _ { z _ { t - 1 } \in \mathcal E } \operatorname* { P r } ( z _ { t } | q , z _ { t - 1 } ) \operatorname* { P r } ( z _ { t - 1 } | q )
+$$
+
+Here $\operatorname* { P r } ( z _ { 0 } | q )$ is the output of an entity linking system over the question, and $\operatorname* { P r } \bigl ( z _ { t } | q , z _ { t - 1 } \bigr )$ corresponds to a single-hop model which answers the $t$ -th hop, given the entity from the previous hop $z _ { t - 1 }$ , by following the appropriate relation. Eq. 1 models reasoning over a chain of latent entities, but when answering questions over a text corpus, we must reason over entity mentions, rather than entities themselves. Hence $\operatorname* { P r } \bigl ( z _ { t } | q , z _ { t - 1 } \bigr )$ needs to be aggregated over all mentions of $z _ { t }$ , which yields
+
+$$
+\operatorname* { P r } ( z _ { t } | q ) = \sum _ { m \in \mathcal { M } } \sum _ { z _ { t - 1 } \in \mathcal { E } } \operatorname* { P r } ( z _ { t } | m ) \operatorname* { P r } ( m | q , z _ { t - 1 } ) \operatorname* { P r } ( z _ { t - 1 } | q )
+$$
+
+The interesting term to model in the above equation is $P r ( m | q , z _ { t - 1 } )$ , which represents the relevance of mention $m$ given the question and entity $z _ { t - 1 }$ . Following the analogy of a KB, we first expand the entity $z _ { t - 1 }$ to co-occuring mentions $m$ and use a learned scoring function to find the relevance of these mentions. Formally, let $F ( m )$ denote a TFIDF vector for the document containing $m$ , $G ( z _ { t - 1 } )$ be the TFIDF vector of the surface form of the entity from the previous hop, and $s _ { t } ( m , z , q )$ be a learnt scoring function (different for each hop). Thus, we model $\mathrm { P r } ( m | q , z _ { t - 1 } )$ as
+
+$$
+\operatorname* { P r } ( m | q , z _ { t - 1 } ) \propto \underbrace { \mathbb { 1 } \left\{ G ( z _ { t - 1 } ) \cdot F ( m ) > \epsilon \right\} } _ { \mathrm { e x p a n s i o n ~ t o ~ c o c c u r r i n g ~ m e n t i o n s } } \times \underbrace { s _ { t } ( m , z _ { t - 1 } , q ) } _ { \mathrm { r e l e v a n c e ~ f l t e r i n g } }
+$$
+
+Another equivalent way to look at our model in Eq. 3 is that the second term retrieves mentions of the correct type requested by the question in the $t$ -th hop, and the first term filters these based on co-occurrence with $z _ { t - 1 }$ . When dealing with a large set of mentions $m$ , we will typically retain only the top- $K$ relevant mentions. We will show that this joint modelling of co-occurrence and relevance is important for good performance, as was also observed by Seo et al. (2019).
+
+The other term left in Eq. 2 is $\Pr ( z | m )$ , which is 1 if mention $m$ refers to the entity $z$ else 0, based on the entity linking system. In general, to compute Eq. 2 the mention scoring of Eq. 3 needs to be evaluated for all latent entity and mention pairs, which is prohibitively expensive. However, by restricting $s _ { t }$ to be an inner product we can implement this efficiently (§2.2).
+
+To highlight the differentiability of the proposed overall scheme, we can represent the computation in Eq. 2 as matrix operations. We pre-compute the TFIDF term for all entities and mentions into a sparse matrix, which we denote as $A _ { E \to \bar { M } } [ e , m ] = \mathbb { 1 } \left( G ( e ) \cdot F ( m ) > \epsilon \right)$ . Then entity expansion to co-occuring mentions can be done using a sparse-matrix by sparse-vector multiplication between $A _ { E  M }$ and $z _ { t - 1 }$ . For the relevance scores, let $\mathbb { T } _ { K } \big ( s _ { t } ( m , z _ { t - 1 } , q ) \big )$ denote the top- $K$ relevant mentions encoded as a sparse vector in $\mathbb { R } ^ { | \mathcal { M } | }$ . Finally, the aggregation of mentions to entities can be formulated as multiplication with another sparse-matrix $B _ { M  E }$ , which encodes coreference, i.e. mentions corresponding to the same entity. Putting all these together, using $\odot$ to denote elementwise product, and defining $Z _ { t } = [ \operatorname* { P r } ( z _ { t } = { \mathsf { \bar { e } } } _ { 1 } | q ) ; \ldots ; \operatorname* { P r } ( z _ { t } = { \mathsf { e } } _ { | \xi | } ^ { - } | q ) ]$ , we can observe that for large $K$ (i.e., as $K  | { \mathcal { M } } | )$ , Eq. 2 becomes equivalent to:
+
+$$
+Z _ { t } = \mathrm { s o f t m a x } ( [ Z _ { t - 1 } ^ { T } A _ { E  M } \odot \mathbb { T } _ { K } ( s _ { t } ( m , z _ { t - 1 } , q ) ) ] B _ { M  E } ) .
+$$
+
+Note that every operation in above equation is differentiable and between sparse matrices and vectors: we will discuss efficient implementations in $\ S 2 . 2$ . Further, the number of non-zero entries in $Z _ { t }$ is bounded by $K$ , since we filtered (the element-wise product in Eq. 4) to top- $K$ relevant mentions among TFIDF based expansion and since each mention can only point to a single entity in $B _ { M  E }$ . This is important, as it prevents the number of entries in $Z _ { t }$ from exploding across hops (which might happen if, for instance, we added the relevance and TFIDF scores instead).
+
+We can view $Z _ { t - 1 } , Z _ { t }$ as weighted multisets of entities, and $s _ { t } ( m , z , q )$ as implicitly selecting mentions which correspond to a relation $R$ . Then Eq. 4 becomes a differentiable implementation of $Z _ { t } = Z _ { t - 1 }$ .follow $( R )$ , i.e. mimicking the graph traversal in a traditional KB. We thus call Eq. 4 a textual follow operation.
+
+Training and Inference. The model is trained end-to-end by optimizing the cross-entropy loss between $Z _ { T }$ , the weighted set of entities after $T$ hops, and the ground truth answer set $A$ . We use a temperature coefficient $\lambda$ when computing the softmax in Eq, 4 since the inner product scores of the top- $K$ retrieved mentions are typically high values, which would otherwise result in very peaked distributions of $Z _ { t }$ . We also found that taking a maximum over the mention set of an entity $M _ { z _ { t } }$ in Eq. 2 works better than taking a sum. This corresponds to optimizing only over the most confident mention of each entity, which works for corpora like Wikipedia that do not have much redundancy. A similar observation was made by Min et al. (2019) in weakly supervised settings.
+
+# 2.2 EFFICIENT IMPLEMENTATION
+
+Sparse TFIDF Mention Encoding. To compute the sparse-matrix $A _ { E  M }$ for entity-mention expansion in Eq. 4, the TFIDF vectors $F ( m )$ and $G ( z _ { t - 1 } )$ are constructed over unigrams and bigrams, hashed to a vocabulary of $1 6 M$ buckets. While $F$ computes the vector from the whole passage around $m$ , $G$ only uses the surface form of $z _ { t - 1 }$ . This corresponds to retrieving all mentions in a document using $z _ { t - 1 }$ as the query. We limit the number of retrieved mentions per entity to a maximum of $\mu$ , which leads to a $| { \mathcal { E } } | \times | { \mathcal { M } } |$ sparse-matrix.
+
+Efficient Entity-Mention expansion. The expansion from a set of entities to mentions occurring around them can be computed using the sparse-matrix by sparse-vector product $Z _ { t - 1 } ^ { T ^ { \prime } } A _ { E \to M }$ . A simple lower bound for multiplying a sparse $| \mathcal { E } | \times | \mathcal { M } |$ matrix, with maximum $\mu$ nonzeros in each row, by a sparse $| \mathcal { E } | \times 1$ vector with $K$ non-zeros is $\Omega ( K \mu )$ . Note that this lower bound is independent of the size of matrix $A _ { E  M }$ , or in other words independent of the number of entities or mentions. To attain the lower bound, the multiplication algorithm must be vector driven, because any matrix-driven algorithms need to at least iterate over all the rows. Instead we slice out the relevant rows from $A _ { E  M }$ . To enable this our solution is to represent the sparse-matrix $A _ { E  M }$ as two row-wise lists of variable-sized lists of the indices and values of the non-zero elements, respectively. This results in a “ragged” representation of the matrix (tf.RaggedTensors, 2018) which can be easily sliced corresponding to the non-zero entries in the vector in $O ( \log | \mathcal { E } | )$ time. We are now left with $K$ sparse-vectors with at most $\mu$ non-zero elements in each. We can add these $K$ sparse-vectors weighted by corresponding values from the vector $Z _ { t - 1 } ^ { T }$ in $O ( K \operatorname* { m a x } \{ K , \mu \} )$ time. Moreover, such an implementation is feasible with deep learning frameworks such as TensorFlow. We tested the scalability of our approach by varying the number of entities for a fixed density of mentions $\mu$ (from Wikipedia). Figure 2 compares our approach to the default sparse-matrix times dense-vector product available in TensorFlow.
+
+![](images/6dcd3f357abcbe05879deb8da50dfb09d6d7f3ee5b37f9f990c10e51134b1892.jpg)  
+Figure 2: Runtime on a single K80 GPU when using ragged representations for implementing sparse-matrix vector product, vs the default sparse-matrix times dense vector product available in TensorFlow. $\vert \mathcal { E } \vert > 1 0 ^ { 5 }$ leads to OOM for the latter.
+
+Efficient top- $K$ mention relevance filtering: To make computation of Eq. 4 feasible, we need an efficient way to get top- $K$ relevant mentions related to an entity in $z _ { t - 1 }$ for a given question $q$ , without enumerating all possibilities. A key insight is that by restricting the scoring function $s _ { t } ( m , z _ { t - 1 } , q )$ to an inner product, we can easily approximate a parallel version of this computation, across all mentions $m$ . To do this, let $f ( m )$ be a dense encoding of $m$ , and $g _ { t } \big ( q , z _ { t - 1 } \big )$ be a dense encoding of the question $q$ for the $t$ -th hop, both in $\mathbb { R } ^ { p }$ (the details of the dense encoding is provided in next paragraph), then the scoring function $s _ { t } ( m , z _ { t - 1 } , q )$ becomes
+
+$$
+s _ { t } ( m , z _ { t - 1 } , q ) \propto \exp \left\{ f ( m ) \cdot g _ { t } ( q , z _ { t - 1 } ) \right\} ,
+$$
+
+which can be computed in parallel by multiplying a matrix $f ( { \cal M } ) = [ f ( m _ { 1 } ) ; f ( m _ { 2 } ) ; . . . ]$ with $g _ { t } \big ( q , z _ { t - 1 } \big )$ . Although this matrix will be very large for a realistic corpus, since eventually we are only interested in the top- $K$ values, we can use an approximate algorithm for Maximum Inner Product Search (MIPS) (Andoni et al., 2015; Shrivastava $\&$ Li, 2014) to find the $K$ top-scoring elements. The complexity of this filtering step using MIPS is roughly $O ( K p \mathrm { p o l y l o g } | \mathcal { M } | )$ .
+
+Mention and Question Encoders. Mentions are encoded by passing the passages they are contained in through a BERT-large (Devlin et al., 2019) model (trained as described in $\ S 2 . 3 )$ . Suppose mention $m$ appears in passage $d$ , starting at position $i$ and ending at position $j$ . Then $\dot { \boldsymbol { f } } ( \boldsymbol { m } ) = \boldsymbol { W } ^ { T } [ \boldsymbol { H } _ { i } ^ { d } ; \dot { \boldsymbol { H } } _ { j } ^ { d } ]$ , where $H ^ { d }$ is the sequence of embeddings output from BERT, and $W$ is a linear projection to size $p$ . The queries are encoded with a smaller BERT-like model: specifically, they are tokenized with WordPieces (Schuster & Nakajima, 2012), appended to a special [CLS] token, and then passed through a 4-layer Transformer network (Vaswani et al., 2017) with the same architecture as BERT, producing an output sequence $H ^ { q }$ . The $g _ { t }$ functions are defined similarly to the BERT model used for SQuAD-style QA. For each hop $t = 1 , \dots , T$ , we add two additional Transformer layers on top of $H ^ { q }$ , which will be trained to produce MIPS queries from the [CLS] encoding; the first added layer produces a MIPS query $H _ { s t } ^ { \bar { q } }$ to retrieve a start token, and the second added layer a MIPS query $H _ { e n } ^ { q }$ to retrieve an end token. We concatenate the two and define $\tilde { g } _ { t } ( q ) = V ^ { T } [ H _ { s t } ^ { q } ; H _ { e n } ^ { q } ]$ . Finally, to condition on current progress we add the embeddings of $z _ { t - 1 }$ . Specifically, we use entity embeddings $\boldsymbol { E } \in \mathbb { R } ^ { | \mathcal { E } | \times p }$ , to construct an average embedding of the set $Z _ { t - 1 }$ , as $\dot { Z } _ { t - 1 } ^ { T } E$ , and define $g _ { t } ( q , z _ { t - 1 } ) \equiv \tilde { g } _ { t } ( q ) + Z _ { t - 1 } ^ { T } E$ . To avoid a large number of parameters in the model, we compute the entity embeddings as an average over the word embeddings of the tokens in the entity’s surface form. The computational cost of the question encoder $g _ { t } ( q )$ is $O ( p ^ { 2 } )$ .
+
+Thus our total computational complexity to answer a query is $\tilde { O } ( K \operatorname* { m a x } \{ K , \mu \} + K p + p ^ { 2 } )$ (almost independent to number of entities or mentions!), with $O ( \mu | \mathcal { E } | + p | \mathcal { M } | )$ memory to store the precomputed matrices and mention index.2
+
+# 2.3 PRETRAINING THE INDEX
+
+Ideally, we would like to train the mention encoder $f ( m )$ end-to-end using labeled QA data only. However, this poses a challenge when combined with approximate nearest neighbor search—since after every update to the parameters of $f$ , one would need to recompute the embeddings of all mentions in $\mathcal { M }$ . We thus adopt a staged training approach: we first pre-train a mention encoder $f ( m )$ , then compute and index embeddings for all mentions once, keeping these embeddings fixed when training the downstream QA task. Empirically, we observed that using BERT representations “out of the box” do not capture the kind of information our task requires (Appendix C), and thus, pretraining the encoder to capture better mention understanding is a crucial step.
+
+One option adopted by previous researchers (Seo et al., 2018) is to fine-tune BERT on SQuAD (Rajpurkar et al., 2016). However, SQuAD is limited to only 536 articles from Wikipedia, leading to
+
+Table 1: (Left) MetaQA and (Right) WikiData Hits $@ 1$ for 1-3 hop sub-tasks. ots: off-the-shelf without retraining. $\dagger$ : obtained from Sun et al. (2019). cascade: adapted to multi-hop setting by repeatedly applying Eq. 2. pre: pre-trained on slot-filling. e2e: end-to-end trained on single-hop and multi-hop queries.   
+
+<table><tr><td colspan="4">MetaQA</td></tr><tr><td>Model</td><td>1hop</td><td>2hop</td><td>3hop</td></tr><tr><td>DrQA (ots)</td><td>0.553</td><td>0.325</td><td>0.197</td></tr><tr><td>KVMemt</td><td>0.762</td><td>0.070</td><td>0.195</td></tr><tr><td>GraftNett</td><td>0.825</td><td>0.362</td><td>0.402</td></tr><tr><td>PullNett</td><td>0.844</td><td>0.810</td><td>0.782</td></tr><tr><td>DrKIT (e2e)</td><td>0.844</td><td>0.860</td><td>0.876</td></tr><tr><td>DrKIT (strong sup.)</td><td>0.845</td><td>0.871</td><td>0.871</td></tr></table>
+
+<table><tr><td colspan="4">WikiData</td></tr><tr><td>Model</td><td>1hop</td><td>2hop</td><td>3hop</td></tr><tr><td>DrQA (ots, cascade)</td><td>0.287</td><td>0.141</td><td>0.070</td></tr><tr><td>PIQA (ots, cascade)</td><td>0.240</td><td>0.118</td><td>0.064</td></tr><tr><td>PIQA (pre, cascade)</td><td>0.670</td><td>0.369</td><td>0.182</td></tr><tr><td>DrKIT( T(pre,cascade)</td><td>0.816</td><td>0.404</td><td>0.198</td></tr><tr><td>DrKIT (e2e)</td><td>0.834</td><td>0.469</td><td>0.244</td></tr><tr><td>-BERT index</td><td>0.643</td><td>0.294</td><td>0.165</td></tr></table>
+
+a very specific distribution of questions, and is not focused on entity- and relation-centric questions.   
+Here we instead train the mention encoder using distant supervision from a KB.
+
+Specifically, assume we are given an open-domain KB consisting of facts $( e _ { 1 } , R , e _ { 2 } )$ specifying that the relation $R$ holds between the subject $e _ { 1 }$ and the object $e _ { 2 }$ . Then for a corpus of entity-linked text passages $\{ d _ { k } \}$ , we automatically identify tuples $( d , ( e _ { 1 } , R , e _ { 2 } ) )$ such that $d$ mentions both $e _ { 1 }$ and $e _ { 2 }$ . Using this data, we learn to answer slot-filling queries in a reading comprehension setup, where the query $q$ is constructed from the surface form of the subject entity $e _ { 1 }$ and a natural language description of $R$ (e.g. “Jerry Garcia, birth place, ?”), and the answer $e _ { 2 }$ needs to be extracted from the passage $d$ . Using string representations in $q$ ensures our pre-training setup is similar to the downstream task. In pretraining, we use the same scoring function as in previous section, but over all spans $m$ in the passage:
+
+$$
+s ( m , e _ { 1 } , q ) \propto \exp \left\{ f ( s ) \cdot g ( q , e _ { 1 } ) \right\} .
+$$
+
+Following Seo et al. (2016), we normalize start and end probabilities of the span separately.
+
+For effective transfer to the full corpus setting, we must also provide negative instances during pretraining, i.e. query and passage pairs where the answer is not contained in the passage. We consider three types of hard negatives: (1) shared-entity negatives, which pair a query $( e _ { 1 } , R , ? )$ with a passage which mentions $e _ { 1 }$ but not the correct tail answer; (2) shared-relation negative, which pair a query $( e _ { 1 } , R , ? )$ with a passage mentioning two other entities $e _ { 1 } ^ { \prime }$ and $e _ { 2 } ^ { \prime }$ in the same relation $R$ ; and (3) random negatives, which pair queries with random passages from the corpus.
+
+For the multi-hop slot-filling experiments below, we used WikiData (Vrandeciˇ c & Kr´ otzsch, 2014)¨ as our KB, Wikipedia as the corpus, and SLING (Ringgaard et al., 2017) to identify entity mentions. We restrict $d$ be from the Wikipedia article of the subject entity to reduce noise. Overall we collected $9 5 0 K$ pairs over $5 5 0 K$ articles. For the experiments with MetaQA, we supplemented this data with the corpus and KB provided with MetaQA, and string matching for entity linking.
+
+# 3 EXPERIMENTS
+
+# 3.1 METAQA: MULTI-HOP QUESTION ANSWERING WITH TEXT
+
+Dataset. We first evaluate DrKIT on the MetaQA benchmark for multi-hop question answering (Zhang et al., 2018). MetaQA consists of around $4 0 0 K$ questions ranging from 1 to 3 hops constructed by sampling relation paths from a movies KB (Miller et al., 2016) and converting them to natural language using templates. The questions cover 8 relations and their inverses, around $4 3 K$ entities, and are paired with a corpus consisting of $1 8 K$ Wikipedia passages about those entities. The questions are all designed to be answerable using either the KB or the corpus, which makes it possible to compare the performance of our “virtual KB” QA system to a plausible upper bound system that has access to a complete KB. We used the same version of the data as Sun et al. (2019). Details of the implementation are in Appendix A.
+
+Results. Table 1 shows the accuracy of the top-most retrieved entity $\left( \mathrm { H i t s } @ 1 \right)$ for the sub-tasks ranging from 1-3 hops, and compares to the state-of-the-art systems for the text-only setting on these tasks. DrKIT outperforms the prior state-of-the-art by a large margin in the 2-hop and 3-hop cases. The strongest prior method, PullNet (Sun et al., 2019; 2018), uses a graph neural network model with learned iterative retrieval from the corpus to answer multi-hop questions. It uses the MetaQA KB during training to identify shortest paths between the question entity and answer entity, which are used to supervise the text retrieval and reading modules. DrKIT, on the other hand, has strong performance without such supervision, demonstrating its capability for end-to-end learning. (Adding the same intermediate supervision to DrKIT does not even consistently improve performance—it gives DrKIT a small lift on 1- and 2-hop questions but does not help for 3-hop questions.)
+
+![](images/d75add69af3c9dd8716a7d92fdb4349616323f9c06444e99de22e8556adfc883.jpg)  
+Figure 3: Hits $@ 1$ vs Queries/sec during inference on (Left) MetaQA and (Middle) WikiData tasks, measured on a single CPU server with 6 cores. MSR: Multi-step Retriever model from Das et al. (2019a) (we only show Q/sec). (Right) Effect of varying number of nearest neighbors $K$ during MIPS.
+
+DrKIT’s architecture is driven, in part, by efficiency considerations: unlike PullNet, it is designed to answer questions with minimal processing at query time. Figure 3 compares the tradeoffs between accuracy and inference time of DrKIT with PullNet as we vary $K$ , the number of dense nearest neighbors retrieved. The runtime gains of DrKIT over PullNet range between 5x-15x.
+
+Analysis. We perform ablations on DrKIT for the MetaQA data. First, we empirically confirm that taking a sum instead of max over the mentions of an entity hurts performance. So does removing the softmax temperature (by setting $\lambda = 1$ ). Removing the TFIDF component from Eq. 3, leads a large decrease in performance for 2-hop and 3-hop questions. This is because the TFIDF component constrains the end-to-end learning to be along reasonable paths of co-occurring mentions, preventing the search space from exploding. The results also highlight the importance of the pretraining method of $\ S 2 . 3$ , as DrKIT over an index of BERT representations without pretraining is 23 points worse in the 3-hop case. We also check the performance when the KB used for pre-training is incomplete. Even with only $5 0 \%$ edges retained, we see good performance—better than PullNet and the state-of-the-art for a KB-only method (in italics).
+
+<table><tr><td>Ablations</td><td>1hop</td><td>2hop</td><td>3hop</td></tr><tr><td>DrKIT</td><td>0.844</td><td>0.860</td><td>0.876</td></tr><tr><td>-Sum over Mzt</td><td>0.837</td><td>0.823</td><td>0.797</td></tr><tr><td>-入=1</td><td>0.836</td><td>0.752</td><td>0.799</td></tr><tr><td>-w/o TFIDF</td><td>0.845</td><td>0.548</td><td>0.488</td></tr><tr><td>-BERTindex</td><td>0.634</td><td>0.610</td><td>0.555</td></tr><tr><td colspan="4"> Incomplete KB for pretraining</td></tr><tr><td>25% KB</td><td>0.839</td><td>0.804</td><td>0.830</td></tr><tr><td>50% KB</td><td>0.843</td><td>0.834</td><td>0.834</td></tr><tr><td>(50% KB-only)</td><td>0.680</td><td>0.521</td><td>0.597</td></tr></table>
+
+We analyzed 100 2-hop questions correctly answered by DrKIT and found that for 83, the intermediate answers were also correct. The other 17 cases were all where the second hop asked about genre, e.g. “What are the genres of the films directed by Justin Simien?”. We found that in these cases the intermediate answer was the same as the correct final answer—essentially the model learned to answer the question in 1 hop and copy it over for the second hop. Among incorrectly answered questions, the intermediate accuracy was only $4 7 \%$ , so the mistakes were evenly distributed across the two hops.
+
+# 3.2 WIKIDATA: MULTI-HOP SLOT-FILLING
+
+The MetaQA dataset has been fairly well-studied, but has limitations since it is constructed over a small KB. In this section we consider a new task, in a larger scale setting with many more relations, entities and text passages. The new dataset also lets us evaluate performance in a setting where the test set contains documents and entities not seen at training time, an important issue when devising a QA system that will be used in a real-world setting, where the corpus and entities in the discourse change over time, and lets us perform analyses not possible with MetaQA, such as extrapolating from single-hop to multi-hop settings without retraining.
+
+Dataset. We sample two subsets of Wikipedia articles, one for pre-training (§2.3) and end-to-end training, and one for testing. For each subset we consider the set of WikiData entities mentioned in the articles, and sample paths of 1-3 hop relations among them, ensuring that any intermediate entity has an in-degree of no more than 100. Then we construct a semi-structured query by concatenating the surface forms of the head entity with the path of relations (e.g. “Helene Gayle, employer, founded by, ?”). The answer is the tail entity at the end of the path, and the task is to extract it from the Wikipedia articles. Existing slot-filling tasks (Levy et al., 2017; Surdeanu, 2013) focus on a singlehop, static corpus setting, whereas our task considers a dynamic setting which requires the system to traverse the corpus. For each setting, we create a dataset with $1 0 K$ articles, $1 2 0 K$ passages, $> 2 0 0 K$ entities and $1 . 5 M$ mentions, resulting in an index of size about $2 \mathrm { g b }$ . We include example queries in Appendix B.
+
+Baselines. We adapt two publicly available open-domain QA systems for this task – $- \mathrm { D r } \mathbf { Q } \mathbf { A } ^ { 3 }$ (Chen et al., 2017) and $\mathrm { P I Q } \mathrm { \dot { A } } ^ { 4 }$ (Seo et al., 2019). While DrQA is relatively mature and widely used, PIQA is recent, and similar to our setup since it also answers questions with minimal computation at query time. It is broadly similar to a single textual follow operation in DrKIT, but is not constructed to allow retrieved answers to be converted to entities and then used in subsequent processing, so it is not directly applicable to multi-hop queries. We thus also consider a cascaded architecture which repeatedly applies Eq. 2, using either of PIQA or $\mathrm { D r Q A }$ to compute $\operatorname* { P r } \bigl ( z _ { t } | q , z _ { t - 1 } \bigr )$ against the corpus, retaining at most $k$ intermediate answers in each step. We tune $k$ in the range of 1-10, since larger values make the runtime infeasible. Further, since these models were trained on natural language questions, we use the templates released by Levy et al. (2017) to convert intermediate questions into natural text.5 We test off-the-shelf versions of these systems, as well as a version of PIQA re-trained on our our slot-filling data.6 We compare to a version of DrKIT trained only on single-hop queries (§2.3) and similarly cascaded, and one version trained end-to-end on the multi-hop queries.
+
+Results. Table 1 (right) lists the Hits $@ 1$ performance on this task. Off-the-shelf open-domain QA systems perform poorly, showing the challenging nature of the task. Re-training PIQA on the slotfilling data improves performance considerably, but DrKIT trained on the same data improves on it. A large improvement over these cascaded architectures is seen with end-to-end training, which is made possible by the differentiable operation introduced in this paper. We also list the performance of DrKIT when trained against an index of fixed BERT-large mention representations. While this is comparable to the re-trained version of PIQA, it lags behind DrKIT pre-trained using the KB, once again highlighting the importance of the scheme outlined in $\ S 2 . 3$ . We also plot the Hits $@ 1$ against Queries/sec for cascaded versions of PIQA and DrKIT in Figure 3 (middle). We observe runtime gains of $2 \mathrm { x } \mathrm { - } 3 \mathrm { x }$ to DrKIT due to the efficient implementation of entity-mention expansion of $\ S 2 . 2$ .
+
+Analysis. In order to understand where the accuracy gains for DrKIT come from, we conduct experiments on the dataset of slot-filling queries released by Levy et al. (2017). We construct an open version of the task by collecting Wikipedia articles of all subject entities in the data. A detailed discussion is in Appendix C, and here we note the main findings. PIQA trained on SQuAD only gets $3 0 \%$ macro-avg accuracy on this data, but this improves to $4 6 \%$ when re-trained on our slot-filling data. Interestingly, a version of DrKIT which selects from all spans in the corpus performs similarly to PIQA $( 5 0 \% )$ , but when using entity linking it significantly improves to $6 6 \%$ . It also has $5 5 \%$ accuracy in answering queries about rare relations, i.e. those observed $< 5$ times in its training data. We also conduct probing experiments comparing the representations learned using slot-filling to those by vanilla BERT. We found that while the two are comparable in detecting fine-grained entity types, the slot-filling version is significantly better at encoding entity co-occurrence information.
+
+3.3 HOTPOTQA: MULTI-HOP INFORMATION RETRIEVAL
+
+Dataset. HotpotQA (Yang et al., 2018) is a recent dataset of over 100K crowd-sourced multi-hop questions and answers over introductory Wikipedia passages. We focus on the open-domain fullwiki setting where the two gold passages required to answer the question are not known in advance. The answers are free-form spans of text in the passages, not necessarily entities, and hence our model which selects entities is not directly applicable here. Instead, inspired by recent works (Das et al., 2019b; Qi et al., 2019), we look at the challenging sub-task of retrieving the passages required to answer the questions from a pool of 5.23M. This is a multi-hop IR task, since for many questions at least one passage may be 1-2 hops away from the entities in the question. Further, each passage is about an entity (the title entity of that Wikipedia page), and hence retrieving passages is the same as identifying the title entities of those passages. We apply DrKIT to this task of identifying the two entities for each question, whose passages contain the information needed to answer that question. Then we pass the top 10 passages identified this way to a standard reading comprehension architecture from Yang et al. (2018) to select the answer span.
+
+<table><tr><td>Model</td><td>EM</td><td>F1</td></tr><tr><td>Baselinet</td><td>0.288</td><td>0.381</td></tr><tr><td>+EC IR</td><td>0.354</td><td>0.462</td></tr><tr><td>+Golden Ret</td><td>0.379</td><td>0.486</td></tr><tr><td>+DrKIT†</td><td>0.357</td><td>0.466</td></tr></table>
+
+Table 2: (Left) Retrieval performance on the HotpotQA benchmark dev set. $\mathrm { Q } / \mathrm { s }$ denotes the number of queries per second during inference on a single 16-core CPU. Accuracy $@ k$ is the fraction where both the correct passages are retrieved in the top $k$ . †: Baselines obtained from Das et al. (2019b). For DrKIT, we report the performance when the index is pretrained using the WikiData KB alone, the HotpotQA training questions alone, or using both. ∗: Measured on different machines with similar specs. (Right) Overall performance on the HotpotQA task, when passing 10 retrieved passages to a downstream reading comprehension model (Yang et al., 2018). $^ \ddag$ : From Das et al. (2019b). : From Qi et al. (2019). $^ \dagger$ : Results on the dev set.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Q/s</td><td colspan="4">Accuracy</td></tr><tr><td>@2</td><td>@5</td><td>@10</td><td>@20</td></tr><tr><td>BM25†</td><td>1</td><td>0.093</td><td>0.191</td><td>0.259</td><td>0.324</td></tr><tr><td>PRF-Task†</td><td>1</td><td>0.097</td><td>0.198</td><td>0.267</td><td>0.330</td></tr><tr><td>BERT re-ranker†</td><td>1</td><td>0.146</td><td>0.271</td><td>0.347</td><td>0.409</td></tr><tr><td>Entity Centric IR+</td><td>0.32*</td><td>0.230</td><td>0.482</td><td>0.612</td><td>0.674</td></tr><tr><td>DrKIT (WikiData)</td><td></td><td>0.355</td><td>0.588</td><td>0.671</td><td>0.710</td></tr><tr><td>DrKIT (Hotpot)</td><td>4.26*</td><td>0.385</td><td>0.595</td><td>0.663</td><td>0.703</td></tr><tr><td>DrKIT (Combined)</td><td></td><td>0.383</td><td>0.603</td><td>0.672</td><td>0.710</td></tr></table>
+
+Setup. We use the Wikipedia abstracts released by Yang et al. (2018) as the text corpus.7 The total number of entities is the same as the number of abstracts, 5.23M, and we consider hyperlinks in the text as mentions of the entities to whose pages they point to, leading to 22.8M total mentions in an index of size 34GB. For pretraining the mention representations, we compare using the WikiData KB as described in $\ S 2 . 3$ to directly using the HotpotQA training questions, with TFIDF based retrieved passages as negative examples. We set $A _ { E  M } [ e , m ] = 1$ if either the entity $e$ is mentioned on the page of the entity denoted by $m$ , or vice versa. For entity linking over the questions, we retrieve the top 20 entities based on the match between a bigram based TFIDF vector of the question with a similar vector derived from the surface form of the entity (same as the title of the Wiki article). We found that the gold entities that need to be retrieved are within 2 hops of the entities linked in this manner for $8 7 \%$ of the dev examples.
+
+Unlike the MetaQA and WikiData datasets, however, for HotpotQA we do not know the number of hops required for each question in advance. Instead, we run DrKIT for 2 hops for each question, and then take a weighted average of the distribution over entities after each hop $Z ^ { * } = \pi _ { 0 } Z _ { 0 } + \pi _ { 1 } Z _ { 1 } +$ $\pi _ { 2 } Z _ { 2 }$ . $Z _ { 0 }$ consists of the entities linked to the question itself, rescored based on an encoding of the question, since in some cases one or both the entities to be retrieved are in this set.8 $Z _ { 1 }$ and $Z _ { 2 }$ are given by Eq. 4. The mixing weights $\pi _ { i }$ are the softmax outputs of a classifier on top of another encoding of the question, learnt end-to-end on the retrieval task. This process can be viewed as soft mixing of different templates ranging from 0 to 2 hops for answering a question, similar to NQL (Cohen et al., 2019).
+
+Results. We compare our retrieval results to those presented in Das et al. (2019b) in Table 2 (Left). We measure the accuracy $@ k$ retrievals, which is the fraction of questions for which both the required passages are in the top $k$ retrieved ones. We see an improvement in accuracy across the board, with much higher gains $@ 2$ and $@ 5 .$ The main baseline is the entity-centric IR approach which runs a BERT-based re-ranker on 200 pairs of passages for each question. Importantly, DrKIT also improves by over $1 0 \mathrm { x }$ in terms of queries per second during inference. Note that the inference time is measured using a batch size of 1 for both models for fair comparison. DrKIT can be easily run with batch sizes up to 40, but the entity centric IR baseline cannot due to the large number of runs of BERT for each query. When comparing different datasets for pretraining the index, there is not much difference between using the WikiData KB, or the HotpotQA questions. The latter has a better accuracy $@ 2$ , but overall the best performance is when using a combination of both.
+
+Table 3: Official leaderboard evaluation on the test set of HotpotQA. #Bert refers to the number of calls to BERT (Devlin et al., 2019) in the model. $\mathrm { s } / \mathrm { Q }$ denotes seconds per query (using batch size 1) for inference on a single 16-core CPU. Answer, Sup Fact and Joint are the official evaluation metrics for HotpotQA. ∗: This is the minimum number of BERT calls based on model and hyperparameter descriptions in the respective papers. †: Computed using code released by authors, using a batch size of 1. ‡: Estimated based on the number of BERT calls, using 0.8s as the time for one call (without considering overhead due to other computation in the model). : One call to a 5-layer Transformer, and one call to BERT.   
+
+<table><tr><td rowspan="2">System</td><td colspan="2">Runtime</td><td colspan="2">Answer</td><td colspan="2">Sup Fact</td><td colspan="2">Joint</td></tr><tr><td>#Bert</td><td>s/Q</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>Baseline (Yang et al., 2018)</td><td>1</td><td>1</td><td>25.23</td><td>34.40</td><td>5.07</td><td>40.69</td><td>2.63</td><td>17.85</td></tr><tr><td>Golden Ret (Qi et al.,2019)</td><td>1</td><td>1.4†</td><td>37.92</td><td>48.58</td><td>30.69</td><td>64.24</td><td>18.04</td><td>39.13</td></tr><tr><td>Semantic Ret (Nie et al., 2019)</td><td>50*</td><td>40.0</td><td>45.32</td><td>57.34</td><td>38.67</td><td>70.83</td><td>25.14</td><td>47.60</td></tr><tr><td>HGN (Fang et al., 2019)</td><td>50*</td><td>40.0t</td><td>56.71</td><td>69.16</td><td>49.97</td><td>76.39</td><td>35.63</td><td>59.86</td></tr><tr><td>Rec Ret (Asai et al., 2020)</td><td>500*</td><td>133.2†</td><td>60.04</td><td>72.96</td><td>49.08</td><td>76.41</td><td>35.35</td><td>61.18</td></tr><tr><td>DrKIT +BERT</td><td>1.20</td><td>1.3</td><td>42.13</td><td>51.72</td><td>37.05</td><td>59.84</td><td>24.69</td><td>42.88</td></tr></table>
+
+In Table 2 (Right), we check the performance of the baseline reading comprehension model from Yang et al. (2018), when given the passages retrieved by DrKIT. While there is a significant improvement over the baseline which uses a TFIDF based retrieval, we see only a small improvement over the passages retrieved by the entity-centric IR baseline, despite the significantly improved accuracy $@ 1 0$ of DrKIT. Among the $3 3 \%$ questions where the top 10 passages do not contain both the correct passages, for around $2 0 \%$ the passage containing the answer is also missing. We conjecture this percentage is lower for the entity-centric IR baseline, and the downstream model is able to answer some of these questions without the other supporting passage.
+
+Lastly, we feed the top 5 passages retrieved by DrKIT to an improved answer span extraction model based on BERT. This model implements a standard architecture for extracting answers from text, and is trained to predict both the answers and the supporting facts. Details are included in Appendix D. Table 3 shows the performance of this system on the HotpotQA test set, compared with other recently published models on the leaderboard.9 In terms of accuracy, DrKIT $^ +$ BERT reaches a modest score of 42.88 joint F1, but is considerably faster (up to 100x) than the models which outperform it.
+
+# 4 RELATED WORK
+
+Neural Query Language (NQL) (Cohen et al., 2019) defines differentiable templates for multi-step access to a symbolic KB, in which relations between entities are explicitly enumerated. Here, we focus on the case where the relations are implicit in mention representations derived from text. Knowledge Graph embeddings (Bordes et al., 2013; Yang et al., 2014; Dettmers et al., 2018) attach continuous representations to discrete symbols which allow them to be incorporated in deep networks (Yang & Mitchell, 2017). Embeddings often allow generalization to unseen facts using relation patterns, but text corpora are more complete in the information they contain.
+
+Talmor & Berant (2018) also examined answering compositional questions by treating a text corpus (in their case the entire web) as a KB. However their approach consists of parsing the query into a computation tree, and running a black-box QA model on its leaves separately, which cannot be trained end-to-end. Recent papers have also looked at complex QA using graph neural networks (Sun et al., 2018; Cao et al., 2019; Xiao et al., 2019) or by identifying paths of entities in text (Jiang et al., 2019; Kundu et al., 2019; Dhingra et al., 2018). These approaches rely on identifying a small relevant pool of evidence documents containing the information required for multi-step QA. Hence, Sun et al. (2019) and Ding et al. (2019), incorporate a dynamic retrieval process to add text about entities identified as relevant in the previous layer of the model. Since the evidence text is processed in a query-dependent manner, the inference speed is slower than when it is pre-processed into an indexed representation (see Figure 3). The same limitation is shared by methods which perform multi-step retrieval interleaved with a reading comprehension model (Das et al., 2019a; Feldman & El-Yaniv, 2019; Lee et al., 2019).
+
+# 5 CONCLUSION
+
+We present DrKIT, a differentiable module that is capable of answering multi-hop questions directly using a large entity-linked text corpus. DrKIT is designed to imitate traversal in KB over the text corpus, providing ability to follow relations in the “virtual” KB over text. We achieve state-of-the-art results on the MetaQA dataset for answering natural language questions, with a 9 point increase in the 3-hop case. We also developed an efficient implementation using sparse operations and inner product search, which led to a 10-100x increase in Queries/sec over baseline approaches.
+
+# ACKNOWLEDGMENTS
+
+Bhuwan Dhingra was supported by a Siemens fellowship during this project. This work was supported in part by ONR Grant N000141812861, Google, Apple, and grants from NVIDIA.
+
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+
+Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William W. Cohen, Ruslan Salakhutdinov, and Christopher D. Manning. HotpotQA: A dataset for diverse, explainable multi-hop question answering. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP), 2018.
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+
+# A METAQA: IMPLEMENTATION DETAILS
+
+We use $p = 4 0 0$ dimensional embeddings for the mentions and queries, and 200-dimensional embeddings each for the start and end positions. This results in an index of size 750MB. When computing $A _ { E  M }$ , the entity to mention co-occurrence matrix, we only retain mentions in the top 50 paragraphs matched with an entity, to ensure sparsity. Further we initialize the first 4 layers of the question encoder with the Transformer network from pre-training. For the first hop, we assign $Z _ { 0 }$ as a 1-hot vector for the least frequent entity detected in the question using an exact match. The number of nearest neighbors $K$ and the softmax temperature $\lambda$ were tuned on the dev set of each task, and we found $\bar { K \mathrm { ~ = ~ } } 1 0 0 0 0$ and $\lambda = 4$ to work best. We pretrain the index on a combination of the MetaQA corpus, using the KB provided with MetaQA for distance data, and the WikiData corpus.
+
+# B WIKIDATA DATASET STATISTICS
+
+Table 4: WikiData dataset   
+
+<table><tr><td>Task</td><td>#train</td><td>#dev</td><td>#test</td><td>Etest|</td><td>|Mtestl</td><td>|Dtest|</td><td>Example</td></tr><tr><td>1hop</td><td>16901</td><td>2467</td><td>10000</td><td>216K</td><td>1.2M</td><td>120K</td><td>Q. Mendix, industry? A.Enterprise Software</td></tr><tr><td>2hop</td><td>163607</td><td>398</td><td>9897</td><td>342K</td><td>1.9M</td><td>120K</td><td>Q.2000 Hel van het Mergelland, winner, place of birth? A.Bert Grabsch → Lutherstadt Wittenberg</td></tr><tr><td>3hop</td><td>36061</td><td>453</td><td>9899</td><td>261K</td><td>1.8M</td><td>120K</td><td>Q. Magnificent!, record label,founded by, date of death? A.Prestige →Bob Weinstock → 14 Jan 2006</td></tr></table>
+
+Details of the collected WikiData dataset are shown in Table 4.
+
+# C INDEX ANALYSIS
+
+Single-hop questions and relation extraction. Levy et al. (2017) released a dataset of $1 M$ slotfilling queries of the form $( e _ { 1 } , R , ? )$ paired with Wikipedia sentences mentioning $e _ { 1 }$ , which was used for training systems that answered single-step slot-filling questions based on a small set of candidate passages. Here we consider an open version of the same task, where answers to the queries must be extracted from a corpus rather than provided candidates. We construct the corpus by collecting and entity-linking all paragraphs in the Wikipedia articles of all $8 K$ subject entities in the dev and test sets, leading to a total of $1 0 9 K$ passages. After constructing the TFIDF $A _ { E  M }$ and coreference $B _ { M  E }$ matrices for this corpus, we directly use our pre-trained index to answer the test set queries.
+
+Figure 4 (Right) shows the $\mathrm { H i t s } @ 1$ performance of the Levy et al. (2017) slot-filling dataset. We report results on 2 subsets of relations in addition to all relations. The Rare subset comprises of relations with frequencies $< 5$ in the training data while the ’Frequent’ subset contains the rest. DrKIT on entity-mentions consistently outperforms the other phrase-based models showing the benefit of indexing only entity-mentions in single-hop questions over all spans. Note that DrKit-entities has a high Hits $@ 1$ performance on the Rare relations subset, showing that there is generalization to less frequent data due to the natural language representations of entities and relations.
+
+![](images/fbb07c2868d546ce61ff70eded30b346c15925fb54cd790fd3cf2edf449776d4.jpg)
+
+Figure 4: Left: F1 scores on Shared Entity and Shared Relation negatives. The negative examples are for the Query $:$ (Neil Herron, occupation, ?). Right: Macro-avg accuracy on the Levy et al. (2017) relation extraction dataset. We split the results based on frequency of the relations in our WikiData training data. DrKIT-all spans refers to a variant of our model which selects from all spans in the corpus, instead of only entity-linked mentions.   
+
+<table><tr><td>Probing Task</td><td>Negative Example</td><td>BERT</td><td>DrKIT</td></tr><tr><td>Shared Entity</td><td>Neil Herron played for West of Scotland.</td><td>0.850</td><td>0.876</td></tr><tr><td>Shared Relation</td><td>William Paston was a British politician.</td><td>0.715</td><td>0.846</td></tr></table>
+
+Probing Experiments Finally, to compare the representations learned by the BERT model finetuned on the WikiData slot-filling task, we design two probing experiments. In each experiment, we keep the parameters of the BERT model (mention encoders) being probed fixed and only train the query encoders. Similar to Tenney et al. (2019), we use a weighted average of the layers of BERT here rather than only the top-most layer, where the weights are learned on the probing task.
+
+In the first experiment, we train and test on shared-entity negatives. Good performance here means the BERT model being probed encodes fine-grained entity-type information reliably10. As shown in Table 4, BERT performs well on this task, suggesting it encodes fine-grained types well.
+
+In the second experiment, we train and test only on shared-relation negatives. Good performance here means that the BERT model encodes entity co-occurrence information reliably. In this probe task, we see a large performance drop for BERT, suggesting it does not encode entity co-occurrence information well. The good performance of the DrKIT model on both experiments suggests that fine-tuning on the slot-filling task primarily helps the contextual representations to also encode entity co-occurrence information, in addition to entity type information.
+
+# D HOTPOTQA ANSWER EXTRACTION
+
+On HotpotQA, we use DrKIT to identify the top passages which are likely to contain the answer to a question. We then train a separate model to extract the answer from a concatenation of these passages. This model is a standard BERT-based architecture used for SQuAD (see Devlin et al. (2019) for details), with a few modifications. First, to handle boolean questions, we train a 3-way classifier on top of the [CLS] representation from BERT to decide whether the question has a “span”, “yes” or “no” answer, respectively. During inference, if this classifier has the highest probability on “span” we extract a start and end position similar to Devlin et al. (2019), else we directly answer as “yes” or “no”.
+
+Second, to handle supporting fact prediction, we prepend each sentence in the concatenated passages passed to BERT with a special symbol [unused0], and train a binary classifier on top of the representation of each of these symbols output from BERT. The binary classifier is trained to predict 1 for sentences which are supporting facts and 0 for sentences which are not. During inference, we take all sentences for which the output probability of this classifier is $> 0 . 5$ as supporting facts.
+
+The training loss is an average of the loss for the 3-way classifier $( \mathcal { L } _ { c l s } )$ , the sum of the losses for the supporting fact classifiers $( \mathcal { L } _ { s p } )$ , and the losses for the start and end positions of span answers $( \mathcal { L } _ { s t } , \mathcal { L } _ { e n } )$ :
+
+$$
+\mathcal { L } = ( \mathcal { L } _ { c l s } + \mathcal { L } _ { s p } + \mathcal { L } _ { s t } + \mathcal { L } _ { e n } ) / 4
+$$
+
+We train the system on 5 passages per question, provided in the distractor setting of HotpotQA— 2 gold ones and 3 negatives from a TFIDF retriever. We keep the gold passages at the beginning for $6 0 \%$ of the examples, and randomly shuffle all passages for the rest, since during inference the correct passages are likely to be retrieved at the top by DrKIT. Other hyperparameters include— batch size 32, learning rate $5 \times 1 0 ^ { - 5 }$ , number of training epochs 5, and a maximum combined passage length 512.

parse/train/SkEqro0ctQ/SkEqro0ctQ.md

@@ -0,0 +1,410 @@
+# HIERARCHICAL INTERPRETATIONS FOR NEURAL NETWORK PREDICTIONS
+
+Chandan Singh∗   
+Department of EECS   
+UC Berkeley   
+c singh@berkeley.edu
+
+W. James Murdoch∗ Department of Statistics UC Berkeley jmurdoch@berkeley.edu
+
+Bin Yu   
+Department of Statistics, EECS   
+UC Berkeley   
+binyu@berkeley.edu
+
+# ABSTRACT
+
+Deep neural networks (DNNs) have achieved impressive predictive performance due to their ability to learn complex, non-linear relationships between variables. However, the inability to effectively visualize these relationships has led to DNNs being characterized as black boxes and consequently limited their applications. To ameliorate this problem, we introduce the use of hierarchical interpretations to explain DNN predictions through our proposed method: agglomerative contextual decomposition (ACD). Given a prediction from a trained DNN, ACD produces a hierarchical clustering of the input features, along with the contribution of each cluster to the final prediction. This hierarchy is optimized to identify clusters of features that the DNN learned are predictive. We introduce ACD using examples from Stanford Sentiment Treebank and ImageNet, in order to diagnose incorrect predictions, identify dataset bias, and extract polarizing phrases of varying lengths. Through human experiments, we demonstrate that ACD enables users both to identify the more accurate of two DNNs and to better trust a DNN’s outputs. We also find that ACD’s hierarchy is largely robust to adversarial perturbations, implying that it captures fundamental aspects of the input and ignores spurious noise.
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs) have recently demonstrated impressive predictive performance due to their ability to learn complex, non-linear, relationships between variables. However, the inability to effectively visualize these relationships has led DNNs to be characterized as black boxes. Consequently, their use has been limited in fields such as medicine (e.g. medical image classification (Litjens et al., 2017)), policy-making (e.g. classification aiding public policy makers (Brennan & Oliver, 2013)), and science (e.g. interpreting the contribution of a stimulus to a biological measurement (Angermueller et al., 2016)). Moreover, the use of black-box models like DNNs in industrial settings has come under increasing scrutiny as they struggle with issues such as fairness (Dwork et al., 2012) and regulatory pressure (Goodman & Flaxman, 2016).
+
+To ameliorate these problems, we introduce the use of hierarchical interpretations to explain DNN predictions. Our proposed method, agglomerative contextual decomposition $( \mathsf { A C D } ) ^ { 1 }$ , is a general technique that can be applied to a wide range of DNN architectures and data types. Given a prediction from a trained DNN, ACD produces a hierarchical clustering of the input features, along with the contribution of each cluster to the final prediction. This hierarchy is optimized to identify clusters of features that the DNN learned are predictive (see Fig 1).
+
+The development of ACD consists of two novel contributions. First, importance scores for groups of features are obtained by generalizing contextual decomposition (CD), a previous method for obtaining importance scores for LSTMs (Murdoch et al., 2018). This work extends CD to arbitrary DNN architectures, including convolutional neural networks (CNNs). Second, most importantly, we introduce the idea of hierarchical saliency, where a group-level importance measure, in this case CD, is used as a joining metric in an agglomerative clustering procedure. While we focus on DNNs and use CD as our importance measure, this concept is general, and could be readily applied to any model with a suitable measure for computing importances of groups of variables.
+
+We demonstrate the utility of ACD on both long short term memory networks (LSTMs) (Hochreiter & Schmidhuber, 1997) trained on the Stanford Sentiment Treebank (SST) (Socher et al., 2013) and CNNs trained on MNIST (LeCun, 1998) and ImageNet (Russakovsky et al., 2015). Through human experiments, we show that ACD produces intuitive visualizations that enable users to better reason about and trust DNNs. In particular, given two DNN models, we show that users can use the output of ACD to select the model with higher predictive accuracy, and that overall they rank ACD as more trustworthy than prior interpretation methods. In addition, we demonstrate that ACD’s hierarchy is robust to adversarial perturbations (Szegedy et al., 2013) in CNNs.
+
+![](images/43a9adeb38b67f264edc357fd7cfe4d70298cfe4c47973b9f4a1011ef7d0544f.jpg)  
+Figure 1: ACD illustrated through the toy example of predicting the phrase “not very good” as negative. Given the network and prediction, ACD constructs a hierarchy of meaningful phrases and provides importance scores for each identified phrase. In this example, ACD identifies that “very” modifies “good” to become the very positive phrase “very good”, which is subsequently negated by ”not” to produce the negative phrase “not very good”. Best viewed in color.
+
+# 2 BACKGROUND
+
+Interpreting DNNs is a growing field (Murdoch et al., 2019) spanning a range of techniques including feature visualization (Olah et al., 2017; Yosinski et al., 2015), analyzing learned weights (Tsang et al., 2017) and others (Frosst & Hinton, 2017; Andreas et al., 2016; Zhang et al., 2017). Our work focuses on local interpretations, where the task is to interpret individual predictions made by a DNN.
+
+Local interpretation Most prior work has focused on assigning importance to individual features, such as pixels in an image or words in a document. There are several methods that give feature-level importance for different architectures. They can be categorized as gradient-based (Springenberg et al., 2014; Sundararajan et al., 2017; Selvaraju et al., 2016; Baehrens et al., 2010), decompositionbased (Murdoch & Szlam, 2017; Shrikumar et al., 2016; Bach et al., 2015) and others (Dabkowski & Gal, 2017; Fong & Vedaldi, 2017; Ribeiro et al., 2016; Zintgraf et al., 2017), with many similarities among the methods (Ancona et al., 2018; Lundberg & Lee, 2017).
+
+By contrast, there are relatively few methods that can extract the interactions between features that a DNN has learned. In the case of LSTMs, Murdoch et al. (2018) demonstrated the limitations of prior work on interpretation using word-level scores, and introduced contextual decomposition (CD), an algorithm for producing phrase-level importance scores from LSTMs. Another simple baseline is occlusion, where a group of features is set to some reference value, such as zero, and the importance of the group is defined to be the resulting decrease in the prediction value (Zeiler & Fergus, 2014; Li et al., 2016). Given an importance score for groups of features, no existing work addresses how to search through the many possible groups of variables in order to find a small set to show to users. To address this problem, this work introduces hierarchical interpretations as a principled way to search for and display important groups.
+
+Hierarchical importance Results from psychology and philosophy suggest that people prefer explanations that are simple but informative (Harman, 1965; Read & Marcus-Newhall, 1993) and include the appropriate amount of detail (Keil, 2006). However, there is no existing work that is both powerful enough to capture interactions between features, and simple enough to not require a user to manually search through the large number of available feature groups. To remedy this, we propose a hierarchical clustering procedure to identify and visualize, out of the considerable number of feature groups, which ones contain meaningful interactions and should be displayed to the end user. In doing so, ACD aims to be informative enough to capture meaningful feature interactions while displaying a sufficiently small subset of all feature groups to maintain simplicity.
+
+# 3 METHOD
+
+This section introduces ACD through two contributions: Sec 3.1 proposes a generalization of CD from LSTMs to arbitrary DNNs, and Sec 3.2 explains the main contribution: how to combine these CD scores with hierarchical clustering to produce ACD.
+
+# 3.1 CONTEXTUAL DECOMPOSITION (CD) IMPORTANCE SCORES FOR GENERAL DNNS
+
+In order to generalize CD to a wider range of DNNs, we first reformulate the original CD algorithm into a more generic setting than originally presented. For a given DNN $f ( x )$ , we can represent its output as a SoftMax operation applied to logits $g ( x )$ . These logits, in turn, are the composition of $L$ layers $g _ { i }$ , such as convolutional operations or ReLU non-linearities.
+
+$$
+f ( x ) = { \mathrm { S o f t M a x } } { \big ( } g ( x ) { \big ) } = { \mathrm { S o f t M a x } } { \big ( } g _ { L } { \big ( } g _ { L - 1 } ( \ldots ( g _ { 2 } ( g _ { 1 } ( x ) ) ) ) { \big ) } { \big ) }
+$$
+
+Given a group of features $\{ x _ { j } \} _ { j \in S }$ , our generalized CD algorithm, $g ^ { C D } ( x )$ , decomposes the logits $g ( x )$ into a sum of two terms, $\beta ( x )$ and $\gamma ( x ) . ~ \beta ( x )$ is the importance measure of the feature group $\{ x _ { j } \} _ { j \in S }$ , and $\gamma ( x )$ captures contributions to $g ( x )$ not included in $\beta ( x )$ .
+
+$$
+\begin{array} { c } { { g ^ { C D } ( x ) = ( \beta ( x ) , \gamma ( x ) ) } } \\ { { \beta ( x ) + \gamma ( x ) = g ( x ) } } \end{array}
+$$
+
+To compute the CD decomposition for $g ( x )$ , we define layer-wise CD decompositions $g _ { i } ^ { C D } ( x ) =$ $( \beta _ { i } , \gamma _ { i } )$ for each layer $g _ { i } ( x )$ . Here, $\beta _ { i }$ corresponds to the importance measure of $\{ x _ { j } \} _ { j \in S }$ to layer $i$ , and $\gamma _ { i }$ corresponds to the contribution of the rest of the input to layer $i$ . To maintain the decomposition we require $\beta _ { i } + \gamma _ { i } = g _ { i } ( x )$ for each $i$ . We then compute CD scores for the full network by composing these decompositions.
+
+$$
+g ^ { C D } ( x ) = g _ { L } ^ { C D } ( g _ { L - 1 } ^ { C D } ( . . . ( g _ { 2 } ^ { C D } ( g _ { 1 } ^ { C D } ( x ) ) ) ) )
+$$
+
+Previous work (Murdoch et al., 2018) introduced decompositions $g _ { i } ^ { C D }$ for layers used in LSTMs. The generalized CD described here extends CD to other widely used DNNs, by introducing layerwise CD decompositions for convolutional, max-pooling, ReLU non-linearity and dropout layers. Doing so generalizes CD scores from LSTMs to a wide range of neural architectures, including CNNs with residual and recurrent architectures.
+
+At first, these decompositions were chosen through an extension of the CD rules detailed in Murdoch et al. (2018), yielding a similar algorithm to that developed concurrently by Godin et al. (2018). However, we found that this algorithm did not perform well on deeper, ImageNet CNNs. We subsequently modified our CD algorithm by partitioning the biases in the convolutional layers between $\gamma _ { i }$ and $\beta _ { i }$ in Equation 5, and modifying the decomposition used for ReLUs in Equation 10. We show the effects of these two changes in Supplement S7, and give additional intuition in Supplement S1.
+
+When $g _ { i }$ is a convolutional or fully connected layer, the layer operation consists of a weight matrix $W$ and a bias $b$ . The weight matrix can be multiplied with $\beta _ { i - 1 }$ and $\gamma _ { i - 1 }$ individually, but the bias must be partitioned between the two. We partition the bias proportionally based on the absolute value of the layer activations. For the convolutional layer, this equation yields only one activation of the output; it must be repeated for each activation.
+
+$$
+\begin{array} { r } { \beta _ { i } = W \beta _ { i - 1 } + \frac { \left| W \beta _ { i - 1 } \right| } { \left| W \beta _ { i - 1 } \right| + \left| W \gamma _ { i - 1 } \right| } \cdot b } \\ { \gamma _ { i } = W \gamma _ { i - 1 } + \frac { \left| W \gamma _ { i - 1 } \right| } { \left| W \beta _ { i - 1 } \right| + \left| W \gamma _ { i - 1 } \right| } \cdot b } \end{array}
+$$
+
+When $g _ { i }$ is a max-pooling layer, we identify the indices, or channels, selected by max-pool when run by $g _ { i } ( x )$ , denoted max idxs below, and use the decompositions for the corresponding channels.
+
+$$
+\begin{array} { c } { { m a x \_ i d x s = \underset { i d x s } { \mathrm { a r g m a x \ [ m a x p o o l ( } \beta _ { i - 1 } + \gamma _ { i - 1 } ; i d x s ) ] } } } \\ { { \beta _ { i } = \beta _ { i - 1 } [ m a x \_ i d x s ] } } \\ { { \gamma _ { i } = \gamma _ { i - 1 } [ m a x \_ i d x s ] } } \end{array}
+$$
+
+Finally, for the ReLU, we update our importance score $\beta _ { i }$ by computing the activation of $\beta _ { i - 1 }$ alone and then update $\gamma _ { i }$ by subtracting this from the total activation.
+
+$$
+\begin{array} { r l } & { \beta _ { i } = \mathrm { R e L U } ( \beta _ { i - 1 } ) } \\ & { \gamma _ { i } = \mathrm { R e L U } ( \beta _ { i - 1 } + \gamma _ { i - 1 } ) - \mathrm { R e L U } ( \beta _ { i - 1 } ) } \end{array}
+$$
+
+For a dropout layer, we simply apply dropout to $\beta _ { i - 1 }$ and $\gamma _ { i - 1 }$ individually, or multiplying each by a scalar. Computationally, a CD call is comparable to a forward pass through the network $f$ .
+
+# 3.2 AGGLOMERATIVE CONTEXTUAL DECOMPOSITION (ACD)
+
+Given the generalized CD scores introduced above, we now introduce the clustering procedure used to produce ACD interpretations. At a high-level, our method is equivalent to agglomerative hierarchical clustering, where the CD interaction is used as the joining metric to determine which clusters to join at each step. This procedure builds the hierarchy by starting with individual features and iteratively combining them based on the interaction scores provided by CD. The displayed ACD interpretation is the hierarchy, along with the CD importance score at each node.
+
+More precisely, algorithm 1 describes the exact steps in the clustering procedure. After initializing by computing the CD scores of each feature individually, the algorithm iteratively selects all groups of features within ${ \mathrm { k } } \%$ of the highest-scoring group (where $k$ is a hyperparameter, fixed at 95 for images and 90 for text) and adds them to the hierarchy.
+
+Each time a new group is added to the hierarchy, a corresponding set of candidate groups is generated by adding individual contiguous features to the original group. For text, the candidate groups correspond to adding one adjacent word onto the current phrase, and for images adding any adjacent pixel onto the current image patch. Candidate groups are ranked according to the CD interaction score, which is the difference between the score of the candidate and original groups.
+
+ACD terminates after an application-specific criterion is met. For sentiment classification, we stop once all words are selected. For images, we stop after some predefined number of iterations and then merge the remaining groups one by one using the same selection criteria described above.
+
+Algorithm 1 is not specific to DNNs; it requires only a method to obtain importance scores for groups of input features. Here, we use CD scores to arrive at the ACD algorithm, which makes the method specific to DNNs, but given a feature group scoring function, Algorithm 1 can yield interpretations for any predictive model. CD is a natural score to use for DNNs as it aggregates saliency at different scales and converges to the final prediction once all the units have been selected.
+
+# Algorithm 1 Agglomeration algorithm.
+
+<table><tr><td colspan="2">AigontnmnrTAggionerationargorunn.</td></tr><tr><td>ACD(Example x, model, hyperparameter k, function CD(x, blob; model)) # initialize</td><td></td></tr><tr><td>tree = Tree()</td><td># tree to output</td></tr><tr><td>scoresQueue =PriorityQueue()</td><td># scores,sorted by importance</td></tr><tr><td>for feature in x :</td><td></td></tr><tr><td>scoresQueue.push(feature,priority=CD(x, feature; model))</td><td></td></tr><tr><td></td><td></td></tr><tr><td># iteratively build up tree while scoresQueue is not empty :</td><td></td></tr><tr><td>selectedGroups = scoresQueue.popTopKPercentile(k)</td><td># pop off top k elements</td></tr><tr><td>tree.add(selectedGroups)</td><td># Add top k elements to the tree</td></tr><tr><td></td><td></td></tr><tr><td colspan="2"># generate new groups of features based on current groups and add them to the queue</td></tr><tr><td colspan="2">for selectedGroup in selectedGroups :</td></tr><tr><td colspan="2"></td></tr><tr><td colspan="2">candidateGroups = getCandidateGroups(selectedGroup)</td></tr><tr><td colspan="2">for candidateGroup in candidateGroups :</td></tr><tr><td colspan="2">scoresQueue.add(candidateGroup, priority=CD(x,candidateGroup;model)-CD(x,selectedGroup;</td></tr><tr><td colspan="2">model))</td></tr><tr><td colspan="2">return tree</td></tr></table>
+
+# 4 RESULTS
+
+We now present empirical validation of ACD on both LSTMs trained on SST and CNNs trained on MNIST and ImageNet. First, we introduce the reader to our visualization in Sec 4.2, and how it can (anecdotally) be used to understand models in settings such as diagnosing incorrect predictions, identifying dataset bias, and identifying representative phrases of differing lengths. We then provide quantitative evidence of the benefits of ACD in Sec 4.3 through human experiments and demonstrating the stability of ACD to adversarial perturbations.
+
+# 4.1 EXPERIMENTAL DETAILS
+
+We first describe the process for training the models from which we produce interpretations. As the objective of this paper is to interpret the predictions of models, rather than increase their predictive accuracy, we use standard best practices to train our models. All models are implemented using PyTorch. For SST, we train a standard binary classification LSTM model2, which achieves $8 6 . 2 \%$ accuracy. On MNIST, we use the standard PyTorch example3, which attains accuracy of $9 7 . 7 \%$ . On ImageNet, we use a pre-trained VGG-16 DNN architecture Simonyan & Zisserman (2014) which attains top-1 accuracy of $4 2 . 8 \%$ . When using ACD on ImageNet, for computational reasons, we start the agglomeration process with 14-by-14 superpixels instead of individual pixels. We also smooth the computed image patches by adding pixels surrounded by the patch. The weakened models for the human experiments are constructed from the original models by randomly permuting a small percentage of their weights. For SST/MNIST/ImageNet, $2 5 / 2 5 / 0 . 8 \%$ of weights are randomized, reducing test accuracy from $8 5 . 8 / 9 7 . 7 / 4 2 . 8 \%$ to $7 9 . 8 / 7 9 . 6 / 3 2 . 3 \%$ .
+
+# 4.2 QUALITATIVE EXPERIMENTS
+
+Before providing quantitative evidence of the benefits of ACD, we first introduce the visualization and demonstrate its utility in interpreting a predictive model’s behavior. To qualitatively evaluate ACD, in Supplement S3 we show the results of several more examples selected using the same criterion as in our human experiments described below.
+
+# 4.2.1 UNDERSTANDING PREDICTIVE MODELS USING ACD
+
+In the following examples, we demonstrate the use of ACD to diagnose incorrect predictions in SST and identify dataset bias in ImageNet. These examples are only a few of the potential uses of ACD.
+
+Table 1: Top-scoring phrases of different lengths extracted by ACD on SST’s validation set. The positive/negative phrases identified by ACD are all indeed positive/negative.   
+
+<table><tr><td>Length</td><td>Positive</td><td>Negative</td></tr><tr><td>1</td><td>pleasurable, sexy, glorious</td><td>nowhere, grotesque, sleep</td></tr><tr><td>3</td><td> amazing accomplishment., great fun.</td><td>bleak and desperate, conspicuously lacks.</td></tr><tr><td>5</td><td>a pretty amazing accomplishment.</td><td>ultimately a pointless endeavour.</td></tr><tr><td>8</td><td>presents it with an unforgettable visual panache.</td><td>my reaction in a word: disappointment.</td></tr></table>
+
+Text example - diagnosing incorrect predictions In the first example, we show the result of running ACD for our SST LSTM model in Figure 2. We can use this ACD visualization to quickly diagnose why the LSTM made an incorrect prediction. In particular, note that the ACD summary of the LSTM correctly identifies two longer phrases and their corresponding sentiment a great ensemble cast (positive) and n’t lift this heartfelt enterprise out of the ordinary (negative). It is only when these two phrases are joined that the LSTM inaccurately predicts a positive sentiment. This suggests that the LSTM has erroneously learned a positive interaction between these two phrases. Prior methods would not be capable of detecting this type of useful information.
+
+![](images/70858e0bb301c5010c159143f2b09c724bc80b179d2d7bf05e27772a81507ca6.jpg)  
+Figure 2: ACD interpretation of an LSTM predicting sentiment. Blue is positive sentiment, white is neutral, red is negative. The bottom row displays CD scores for individual words in the sentence. Higher rows display important phrases identified by ACD, along with their CD scores, converging to the model’s (incorrect) prediction in the top row. (Best viewed in color)
+
+Vision example - identifying dataset bias Fig 3 shows an example using ACD for an ImageNet VGG model. Using ACD, we can see that to predict “puck”, the CNN is not just focusing on the puck in the image, but also on the hockey player’s skates. Moreover, by comparing the fifth and sixth plots in the third row, we can see that the network is only able to distinguish between the class “puck” and the other top classes when the orange skate and green puck patches merge into a single orange patch. This suggests that the CNN has learned that skates are a strong corroborating features for pucks. While intuitively reasonable in the context of ImageNet, this may not be desirable behavior if the model were used in other domains.
+
+# 4.2.2 IDENTIFYING TOP-SCORING PHRASES
+
+When feasible, a common means of scrutinizing what a model has learned is to inspect its most important features, and interactions. In Table 1, we use ACD to show the top-scoring phrases of different lengths for our LSTM trained on SST. These phrases were extracted by running ACD separately on each sample in SST’s validation set. The score of each phrase was then computed by averaging over the score it received in each occurrence in a ACD hierarchy. The extracted phrases are clearly reflective of the corresponding sentiment, providing additional evidence that ACD is able to capture meaningful positive and negative phrases. Additional phrases are given in Supplement S2.
+
+![](images/3924c07f33cdf890a66a501de3e1ece64184f84c05ed93519d85598ea7d45287.jpg)  
+Figure 3: ACD interpretation for a VGG network prediction, described in 4.2.1. ACD shows that the CNN is focusing on skates to predict the class “puck”, indicating that the model has captured dataset bias. The top row shows the original image, logits for the five top-predicted classes, and the CD superpixel-level scores for those classes. The second row shows separate image patches ACD has identified as being independently predictive of the class “puck”. Starting from the left, each image shows a successive iteration in the agglomeration procedure. The third row shows the CD scores for each of these patches, where patch colors in the second row correspond to line colors in the third row. ACD successfully finds important regions for the target class (such as the puck), and this importance increases as more pixels are selected. Best viewed in color.
+
+# 4.3 QUANTITATIVE EXPERIMENTS
+
+Having introduced our visualization and provided qualitative evidence of its uses, we now provide quantitative evidence of the benefits of ACD.
+
+# 4.3.1 HUMAN EXPERIMENTS
+
+We now demonstrate through human experiments that ACD allows users to better trust and reason about the accuracy of DNNs. Human subjects consist of eleven graduate students at the author’s institution, each of whom has taken a class in machine learning. Each subject was asked to fill out a survey with two types of questions: whether, using ACD, they could identify the more accurate of two models and whether they trusted a models output. In both cases, similar questions were asked on three datasets (SST, MNIST and ImageNet), and ACD was compared against three baselines: CD (Murdoch et al., 2018), Integrated Gradients (IG) (Sundararajan et al., 2017), and occlusion (Li et al., 2016; Zeiler & Fergus, 2014). The exact survey prompts are provided in Supplement S4.
+
+Identifying an accurate model The objective of this section was to determine if subjects could use a small number of interpretations produced by ACD in order to identify the more accurate of two models. For each question in this section, two example predictions were chosen. For each of these two predictions, subjects were given interpretations from two different models (four total), and asked to identify which of the two models had a higher predictive accuracy. Each subject was asked to make this comparison using three different sets of examples for each combination of dataset and interpretation method, for 36 total comparisons. To remove variance due to examples, the same three sets of examples were used across all four interpretation methods.
+
+The predictions shown were chosen to maximize disagreement between models, with SST also being restricted to sentences between five and twenty words, for ease of visualization. To prevent subjects from simply picking the model that predicts more accurately for the given example, for each question a user is shown two examples: one where only the first model predicts correctly and one where only the second model predicts correctly. The two models considered were the accurate models of the previous section and a weakened version of that same model (details given in Sec 4.1).
+
+![](images/dbda9e49f3ac52adc6d32ec1c19bc8431237051a0362e5dedba7ef347f9bc081.jpg)  
+Figure 4: Results for human studies. A. Binary accuracy for whether a subject correctly selected the more accurate model using different interpretation techniques B. Average rank (from 1 to 4) of how much different interpretation techniques helped a subject to trust a model, higher ranks are better.
+
+Fig 4A shows the results of the survey. For SST, humans were better able to identify the strongly predictive model using ACD compared to other baselines, with only ACD and CD outperforming random selection $( 5 0 \% )$ . Based on a one-sided two-sample t-test, the gaps between ACD and IG/Occlusion are significant, but not the gap between ACD and CD. In the simple setting of MNIST, ACD performs similarly to other methods. When applied to ImageNet, a more complex dataset, ACD substantially outperforms prior, non-hierarchical methods, and is the only method to outperform random chance, although the gaps between ACD and other methods are only statistically suggestive (p-values fall between 0.15 and 0.07).
+
+Evaluating trust in a model In this section, the goal is to gauge whether ACD helps a subject to better trust a model’s predictions, relative to prior techniques. For each question, subjects were shown interpretations of the same prediction using four different interpretation methods, and were asked to rank the interpretations from one to four based on how much they instilled trust in trust the model. Subjects were asked to do this ranking for three different examples in each dataset, for nine total rankings. The interpretations were produced from the more accurate model from the previous section, and the examples were chosen using the same criteria as the previous section, except they were restricted to examples correctly predicted by the more accurate model.
+
+Fig 4B shows the average ranking received by each method/dataset pair. ACD substantially outperforms other baselines, particularly for ImageNet, achieving an average rank of 3.5 out of 4, where higher ranks are better. As in the prior question, we found that the hierarchy only provided benefits in the more complicated ImageNet setting, with results on MNIST inconclusive. For both SST and ImageNet, the difference in mean ranks between ACD and all other methods is statistically significant (p-value less than 0.005) based on a permutation test, while on MNIST only the difference between ACD and occlusion is significant.
+
+# 4.3.2 ACD HIERARCHY IS ROBUST TO ADVERSARIAL PERTURBATIONS
+
+While there has been a considerable amount of work on adversarial attacks, little effort has been devoted to qualitatively understanding this phenomenon. In this section, we provide evidence that, on MNIST, the hierarchical clustering produced by ACD is largely robust to adversarial perturbations. This suggests that ACD’s hierarchy captures fundamental features of an image, and is largely immune to the spurious noise favored by adversarial examples.
+
+To measure the robustness of ACD’s hierarchy, we first qualitatively compare the interpretations produced by ACD on both an unaltered image and an adversarially perturbed version of that image. Empirically, we found that the extracted hierarchies are often very similar, see Supplement S5. To generalize these observations, we introduce a metric to quantify the similarity between two ACD hierarchies. This metric allows us to make quantitative, dataset-level statements about the stability of ACD feature hierarchies with respect to adversarial inputs. Given an ACD hierarchy, we compute a ranking of the input image’s pixels according to the order in which they were added to the hierarchy. To measure the similarity between the ACD hierarchies for original and adversarial images, we compute the correlation between their corresponding rankings. As ACD hierarchies are class-specific, we average the correlations for the original and adversarially altered predictions.
+
+Table 2: Correlation between pixel ranks for different adversarial attacks. ACD achieves consistently high correlation across different attack types, indicating that ACD hierarchies are largely robust to adversarial attacks. Using occlusion in place of CD produces substantially less stable hierarchies.   
+
+<table><tr><td>Attack Type</td><td>ACD</td><td>Agglomerative Occlusion</td></tr><tr><td>Saliency (Papernot et al., 2016)</td><td>0.762</td><td>0.259</td></tr><tr><td>Gradient attack</td><td>0.662</td><td>0.196</td></tr><tr><td>FGSM (Goodfellow et al., 2014)</td><td>0.590</td><td>0.131</td></tr><tr><td>Boundary (Brendel et al.,2017)</td><td>0.684</td><td>0.155</td></tr><tr><td>DeepFool (Moosavi Dezfooli et al., 2016)</td><td>0.694</td><td>0.202</td></tr></table>
+
+We display the correlations for five different attacks (computed using the Foolbox package Rauber et al. (2017), examples shown in Supplement S6), each averaged over 100 randomly chosen predictions, in Table 2. As ACD is the first local interpretation technique to compute a hierarchy, there is little prior work available for comparison. As a baseline, we use our agglomeration algorithm with occlusion in place of CD. The resulting correlations are substantially lower, indicating that features detected by ACD are more stable to adversarial attacks than comparable methods. These results provide evidence that ACD’s hierarchy captures fundamental features of an image, and is largely immune to the spurious noise favored by adversarial examples.
+
+# 5 CONCLUSION
+
+In this work, we introduce agglomerative contextual decomposition (ACD), a novel hierarchical interpretation algorithm. ACD is the first method to use a hierarchy to interpret individual neural network predictions. Doing so enables ACD to automatically detect and display non-linear contributions to individual DNN predictions, something prior interpretation methods are unable to do. The benefits of capturing the non-linearities inherent in DNNs are demonstrated through human experiments and examples of diagnosing incorrect predictions and dataset bias. We also demonstrate that ACD’s hierarchy is robust to adversarial perturbations in CNNs, implying that it captures fundamental aspects of the input and ignores spurious noise.
+
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+
+# ACD SUPPLEMENT
+
+![](images/114f381df6fe9d65332fbb65fbf954156c8ee8be6580594e548c781875982e7a.jpg)
+
+![](images/847e5a33bb996a529bb5e1d0d673e4fcd22521a79da77e6d76d024755ae1ef98.jpg)  
+Figure S1: Intuition for CD run on a corner-shaped blob compared to build-up and occlusion. CD decomposes a DNN’s feedforward pass into a part from the blob of interest (top row) and everything else (second row). Left column shows original image with overlaid blob. Other columns show DNN activations summed over the filter dimension. Top and third rows are on same color scale. Second and bottom rows are   
+Figure S2: Comparing unit-level CD scores for the correct class to scores from baseline methods. In each case, the model correctly predicts the label, shown on the y axis. Blue is positive, white is neutral, and red is negative. Best viewed in color.
+
+Fig S1 gives intuition for CD on the VGG-16 ImageNet model described in Sec 4. CD keeps track of the contributions of the blob and non-blob throughout the network. This is intuitively similar to the occlusion and build-up methods, shown in the bottom two rows. The build-up method sets everything but the patch of interest to a references value (often zero). These rows compare the CD decomposition to perturbing the input as in the occlusion and build-up methods. They are similar in early layers, but differences become apparent in later layers.
+
+Table S1: Top-scoring phrases of different lengths extracted by ACD on SST’s validation set. The positive/negative phrases identified by ACD are all indeed positive/negative Fig S2 compares the $7 \mathbf { x } 7$ superpixel-level scores for four images comparing different methods for obtaining importance scores. CD scores better find information relevant to predicting the correct class.   
+
+<table><tr><td>Length</td><td>Positive</td><td>Negative</td></tr><tr><td>1</td><td>&#x27;pleasurable&#x27;,&#x27;sexy’,&#x27;glorious&#x27;,&#x27;delight&#x27;, &#x27;unforgettable&#x27;</td><td>&#x27;nowhere&#x27;，&#x27;grotesque’，&#x27;sleep’，&#x27;mun- dane&#x27;,&#x27;clich&#x27;</td></tr><tr><td>3</td><td>&#x27;amazing accomplishment .,&#x27;great fun .&#x27;, &#x27;good fun &#x27;, &#x27;language sexy :,&#x27;are mag- nificent &#x27;</td><td>&#x27;very bad .&#x27;,&#x27;: disappointment .&#x27;,&#x27;quite bad ’,conspicuously lacks &#x27;,&#x27;bleak and des- perate&#x27;</td></tr><tr><td>5</td><td>&#x27;apretty amazing accomplishment&#x27;, &#x27;clearly，great fun.,&#x27;richness of its per- formances .,&#x27;a delightful coming-of-age story.&#x27;,&#x27;an unforgettable visual panache .</td><td>&#x27;ultimately a pointless endeavor &#x27;,&#x27;this is so bad .,&#x27;emotion closer to pity ?, &#x27;fat waste of time .&#x27;,&#x27;sketch gone horribly wrong.&#x27;</td></tr><tr><td>8</td><td>&#x27;presents it with an unforgettable visual panache .&#x27;,&#x27;film is packed with informa- tion and impressions .&#x27;,&#x27;entertains by pro- viding good,lively company .</td><td>&#x27;myreaction in a word :disappointment ’,&quot;s slow - very，very slow ”,&#x27;a dull , ridiculous attempt at heart-tugging .</td></tr><tr><td>12</td><td>&#x27;in delicious colors，and the costumes and sets are grand .，&#x27;part stevens glides through on some solid performances and witty dialogue .&#x27;,&#x27;mamet enthusiast and for anyone who appreciates intelligent , stylish moviemaking.</td><td>&quot;actors provide scant reason to care in this crude ’7Os throwback .”, &#x27;more often just feels generic,derivative and done to death .&#x27;,&#x27;its storyline with glitches casual fans could correct in their sleep .</td></tr><tr><td>15</td><td>&#x27;serry shows a remarkable gift for story- telling with this moving ，effective little film .&#x27;,&#x27;,lathan and diggs are charming and have chemistry both as friends and lovers .&#x27;</td><td>&#x27;level that one enjoysa bad slasher flick, primarily because it is dull .,&#x27;technicality that strains credulity and leaves the viewer haunted by the waste of potential .</td></tr></table>
+
+# S2 TOP SCORING ACD PHRASES
+
+Here we provide an extended version of Table S1, containing the top 5 phrases of each length for positive/negative polarities. These were extracted using ACD from an LSTM trained on SST.
+
+# S3 ACD EXAMPLES
+
+We provide additional, automatically selected, visualizations produced by ACD. These examples were chosen using the same criteria as the human experiments describes in Sec 4.3.1. All examples are best viewed in color.
+
+SST top-predicted examples. Here, the model used and figure produced correspond to Fig 2.   
+
+<table><tr><td rowspan=1 colspan=1>it     offers</td><td rowspan=1 colspan=1>little</td><td rowspan=1 colspan=1>beyond</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>momentary</td><td rowspan=1 colspan=1>joys</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>pretty    and</td><td rowspan=1 colspan=1>weightless</td><td rowspan=1 colspan=2>intellectualentertainment</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=4 colspan=1></td><td rowspan=1 colspan=1>little</td><td rowspan=1 colspan=1>beyond</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>momentary</td><td rowspan=1 colspan=1>joys</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>pretty    and</td><td rowspan=1 colspan=1>weightless</td><td rowspan=1 colspan=2>intellectual entertainment</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>momentary  joys</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>pretty    and</td><td rowspan=1 colspan=1>weightless</td><td rowspan=1 colspan=3>intellectualentertainment</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>weightless</td><td rowspan=1 colspan=3>intellectual entertainment</td></tr><tr><td rowspan=1 colspan=2>little   beyond</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>momentary</td><td rowspan=1 colspan=1>joys</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>entertainment</td></tr><tr><td rowspan=1 colspan=1>it     offers</td><td rowspan=1 colspan=2>little   beyond</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>momentary</td><td rowspan=1 colspan=1>joys</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>pretty   and</td><td rowspan=1 colspan=1>weightless</td><td rowspan=1 colspan=1>intellectual</td><td rowspan=1 colspan=1>entertainment</td><td rowspan=1 colspan=1></td></tr></table>
+
+![](images/bbffe188c85814a03a70861711afb4ad8256c2fa74c2eb57958a77539db4115f.jpg)  
+SST lowest-predicted examples. Here, the model used and figure produced correspond to Fig 2.
+
+<table><tr><td>the</td><td>primitive</td><td>force</td><td>of</td><td>this</td><td>film seems</td><td>to</td><td>bubble</td><td>up</td><td>from</td><td>the</td><td>vast</td><td></td><td>collective memory</td><td>of</td><td>the</td><td>combatants</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>from</td><td>the</td><td></td><td>vast collective</td><td>memory</td><td>of</td><td>the</td><td>combatants</td></tr><tr><td>the</td><td>primitive</td><td>force</td><td>of</td><td>this</td><td>film</td><td>seems</td><td>bubble</td><td></td><td></td><td></td><td>vast</td><td>collective</td><td>memory</td><td>of</td><td>the</td><td>combatants</td></tr><tr><td>the</td><td>primitive</td><td>force</td><td>of</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>collective</td><td>memory</td><td>of</td><td>the</td><td>combatants</td></tr><tr><td></td><td>primitive</td><td>force</td><td>of</td><td></td><td>film</td><td>seems</td><td>bubble</td><td></td><td></td><td></td><td></td><td></td><td>memory</td><td>of</td><td>the</td><td>combatants</td></tr><tr><td></td><td>primitive</td><td>force</td><td></td><td></td><td>film</td><td>seems</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>combatants</td></tr><tr><td>the</td><td>primitive</td><td>force</td><td>of</td><td>this</td><td>film</td><td>seems</td><td>bubble</td><td>up</td><td>from</td><td>the</td><td>vast</td><td>collective</td><td>memory</td><td>of</td><td>the</td><td>combatants</td></tr><tr><td>50</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>50</td><td>much</td><td></td><td>facile</td><td>technique</td><td></td><td></td><td>such</td><td>cute</td><td></td><td>ideas</td><td></td><td>50</td><td></td><td>little</td><td>movie</td><td>：</td></tr><tr><td></td><td>much</td><td></td><td>facile</td><td>technique</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>much</td><td></td><td>facile</td><td>technique</td><td></td><td></td><td></td><td>cute</td><td></td><td>ideas</td><td>，</td><td>50</td><td></td><td>little</td><td>movie</td><td>，</td></tr><tr><td></td><td>much</td><td></td><td>facile</td><td>technique</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>50</td><td></td><td>little</td><td>movie</td><td></td></tr><tr><td></td><td></td><td></td><td>facile</td><td></td><td>technique</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>little</td><td>movie</td><td>，</td></tr><tr><td>50</td><td>much</td><td></td><td>facile</td><td>technique</td><td></td><td>4</td><td>such</td><td>cute</td><td></td><td>ideas</td><td></td><td>s0</td><td></td><td>little</td><td>movie</td><td></td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>manages</td><td rowspan=1 colspan=1>to</td><td rowspan=1 colspan=1>show</td><td rowspan=1 colspan=1>life</td><td rowspan=1 colspan=1>in</td><td rowspan=1 colspan=1>all</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>its</td><td rowspan=1 colspan=1>banality</td><td rowspan=1 colspan=1>when</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>intention</td><td rowspan=5 colspan=1>is   quite   the  opposite</td></tr><tr><td rowspan=1 colspan=1>manages</td><td rowspan=1 colspan=1>to</td><td rowspan=1 colspan=1>show</td><td rowspan=1 colspan=1>life</td><td rowspan=1 colspan=1>in</td><td rowspan=1 colspan=1>all</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>its</td><td rowspan=1 colspan=1>banality</td><td rowspan=1 colspan=1>when</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>intention</td></tr><tr><td rowspan=4 colspan=3></td><td rowspan=1 colspan=1>life</td><td rowspan=1 colspan=1>in</td><td rowspan=1 colspan=1>all</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>its</td><td rowspan=1 colspan=1>banality</td><td rowspan=1 colspan=1>when</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>intention</td></tr><tr><td rowspan=3 colspan=3></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>its</td><td rowspan=1 colspan=1>banality</td><td rowspan=1 colspan=1>when</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>intention</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>banality</td><td rowspan=1 colspan=1>when</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1>intention</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>banality when</td><td rowspan=1 colspan=1>the</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>quite   the  opposite</td></tr><tr><td rowspan=1 colspan=2>manages  to</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>life    in</td><td rowspan=1 colspan=1>all</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>banality</td><td rowspan=1 colspan=1>when</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2></td></tr><tr><td rowspan=1 colspan=2>manages  to</td><td rowspan=1 colspan=1>show</td><td rowspan=1 colspan=2>life    in</td><td rowspan=1 colspan=1>all</td><td rowspan=1 colspan=1>of</td><td rowspan=1 colspan=1>its</td><td rowspan=1 colspan=1>banality</td><td rowspan=1 colspan=1>when</td><td rowspan=1 colspan=3>the  intention  is   quite  the</td></tr></table>
+
+MNIST top-predicted examples. Here, the model used is the same as in $\mathrm { S e c } 4 . 3 . 2 $ and the interpretation of the figure produced is the same as in Fig 3.
+
+![](images/b0a4576a05871421ac183fc4f838e1d5800f179935706140ecaa36827263982a.jpg)
+
+![](images/2477fa0b9b948ec6fe9017d9d0becfb532bb3abd5bf71d9608ff4351314e2395.jpg)
+
+MNIST lowest-predicted examples. Here, the model used is the same as in Sec 4.3.2 and the interpretation of the figure produced is the same as in Fig 3.
+
+![](images/045180043b8e500b0b82410accc56a634117b3c506db7bc88c867f4ada818e59.jpg)
+
+Imagenet top-predicted examples. Here, the model used and figure produced correspond to that in Fig 3.
+
+![](images/f46266f117f6103086e6849896814887d5286dc80e49f03c47a0bede5e1041e1.jpg)
+
+![](images/9f306bc4da198291fe9d480558ae5b0717c16c4706ac455245866765324e9fe5.jpg)  
+predictionlogits
+
+CD (prairie ch） CD (ruffed gro） CD (partridge） CD (black grou) CD (robin,Ame)
+
+![](images/ce73e8b60cfceffb06cada34bae09de76684269bfd9f91e592692bcbd0d01eb8.jpg)
+
+![](images/5b3c9afef6d7dee6640e42bb403c27a355ba40164ec5f73529801b2dbb255b53.jpg)
+
+![](images/55a015ea8fc8d0200828d7a5d2e15b8067d4e64ced4272c156e960903ec0b490.jpg)
+
+![](images/fc021be5d0cb6e237fe29b04dc3035f9d380c403dc16962df4d302dda8525ac6.jpg)
+
+![](images/c18f17cc97601a941b9c56cdb7c2d0b672cc2b6497e1f7a8b714ec383dc2dbdf.jpg)
+
+![](images/5a8007675179efd6bcd8386558c3203a51a26482f46f2dfc700a581e3e880afd.jpg)
+
+Imagenet lowest-predicted examples. Here, the model used and figure produced correspond to that in Fig 3.
+
+![](images/311319399f94b5f1894bb705863c0b3e7f9b13e022fab5cb823df98111c815b9.jpg)
+
+![](images/ca0ecc017e38eb8f49c8b7074888ecf98c6d48383c7534f32b9eeb9b1999df30.jpg)
+
+![](images/8b4d205208f7b5248f3ddc4b03affdb8b29f0a1e18acd5a698e6d1d7377074a6.jpg)
+
+# S4 HUMAN EXPERIMENTS EXPERIMENTAL SETUP
+
+Order of questions is randomized for each subject. Below are the instructions and questions given to the user (for brevity, the actual visualizations are omitted, but are similar to the visualizations shown in Supplement S3).
+
+<table><tr><td>This survey aims to compare different interpretation techniques. In what follows, blue is positive, white is neutral,and red is negative.</td></tr></table>
+
+# S4.1 SENTIMENT CLASSIFICATION
+
+# S4.1.1 CHOOSING THE BETTER MODEL
+
+In this section, the task is to compare two models that classify movie reviews as either positive (good movie) or negative (bad movie). One model has better predictive accuracy than the other.
+
+In what follows, you will see visualizations of what both models have learned. These visualizations use different methods of identifying contributions to the final prediction of either individual words or groups of them. For each model, we show visualizations of two different examples.
+
+In these visualizations, the color shows what the model thinks for individual words / groups of words. Blue is positive sentiment (e.g. ”great”, ”fantastic”) and red is negative sentiment (e.g. ”terrible”, ”miserable”).
+
+Using these visualizations, please write A or B to select which model you think has higher predictive accuracy.
+
+# S4.1.2 GAUGING TRUST
+
+Now, we show results only from the good model. Your task is to compare different visualizations. For the following predictions, please select which visualization method leads you to trust the model the most.
+
+Put a number next to each of the following letters ranking them in the order of how much they make you trust the model (1-4, 1 is the most trustworthy).
+
+# S4.2 MNIST
+
+# S4.2.1 CHOOSING THE BETTER MODEL
+
+Now we will perform a similar challenge for vision. Your task is to compare two models that classify images into classes, in this case digits from 0-9. One model has higher predictive accuracy than the other.
+
+In what follows, you will see visualizations of what both models have learned. These visualizations use different methods of identifying contributions to the final prediction of either individual pixels or groups of them. Using these visualizations, please select the model you think has higher accuracy.
+
+For each prediction, the top row contains the raw image followed by five heat maps, and the title shows the predicted class. Each heatmap corresponds to a different class, with blue pixels indicating a pixel is a positive signal for that class, and red pixels indicating a negative signal. The first heatmap title shows the predicted class of the network - this is wrong half the time. In some cases, each visualization has an extra row, which shows groups of pixels, at multiple levels of granularity, that contribute to the predicted class.
+
+Using these visualizations, please select which model you think has higher predictive accuracy, A or B.
+
+# S4.2.2 GAUGING TRUST
+
+Now, we show results only from the good model. Your task is to compare different visualizations. For the following predictions, please select which visualization method leads you to trust the model the most.
+
+Put a number next to each of the following letters ranking them in the order of how much they make you trust the model (1-4, 1 is the most trustworthy).
+
+# S4.2.3 CHOOSING THE MORE ACCURATE MODEL
+
+Now we will perform a similar challenge for vision. Your task is to compare two models that classify images into classes (ex. balloon, bee, pomegranate). One model is better than the other in terms of predictive accuracy.
+
+In what follows, you will see visualizations of what both models have learned. These visualizations use different methods of identifying contributions to the final prediction of either individual pixels or groups of them.
+
+For each prediction, the top row contains the raw image followed by five heat maps, and the title shows the predicted class. Each heatmap corresponds to a different class, with blue pixels indicating a pixel is a positive signal for that class, and red pixels indicating a negative signal. The first heatmap title shows the predicted class of the network - this is wrong half the time. In some cases, each visualization has an extra row, which shows groups of pixels, at multiple levels of granularity, that contribute to the predicted class.
+
+Using these visualizations, please select which model you think has higher predictive accuracy, A or B.
+
+# S4.2.4 GAUGING TRUST
+
+Now, we show results only from the more accurate model. Your task is to compare different visualizations. For the following predictions, please select which visualization method leads you to trust the model’s decision the most.
+
+Put a number next to each of the following letters ranking them in the order of how much they make you trust the model (1-4, 1 is the most trustworthy).
+
+S5 ACD ON ADVERSARIAL EXAMPLES
+
+The hierarchies constructed by ACD to explain a prediction of 0 are substantially similar for both the original image and an adversarially perturbed image predicted to be a 6. Original image
+
+![](images/0766be086aaf891e1454c32cf61fd3eaf2abb62d616b821a38d4c11143bcc711.jpg)  
+Figure S3: Example of ACD run on an image of class 0 before and after an adversarial perturbation (a DeepFool attack). Best viewed in color.
+
+![](images/8c8366419730954d85012c48cd6f39038cea61b3b99120e5ec9a3162223cb4c5.jpg)  
+Figure S4: Examples of attacks for one image. Original image (left column) is correctly predicted as class 0. After each adversarial perturbation (middle column), the predicted class for the adversarial image (right column) is now altered.
+
+# S7 GENERALIZING CD TO CNNS
+
+Fig S5 qualitatively shows the change in behavior as the result of two modifications made to the naive extension of CD to CNNs, which was independently developed by Godin et al. (2018). During development of our general CD, two changes were made. First, we partitioned the bias between $\gamma _ { i }$ and $\beta _ { i }$ , as described in Equation 5. As can be seen in the second column, this qualitatively reduces the noise in the heat maps. Next, we replace the ReLU Shapely decomposition by the decomposition provided in Equation 10. In the third column, you can see that this effectively prevents the CD scores from becoming unrealistically large in areas that should not be influencing the model’s decision. When these two approaches are combined in the fourth column, they provide qualitatively sensible heatmaps with reasonably valued CD scores. When applied to the smaller models used on SST and MNIST, these changes don’t have large effects on the interpretations.
+
+![](images/de97701982e61b4adc1d445b31599763464bc1f8fd4ffd4e6a35480367a752a2.jpg)  
+Figure S5: Comparing unit-level CD scores to CD scores from the naive extension of CD to CNNs, independently developed by Godin et al. (2018). Labels under the bottom row signify the minimum and maximum scores from each column. Altering the bias partition and ReLU decomposition qualitatively improves scores (e.g. see scores in bottom row corresponding to the location of the crane), and avoids extremely large magnitudes (see values under left two columns). Blue is positive, white is neutral, and red is negative. In each case, scores are for the correct class, which the model predicts correctly (shown on the y axis).

parse/train/i_Q1yrOegLY/i_Q1yrOegLY.md

@@ -0,0 +1,318 @@
+# Revisiting Deep Learning Models for Tabular Data
+
+Yury Gorishniy∗†‡ Ivan Rubachev†♣
+
+Valentin Khrulkov† Artem Babenko†♣
+
+Yandex, Russia Moscow Institute of Physics and Technology, Russia National Research University Higher School of Economics, Russia
+
+# Abstract
+
+The existing literature on deep learning for tabular data proposes a wide range of novel architectures and reports competitive results on various datasets. However, the proposed models are usually not properly compared to each other and existing works often use different benchmarks and experiment protocols. As a result, it is unclear for both researchers and practitioners what models perform best. Additionally, the field still lacks effective baselines, that is, the easy-to-use models that provide competitive performance across different problems.
+
+In this work, we perform an overview of the main families of DL architectures for tabular data and raise the bar of baselines in tabular DL by identifying two simple and powerful deep architectures. The first one is a ResNet-like architecture which turns out to be a strong baseline that is often missing in prior works. The second model is our simple adaptation of the Transformer architecture for tabular data, which outperforms other solutions on most tasks. Both models are compared to many existing architectures on a diverse set of tasks under the same training and tuning protocols. We also compare the best DL models with Gradient Boosted Decision Trees and conclude that there is still no universally superior solution. The source code is available at https://github.com/yandex-research/rtdl.
+
+# 1 Introduction
+
+Due to the tremendous success of deep learning on such data domains as images, audio and texts (Goodfellow et al., 2016), there has been a lot of research interest to extend this success to problems with data stored in tabular format. In these problems, data points are represented as vectors of heterogeneous features, which is typical for industrial applications and ML competitions, where neural networks have a strong non-deep competitor in the form of GBDT (Chen and Guestrin, 2016; Ke et al., 2017; Prokhorenkova et al., 2018). Along with potentially higher performance, using deep learning for tabular data is appealing as it would allow constructing multi-modal pipelines for problems, where only one part of the input is tabular, and other parts include images, audio and other DL-friendly data. Such pipelines can then be trained end-to-end by gradient optimization for all modalities. For these reasons, a large number of DL solutions were recently proposed, and new models continue to emerge (Arik and Pfister, 2020; Badirli et al., 2020; Hazimeh et al., 2020; Huang et al., 2020; Klambauer et al., 2017; Popov et al., 2020; Song et al., 2019; Wang et al., 2017, 2020).
+
+Unfortunately, due to the lack of established benchmarks (such as ImageNet (Deng et al., 2009) for computer vision or GLUE (Wang et al., 2019a) for NLP), existing papers use different datasets for evaluation and proposed DL models are often not adequately compared to each other. Therefore, from the current literature, it is unclear what DL model generally performs better than others and whether GBDT is surpassed by DL models. Additionally, despite the large number of novel architectures, the field still lacks simple and reliable solutions that allow achieving competitive performance with moderate effort and provide stable performance across many tasks. In that regard, Multilayer
+
+Perceptron (MLP) remains the main simple baseline for the field, however, it does not always represent a significant challenge for other competitors.
+
+The described problems impede the research process and make the observations from the papers not conclusive enough. Therefore, we believe it is timely to review the recent developments from the field and raise the bar of baselines in tabular DL. We start with a hypothesis that well-studied DL architecture blocks may be underexplored in the context of tabular data and may be used to design better baselines. Thus, we take inspiration from well-known battle-tested architectures from other fields and obtain two simple models for tabular data. The first one is a ResNet-like architecture (He et al., 2015) and the second one is FT-Transformer — our simple adaptation of the Transformer architecture (Vaswani et al., 2017) for tabular data. Then, we compare these models with many existing solutions on a diverse set of tasks under the same protocols of training and hyperparameters tuning. First, we reveal that none of the considered DL models can consistently outperform the ResNet-like model. Given its simplicity, it can serve as a strong baseline for future work. Second, FT-Transformer demonstrates the best performance on most tasks and becomes a new powerful solution for the field. Interestingly, FT-Transformer turns out to be a more universal architecture for tabular data: it performs well on a wider range of tasks than the more “conventional” ResNet and other DL models. Finally, we compare the best DL models to GBDT and conclude that there is still no universally superior solution.
+
+We summarize the contributions of our paper as follows:
+
+1. We thoroughly evaluate the main models for tabular DL on a diverse set of tasks to investigate their relative performance.   
+2. We demonstrate that a simple ResNet-like architecture is an effective baseline for tabular DL, which was overlooked by existing literature. Given its simplicity, we recommend this baseline for comparison in future tabular DL works.   
+3. We introduce FT-Transformer — a simple adaptation of the Transformer architecture for tabular data that becomes a new powerful solution for the field. We observe that it is a more universal architecture: it performs well on a wider range of tasks than other DL models.   
+4. We reveal that there is still no universally superior solution among GBDT and deep models.
+
+# 2 Related work
+
+The “shallow” state-of-the-art for problems with tabular data is currently ensembles of decision trees, such as GBDT (Gradient Boosting Decision Tree) (Friedman, 2001), which are typically the top-choice in various ML competitions. At the moment, there are several established GBDT libraries, such as XGBoost (Chen and Guestrin, 2016), LightGBM (Ke et al., 2017), CatBoost (Prokhorenkova et al., 2018), which are widely used by both ML researchers and practitioners. While these implementations vary in detail, on most of the tasks, their performances do not differ much (Prokhorenkova et al., 2018).
+
+During several recent years, a large number of deep learning models for tabular data have been developed (Arik and Pfister, 2020; Badirli et al., 2020; Hazimeh et al., 2020; Huang et al., 2020; Klambauer et al., 2017; Popov et al., 2020; Song et al., 2019; Wang et al., 2017). Most of these models can be roughly categorized into three groups, which we briefly describe below.
+
+Differentiable trees. The first group of models is motivated by the strong performance of decision tree ensembles for tabular data. Since decision trees are not differentiable and do not allow gradient optimization, they cannot be used as a component for pipelines trained in the end-to-end fashion. To address this issue, several works (Hazimeh et al., 2020; Kontschieder et al., 2015; Popov et al., 2020; Yang et al., 2018) propose to “smooth” decision functions in the internal tree nodes to make the overall tree function and tree routing differentiable. While the methods of this family can outperform GBDT on some tasks (Popov et al., 2020), in our experiments, they do not consistently outperform ResNet.
+
+Attention-based models. Due to the ubiquitous success of attention-based architectures for different domains (Dosovitskiy et al., 2021; Vaswani et al., 2017), several authors propose to employ attentionlike modules for tabular DL as well (Arik and Pfister, 2020; Huang et al., 2020; Song et al., 2019). In our experiments, we show that the properly tuned ResNet outperforms the existing attention-based models. Nevertheless, we identify an effective way to apply the Transformer architecture (Vaswani et al., 2017) to tabular data: the resulting architecture outperforms ResNet on most of the tasks.
+
+Explicit modeling of multiplicative interactions. In the literature on recommender systems and click-through-rate prediction, several works criticize MLP since it is unsuitable for modeling multiplicative interactions between features (Beutel et al., 2018; Qin et al., 2021; Wang et al., 2017). Inspired by this motivation, some works (Beutel et al., 2018; Wang et al., 2017, 2020) have proposed different ways to incorporate feature products into MLP. In our experiments, however, we do not find such methods to be superior to properly tuned baselines.
+
+The literature also proposes some other architectural designs (Badirli et al., 2020; Klambauer et al., 2017) that cannot be explicitly assigned to any of the groups above. Overall, the community has developed a variety of models that are evaluated on different benchmarks and are rarely compared to each other. Our work aims to establish a fair comparison of them and identify the solutions that consistently provide high performance.
+
+# 3 Models for tabular data problems
+
+In this section, we describe the main deep architectures that we highlight in our work, as well as the existing solutions included in the comparison. Since we argue that the field needs strong easy-to-use baselines, we try to reuse well-established DL building blocks as much as possible when designing ResNet (section 3.2) and FT-Transformer (section 3.3). We hope this approach will result in conceptually familiar models that require less effort to achieve good performance. Additional discussion and technical details for all the models are provided in supplementary.
+
+Notation. In this work, we consider supervised learning problems. $D { = } \{ ( x _ { i } , ~ y _ { i } ) \} _ { i { = } 1 } ^ { n }$ denotes a dataset, where $x _ { i } = ( x _ { i } ^ { ( n u m ) } , x _ { i } ^ { ( c a t ) } ) \in \mathbb { X }$ )) ∈ X represents numerical x(nuij $x _ { i j } ^ { ( n u m ) }$ and categorical $x _ { i j } ^ { ( c a t ) }$ features of an object and $y _ { i } \in \mathbb { Y }$ denotes the corresponding object label. The total number of features is denoted as $k$ . The dataset is split into three disjoint subsets: $D = D _ { t r a i n } \cup D _ { v a l } \cup D _ { t e s t }$ , where $D _ { t r a i n }$ is used for training, $D _ { v a l }$ is used for early stopping and hyperparameter tuning, and $D _ { t e s t }$ is used for the final evaluation. We consider three types of tasks: binary classification $\mathbb { Y } = \{ 0 , 1 \}$ , multiclass classification $\mathbb { Y } = \{ 1 , \ . . . , C \}$ and regression $\mathbb { Y } = \mathbb { R }$ .
+
+# 3.1 MLP
+
+We formalize the “MLP” architecture in Equation 1.
+
+$$
+\begin{array} { c } { \mathtt { M L P } ( x ) = \mathtt { L i n e a r } \left( \mathtt { M L P B l o c k } \left( \dots \left( \mathtt { M L P B l o c k } ( x ) \right) \right) \right) } \\ { \mathtt { M L P B l o c k } ( x ) = \mathtt { D r o p o u t } \left( \mathtt { R e L U } \left( \mathtt { L i n e a r } ( x ) \right) \right) } \end{array}
+$$
+
+# 3.2 ResNet
+
+We are aware of one attempt to design a ResNet-like baseline (Klambauer et al., 2017) where the reported results were not competitive. However, given ResNet’s success story in computer vision (He et al., 2015) and its recent achievements on NLP tasks (Sun and Iyyer, 2021), we give it a second try and construct a simple variation of ResNet as described in Equation 2. The main building block is simplified compared to the original architecture, and there is an almost clear path from the input to output which we find to be beneficial for the optimization. Overall, we expect this architecture to outperform MLP on tasks where deeper representations can be helpful.
+
+ResNet $( x ) =$ Prediction (ResNetBlock (. . . (ResNetBlock (Linear(x))))) ResNetBlock(x) = x + Dropout(Linear(Dropout(ReLU(Linear(BatchNorm(x)))))) Prediction $( x ) =$ Linear (ReLU (BatchNorm (x)))
+
+# 3.3 FT-Transformer
+
+In this section, we introduce FT-Transformer (Feature Tokenizer $^ +$ Transformer) — a simple adaptation of the Transformer architecture (Vaswani et al., 2017) for the tabular domain. Figure 1 demonstrates the main parts of FT-Transformer. In a nutshell, our model transforms all features (categorical and numerical) to embeddings and applies a stack of Transformer layers to the embeddings. Thus, every Transformer layer operates on the feature level of one object. We compare FT-Transformer to conceptually similar AutoInt in section 5.2.
+
+![](images/df9999ef23ac55d1399747a5900f3eac717d0c6a4d8ac680caf051f5bf2bddda.jpg)  
+Figure 1: The FT-Transformer architecture. Firstly, Feature Tokenizer transforms features to embeddings. The embeddings are then processed by the Transformer module and the final representation of the [CLS] token is used for prediction.
+
+![](images/a59f8fc74fd228878c534e25ebbe4e2ef3f34dca0d1a9f7e4aafb3653f67b0ac.jpg)  
+Figure 2: (a) Feature Tokenizer; in the example, there are three numerical and two categorical features; (b) One Transformer layer.
+
+Feature Tokenizer. The Feature Tokenizer module (see Figure 2) transforms the input features $x$ to embeddings $T \in \mathbb { R } ^ { k \times d }$ . The embedding for a given feature $x _ { j }$ is computed as follows:
+
+$$
+T _ { j } = b _ { j } + f _ { j } ( x _ { j } ) \in \mathbb { R } ^ { d } \qquad f _ { j } : \mathbb { X } _ { j } \to \mathbb { R } ^ { d } .
+$$
+
+where $b _ { j }$ is the $j$ -th feature bias, $f _ { j } ^ { ( n u m ) }$ is implemented as the element-wise multiplication with the vector $W _ { j _ { \lambda } } ^ { ( n u m ) } \in \mathbb { R } ^ { d }$ and $f _ { j } ^ { ( c a t ) }$ is implemented as the lookup table $W _ { j } ^ { ( c a t ) } \in \mathbb { R } ^ { S _ { j } \times d }$ for categorical features. Overall:
+
+$$
+\begin{array} { r l r } & { T _ { j } ^ { ( n u m ) } = b _ { j } ^ { ( n u m ) } + x _ { j } ^ { ( n u m ) } \cdot W _ { j } ^ { ( n u m ) } } & { \in \mathbb { R } ^ { d } , } \\ & { T _ { j } ^ { ( c a t ) } = b _ { j } ^ { ( c a t ) } + e _ { j } ^ { T } W _ { j } ^ { ( c a t ) } } & { \in \mathbb { R } ^ { d } , } \\ & { T = \mathsf { s t a c k } \left[ T _ { 1 } ^ { ( n u m ) } , \ldots , T _ { k ^ { ( n u m ) } } ^ { ( n u m ) } , T _ { 1 } ^ { ( c a t ) } , \ldots , T _ { k ^ { ( c a t ) } } ^ { ( c a t ) } \right] \in \mathbb { R } ^ { k \times d } . } \end{array}
+$$
+
+where $e _ { j } ^ { T }$ is a one-hot vector for the corresponding categorical feature.
+
+Transformer. At this stage, the embedding of the [CLS] token (or “classification token”, or “output token”, see Devlin et al. (2019)) is appended to $T$ and $L$ Transformer layers $F _ { 1 }$ , . . . , $F _ { L }$ are applied:
+
+$$
+T _ { 0 } = { \tt s t a c k } \left[ \left[ { \tt C L S } \right] , T \right] \qquad T _ { i } = F _ { i } ( T _ { i - 1 } ) .
+$$
+
+We use the PreNorm variant for easier optimization (Wang et al., 2019b), see Figure 2. In the PreNorm setting, we also found it to be necessary to remove the first normalization from the first Transformer layer to achieve good performance. See the original paper (Vaswani et al., 2017) for the background on Multi-Head Self-Attention (MHSA) and the Feed Forward module. See supplementary for details such as activations, placement of normalizations and dropout modules (Srivastava et al., 2014).
+
+Prediction. The final representation of the [CLS] token is used for prediction:
+
+$$
+\begin{array} { r } { \hat { y } = \mathtt { L i n e a r } \big ( \mathtt { R e L U } \big ( \mathtt { L a y e r N o r m } ( T _ { L } ^ { \mathtt { L C L S } } ) \big ) \big ) . } \end{array}
+$$
+
+Limitations. FT-Transformer requires more resources (both hardware and time) for training than simple models such as ResNet and may not be easily scaled to datasets when the number of features is “too large” (it is determined by the available hardware and time budget). Consequently, widespread usage of FT-Transformer for solving tabular data problems can lead to greater CO2 emissions produced by ML pipelines, since tabular data problems are ubiquitous. The main cause of the described problem lies in the quadratic complexity of the vanilla MHSA with respect to the number of features. However, the issue can be alleviated by using efficient approximations of MHSA (Tay et al., 2020). Additionally, it is still possible to distill FT-Transformer into simpler architectures for better inference performance. We report training times and the used hardware in supplementary.
+
+# 3.4 Other models
+
+In this section, we list the existing models designed specifically for tabular data that we include in the comparison.
+
+• SNN (Klambauer et al., 2017). An MLP-like architecture with the SELU activation that enables training deeper models.   
+• NODE (Popov et al., 2020). A differentiable ensemble of oblivious decision trees.   
+• TabNet (Arik and Pfister, 2020). A recurrent architecture that alternates dynamical reweighing of features and conventional feed-forward modules. GrowNet (Badirli et al., 2020). Gradient boosted weak MLPs. The official implementation supports only classification and regression problems. DCN V2 (Wang et al., 2020). Consists of an MLP-like module and the feature crossing module (a combination of linear layers and multiplications).   
+• AutoInt (Song et al., 2019). Transforms features to embeddings and applies a series of attention-based transformations to the embeddings.   
+• XGBoost (Chen and Guestrin, 2016). One of the most popular GBDT implementations.   
+• CatBoost (Prokhorenkova et al., 2018). GBDT implementation that uses oblivious decision trees (Lou and Obukhov, 2017) as weak learners.
+
+# 4 Experiments
+
+In this section, we compare DL models to each other as well as to GBDT. Note that in the main text, we report only the key results. In supplementary, we provide: (1) the results for all models on all datasets; (2) information on hardware; (3) training times for ResNet and FT-Transformer.
+
+# 4.1 Scope of the comparison
+
+In our work, we focus on the relative performance of different architectures and do not employ various model-agnostic DL practices, such as pretraining, additional loss functions, data augmentation, distillation, learning rate warmup, learning rate decay and many others. While these practices can potentially improve the performance, our goal is to evaluate the impact of inductive biases imposed by the different model architectures.
+
+# 4.2 Datasets
+
+We use a diverse set of eleven public datasets (see supplementary for the detailed description). For each dataset, there is exactly one train-validation-test split, so all algorithms use the same splits. The datasets include: California Housing (CA, real estate data, Kelley Pace and Barry (1997)), Adult (AD, income estimation, Kohavi (1996)), Helena (HE, anonymized dataset, Guyon et al. (2019)),
+
+Jannis (JA, anonymized dataset, Guyon et al. (2019)), Higgs (HI, simulated physical particles, Baldi et al. (2014); we use the version with 98K samples available at the OpenML repository (Vanschoren et al., 2014)), ALOI (AL, images, Geusebroek et al. (2005)), Epsilon (EP, simulated physics experiments), Year (YE, audio features, Bertin-Mahieux et al. (2011)), Covertype (CO, forest characteristics, Blackard and Dean. (2000)), Yahoo (YA, search queries, Chapelle and Chang (2011)), Microsoft (MI, search queries, Qin and Liu (2013)). We follow the pointwise approach to learning-to-rank and treat ranking problems (Microsoft, Yahoo) as regression problems. The dataset properties are summarized in Table 1.
+
+Table 1: Dataset properties. Notation: “RMSE” $\sim$ root-mean-square error, “Acc.” $\sim$ accuracy.   
+
+<table><tr><td></td><td>CA</td><td>AD</td><td>HE</td><td>JA</td><td>HI</td><td>AL</td><td>EP</td><td>YE</td><td>Co</td><td>YA</td><td>MI</td></tr><tr><td>#objects</td><td>20640</td><td>48842</td><td>65196</td><td>83733</td><td>98050</td><td>108000</td><td>500000</td><td>515345</td><td>581012</td><td>709877</td><td>1200192</td></tr><tr><td>#num. features #cat. features</td><td>8</td><td>6</td><td>27</td><td>54</td><td>28</td><td>128</td><td>2000</td><td>90</td><td>54</td><td>699</td><td>136</td></tr><tr><td>metric</td><td>0</td><td>8</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td>RMSE</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>RMSE</td><td>Acc.</td><td>RMSE</td><td>RMSE</td></tr><tr><td>#classes</td><td>1</td><td>2</td><td>100</td><td>4</td><td>2</td><td>1000</td><td>2</td><td>1</td><td>7</td><td>1</td><td>1</td></tr></table>
+
+# 4.3 Implementation details
+
+Data preprocessing. Data preprocessing is known to be vital for DL models. For each dataset, the same preprocessing was used for all deep models for a fair comparison. By default, we used the quantile transformation from the Scikit-learn library (Pedregosa et al., 2011). We apply standardization (mean subtraction and scaling) to Helena and ALOI. The latter one represents image data, and standardization is a common practice in computer vision. On the Epsilon dataset, we observed preprocessing to be detrimental to deep models’ performance, so we use the raw features on this dataset. We apply standardization to regression targets for all algorithms.
+
+Tuning. For every dataset, we carefully tune each model’s hyperparameters. The best hyperparameters are the ones that perform best on the validation set, so the test set is never used for tuning. For most algorithms, we use the Optuna library (Akiba et al., 2019) to run Bayesian optimization (the Tree-Structured Parzen Estimator algorithm), which is reported to be superior to random search (Turner et al., 2021). For the rest, we iterate over predefined sets of configurations recommended by corresponding papers. We provide parameter spaces and grids in supplementary. We set the budget for Optuna-based tuning in terms of iterations and provide additional analysis on setting the budget in terms of time in supplementary.
+
+Evaluation. For each tuned configuration, we run 15 experiments with different random seeds and report the performance on the test set. For some algorithms, we also report the performance of default configurations without hyperparameter tuning.
+
+Ensembles. For each model, on each dataset, we obtain three ensembles by splitting the 15 single models into three disjoint groups of equal size and averaging predictions of single models within each group.
+
+Neural networks. We minimize cross-entropy for classification problems and mean squared error for regression problems. For TabNet and GrowNet, we follow the original implementations and use the Adam optimizer (Kingma and Ba, 2017). For all other algorithms, we use the AdamW optimizer (Loshchilov and Hutter, 2019). We do not apply learning rate schedules. For each dataset, we use a predefined batch size for all algorithms unless special instructions on batch sizes are given in the corresponding papers (see supplementary). We continue training until there are patience $+ 1$ consecutive epochs without improvements on the validation set; we set patience $= 1 6$ for all algorithms.
+
+Categorical features. For XGBoost, we use one-hot encoding. For CatBoost, we employ the built-in support for categorical features. For Neural Networks, we use embeddings of the same dimensionality for all categorical features.
+
+Table 2: Results for DL models. The metric values averaged over 15 random seeds are reported. See supplementary for standard deviations. For each dataset, top results are in bold. “Top” means “the gap between this result and the result with the best score is not statistically significant”. For each dataset, ranks are calculated by sorting the reported scores; the “rank” column reports the average rank across all datasets. Notation: FT-T \~ FT-Transformer, $\downarrow \sim \mathrm { R M S E }$ , $\uparrow$ \~ accuracy   
+
+<table><tr><td></td><td>CA↓</td><td>AD↑</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>EP↑</td><td>YE↓</td><td>CO↑</td><td>YA↓</td><td>MI↓</td><td>rank (std)</td></tr><tr><td>TabNet</td><td>0.510</td><td>0.850</td><td>0.378</td><td>0.723</td><td>0.719</td><td>0.954</td><td>0.8896</td><td>8.909</td><td>0.957</td><td>0.823</td><td>0.751</td><td>7.5 (2.0)</td></tr><tr><td>SNN</td><td>0.493</td><td>0.854</td><td>0.373</td><td>0.719</td><td>0.722</td><td>0.954</td><td>0.8975</td><td>8.895</td><td>0.961</td><td>0.761</td><td>0.751</td><td>6.4 (1.4)</td></tr><tr><td>AutoInt</td><td>0.474</td><td>0.859</td><td>0.372</td><td>0.721</td><td>0.725</td><td>0.945</td><td>0.8949</td><td>8.882</td><td>0.934</td><td>0.768</td><td>0.750</td><td>5.7 (2.3)</td></tr><tr><td>GrowNet</td><td>0.487</td><td>0.857</td><td></td><td></td><td>0.722</td><td>1</td><td>0.8970</td><td>8.827</td><td>1</td><td>0.765</td><td>0.751</td><td>5.7 (2.2)</td></tr><tr><td>MLP</td><td>0.499</td><td>0.852</td><td>0.383</td><td>0.719</td><td>0.723</td><td>0.954</td><td>0.8977</td><td>8.853</td><td>0.962</td><td>0.757</td><td>0.747</td><td>4.8 (1.9)</td></tr><tr><td>DCN2</td><td>0.484</td><td>0.853</td><td>0.385</td><td>0.716</td><td>0.723</td><td>0.955</td><td>0.8977</td><td>8.890</td><td>0.965</td><td>0.757</td><td>0.749</td><td>4.7 (2.0)</td></tr><tr><td>NODE</td><td>0.464</td><td>0.858</td><td>0.359</td><td>0.727</td><td>0.726</td><td>0.918</td><td>0.8958</td><td>8.784</td><td>0.958</td><td>0.753</td><td>0.745</td><td>3.9 (2.8)</td></tr><tr><td>ResNet</td><td>0.486</td><td>0.854</td><td>0.396</td><td>0.728</td><td>0.727</td><td>0.963</td><td>0.8969</td><td>8.846</td><td>0.964</td><td>0.757</td><td>0.748</td><td>3.3 (1.8)</td></tr><tr><td>FT-T</td><td></td><td>0.459 0.859</td><td>0.391</td><td>0.732</td><td>0.729</td><td>0.960</td><td>0.8982</td><td>8.855</td><td>0.970</td><td>0.756</td><td>0.746</td><td>1.8 (1.2)</td></tr></table>
+
+# 4.4 Comparing DL models
+
+Table 2 reports the results for deep architectures.
+
+# The main takeaways:
+
+• MLP is still a good sanity check   
+• ResNet turns out to be an effective baseline that none of the competitors can consistently outperform.   
+• FT-Transformer performs best on most tasks and becomes a new powerful solution for the field.   
+• Tuning makes simple models such as MLP and ResNet competitive, so we recommend tuning baselines when possible. Luckily, today, it is more approachable with libraries such as Optuna (Akiba et al., 2019).
+
+Among other models, NODE (Popov et al., 2020) is the only one that demonstrates high performance on several tasks. However, it is still inferior to ResNet on six datasets (Helena, Jannis, Higgs, ALOI, Epsilon, Covertype), while being a more complex solution. Moreover, it is not a truly “single” model; in fact, it often contains significantly more parameters than ResNet and FT-Transformer and has an ensemble-like structure. We illustrate that by comparing ensembles in Table 3. The results indicate that FT-Transformer and ResNet benefit more from ensembling; in this regime, FT-Transformer outperforms NODE and the gap between ResNet and NODE is significantly reduced. Nevertheless, NODE remains a prominent solution among tree-based approaches.
+
+Table 3: Results for ensembles of DL models with the highest ranks (see Table 2). For each model-dataset pair, the metric value averaged over three ensembles is reported. See supplementary for standard deviations. Depending on the dataset, the highest accuracy or the lowest RMSE is in bold. Due to the limited precision, some different values are represented with the same figures. Notation: $\downarrow \sim \mathrm { R M S E }$ , $\uparrow$ \~ accuracy.   
+
+<table><tr><td></td><td>CA↓</td><td>AD↑</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>EP↑</td><td>YE↓</td><td>CO→</td><td>YA↓</td><td>MI↓</td></tr><tr><td>NODE</td><td>0.461</td><td>0.860</td><td>0.361</td><td>0.730</td><td>0.727</td><td>0.921</td><td>0.8970</td><td>8.716</td><td>0.965</td><td>0.750</td><td>0.744</td></tr><tr><td>ResNet</td><td>0.478</td><td>0.857</td><td>0.398</td><td>0.734</td><td>0.731</td><td>0.966</td><td>0.8976</td><td>8.770</td><td>0.967</td><td>0.751</td><td>0.745</td></tr><tr><td>FT-Transformer</td><td>0.448</td><td>0.860</td><td>0.398</td><td>0.739</td><td>0.731</td><td>0.967</td><td>0.8984</td><td>8.751</td><td>0.973</td><td>0.747</td><td>0.743</td></tr></table>
+
+# 4.5 Comparing DL models and GBDT
+
+In this section, our goal is to check whether DL models are conceptually ready to outperform GBDT. To this end, we compare the best possible metric values that one can achieve using GBDT or DL models, without taking speed and hardware requirements into account (undoubtedly, GBDT is a more lightweight solution). We accomplish that by comparing ensembles instead of single models since
+
+GBDT is essentially an ensembling technique and we expect that deep architectures will benefit more from ensembling (Fort et al., 2020). We report the results in Table 4.
+
+Table 4: Results for ensembles of GBDT and the main DL models. For each model-dataset pair, the metric value averaged over three ensembles is reported. See supplementary for standard deviations. Notation follows Table 3.   
+
+<table><tr><td></td><td>CA←</td><td>AD个</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>EP个</td><td>YE↓</td><td>CO ↑</td><td>YA↓</td><td>MI↓</td></tr><tr><td colspan="10">Default hyperparameters</td></tr><tr><td>XGBoost</td><td>0.462 0.874 0.348</td><td></td><td></td><td>0.711</td><td>0.717</td><td>0.924</td><td>0.8799</td><td>9.192</td><td>0.964 0.761</td><td></td><td>0.751</td></tr><tr><td>CatBoost</td><td>0.428 0.873</td><td></td><td>0.386</td><td>0.724</td><td>0.728</td><td>0.948</td><td>0.8893</td><td>8.885</td><td>0.910</td><td>0.749</td><td>0.744</td></tr><tr><td>FT-Transformer 0.454</td><td></td><td>0.860</td><td>0.395</td><td>0.734</td><td>0.731</td><td></td><td>0.966 0.8969 8.727 0.973</td><td></td><td></td><td></td><td>0.747 0.742</td></tr><tr><td colspan="10">Tuned hyperparameters</td></tr><tr><td>XGBoost</td><td>0.431</td><td>0.872</td><td>0.377</td><td>0.724</td><td>0.728</td><td>一</td><td>0.8861</td><td>8.819</td><td>0.969</td><td>0.732</td><td>0.742</td></tr><tr><td>CatBoost</td><td>0.423 0.874</td><td></td><td>0.388</td><td>0.727</td><td>0.729</td><td></td><td>0.8898</td><td>8.837</td><td>0.968</td><td>0.740</td><td>0.741</td></tr><tr><td>ResNet</td><td>0.478</td><td>0.857</td><td>0.398</td><td>0.734</td><td>0.731</td><td>0.966</td><td>0.8976</td><td>8.770</td><td>0.967</td><td>0.751</td><td>0.745</td></tr><tr><td>FT-Transformer</td><td>0.448</td><td>0.860</td><td>0.398</td><td>0.739</td><td>0.731</td><td>0.967</td><td>0.8984</td><td>8.751</td><td>0.973</td><td>0.747</td><td>0.743</td></tr></table>
+
+Default hyperparameters. We start with the default configurations to check the “out-of-the-box” performance, which is an important practical scenario. The default FT-Transformer implies a configuration with all hyperparameters set to some specific values that we provide in supplementary. Table 4 demonstrates that the ensemble of FT-Transformers mostly outperforms the ensembles of GBDT, which is not the case for only two datasets (California Housing, Adult). Interestingly, the ensemble of default FT-Transformers performs quite on par with the ensembles of tuned FT-Transformers.
+
+The main takeaway: FT-Transformer allows building powerful ensembles out of the box.
+
+Tuned hyperparameters. Once hyperparameters are properly tuned, GBDTs start dominating on some datasets (California Housing, Adult, Yahoo; see Table 4). In those cases, the gaps are significant enough to conclude that DL models do not universally outperform GBDT. Importantly, the fact that DL models outperform GBDT on most of the tasks does not mean that DL solutions are “better” in any sense. In fact, it only means that the constructed benchmark is slightly biased towards “DL-friendly” problems. Admittedly, GBDT remains an unsuitable solution to multiclass problems with a large number of classes. Depending on the number of classes, GBDT can demonstrate unsatisfactory performance (Helena) or even be untunable due to extremely slow training (ALOI).
+
+# The main takeaways:
+
+• there is still no universal solution among DL models and GBDT   
+• DL research efforts aimed at surpassing GBDT should focus on datasets where GBDT outperforms state-of-the-art DL solutions. Note that including “DL-friendly” problems is still important to avoid degradation on such problems.
+
+# 4.6 An intriguing property of FT-Transformer
+
+Table 4 tells one more important story. Namely, FT-Transformer delivers most of its advantage over the “conventional” DL model in the form of ResNet exactly on those problems where GBDT is superior to ResNet (California Housing, Adult, Covertype, Yahoo, Microsoft) while performing on par with ResNet on the remaining problems. In other words, FT-Transformer provides competitive performance on all tasks, while GBDT and ResNet perform well only on some subsets of the tasks. This observation may be the evidence that FT-Transformer is a more “universal” model for tabular data problems. We develop this intuition further in section 5.1. Note that the described phenomenon is not related to ensembling and is observed for single models too (see supplementary).
+
+# 5 Analysis
+
+# 5.1 When FT-Transformer is better than ResNet?
+
+In this section, we make the first step towards understanding the difference in behavior between FT-Transformer and ResNet, which was first observed in section 4.6. To achieve that, we design a sequence of synthetic tasks where the difference in performance of the two models gradually changes from negligible to dramatic. Namely, we generate and $\mathit { \Omega } \mathcal { f } x$ objects $\{ x _ { i } \} _ { i = 1 } ^ { n }$ , perform the train-val-test split once and interpolate between two regression targets: $f _ { G B D T }$ , which is supposed to be easier for GBDT and $f _ { D L }$ , which is expected to be easier for ResNet. Formally, for one object:
+
+$$
+x \sim \mathcal { N } ( 0 , I _ { k } ) , \qquad y = \alpha \cdot f _ { G B D T } ( x ) + ( 1 - \alpha ) \cdot f _ { D L } ( x ) .
+$$
+
+where $f _ { G B D T } ( x )$ is an average prediction of 30 randomly constructed decision trees, and $f _ { D L } ( x )$ is an MLP with three randomly initialized hidden layers. Both $f _ { G B D T }$ and $f _ { D L }$ are generated once, i.e. the same functions are applied to all objects (see supplementary for details). The resulting targets are standardized before training. The results are visualized in Figure 3. ResNet and FT-Transformer perform similarly well on the ResNet-friendly tasks and outperform CatBoost on those tasks. However, the ResNet’s relative performance drops significantly when the target becomes more GBDT friendly. By contrast, FT-Transformer yields competitive performance across the whole range of tasks.
+
+The conducted experiment reveals a type of functions that are better approximated by FT-Transformer than by ResNet. Additionally,
+
+![](images/5b076169d8fc7cd8436cac63dd25ffd01fe43409ea925556c824fbd55e8c9eba.jpg)  
+Figure 3: Test RMSE averaged over five seeds (shadows represent std. dev.). One $\alpha$ corresponds to one task; each task has the same set of train, validation and test features, but different targets.
+
+the fact that these functions are based on decision trees correlates with the observations in section 4.6 and the results in Table 4, where FT-Transformer shows the most convincing improvements over ResNet exactly on those datasets where GBDT outperforms ResNet.
+
+# 5.2 Ablation study
+
+In this section, we test some design choices of FT-Transformer.
+
+First, we compare FT-Transformer with AutoInt (Song et al., 2019), since it is the closest competitor in its spirit. AutoInt also converts all features to embeddings and applies self-attention on top of them. However, in its details, AutoInt significantly differs from FT-Transformer: its embedding layer does not include feature biases, its backbone significantly differs from the vanilla Transformer (Vaswani et al., 2017), and the inference mechanism does not use the [CLS] token.
+
+Second, we check whether feature biases in Feature Tokenizer are essential for good performance.
+
+We tune and evaluate FT-Transformer without feature biases following the same protocol as in section 4.3 and reuse the remaining numbers from Table 2. The results averaged over 15 runs are reported in Table 5 and demonstrate both the superiority of the Transformer’s backbone to that of AutoInt and the necessity of feature biases.
+
+Table 5: The results of the comparison between FT-Transformer and two attention-based alternatives: AutoInt and FT-Transformer without feature biases. Notation follows Table 2.   
+
+<table><tr><td></td><td>CA↓</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>YE↓</td><td>Co↑</td><td>MI↓</td></tr><tr><td>AutoInt</td><td>0.474</td><td>0.372</td><td>0.721</td><td>0.725</td><td>0.945</td><td>8.882</td><td>0.934</td><td>0.750</td></tr><tr><td>FT-Transformer (w/o feature biases)</td><td>0.470</td><td>0.381</td><td>0.724</td><td>0.727</td><td>0.958</td><td>8.843</td><td>0.964</td><td>0.751</td></tr><tr><td>FT-Transformer</td><td>0.459</td><td>0.391</td><td>0.732</td><td>0.729</td><td>0.960</td><td>8.855</td><td>0.970</td><td>0.746</td></tr></table>
+
+# 5.3 Obtaining feature importances from attention maps
+
+In this section, we evaluate attention maps as a source of information on feature importances for FT-Transformer for a given set of samples. For the $i$ -th sample, we calculate the average attention map $p _ { i }$ for the [CLS] token from Transformer’s forward pass. Then, the obtained individual distributions are averaged into one distribution $p$ that represents the feature importances:
+
+$$
+p = \frac { 1 } { n _ { s a m p l e s } } \sum _ { i } p _ { i } \qquad p _ { i } = \frac { 1 } { n _ { h e a d s } \times L } \sum _ { h , l } p _ { i h l } .
+$$
+
+where $p _ { i h l }$ is the $h$ -th head’s attention map for the [CLS] token from the forward pass of the $l$ -th layer on the $i$ -th sample. The main advantage of the described heuristic technique is its efficiency: it requires a single forward for one sample.
+
+In order to evaluate our approach, we compare it with Integrated Gradients (IG, Sundararajan et al. (2017)), a general technique applicable to any differentiable model. We use permutation test (PT, Breiman (2001)) as a reasonable interpretable method that allows us to establish a constructive metric, namely, rank correlation. We run all the methods on the train set and summarize results in Table 6. Interestingly, the proposed method yields reasonable feature importances and performs similarly to IG (note that this does not imply similarity to IG’s feature importances). Given that IG can be orders of magnitude slower and the “baseline” in the form of PT requires $( n _ { f e a t u r e s } + 1 )$ forward passes (versus one for the proposed method), we conclude that the simple averaging of attention maps can be a good choice in terms of cost-effectiveness.
+
+Table 6: Rank correlation (takes values in $[ - 1 , \ 1 ] ,$ ) between permutation test’s feature importances ranking and two alternative rankings: Attention Maps (AM) and Integrated Gradients (IG). Means and standard deviations over five runs are reported.   
+
+<table><tr><td>CA</td><td></td><td>HE</td><td>JA</td><td>HI</td><td>AL</td><td>YE</td><td>Co</td><td>MI</td></tr><tr><td>AM 0.81 (0.05)</td><td></td><td>0.77 (0.03)</td><td>0.78 (0.05)</td><td>0.91 (0.03)</td><td>0.84 (0.01)</td><td>0.92 (0.01)</td><td>0.84 (0.04)</td><td>0.86 (0.02)</td></tr><tr><td>IG 0.84 (0.08)</td><td></td><td>0.74 (0.03)</td><td>0.75 (0.04)</td><td>0.72 (0.03)</td><td>0.89 (0.01)</td><td>0.50 (0.03)</td><td>0.90 (0.02)</td><td>0.56 (0.02)</td></tr></table>
+
+# 6 Conclusion
+
+In this work, we have investigated the status quo in the field of deep learning for tabular data and improved the state of baselines in tabular DL. First, we have demonstrated that a simple ResNet-like architecture can serve as an effective baseline. Second, we have proposed FT-Transformer — a simple adaptation of the Transformer architecture that outperforms other DL solutions on most of the tasks. We have also compared the new baselines with GBDT and demonstrated that GBDT still dominates on some tasks. The code and all the details of the study are open-sourced 1, and we hope that our evaluation and two simple models (ResNet and FT-Transformer) will serve as a basis for further developments on tabular DL.
+
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parse/train/i_Q1yrOegLY/i_Q1yrOegLY_content_list.json

@@ -0,0 +1,1512 @@
+[
+    {
+        "type": "text",
+        "text": "Revisiting Deep Learning Models for Tabular Data ",
+        "text_level": 1,
+        "bbox": [
+            189,
+            122,
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+            148
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yury Gorishniy∗†‡ Ivan Rubachev†♣ ",
+        "bbox": [
+            194,
+            199,
+            475,
+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Valentin Khrulkov† Artem Babenko†♣ ",
+        "bbox": [
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+            199,
+            800,
+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yandex, Russia Moscow Institute of Physics and Technology, Russia National Research University Higher School of Economics, Russia ",
+        "bbox": [
+            281,
+            228,
+            736,
+            270
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            286,
+            535,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The existing literature on deep learning for tabular data proposes a wide range of novel architectures and reports competitive results on various datasets. However, the proposed models are usually not properly compared to each other and existing works often use different benchmarks and experiment protocols. As a result, it is unclear for both researchers and practitioners what models perform best. Additionally, the field still lacks effective baselines, that is, the easy-to-use models that provide competitive performance across different problems. ",
+        "bbox": [
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+            316,
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+            414
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we perform an overview of the main families of DL architectures for tabular data and raise the bar of baselines in tabular DL by identifying two simple and powerful deep architectures. The first one is a ResNet-like architecture which turns out to be a strong baseline that is often missing in prior works. The second model is our simple adaptation of the Transformer architecture for tabular data, which outperforms other solutions on most tasks. Both models are compared to many existing architectures on a diverse set of tasks under the same training and tuning protocols. We also compare the best DL models with Gradient Boosted Decision Trees and conclude that there is still no universally superior solution. The source code is available at https://github.com/yandex-research/rtdl. ",
+        "bbox": [
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+            416,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            579,
+            310,
+            595
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Due to the tremendous success of deep learning on such data domains as images, audio and texts (Goodfellow et al., 2016), there has been a lot of research interest to extend this success to problems with data stored in tabular format. In these problems, data points are represented as vectors of heterogeneous features, which is typical for industrial applications and ML competitions, where neural networks have a strong non-deep competitor in the form of GBDT (Chen and Guestrin, 2016; Ke et al., 2017; Prokhorenkova et al., 2018). Along with potentially higher performance, using deep learning for tabular data is appealing as it would allow constructing multi-modal pipelines for problems, where only one part of the input is tabular, and other parts include images, audio and other DL-friendly data. Such pipelines can then be trained end-to-end by gradient optimization for all modalities. For these reasons, a large number of DL solutions were recently proposed, and new models continue to emerge (Arik and Pfister, 2020; Badirli et al., 2020; Hazimeh et al., 2020; Huang et al., 2020; Klambauer et al., 2017; Popov et al., 2020; Song et al., 2019; Wang et al., 2017, 2020). ",
+        "bbox": [
+            174,
+            609,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, due to the lack of established benchmarks (such as ImageNet (Deng et al., 2009) for computer vision or GLUE (Wang et al., 2019a) for NLP), existing papers use different datasets for evaluation and proposed DL models are often not adequately compared to each other. Therefore, from the current literature, it is unclear what DL model generally performs better than others and whether GBDT is surpassed by DL models. Additionally, despite the large number of novel architectures, the field still lacks simple and reliable solutions that allow achieving competitive performance with moderate effort and provide stable performance across many tasks. In that regard, Multilayer ",
+        "bbox": [
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+            781,
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+            878
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Perceptron (MLP) remains the main simple baseline for the field, however, it does not always represent a significant challenge for other competitors. ",
+        "bbox": [
+            173,
+            90,
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+            119
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The described problems impede the research process and make the observations from the papers not conclusive enough. Therefore, we believe it is timely to review the recent developments from the field and raise the bar of baselines in tabular DL. We start with a hypothesis that well-studied DL architecture blocks may be underexplored in the context of tabular data and may be used to design better baselines. Thus, we take inspiration from well-known battle-tested architectures from other fields and obtain two simple models for tabular data. The first one is a ResNet-like architecture (He et al., 2015) and the second one is FT-Transformer — our simple adaptation of the Transformer architecture (Vaswani et al., 2017) for tabular data. Then, we compare these models with many existing solutions on a diverse set of tasks under the same protocols of training and hyperparameters tuning. First, we reveal that none of the considered DL models can consistently outperform the ResNet-like model. Given its simplicity, it can serve as a strong baseline for future work. Second, FT-Transformer demonstrates the best performance on most tasks and becomes a new powerful solution for the field. Interestingly, FT-Transformer turns out to be a more universal architecture for tabular data: it performs well on a wider range of tasks than the more “conventional” ResNet and other DL models. Finally, we compare the best DL models to GBDT and conclude that there is still no universally superior solution. ",
+        "bbox": [
+            174,
+            126,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We summarize the contributions of our paper as follows: ",
+        "bbox": [
+            173,
+            353,
+            544,
+            367
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We thoroughly evaluate the main models for tabular DL on a diverse set of tasks to investigate their relative performance.   \n2. We demonstrate that a simple ResNet-like architecture is an effective baseline for tabular DL, which was overlooked by existing literature. Given its simplicity, we recommend this baseline for comparison in future tabular DL works.   \n3. We introduce FT-Transformer — a simple adaptation of the Transformer architecture for tabular data that becomes a new powerful solution for the field. We observe that it is a more universal architecture: it performs well on a wider range of tasks than other DL models.   \n4. We reveal that there is still no universally superior solution among GBDT and deep models. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            541,
+            316,
+            558
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The “shallow” state-of-the-art for problems with tabular data is currently ensembles of decision trees, such as GBDT (Gradient Boosting Decision Tree) (Friedman, 2001), which are typically the top-choice in various ML competitions. At the moment, there are several established GBDT libraries, such as XGBoost (Chen and Guestrin, 2016), LightGBM (Ke et al., 2017), CatBoost (Prokhorenkova et al., 2018), which are widely used by both ML researchers and practitioners. While these implementations vary in detail, on most of the tasks, their performances do not differ much (Prokhorenkova et al., 2018). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "During several recent years, a large number of deep learning models for tabular data have been developed (Arik and Pfister, 2020; Badirli et al., 2020; Hazimeh et al., 2020; Huang et al., 2020; Klambauer et al., 2017; Popov et al., 2020; Song et al., 2019; Wang et al., 2017). Most of these models can be roughly categorized into three groups, which we briefly describe below. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Differentiable trees. The first group of models is motivated by the strong performance of decision tree ensembles for tabular data. Since decision trees are not differentiable and do not allow gradient optimization, they cannot be used as a component for pipelines trained in the end-to-end fashion. To address this issue, several works (Hazimeh et al., 2020; Kontschieder et al., 2015; Popov et al., 2020; Yang et al., 2018) propose to “smooth” decision functions in the internal tree nodes to make the overall tree function and tree routing differentiable. While the methods of this family can outperform GBDT on some tasks (Popov et al., 2020), in our experiments, they do not consistently outperform ResNet. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Attention-based models. Due to the ubiquitous success of attention-based architectures for different domains (Dosovitskiy et al., 2021; Vaswani et al., 2017), several authors propose to employ attentionlike modules for tabular DL as well (Arik and Pfister, 2020; Huang et al., 2020; Song et al., 2019). In our experiments, we show that the properly tuned ResNet outperforms the existing attention-based models. Nevertheless, we identify an effective way to apply the Transformer architecture (Vaswani et al., 2017) to tabular data: the resulting architecture outperforms ResNet on most of the tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            119
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Explicit modeling of multiplicative interactions. In the literature on recommender systems and click-through-rate prediction, several works criticize MLP since it is unsuitable for modeling multiplicative interactions between features (Beutel et al., 2018; Qin et al., 2021; Wang et al., 2017). Inspired by this motivation, some works (Beutel et al., 2018; Wang et al., 2017, 2020) have proposed different ways to incorporate feature products into MLP. In our experiments, however, we do not find such methods to be superior to properly tuned baselines. ",
+        "bbox": [
+            173,
+            126,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The literature also proposes some other architectural designs (Badirli et al., 2020; Klambauer et al., 2017) that cannot be explicitly assigned to any of the groups above. Overall, the community has developed a variety of models that are evaluated on different benchmarks and are rarely compared to each other. Our work aims to establish a fair comparison of them and identify the solutions that consistently provide high performance. ",
+        "bbox": [
+            174,
+            215,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Models for tabular data problems ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            311,
+            486,
+            329
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe the main deep architectures that we highlight in our work, as well as the existing solutions included in the comparison. Since we argue that the field needs strong easy-to-use baselines, we try to reuse well-established DL building blocks as much as possible when designing ResNet (section 3.2) and FT-Transformer (section 3.3). We hope this approach will result in conceptually familiar models that require less effort to achieve good performance. Additional discussion and technical details for all the models are provided in supplementary. ",
+        "bbox": [
+            173,
+            343,
+            825,
+            426
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Notation. In this work, we consider supervised learning problems. $D { = } \\{ ( x _ { i } , ~ y _ { i } ) \\} _ { i { = } 1 } ^ { n }$ denotes a dataset, where $x _ { i } = ( x _ { i } ^ { ( n u m ) } , x _ { i } ^ { ( c a t ) } ) \\in \\mathbb { X }$ )) ∈ X represents numerical x(nuij $x _ { i j } ^ { ( n u m ) }$ and categorical $x _ { i j } ^ { ( c a t ) }$ features of an object and $y _ { i } \\in \\mathbb { Y }$ denotes the corresponding object label. The total number of features is denoted as $k$ . The dataset is split into three disjoint subsets: $D = D _ { t r a i n } \\cup D _ { v a l } \\cup D _ { t e s t }$ , where $D _ { t r a i n }$ is used for training, $D _ { v a l }$ is used for early stopping and hyperparameter tuning, and $D _ { t e s t }$ is used for the final evaluation. We consider three types of tasks: binary classification $\\mathbb { Y } = \\{ 0 , 1 \\}$ , multiclass classification $\\mathbb { Y } = \\{ 1 , \\ . . . , C \\}$ and regression $\\mathbb { Y } = \\mathbb { R }$ . ",
+        "bbox": [
+            173,
+            433,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 MLP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            558,
+            250,
+            573
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We formalize the “MLP” architecture in Equation 1. ",
+        "bbox": [
+            174,
+            584,
+            514,
+            599
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/70035fc4f2b2f333456a3079ca16a07e9b529e54f179bc531a36e96f705c885c.jpg",
+        "text": "$$\n\\begin{array} { c } { \\mathtt { M L P } ( x ) = \\mathtt { L i n e a r } \\left( \\mathtt { M L P B l o c k } \\left( \\dots \\left( \\mathtt { M L P B l o c k } ( x ) \\right) \\right) \\right) } \\\\ { \\mathtt { M L P B l o c k } ( x ) = \\mathtt { D r o p o u t } \\left( \\mathtt { R e L U } \\left( \\mathtt { L i n e a r } ( x ) \\right) \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 ResNet ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            684,
+            263,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We are aware of one attempt to design a ResNet-like baseline (Klambauer et al., 2017) where the reported results were not competitive. However, given ResNet’s success story in computer vision (He et al., 2015) and its recent achievements on NLP tasks (Sun and Iyyer, 2021), we give it a second try and construct a simple variation of ResNet as described in Equation 2. The main building block is simplified compared to the original architecture, and there is an almost clear path from the input to output which we find to be beneficial for the optimization. Overall, we expect this architecture to outperform MLP on tasks where deeper representations can be helpful. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "ResNet $( x ) =$ Prediction (ResNetBlock (. . . (ResNetBlock (Linear(x))))) ResNetBlock(x) = x + Dropout(Linear(Dropout(ReLU(Linear(BatchNorm(x)))))) Prediction $( x ) =$ Linear (ReLU (BatchNorm (x))) ",
+        "bbox": [
+            194,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3 FT-Transformer ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            326,
+            106
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce FT-Transformer (Feature Tokenizer $^ +$ Transformer) — a simple adaptation of the Transformer architecture (Vaswani et al., 2017) for the tabular domain. Figure 1 demonstrates the main parts of FT-Transformer. In a nutshell, our model transforms all features (categorical and numerical) to embeddings and applies a stack of Transformer layers to the embeddings. Thus, every Transformer layer operates on the feature level of one object. We compare FT-Transformer to conceptually similar AutoInt in section 5.2. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/df9999ef23ac55d1399747a5900f3eac717d0c6a4d8ac680caf051f5bf2bddda.jpg",
+        "image_caption": [
+            "Figure 1: The FT-Transformer architecture. Firstly, Feature Tokenizer transforms features to embeddings. The embeddings are then processed by the Transformer module and the final representation of the [CLS] token is used for prediction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/a59f8fc74fd228878c534e25ebbe4e2ef3f34dca0d1a9f7e4aafb3653f67b0ac.jpg",
+        "image_caption": [
+            "Figure 2: (a) Feature Tokenizer; in the example, there are three numerical and two categorical features; (b) One Transformer layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            378,
+            794,
+            611
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Feature Tokenizer. The Feature Tokenizer module (see Figure 2) transforms the input features $x$ to embeddings $T \\in \\mathbb { R } ^ { k \\times d }$ . The embedding for a given feature $x _ { j }$ is computed as follows: ",
+        "bbox": [
+            173,
+            659,
+            825,
+            689
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0dbd4df6ae5a290152d66e9297ea86d18473f700be95e1e3110ef70cbcaef4ae.jpg",
+        "text": "$$\nT _ { j } = b _ { j } + f _ { j } ( x _ { j } ) \\in \\mathbb { R } ^ { d } \\qquad f _ { j } : \\mathbb { X } _ { j } \\to \\mathbb { R } ^ { d } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            691,
+            642,
+            712
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $b _ { j }$ is the $j$ -th feature bias, $f _ { j } ^ { ( n u m ) }$ is implemented as the element-wise multiplication with the vector $W _ { j _ { \\lambda } } ^ { ( n u m ) } \\in \\mathbb { R } ^ { d }$ and $f _ { j } ^ { ( c a t ) }$ is implemented as the lookup table $W _ { j } ^ { ( c a t ) } \\in \\mathbb { R } ^ { S _ { j } \\times d }$ for categorical features. Overall: ",
+        "bbox": [
+            176,
+            715,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e8123ce902bf5134b8163ff012a1a2b3bb49c56140e3bf5cd876e98958b0eaa.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { T _ { j } ^ { ( n u m ) } = b _ { j } ^ { ( n u m ) } + x _ { j } ^ { ( n u m ) } \\cdot W _ { j } ^ { ( n u m ) } } & { \\in \\mathbb { R } ^ { d } , } \\\\ & { T _ { j } ^ { ( c a t ) } = b _ { j } ^ { ( c a t ) } + e _ { j } ^ { T } W _ { j } ^ { ( c a t ) } } & { \\in \\mathbb { R } ^ { d } , } \\\\ & { T = \\mathsf { s t a c k } \\left[ T _ { 1 } ^ { ( n u m ) } , \\ldots , T _ { k ^ { ( n u m ) } } ^ { ( n u m ) } , T _ { 1 } ^ { ( c a t ) } , \\ldots , T _ { k ^ { ( c a t ) } } ^ { ( c a t ) } \\right] \\in \\mathbb { R } ^ { k \\times d } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            763,
+            722,
+            840
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $e _ { j } ^ { T }$ is a one-hot vector for the corresponding categorical feature. ",
+        "bbox": [
+            173,
+            843,
+            635,
+            859
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Transformer. At this stage, the embedding of the [CLS] token (or “classification token”, or “output token”, see Devlin et al. (2019)) is appended to $T$ and $L$ Transformer layers $F _ { 1 }$ , . . . , $F _ { L }$ are applied: ",
+        "bbox": [
+            171,
+            863,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1bedecfcb293d601e2d8cbf26c882d109591dd1cb861d5ab5e1e762cbdaf8ff5.jpg",
+        "text": "$$\nT _ { 0 } = { \\tt s t a c k } \\left[ \\left[ { \\tt C L S } \\right] , T \\right] \\qquad T _ { i } = F _ { i } ( T _ { i - 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            896,
+            648,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use the PreNorm variant for easier optimization (Wang et al., 2019b), see Figure 2. In the PreNorm setting, we also found it to be necessary to remove the first normalization from the first Transformer layer to achieve good performance. See the original paper (Vaswani et al., 2017) for the background on Multi-Head Self-Attention (MHSA) and the Feed Forward module. See supplementary for details such as activations, placement of normalizations and dropout modules (Srivastava et al., 2014). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Prediction. The final representation of the [CLS] token is used for prediction: ",
+        "bbox": [
+            173,
+            166,
+            687,
+            181
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/606afbc00e4d2c121cf02ebfa3526300a73cebe517d223df148b626fcfa4af10.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { y } = \\mathtt { L i n e a r } \\big ( \\mathtt { R e L U } \\big ( \\mathtt { L a y e r N o r m } ( T _ { L } ^ { \\mathtt { L C L S } } ) \\big ) \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            638,
+            205
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Limitations. FT-Transformer requires more resources (both hardware and time) for training than simple models such as ResNet and may not be easily scaled to datasets when the number of features is “too large” (it is determined by the available hardware and time budget). Consequently, widespread usage of FT-Transformer for solving tabular data problems can lead to greater CO2 emissions produced by ML pipelines, since tabular data problems are ubiquitous. The main cause of the described problem lies in the quadratic complexity of the vanilla MHSA with respect to the number of features. However, the issue can be alleviated by using efficient approximations of MHSA (Tay et al., 2020). Additionally, it is still possible to distill FT-Transformer into simpler architectures for better inference performance. We report training times and the used hardware in supplementary. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 Other models ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we list the existing models designed specifically for tabular data that we include in the comparison. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• SNN (Klambauer et al., 2017). An MLP-like architecture with the SELU activation that enables training deeper models.   \n• NODE (Popov et al., 2020). A differentiable ensemble of oblivious decision trees.   \n• TabNet (Arik and Pfister, 2020). A recurrent architecture that alternates dynamical reweighing of features and conventional feed-forward modules. GrowNet (Badirli et al., 2020). Gradient boosted weak MLPs. The official implementation supports only classification and regression problems. DCN V2 (Wang et al., 2020). Consists of an MLP-like module and the feature crossing module (a combination of linear layers and multiplications).   \n• AutoInt (Song et al., 2019). Transforms features to embeddings and applies a series of attention-based transformations to the embeddings.   \n• XGBoost (Chen and Guestrin, 2016). One of the most popular GBDT implementations.   \n• CatBoost (Prokhorenkova et al., 2018). GBDT implementation that uses oblivious decision trees (Lou and Obukhov, 2017) as weak learners. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare DL models to each other as well as to GBDT. Note that in the main text, we report only the key results. In supplementary, we provide: (1) the results for all models on all datasets; (2) information on hardware; (3) training times for ResNet and FT-Transformer. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 Scope of the comparison ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In our work, we focus on the relative performance of different architectures and do not employ various model-agnostic DL practices, such as pretraining, additional loss functions, data augmentation, distillation, learning rate warmup, learning rate decay and many others. While these practices can potentially improve the performance, our goal is to evaluate the impact of inductive biases imposed by the different model architectures. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Datasets ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use a diverse set of eleven public datasets (see supplementary for the detailed description). For each dataset, there is exactly one train-validation-test split, so all algorithms use the same splits. The datasets include: California Housing (CA, real estate data, Kelley Pace and Barry (1997)), Adult (AD, income estimation, Kohavi (1996)), Helena (HE, anonymized dataset, Guyon et al. (2019)), ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Jannis (JA, anonymized dataset, Guyon et al. (2019)), Higgs (HI, simulated physical particles, Baldi et al. (2014); we use the version with 98K samples available at the OpenML repository (Vanschoren et al., 2014)), ALOI (AL, images, Geusebroek et al. (2005)), Epsilon (EP, simulated physics experiments), Year (YE, audio features, Bertin-Mahieux et al. (2011)), Covertype (CO, forest characteristics, Blackard and Dean. (2000)), Yahoo (YA, search queries, Chapelle and Chang (2011)), Microsoft (MI, search queries, Qin and Liu (2013)). We follow the pointwise approach to learning-to-rank and treat ranking problems (Microsoft, Yahoo) as regression problems. The dataset properties are summarized in Table 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/8afe97a77af1cca0053e4d079315d2534903478433496e006956ce35ba3ac9dd.jpg",
+        "table_caption": [
+            "Table 1: Dataset properties. Notation: “RMSE” $\\sim$ root-mean-square error, “Acc.” $\\sim$ accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CA</td><td>AD</td><td>HE</td><td>JA</td><td>HI</td><td>AL</td><td>EP</td><td>YE</td><td>Co</td><td>YA</td><td>MI</td></tr><tr><td>#objects</td><td>20640</td><td>48842</td><td>65196</td><td>83733</td><td>98050</td><td>108000</td><td>500000</td><td>515345</td><td>581012</td><td>709877</td><td>1200192</td></tr><tr><td>#num. features #cat. features</td><td>8</td><td>6</td><td>27</td><td>54</td><td>28</td><td>128</td><td>2000</td><td>90</td><td>54</td><td>699</td><td>136</td></tr><tr><td>metric</td><td>0</td><td>8</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td>RMSE</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>RMSE</td><td>Acc.</td><td>RMSE</td><td>RMSE</td></tr><tr><td>#classes</td><td>1</td><td>2</td><td>100</td><td>4</td><td>2</td><td>1000</td><td>2</td><td>1</td><td>7</td><td>1</td><td>1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Implementation details ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Data preprocessing. Data preprocessing is known to be vital for DL models. For each dataset, the same preprocessing was used for all deep models for a fair comparison. By default, we used the quantile transformation from the Scikit-learn library (Pedregosa et al., 2011). We apply standardization (mean subtraction and scaling) to Helena and ALOI. The latter one represents image data, and standardization is a common practice in computer vision. On the Epsilon dataset, we observed preprocessing to be detrimental to deep models’ performance, so we use the raw features on this dataset. We apply standardization to regression targets for all algorithms. ",
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+    },
+    {
+        "type": "text",
+        "text": "Tuning. For every dataset, we carefully tune each model’s hyperparameters. The best hyperparameters are the ones that perform best on the validation set, so the test set is never used for tuning. For most algorithms, we use the Optuna library (Akiba et al., 2019) to run Bayesian optimization (the Tree-Structured Parzen Estimator algorithm), which is reported to be superior to random search (Turner et al., 2021). For the rest, we iterate over predefined sets of configurations recommended by corresponding papers. We provide parameter spaces and grids in supplementary. We set the budget for Optuna-based tuning in terms of iterations and provide additional analysis on setting the budget in terms of time in supplementary. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Evaluation. For each tuned configuration, we run 15 experiments with different random seeds and report the performance on the test set. For some algorithms, we also report the performance of default configurations without hyperparameter tuning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ensembles. For each model, on each dataset, we obtain three ensembles by splitting the 15 single models into three disjoint groups of equal size and averaging predictions of single models within each group. ",
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Neural networks. We minimize cross-entropy for classification problems and mean squared error for regression problems. For TabNet and GrowNet, we follow the original implementations and use the Adam optimizer (Kingma and Ba, 2017). For all other algorithms, we use the AdamW optimizer (Loshchilov and Hutter, 2019). We do not apply learning rate schedules. For each dataset, we use a predefined batch size for all algorithms unless special instructions on batch sizes are given in the corresponding papers (see supplementary). We continue training until there are patience $+ 1$ consecutive epochs without improvements on the validation set; we set patience $= 1 6$ for all algorithms. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Categorical features. For XGBoost, we use one-hot encoding. For CatBoost, we employ the built-in support for categorical features. For Neural Networks, we use embeddings of the same dimensionality for all categorical features. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/35cb8aa9ff784351b356f365e5c0357c6a6b562707bc3486b1d1851c2946a5fb.jpg",
+        "table_caption": [
+            "Table 2: Results for DL models. The metric values averaged over 15 random seeds are reported. See supplementary for standard deviations. For each dataset, top results are in bold. “Top” means “the gap between this result and the result with the best score is not statistically significant”. For each dataset, ranks are calculated by sorting the reported scores; the “rank” column reports the average rank across all datasets. Notation: FT-T \\~ FT-Transformer, $\\downarrow \\sim \\mathrm { R M S E }$ , $\\uparrow$ \\~ accuracy "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CA↓</td><td>AD↑</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>EP↑</td><td>YE↓</td><td>CO↑</td><td>YA↓</td><td>MI↓</td><td>rank (std)</td></tr><tr><td>TabNet</td><td>0.510</td><td>0.850</td><td>0.378</td><td>0.723</td><td>0.719</td><td>0.954</td><td>0.8896</td><td>8.909</td><td>0.957</td><td>0.823</td><td>0.751</td><td>7.5 (2.0)</td></tr><tr><td>SNN</td><td>0.493</td><td>0.854</td><td>0.373</td><td>0.719</td><td>0.722</td><td>0.954</td><td>0.8975</td><td>8.895</td><td>0.961</td><td>0.761</td><td>0.751</td><td>6.4 (1.4)</td></tr><tr><td>AutoInt</td><td>0.474</td><td>0.859</td><td>0.372</td><td>0.721</td><td>0.725</td><td>0.945</td><td>0.8949</td><td>8.882</td><td>0.934</td><td>0.768</td><td>0.750</td><td>5.7 (2.3)</td></tr><tr><td>GrowNet</td><td>0.487</td><td>0.857</td><td></td><td></td><td>0.722</td><td>1</td><td>0.8970</td><td>8.827</td><td>1</td><td>0.765</td><td>0.751</td><td>5.7 (2.2)</td></tr><tr><td>MLP</td><td>0.499</td><td>0.852</td><td>0.383</td><td>0.719</td><td>0.723</td><td>0.954</td><td>0.8977</td><td>8.853</td><td>0.962</td><td>0.757</td><td>0.747</td><td>4.8 (1.9)</td></tr><tr><td>DCN2</td><td>0.484</td><td>0.853</td><td>0.385</td><td>0.716</td><td>0.723</td><td>0.955</td><td>0.8977</td><td>8.890</td><td>0.965</td><td>0.757</td><td>0.749</td><td>4.7 (2.0)</td></tr><tr><td>NODE</td><td>0.464</td><td>0.858</td><td>0.359</td><td>0.727</td><td>0.726</td><td>0.918</td><td>0.8958</td><td>8.784</td><td>0.958</td><td>0.753</td><td>0.745</td><td>3.9 (2.8)</td></tr><tr><td>ResNet</td><td>0.486</td><td>0.854</td><td>0.396</td><td>0.728</td><td>0.727</td><td>0.963</td><td>0.8969</td><td>8.846</td><td>0.964</td><td>0.757</td><td>0.748</td><td>3.3 (1.8)</td></tr><tr><td>FT-T</td><td></td><td>0.459 0.859</td><td>0.391</td><td>0.732</td><td>0.729</td><td>0.960</td><td>0.8982</td><td>8.855</td><td>0.970</td><td>0.756</td><td>0.746</td><td>1.8 (1.2)</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.4 Comparing DL models ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2 reports the results for deep architectures. ",
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+    },
+    {
+        "type": "text",
+        "text": "The main takeaways: ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "• MLP is still a good sanity check   \n• ResNet turns out to be an effective baseline that none of the competitors can consistently outperform.   \n• FT-Transformer performs best on most tasks and becomes a new powerful solution for the field.   \n• Tuning makes simple models such as MLP and ResNet competitive, so we recommend tuning baselines when possible. Luckily, today, it is more approachable with libraries such as Optuna (Akiba et al., 2019). ",
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+    },
+    {
+        "type": "text",
+        "text": "Among other models, NODE (Popov et al., 2020) is the only one that demonstrates high performance on several tasks. However, it is still inferior to ResNet on six datasets (Helena, Jannis, Higgs, ALOI, Epsilon, Covertype), while being a more complex solution. Moreover, it is not a truly “single” model; in fact, it often contains significantly more parameters than ResNet and FT-Transformer and has an ensemble-like structure. We illustrate that by comparing ensembles in Table 3. The results indicate that FT-Transformer and ResNet benefit more from ensembling; in this regime, FT-Transformer outperforms NODE and the gap between ResNet and NODE is significantly reduced. Nevertheless, NODE remains a prominent solution among tree-based approaches. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f924d58e3a8b16072b931f4d288c264254c46582bd76358b6b17d9ea6db6539a.jpg",
+        "table_caption": [
+            "Table 3: Results for ensembles of DL models with the highest ranks (see Table 2). For each model-dataset pair, the metric value averaged over three ensembles is reported. See supplementary for standard deviations. Depending on the dataset, the highest accuracy or the lowest RMSE is in bold. Due to the limited precision, some different values are represented with the same figures. Notation: $\\downarrow \\sim \\mathrm { R M S E }$ , $\\uparrow$ \\~ accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CA↓</td><td>AD↑</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>EP↑</td><td>YE↓</td><td>CO→</td><td>YA↓</td><td>MI↓</td></tr><tr><td>NODE</td><td>0.461</td><td>0.860</td><td>0.361</td><td>0.730</td><td>0.727</td><td>0.921</td><td>0.8970</td><td>8.716</td><td>0.965</td><td>0.750</td><td>0.744</td></tr><tr><td>ResNet</td><td>0.478</td><td>0.857</td><td>0.398</td><td>0.734</td><td>0.731</td><td>0.966</td><td>0.8976</td><td>8.770</td><td>0.967</td><td>0.751</td><td>0.745</td></tr><tr><td>FT-Transformer</td><td>0.448</td><td>0.860</td><td>0.398</td><td>0.739</td><td>0.731</td><td>0.967</td><td>0.8984</td><td>8.751</td><td>0.973</td><td>0.747</td><td>0.743</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 Comparing DL models and GBDT ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, our goal is to check whether DL models are conceptually ready to outperform GBDT. To this end, we compare the best possible metric values that one can achieve using GBDT or DL models, without taking speed and hardware requirements into account (undoubtedly, GBDT is a more lightweight solution). We accomplish that by comparing ensembles instead of single models since ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "GBDT is essentially an ensembling technique and we expect that deep architectures will benefit more from ensembling (Fort et al., 2020). We report the results in Table 4. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/9f5b0b1b904de621fabc87e7d7c2473bc157673b6a90523da40dd19da774fc08.jpg",
+        "table_caption": [
+            "Table 4: Results for ensembles of GBDT and the main DL models. For each model-dataset pair, the metric value averaged over three ensembles is reported. See supplementary for standard deviations. Notation follows Table 3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CA←</td><td>AD个</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>EP个</td><td>YE↓</td><td>CO ↑</td><td>YA↓</td><td>MI↓</td></tr><tr><td colspan=\"10\">Default hyperparameters</td></tr><tr><td>XGBoost</td><td>0.462 0.874 0.348</td><td></td><td></td><td>0.711</td><td>0.717</td><td>0.924</td><td>0.8799</td><td>9.192</td><td>0.964 0.761</td><td></td><td>0.751</td></tr><tr><td>CatBoost</td><td>0.428 0.873</td><td></td><td>0.386</td><td>0.724</td><td>0.728</td><td>0.948</td><td>0.8893</td><td>8.885</td><td>0.910</td><td>0.749</td><td>0.744</td></tr><tr><td>FT-Transformer 0.454</td><td></td><td>0.860</td><td>0.395</td><td>0.734</td><td>0.731</td><td></td><td>0.966 0.8969 8.727 0.973</td><td></td><td></td><td></td><td>0.747 0.742</td></tr><tr><td colspan=\"10\">Tuned hyperparameters</td></tr><tr><td>XGBoost</td><td>0.431</td><td>0.872</td><td>0.377</td><td>0.724</td><td>0.728</td><td>一</td><td>0.8861</td><td>8.819</td><td>0.969</td><td>0.732</td><td>0.742</td></tr><tr><td>CatBoost</td><td>0.423 0.874</td><td></td><td>0.388</td><td>0.727</td><td>0.729</td><td></td><td>0.8898</td><td>8.837</td><td>0.968</td><td>0.740</td><td>0.741</td></tr><tr><td>ResNet</td><td>0.478</td><td>0.857</td><td>0.398</td><td>0.734</td><td>0.731</td><td>0.966</td><td>0.8976</td><td>8.770</td><td>0.967</td><td>0.751</td><td>0.745</td></tr><tr><td>FT-Transformer</td><td>0.448</td><td>0.860</td><td>0.398</td><td>0.739</td><td>0.731</td><td>0.967</td><td>0.8984</td><td>8.751</td><td>0.973</td><td>0.747</td><td>0.743</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Default hyperparameters. We start with the default configurations to check the “out-of-the-box” performance, which is an important practical scenario. The default FT-Transformer implies a configuration with all hyperparameters set to some specific values that we provide in supplementary. Table 4 demonstrates that the ensemble of FT-Transformers mostly outperforms the ensembles of GBDT, which is not the case for only two datasets (California Housing, Adult). Interestingly, the ensemble of default FT-Transformers performs quite on par with the ensembles of tuned FT-Transformers. ",
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+    {
+        "type": "text",
+        "text": "The main takeaway: FT-Transformer allows building powerful ensembles out of the box. ",
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+        "type": "text",
+        "text": "Tuned hyperparameters. Once hyperparameters are properly tuned, GBDTs start dominating on some datasets (California Housing, Adult, Yahoo; see Table 4). In those cases, the gaps are significant enough to conclude that DL models do not universally outperform GBDT. Importantly, the fact that DL models outperform GBDT on most of the tasks does not mean that DL solutions are “better” in any sense. In fact, it only means that the constructed benchmark is slightly biased towards “DL-friendly” problems. Admittedly, GBDT remains an unsuitable solution to multiclass problems with a large number of classes. Depending on the number of classes, GBDT can demonstrate unsatisfactory performance (Helena) or even be untunable due to extremely slow training (ALOI). ",
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+        "type": "text",
+        "text": "The main takeaways: ",
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+        "type": "text",
+        "text": "• there is still no universal solution among DL models and GBDT   \n• DL research efforts aimed at surpassing GBDT should focus on datasets where GBDT outperforms state-of-the-art DL solutions. Note that including “DL-friendly” problems is still important to avoid degradation on such problems. ",
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+    {
+        "type": "text",
+        "text": "4.6 An intriguing property of FT-Transformer ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 4 tells one more important story. Namely, FT-Transformer delivers most of its advantage over the “conventional” DL model in the form of ResNet exactly on those problems where GBDT is superior to ResNet (California Housing, Adult, Covertype, Yahoo, Microsoft) while performing on par with ResNet on the remaining problems. In other words, FT-Transformer provides competitive performance on all tasks, while GBDT and ResNet perform well only on some subsets of the tasks. This observation may be the evidence that FT-Transformer is a more “universal” model for tabular data problems. We develop this intuition further in section 5.1. Note that the described phenomenon is not related to ensembling and is observed for single models too (see supplementary). ",
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+    {
+        "type": "text",
+        "text": "5 Analysis ",
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+        "type": "text",
+        "text": "5.1 When FT-Transformer is better than ResNet? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we make the first step towards understanding the difference in behavior between FT-Transformer and ResNet, which was first observed in section 4.6. To achieve that, we design a sequence of synthetic tasks where the difference in performance of the two models gradually changes from negligible to dramatic. Namely, we generate and $\\mathit { \\Omega } \\mathcal { f } x$ objects $\\{ x _ { i } \\} _ { i = 1 } ^ { n }$ , perform the train-val-test split once and interpolate between two regression targets: $f _ { G B D T }$ , which is supposed to be easier for GBDT and $f _ { D L }$ , which is expected to be easier for ResNet. Formally, for one object: ",
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+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/377b33b8c703ceeaa2f002ba0942b300b2d30b13c7711551225e7dd7fc0441d0.jpg",
+        "text": "$$\nx \\sim \\mathcal { N } ( 0 , I _ { k } ) , \\qquad y = \\alpha \\cdot f _ { G B D T } ( x ) + ( 1 - \\alpha ) \\cdot f _ { D L } ( x ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $f _ { G B D T } ( x )$ is an average prediction of 30 randomly constructed decision trees, and $f _ { D L } ( x )$ is an MLP with three randomly initialized hidden layers. Both $f _ { G B D T }$ and $f _ { D L }$ are generated once, i.e. the same functions are applied to all objects (see supplementary for details). The resulting targets are standardized before training. The results are visualized in Figure 3. ResNet and FT-Transformer perform similarly well on the ResNet-friendly tasks and outperform CatBoost on those tasks. However, the ResNet’s relative performance drops significantly when the target becomes more GBDT friendly. By contrast, FT-Transformer yields competitive performance across the whole range of tasks. ",
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+        "type": "text",
+        "text": "The conducted experiment reveals a type of functions that are better approximated by FT-Transformer than by ResNet. Additionally, ",
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+    {
+        "type": "image",
+        "img_path": "images/5b076169d8fc7cd8436cac63dd25ffd01fe43409ea925556c824fbd55e8c9eba.jpg",
+        "image_caption": [
+            "Figure 3: Test RMSE averaged over five seeds (shadows represent std. dev.). One $\\alpha$ corresponds to one task; each task has the same set of train, validation and test features, but different targets. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "the fact that these functions are based on decision trees correlates with the observations in section 4.6 and the results in Table 4, where FT-Transformer shows the most convincing improvements over ResNet exactly on those datasets where GBDT outperforms ResNet. ",
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+        "type": "text",
+        "text": "5.2 Ablation study ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we test some design choices of FT-Transformer. ",
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+        "type": "text",
+        "text": "First, we compare FT-Transformer with AutoInt (Song et al., 2019), since it is the closest competitor in its spirit. AutoInt also converts all features to embeddings and applies self-attention on top of them. However, in its details, AutoInt significantly differs from FT-Transformer: its embedding layer does not include feature biases, its backbone significantly differs from the vanilla Transformer (Vaswani et al., 2017), and the inference mechanism does not use the [CLS] token. ",
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+    {
+        "type": "text",
+        "text": "Second, we check whether feature biases in Feature Tokenizer are essential for good performance. ",
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+    {
+        "type": "text",
+        "text": "We tune and evaluate FT-Transformer without feature biases following the same protocol as in section 4.3 and reuse the remaining numbers from Table 2. The results averaged over 15 runs are reported in Table 5 and demonstrate both the superiority of the Transformer’s backbone to that of AutoInt and the necessity of feature biases. ",
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+    {
+        "type": "table",
+        "img_path": "images/00f88508dd446efb5c92d7dae9e6288c07d678a50518e352325c73bce2328e69.jpg",
+        "table_caption": [
+            "Table 5: The results of the comparison between FT-Transformer and two attention-based alternatives: AutoInt and FT-Transformer without feature biases. Notation follows Table 2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CA↓</td><td>HE↑</td><td>JA↑</td><td>HI↑</td><td>AL↑</td><td>YE↓</td><td>Co↑</td><td>MI↓</td></tr><tr><td>AutoInt</td><td>0.474</td><td>0.372</td><td>0.721</td><td>0.725</td><td>0.945</td><td>8.882</td><td>0.934</td><td>0.750</td></tr><tr><td>FT-Transformer (w/o feature biases)</td><td>0.470</td><td>0.381</td><td>0.724</td><td>0.727</td><td>0.958</td><td>8.843</td><td>0.964</td><td>0.751</td></tr><tr><td>FT-Transformer</td><td>0.459</td><td>0.391</td><td>0.732</td><td>0.729</td><td>0.960</td><td>8.855</td><td>0.970</td><td>0.746</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.3 Obtaining feature importances from attention maps ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we evaluate attention maps as a source of information on feature importances for FT-Transformer for a given set of samples. For the $i$ -th sample, we calculate the average attention map $p _ { i }$ for the [CLS] token from Transformer’s forward pass. Then, the obtained individual distributions are averaged into one distribution $p$ that represents the feature importances: ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa4510df0122c0a0dda024c74c30f370f5157cf1f60220a44f617c6191c42dca.jpg",
+        "text": "$$\np = \\frac { 1 } { n _ { s a m p l e s } } \\sum _ { i } p _ { i } \\qquad p _ { i } = \\frac { 1 } { n _ { h e a d s } \\times L } \\sum _ { h , l } p _ { i h l } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { i h l }$ is the $h$ -th head’s attention map for the [CLS] token from the forward pass of the $l$ -th layer on the $i$ -th sample. The main advantage of the described heuristic technique is its efficiency: it requires a single forward for one sample. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In order to evaluate our approach, we compare it with Integrated Gradients (IG, Sundararajan et al. (2017)), a general technique applicable to any differentiable model. We use permutation test (PT, Breiman (2001)) as a reasonable interpretable method that allows us to establish a constructive metric, namely, rank correlation. We run all the methods on the train set and summarize results in Table 6. Interestingly, the proposed method yields reasonable feature importances and performs similarly to IG (note that this does not imply similarity to IG’s feature importances). Given that IG can be orders of magnitude slower and the “baseline” in the form of PT requires $( n _ { f e a t u r e s } + 1 )$ forward passes (versus one for the proposed method), we conclude that the simple averaging of attention maps can be a good choice in terms of cost-effectiveness. ",
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+    {
+        "type": "table",
+        "img_path": "images/4aaba7ee3bdbfddc6bdfa51a33e1251bf196faef7e8e4fd7fb5f38df34a35c32.jpg",
+        "table_caption": [
+            "Table 6: Rank correlation (takes values in $[ - 1 , \\ 1 ] ,$ ) between permutation test’s feature importances ranking and two alternative rankings: Attention Maps (AM) and Integrated Gradients (IG). Means and standard deviations over five runs are reported. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CA</td><td></td><td>HE</td><td>JA</td><td>HI</td><td>AL</td><td>YE</td><td>Co</td><td>MI</td></tr><tr><td>AM 0.81 (0.05)</td><td></td><td>0.77 (0.03)</td><td>0.78 (0.05)</td><td>0.91 (0.03)</td><td>0.84 (0.01)</td><td>0.92 (0.01)</td><td>0.84 (0.04)</td><td>0.86 (0.02)</td></tr><tr><td>IG 0.84 (0.08)</td><td></td><td>0.74 (0.03)</td><td>0.75 (0.04)</td><td>0.72 (0.03)</td><td>0.89 (0.01)</td><td>0.50 (0.03)</td><td>0.90 (0.02)</td><td>0.56 (0.02)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we have investigated the status quo in the field of deep learning for tabular data and improved the state of baselines in tabular DL. First, we have demonstrated that a simple ResNet-like architecture can serve as an effective baseline. Second, we have proposed FT-Transformer — a simple adaptation of the Transformer architecture that outperforms other DL solutions on most of the tasks. We have also compared the new baselines with GBDT and demonstrated that GBDT still dominates on some tasks. The code and all the details of the study are open-sourced 1, and we hope that our evaluation and two simple models (ResNet and FT-Transformer) will serve as a basis for further developments on tabular DL. ",
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+        "type": "text",
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parse/train/ryloogSKDS/ryloogSKDS.md

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+# DEEP ORIENTATION UNCERTAINTY LEARNING BASED ON A BINGHAM LOSS
+
+Igor Gilitschenski1, Roshni Sahoo1, Wilko Schwarting1, Alexander Amini1,   
+Sertac Karaman2, Daniela Rus1   
+1 Computer Science and Artificial Intelligence Lab, MIT   
+2 Laboratory for Information and Decision Systems, MIT   
+{igilitschenski, rsahoo, wilkos, amini, sertac, rus}@mit.edu
+
+# ABSTRACT
+
+Reasoning about uncertain orientations is one of the core problems in many perception tasks such as object pose estimation or motion estimation. In these scenarios, poor illumination conditions, sensor limitations, or appearance invariance may result in highly uncertain estimates. In this work, we propose a novel learningbased representation for orientation uncertainty. By characterizing uncertainty over unit quaternions with the Bingham distribution, we formulate a loss that naturally captures the antipodal symmetry of the representation. We discuss the interpretability of the learned distribution parameters and demonstrate the feasibility of our approach on several challenging real-world pose estimation tasks involving uncertain orientations.
+
+# 1 INTRODUCTION
+
+Reasoning about uncertain poses and orientations, specifically 3-dimensional (3d) positions and 3-axes orientations, is one of the main inference tasks in computer vision (Sattler et al., 2019), robotics (Glover et al., 2011), aerospace (Crassidis & Markley, 2003), and other fields.
+
+Proper representation and estimation of uncertainty is important, e.g. when dealing with structural ambiguities in object pose estimation or coping with sensor corruption.
+
+In vision and robotics tasks, high levels of pose uncertainty may occur due to potentially adversarial conditions that arise in real-world scenarios. A principled approach to uncertainty quantification allows for better execution of planning and situation-awareness tasks such as grasping, tracking, and motion estimation.
+
+When representing uncertainties over poses, the position can be modeled using a Gaussian distribution. This approach is well-motivated by the Central Limit Theorem and widely used in probabilistic deep learning models. However, this paradigm cannot be as easily applied to modeling periodic quantities, such as the orientation of an object. Therefore, Gaussian models become unsuitable particularly in learning regimes involving high uncertainties where one cannot assume local linearity of the underlying space. In this work, we set out to develop a principled probabilistic deep learning approach capable of coping with uncertain orientations.
+
+Currently, most deep learning approaches that predict poses or rigid-body motions suffer from at least one of three drawbacks: 1) they do not model the uncertainty at all and merely focus on the accuracy of the predicted pose, 2) they make simplifying assumptions not taking into account that the orientation is defined on a periodic manifold, making the approach only suitable in low-noise regimes, or 3) even when trying to account for periodicity, no dependency is assumed between the orientation axes and usually an Euler angle-based representation is required. To this point, there are no probabilistic deep learning models for uncertainty of orientations that take the geometry of the underlying domain into account.
+
+![](images/8dd3507a6df732500949be3524c760ebc91aad24058d2ab9bcc69707af5af2e8.jpg)  
+Figure 1: Objects from the T-LESS dataset and the corresponding orientation uncertainty predicted by the model trained on the newly proposed Bingham loss, which is capable of capturing rotational symmetries.
+
+In this work, we close this research gap by proposing a probabilistic deep learning model inspired by Directional Statistics (Mardia & Jupp, 1999). We present a loss based on the Bingham distribution (Bingham, 1974), an antipodally symmetric distribution on the sphere. With this loss, we represent uncertain orientations by modeling uncertainty over unit quaternions. Our contributions involve Bingham parameter learning using backpropagation through a Gram-Schmidt method to ensure orthonormalization, efficient approximate evaluation of the normalization constant of the Bingham distribution from a lookup table, and backpropagating through an interpolation scheme during learning. We also discuss interpretability of the Bingham distribution parameters and establish the feasibility of the approach through extensive evaluations.
+
+In summary, this work makes the following contributions: 1) We propose the Bingham loss, a novel loss function for deep learning-based predictions of orientations and their uncertainty. 2) We provide a methodology for making the newly proposed loss and its normalization constant computationally tractable in a deep learning pipeline. 3) We demonstrate multi-modal orientation prediction using a Bingham variant of Mixture Density Networks. 4) We demonstrate how our approach outperforms the state-of-the-art on challenging pose and orientation estimation tasks1.
+
+# 2 BACKGROUND: BINGHAM DISTRIBUTION FOR UNCERTAIN ORIENTATIONS
+
+Unit quaternions are a widely used representation for object orientation in 3d space. They are more compact than rotation matrices, and unlike Euler angles, do not suffer from degeneracies such as Gimbal lock. Additionally, quaternions provide a convenient mathematical notation where the quaternion product, ${ \bf q } _ { 1 } \odot { \bf q } _ { 2 }$ , of two unit quaternions $\mathbf { q } _ { 1 }$ , $\mathbf { q } _ { 2 } \in \mathbb { H } _ { 1 }$ results in a concatenation of the rotations represented by each of the quaternions individually. A full introduction to this representation by given in Kuipers (1999) and notational aspects are discussed by Sommer et al. (2018). In this work, a quaternion $q _ { 1 } i + q _ { 2 } j + q _ { 3 } k + q _ { 4 }$ will be interpreted as a vector $\mathbf { q } \in \mathbb { R } ^ { 4 }$ . It is important to note that the definition of unit quaternions is equivalent to the vector q being of unit length $| | \mathbf { q } | | = 1$ . Furthermore, the quaternions $\mathbf { q }$ and $\mathbf { - q }$ represent the same orientation. Therefore, representing uncertain orientations using quaternions requires a probability distribution on the 4d hypersphere that exhibits antipodal symmetry, i.e. for the density function $f ( \cdot )$ of this distribution ${ \dot { f } } ( \mathbf { \bar { q } } ) = f ( - \mathbf { q } )$ has to hold.
+
+A probability distribution exhibiting these properties was proposed by Bingham (1974). It arises by conditioning a zero mean Gaussian to unit length. The Bingham distribution is given in terms of its p.d.f. as $\bar { p } ( \mathbf { x } ; \mathbf { M } , \mathbf { Z } ) = N ( \mathbf { M } \mathbf { Z } \mathbf { M } ^ { \top } ) ^ { - 1 } \exp ( \mathbf { x } ^ { \top } \mathbf { M } \mathbf { Z } \mathbf { M } ^ { \top } \mathbf { x } )$ ,where $\mathbf { x } \in \mathbb { R } ^ { 4 }$ with $| | \mathbf { x } | | = 1$ , $N ( \mathbf { M } \mathbf { \bar { Z } } \mathbf { M } ^ { \top } )$ is a normalization constant, $\dot { \mathbf { M } } \in \mathbb { R } ^ { 4 \times 4 }$ orthogonal, and $\mathbf { Z } = \mathrm { d i a g } ( z _ { 1 } , z _ { 2 } , z _ { 3 } , 0 ) \in$ $\mathbb { R } ^ { 4 \times 4 }$ diagonal, with diagonal entries $z _ { i } < = 0$ and the last entry being zero. We use the notation $\mathrm { B i n g h a m } ( \mathbf { M } , \mathbf { Z } )$ . The restriction on the range of the diagonal entries in $\mathbf { Z }$ has numerical and representational convenience reasons. It can be shown that Bingham $\mathbf { \tau } _ { \mathrm { l } } ( \mathbf { M } , \mathbf { Z } ) = \mathrm { B i n g h a m } ( \mathbf { M } , \mathbf { Z } + c \mathbf { \bar { I } } )$ for all $c \in \mathbb { R }$ with $\mathbf { I } \in \mathbb { R } ^ { 4 \times 4 }$ denoting the identity matrix. Similarly, changing the order of diagonal entries in $\mathbf { Z }$ has no effect on the distribution as long as the columns in $\mathbf { M }$ are permuted accordingly.
+
+In the definition above, the parameters $\mathbf { M }$ and $\mathbf { Z }$ bear some similarity to the mean and variance of a Gaussian. The density obtains its maxima at $\pm \mathbf { M } _ { : , 4 }$ (the fourth column of $\mathbf { M }$ ) which can be thought of as a mean orientation respecting the manifold structure. The diagonal entries of $\mathbf { Z }$ can be interpreted as dispersion parameters, and the first three columns of $\mathbf { M }$ can be interpreted as the directions of the dispersion (the Gaussian analog is the orientation of the covariance ellipsoid). Bingham distributions allow for representation of uniform priors over individual axes or even the entire space, making them superior to Gaussians in any of the usual orientation representations.
+
+![](images/9e1844f2c30eb42b33cba4e89d8c366ffb27182986c158e5eefb3a2605d67764.jpg)  
+Figure 2: Densities of the Bingham distribution represented for different dimensionality. For the circular case (a), the density is shown as a function of unit vectors on the plane. For the spherical case (b), it is shown as a heatmap on a 3d unit sphere. For the 4d case (c), which is of our particular interest, we visualize the mode of the Bingham in terms of the coordinate system orientation represented by the corresponding quaternion. Then, we draw samples from the distribution and visualize each sample as a potential coordinate arrow endpoint for each axis (i.e. each sample drawn from the Bingham distribution is represented by three points in the plot). This representation allows us to simultaneously represent the orientation and the corresponding uncertainty.
+
+One of the main challenges of using the Bingham distribution is the computation of its normalization constant
+
+$$
+N ( \mathbf { M Z M } ^ { \top } ) = \int _ { | | q | | = 1 } \exp ( \mathbf { q } ^ { \top } \mathbf { M Z M } ^ { \top } \mathbf { q } ) \mathrm { d } \mathbf { q } ,
+$$
+
+which is a Hypergeometric function of matrix argument (Herz, 1955). Evaluating these functions imposes a high computational burden and is still an area of active research (Koev & Edelman, 2006; Kume et al., 2013; Koyama et al., 2014; Kume & Sei, 2018). Using the transformation theorem and the fact that M is orthogonal, the normalization constant can be simplified as $N ( \mathbf { M Z M } ^ { \top } ) = N ( \mathbf { Z } )$ , making it merely a function of the three parameters $z _ { i }$ $( i = 1 , 2 , 3$ ) and motivating the use of precomputed lookup tables in practice.
+
+Furthermore, to make the uncertainty of a Bingham Distribution more interpretable in practice, we propose the use of Expected Absolute Angular Deviation (EAAD) which is defined as
+
+$$
+\mathrm { E A A D } ( \mathbf { Z } ) = \int _ { | | q | | = 1 } \theta ( \mathbf { q } , \mathbf { e } ) \cdot p ( \mathbf { q } ; \ \mathbf { I } , \mathbf { Z } ) \mathrm { d } \mathbf { q } ,
+$$
+
+where $p ( \cdot )$ is the $\mathrm { B i n g h a m } ( \mathbf { I } , \mathbf { Z } )$ density, $\mathbf { I }$ is the identity matrix, $\mathbf { e } ~ = ~ [ 0 , 0 , 0 , 1 ]$ is the vector corresponding to the unit quaternion representing the identity and $\theta ( \mathbf { q } , \mathbf { e } ) = 2 \cdot \operatorname { a r c c o s } ( | \langle \mathbf { q } , \mathbf { e } \rangle | )$ denotes the angular distance between $\mathbf { q }$ and $\mathbf { e }$ . The EAAD describes the expected angular deviation from the “mean” orientation. It can be loosely thought of as the orientation counterpart to the standard deviation in Euclidean space. For the same reason as in the normalization constant, the EAAD computation does not involve the parameter $\mathbf { M }$ .
+
+# 3 DEEP ORIENTATION UNCERTAINTY LEARNING
+
+The Bingham distribution is the main component of the proposed probabilistic framework for representing deep learned uncertain orientations. Drawing inspiration from Mixture Density Networks (Bishop, 1994), we propose using the Bingham distribution’s negative log-likelihood as a loss function
+
+$$
+\begin{array} { r } { L ( \mathbf { y } , \mathbf { M } , \mathbf { Z } ) = - \log p ( \mathbf { y } ; \mathbf { M } , \mathbf { Z } ) = - \mathbf { y } ^ { \top } \mathbf { M } \mathbf { Z } \mathbf { M } ^ { \top } \mathbf { y } + \log N ( \mathbf { Z } ) , } \end{array}
+$$
+
+with M, $\mathbf { Z }$ as defined above and y being the orientation label given in the training data. We use a neural network to learn $\mathbf { M }$ and $\mathbf { Z }$ , end-to-end, directly from the input data (e.g. RGB images). From this prediction, the point estimate of $\mathbf { y }$ is obtained as $\hat { \mathbf { y } } = \mathbf { M } _ { : , 4 }$ as the last column corresponds to the highest diagonal entry of $\mathbf { Z }$ and thus represents one of the modes of the distribution (the other being $- \hat { \mathbf { y } }$ due to antipodal symmetry).
+
+![](images/93d820669a8e990c2177fda9970ffe82317decc83b8b47b89262ba82eb327bfe.jpg)  
+Figure 3: The proposed orientation uncertainty estimation pipeline predicts the parameters of a Bingham distribution for representing uncertain unit quaternions. Backpropagation through an interpolator and use of a lookup table allows for avoiding evaluations of the computationally expensive Bingham normalization constant.
+
+No costly evaluation of the normalization constant is required and no major computational challenges arise in the special case where the dispersion parameter $\mathbf { Z }$ is known and not predicted by a neural network. However, as our goal is the modeling of uncertainty, we propose methods for modeling M and $\mathbf { Z }$ as well as backpropagating through $N ( \mathbf { Z } )$ .
+
+# 3.1 MODELING OF DISTRIBUTION PARAMETERS
+
+In order to obtain predictions $\hat { \textbf { M } }$ and $\hat { \mathbf { Z } }$ , we require a 19 dimensional output $\mathbf { \tau } ( \mathbf { o } \in \mathbb { R } ^ { 1 9 } .$ ) of the predictor network (3 outputs for $\mathbf { Z }$ , 16 outputs for $\mathbf { M }$ ). On its own, these outputs do not satisfy the above-mentioned constraints on the Bingham distribution parameters. Thus, we define the differentiable transforms $T _ { \mathbf { M } } : \mathbb { R } ^ { 1 6 }  \mathbb { R } ^ { 4 \times 4 }$ and $T _ { \mathbf { Z } } : \mathbb { R } ^ { 3 }  \mathbb { R } ^ { 4 \times 4 }$ that transform these outputs such that the constraints are satisfied.
+
+The transform $T _ { \mathbf { Z } }$ is obtained as $T _ { \mathbf { Z } } ( o _ { 1 } , o _ { 2 } , o _ { 3 } ) = \mathrm { d i a g } ( \hat { z } _ { 1 } , \hat { z } _ { 2 } , \hat { z } _ { 3 } , 0 )$ with $\hat { z } _ { i } = - \exp ( o _ { i } )$ . For computing $\hat { \textbf { M } }$ , we first subdivide $O _ { 4 } , \ldots , O _ { 1 9 }$ into four vectors $\mathbf { v } _ { i } ~ \in ~ \mathbb { R } ^ { 4 }$ $( i = 1 , \dots , 4 )$ . Then, we apply the Gram-Schmidt orthonormalization method to these vectors according to $\begin{array} { r l } { \hat { \mathbf { m } } _ { i } } & { { } = } \end{array}$ N $\begin{array} { r } { \mathrm { o r m a l i z e } ( \mathbf { v } _ { i } - \sum _ { k = 1 } ^ { i - 1 } \langle \hat { \mathbf { m } } _ { k } , \mathbf { v } _ { i } \rangle \cdot \hat { \mathbf { m } } _ { k } ) } \end{array}$ with $i \in \{ 1 , 2 , 3 , 4 \}$ and $\mathrm { N o r m a l i z e } ( \mathbf { x } ) = \mathbf { x } / \left| \left| \mathbf { x } \right| \right|$ . Finally, the prediction $\dot { \bf M }$ is obtained as $T _ { \mathbf { M } } ( o _ { 3 } , \ldots , o _ { 1 9 } ) = [ \hat { \mathbf { m } } _ { 1 } , \ldots , \hat { \mathbf { m } } _ { 4 } ]$ , and $\hat { \textbf { M } }$ is orthogonal by construction.
+
+# 3.2 BACKPROPAGATION THROUGH THE BINGHAM NORMALIZATION CONSTANT
+
+As mentioned earlier, computation of the Bingham normalization constant is numerically burdensome. This is also true for its derivatives which can be shown to be proportional to the normalization constant of Bingham distributions of higher dimension (Kume & Wood, 2007). A forward-backward pass for one single data point requires 4 evaluations of hypergeometric functions of matrix argument.
+
+We avoid this by precomputing a lookup table for $N ( \mathbf { Z } )$ at $\mathrm { L }$ different locations $\mathbf { t } _ { i }$ (with $\mathbf { Z } _ { i } \mathbf { \Psi } =$ $\mathrm { d i a g } ( [ \mathbf { t } _ { i } ^ { \top } , 0 ] )$ . This table is then used to build an interpolator $\begin{array} { r } { f _ { N } ( \mathbf { z } ) = \sum _ { i - 1 } ^ { L } w _ { i } \phi ( | | \mathbf { z } - \mathbf { t } _ { i } | | ) } \end{array}$ with $\textbf { z } \in \mathbb { R } ^ { 3 }$ and $\phi$ denoting a radial basis function. The weights $w _ { i }$ can also be precomputed during generation of the interpolator. Thus, we can approximate ${ \cal N } ( { \bf Z } ) \approx f _ { N } ( { \bf z } )$ and $\nabla _ { \mathbf { z } } N ( \mathbf { Z } ) \approx \nabla _ { \mathbf { z } } f _ { N } ( \mathbf { z } )$ . To the best of our knowledge, this is the first time that a lookup table based interpolation mechanism has been included in the computation graph of a neural network.
+
+# 3.3 MULTI-MODAL PREDICTION
+
+A Bingham variant of Mixture Density Networks can be used to obtain multi-modal predictions. However, MDNs are hard to train even in the Gaussian case. Following the discussion in Makansi et al. (2019), we separate the training in two stages. In the first stage, we only learn to predict $\mathbf { M }$ and assume the dispersion to be fixed with $\mathbf { Z } = \mathrm { d i a g } ( - a , - a , - a , 0 )$ . In practice $a \in \mathbb { R } ^ { + }$ can usually be set to 1 as it merely scales the cost term. In the second stage, we train to predict M and $\mathbf { Z }$ jointly. Our evaluation will show that in high uncertainty regimes, this training method is also helpful for the unimodal case.
+
+# 4 EXPERIMENTS
+
+In this section we evaluate the proposed Bingham loss on its ability to learn calibrated uncertainty estimates for orientations. This goes beyond comparing point estimates of orientations; we evaluate how well the estimated distribution of orientations can explain the data. We will also show that the Bingham distribution representation is capable of capturing ambiguity and uncertainty in SO(3) better than state-of-the-art approaches.
+
+We investigate characteristics and behaviors by training neural networks on two head-pose datasets, IDIAP (Odobez, 2003) and UPNA (Ariz et al., 2016), as well as the object pose dataset TLESS (Hodan et al., 2017). We show the capability of calibrated uncertainty estimation by applying ˇ artificial label-noise to IDIAP and UPNA and observing that the Bingham parametrization allows for accurate prediction of uncertainty. In addition to calibrated uncertainty estimation, we demonstrate advanced capabilities in the face of object orientation ambiguity on the T-LESS dataset by visualizing the predicted distributions for different orientation ambiguous objects, e.g. symmetric, and comparing to objects with clear orientation.
+
+# 4.1 ARCHITECTURE AND EXPERIMENTAL SETUP
+
+We seek to estimate the Bingham distribution parameters directly from image data. Our pipeline is shown in Figure 3 and begins by passing an image input to a convolutional encoder, in our case a standard ResNet-18 network followed by a fully connected layer, populating the entries of $o _ { 1 } , o _ { 2 } , o _ { 3 }$ and $v _ { 1 } , v _ { 2 } , v _ { 3 } , v _ { 4 }$ . Subsequently, $\mathbf { Z }$ is computed by constrained diagonalization of $O 1 , O 2 , O 3$ , and Gram-Schmidt orthonormalization of $v _ { 1 } , v _ { 2 } , v _ { 3 } , v _ { 4 }$ yields $\mathbf { M }$ , as described in Section 3.1. To evaluate the Bingham loss, the normalizer $N ( \mathbf { Z } )$ needs to be queried from the RBF lookup table, Section 3.2. Differentiation of the interpolator via finite differences enables us to back-propagate through the entire pipeline. All models were implemented in PyTorch and optimized with the Adam optimizer.
+
+We create the lookup table by numerical integration. More precisely, we use Scipy’s tplquad method to compute a triple integral for each $\mathbf { Z }$ in the table. We set the relative error tolerance to 1e-3 and the absolute error tolerance to 1e-7. The actual computed integral is
+
+$$
+N ( \mathbf { Z } ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { \pi } \exp \big ( t ( \phi _ { 1 } , \phi _ { 2 } , \phi _ { 3 } ) ^ { \top } \mathbf { Z } t ( \phi _ { 1 } , \phi _ { 2 } , \phi _ { 3 } ) \big ) \cdot \sin ( \phi _ { 1 } ) ^ { 2 } \cdot \sin ( \phi _ { 2 } ) \mathrm { d } \phi _ { 1 } \mathrm { d } \phi _ { 2 } \mathrm { d } \phi _ { 3 } ,
+$$
+
+with
+
+$$
+t ( \phi _ { 1 } , \phi _ { 2 } , \phi _ { 3 } ) = \left[ { \sin ( \phi _ { 1 } ) \cdot \sin ( \phi _ { 2 } ) \cdot \cos ( \phi _ { 3 } ) } \right]
+$$
+
+to account for a transformation of coordinates from unit quaternions to 4d spherical coordinates. Because we use the Bingham log likelihood as our optimization objective, we compute the logarithm before the interpolation to avoid failure at locations where the interpolator wrongly outputs negative values.
+
+# 4.2 BASELINES
+
+We compare our work with the approach proposed by Prokudin et al. (2018). It also uses a loss based on directional statistics, specifically the Von Mises distribution. The Von Mises distribution can be thought of as a circular analog of the Normal distribution. In order to apply this approach to our setting, orientations are modeled with Euler angles. The loss then consists of the sum of log-likelihoods for each angle. While this approach can properly account for periodicity of the underlying data, we expect it to fail in cases where the underlying uncertainty is not axis aligned because it does not account for dependencies between uncertain rotation axes.
+
+Furthermore, we also evaluate several different representations of the parameter matrix M. We consider the classical Gram-Schmidt (CGS), modified Gram-Schmidt (MGS), and the matrix representation of the quaternion (QM) used by Birdal et al. (2018). Finally, we also include two nonprobabilistic orientation prediction baselines. The first one is based on a Mean Square Error (MSE) between the predicted and ground truth quaternion. The second one is based on a cosine loss applied to each angle’s biternion as discussed by Prokudin et al. (2018).
+
+# 4.3 EVALUATION METRICS
+
+To evaluate error metrics over predicted orientations, it is unsuitable to compute the RMSE over angles, since it does not sufficiently consider the spherical nature of the underlying data. Instead, we make use of the Mean Absolute Angular Deviation (MAAD) which has also been used by Prokudin et al. (2018). It is based on the angular distance between two angles defined above. We also compute the EAAD to assess the quality of the results. Additionally, the difference between EAAD and MAAD serves as an indicator of the quality of the predicted uncertainty. The acceptable difference in practice is application dependent. For the cases of the Von Mises distribution parameters, EAAD computation is carried out in a similar way as for the Bingham defined above. EAAD is calculated over the learned dispersion parameters for each example and averaged. The quality of the respective model is measured in terms of log-likelihood to indicate the goodness of an individual fit. For MDNs, we additionally report a Mean Minimum Absolute Angular Deviation (MMAAD), which uses the component closest to ground-truth for absolute angular deviation computation. The MAAD and EAAD for MDNs are computed in a per-component fashion and then weighted using the predicted mixture weights.
+
+# 4.4 CALIBRATED UNCERTAINTY ESTIMATION
+
+We evaluate the distribution fit on the head pose datasets UPNA and IDIAP, which consist of head images from a video of several people inside a room. Each image is annotated with head orientation given by pan, tilt and roll angles. We use these datasets as they provide accurate labels and allow for carrying out experiments involving artificial label noise.
+
+The results on the raw dataset are shown in Table 1. They demonstrate that the general performance for point estimates, indicated by MAAD, of the Bingham distribution remains on a similar level as the Von Mises distribution and the non-probabilistic approaches. In this setting, most motions of the subjects’ heads are aligned with the gravity axis allowing both distributions to successfully capture
+
+Table 1: Bingham (BD), Von Mises (VM), Mean Square Error (MSE), and cosine based loss prediction performance on raw UPNA and IDIAP datasets.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">UPNA</td><td colspan="3">IDIAP</td></tr><tr><td>EAAD</td><td>MAAD</td><td>LL</td><td>EAAD</td><td>MAAD</td><td>LL</td></tr><tr><td>BD-CGS</td><td>0.10</td><td>0.11</td><td>4.70</td><td>0.10</td><td>0.09</td><td>4.49</td></tr><tr><td>BD-MGS</td><td>0.10</td><td>0.13</td><td>3.87</td><td>0.10</td><td>0.10</td><td>4.58</td></tr><tr><td>BD-QM</td><td>0.10</td><td>0.16</td><td>0.31</td><td>0.10</td><td>0.09</td><td>4.74</td></tr><tr><td>VM</td><td>0.13</td><td>0.11</td><td>3.69</td><td>0.12</td><td>0.09</td><td>2.08</td></tr><tr><td>MSE</td><td>=</td><td>0.12</td><td>-</td><td></td><td>0.10</td><td></td></tr><tr><td>Cosine</td><td>=</td><td>0.12</td><td>=</td><td>=</td><td>0.10</td><td>=</td></tr></table>
+
+the noise. However, the Bingham still attains a higher log-likelihood and a smaller gap between MAAD and EAAD. Similarly, the parametrization of the concentration matrix M has a relatively small impact on the estimation performance. Although MGS has stronger robustness guarantees than CGS (the latter has a quadratic dependency on the condition number of the input matrix, see Giraud et al. (2005) for a discussion of both), the condition of the input is not poor enough to impact performance. While the quaternion matrix approach is easier to train, it also loses some of the expressiveness of the Bingham distribution because the underlying mapping (from quaternions to the space of orthogonal matrices) is not surjective.
+
+To estimate how well the predicted uncertainties are calibrated, we add artificial noise by drawing random perturbations from the Bingham distribution with varying $z _ { 1 } , z _ { 2 }$ , and $z _ { 3 }$ parameters and applying them to the quaternion labels before training. Both UPNA and IDIAP contain negligible amounts of noise, so the dispersion of the noise distribution should be captured by the learned $\mathbf { Z }$ to high accuracy. An evaluation of uncertainty and label noise is shown in Table 2. For the case of no noise, the Bingham uncertainty parameters approximate the highest certainty levels represented in the lookup table. Thus, the maximum and minimum values in the lookup table automatically become the bounds of what certainty levels can be represented by the proposed loss. When noise is applied to the training labels, the learned uncertainty parameters closely match the dispersion of label noise, so the predicted EAAD accurately captures the EAAD corresponding to the dispersion of the label noise distribution. We note that the MAAD is slightly higher than the true and estimated EAAD values. This overconfidence effect is typical in probabilistic deep learning and also arises when predicting the parameters of a Gaussian (Amini et al., 2019). In addition, we evaluated a scenario where the noise is newly sampled and applied to the true labels in each iteration (rather than corrupting the labels with the sampled noise prior to training). In this scenario, the EAAD computed from the learned dispersion parameters, the true EAAD, and the MAAD are approximately equal in value. While this scenario is less realistic in practice (and thus not visualized), it provides further evidence for representational consistency of the loss.
+
+![](images/a3018f6474f4a75f305d55ffc845593d9fb2b5503241b32b8d774c1fa3b4a48c.jpg)
+
+Table 2: Testing accuracy of uncertainty calibration. Prior to training, we perturb the labels with noise sampled from the Bingham distribution with M equal to the identity and varying $z _ { 1 } , z _ { 2 } , z _ { 3 }$ . The figures represent the different noise distributions.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">-z1</td><td rowspan="2">-22 -23</td><td rowspan="2">EAAD</td><td rowspan="2"></td><td rowspan="2">-z1</td><td rowspan="2">-z2 -23</td><td rowspan="2"></td><td rowspan="2">EAAD MAAD</td><td rowspan="2">-21</td><td rowspan="2">-22 -z3</td><td rowspan="2"></td><td rowspan="2">EAAD MAAD</td><td rowspan="2">-21 -22</td><td rowspan="2">-z3</td><td rowspan="2"></td><td rowspan="2">EAAD MAAD</td><td rowspan="2">-z1</td><td rowspan="2">-22</td><td rowspan="2">-23</td><td rowspan="2">EAAD MAAD</td></tr><tr><td></td></tr><tr><td>Label noise</td><td></td><td>No noise</td><td>0</td><td></td><td>20</td><td>20</td><td>20</td><td>0.52</td><td>250</td><td>150</td><td>50</td><td>0.22</td><td>150</td><td>100</td><td>75</td><td>0.23</td><td>300</td><td>300</td><td>300</td><td>0.13</td></tr><tr><td>UPNA</td><td>497497</td><td>497</td><td>0.10</td><td>19</td><td></td><td>19</td><td>19</td><td>0.54</td><td>186</td><td>105</td><td>63</td><td>0.23</td><td>130</td><td>114</td><td>74</td><td>0.23</td><td>303</td><td>300</td><td>295</td><td>0.13</td></tr><tr><td rowspan="2">IDIAP</td><td></td><td></td><td></td><td></td><td>±0.4</td><td>±0.4</td><td>±0.5</td><td>0.69</td><td>±78</td><td>±30</td><td>±15</td><td>0.29</td><td>±35</td><td>±10</td><td>±14</td><td>0.28</td><td>±16</td><td>±16</td><td>±17</td><td>0.20</td></tr><tr><td></td><td>499499 499</td><td>0.10</td><td></td><td>19</td><td>19</td><td>18</td><td>0.55</td><td>167</td><td>164</td><td>47</td><td>0.24</td><td>93</td><td>87</td><td>76</td><td>0.25</td><td>300</td><td>294</td><td>280</td><td>0.13</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>±0.5</td><td>±0.5 ±0.3</td><td></td><td>0.59</td><td>±17</td><td>±20</td><td>士3</td><td>0.29</td><td>±8</td><td>±8</td><td>±7</td><td>0.28</td><td>±24</td><td>±25</td><td>±35 0.20</td><td></td></tr></table>
+
+# 4.5 HANDLING AMBIGUOUS DATA
+
+We use the T-LESS dataset for evaluating the proposed model using ambiguous data. It contains images of 30 different textureless objects taken from different cameras. We use the Kinect RGB single-object images all of which are split into training, test, and validation sets. At a coarse scale most of the objects in the dataset exhibit rotational or other symmetries. At a finer scale some of these ambiguities disappear due to smaller structures. On the one hand, we expect those to be more challenging to learn. On the other hand, capturing these structures allows for very precise orientation estimation. To be able to disregard these structures, we create a variant of T-LESS where we add blur to each image using a uniform $1 0 \mathrm { p x } \times 1 0 \mathrm { p x }$ kernel.
+
+We carry out two sets of experiments. In the first set of experiments, we train orientation estimation models for 5 epochs using the Bingham loss (BD-5) and the Von Mises loss (VM-5) on the blurred and original set of images. This allows to investigate the uncertainty estimation properties before the network captures the finer grained structures. In the second set of experiments, we use the original set of images to evaluate multi-modal orientation prediction using the two-stage training approach for models with 1 (BD-MDN-1), 2 (BD-MDN-2), and 4 (BD-MDN-4) mixture components. Each stage is carried out for 30 epochs. The comparison methods use Von Mises (VM), Mean Square Error (MSE), and Cosine losses with an overall training duration of 60 epochs (or until convergence if that is earlier).
+
+Table 3: Results on the T-LESS dataset in the high uncertainty regime.   
+
+<table><tr><td>Method</td><td>[Log-likelihood MAAD</td><td>EAAD</td></tr><tr><td>VM-5</td><td>-0.12</td><td>0.48 0.33</td></tr><tr><td>BD-5</td><td>2.82 1.57</td><td>1.58</td></tr><tr><td>VM-5 w. blur</td><td>-0.03 0.56</td><td>0.44</td></tr><tr><td>BD-5 w. blur</td><td>2.71 1.59</td><td>1.58</td></tr></table>
+
+The results for the first set of experiments are visualized in Table 3. As expected, both approaches are on average far off in terms of the true orientation. While Von Mises performs better on the MAAD, we observe that there is a larger difference between the MAAD and EAAD values for the
+
+Von Mises distribution than the Bingham distribution. This indicates that the uncertainty estimates of the Von Mises distribution may be overconfident. On the other hand the Bingham distribution better captures the uncertainty over individual axes. One interesting insight is that allowing for uniform distributions over individual non-aligned periodic axes can make it hard for the learning method to pick up on the proper pose and thus may require pre-training on the pure pose estimation task in such regimes.
+
+In the second set of experiments, as visualized in Table 4, we use this training strategy for all Bingham MDN models resulting in robust convergence behavior. However, the unimodal Bingham (BD-MDN1) converges slower than Von Mises (VM) thus achieving a higher MAAD, which is adequately captured by the Bingham’s EAAD. For multiple mixture components, we obtain a very low MAAD and can observe again the phenomenon of the lookup table limitations in the EAAD. Thus, the
+
+Table 4: Results on the T-LESS dataset involving multi modal prediction.   
+
+<table><tr><td>Method</td><td colspan="4">Log-likelihood MAAD MMAAD EAAD</td></tr><tr><td>VM</td><td>3.73</td><td>0.10</td><td>-</td><td>0.17</td></tr><tr><td>BD-MDN-1</td><td>5.00</td><td>0.20</td><td>=</td><td>0.21</td></tr><tr><td>BD-MDN-2</td><td>6.17</td><td>0.07</td><td>0.06</td><td>0.12</td></tr><tr><td>BD-MDN-4</td><td>6.19</td><td>0.06</td><td>0.05</td><td>0.10</td></tr><tr><td>MSE</td><td>-</td><td>0.22</td><td>-</td><td>1</td></tr><tr><td>Cosine</td><td>=</td><td>0.10</td><td>=</td><td>=</td></tr></table>
+
+MAAD achieved during the first training stage can not only be used for inspecting the network’s accuracy but also for determining the minimum Z parameter values stored in the lookup table. Another interesting phenomenon can be observed in the EAAD and MAAD of the VM loss. As the representation required by Von Mises assumes that each axis is independent, EAAD is computed per rotation axis. This results in an overapproximation of the uncertainty overall. For the nonprobabilistic losses, the cosine loss achieves better performance which is probably due to better consideration of the underlying geometry. In summary, while the proposed Bingham loss shares the general challenges of training Mixture Density Networks, it better captures the underlying noise structure by explicitly modeling dependencies between rotation axes.
+
+# 5 DISCUSSION AND RELATED WORK
+
+Quantifying and representing uncertainty by and in neural networks has been a subject of extensive research initially focused on modeling probability distribution parameters (Nix & Weigend, 1994) and mixture distributions (Bishop, 1994) as neural network outputs. More recent approaches focus on improving understanding of the underlying uncertainties (Kendall & Gal, 2017), providing scalable techniques for estimating predictive uncertainty (Lakshminarayanan et al., 2017), and stabilizing training to avoid mode collapse (Makansi et al., 2019). The present work is orthogonal to these approaches in the sense that it focuses on proper modeling of the underlying geometric domain and coping with a computationally demanding normalization constant.
+
+Handling of poses and orientations has been extensively studied in the context of Bayesian filtering for applications such as spacecraft attitude estimation (Crassidis & Markley, 2003) and ego-motion estimation (Bloesch et al., 2015), where one can often assume the underlying uncertainties to be small. This allows for leveraging local-linearity and using the Gaussian distribution. Recently, methods based on directional statistics enabled modeling of high uncertainty levels for inferring orientations (Gilitschenski et al., 2016) and full poses (Glover et al., 2011; Glover & Kaelbling, 2014; Srivatsan et al., 2016) by using the Bingham distribution. Drawing inspiration from these results, this work extends the applicability of these approaches to probabilistic deep learning models.
+
+Particularly in computer vision, deep learning has been applied to spherical regression and pose estimation problems (Liao et al., 2019; Huang et al., 2018). These applications involve inferring object (Brachmann et al., 2014; Hodan et al., 2018; Li et al., 2018b;a; Manhardt et al., 2019; Sun- ˇ dermeyer et al., 2018; Tekin et al., 2018; Wang et al., 2019b;a), body (Yang et al., 2019), and camera poses (Clark et al., 2017; Sattler et al., 2019; Wang et al., 2017; 2018). In all of these scenarios there is a multitude of sources for potentially high uncertainties such as the use of low-resolution data (e.g. tracking pose of distant pedestrians), absence of textures (e.g. when operating on depth data), or motion blur (e.g. due to high speeds in ego-motion estimation). However, most of the existing approaches merely focus on inferring the pose but do not account for the underlying uncertainty.
+
+The representation proposed in our work closes this gap by allowing for neural networks to output well-calibrated orientation uncertainty estimates.
+
+Only a few approaches consider modeling the uncertainty of orientations for deep learning based pose estimation. PoseRBPF by Deng et al. (2019) discretizes the orientation space into over 190 000 bins and learns a codebook to allow for tractable inference. In contrast to that approach, we do not require an a priori discretization and can directly obtain interpretable estimates. Similarly to us, Prokudin et al. (2018) propose a loss based on directional statistics. By making use of the Von Mises distribution, their work can properly account for periodicity of circular data. However, as we have shown in our evaluations, this approach cannot properly account for dependencies between different axes and thus, struggles when the underlying uncertainty is not axis aligned.
+
+# 6 CONCLUSION
+
+In this work, we introduced the Bingham loss, a loss function based on the Bingham distribution that enables neural networks to predict uncertainty over unit quaternions and thus uncertain orientations. This allows for using (rotation-)symmetric objects and ambiguous sensor data in the context of pose and orientation estimation. In addition, we demonstrate how to cope with intractable likelihoods in deep learning pipelines by using non-linear interpolation and lookup tables as part of the computation graph.
+
+The presented approach is directly usable in existing probabilistic deep learning techniques. Moreover, we demonstrate its applicability for mixture density models. The choice of parametrization remains one of the main design decisions in pose and orientation estimation pipelines. Our work supports the case for using quaternions over other parametrizations for deep learning. It also motivates further research on how to properly model dependencies between uncertain periodic and non-periodic quantities.
+
+# ACKNOWLEDGMENTS
+
+This work was supported in part by NSF Grant 1723943, the Office of Naval Research (ONR) Grant N00014-18-1-2830, and Toyota Research Institute (TRI). This article solely reflects the opinions and conclusions of its authors and not TRI, Toyota, or any other Toyota entity. Their support is gratefully acknowledged.
+
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